**Meet the editor**

Professor Ramesh K. Agarwal is the William Palm Professor of Engineering and the director of Aerospace Research and Education Center at Washington University in St. Louis, USA. From 1994 to 2001, he was the Sam Bloomfield Distinguished Professor and Executive Director of the National Institute for Aviation Research at Wichita State University in Kansas. From 1978 to 1994,

he worked in various scientific and managerial positions at McDonnell Douglas Research Laboratories in St. Louis; he became the Program Director and McDonnell Douglas Fellow in 1990. Dr. Agarwal received Ph.D in Aeronautical Sciences from Stanford University in 1975, M.S. in Aeronautical Engineering from the University of Minnesota in 1969 and B.S. in Mechanical Engineering from Indian Institute of Technology, Kharagpur, India in 1968. Professor Agarwal has worked in Computational Fluid Dynamics, Rarefied Gas Dynamics and Hypersonic Flows, Flow Control, and more recently in Sustainable Air and Ground Transportation.

Contents

**Preface IX** 

Chapter 1 **One Dimensional Morphing** 

Juraj Belan

Chapter 4 **ALLVAC 718 Plus™ Superalloy** 

**Gas Turbine Engines 97** 

**Part 2 Aircraft Control Systems 117** 

Chapter 6 **An Algorithm for Parameters** 

I. A. Boguslavsky

Chapter 7 **Influence of Forward** 

**Part 1 Aircraft Structures and Advanced Materials 1** 

**Structures for Advanced Aircraft 3** 

Giorgio Cavallini and Roberta Lazzeri

**for Aircraft Engine Applications 75**  Melih Cemal Kushan, Sinem Cevik Uzgur, Yagiz Uzunonat and Fehmi Diltemiz

Melih Cemal Kushan, Yagiz Uzunonat, Sinem Cevik Uzgur and Fehmi Diltemiz

Matko Orsag and Stjepan Bogdan

Chapter 3 **Study of Advanced Materials for Aircraft** 

Edward A. Bubert and Norman M. Wereley

Chapter 2 **A Probabilistic Approach to Fatigue Design of Aerospace** 

**Jet Engines Using Quantitative Metallography 49** 

Chapter 5 **Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft** 

**Identificationof an Aircraft's Dynamics 119** 

**and Descent Flight on Quadrotor Dynamics 141** 

Robert D. Vocke III, Curt S. Kothera, Benjamin K.S. Woods,

**Components by Using the Risk Assessment Evaluation 17** 

## Contents

#### **Preface** XIII

	- **Part 2 Aircraft Control Systems 117**

X Contents


Chapter 9 **GNSS Carrier Phase-Based Attitude Determination 193**  Gabriele Giorgi and Peter J. G. Teunissen

Contents VII

Chapter 20 **Synthetic Aperture Radar Systems** 

Chapter 21 **Avionics Design for a Sub-Scale** 

Chapter 22 **Study of Effects** 

and Marcello Napolitano

**for Small Aircrafts: Data Processing Approaches 465**

N.I. Petrov, A. Haddad, G.N. Petrova, H. Griffiths and R.T. Waters

Oleksandr O. Bezvesilniy and Dmytro M. Vavriv

**Fault-Tolerant Flight Control Test-Bed 499**  Yu Gu, Jason Gross, Francis Barchesky, Haiyang Chao

**of Lightning Strikes to an Aircraft 523**

	- **Part 3 Aircraft Electrical Systems 287**
	- **Part 4 Aircraft Inspection and Maintenance 381**
	- **Part 5 Miscellaneous Topics 425**

Chapter 20 **Synthetic Aperture Radar Systems for Small Aircrafts: Data Processing Approaches 465**  Oleksandr O. Bezvesilniy and Dmytro M. Vavriv

#### Chapter 21 **Avionics Design for a Sub-Scale Fault-Tolerant Flight Control Test-Bed 499**  Yu Gu, Jason Gross, Francis Barchesky, Haiyang Chao and Marcello Napolitano

VI Contents

Chapter 8 **Advanced Graph Search Algorithms**

Chapter 10 **A Variational Approach to** 

Jozsef Rohacs

**for Path Planning of Flight Vehicles 157** 

Chapter 9 **GNSS Carrier Phase-Based Attitude Determination 193**  Gabriele Giorgi and Peter J. G. Teunissen

**the Fuel Optimal Control Problem for UAV Formations 221**

**Fusion for Application to Aircraft Identification System 249**

Luca De Filippis and Giorgio Guglieri

Andrea L'Afflitto and Wassim M. Haddad

Peter Pong and Subhash Challa

Chapter 12 **Subjective Factors in Flight Safety 263**

Chapter 13 **Power Generation and Distribution System**

**Part 3 Aircraft Electrical Systems 287**

Ahmed Abdel-Hafez

Chapter 15 **Key Factors in Designing** 

Marco Leo

Mariusz Wazny

Chapter 19 **Review of Technologies**

Chapter 18 **The Analysis of the Maintenance** 

**Part 5 Miscellaneous Topics 425** 

Ramesh K. Agarwal

Mohamad Hussien Taha

Chapter 16 **Methods for Analyzing the Reliability** 

Nicolae Jula and Cepisca Costin

**Part 4 Aircraft Inspection and Maintenance 381** 

Chapter 17 **Automatic Inspection of Aircraft Components** 

**Process of the Military Aircraft 399**

**to Achieve Sustainable (Green) Aviation 427**

Chapter 11 **Measuring and Managing Uncertainty Through Data** 

**for a More Electric Aircraft - A Review 289**

**In-Flight Entertainment Systems 331** Ahmed Akl, Thierry Gayraud and Pascal Berthou

Chapter 14 **Power Electronics Application for More Electric Aircraft 309**

**of Electrical Systems Used Inside Aircrafts 361**

**Using Thermographic and Ultrasonic Techniques 383** 

Chapter 22 **Study of Effects of Lightning Strikes to an Aircraft 523**  N.I. Petrov, A. Haddad, G.N. Petrova, H. Griffiths and R.T. Waters

Preface

divided into five sections.

aerospace applications.

of aircrafts.

and subjective factors in flight safety.

deals with the analysis of the maintenance process.

The book is a compilation of research articles and review articles describing the state of the art and latest advancements in technologies for various areas of aircraft system. The authors contributing to this volume are leading experts in their fields. The book is

Section one is titled "Aircraft Structure and Advanced Materials". It has five papers, dealing with aircraft structures and advanced materials. In the area of aircraft structures, the topics such as morphing structures and probabilistic approach to fatigue design are covered, while the chapters on advanced materials include the study of advanced materials for jet engines using quantitative metallography, the innovative approaches to gas turbine engine applications and superalloys for

Section two is titled "Aircraft Control Systems". It contains seven papers dealing with a wide variety of topics. The topics include algorithms for parameter identification of the aircraft dynamics, quadrotor dynamics, graph search algorithms for path planning, GNSS carrier phase-based attitude determination, fuel optimal control problem for UAV formations, measurement and management of uncertainty through data fusion,

Section three is titled "Aircraft Electrical Systems". It has four papers dealing with a wide variety of topics. The topics include a review of power generation system for a more electric aircraft, power electronics for a more electric aircraft, design of an inflight entertainment system, and methods for reliability analysis of electrical systems

Section four deals with inspection and maintenance of an aircraft. It has two papers dealing with a number of topics concerning techniques for inspection and maintenance. The first chapter describes the automatic inspection of aircraft components using thermographic and ultrasonic techniques, while the second chapter

The last section of the book contains chapters on miscellaneous topics. One chapter reviews the technologies for sustainable green aviation, another chapter describes the synthetic aperture radar systems for small aircraft, the third chapter describes the

## Preface

The book is a compilation of research articles and review articles describing the state of the art and latest advancements in technologies for various areas of aircraft system. The authors contributing to this volume are leading experts in their fields. The book is divided into five sections.

Section one is titled "Aircraft Structure and Advanced Materials". It has five papers, dealing with aircraft structures and advanced materials. In the area of aircraft structures, the topics such as morphing structures and probabilistic approach to fatigue design are covered, while the chapters on advanced materials include the study of advanced materials for jet engines using quantitative metallography, the innovative approaches to gas turbine engine applications and superalloys for aerospace applications.

Section two is titled "Aircraft Control Systems". It contains seven papers dealing with a wide variety of topics. The topics include algorithms for parameter identification of the aircraft dynamics, quadrotor dynamics, graph search algorithms for path planning, GNSS carrier phase-based attitude determination, fuel optimal control problem for UAV formations, measurement and management of uncertainty through data fusion, and subjective factors in flight safety.

Section three is titled "Aircraft Electrical Systems". It has four papers dealing with a wide variety of topics. The topics include a review of power generation system for a more electric aircraft, power electronics for a more electric aircraft, design of an inflight entertainment system, and methods for reliability analysis of electrical systems of aircrafts.

Section four deals with inspection and maintenance of an aircraft. It has two papers dealing with a number of topics concerning techniques for inspection and maintenance. The first chapter describes the automatic inspection of aircraft components using thermographic and ultrasonic techniques, while the second chapter deals with the analysis of the maintenance process.

The last section of the book contains chapters on miscellaneous topics. One chapter reviews the technologies for sustainable green aviation, another chapter describes the synthetic aperture radar systems for small aircraft, the third chapter describes the

#### XIV Preface

avionics design for a fault-tolerant flight control test-bed and the final chapter in this section discusses the lightening strike effects to a radar dome.

Thus the book covers a wide variety of topics related to aircraft technologies in twenty two chapters in a single volume. There is hardly another book that covers such a wide range of topics in a single volume. Therefore, it can serve as a useful source of reference to both researchers and students interested in learning about specific aircraft technologies, as well as obtaining a general overview of the state of the art of many technologies relevant to aircraft systems and their improvement.

> **Ramesh K. Agarwal**  Washington University in St. Louis, USA

## **Part 1**

## **Aircraft Structures and Advanced Materials**

**1** 

*USA* 

**One Dimensional Morphing Structures** 

Robert D. Vocke III1, Curt S. Kothera2, Benjamin K.S. Woods1,

Since the Wright Brothers' first flight, the idea of "morphing" an airplane's characteristics through continuous, rather than discrete, movable aerodynamic surfaces has held the promise of more efficient flight control. While the Wrights used a technique known as wing warping, or twisting the wings to control the roll of the aircraft (Wright and Wright, 1906), any number of possible morphological changes could be undertaken to modify an aircraft's flight path or overall performance. Some notable examples include the Parker Variable Camber Wing used for increased forward speed (Parker, 1920), the impact of a variable dihedral wing on aircraft stability (Munk, 1924), the high speed dash/low speed cruise abilities associated with wings of varying sweep (Buseman, 1935), and the multiple benefits of cruise/dash performance and efficient roll control gained through telescopic wingspan

While the aforementioned concepts focused on large-scale, manned aircraft, morphing technology is certainly not limited to vehicles of this size. In fact, the development of a new generation of unmanned aerial vehicles (UAVs), combined with advances in actuator and materials technology, has spawned renewed interest in radical morphing configurations capable of matching multiple mission profiles through shape change – this class has come to be referred to as "morphing aircraft" (Barbarino *et al*., 2011). Gomez and Garcia (2011) presented a comprehensive review of morphing UAVs. Contemporary research is primarily dedicated to various conformal changes, namely, twist, camber, span, and sweep. It has been shown that morphing adjustments in the planform of a wing without hinged surfaces lead to improved roll performance, which can expand the flight envelope of an aircraft (Gern *et al*., 2002), and more specifically, morphing to increase the span of a wing results in a reduction in induced drag, allowing for increased range or endurance (Bae *et al*., 2005). The work presented here is intended for just such a one dimensional (1-D) span-morphing application, for example a UAV with span-morphing wingtips depicted in Figure 1. By achieving large deformations in the span dimension over a small section of wing, the wingspan can be altered during flight to optimize aspect ratio for different roles. Furthermore, differential span change between wingtips can generate a roll moment, replacing the use of ailerons on the aircraft (Hetrick *et al*., 2007). This one dimensional

changes (Sarh, 1991; Gevers, 1997; Samuel and Pines, 2007).

**1. Introduction** 

**for Advanced Aircraft** 

*1University of Maryland, College Park, MD 2Techno-Sciences, Inc., Beltsville, MD,* 

Edward A. Bubert1 and Norman M. Wereley1

## **One Dimensional Morphing Structures for Advanced Aircraft**

Robert D. Vocke III1, Curt S. Kothera2, Benjamin K.S. Woods1, Edward A. Bubert1 and Norman M. Wereley1 *1University of Maryland, College Park, MD 2Techno-Sciences, Inc., Beltsville, MD, USA* 

## **1. Introduction**

Since the Wright Brothers' first flight, the idea of "morphing" an airplane's characteristics through continuous, rather than discrete, movable aerodynamic surfaces has held the promise of more efficient flight control. While the Wrights used a technique known as wing warping, or twisting the wings to control the roll of the aircraft (Wright and Wright, 1906), any number of possible morphological changes could be undertaken to modify an aircraft's flight path or overall performance. Some notable examples include the Parker Variable Camber Wing used for increased forward speed (Parker, 1920), the impact of a variable dihedral wing on aircraft stability (Munk, 1924), the high speed dash/low speed cruise abilities associated with wings of varying sweep (Buseman, 1935), and the multiple benefits of cruise/dash performance and efficient roll control gained through telescopic wingspan changes (Sarh, 1991; Gevers, 1997; Samuel and Pines, 2007).

While the aforementioned concepts focused on large-scale, manned aircraft, morphing technology is certainly not limited to vehicles of this size. In fact, the development of a new generation of unmanned aerial vehicles (UAVs), combined with advances in actuator and materials technology, has spawned renewed interest in radical morphing configurations capable of matching multiple mission profiles through shape change – this class has come to be referred to as "morphing aircraft" (Barbarino *et al*., 2011). Gomez and Garcia (2011) presented a comprehensive review of morphing UAVs. Contemporary research is primarily dedicated to various conformal changes, namely, twist, camber, span, and sweep. It has been shown that morphing adjustments in the planform of a wing without hinged surfaces lead to improved roll performance, which can expand the flight envelope of an aircraft (Gern *et al*., 2002), and more specifically, morphing to increase the span of a wing results in a reduction in induced drag, allowing for increased range or endurance (Bae *et al*., 2005). The work presented here is intended for just such a one dimensional (1-D) span-morphing application, for example a UAV with span-morphing wingtips depicted in Figure 1. By achieving large deformations in the span dimension over a small section of wing, the wingspan can be altered during flight to optimize aspect ratio for different roles. Furthermore, differential span change between wingtips can generate a roll moment, replacing the use of ailerons on the aircraft (Hetrick *et al*., 2007). This one dimensional

One Dimensional Morphing Structures for Advanced Aircraft 5

SMP skin was abandoned as a high-risk option. Additionally, the state-of-the-art of SMP technology does not appear to be well-suited for dynamic control morphing objectives.

With maximum strains above 100%, low stiffness, and a lower degree of risk due to their passive operation, elastomeric materials are ideal candidates for a morphing skin. Isotropic elastomer morphing skins have been successfully implemented on the MFX-1 UAV (Flanagan *et al*., 2007). This UAV employed a mechanized sliding spar wing structure capable of altering the sweep, wing area, and aspect ratio during flight. Sheets of silicone elastomer connect rigid leading and trailing edge spars, forming the upper and lower surfaces of the wing. The elastomer skin is reinforced against out-of-plane loads by ribbons stretched taught immediately underneath the skin, which proved effective for wind tunnel testing and flight testing. Morphing sandwich structures capable of high global strains have also been investigated (Joo *et al.,* 2009; Bubert *et. al.,* 2010; Olympio *et al.,* 2010). However, suitable improvements over these structures, such as anisotropic fiber reinforcement and a better developed substructure for out-of-plane reinforcement, are desired for a fully

The present research therefore focuses on the development of a passive anisotropic elastomer composite skin with potential for use in a 1-D span-morphing UAV wingtip. The skin should be capable of sustaining 100% active strain with negligible major axis Poisson's ratio effects, giving a 100% change in surface area, and should also be able to withstand typical aerodynamic loads, assumed to range up to 200 psf (9.58 kPa) for a maneuvering flight surface, with minimal out-of-plane deflection. The following will describe the process of designing, building, and testing a morphing skin with these goals in mind, and will

The primary challenge in developing a morphing skin suitable as an aerodynamic surface is balancing the competing goals of low in-plane actuation requirements and high out-of-plane stiffness. In order to make the skin viable, actuation requirements must be low enough that a reasonable actuation system within the aircraft can stretch the skin to the desired shape and hold it for the required morphing duration. At the same time, the skin must withstand typical aerodynamic loads without deforming excessively (e.g., rippling or bowing), which

To achieve these design goals, a soft, thin silicone elastomer sheet with highly anisotropic carbon fiber reinforcement, called an elastomeric matrix composite (EMC), would be oriented such that the fiber-dominated direction runs chordwise at the wingtip, and the matrix-dominated direction runs spanwise (Figure 2a). Reinforcing carbon fibers controlling the major axis Poisson's ratio of the sheet would limit the EMC to 1-D spanwise shape change (Figure 2b). For a given skin stiffness, actuation requirements will increase in proportion to the skin thickness, *ts*, while out-of-plane stiffness will be proportional to *ts*

the second moment of the area. To alleviate these competing factors, a flexible substructure is desired (Figure 2c) that would be capable of handling out-of-plane loads without greatly adding to the in-plane stiffness. This allows a thinner skin which, in turn, reduces actuation requirements. The combined EMC sheet and substructure form a continuous span-

3 by

would result in degradation to the aerodynamic characteristics of the airfoil surface.

compare the performance of the final article to the initial design objectives.

functional morphing skin.

**2. Conceptual development** 

morphing skin.

morphing could also be used in the chordwise direction, and is not limited in application to fixed-wing aircraft, as rotorcraft would also benefit from a variable diameter or chord rotor.

Fig. 1. Illustration of span-morphing UAV showing 1-D morphing wingtips.

A key challenge in developing a one dimensional morphing structure is the development of a useful morphing skin, defined here as a continuous layer of material that would stretch over the morphing structure and mechanism to form a smooth aerodynamic skin surface. For a span-morphing wingtip in particular, the necessity of a high degree of surface area change, large strain capability in the span direction, and little to no strain in the chordwise direction all impose difficult requirements on any proposed morphing skin. The goal of this effort was a 100% increase in both the span and area of a morphing wingtip, or "morphing cell."

Reviews of contemporary morphing skin technology (Thill *et al*., 2008; Wereley and Gandhi, 2010) yield three major areas of research being pursued: compliant structures, shape memory polymers, and anisotropic elastomeric skins. Compliant structures, such as the FlexSys Inc. Mission Adaptive Compliant Wing (MACW), rely on a highly tailored internal structure and a conventional skin material to allow small amounts of trailing edge camber change (Perkins *et al*., 2004). Due to the large geometrical changes required for a spanmorphing wingtip as envisioned here, metal or resin-matrix-composite skin materials are unsuitable because they are simply unable to achieve the desired goal of 100% increases in morphing cell span and area.

Shape memory polymer (SMP) skin materials are relatively new and have recently received attention for morphing aircraft concepts. They may at first glance seem highly suited to a span-morphing wingtip: shape memory polymers made by Cornerstone Research Group exhibit an order of magnitude change in modulus and up to 200% strain capability when heated past a transition temperature, yet return to their original modulus upon cooling. There have been attempts to capitalize on the capabilities of SMP skins, such as Lockheed Martin's Z-wing morphing UAV concept (Bye and McClure, 2007) and a reconfigurable segmented variable stiffness skin composed of rigid disks and shape memory polymer proposed by McKnight *et al.* (2010). However, electrical heating of the SMP skin to reach transition temperature proved difficult to implement in the wind tunnel test article and the

morphing could also be used in the chordwise direction, and is not limited in application to fixed-wing aircraft, as rotorcraft would also benefit from a variable diameter or chord rotor.

Fig. 1. Illustration of span-morphing UAV showing 1-D morphing wingtips.

cell."

morphing cell span and area.

A key challenge in developing a one dimensional morphing structure is the development of a useful morphing skin, defined here as a continuous layer of material that would stretch over the morphing structure and mechanism to form a smooth aerodynamic skin surface. For a span-morphing wingtip in particular, the necessity of a high degree of surface area change, large strain capability in the span direction, and little to no strain in the chordwise direction all impose difficult requirements on any proposed morphing skin. The goal of this effort was a 100% increase in both the span and area of a morphing wingtip, or "morphing

Reviews of contemporary morphing skin technology (Thill *et al*., 2008; Wereley and Gandhi, 2010) yield three major areas of research being pursued: compliant structures, shape memory polymers, and anisotropic elastomeric skins. Compliant structures, such as the FlexSys Inc. Mission Adaptive Compliant Wing (MACW), rely on a highly tailored internal structure and a conventional skin material to allow small amounts of trailing edge camber change (Perkins *et al*., 2004). Due to the large geometrical changes required for a spanmorphing wingtip as envisioned here, metal or resin-matrix-composite skin materials are unsuitable because they are simply unable to achieve the desired goal of 100% increases in

Shape memory polymer (SMP) skin materials are relatively new and have recently received attention for morphing aircraft concepts. They may at first glance seem highly suited to a span-morphing wingtip: shape memory polymers made by Cornerstone Research Group exhibit an order of magnitude change in modulus and up to 200% strain capability when heated past a transition temperature, yet return to their original modulus upon cooling. There have been attempts to capitalize on the capabilities of SMP skins, such as Lockheed Martin's Z-wing morphing UAV concept (Bye and McClure, 2007) and a reconfigurable segmented variable stiffness skin composed of rigid disks and shape memory polymer proposed by McKnight *et al.* (2010). However, electrical heating of the SMP skin to reach transition temperature proved difficult to implement in the wind tunnel test article and the SMP skin was abandoned as a high-risk option. Additionally, the state-of-the-art of SMP technology does not appear to be well-suited for dynamic control morphing objectives.

With maximum strains above 100%, low stiffness, and a lower degree of risk due to their passive operation, elastomeric materials are ideal candidates for a morphing skin. Isotropic elastomer morphing skins have been successfully implemented on the MFX-1 UAV (Flanagan *et al*., 2007). This UAV employed a mechanized sliding spar wing structure capable of altering the sweep, wing area, and aspect ratio during flight. Sheets of silicone elastomer connect rigid leading and trailing edge spars, forming the upper and lower surfaces of the wing. The elastomer skin is reinforced against out-of-plane loads by ribbons stretched taught immediately underneath the skin, which proved effective for wind tunnel testing and flight testing. Morphing sandwich structures capable of high global strains have also been investigated (Joo *et al.,* 2009; Bubert *et. al.,* 2010; Olympio *et al.,* 2010). However, suitable improvements over these structures, such as anisotropic fiber reinforcement and a better developed substructure for out-of-plane reinforcement, are desired for a fully functional morphing skin.

The present research therefore focuses on the development of a passive anisotropic elastomer composite skin with potential for use in a 1-D span-morphing UAV wingtip. The skin should be capable of sustaining 100% active strain with negligible major axis Poisson's ratio effects, giving a 100% change in surface area, and should also be able to withstand typical aerodynamic loads, assumed to range up to 200 psf (9.58 kPa) for a maneuvering flight surface, with minimal out-of-plane deflection. The following will describe the process of designing, building, and testing a morphing skin with these goals in mind, and will compare the performance of the final article to the initial design objectives.

## **2. Conceptual development**

The primary challenge in developing a morphing skin suitable as an aerodynamic surface is balancing the competing goals of low in-plane actuation requirements and high out-of-plane stiffness. In order to make the skin viable, actuation requirements must be low enough that a reasonable actuation system within the aircraft can stretch the skin to the desired shape and hold it for the required morphing duration. At the same time, the skin must withstand typical aerodynamic loads without deforming excessively (e.g., rippling or bowing), which would result in degradation to the aerodynamic characteristics of the airfoil surface.

To achieve these design goals, a soft, thin silicone elastomer sheet with highly anisotropic carbon fiber reinforcement, called an elastomeric matrix composite (EMC), would be oriented such that the fiber-dominated direction runs chordwise at the wingtip, and the matrix-dominated direction runs spanwise (Figure 2a). Reinforcing carbon fibers controlling the major axis Poisson's ratio of the sheet would limit the EMC to 1-D spanwise shape change (Figure 2b). For a given skin stiffness, actuation requirements will increase in proportion to the skin thickness, *ts*, while out-of-plane stiffness will be proportional to *ts* 3 by the second moment of the area. To alleviate these competing factors, a flexible substructure is desired (Figure 2c) that would be capable of handling out-of-plane loads without greatly adding to the in-plane stiffness. This allows a thinner skin which, in turn, reduces actuation requirements. The combined EMC sheet and substructure form a continuous spanmorphing skin.

One Dimensional Morphing Structures for Advanced Aircraft 7

tailoring the EMC to the application, including elastomer stiffness, durometer, ease of handling during manufacturing, and the quantity, thickness, and angle of carbon fiber

Initially, a large number of silicone elastomers were tested for viability as matrix material. Desired properties included maximum elongation well over 100%, a low stiffness to minimize actuation forces, moderate durometer to avoid having too soft a skin surface, and good working properties. Workability became a primary challenge to overcome, as two-part elastomers with high viscosities or very short work times would not fully wet out the carbon fiber layers. While over a dozen candidate elastomer samples were examined, only four were selected for further testing. Table 1 details the silicone elastomers tested as matrix

DC 3-4207 130 430 100+ difficult to demold Sylgard-186 410 65,000 100+ too viscous V-330, CA-45 570 10,000 500 excellent workability V-330, CA-35 330 10,000 510 excellent workability

The most promising compositions tested were Dow Corning 3-4207 series and the Rhodorsil V-330 series. Both exhibited the desired low stiffness and greater than 100% elongation, but DC 3-4207 suffered from poor working qualities and lower maximum elongation and was not down-selected. Rhodorsil's V-330 series two-part room temperature vulcanization (RTV) silicone elastomer had the desired combination of low viscosity, long working time, and easy demolding to enable effective EMC manufacture, and also demonstrated very high maximum elongation and tear strength. V-330 with CA-35 had the lowest stiffness of the two V-330

Concurrently, using classical laminated plate theory (CLPT), a simple model of the EMC laminate was developed to study the effects of changing composite configuration on performance. The skin lay-up shown in Figure 4a was examined: two silicone elastomer face sheets sandwiching two symmetric unidirectional carbon fiber/elastomer composite laminae. The unidirectional fiber layers are offset by an angle *θ*f from the **1**-axis, which corresponds to the chordwise direction. Orienting the fiber-dominated direction along the wing chord controls minor Poisson's ratio effects while retaining low stiffness and high

In order to determine directional properties of the EMC laminate, directional properties of each lamina must first be found. The following micromechanics derivation comes from Agarwal *et al*. (2006). For a unidirectional sheet with the material longitudinal (**L**) and transverse (**T**) axes oriented along the fiber direction as shown in Figure 4b, we assume that

elastomers tested. This led to selecting V-330, CA-35 for use in test article fabrication.

strain capability in the **2**-axis, which corresponds to the spanwise direction.

**at Break Comments** 

**Material Modulus (kPa) Viscosity (cP) % Elongation**

reinforcement.

candidates.

**3.1 Elastomer selection** 

Table 1. Elastomer properties.

**3.2 CLPT predictions and validation** 

Fig. 2. Design concept as a span morphing wingtip. (Bubert *et al*., 2010)

To motivate the goal of low in-plane stiffness for this research, the skin prototype was designed to be actuated by a span-morphing pneumatic artificial muscle (PAM) scissor mechanism described separately by Wereley and Kothera (2007). The PAM scissor mechanism shown in Figure 3 was designed to transform contraction of the PAM actuator into extensile force necessary in a span-morphing wing. Based upon the maximum performance of the PAM and the kinematics of the scissor frame, the maximum force output of the actuation system was predicted and a skin stiffness goal was determined such that 100% active strain could be achieved, with the skin simplified as having linear stiffness. A margin of 15% was added to the 100% strain goal to account for anticipated losses due to friction or manufacturing shortcomings in the skin or actuation system.

Fig. 3. Morphing skin demonstrator including PAM actuation system.

In addition, minimal out-of-plane deflection of the skin surface under aerodynamic loading was desired. No specific out-of-plane deflection goal was set or designed for, but out-ofplane stiffness of the substructure was kept in mind during the design process. Deflection due to distributed loads was included as a final test to ensure that the aerodynamic shape of a UAV wing morphing structure could be maintained during flight.

## **3. Skin development**

The primary phase of the morphing skin development was to fabricate the EMC sheet that would make up the skin or face sheet. A number of design variables were available for tailoring the EMC to the application, including elastomer stiffness, durometer, ease of handling during manufacturing, and the quantity, thickness, and angle of carbon fiber reinforcement.

### **3.1 Elastomer selection**

6 Recent Advances in Aircraft Technology

a) (b) (c)

To motivate the goal of low in-plane stiffness for this research, the skin prototype was designed to be actuated by a span-morphing pneumatic artificial muscle (PAM) scissor mechanism described separately by Wereley and Kothera (2007). The PAM scissor mechanism shown in Figure 3 was designed to transform contraction of the PAM actuator into extensile force necessary in a span-morphing wing. Based upon the maximum performance of the PAM and the kinematics of the scissor frame, the maximum force output of the actuation system was predicted and a skin stiffness goal was determined such that 100% active strain could be achieved, with the skin simplified as having linear stiffness. A margin of 15% was added to the 100% strain goal to account for anticipated losses due to

Fig. 2. Design concept as a span morphing wingtip. (Bubert *et al*., 2010)

friction or manufacturing shortcomings in the skin or actuation system.

Fig. 3. Morphing skin demonstrator including PAM actuation system.

a UAV wing morphing structure could be maintained during flight.

**3. Skin development** 

In addition, minimal out-of-plane deflection of the skin surface under aerodynamic loading was desired. No specific out-of-plane deflection goal was set or designed for, but out-ofplane stiffness of the substructure was kept in mind during the design process. Deflection due to distributed loads was included as a final test to ensure that the aerodynamic shape of

The primary phase of the morphing skin development was to fabricate the EMC sheet that would make up the skin or face sheet. A number of design variables were available for Initially, a large number of silicone elastomers were tested for viability as matrix material. Desired properties included maximum elongation well over 100%, a low stiffness to minimize actuation forces, moderate durometer to avoid having too soft a skin surface, and good working properties. Workability became a primary challenge to overcome, as two-part elastomers with high viscosities or very short work times would not fully wet out the carbon fiber layers. While over a dozen candidate elastomer samples were examined, only four were selected for further testing. Table 1 details the silicone elastomers tested as matrix candidates.


Table 1. Elastomer properties.

The most promising compositions tested were Dow Corning 3-4207 series and the Rhodorsil V-330 series. Both exhibited the desired low stiffness and greater than 100% elongation, but DC 3-4207 suffered from poor working qualities and lower maximum elongation and was not down-selected. Rhodorsil's V-330 series two-part room temperature vulcanization (RTV) silicone elastomer had the desired combination of low viscosity, long working time, and easy demolding to enable effective EMC manufacture, and also demonstrated very high maximum elongation and tear strength. V-330 with CA-35 had the lowest stiffness of the two V-330 elastomers tested. This led to selecting V-330, CA-35 for use in test article fabrication.

## **3.2 CLPT predictions and validation**

Concurrently, using classical laminated plate theory (CLPT), a simple model of the EMC laminate was developed to study the effects of changing composite configuration on performance. The skin lay-up shown in Figure 4a was examined: two silicone elastomer face sheets sandwiching two symmetric unidirectional carbon fiber/elastomer composite laminae. The unidirectional fiber layers are offset by an angle *θ*f from the **1**-axis, which corresponds to the chordwise direction. Orienting the fiber-dominated direction along the wing chord controls minor Poisson's ratio effects while retaining low stiffness and high strain capability in the **2**-axis, which corresponds to the spanwise direction.

In order to determine directional properties of the EMC laminate, directional properties of each lamina must first be found. The following micromechanics derivation comes from Agarwal *et al*. (2006). For a unidirectional sheet with the material longitudinal (**L**) and transverse (**T**) axes oriented along the fiber direction as shown in Figure 4b, we assume that

One Dimensional Morphing Structures for Advanced Aircraft 9

Recall it was assumed that perfect bonding between fiber and matrix occurred, as illustrated in Figure 6a. This implies stress was equally shared between matrix and fiber under transverse loading. Close visual examination of the EMC samples during testing revealed that the fiber/matrix bond was actually very poor, and the matrix pulled away from individual fibers under transverse loading as illustrated in Figure 6b. Thus, the fibers carry no stress in the transverse direction, and the effective cross-sectional area of matrix left to carry transverse force in the lamina is reduced by the fiber volume fraction. For the case of poor transverse bonding exhibited in the fiber laminae, the transverse modulus in Eq. (2)

Using Eq. (3) to calculate transverse modulus for the fiber laminae, new CLPT predictions for EMC non-dimensionalized transverse modulus and minor Poisson's ratio are also plotted in Figure 5a and 5b, respectively. Much better agreement is seen between the analytical and experimental values for *E*2/*E*m. In spite of the poor bond between fiber and matrix material in the EMCs, the fiber stiffness still appears to contribute to the transverse stiffness at higher fiber offset angles. The minor Poisson's ratio is also influenced by the fiber offset angle. The EMC's longitudinal modulus, not shown, also remains high. These findings clearly indicate the fiber continues to contribute to the longitudinal stiffness of the fiber

To explain this contribution, it is hypothesized that friction between fiber and matrix help share load between the two materials in the longitudinal direction, while the matrix is free to pull away from the fiber in the transverse direction. This would explain the stiffening effect seen in the transverse modulus at increased offset angles and the controlling effect the

(a) (b)


21

Fig. 5. Comparison of CLPT predictions with experimental data for three different fiber angles (a) non-dimensionalized transverse elastic modulus *E2*/*Em,* (b) minor Poisson's ratio

laminae even when bonding between matrix and fiber is poor.

fiber appears to have on Poisson's ratio at very low offset angles.


Transverse Modulus vs Fiber Angle

CLPT, perfect bonding CLPT poor bonding Experiment

Offset, deg.

*EE V* <sup>T</sup> *m f* /(1 ) (3)


Poissons Ratio vs Fiber Angle

CLPT, perfect bonding CLPT poor bonding Experiment

Offset, deg.

can thus be simplified to:

21. 0.8 1 1.2 1.4 1.6 1.8 2 2.2

E2/Em

Fig. 4*.* (a) EMC lay-up used in CLPT predictions. (b) Unidirectional composite layer showing fiber orientation.

perfect bonding occurs between the fiber and matrix material such that equal strain is experienced by both fiber and matrix in the **L** direction. Based upon these assumptions, the longitudinal elastic modulus is given by the rule of mixtures:

$$E\_{\rm L} = E\_f V\_f + E\_m (1 - V\_f) \tag{1}$$

Here *E***L** is the longitudinal elastic modulus for the layer, *E*f is the fiber elastic modulus, *E*m is the matrix elastic modulus, and *V*f is the fiber volume fraction. To find the elastic modulus in the transverse direction, it is assumed that stress is uniform through the matrix and fiber. The equation for the transverse modulus, *E***T**, is:

$$E\_\mathrm{T} = \mathbf{1} \left< \left( V\_f \, \right| \, E\_f + \left( \mathbf{1} - V\_f \right) / E\_\mathrm{m} \right> \tag{2}$$

Calculations based on these micromechanics assumptions supported the intuitive conclusion that thinner EMC skins would have a lower in-plane stiffness modulus in the spanwise direction, *E*2. Predictions for the transverse elastic modulus and the minor Poisson's ratio are plotted versus fiber offset angle in Figure 5a and Figure 5b, respectively, as solid lines. In order to provide some validation for the CLPT predictions, three EMC sample coupons were manufactured, consisting of 0.5 mm elastomer face sheets sandwiching two 0.2-0.3 mm composite lamina with a fiber volume fraction of 0.7. Nominal fiber axis offset angles of 0°, 10°, and 20° were used. The measured transverse modulus and minor Poisson's ratio are plotted as circles in their respective figure. As expected, increasing fiber offset angle increases the in-plane stiffness of the EMC, requiring greater actuation forces. Also, it is noteworthy that the inclusion of unidirectional fiber reinforcement at 0° offset angle nearly eliminates minor Poisson's ratio effects as predicted by CLPT theory.

It is of critical importance to note that, according to the assumptions used in deriving the lamina transverse modulus in Eq. (2), the transverse modulus has a lower bound equal to the matrix modulus. This lower bound is shown in Figure 5a as a horizontal black line at *E*2/*E*m = 1. However, the experimental data is close to this lower bound for the 10° and 20° samples, and the modulus is actually below the lower bound for the 0° case. Clearly in this case there is a problem in the micromechanics from which the transverse modulus prediction was derived.

(a) (b) Fig. 4*.* (a) EMC lay-up used in CLPT predictions. (b) Unidirectional composite layer showing

perfect bonding occurs between the fiber and matrix material such that equal strain is experienced by both fiber and matrix in the **L** direction. Based upon these assumptions, the

Here *E***L** is the longitudinal elastic modulus for the layer, *E*f is the fiber elastic modulus, *E*m is the matrix elastic modulus, and *V*f is the fiber volume fraction. To find the elastic modulus in the transverse direction, it is assumed that stress is uniform through the matrix and fiber.

Calculations based on these micromechanics assumptions supported the intuitive conclusion that thinner EMC skins would have a lower in-plane stiffness modulus in the spanwise direction, *E*2. Predictions for the transverse elastic modulus and the minor Poisson's ratio are plotted versus fiber offset angle in Figure 5a and Figure 5b, respectively, as solid lines. In order to provide some validation for the CLPT predictions, three EMC sample coupons were manufactured, consisting of 0.5 mm elastomer face sheets sandwiching two 0.2-0.3 mm composite lamina with a fiber volume fraction of 0.7. Nominal fiber axis offset angles of 0°, 10°, and 20° were used. The measured transverse modulus and minor Poisson's ratio are plotted as circles in their respective figure. As expected, increasing fiber offset angle increases the in-plane stiffness of the EMC, requiring greater actuation forces. Also, it is noteworthy that the inclusion of unidirectional fiber reinforcement at 0° offset angle nearly eliminates minor Poisson's ratio effects as predicted by CLPT theory.

It is of critical importance to note that, according to the assumptions used in deriving the lamina transverse modulus in Eq. (2), the transverse modulus has a lower bound equal to the matrix modulus. This lower bound is shown in Figure 5a as a horizontal black line at *E*2/*E*m = 1. However, the experimental data is close to this lower bound for the 10° and 20° samples, and the modulus is actually below the lower bound for the 0° case. Clearly in this case there is a problem in the micromechanics from which the transverse modulus

<sup>L</sup> (1 ) *E EV E V ff m f* (1)

<sup>T</sup> 1 /( / (1 )/ ) *E VE V E ff f m* (2)

longitudinal elastic modulus is given by the rule of mixtures:

The equation for the transverse modulus, *E***T**, is:

fiber orientation.

prediction was derived.

Recall it was assumed that perfect bonding between fiber and matrix occurred, as illustrated in Figure 6a. This implies stress was equally shared between matrix and fiber under transverse loading. Close visual examination of the EMC samples during testing revealed that the fiber/matrix bond was actually very poor, and the matrix pulled away from individual fibers under transverse loading as illustrated in Figure 6b. Thus, the fibers carry no stress in the transverse direction, and the effective cross-sectional area of matrix left to carry transverse force in the lamina is reduced by the fiber volume fraction. For the case of poor transverse bonding exhibited in the fiber laminae, the transverse modulus in Eq. (2) can thus be simplified to:

$$E\_T = E\_m \left/ \left(1 - V\_f\right)\right.\tag{3}$$

Using Eq. (3) to calculate transverse modulus for the fiber laminae, new CLPT predictions for EMC non-dimensionalized transverse modulus and minor Poisson's ratio are also plotted in Figure 5a and 5b, respectively. Much better agreement is seen between the analytical and experimental values for *E*2/*E*m. In spite of the poor bond between fiber and matrix material in the EMCs, the fiber stiffness still appears to contribute to the transverse stiffness at higher fiber offset angles. The minor Poisson's ratio is also influenced by the fiber offset angle. The EMC's longitudinal modulus, not shown, also remains high. These findings clearly indicate the fiber continues to contribute to the longitudinal stiffness of the fiber laminae even when bonding between matrix and fiber is poor.

To explain this contribution, it is hypothesized that friction between fiber and matrix help share load between the two materials in the longitudinal direction, while the matrix is free to pull away from the fiber in the transverse direction. This would explain the stiffening effect seen in the transverse modulus at increased offset angles and the controlling effect the fiber appears to have on Poisson's ratio at very low offset angles.

Fig. 5. Comparison of CLPT predictions with experimental data for three different fiber angles (a) non-dimensionalized transverse elastic modulus *E2*/*Em,* (b) minor Poisson's ratio 21.

One Dimensional Morphing Structures for Advanced Aircraft 11

for the three EMC samples. EMC #3 was not intended to be used in the final morphing skin demonstrator, but instead was an academic exercise intended to increase out-of-plane

(a) (b)

Sample strips measuring 51 mm x 152 mm were cut from the three EMCs and tested on a Material Test System (MTS) machine. Each sample was strained to 100% of its original length and then returned to its resting position. The test setup is depicted in Figure 8a and data from these tests are presented in Figure 8b. Notice the visibly low Poisson's ratio effects as the EMC is stretched to 100% strain in Figure 8a – there is little measurable reduction in width. It is also important to note that the stress-strain curves measured for each EMC reflect not only the impact of their lay-ups on stiffness, but also improvements in

Fig. 8. In-plane skin testing, (a) EMC sample taken to 100% strain; (b) data from EMC

Fiber layer thickness (mm)

Total Thickness (mm)

Fiber orientation (deg)

EMC #1 0.5 0 0.4 1.4 EMC #2 0.5 0 0.7 1.7 EMC #3 0.5 +15, 0, -15 deg 0.8 1.8

Fig. 7. Progression of skin manufacturing process.

stiffness at the expense of in-plane stiffness.

Table 2. Summary of EMC sample properties.

Sheet thickness (mm)

samples.

Fig. 6. Fiber/matrix bond (a) assumed perfect bonding and equal transverse stress sharing in CLPT, (b) actual condition with poor fiber/matrix bond and no fiber stress under transverse loading.

Based upon these CLPT results, a fiber offset angle of 0° was selected to minimize transverse stiffness and also to minimize the minor Poisson's ratio. As the analytical and experimental results in Figure 5b indicate, a 0° fiber offset angle can resist chordwise shape change during spanwise morphing. While this conclusion appears obvious, the results demonstrate that with the appropriate correction to micromechanics assumptions in the transverse direction, simple CLPT analysis can be more confidently used to predict EMC directional properties. This simplifies the morphing skin design procedure by allowing in-plane EMC stiffness to be predicted by analytical methods.

#### **3.3 EMC fabrication and testing**

A key issue in this study was developing a dependable and repeatable skin manufacturing process. The final manufacturing process involved a multi-step lay-up process, building the skin up through its thickness (Figure 7). First, a sheet of elastomer was cast between two aluminum caul plates using shim stock to enforce the desired thickness. Secondly, unidirectional carbon fiber was applied to the cured elastomer sheet, with particular attention paid to the alignment of the fibers to ensure that they maintained their uniform spacing and unidirectional orientation (or angular displacement, depending on the sample). Enough additional liquid elastomer was then spread on top of the carbon to wet out all of the fibers. An aluminum caul plate was placed on top of the lay-up, compressing the carbon/elastomer layer while the elastomer cured. The third and final step in the skin layup process was to build the skin up to its final thickness. The bottom sheet of skin with attached carbon fiber was laid out on a caul plate. As in the first step, shim stock was used to enforce the desired thickness (now the full thickness of the skin) and liquid elastomer was poured over the existing sheet. A caul plate was then placed on top of this uncured elastomer and left for at least 4 hours. Once cured, the completed skin was removed from the plates, trimmed of excess material, and inspected for flaws. A successfully manufactured skin had a consistent cross-section and no air bubbles or visible flaws.

Several EMC sheets were originally manufactured in an effort to experimentally test the effect of fiber thickness and orientation on in-plane and out-of-plane characteristics and to attempt to optimize both. Table 2 describes the nominal dimensions and fiber angle values

(a) (b) Fig. 6. Fiber/matrix bond (a) assumed perfect bonding and equal transverse stress sharing in CLPT, (b) actual condition with poor fiber/matrix bond and no fiber stress under

Based upon these CLPT results, a fiber offset angle of 0° was selected to minimize transverse stiffness and also to minimize the minor Poisson's ratio. As the analytical and experimental results in Figure 5b indicate, a 0° fiber offset angle can resist chordwise shape change during spanwise morphing. While this conclusion appears obvious, the results demonstrate that with the appropriate correction to micromechanics assumptions in the transverse direction, simple CLPT analysis can be more confidently used to predict EMC directional properties. This simplifies the morphing skin design procedure by allowing in-plane EMC stiffness to

A key issue in this study was developing a dependable and repeatable skin manufacturing process. The final manufacturing process involved a multi-step lay-up process, building the skin up through its thickness (Figure 7). First, a sheet of elastomer was cast between two aluminum caul plates using shim stock to enforce the desired thickness. Secondly, unidirectional carbon fiber was applied to the cured elastomer sheet, with particular attention paid to the alignment of the fibers to ensure that they maintained their uniform spacing and unidirectional orientation (or angular displacement, depending on the sample). Enough additional liquid elastomer was then spread on top of the carbon to wet out all of the fibers. An aluminum caul plate was placed on top of the lay-up, compressing the carbon/elastomer layer while the elastomer cured. The third and final step in the skin layup process was to build the skin up to its final thickness. The bottom sheet of skin with attached carbon fiber was laid out on a caul plate. As in the first step, shim stock was used to enforce the desired thickness (now the full thickness of the skin) and liquid elastomer was poured over the existing sheet. A caul plate was then placed on top of this uncured elastomer and left for at least 4 hours. Once cured, the completed skin was removed from the plates, trimmed of excess material, and inspected for flaws. A successfully manufactured

Several EMC sheets were originally manufactured in an effort to experimentally test the effect of fiber thickness and orientation on in-plane and out-of-plane characteristics and to attempt to optimize both. Table 2 describes the nominal dimensions and fiber angle values

skin had a consistent cross-section and no air bubbles or visible flaws.

transverse loading.

be predicted by analytical methods.

**3.3 EMC fabrication and testing** 

Fig. 7. Progression of skin manufacturing process.

for the three EMC samples. EMC #3 was not intended to be used in the final morphing skin demonstrator, but instead was an academic exercise intended to increase out-of-plane stiffness at the expense of in-plane stiffness.


Table 2. Summary of EMC sample properties.

Fig. 8. In-plane skin testing, (a) EMC sample taken to 100% strain; (b) data from EMC samples.

Sample strips measuring 51 mm x 152 mm were cut from the three EMCs and tested on a Material Test System (MTS) machine. Each sample was strained to 100% of its original length and then returned to its resting position. The test setup is depicted in Figure 8a and data from these tests are presented in Figure 8b. Notice the visibly low Poisson's ratio effects as the EMC is stretched to 100% strain in Figure 8a – there is little measurable reduction in width. It is also important to note that the stress-strain curves measured for each EMC reflect not only the impact of their lay-ups on stiffness, but also improvements in

One Dimensional Morphing Structures for Advanced Aircraft 13

Fig. 9*.* Comparison of standard, auxetic, and modified zero-Poisson cellular structures

(a) (b) Fig. 10. (a) Geometry of zero-Poisson honeycomb cell, (b) Forces and moments on bending

Bernoulli theory, the cosine component of the force *F* will cause a bending deflection

dimensionalizing the v-shaped member's bending deflection 2

3

(4)

by the cell height *h*. These

0 cos 12 *F*

*E I* 

Here *E*0 is the Young's Modulus of the honeycomb material and *I* is the second moment of the area of the bending member; in this case *I* = *bt*3/12. In order to determine an effective tensile modulus for the honeycomb substructure, the relationship in Eq. (4) between force and displacement needs to be transformed into an equivalent stress-strain relationship. The equivalent stress through one cell can be found by using the cell width *c* and honeycomb depth *b* to establish a reference area, and the global equivalent strain is determined by non-

showing strain relationships.

member leg.

(Shigley *et al*., 2004):

manufacturing ability. Thus, due to improved control of carbon fiber angles and the thickness of elastomer matrix, EMC #3 has roughly the same stiffness as EMC #2, in spite of the larger amount of carbon fiber present and higher fiber angles. EMC #1 exhibited high quality control and linearity of fiber arrangement and has the lowest stiffness of all, regardless of its nominal similarity to EMC #2. Based upon these tests, EMC #1 and EMC #2 were selected for incorporation into integrated test articles. EMC #1 displayed the lowest inplane stiffness, while EMC #2 had the second lowest stiffness, making them the most attractive candidates for a useful morphing skin.

## **4. Substructure development**

The most challenging aspect of the morphing skin to design was the substructure. Structural requirements necessitated high out-of-plane stiffness to help support the aerodynamic pressure load while still maintaining low in-plane stiffness and high strain capability.

## **4.1 Honeycomb design**

The substructure concept originally evolved from the use of honeycomb core reinforcement in composite structures such as rotor blades. Honeycomb structures are naturally suited for high out-of-plane stiffness, and if properly designed can have tailored in-plane stiffness as well (Gibson and Ashby, 1988). By modification of the arrangement of a cellular structure, the desired shape change properties can be incorporated.

In order to create a honeycomb structure with a Poisson's ratio of zero, a negative Poisson's ratio cellular design presented by Chavez *et al*. (2003), or so-called auxetic structure (Evans *et al*., 1991), was rearranged to resemble a series of v-shaped members connecting parallel rib-like members, as seen in Figure 9. This arrangement gives large strains in one direction with no deflection at all in the other by means of extending or compressing the v-shaped members, which essentially act as spring elements. The chordwise rib members act as ribs in a conventional airplane wing by defining the shape of the EMC face sheet and supporting against out-of-plane loads. The v-shaped members connect the ribs into a single deformable substructure which can then be bonded to the EMC face sheet as a unit, with the v-shaped bending members controlling the rib spacing.

For a standard honeycomb, Gibson and Ashby (1988) describe the in-plane stiffness as a ratio of in-plane modulus to material modulus, given in terms of the geometric properties of the honeycomb cells. By modifying this standard equation, it is possible to describe the inplane stiffness of a zero-Poisson honeycomb structure with cell geometric properties as illustrated in Figure 10a. Here *t* is the thickness of the bending (v-shaped) members, *ℓ* is the length of the bending members, *h* is the cell height, *c* is the cell width, and is the angle between the rib members and the bending members. Note that in the figure the cell is being stretched vertically and *F* is the force carried by a bending member under tension. Also note that the depth of the cell, denoted as *b*, is not represented in Figure 10.

With the geometry of the cell defined, an expression can be found for the honeycomb's equivalent of a stress-strain relationship. For small deflections, the bending member between points **1** and **2** can be considered an Euler-Bernoulli beam as shown in Figure 10b, with the forces causing a second mode deflection similar to a pure moment. From Euler-

manufacturing ability. Thus, due to improved control of carbon fiber angles and the thickness of elastomer matrix, EMC #3 has roughly the same stiffness as EMC #2, in spite of the larger amount of carbon fiber present and higher fiber angles. EMC #1 exhibited high quality control and linearity of fiber arrangement and has the lowest stiffness of all, regardless of its nominal similarity to EMC #2. Based upon these tests, EMC #1 and EMC #2 were selected for incorporation into integrated test articles. EMC #1 displayed the lowest inplane stiffness, while EMC #2 had the second lowest stiffness, making them the most

The most challenging aspect of the morphing skin to design was the substructure. Structural requirements necessitated high out-of-plane stiffness to help support the aerodynamic pressure load while still maintaining low in-plane stiffness and high strain capability.

The substructure concept originally evolved from the use of honeycomb core reinforcement in composite structures such as rotor blades. Honeycomb structures are naturally suited for high out-of-plane stiffness, and if properly designed can have tailored in-plane stiffness as well (Gibson and Ashby, 1988). By modification of the arrangement of a cellular structure,

In order to create a honeycomb structure with a Poisson's ratio of zero, a negative Poisson's ratio cellular design presented by Chavez *et al*. (2003), or so-called auxetic structure (Evans *et al*., 1991), was rearranged to resemble a series of v-shaped members connecting parallel rib-like members, as seen in Figure 9. This arrangement gives large strains in one direction with no deflection at all in the other by means of extending or compressing the v-shaped members, which essentially act as spring elements. The chordwise rib members act as ribs in a conventional airplane wing by defining the shape of the EMC face sheet and supporting against out-of-plane loads. The v-shaped members connect the ribs into a single deformable substructure which can then be bonded to the EMC face sheet as a unit, with the v-shaped

For a standard honeycomb, Gibson and Ashby (1988) describe the in-plane stiffness as a ratio of in-plane modulus to material modulus, given in terms of the geometric properties of the honeycomb cells. By modifying this standard equation, it is possible to describe the inplane stiffness of a zero-Poisson honeycomb structure with cell geometric properties as illustrated in Figure 10a. Here *t* is the thickness of the bending (v-shaped) members, *ℓ* is the

between the rib members and the bending members. Note that in the figure the cell is being stretched vertically and *F* is the force carried by a bending member under tension. Also note

With the geometry of the cell defined, an expression can be found for the honeycomb's equivalent of a stress-strain relationship. For small deflections, the bending member between points **1** and **2** can be considered an Euler-Bernoulli beam as shown in Figure 10b, with the forces causing a second mode deflection similar to a pure moment. From Euler-

is the angle

length of the bending members, *h* is the cell height, *c* is the cell width, and

that the depth of the cell, denoted as *b*, is not represented in Figure 10.

attractive candidates for a useful morphing skin.

the desired shape change properties can be incorporated.

bending members controlling the rib spacing.

**4. Substructure development** 

**4.1 Honeycomb design** 

Fig. 9*.* Comparison of standard, auxetic, and modified zero-Poisson cellular structures showing strain relationships.

Fig. 10. (a) Geometry of zero-Poisson honeycomb cell, (b) Forces and moments on bending member leg.

Bernoulli theory, the cosine component of the force *F* will cause a bending deflection (Shigley *et al*., 2004):

$$\mathcal{S} = \frac{F \cos \theta \ell^3}{12E\_0 I} \tag{4}$$

Here *E*0 is the Young's Modulus of the honeycomb material and *I* is the second moment of the area of the bending member; in this case *I* = *bt*3/12. In order to determine an effective tensile modulus for the honeycomb substructure, the relationship in Eq. (4) between force and displacement needs to be transformed into an equivalent stress-strain relationship. The equivalent stress through one cell can be found by using the cell width *c* and honeycomb depth *b* to establish a reference area, and the global equivalent strain is determined by nondimensionalizing the v-shaped member's bending deflection 2by the cell height *h*. These

One Dimensional Morphing Structures for Advanced Aircraft 15

linear least squares regression to the data in Figure 12a. The resulting stiffnesses were then

The strong correlation between the analytical predictions and measured behaviour suggests the assumptions made in the modified Gibson-Ashby equation are accurate over the intended operating range of the honeycomb substructure, and local strains are indeed relatively low. Having low local strain is a benefit as it will increase the fatigue life of the substructure. The low local strains were verified with a finite element analysis that predicted a maximum local strain of 1.5% while undergoing 30% compression globally, a 20:1 ratio. This offers hope that a honeycomb substructure capable of high global strains with a long fatigue life can be designed by minimizing local strain, an area which should be a topic of further research. Further details regarding this structure can be found by

(a) (b)

(a) (b)

To minimize the in-plane stiffness of the substructure, the lowest manufacturable bending member angle, 14°, was selected for integration into complete morphing skin prototypes.

Fig. 12. (a) Stress-strain curves of substructures of various interior angles, (b) In-plane

Fig. 11. (a) Example of Objet PolyJet rapid-prototyped zero-Poisson honeycomb,

plotted with the analytical predictions from Eq. (8) in Figure 12b.

consulting Kothera *et al*. (2011).

(b) morphing substructure on MTS machine.

substructure stiffness, analytical versus experiment.

equivalent stresses and strains are used to determine a transverse stiffness modulus for the honeycomb, *E*2:

$$
\sigma\_2 = \frac{F}{cb},
\tag{5}
$$

$$
\varepsilon\_2 = \frac{\delta \cos \theta}{\hbar / 2},
\tag{6}
$$

$$E\_2 = \frac{\sigma\_{\frac{\nu}{2}}}{\varepsilon\_2} \tag{7}$$

Substituting Eqns. (5) through (7) into Eq. (4) and simplifying yields the following expression for the stiffness of the overall honeycomb relative to the material modulus:

$$\frac{E\_2}{E\_0} = \left(\frac{t}{l}\right)^3 \frac{\sin\theta}{\frac{c}{l}\cos^2\theta} \tag{8}$$

Because this modified Gibson-Ashby model assumes the bending member legs to be beams with low deflection angles and low local strains, Eq. (8) should only be valid for global strains that result in small local deflections. However, it will be shown that due to the nature of the honeycomb design, relatively large global strains are achievable with only small local strains.

With this fairly simple equation, the cell design parameters can easily be varied and their effect on the overall in-plane stiffness of the structure can be studied. For fixed values of *t*, *h*, *c*, and *b*, the modulus ratio of the structure, *E*2/*E*0, increases with the angle . Noting the definitions in Figure 10a, it can be seen that decreasing consequently affects the bending member length *l*, as the upper and lower ends must meet to form a viable structure. Thus, for a given cell height *h*, minimum stiffness limitations are introduced into the design from a practicality standpoint in that the bending members must connect to the structure and cannot intersect one another. Lower in-plane stiffness can be achieved by increasing cell width to accommodate lower bending member angles.

In Figure 11a, an example is given of a zero-Poisson substructure designed in a commercial CAD software and produced on a rapid prototyping machine out of a photocure polymer. Using this method, a large number of samples could be fabricated with variations in bending member angle, . By testing these structures on an MTS machine (Figure 11b), a comparison could be made between the predicted effect of bending member angle on inplane stiffness and the actual observed effect.

The stress-strain test data from a series of rapid prototyped honeycombs is presented in Figure 12a. Each honeycomb was tested over the intended operating range, starting at a reference length of 67% of resting length (pre-compressed) and extending to 133% of resting length to achieve 100% total length change. To test the validity of the modified Gibson-Ashby model, comparisons of experimental data and analytical predictions were made. The stiffness modulus of each experimentally tested honeycomb was determined by applying a

equivalent stresses and strains are used to determine a transverse stiffness modulus for the

<sup>2</sup> , *<sup>F</sup> cb*

> cos , *<sup>h</sup>* /2

> > 2

Substituting Eqns. (5) through (7) into Eq. (4) and simplifying yields the following expression for the stiffness of the overall honeycomb relative to the material modulus:

3

2 0

Because this modified Gibson-Ashby model assumes the bending member legs to be beams with low deflection angles and low local strains, Eq. (8) should only be valid for global strains that result in small local deflections. However, it will be shown that due to the nature of the honeycomb design, relatively large global strains are achievable with only small local

With this fairly simple equation, the cell design parameters can easily be varied and their effect on the overall in-plane stiffness of the structure can be studied. For fixed values of *t*, *h*,

member length *l*, as the upper and lower ends must meet to form a viable structure. Thus, for a given cell height *h*, minimum stiffness limitations are introduced into the design from a practicality standpoint in that the bending members must connect to the structure and cannot intersect one another. Lower in-plane stiffness can be achieved by increasing cell

In Figure 11a, an example is given of a zero-Poisson substructure designed in a commercial CAD software and produced on a rapid prototyping machine out of a photocure polymer. Using this method, a large number of samples could be fabricated with variations in

comparison could be made between the predicted effect of bending member angle on in-

The stress-strain test data from a series of rapid prototyped honeycombs is presented in Figure 12a. Each honeycomb was tested over the intended operating range, starting at a reference length of 67% of resting length (pre-compressed) and extending to 133% of resting length to achieve 100% total length change. To test the validity of the modified Gibson-Ashby model, comparisons of experimental data and analytical predictions were made. The stiffness modulus of each experimentally tested honeycomb was determined by applying a

*l*

2

sin cos

. By testing these structures on an MTS machine (Figure 11b), a

(5)

(6)

(7)

(8)

consequently affects the bending

. Noting the

2

2

*E*

2

*c*, and *b*, the modulus ratio of the structure, *E*2/*E*0, increases with the angle

definitions in Figure 10a, it can be seen that decreasing

width to accommodate lower bending member angles.

plane stiffness and the actual observed effect.

*E t E l c*

honeycomb, *E*2:

strains.

bending member angle,

linear least squares regression to the data in Figure 12a. The resulting stiffnesses were then plotted with the analytical predictions from Eq. (8) in Figure 12b.

The strong correlation between the analytical predictions and measured behaviour suggests the assumptions made in the modified Gibson-Ashby equation are accurate over the intended operating range of the honeycomb substructure, and local strains are indeed relatively low. Having low local strain is a benefit as it will increase the fatigue life of the substructure. The low local strains were verified with a finite element analysis that predicted a maximum local strain of 1.5% while undergoing 30% compression globally, a 20:1 ratio. This offers hope that a honeycomb substructure capable of high global strains with a long fatigue life can be designed by minimizing local strain, an area which should be a topic of further research. Further details regarding this structure can be found by consulting Kothera *et al*. (2011).

Fig. 11. (a) Example of Objet PolyJet rapid-prototyped zero-Poisson honeycomb, (b) morphing substructure on MTS machine.

Fig. 12. (a) Stress-strain curves of substructures of various interior angles, (b) In-plane substructure stiffness, analytical versus experiment.

To minimize the in-plane stiffness of the substructure, the lowest manufacturable bending member angle, 14°, was selected for integration into complete morphing skin prototypes.

One Dimensional Morphing Structures for Advanced Aircraft 17

honeycomb core were designed with raised edges on one side, as shown in Figure 13a. This figure shows a side view of the zero-Poisson honeycomb, where it can be seen that the top surface has the ribs extended taller than the bending members. Therefore, the bonding layer can be applied to the raised rib surfaces and pressed onto the EMC without bonding the bending members to the EMC. A sectional side view of a single honeycomb cell, shown in Figure 13b, illustrates conceptually how the bonded morphing skin looks. A thin layer of adhesive is shown between the EMC and the ribs of the honeycomb, but it does not affect the movement of the bending members. The outermost two ribs on the substructure were each 26 mm wide, providing large bonding areas to carry the load of the skin under strain.

(a) (b)

prototyped VeroBlue DC-700 100 mm

This left 100 mm of active length capable of undergoing high strain deformation.

Fig. 13. EMC-structure bonding method – (a) honeycomb core; (b) single cell diagram.

with the PAM actuation system described in Section 2.

EMC #1, 1.4 mm thick, two

Table 3. Morphing skin configuration.

for the dips in force seen in the figure.

CF layers at 0o

**5.4 In-plane testing** 

The configuration of the morphing skin design is summarized in Table 3. The assembled morphing skin sample was used to assess in-plane and out-of-plane stiffness before fabricating a final 165 mm x 330 mm full scale test article for combination and evaluation

**EMC Honeycomb Adhesive Active Length** 

The morphing skin sample was tested on an MTS machine to 50% strain. The level of strain was limited in order to prevent unforeseen damage to the morphing skin before it could be tested for out-of-plane stiffness as well. In Figure 14a, the morphing skin is shown undergoing in-plane testing, with results presented in Figure 14b. Note that the test procedure strained the specimen incrementally to measure quasi-static stiffness, holding the position briefly before starting with the next stage. Relaxation of the EMC sheet is the cause

Based upon the individually measured stiffnesses of the EMC and substructure components used in the morphing skin and the stiffness of the skin overall, the energy required to strain each structural element can be determined (Figure 15), with the adhesive strain energy found by subtracting the strain energy of the other two components from the total for the morphing skin. The strain energy contribution of each element is broken down in energy

per unit width required to strain the sample from 10 cm to 20 cm.

14o zero-Poisson rapid

Furthermore, this testing demonstrated the usefulness of the modified Gibson-Ashby equation for future honeycomb substructure design efforts. The in-plane stiffness of zero-Poisson honeycomb structures can be predicted.

## **5. One dimensional morphing demonstrator**

## **5.1 Carbon fiber stringers**

One unfortunate aspect of the zero-Poisson honeycomb described above is the lack of bending stiffness about the in-plane axis perpendicular to the rib members. Another structural element is needed to reinforce the substructure for out-of-plane loads. In order to reinforce the substructure, carbon fiber "stringers" were added perpendicular to the rib members. Simply comprised of carbon fiber rods sliding into holes in the substructure, the stringers reinforce the honeycomb against bending about the transverse axis.

The impact of the stringers on the in-plane stiffness of the combined skin was imperceptible. Fit of the stringer through the holes in the substructure was loose and thus the assembly had low friction. Additionally, the EMC sheet and bending members of the substructure kept the substructure ribs stable and vertical, preventing any binding while sliding along the stringers.

## **5.2 EMC/substructure adhesive**

In order to integrate the EMC face sheets with the honeycomb substructure and carry inplane loads, a suitable bonding agent was necessary. The desired adhesive was required to bond the silicone EMC to the plastic rapid-prototyped honeycomb sufficiently to withstand the shear forces generated while deforming the structure. In addition, the adhesive also needed to be capable of high strain levels in order to match the local strain of the EMC at the bond site. Loads imposed on the adhesive by distributed loads (such as aerodynamic loads on the upper surface of a wing) were not taken into account in this preliminary study.

Due to the fact that the substructure, and not the EMC itself, would be attached to the actuation mechanism, the adhesive was required to transfer all the force necessary to strain the EMC sheet. Based upon the known stiffness of the EMCs selected for integration into the morphing skin prototype, the adhesive was required to withstand up to 10.5 N/cm of skin width for 100% area change. The adhesive was to bond the EMC along a strip of plastic 2.54 cm deep, so the equivalent shear strength required was 41.4 kPa. A couple silicone-based candidate adhesives were selected for lap shear evaluation, all of which were capable of high levels of strain. Test results indicated that Dow Corning (DC) 700, Industrial Grade Silicone Sealant, a one-part silicone rubber that is resistant to weathering and withstands temperature extremes, was most capable of bonding the EMC skin to the substructure, as it had a safety factor of 2.

#### **5.3 Morphing structure assembly**

A 152 mm x 152 mm morphing skin sample was fabricated from EMC #1. A 14° angle honeycomb was used for the substructure, and DC 700 adhesive was used to bond the EMC to the honeycomb substructure. To assist in the attachment, the rib members of the

Furthermore, this testing demonstrated the usefulness of the modified Gibson-Ashby equation for future honeycomb substructure design efforts. The in-plane stiffness of zero-

One unfortunate aspect of the zero-Poisson honeycomb described above is the lack of bending stiffness about the in-plane axis perpendicular to the rib members. Another structural element is needed to reinforce the substructure for out-of-plane loads. In order to reinforce the substructure, carbon fiber "stringers" were added perpendicular to the rib members. Simply comprised of carbon fiber rods sliding into holes in the substructure, the

The impact of the stringers on the in-plane stiffness of the combined skin was imperceptible. Fit of the stringer through the holes in the substructure was loose and thus the assembly had low friction. Additionally, the EMC sheet and bending members of the substructure kept the substructure ribs stable and vertical, preventing any binding while sliding along the

In order to integrate the EMC face sheets with the honeycomb substructure and carry inplane loads, a suitable bonding agent was necessary. The desired adhesive was required to bond the silicone EMC to the plastic rapid-prototyped honeycomb sufficiently to withstand the shear forces generated while deforming the structure. In addition, the adhesive also needed to be capable of high strain levels in order to match the local strain of the EMC at the bond site. Loads imposed on the adhesive by distributed loads (such as aerodynamic loads on the upper surface of a wing) were not taken into account in this

Due to the fact that the substructure, and not the EMC itself, would be attached to the actuation mechanism, the adhesive was required to transfer all the force necessary to strain the EMC sheet. Based upon the known stiffness of the EMCs selected for integration into the morphing skin prototype, the adhesive was required to withstand up to 10.5 N/cm of skin width for 100% area change. The adhesive was to bond the EMC along a strip of plastic 2.54 cm deep, so the equivalent shear strength required was 41.4 kPa. A couple silicone-based candidate adhesives were selected for lap shear evaluation, all of which were capable of high levels of strain. Test results indicated that Dow Corning (DC) 700, Industrial Grade Silicone Sealant, a one-part silicone rubber that is resistant to weathering and withstands temperature extremes, was most capable of bonding the EMC skin to the substructure, as it

A 152 mm x 152 mm morphing skin sample was fabricated from EMC #1. A 14° angle honeycomb was used for the substructure, and DC 700 adhesive was used to bond the EMC to the honeycomb substructure. To assist in the attachment, the rib members of the

stringers reinforce the honeycomb against bending about the transverse axis.

Poisson honeycomb structures can be predicted.

**5.1 Carbon fiber stringers** 

**5.2 EMC/substructure adhesive** 

stringers.

preliminary study.

had a safety factor of 2.

**5.3 Morphing structure assembly** 

**5. One dimensional morphing demonstrator** 

honeycomb core were designed with raised edges on one side, as shown in Figure 13a. This figure shows a side view of the zero-Poisson honeycomb, where it can be seen that the top surface has the ribs extended taller than the bending members. Therefore, the bonding layer can be applied to the raised rib surfaces and pressed onto the EMC without bonding the bending members to the EMC. A sectional side view of a single honeycomb cell, shown in Figure 13b, illustrates conceptually how the bonded morphing skin looks. A thin layer of adhesive is shown between the EMC and the ribs of the honeycomb, but it does not affect the movement of the bending members. The outermost two ribs on the substructure were each 26 mm wide, providing large bonding areas to carry the load of the skin under strain. This left 100 mm of active length capable of undergoing high strain deformation.

Fig. 13. EMC-structure bonding method – (a) honeycomb core; (b) single cell diagram.

The configuration of the morphing skin design is summarized in Table 3. The assembled morphing skin sample was used to assess in-plane and out-of-plane stiffness before fabricating a final 165 mm x 330 mm full scale test article for combination and evaluation with the PAM actuation system described in Section 2.


Table 3. Morphing skin configuration.

### **5.4 In-plane testing**

The morphing skin sample was tested on an MTS machine to 50% strain. The level of strain was limited in order to prevent unforeseen damage to the morphing skin before it could be tested for out-of-plane stiffness as well. In Figure 14a, the morphing skin is shown undergoing in-plane testing, with results presented in Figure 14b. Note that the test procedure strained the specimen incrementally to measure quasi-static stiffness, holding the position briefly before starting with the next stage. Relaxation of the EMC sheet is the cause for the dips in force seen in the figure.

Based upon the individually measured stiffnesses of the EMC and substructure components used in the morphing skin and the stiffness of the skin overall, the energy required to strain each structural element can be determined (Figure 15), with the adhesive strain energy found by subtracting the strain energy of the other two components from the total for the morphing skin. The strain energy contribution of each element is broken down in energy per unit width required to strain the sample from 10 cm to 20 cm.

One Dimensional Morphing Structures for Advanced Aircraft 19

layer of sand directly to the surface of the skin, the weight of the lead shot was distributed relatively evenly over the surface of the EMC. Moreover, as the skin deflected under load, the sand would adjust to conform to the surface and continue to spread the weight of the lead. A single-point laser position sensor was also placed underneath to measure the

(a) (b) Fig. 16. (a) Out-of-plane deflection test apparatus design. (b) Out-of-plane deflection results

The test procedure for each morphing skin covered the full range of operation, from resting (neutral position) to 100% area change. Lead-screws were used to set the skin to a nominal strain condition between 0% and 100% of the resting length. The laser position sensor shown below the skin in the figure was positioned in the center of a honeycomb cell at the center of the morphing skin, where the greatest deflection is seen. This positioning was achieved using a small two-axis adjustable table. The laser was zeroed on the under-surface of the EMC, and the relative distance to the bottom of an adjacent rib was measured. This established a zero measurement for rib deflection as well. A layer of sand with known weight was poured onto the surface of the EMC, and lead shot sufficient to load the skin to one of the three desired distributed loads was added to the top of the sand. Wing loadings of 40 psf (1.92 kPa), 100 psf (4.79 kPa), and 200 psf (1.92 kPa) were simulated. Once the load had been applied, measurements were taken at the same points on the EMC and the adjacent rib to determine deflection. These measurements were repeated for four different

strain conditions (0, 25%, 66%, 100%) and the three different noted distributed loads.

morphing skins will require stiffer substructures to withstand out-of-plane loads.

Experimental results from the morphing skins is provided in Figure 16b. It was observed that, relative to the rib deflections, the EMC sheet itself deflected very little (less than 0.25 mm). The results therefore ignore the small EMC deflections and show only the maximum deflection measured on the rib at the midpoint of each morphing skin. Overall, the morphing skin deflections show that as the skin is strained and unsupported length increases, the out-of-plane deflection increases. Naturally, the deflection increases with load as well. Based on observation and on these results, the EMC sheets appeared to carry a greater out-of-plane load than expected, probably due to tension in the skin. EMC deflections between ribs remained low at all loading and strain conditions, while the substructure experienced deflections an order of magnitude greater. Future iterations of

maximum deflections at the center of the skin, between the rib members.

EMC Sheet

as measured on the center rib.

It can be seen that the adhesive had a considerable strain energy requirement, more than double that of the honeycomb substructure. When designing future morphing skins, the energy to strain the adhesive layer must be taken into account to ensure sufficient actuation force is available to meet strain requirements. More careful attention to minimizing the amount of adhesive used to bond the skin and substructure would also likely reduce the inplane stiffness of the morphing skin by a non-trivial amount.

Fig. 14. Morphing skin sample in-plane testing – (a) Skin #1 on MTS; (b) Data from morphing skin in-plane testing.

Fig. 15. Contributions to morphing skin strain energy.

#### **5.5 Out-of-plane testing**

The final phase of evaluation for morphing skin sample required measuring out-of-plane deflection under distributed loadings, approximating aerodynamic forces. A number of testing protocols were investigated, including ASTM standard D 6416/D 6416M for testing simply supported composite plates subject to a distributed load. This particular test protocol is intended for very stiff composites, not flexible or membrane-like composites. A simpler approach to the problem was adopted wherein acrylic retaining walls were placed above the morphing skin sample into which a distributed load of lead shot and sand could be poured. The final configuration of the out-of-plane deflection testing apparatus can be seen in Figure 16a. A set of lead-screws stretched the morphing skin sample from rest to 100% strain. The acrylic retaining walls could be adjusted to match the active skin area, and were tall enough to contain lead shot equivalent to a distributed load of 200 psf (9.58 kPa). By applying a thin

It can be seen that the adhesive had a considerable strain energy requirement, more than double that of the honeycomb substructure. When designing future morphing skins, the energy to strain the adhesive layer must be taken into account to ensure sufficient actuation force is available to meet strain requirements. More careful attention to minimizing the amount of adhesive used to bond the skin and substructure would also likely reduce the in-

(a) (b)

Honeycomb Adhesive Skin

The final phase of evaluation for morphing skin sample required measuring out-of-plane deflection under distributed loadings, approximating aerodynamic forces. A number of testing protocols were investigated, including ASTM standard D 6416/D 6416M for testing simply supported composite plates subject to a distributed load. This particular test protocol is intended for very stiff composites, not flexible or membrane-like composites. A simpler approach to the problem was adopted wherein acrylic retaining walls were placed above the morphing skin sample into which a distributed load of lead shot and sand could be poured. The final configuration of the out-of-plane deflection testing apparatus can be seen in Figure 16a. A set of lead-screws stretched the morphing skin sample from rest to 100% strain. The acrylic retaining walls could be adjusted to match the active skin area, and were tall enough to contain lead shot equivalent to a distributed load of 200 psf (9.58 kPa). By applying a thin

**Components**

Fig. 14. Morphing skin sample in-plane testing – (a) Skin #1 on MTS; (b) Data from

plane stiffness of the morphing skin by a non-trivial amount.

morphing skin in-plane testing.

**5.5 Out-of-plane testing** 

**Energy/width** 

**(J/m)**

Fig. 15. Contributions to morphing skin strain energy.

layer of sand directly to the surface of the skin, the weight of the lead shot was distributed relatively evenly over the surface of the EMC. Moreover, as the skin deflected under load, the sand would adjust to conform to the surface and continue to spread the weight of the lead. A single-point laser position sensor was also placed underneath to measure the maximum deflections at the center of the skin, between the rib members.

Fig. 16. (a) Out-of-plane deflection test apparatus design. (b) Out-of-plane deflection results as measured on the center rib.

The test procedure for each morphing skin covered the full range of operation, from resting (neutral position) to 100% area change. Lead-screws were used to set the skin to a nominal strain condition between 0% and 100% of the resting length. The laser position sensor shown below the skin in the figure was positioned in the center of a honeycomb cell at the center of the morphing skin, where the greatest deflection is seen. This positioning was achieved using a small two-axis adjustable table. The laser was zeroed on the under-surface of the EMC, and the relative distance to the bottom of an adjacent rib was measured. This established a zero measurement for rib deflection as well. A layer of sand with known weight was poured onto the surface of the EMC, and lead shot sufficient to load the skin to one of the three desired distributed loads was added to the top of the sand. Wing loadings of 40 psf (1.92 kPa), 100 psf (4.79 kPa), and 200 psf (1.92 kPa) were simulated. Once the load had been applied, measurements were taken at the same points on the EMC and the adjacent rib to determine deflection. These measurements were repeated for four different strain conditions (0, 25%, 66%, 100%) and the three different noted distributed loads.

Experimental results from the morphing skins is provided in Figure 16b. It was observed that, relative to the rib deflections, the EMC sheet itself deflected very little (less than 0.25 mm). The results therefore ignore the small EMC deflections and show only the maximum deflection measured on the rib at the midpoint of each morphing skin. Overall, the morphing skin deflections show that as the skin is strained and unsupported length increases, the out-of-plane deflection increases. Naturally, the deflection increases with load as well. Based on observation and on these results, the EMC sheets appeared to carry a greater out-of-plane load than expected, probably due to tension in the skin. EMC deflections between ribs remained low at all loading and strain conditions, while the substructure experienced deflections an order of magnitude greater. Future iterations of morphing skins will require stiffer substructures to withstand out-of-plane loads.

One Dimensional Morphing Structures for Advanced Aircraft 21

performance data, while not perfectly linear, approximately matches the slope of the experimental skin stiffness and intersects the actuation system experimental data near 100% extension. Furthermore, although the performance data falls roughly 15% short of original predictions, the morphing skin meets the design goal, validating the analytical design process. Losses were not included in the original system predictions. However, the margin of error included in the original design for friction, increased skin stiffness, and other losses enabled the final morphing cell prototype to achieve 100% strain. It should also be noted that 100% area increases could be achieved repeatedly at 1 Hz using manual actuator

Building on the success of the 1-D morphing demonstrator, a wind tunnel-ready morphing wing was designed and tested. A key technical issue addressed here was determining the scalability of the skin and substructure manufacturing processes for use on a real UAV. Thus, the prototype airfoil system was designed such that future integration with a candidate UAV is feasible, and experimentally evaluated as a wind tunnel prototype. Nominal design parameters for the prototype are a 30.5 cm chord wing section capable of 100% span extension over a 61.0 cm active morphing section with less than 2.54 mm of outof-plane deflection between ribs due to dynamic pressures consistent with a 130 kph

Initially, the planar core design was extruded and cut into the form of a NACA 633-618 airfoil with a chord of approximately 30.5 cm and span of 91.4 cm. A segment of the resulting morphing airfoil core appears in Figure 19a. While this morphing structure is capable of achieving greater than 100% length change itself, it has insufficient spanwise bending and torsional stiffness and so does not constitute a viable wing structure. The structure was therefore augmented with continuous sliding spars. Additionally, the center of the wing structure was hollowed out to potentially accommodate an actuation system for

pressurization.

Fig. 18. Morphing cell data comparison with predictions.

**6. Wind tunnel prototype** 

maximum speed.

the span extension.

**6.1 Structure development** 

#### **5.6 Full scale integration and evaluation**

After proving capable of reaching over 100% strain with largely acceptable out-of-plane performance, the morphing skin sample from the previous subsection was used as the basis for a larger test article. A 34.3 cm x 14 cm morphing skin, nominally identical to the morphing skin sample in configuration, was fabricated and attached to the actuation assembly. The actuation assembly, honeycomb substructure, and completed morphing cell can be seen in Figure 17. Individual components of the system are pictured in Figure 17a, while the assembled morphing skin test article appears in Figure 17b. The active region stretches from 9.1 cm to 18.3 cm with no transverse contraction, thus, producing a 1-D, 100% increase in surface area with zero Poisson's ratio.

Fig. 17. Integration of morphing cell – (a) actuation and substructure components; (b) complete morphing cell exhibiting 100% area change.

To characterize the static performance of the morphing cell, input pressure to the PAM actuators was increased incrementally and the strain of the active region was recorded at each input pressure, and a load cell in line with one PAM recorded actuator force for comparison to predicted values. This measurement process was repeated three times, recording strain, input pressure, and actuator force at each point. Note that the entire upper surface of the EMC is not the active region: each of the fixed-length ends of the honeycomb was designed and manufactured with 25.4 mm of excess material to allow adequate EMC bonding area and an attachment point to the mechanism. This inactive region can be seen on the top and bottom of the honeycomb shown in Figure 17a. The two extremities of the arrows in Figure 17b also account for the inactive region at both ends of the morphing skin.

The static strain response to input actuator pressure is displayed in Figure 18. Strain is seen to level off with increasing pressure due to a combination of mechanism kinematics and the PAM actuator characteristics, but the system was measured to achieve 100% strain with the PAMs pressurized to slightly over 620 kPa.

The measured system performance matches analytical predictions very closely. The previously mentioned analytical predictions and associated experimental data for the actuation system and skin performance are also repeated in this figure. The morphing cell

After proving capable of reaching over 100% strain with largely acceptable out-of-plane performance, the morphing skin sample from the previous subsection was used as the basis for a larger test article. A 34.3 cm x 14 cm morphing skin, nominally identical to the morphing skin sample in configuration, was fabricated and attached to the actuation assembly. The actuation assembly, honeycomb substructure, and completed morphing cell can be seen in Figure 17. Individual components of the system are pictured in Figure 17a, while the assembled morphing skin test article appears in Figure 17b. The active region stretches from 9.1 cm to 18.3 cm with no transverse contraction, thus, producing a 1-D, 100%

(a) (b)

Fig. 17. Integration of morphing cell – (a) actuation and substructure components; (b)

To characterize the static performance of the morphing cell, input pressure to the PAM actuators was increased incrementally and the strain of the active region was recorded at each input pressure, and a load cell in line with one PAM recorded actuator force for comparison to predicted values. This measurement process was repeated three times, recording strain, input pressure, and actuator force at each point. Note that the entire upper surface of the EMC is not the active region: each of the fixed-length ends of the honeycomb was designed and manufactured with 25.4 mm of excess material to allow adequate EMC bonding area and an attachment point to the mechanism. This inactive region can be seen on the top and bottom of the honeycomb shown in Figure 17a. The two extremities of the arrows in Figure 17b also account for the inactive region at both ends of the morphing skin. The static strain response to input actuator pressure is displayed in Figure 18. Strain is seen to level off with increasing pressure due to a combination of mechanism kinematics and the PAM actuator characteristics, but the system was measured to achieve 100% strain with the

The measured system performance matches analytical predictions very closely. The previously mentioned analytical predictions and associated experimental data for the actuation system and skin performance are also repeated in this figure. The morphing cell

**5.6 Full scale integration and evaluation** 

increase in surface area with zero Poisson's ratio.

complete morphing cell exhibiting 100% area change.

PAMs pressurized to slightly over 620 kPa.

performance data, while not perfectly linear, approximately matches the slope of the experimental skin stiffness and intersects the actuation system experimental data near 100% extension. Furthermore, although the performance data falls roughly 15% short of original predictions, the morphing skin meets the design goal, validating the analytical design process. Losses were not included in the original system predictions. However, the margin of error included in the original design for friction, increased skin stiffness, and other losses enabled the final morphing cell prototype to achieve 100% strain. It should also be noted that 100% area increases could be achieved repeatedly at 1 Hz using manual actuator pressurization.

Fig. 18. Morphing cell data comparison with predictions.

## **6. Wind tunnel prototype**

Building on the success of the 1-D morphing demonstrator, a wind tunnel-ready morphing wing was designed and tested. A key technical issue addressed here was determining the scalability of the skin and substructure manufacturing processes for use on a real UAV. Thus, the prototype airfoil system was designed such that future integration with a candidate UAV is feasible, and experimentally evaluated as a wind tunnel prototype. Nominal design parameters for the prototype are a 30.5 cm chord wing section capable of 100% span extension over a 61.0 cm active morphing section with less than 2.54 mm of outof-plane deflection between ribs due to dynamic pressures consistent with a 130 kph maximum speed.

#### **6.1 Structure development**

Initially, the planar core design was extruded and cut into the form of a NACA 633-618 airfoil with a chord of approximately 30.5 cm and span of 91.4 cm. A segment of the resulting morphing airfoil core appears in Figure 19a. While this morphing structure is capable of achieving greater than 100% length change itself, it has insufficient spanwise bending and torsional stiffness and so does not constitute a viable wing structure. The structure was therefore augmented with continuous sliding spars. Additionally, the center of the wing structure was hollowed out to potentially accommodate an actuation system for the span extension.

One Dimensional Morphing Structures for Advanced Aircraft 23

Spanwise bending and torsional stiffness was provided by two 1.91 cm diameter carbon fiber spars. The spars were anchored at the leftmost outboard portion of the wing but were free to slide through the inboard end plates, thus allowing the wing to extend while maintaining structural integrity. The spars were sized in bending to deflect less than 2.54 cm at 100% extension under the maximum expected aerodynamic loads. Note that the spars are also capable of resolving torsional pitching moments, but as the express purpose of the present work was to demonstrate the feasibility of a span morphing wing, these torsional

(a)

(b)

Fig. 20. Assembled core with spars and end plates – (a) contracted state; (b) extended state.

The skin was bonded to the morphing substructure using DC-700. The skin was attached to each rib member, but not to the v-shaped bending members. Particular caution was used when bonding the skin to the end plates, as all of the tensile stress in the skin was resolved

At the resting condition with no elastic energy stored in the skin, Figure 21 shows the 0% morphing state with a 61.0 cm span. Increasing the span by another 61.0 cm highlights the full potential of this morphing system as the prototype wing section doubles its initial span, which has gone from 61.0 cm to 122 cm to show the 100% morphing capability (Figure 21b). Recall from the design that the wing section chord stays constant during these span

properties were not directly evaluated.

**6.2 Prototype integration** 

through its connection to the end plates.

The final form of the morphing airfoil core is shown in Figure 19b. This figure shows a shelllike section mostly around the center of the airfoil, where an actuator could be located. Both the leading and trailing edges feature circular cut-outs to accommodate the carbon fiber spars, and near the trailing edge is a solid thickness airfoil shape for more rigidity where the airfoil is thinnest. The spars were sized using simple Euler-Bernoulli beam approximations and a desired tip deflection of less than 6.4 mm at full extension.

Fig. 19. (a) Final substructure design, cross-section view (b) Manufactured substructure, side view.

Due to the complex geometry of the morphing core and the desire for rapid part turn around, a stereo lithographic rapid-prototyping machine was again used to manufacture the morphing core sections from an acrylic-based photopolymer. The viability of this approach for flyable aircraft applications would have to be studied, but the material/manufacturing approach was sufficient for this proof-of-concept structural demonstrator. Other fabrication techniques such as investment casting, electrical discharge machining, etc. could be considered when fabricating this structure to meet full scale aircraft requirements. It should also be noted that the prototype will feature three of the core segments shown in Figure 19b. They will be pre-compressed when the EMC skin is bonded to allow for more expansion capability and introduce a nominal amount of tension in the EMC skin.

Figure 20a shows the core sections together between two aluminum end plates, with the leading edge and trailing edge support spars. The end plates were sized to provide a suitably large bonding surface for attaching the skin on the tip and root of the morphing section. In this configuration, the core sections are initially contracted such that the active span length is 61.0 cm. In terms of the aircraft, this contracted state will be considered the neutral, resting state because the EMC skin will not be stretched here and a potential actuation system would not be engaged. Hence, this is the condition in which the skin would be bonded to the core. Also shown in Figure 20b is the same arrangement in the fully extended (100% span increase) state with a span of 122.0 cm. The figure shows that the spacing between each of the rib-like members has nearly doubled from what was shown in the contracted state. This figure helps illustrate the large area morphing potential of this technological development in a way that could not be seen once the skin was attached.

The final form of the morphing airfoil core is shown in Figure 19b. This figure shows a shelllike section mostly around the center of the airfoil, where an actuator could be located. Both the leading and trailing edges feature circular cut-outs to accommodate the carbon fiber spars, and near the trailing edge is a solid thickness airfoil shape for more rigidity where the airfoil is thinnest. The spars were sized using simple Euler-Bernoulli beam approximations

(a) (b) Fig. 19. (a) Final substructure design, cross-section view (b) Manufactured substructure, side

Due to the complex geometry of the morphing core and the desire for rapid part turn around, a stereo lithographic rapid-prototyping machine was again used to manufacture the morphing core sections from an acrylic-based photopolymer. The viability of this approach for flyable aircraft applications would have to be studied, but the material/manufacturing approach was sufficient for this proof-of-concept structural demonstrator. Other fabrication techniques such as investment casting, electrical discharge machining, etc. could be considered when fabricating this structure to meet full scale aircraft requirements. It should also be noted that the prototype will feature three of the core segments shown in Figure 19b. They will be pre-compressed when the EMC skin is bonded to allow for more expansion

Figure 20a shows the core sections together between two aluminum end plates, with the leading edge and trailing edge support spars. The end plates were sized to provide a suitably large bonding surface for attaching the skin on the tip and root of the morphing section. In this configuration, the core sections are initially contracted such that the active span length is 61.0 cm. In terms of the aircraft, this contracted state will be considered the neutral, resting state because the EMC skin will not be stretched here and a potential actuation system would not be engaged. Hence, this is the condition in which the skin would be bonded to the core. Also shown in Figure 20b is the same arrangement in the fully extended (100% span increase) state with a span of 122.0 cm. The figure shows that the spacing between each of the rib-like members has nearly doubled from what was shown in the contracted state. This figure helps illustrate the large area morphing potential of this technological development in a way that could not be seen once the skin was attached.

capability and introduce a nominal amount of tension in the EMC skin.

and a desired tip deflection of less than 6.4 mm at full extension.

view.

Spanwise bending and torsional stiffness was provided by two 1.91 cm diameter carbon fiber spars. The spars were anchored at the leftmost outboard portion of the wing but were free to slide through the inboard end plates, thus allowing the wing to extend while maintaining structural integrity. The spars were sized in bending to deflect less than 2.54 cm at 100% extension under the maximum expected aerodynamic loads. Note that the spars are also capable of resolving torsional pitching moments, but as the express purpose of the present work was to demonstrate the feasibility of a span morphing wing, these torsional properties were not directly evaluated.

Fig. 20. Assembled core with spars and end plates – (a) contracted state; (b) extended state.

#### **6.2 Prototype integration**

The skin was bonded to the morphing substructure using DC-700. The skin was attached to each rib member, but not to the v-shaped bending members. Particular caution was used when bonding the skin to the end plates, as all of the tensile stress in the skin was resolved through its connection to the end plates.

At the resting condition with no elastic energy stored in the skin, Figure 21 shows the 0% morphing state with a 61.0 cm span. Increasing the span by another 61.0 cm highlights the full potential of this morphing system as the prototype wing section doubles its initial span, which has gone from 61.0 cm to 122 cm to show the 100% morphing capability (Figure 21b). Recall from the design that the wing section chord stays constant during these span

One Dimensional Morphing Structures for Advanced Aircraft 25

Both the cruise (105 kph) and maximum (130 kph) rated speeds of the candidate UAV were tested. Three angles-of-attack (0o, 2o, 4o) and three wing span morphing conditions (0%, 50%, 100%) were also included in the test matrix. Tests were performed by first setting the morphing condition of the wing section, then positioning the wing for the desired angle-ofattack (AOA). With these values fixed, the tunnel was turned on and the speed was increased incrementally, stopping at the two noted test speeds while experimental observations were made. Tests were completed at each of the conditions in the table indicated with an x-mark. Note that tests were not performed at two of the angles-of-attack at the 100% morphing condition. This was because the skin began to debond near the trailing edge at one of the end plates. This occurred over a section approximately 7.6 cm in span at the 100% morphing condition, though the majority of the prototype remained intact. After removing the wing section from the wind tunnel and inspecting the debonded corner, it was discovered that very little adhesive was on the skin, core, and end plate. Thus, the likely cause for this particular debonding was inconsistent surface preparation, which can easily be rectified in future refinements. Note that the upper surface of the trailing edge experiences relatively small dynamic pressures compared to the rest of the wing, so that this debonding was most likely unrelated to the wind tunnel test. Rather, it was the result of

(a) (b) (c) Fig. 22. Wind tunnel test setup – (a) Overall wind tunnel setup at 100% morphing; (b) Wing installed in wind tunnel – from trailing edge; (c) Picture of wing section leading edge at 130

During execution of the test matrix, digital photographs (Figure 22c) were taken of the leading edge at each test point to determine the amount of skin deflection (e.g., dimpling) that resulted from the dynamic pressure. The leading edge location was chosen as the point to measure because the pressure is highest at the stagnation point. Pictures were taken perpendicular to the air flow direction and angled from the trailing edge, looking forward on the upper skin surface. Grids were taped to the outside of the transparent wall on the opposite side of the test section to provide reference lengths for processing. The

manufacturing inconsistency.

kph, 100% morphing.

extensions, so the morphing percentages indicated (e.g., 100%) are consistent with the increase in wing area. As a fixed point of reference in each of these figures, note that the length of the white poster board underneath the prototype wing section does not change. Note also that this demonstration will use fixed-length internal spreader bars to hold the structure in different morphing lengths. Actuation was achieved by manually stretching the skin/core structure and then attaching the appropriate spreader bar to maintain the stretched distance.

Fig. 21. Prototype morphing wing demonstration – (a) resting length, 0% morphing; (b) 61.0 cm span extension, 100% morphing.

#### **6.3 Wind tunnel testing**

Having shown that the prototype morphing wing section could achieve the goal of 100% span morphing for a total 100% wing area increase, the final test that was performed placed the wing section in a wind tunnel. The purpose of this test was to ensure that the EMC skin and core could maintain a viable airfoil shape at different morphing states under true aerodynamic loading, with minimal out-of-plane deflection between ribs. An open circuit wind tunnel at the University of Maryland with a 50.8 cm tall, 71.1 cm wide test section was used in this test. An overall view of the test section is shown in Figure 22a, with the wing at its extended length, and a close-up view of the test section is shown in Figure 22b looking upstream from the trailing edge.

With only a 50.8 cm tall test section in the wind tunnel, where only this span length of the prototype morphing wing would be placed in the wind flow, while the remaining span and support structure was below the tunnel. This is illustrated in Figure 22a, where the full extension condition (100% morphing) is shown. It should also be noted that while only a 50.8 cm span section of the wing is in the air flow, this is sufficient to determine whether or not the skin and core can maintain a viable airfoil shape in the presence of representative aerodynamic conditions, which was the primary goal of this test. That is, the morphing core motion and skin stretching is consistent and substantially uniform across the span of the prototype, so any characteristics seen in one small section of the wing could similarly be seen or expected elsewhere in the wing, making this 50.8 cm span "sampling" a reasonable measure of system performance.

extensions, so the morphing percentages indicated (e.g., 100%) are consistent with the increase in wing area. As a fixed point of reference in each of these figures, note that the length of the white poster board underneath the prototype wing section does not change. Note also that this demonstration will use fixed-length internal spreader bars to hold the structure in different morphing lengths. Actuation was achieved by manually stretching the skin/core structure and then attaching the appropriate spreader bar to maintain the

(a) (b) Fig. 21. Prototype morphing wing demonstration – (a) resting length, 0% morphing; (b) 61.0

Having shown that the prototype morphing wing section could achieve the goal of 100% span morphing for a total 100% wing area increase, the final test that was performed placed the wing section in a wind tunnel. The purpose of this test was to ensure that the EMC skin and core could maintain a viable airfoil shape at different morphing states under true aerodynamic loading, with minimal out-of-plane deflection between ribs. An open circuit wind tunnel at the University of Maryland with a 50.8 cm tall, 71.1 cm wide test section was used in this test. An overall view of the test section is shown in Figure 22a, with the wing at its extended length, and a close-up view of the test section is shown in Figure 22b looking

With only a 50.8 cm tall test section in the wind tunnel, where only this span length of the prototype morphing wing would be placed in the wind flow, while the remaining span and support structure was below the tunnel. This is illustrated in Figure 22a, where the full extension condition (100% morphing) is shown. It should also be noted that while only a 50.8 cm span section of the wing is in the air flow, this is sufficient to determine whether or not the skin and core can maintain a viable airfoil shape in the presence of representative aerodynamic conditions, which was the primary goal of this test. That is, the morphing core motion and skin stretching is consistent and substantially uniform across the span of the prototype, so any characteristics seen in one small section of the wing could similarly be seen or expected elsewhere in the wing, making this 50.8 cm span "sampling" a reasonable

stretched distance.

cm span extension, 100% morphing.

upstream from the trailing edge.

measure of system performance.

**6.3 Wind tunnel testing** 

Both the cruise (105 kph) and maximum (130 kph) rated speeds of the candidate UAV were tested. Three angles-of-attack (0o, 2o, 4o) and three wing span morphing conditions (0%, 50%, 100%) were also included in the test matrix. Tests were performed by first setting the morphing condition of the wing section, then positioning the wing for the desired angle-ofattack (AOA). With these values fixed, the tunnel was turned on and the speed was increased incrementally, stopping at the two noted test speeds while experimental observations were made. Tests were completed at each of the conditions in the table indicated with an x-mark. Note that tests were not performed at two of the angles-of-attack at the 100% morphing condition. This was because the skin began to debond near the trailing edge at one of the end plates. This occurred over a section approximately 7.6 cm in span at the 100% morphing condition, though the majority of the prototype remained intact. After removing the wing section from the wind tunnel and inspecting the debonded corner, it was discovered that very little adhesive was on the skin, core, and end plate. Thus, the likely cause for this particular debonding was inconsistent surface preparation, which can easily be rectified in future refinements. Note that the upper surface of the trailing edge experiences relatively small dynamic pressures compared to the rest of the wing, so that this debonding was most likely unrelated to the wind tunnel test. Rather, it was the result of manufacturing inconsistency.

Fig. 22. Wind tunnel test setup – (a) Overall wind tunnel setup at 100% morphing; (b) Wing installed in wind tunnel – from trailing edge; (c) Picture of wing section leading edge at 130 kph, 100% morphing.

During execution of the test matrix, digital photographs (Figure 22c) were taken of the leading edge at each test point to determine the amount of skin deflection (e.g., dimpling) that resulted from the dynamic pressure. The leading edge location was chosen as the point to measure because the pressure is highest at the stagnation point. Pictures were taken perpendicular to the air flow direction and angled from the trailing edge, looking forward on the upper skin surface. Grids were taped to the outside of the transparent wall on the opposite side of the test section to provide reference lengths for processing. The

One Dimensional Morphing Structures for Advanced Aircraft 27

Agarwal, B. D., Broutman, L. J., and Chandrashekhara, K. (2006). *Analysis and Performance of* 

Bae, J.S., Seigler, T.M. and Inman, D.J. (2005). ''Aerodynamic and Static Aeroelastic

Barbarino, S., Bilgen, O., Ajaj, R.M., Friswell, M.I., and Inman, D.J. (2011). "A Review of

Bubert, E.A., Woods, B.K.S., Lee, K., Kothera, C.S., and Wereley, N.M. (2010). "Design and

Bye, D.R. and McClure, P.D. (2007). ''Design of a Morphing Vehicle,'' *48th AIAA Structures,* 

Chaves, F. D., Avila, J., and Avila, A. F. (2003). "A morphological study on cellular

Flanagan, J.S., Strutzenberg, R.C., Myers, R.B., and Rodrian, J.E. (2007). "Development and

Gern, F.H., Inman, D.J., and Kapania, R.K. (2002). "Structural and Aeroelastic Modeling of

Gibson, L. J. and Ashby, M. F. (1988). *Cellular Solids: Structure and Properties*, Pergamon

Gomez, J. C., and Garcia, E. (2011). Morphing unmanned aerial vehicles. *Smart Materials and* 

Hetrick, J. A., Osborn, R. F., Kota, S., Flick, P. M., and Paul, D. B. (2007). "Flight Testing of

Joo, J.J. Reich, G.W. and Westfall , J.T. (2009). "Flexible Skin Development for Morphing

Kothera, C.S., Woods, B.K.S., Bubert, E.A., Wereley, N.M., and Chen, P.C. (2011). "Cellular

McKnight, G. Doty, R. Keefe, Herrera, A.G. and Henry, C. (2010). "Segmented

*Structures*, 20(10):103001. doi:10.1088/0964-1726/20/10/103001

*Materials Conference*, Honolulu, HI, Paper No. AIAA 2007-1709.

*Systems and Structures*, 21(17):1699-1717 doi: 10.1177/1045389X10378777 Buseman, A. (1935) "Aerodynamic Lift at Supersonic Speeds," Ae. Techl. 1201, Report No.

2844 (British ARC, February 3, 1937), Bd. 12, Nr. 6: 210-220.

Gevers, D.E. (1997). "*Multi-purpose Aircraft*," US Patent No. 5,645,250.

*Systems and Structures,* 20(16):1969-1985.

7,931,240. Filed: 16 Feb 2007. Issued: 26 Apr 2011.

Characteristics of a Variable-Span Morphing Wing,'' *Journal of Aircraft*, 42(2): 528-

Morphing Aircraft," *Journal of Intelligent Material Systems and Structures,* 22: 823-877.

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*Structural Dynamics, and Materials Conference*, 23-26 April, Honolulu, HI, Paper No.

composites with negative Poisson's ratios,"*44th AIAA Structures, Structural Dynamics, and Materials Conference*, Norfolk, VA, Paper No. AIAA 2003-1951. Evans, K.E., Nkansah, M.A., Hutchinson, I.J., and Rogers, S.C. (1991). "Molecular network

Flight Testing of a Morphing Aircraft, the NextGen MFX-1,"*AIAA Structures, Structural Dynamics and Materials Conference*, Honolulu, HI. Paper No. AIAA-2007-

General Planform Wings with Morphing Airfoils," *AIAA Journal*, 40(4): 628-637.

Mission Adaptive Compliant Wing,"*48th AIAA Structures, Structural Dynamics, and* 

Aircraft Applications Via Topology Optimization." *Journal of Intelligent Material* 

Support Structures for Controlled Actuation of Fluid Contact Surfaces." U.S. Patent

Reinforcement Variable Stiffness Materials for Reconfigurable Surfaces." *Journal of* 

*Fiber Composites*, John Wiley & Sons, Hoboken.

534. doi: 10.2514/1.4397

AIAA-2007-1728.

1707.

design," *Nature*, 353: 124.

doi: 10.2514/2.1719

Press, Oxford.

doi:10.1177/1045389X11414084.

**9. References** 

grids form 12.7 mm squares and are located 35.6 cm behind the airfoil in the frame of view, which is also 35.6 cm from the camera lens. These can be seen in Figure 22c. Using image processing, the maximum error in the measurements was determined to be ±7%. This error can be attributed to vibration of the wind tunnel wall, which the camera lens was pressed against, or deviations in the focus of the pictures. In all the data processed, the maximum discernible out-of-plane deflection was approximately 0.51 ± 0.04 mm, which is well within the goal of less than 2.54 mm. In reference to the 30.5 cm chord and 5.49 cm thickness, this deflection accounts for only 0.17% and 0.93%, respectively. Additionally, in observing this experiment, it can be qualitatively stated the morphing wing held its shape remarkably well under all tested conditions. This can be confirmed through visual inspection of the figures, as well.

## **7. Conclusions**

This work explored the development of a continuous one dimensional morphing structure. For an aircraft, continuous morphing wing surfaces have the capability to improve efficiency in multiple flight regimes. However, material limitations and excessive complexity have generally prevented morphing concepts from being practical. Thus, the goal of the present work was to design a simple morphing system capable of being scaled to UAV or full scale aircraft. To this end, a passive 1-D morphing skin was designed, consisting of an elastomer matrix composite (EMC) skin with a zero-Poisson honeycomb substructure intended to support out-of-plane loads. In-plane stiffness was controlled to match the capabilities of an actuator by careful design and testing of each separate skin component. Complete morphing skins were tested for in-plane and out-of-plane performance and integrated with the actuator to validate the design process on a small-scale morphing cell section.

Design goals of 100% global strain and 100% area change were demonstrated on a laboratory prototype using the combined morphing skin and actuation mechanism. The morphing skin strained smoothly and exhibited a very low in-plane Poisson's ratio. Actuation frequencies of roughly 1 Hz were achieved.

This work was then extended to a full morphing UAV-scale wing suitable for testing in a wind tunnel. The system was assembled as designed and demonstrated its ability to increase span by 100% while maintaining a constant chord. Wind tunnel tests were conducted at cruise (105 kph) and maximum speed (130 kph) conditions of a candidate UAV test platform, at 0o, 2o, and 4o angles-of-attack, and at 0%, 50%, and 100% extensions. At each test point, image processing was used to determine the maximum out-of-plane deflection of the skin between ribs. Across all tests, the maximum discernable out-of-plane deflection was little more than 0.5 mm, indicating that a viable aerodynamic surface was maintained throughout the tested conditions.

## **8. Acknowledgement**

This work was sponsored by the Air Force Research Laboratory (AFRL) through a Phase I STTR (contract number FA9550-06-C-0132), and also by a Phase I SBIR project from NASA Langley Research Center (contract number NNX09CF06P).

#### **9. References**

26 Recent Advances in Aircraft Technology

grids form 12.7 mm squares and are located 35.6 cm behind the airfoil in the frame of view, which is also 35.6 cm from the camera lens. These can be seen in Figure 22c. Using image processing, the maximum error in the measurements was determined to be ±7%. This error can be attributed to vibration of the wind tunnel wall, which the camera lens was pressed against, or deviations in the focus of the pictures. In all the data processed, the maximum discernible out-of-plane deflection was approximately 0.51 ± 0.04 mm, which is well within the goal of less than 2.54 mm. In reference to the 30.5 cm chord and 5.49 cm thickness, this deflection accounts for only 0.17% and 0.93%, respectively. Additionally, in observing this experiment, it can be qualitatively stated the morphing wing held its shape remarkably well under all tested conditions. This can be confirmed

This work explored the development of a continuous one dimensional morphing structure. For an aircraft, continuous morphing wing surfaces have the capability to improve efficiency in multiple flight regimes. However, material limitations and excessive complexity have generally prevented morphing concepts from being practical. Thus, the goal of the present work was to design a simple morphing system capable of being scaled to UAV or full scale aircraft. To this end, a passive 1-D morphing skin was designed, consisting of an elastomer matrix composite (EMC) skin with a zero-Poisson honeycomb substructure intended to support out-of-plane loads. In-plane stiffness was controlled to match the capabilities of an actuator by careful design and testing of each separate skin component. Complete morphing skins were tested for in-plane and out-of-plane performance and integrated with the actuator to validate the design process on a small-scale morphing cell

Design goals of 100% global strain and 100% area change were demonstrated on a laboratory prototype using the combined morphing skin and actuation mechanism. The morphing skin strained smoothly and exhibited a very low in-plane Poisson's ratio.

This work was then extended to a full morphing UAV-scale wing suitable for testing in a wind tunnel. The system was assembled as designed and demonstrated its ability to increase span by 100% while maintaining a constant chord. Wind tunnel tests were conducted at cruise (105 kph) and maximum speed (130 kph) conditions of a candidate UAV test platform, at 0o, 2o, and 4o angles-of-attack, and at 0%, 50%, and 100% extensions. At each test point, image processing was used to determine the maximum out-of-plane deflection of the skin between ribs. Across all tests, the maximum discernable out-of-plane deflection was little more than 0.5 mm, indicating that a viable aerodynamic surface was maintained

This work was sponsored by the Air Force Research Laboratory (AFRL) through a Phase I STTR (contract number FA9550-06-C-0132), and also by a Phase I SBIR project from NASA

through visual inspection of the figures, as well.

Actuation frequencies of roughly 1 Hz were achieved.

Langley Research Center (contract number NNX09CF06P).

throughout the tested conditions.

**8. Acknowledgement** 

**7. Conclusions** 

section.


**2** 

*Italy* 

**A Probabilistic Approach to Fatigue Design** 

Fatigue design of aerospace metallic components is carried out by using two methodologies: damage tolerance and safe-life. At present, regulations mainly recommend the use of the former, which entrusts safety to a suitable inspections plan. Indeed, a crack or a flaw is supposed to have been present in the component since the beginning of its operative life, and it must remain not critical, i.e. it must not cause a catastrophic failure in the life period between two following inspections, (Federal Aviation Administration, 1998; Joint Aviation Authorities, 1994; US Department of Defence, 1998). When a crack is detected, the

If the damage tolerance criterion cannot be applied, the regulations state the safe-life criterion should be used, i.e. components must remain free of crack for their whole operative

Therefore, both methodologies have deterministic bases and a single value (usually the mean value) is associated to each parameter that can influence the fatigue phenomenon,

To take these items into account and in order to protect against unexpected events, it is necessary to introduce safety factors in the fatigue life design (generally equal to 2 or 3 for damage tolerance and equal to 4 or even more for safe-life). They usually produce heavy or expensive structures and, in the past, they were not always able to protect against catastrophic failures, because the real risk level is in any case unknown. On the one hand, indeed, the inspected structures or the substituted components may be still undamaged, with high costs; on the other hand, highly insidious phenomena, such as Multiple Site Damage and Widespread Fatigue Damage (which are typical of ageing aircrafts) cannot be taken into

account very well and in the past they were the causes of some catastrophic accidents.

For these reasons, researchers are hypothesizing the possibility of facing fatigue design in a new way, by using the risk evaluation from a probabilistic point of view. Indeed, the parameters that affect the phenomenon have a statistical behaviour, and this can be

component is repaired or substituted and the structural integrity is so restored.

life and, at their ends, components must be in any case substituted.

which on the contrary has a deep stochastic behaviour.

described by means of statistical distributions.

**1. Introduction** 

**of Aerospace Components by Using** 

**the Risk Assessment Evaluation** 

*University of Pisa-Department of Aerospace Engineering* 

Giorgio Cavallini and Roberta Lazzeri

*Intelligent Material Systems and Structures*, 21:1783-1793, doi:10.1177/ 1045389X10386399


## **A Probabilistic Approach to Fatigue Design of Aerospace Components by Using the Risk Assessment Evaluation**

Giorgio Cavallini and Roberta Lazzeri *University of Pisa-Department of Aerospace Engineering Italy* 

## **1. Introduction**

28 Recent Advances in Aircraft Technology

Munk, M. M. (1924). "Note on the relative Effect of the Dihedral and the Sweep Back of

Olympio, K.R., and Gandhi, F. (2010). "Flexible Skins for Morphing Aircraft Using Cellular

Parker, H.J. (1920). "The Parker Variable Camber Wing," Report #77 Fifth Annual Report,

Perkins, D. A., Reed, J. L., and Havens, E. (2004). "Morphing Wing Structures for Loitering

Samuel, J.B. and Pines, D.J. (2007). "Design and Testing of a Pneumatic Telescopic Wing for Unmanned Aerial Vehicles," *Journal of Aircraft*, 44(4) DOI: 10.2514/1.22205 Shigley, J., Mishke, C., and Budynas, R. (2004). *Mechanical Engineering Design*, McGraw-Hill,

Thill, C., Etches, J., Bond, I., Potter, K., and Weaver, P. (2008). "Morphing Skins," *The* 

Wereley, N. and Gandhi, F. (2010). "Flexible Skins for Morphing Aircraft." *Journal of* 

Wereley, N. M. and Kothera, C. S. (2007). "Morphing Aircraft Using Fluidic Artificial

Wright, O. and Wright, W. (1906). "Flying-Machine" U.S. Patent 821,393. Filed: 23 Mar 1903.

*Intelligent Material Systems and Structures*, 21: 1697-1698,

Muscles," *International Conference on Adaptive Structures and Technologies*, Ottawa,

*National Advisory Committee for Aeronautics,* Washington, D.C.

1045389X10386399

New York.

ON, Paper ID 171.

Issued: 22 May, 1906.

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Palm Springs, CA, Paper No. AIAA 2004-1888.

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1735, doi:10.1177/1045389X09350331

*Aeronautical Journal*, 112(1129):117-139.

doi:10.1177/1045389X10393157.

*Intelligent Material Systems and Structures*, 21:1783-1793, doi:10.1177/

Honeycomb Cores," *Journal of Intelligent Material Systems and Structures*, 21:1719-

Air Vehicles," *45th AIAA Structures, Structural Dynamics & Materials Conference*,

Fatigue design of aerospace metallic components is carried out by using two methodologies: damage tolerance and safe-life. At present, regulations mainly recommend the use of the former, which entrusts safety to a suitable inspections plan. Indeed, a crack or a flaw is supposed to have been present in the component since the beginning of its operative life, and it must remain not critical, i.e. it must not cause a catastrophic failure in the life period between two following inspections, (Federal Aviation Administration, 1998; Joint Aviation Authorities, 1994; US Department of Defence, 1998). When a crack is detected, the component is repaired or substituted and the structural integrity is so restored.

If the damage tolerance criterion cannot be applied, the regulations state the safe-life criterion should be used, i.e. components must remain free of crack for their whole operative life and, at their ends, components must be in any case substituted.

Therefore, both methodologies have deterministic bases and a single value (usually the mean value) is associated to each parameter that can influence the fatigue phenomenon, which on the contrary has a deep stochastic behaviour.

To take these items into account and in order to protect against unexpected events, it is necessary to introduce safety factors in the fatigue life design (generally equal to 2 or 3 for damage tolerance and equal to 4 or even more for safe-life). They usually produce heavy or expensive structures and, in the past, they were not always able to protect against catastrophic failures, because the real risk level is in any case unknown. On the one hand, indeed, the inspected structures or the substituted components may be still undamaged, with high costs; on the other hand, highly insidious phenomena, such as Multiple Site Damage and Widespread Fatigue Damage (which are typical of ageing aircrafts) cannot be taken into account very well and in the past they were the causes of some catastrophic accidents.

For these reasons, researchers are hypothesizing the possibility of facing fatigue design in a new way, by using the risk evaluation from a probabilistic point of view. Indeed, the parameters that affect the phenomenon have a statistical behaviour, and this can be described by means of statistical distributions.

A Probabilistic Approach to Fatigue Design

**cref**

**CRACK SIZE**

**cin**

be taken into account, too.

Fig. 1. TTCI and EIFS distributions.

 the (Equivalent) Initial Flaw Size distribution, the Time To Crack Initiation distribution.

of Aerospace Components by Using the Risk Assessment Evaluation 31

Therefore, a 'tool' able to characterize the component initial condition is necessary, in order

**TTCI**

to predict the fatigue life. At present, two approaches are available, Fig. 1:

**EIFS** 

connected to the crack growth, and so to the loading fatigue spectrum.

**2.1.2 The (Equivalent) Initial Flaw Size distribution (EIFS)** 

**2.1.1 The Time To Crack Initiation distribution (TTCI)** 

**DISTRIBUTION**

The TTCI model can be used to describe fatigue crack nucleation in metallic components. It can be defined as the time (in cycles, flights or flight hours) necessary for an initial defect to grow to a detectable reference crack, cref. In such a way, this method does not reveal the crack dimension during the early steps of the component life. The TTCI can be described by using a Weibull, (Manning et al., 1987), or a lognormal distribution (Liao & Komorowski, 2004). It must be noted that the TTCI distribution is not only a material property, as it is

The EIFS is the (fictitious) dimension of a crack at time t=0. The use of the adjective 'equivalent' indicates that the initial flaw is not the actual one and its size is only 'equivalent' to it because it is very difficult to account for the influence of all the relevant parameters. The distribution can be numerically obtained starting from experimental crack size data by using a 'fictitious' backward integration of the crack growth. It is affected by the material properties and the stress distribution: cold-working, rivet interference, … have to

**EIFS** 

**MASTER CURVE**

**TTCI** 

**DISTRIBUTION**

In such a way, by using a statistical method, such as the Monte Carlo Method (Besuner, 1987; Hammersley & Handscomb, 1983), all the parameter distributions can be managed and each simulated 'event' can be considered as a possible 'event'. So, the computer simulation of the fatigue life of a big amount of components and the evaluation of the real risk level are possible, making this approach extremely useful.

The Authorities are interested in this approach but, before allowing the use of it as a design criterion, they require impartial evidence, first of all about the reliability of the analytical models used for fatigue simulation and for parameter distribution evaluation.

This paper intends to show a computer code – PISA, Probabilistic Investigation for Safe Aircrafts – and how it can be applied to the fatigue design of typical aerospace components, such as riveted joints, making it possible to integrate damage tolerance and the evaluation of the real risk level connected to the chosen inspection plan, (Cavallini et al., 1997; Cavallini & R. Lazzeri, 2007).

## **2. The parameters that mainly affect the fatigue phenomenon**

A metallic component subjected to repeated loads can fail due to the fatigue phenomenon, (Schijve, 2001). Many research activities on this subject are known from both the theoretical and the experimental points of view and it is well known that fatigue has a random behaviour, with a high number of parameters (mechanical behaviour of the material, loads, geometry, manufacturing technologies, etc.) that can affect it.

Almost all these parameters have a statistical behaviour, but some among them play a more important role compared to the others and must be taken into account with their distributions, while the others can be assumed to be constant.

In detail, we can assume four main parameters as statistically distributed:


They are described in the following.

## **2.1 The Initial Fatigue Quality (IFQ)**

Structural components can have, until the end of the manufacturing process, defects due to metallurgical effects, scratches, roughness, inclusions, welding defects, etc.

So, the IFQ can be considered as a property linked to the material and the manufacturing process. Defects can be the starting point for fatigue cracks. For this reason it is extremely important to know their position and size, but, as they are very small, they are very difficult to be measured even if by using very sophisticated inspection methods.

As a consequence, this information can be reached only through an indirect evaluation by means of a 'draw-back' procedure starting from experimental data about detectable cracks.

Therefore, a 'tool' able to characterize the component initial condition is necessary, in order to predict the fatigue life. At present, two approaches are available, Fig. 1:


30 Recent Advances in Aircraft Technology

In such a way, by using a statistical method, such as the Monte Carlo Method (Besuner, 1987; Hammersley & Handscomb, 1983), all the parameter distributions can be managed and each simulated 'event' can be considered as a possible 'event'. So, the computer simulation of the fatigue life of a big amount of components and the evaluation of the real

The Authorities are interested in this approach but, before allowing the use of it as a design criterion, they require impartial evidence, first of all about the reliability of the analytical

This paper intends to show a computer code – PISA, Probabilistic Investigation for Safe Aircrafts – and how it can be applied to the fatigue design of typical aerospace components, such as riveted joints, making it possible to integrate damage tolerance and the evaluation of the real risk level connected to the chosen inspection plan, (Cavallini et al., 1997; Cavallini &

A metallic component subjected to repeated loads can fail due to the fatigue phenomenon, (Schijve, 2001). Many research activities on this subject are known from both the theoretical and the experimental points of view and it is well known that fatigue has a random behaviour, with a high number of parameters (mechanical behaviour of the material, loads,

Almost all these parameters have a statistical behaviour, but some among them play a more important role compared to the others and must be taken into account with their

 the Initial Fatigue Quality (IFQ), described by using the (Equivalent) Initial Flaw Size, (E)IFS, or the Time To Crack Initiation, TTCI, distribution, (Manning et al., 1987);

Structural components can have, until the end of the manufacturing process, defects due to

So, the IFQ can be considered as a property linked to the material and the manufacturing process. Defects can be the starting point for fatigue cracks. For this reason it is extremely important to know their position and size, but, as they are very small, they are very difficult

As a consequence, this information can be reached only through an indirect evaluation by means of a 'draw-back' procedure starting from experimental data about detectable cracks.

models used for fatigue simulation and for parameter distribution evaluation.

**2. The parameters that mainly affect the fatigue phenomenon** 

In detail, we can assume four main parameters as statistically distributed:

the inspection reliability, i.e. the Probability of crack Detection, PoD.

metallurgical effects, scratches, roughness, inclusions, welding defects, etc.

to be measured even if by using very sophisticated inspection methods.

geometry, manufacturing technologies, etc.) that can affect it.

distributions, while the others can be assumed to be constant.

the crack grow rate, CGR (constant C in the Paris law);

the fracture toughness KIc, and

They are described in the following.

**2.1 The Initial Fatigue Quality (IFQ)** 

risk level are possible, making this approach extremely useful.

R. Lazzeri, 2007).

Fig. 1. TTCI and EIFS distributions.

#### **2.1.1 The Time To Crack Initiation distribution (TTCI)**

The TTCI model can be used to describe fatigue crack nucleation in metallic components. It can be defined as the time (in cycles, flights or flight hours) necessary for an initial defect to grow to a detectable reference crack, cref. In such a way, this method does not reveal the crack dimension during the early steps of the component life. The TTCI can be described by using a Weibull, (Manning et al., 1987), or a lognormal distribution (Liao & Komorowski, 2004). It must be noted that the TTCI distribution is not only a material property, as it is connected to the crack growth, and so to the loading fatigue spectrum.

#### **2.1.2 The (Equivalent) Initial Flaw Size distribution (EIFS)**

The EIFS is the (fictitious) dimension of a crack at time t=0. The use of the adjective 'equivalent' indicates that the initial flaw is not the actual one and its size is only 'equivalent' to it because it is very difficult to account for the influence of all the relevant parameters. The distribution can be numerically obtained starting from experimental crack size data by using a 'fictitious' backward integration of the crack growth. It is affected by the material properties and the stress distribution: cold-working, rivet interference, … have to be taken into account, too.

A Probabilistic Approach to Fatigue Design

obtained from its distribution.

Fig. 2. Riveted lap-joint panel.

are necessary.

of Aerospace Components by Using the Risk Assessment Evaluation 33

The Monte Carlo method is based on a very easy assumption: the probability of an event *p*f – in the present paper the component failure – is evaluated by using the analytical expression

Each simulation reproduces only one deterministic event, in which, for each deterministic or random variable a value is assumed; for the stochastic parameters, the value is randomly

The only disadvantage of this method is that it requires a high number of simulations to have a low probability of the event. As an example, if the required probability is 10-6, with a

> 6 <sup>3</sup> 3 10

The PISA code (Cavallini et al., 1997; Cavallini & R. Lazzeri, 2007), developed at the Department of Aerospace Engineering of the University of Pisa, allows the simulation of the whole fatigue life of typical aerospace components, such as simple plane panels, riveted lapjoint panels, Fig. 2, or stiffened panels, subjected to constant amplitude fatigue loading.

10

**4. The PISA code and the simulation of the fatigue phenomenon** 

6

*N x trials* (3)

where *N* is the total simulation number and *n* is the number of positive results.

After a high number of trials, the method converges to the solution.

confidence level of 95%, at least (Grooteman, 2002)

*pf n N*/ (2)

The EIFS can be described by using a lognormal or a Weibull (Manning et al., 1987) distribution. In the present paper, a lognormal distribution is assumed.

#### **2.2 The Crack Growth Rate, CGR (C constant in the Paris law)**

Different models are available to describe the crack growth law according to Linear Elastic Fracture Mechanics and the distribution of the involved parameters. We assumed to use the simple and effective Paris law d*c*/d*N*=C(*K*)m. The parameter *m* is assumed to be constant and all the scatter is consolidated in C, which is assumed to belong to a normal distribution.

#### **2.3 The Fracture Toughness KIc**

Fracture toughness is a very important material property because it identifies a failure criterion (crack instability). Unfortunately, few experimental data are available to characterize its distribution. Anyway, a normal distribution (Hovey et al., 1991), or a lognormal distribution (Johnson, 1983; Schutz, 1980) can be supposed. We assumed the fracture toughness can be described through a lognormal distribution.

#### **2.4 The Probability of crack Detection (PoD)**

Non destructive inspections are among the principal items of the damage tolerance methodology. Indeed, during inspection, it is supposed that cracks are detected and the component can be re-qualified for further use. This action depends on many parameters, included the human factor and so it can be described only by using a probabilistic approach. Usually, (Lincoln, 1998), we define the inspection capability as the 90% probability of crack detection with the 95% of confidence.

Many distributions have been proposed for the Probability of Detection, (Tong, 2001; Ratwani, 1996).

In the present work we assumed a three parameters Weibull distribution, [Lewis et al., 1978]:

$$PoD = 1 - e^{-\left[\frac{c - c\_{\min}}{\lambda - c\_{\min}}\right]^\beta} \tag{1}$$

where, *c*min is the minimum detectable crack size, *c* is the actual crack size and and are parameters connected to the chosen inspection method.

#### **3. The Monte Carlo method**

A tool is necessary to manage all the parameter distributions together and at the same time. Some reliable approaches are available - FORM (First Order Reliability Method), SORM (Second Order Reliability Method) and many others (Madsen et al., 1986) – but we decided to use the Monte Carlo method because of its simplicity and effectiveness, as it can handle high numbers of different distributions for the stochastic variables to simulate many different deterministic situations. In addition, the Monte Carlo method easily allows the introduction of the repairs, that is a non-continuous change in the crack size.

The Monte Carlo method is based on a very easy assumption: the probability of an event *p*f – in the present paper the component failure – is evaluated by using the analytical expression

$$p\_f = n \,/\,\text{N} \tag{2}$$

where *N* is the total simulation number and *n* is the number of positive results.

Each simulation reproduces only one deterministic event, in which, for each deterministic or random variable a value is assumed; for the stochastic parameters, the value is randomly obtained from its distribution.

After a high number of trials, the method converges to the solution.

The only disadvantage of this method is that it requires a high number of simulations to have a low probability of the event. As an example, if the required probability is 10-6, with a confidence level of 95%, at least (Grooteman, 2002)

$$N\_{\text{trials}} > \frac{3}{10^{-6}} = 3 \text{x10}^6\tag{3}$$

are necessary.

32 Recent Advances in Aircraft Technology

The EIFS can be described by using a lognormal or a Weibull (Manning et al., 1987)

Different models are available to describe the crack growth law according to Linear Elastic Fracture Mechanics and the distribution of the involved parameters. We assumed to use the simple and effective Paris law d*c*/d*N*=C(*K*)m. The parameter *m* is assumed to be constant and all the scatter is consolidated in C, which is assumed to belong to a normal distribution.

Fracture toughness is a very important material property because it identifies a failure criterion (crack instability). Unfortunately, few experimental data are available to characterize its distribution. Anyway, a normal distribution (Hovey et al., 1991), or a lognormal distribution (Johnson, 1983; Schutz, 1980) can be supposed. We assumed the

Non destructive inspections are among the principal items of the damage tolerance methodology. Indeed, during inspection, it is supposed that cracks are detected and the component can be re-qualified for further use. This action depends on many parameters, included the human factor and so it can be described only by using a probabilistic approach. Usually, (Lincoln, 1998), we define the inspection capability as the 90% probability of crack

Many distributions have been proposed for the Probability of Detection, (Tong, 2001;

In the present work we assumed a three parameters Weibull distribution, [Lewis et al.,

min 1 *c c*

where, *c*min is the minimum detectable crack size, *c* is the actual crack size and and are

A tool is necessary to manage all the parameter distributions together and at the same time. Some reliable approaches are available - FORM (First Order Reliability Method), SORM (Second Order Reliability Method) and many others (Madsen et al., 1986) – but we decided to use the Monte Carlo method because of its simplicity and effectiveness, as it can handle high numbers of different distributions for the stochastic variables to simulate many different deterministic situations. In addition, the Monte Carlo method easily allows the introduction of the repairs, that is a non-continuous change in the

*<sup>c</sup> PoD e*

min

(1)

distribution. In the present paper, a lognormal distribution is assumed.

fracture toughness can be described through a lognormal distribution.

**2.2 The Crack Growth Rate, CGR (C constant in the Paris law)** 

**2.3 The Fracture Toughness KIc**

**2.4 The Probability of crack Detection (PoD)** 

parameters connected to the chosen inspection method.

detection with the 95% of confidence.

**3. The Monte Carlo method** 

Ratwani, 1996).

1978]:

crack size.

#### **4. The PISA code and the simulation of the fatigue phenomenon**

The PISA code (Cavallini et al., 1997; Cavallini & R. Lazzeri, 2007), developed at the Department of Aerospace Engineering of the University of Pisa, allows the simulation of the whole fatigue life of typical aerospace components, such as simple plane panels, riveted lapjoint panels, Fig. 2, or stiffened panels, subjected to constant amplitude fatigue loading.

Fig. 2. Riveted lap-joint panel.

A Probabilistic Approach to Fatigue Design

propagation phase.

m

(Sampath & Broek, 1991).

solutions are known.

With regard to the open hole

Fig. 4. Crack at an open hole.

*p* on the hole, (Kuo et al., 1986), Fig. 5.

composition approach (Kuo et al., 1986).

d*c*/d*N*=C(*K*)

assigned by taking off a value from their own distributions.

*K* from the beginning of the life to the final failure.

of Aerospace Components by Using the Risk Assessment Evaluation 35

Many deterministic simulations can be run and the risk assessment can be evaluated by using the Monte Carlo method, Fig. 3. In each simulation, the deterministic parameters are

The first phase – crack nucleation at holes – is simulated by using the EIFS distribution, which can be considered as an indication of the initial fatigue quality. In this context, this approach has to be preferred to the TTCI, as it allows to consider the whole life as the only

The second phase – crack growth – is simulated by using the simple, well-known Paris law

The core for the evaluation of the crack growth is the expression of the stress intensity factor

In the stress intensity factor evaluation, suitable corrective factors (Sampath & Broek, 1991; Kuo et al., 1986) have been used to take into account the different boundary conditions, and the load transfer inside the joints has been simulated by using the Broek & Sampath model,

In detail, the effect of different boundary conditions can be taken into account by using the

*K* is analytically evaluated by means of a corrective coefficient which has been found by splitting the complex geometry into simple problems (open hole, finite width, …), whose

1 2

c

( ) *<sup>R</sup> K S cr* 

r

0 0

 

As to the rivet effect, the load *P* on the hole has been approximated with a uniform pressure

*P t p sen r d t p r sen d* 2 *p r t*

 

 (6)

where *S* is the uniform membrane stress and (*c*-*r*) the crack length, Fig. 4.

*<sup>R</sup> K K CR CR CR <sup>n</sup>* (4)

(5)


As far as the panel is concerned, the following hypotheses can be made:

Table 1. Assumed geometrical and physical hypotheses.

The code can simulate crack nucleation, growth, inspection actions and failures in components subjected to uniform loading. The basic idea is that the damage process can be simulated as the continuous growth of an initial defect due to metallurgical effects and/or the manufacturing process, and/or other parameters.

Fig. 3. Structure of the code.

Many deterministic simulations can be run and the risk assessment can be evaluated by using the Monte Carlo method, Fig. 3. In each simulation, the deterministic parameters are assigned by taking off a value from their own distributions.

The first phase – crack nucleation at holes – is simulated by using the EIFS distribution, which can be considered as an indication of the initial fatigue quality. In this context, this approach has to be preferred to the TTCI, as it allows to consider the whole life as the only propagation phase.

The second phase – crack growth – is simulated by using the simple, well-known Paris law d*c*/d*N*=C(*K*) m

The core for the evaluation of the crack growth is the expression of the stress intensity factor *K* from the beginning of the life to the final failure.

In the stress intensity factor evaluation, suitable corrective factors (Sampath & Broek, 1991; Kuo et al., 1986) have been used to take into account the different boundary conditions, and the load transfer inside the joints has been simulated by using the Broek & Sampath model, (Sampath & Broek, 1991).

In detail, the effect of different boundary conditions can be taken into account by using the composition approach (Kuo et al., 1986).

*K* is analytically evaluated by means of a corrective coefficient which has been found by splitting the complex geometry into simple problems (open hole, finite width, …), whose solutions are known.

With regard to the open hole

34 Recent Advances in Aircraft Technology

Plane panel Uniform thickness

Rivets with or without countersunk head Through cracks

> Uniform stress Plane stress

Rivets with extremely high stiffness Fretting and corrosion effects are negligible

FRACTURE TOUGHNESS

KC, [ LogK ]C

**Life**

**Kc**

POD

**POD**

10-9

**a**

amin, ,

MAINTENANCE STRATEGIES

THRESHOLD INSPECTION INTERVALS &

PROBABILITY OF FAILURE

**Inspection Interval**

Life

Cracks on one or both hole sides and orthogonal to the load direction

The code can simulate crack nucleation, growth, inspection actions and failures in components subjected to uniform loading. The basic idea is that the damage process can be simulated as the continuous growth of an initial defect due to metallurgical effects and/or

STATISTICAL INPUT

The PISA Code

**PDF**

MONTECARLO METHOD

SIMULATION

As far as the panel is concerned, the following hypotheses can be made:

**Physical** Uniform pin load in the same row

CRACK GROWTH RATE

**Log (da/ dN)**

K evaluation

**Log( K)**

C, [LogC], m

**a**

acr

**Geometrical** Constant rivet pitch

Table 1. Assumed geometrical and physical hypotheses.

the manufacturing process, and/or other parameters.

**N**

EIFS

Sy, E,

Fig. 3. Structure of the code.

Smax, R Const Ampl

Equivalent Const. Ampl.

Var. Ampl.

**a**

G E O M E T R <sup>Y</sup> Plane and curved

D E T E R M I N I S T I C I N P U T

$$K = K^R \cdot \text{CR}\_1 \cdot \text{CR}\_2 \cdot \text{CR}\_n \tag{4}$$

$$K^{\mathbb{R}} = S \sqrt{\pi (c - r)}\tag{5}$$

where *S* is the uniform membrane stress and (*c*-*r*) the crack length, Fig. 4.

Fig. 4. Crack at an open hole.

As to the rivet effect, the load *P* on the hole has been approximated with a uniform pressure *p* on the hole, (Kuo et al., 1986), Fig. 5.

$$P = \text{t} \cdot \int\_0^\pi p \cdot \text{sen}\theta \cdot r \cdot d\theta = \text{t} \cdot p \cdot r \cdot \int\_0^\pi \text{sen}\theta \cdot d\theta = \text{2} \cdot p \cdot r \cdot \text{t} \tag{6}$$

A Probabilistic Approach to Fatigue Design

by using the superposition approach, Fig. 6.

of Aerospace Components by Using the Risk Assessment Evaluation 37

The Broek & Sampath model joins the solutions related to the open hole and the loaded hole

PPP

Sbypass Sbypass

S S

Sbypass

<sup>I</sup> <sup>=</sup>

PPP

II

Fig. 6. The Broek & Sampath model.

not transferred by the rivets.

*S*max-*S*o for crack propagation.

S - Sbypass

Sbypass

<sup>=</sup> III

S - Sbypass

S - Sbypass

+

+

K = K<sup>I</sup> + KII = K<sup>I</sup> + ½(KIII + KIV)

*S* is the membrane uniform stress and *S*bypass is the bypass stress, i.e. for each row, the stress

Also rivet interference introduces an additional stress distribution. Its main effect is that only a part of the applied load amplitude *S*max-*S*min is effective for crack propagation. This effect can be taken into account by using the Wang model (Wang, 1988) for the evaluation of the lift-off stress *S*o, corresponding to the separation of the rivet from the hole, and then by introducing in the simulations carried out with the PISA code only the effective amplitude

 1 1 2 2 *K CR S S c CP p c bypass* 

Once the stress intensity factor is calculated, crack growth simulation can start.

II

PPP IV <sup>P</sup> <sup>P</sup> <sup>P</sup> -

=

PPP

S - Sbypass

S - Sbypass

PP P

V

(10)

Fig. 5. Approximation of the *P* load with a uniform pressure *p.*

$$K = K^P \cdot \text{CP}\_1 \cdot \text{CP}\_2 \cdots \text{CP}\_n \tag{8}$$

$$K^P = p\sqrt{\pi \cdot c} \tag{9}$$

The main corrective factors for open holes and for filled holes taken into account are summarized in Table 2.

Table 2. The main corrective factors for open holes and for filled holes taken into account in the PISA code.

2 *P*

1 2

The main corrective factors for open holes and for filled holes taken into account are

*<sup>P</sup> K p*

1986), e

Table 2. The main corrective factors for open holes and for filled holes taken into account in

P

P

Fig. 5. Approximation of the *P* load with a uniform pressure *p.*

summarized in Table 2.

Crack at a hole (Kuo et al., 1986),

Two cracks at a hole (Kuo et al., 1986),

Link-up, with one crack (Kuo et al., 1986),

Link-up, with two cracks (Kuo et al., 1986),

Edge crack (Kuo et al., 1986),

Panel finite width (Kuo et al.,

Countersink (Kuo et al., 1986), Secondary bending, (Sampath & Broek, 1991)

the PISA code.

*<sup>p</sup> r t* (7)

p

*<sup>P</sup> K K CP CP CP <sup>n</sup>* (8)

c

c2 c1

c2 c1

c

2c

r

*c* (9)

c

The Broek & Sampath model joins the solutions related to the open hole and the loaded hole by using the superposition approach, Fig. 6.

PPP Sbypass Sbypass S S = + PPP S - Sbypass II Sbypass Sbypass <sup>I</sup> <sup>=</sup> S - Sbypass PPP S - Sbypass II S - Sbypass PP P V <sup>=</sup> III S - Sbypass + PPP IV <sup>P</sup> <sup>P</sup> <sup>P</sup> -

$$\mathsf{K} = \mathsf{K}^{\mathsf{I}} + \mathsf{K}^{\mathsf{II}} = \mathsf{K}^{\mathsf{I}} + \mathsf{N}(\mathsf{K}\_{\mathsf{III}} + \mathsf{K}\_{\mathsf{IV}})^{\mathsf{I}}$$

Fig. 6. The Broek & Sampath model.

$$K = \frac{1}{2} \cdot \text{CR} \cdot \left( S\_{\text{os}} + S\_{\text{bypass}} \right) \cdot \sqrt{\pi \cdot c} + \frac{1}{2} \cdot \text{CP} \cdot p \cdot \sqrt{\pi \cdot c} \tag{10}$$

*S* is the membrane uniform stress and *S*bypass is the bypass stress, i.e. for each row, the stress not transferred by the rivets.

Also rivet interference introduces an additional stress distribution. Its main effect is that only a part of the applied load amplitude *S*max-*S*min is effective for crack propagation. This effect can be taken into account by using the Wang model (Wang, 1988) for the evaluation of the lift-off stress *S*o, corresponding to the separation of the rivet from the hole, and then by introducing in the simulations carried out with the PISA code only the effective amplitude *S*max-*S*o for crack propagation.

Once the stress intensity factor is calculated, crack growth simulation can start.

A Probabilistic Approach to Fatigue Design

10 20

of Aerospace Components by Using the Risk Assessment Evaluation 39

**SIMPLE STRIP LAP-JOINT**

60

2

**SIMPLIFIED MODEL**

**Smembr**

**Smembr**

10

20 20

2

4.8

100

0.75

**Sbypass**

**Privet**

**Privet**

**Smembr**

300

**Sbending**

Fig. 8. Simplified model implemented inside the PISA code in order to evaluate the EIFS

In addition, an iterative positive integration was carried out starting from an initial tentative crack size value and stopping at the same number of cycles of the experimental result. The simulated final crack dimension was compared with the experimental one, and an iterative

In this way, a lognormal distribution for the EIFS was found, with [Log10(*c*0)]=-2.88605, and

**Sbypass**

process was carried out to the required convergence.

**Smembr Privet Interf . effect**

**Sbending**

**ACTUAL MODEL SIMPLIFIED MODEL**

Fig. 7. Specimen geometry, all lengths in mm.

**Sme mbr**

distribution.

**memb**

[Log10(*c*0)]=0.28456, Fig. 9.

**Sme mbr**

**membr**

Two collinear cracks are considered as linked according to the Swift criterion, i.e. when their plastic radii *r*p - evaluated by using the Irving expression - are tangential.

Inspections at planned intervals are simulated through the PoD distribution, applied at each crack at both hole sides. Though the repair of the hole has the same quality as the pristine panel, the repair itself of the detected crack is not immune from the possibility of having some tiny cracks.

The final failure can happen either for crack instability (*K*max higher than the fracture toughness, *K*max*K*Ic) or for static failure (*S*max higher than the yield stress in the net section evaluated without the plastic zones, *S*max *S*02)

## **5. Experimental activity as a support for the evaluation of the statistical distributions**

To support this activity, all the parameter statistical distributions and the coefficients for the used analytical law (for example, the EIFS distribution, *C* and *m* coefficients for the Paris law, etc.) have to be experimentally evaluated. Of course, they cannot be obtained from tests on real components, but we have tested realistic simple specimens and we have demonstrated the applicability of the obtained results to the life evaluation of the actual components.

In this paper the activity carried out to find the distributions of the EIFS and the C constant in Paris law are shown. A similar approach can be used for the definition of the distributions for *KI*c and PoD.

### **5.1 Equivalent Initial Flaw Size distribution evaluation**

For the evaluation of the EIFS distribution it was necessary to use a 'fictitious' negative integration (draw-back) which, starting from the experimental crack data at assigned number of cycles, would be able to find the 'equivalent' initial size, i.e. the crack size at *N*=0.

To support this approach, a wide experimental activity was performed on 29 simple strip lap-joints, Fig. 7, in aluminum alloy 2024-T3 (Cavallini & R. Lazzeri, 2007). They were fatigue tested under a constant amplitude load spectrum with *S*max=120 MPa and *R*=0.1. The tests were stopped at a set number of cycles, the specimens were statically broken and the crack dimensions were carefully measured. The tests confirmed an already well known result: all the cracks were found in the most critical location, i.e. in the first row, at the countersunk side.

At present, several numerical codes are available to simulate the growth of a single crack in the long crack range, but few can manage also the short crack range and none is able to carry out a direct negative integration that starting from the experimental crack data can find the initial dimension. For this reason, we decided to use the PISA code itself and the simplified model of a specimen with a through crack at a lap-joint, taking into account the effects of countersink of the hole, membrane stress, by-pass loading and pin load, Fig. 8, (Cavallini & R. Lazzeri, 2007).

Two collinear cracks are considered as linked according to the Swift criterion, i.e. when their

Inspections at planned intervals are simulated through the PoD distribution, applied at each crack at both hole sides. Though the repair of the hole has the same quality as the pristine panel, the repair itself of the detected crack is not immune from the possibility of having

The final failure can happen either for crack instability (*K*max higher than the fracture toughness, *K*max*K*Ic) or for static failure (*S*max higher than the yield stress in the net section

To support this activity, all the parameter statistical distributions and the coefficients for the used analytical law (for example, the EIFS distribution, *C* and *m* coefficients for the Paris law, etc.) have to be experimentally evaluated. Of course, they cannot be obtained from tests on real components, but we have tested realistic simple specimens and we have demonstrated the applicability of the obtained results to the life evaluation of the actual

In this paper the activity carried out to find the distributions of the EIFS and the C constant in Paris law are shown. A similar approach can be used for the definition of the distributions

For the evaluation of the EIFS distribution it was necessary to use a 'fictitious' negative integration (draw-back) which, starting from the experimental crack data at assigned number of cycles, would be able to find the 'equivalent' initial size, i.e. the crack size at

To support this approach, a wide experimental activity was performed on 29 simple strip lap-joints, Fig. 7, in aluminum alloy 2024-T3 (Cavallini & R. Lazzeri, 2007). They were fatigue tested under a constant amplitude load spectrum with *S*max=120 MPa and *R*=0.1. The tests were stopped at a set number of cycles, the specimens were statically broken and the crack dimensions were carefully measured. The tests confirmed an already well known result: all the cracks were found in the most critical location, i.e. in the first row, at the

At present, several numerical codes are available to simulate the growth of a single crack in the long crack range, but few can manage also the short crack range and none is able to carry out a direct negative integration that starting from the experimental crack data can find the initial dimension. For this reason, we decided to use the PISA code itself and the simplified model of a specimen with a through crack at a lap-joint, taking into account the effects of countersink of the hole, membrane stress, by-pass loading and pin load, Fig. 8,

**5. Experimental activity as a support for the evaluation of the statistical** 

plastic radii *r*p - evaluated by using the Irving expression - are tangential.

some tiny cracks.

**distributions** 

components.

for *KI*c and PoD.

countersunk side.

(Cavallini & R. Lazzeri, 2007).

*N*=0.

evaluated without the plastic zones, *S*max *S*02)

**5.1 Equivalent Initial Flaw Size distribution evaluation** 

Fig. 7. Specimen geometry, all lengths in mm.

Fig. 8. Simplified model implemented inside the PISA code in order to evaluate the EIFS distribution.

In addition, an iterative positive integration was carried out starting from an initial tentative crack size value and stopping at the same number of cycles of the experimental result. The simulated final crack dimension was compared with the experimental one, and an iterative process was carried out to the required convergence.

In this way, a lognormal distribution for the EIFS was found, with [Log10(*c*0)]=-2.88605, and [Log10(*c*0)]=0.28456, Fig. 9.

A Probabilistic Approach to Fatigue Design

Fig. 11. Result for one CCT coupon test.

[(*C*)]=2.8834x10-7, and [(*C*)]=0.036792.

of Aerospace Components by Using the Risk Assessment Evaluation 41

The test results were elaborated splitted for the different R values. For the characterization of the linear portion of the curve, we assumed *m* as a deterministic parameter (equal for

In such a way, for *R*=0.1, we found that *m*=2.555 and *C* is normal distributed with

each test, Fig. 12), and we considered only *C* as normal distributed.

Fig. 12. Elaboration of the results for the tests at the same *R* value.

**5.3 The validation of the analytical models implemented inside the PISA code** 

To validate the approach and to verify the capability of the PISA code to simulate the fatigue behavior of aerospace structural components, further experimental tests were carried out on

Fig. 9. EIFS distribution obtained by using the draw-back procedure.

#### **5.2 The Crack Growth Rate (CGR) (C constant in the Paris law)**

The crack growth law must be experimentally characterized, in order to evaluate the parameters involved in the selected crack growth law (Paris).

For their evaluation, a test campaign on a 2.1 mm thick Center Crack Tension (CCT) specimen with an open hole (4 mm in diameter) in aluminium alloy 2024-T3 was carried out (Imparato & Santini, 1997), Fig. 10.

Fig. 10. CCT specimen in Al 2024-T3.

They were pre-cracked and the further crack propagation in the 5 to 40 mm range was investigated by using the Potential Drop Technique.

Tests were carried out under a constant amplitude (C.A.) spectrum, at the same *S*max=68.7 MPa, with 4 different *R*=*S*min/*S*max values (*R*=0.1, *R*=0.25, *R*=0.4, *R*=0.55). In Fig. 11 the experimental results for one test are shown.

Fig. 11. Result for one CCT coupon test.

**Database a Smax=120 MPa, EIFS Distribution - [Log(c0)] = -2.88605, [Log(c0)] = 0.28456**

Equivalent initial cracks corresponding to

Fig. 9. EIFS distribution obtained by using the draw-back procedure.

**5.2 The Crack Growth Rate (CGR) (C constant in the Paris law)** 

parameters involved in the selected crack growth law (Paris).

experimental data


The crack growth law must be experimentally characterized, in order to evaluate the

For their evaluation, a test campaign on a 2.1 mm thick Center Crack Tension (CCT) specimen with an open hole (4 mm in diameter) in aluminium alloy 2024-T3 was carried out

A

10

4

[L] = mm

Detail A

940

They were pre-cracked and the further crack propagation in the 5 to 40 mm range was

Tests were carried out under a constant amplitude (C.A.) spectrum, at the same *S*max=68.7 MPa, with 4 different *R*=*S*min/*S*max values (*R*=0.1, *R*=0.25, *R*=0.4, *R*=0.55). In Fig. 11 the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(Imparato & Santini, 1997), Fig. 10.

Fig. 10. CCT specimen in Al 2024-T3.

300

investigated by using the Potential Drop Technique.

experimental results for one test are shown.

**P**

The test results were elaborated splitted for the different R values. For the characterization of the linear portion of the curve, we assumed *m* as a deterministic parameter (equal for each test, Fig. 12), and we considered only *C* as normal distributed.

In such a way, for *R*=0.1, we found that *m*=2.555 and *C* is normal distributed with [(*C*)]=2.8834x10-7, and [(*C*)]=0.036792.

Fig. 12. Elaboration of the results for the tests at the same *R* value.

#### **5.3 The validation of the analytical models implemented inside the PISA code**

To validate the approach and to verify the capability of the PISA code to simulate the fatigue behavior of aerospace structural components, further experimental tests were carried out on

A Probabilistic Approach to Fatigue Design

**6. Applications of the PISA code** 

**6.1 The simulation of a single panel** 

distribution with and without the run-outs, Fig. 13.

deterministic simulations for the probabilistic approach.

of Aerospace Components by Using the Risk Assessment Evaluation 43

The agreement between predictions and experimental results can be seen; in detail, the predicted distribution obtained by using PISA is included between the experimental data

In addition, 5 further lap-joint panels were fatigue tested till failure. In Fig. 13 the comparison between PISA simulations and the experimental results is also shown. The

The Pisa code is organized in such a way that all the information about geometry, material characterization, loads, inspection methods, failure criteria are collected in an input file.

The code can be used for the evaluation of the fatigue behaviour of only one component, starting from an initial known crack path, or for the generation of high numbers of

In Fig. 14, the fatigue behaviour of a very simple panel, with only four open holes (diameter=4 mm), in Al 2024-T3, loaded under a constant amplitude load with *S*max=100 MPa and *R*=0.1, is simulated. The initial crack path is extracted from the EIFS distribution, but it would be assigned also as an external input. The assigned life was 100,000 cycles. Inspections were planned every 25,000 cycles. For the probability of detection parameters,

we assumed cmin=0.65 mm, =1.62 mm, =1.35, (extrapolated from Ratwani, 1996).

Fig. 14. Simulation of the fatigue life of an open hole panel (damage tolerance criterion).

As it can be seen in the Figure, during the first inspection (at 25,000 cycles) cracks were too small and were not detected. They grew till the second inspection (at 50,000 cycles), when

agreement is good, though the simulation results are a little conservative.

wide lap-joint panels, in the same aluminum alloy as the simple strips, Fig. 7, loaded under a constant amplitude spectrum (*S*max = 120 MPa, *R*=0.1).

A group of panels was fatigue tested for an assigned number of cycles (1 at 70,000 cycles, 4 at 75,000 cycles, 4 at 80,000 cycles, 3 at 85,000 cycles) and then statically broken to measure the sizes of the nucleated cracks. Also, in these panels the cracks were found only in the most critical row, i.e. the first one, at the countersunk side.

After having statically broken the tested panels, it was not possible, at all the hole sides, to detect a crack and they were considered as run-outs. The run-out effect has been introduced inside the crack size distribution by using the maximum likelihood method (Spindel & Haibach, 1979). Their crack dimensions have been supposed less than 0.1 mm. We supposed that the crack sizes (both with and without the run-outs) belong to two lognormal distributions.

Fig. 13. Comparison between experimental data and PISA simulations (crack dimensions at 80,000 cycles and cycles to failure for the lap-joint panels).

Starting from the EIFS distribution obtained by the simple strip joints and by using the PISA code, the capability of the procedure has been verified by simulating the behavior of the cracks in the most critical row of the lap-joint panels.

The comparison has been made by generating 1000 runs, i.e. by simulating the crack sizes at different number of cycles in the most critical row of 1000 lap-joint panels similar to the tested lap-joint panels (i.e. 15 holes x 2 sides = 30 positions for each panel). In Figure 13 (Cavallini & R. Lazzeri, 2007) the comparison is shown between the predicted crack dimensions and the corresponding experimental results at 80,000 cycles.

The agreement between predictions and experimental results can be seen; in detail, the predicted distribution obtained by using PISA is included between the experimental data distribution with and without the run-outs, Fig. 13.

In addition, 5 further lap-joint panels were fatigue tested till failure. In Fig. 13 the comparison between PISA simulations and the experimental results is also shown. The agreement is good, though the simulation results are a little conservative.

## **6. Applications of the PISA code**

42 Recent Advances in Aircraft Technology

wide lap-joint panels, in the same aluminum alloy as the simple strips, Fig. 7, loaded under

A group of panels was fatigue tested for an assigned number of cycles (1 at 70,000 cycles, 4 at 75,000 cycles, 4 at 80,000 cycles, 3 at 85,000 cycles) and then statically broken to measure the sizes of the nucleated cracks. Also, in these panels the cracks were found only in the

After having statically broken the tested panels, it was not possible, at all the hole sides, to detect a crack and they were considered as run-outs. The run-out effect has been introduced inside the crack size distribution by using the maximum likelihood method (Spindel & Haibach, 1979). Their crack dimensions have been supposed less than 0.1 mm. We supposed that the crack sizes (both with and without the run-outs) belong to two lognormal

**Comparison between experimental data and PISA simulations** 

0 20000 40000 60000 80000 100000 120000 140000

Fig. 13. Comparison between experimental data and PISA simulations (crack dimensions at

Experimental data (failure) PISA code simulation (failure)


Experimental data (crack size at 80000 cycles) Exp. data distr. (crack size at 80000 cycles) + run-out

**Log[c], mm**

**Cycles**

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Starting from the EIFS distribution obtained by the simple strip joints and by using the PISA code, the capability of the procedure has been verified by simulating the behavior of the

The comparison has been made by generating 1000 runs, i.e. by simulating the crack sizes at different number of cycles in the most critical row of 1000 lap-joint panels similar to the tested lap-joint panels (i.e. 15 holes x 2 sides = 30 positions for each panel). In Figure 13 (Cavallini & R. Lazzeri, 2007) the comparison is shown between the predicted crack

80,000 cycles and cycles to failure for the lap-joint panels).

PISA code simulation (crack size at 80000 cycles)

dimensions and the corresponding experimental results at 80,000 cycles.

cracks in the most critical row of the lap-joint panels.

a constant amplitude spectrum (*S*max = 120 MPa, *R*=0.1).

most critical row, i.e. the first one, at the countersunk side.

distributions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

**P**

The Pisa code is organized in such a way that all the information about geometry, material characterization, loads, inspection methods, failure criteria are collected in an input file.

The code can be used for the evaluation of the fatigue behaviour of only one component, starting from an initial known crack path, or for the generation of high numbers of deterministic simulations for the probabilistic approach.

## **6.1 The simulation of a single panel**

In Fig. 14, the fatigue behaviour of a very simple panel, with only four open holes (diameter=4 mm), in Al 2024-T3, loaded under a constant amplitude load with *S*max=100 MPa and *R*=0.1, is simulated. The initial crack path is extracted from the EIFS distribution, but it would be assigned also as an external input. The assigned life was 100,000 cycles. Inspections were planned every 25,000 cycles. For the probability of detection parameters, we assumed cmin=0.65 mm, =1.62 mm, =1.35, (extrapolated from Ratwani, 1996).

Fig. 14. Simulation of the fatigue life of an open hole panel (damage tolerance criterion).

As it can be seen in the Figure, during the first inspection (at 25,000 cycles) cracks were too small and were not detected. They grew till the second inspection (at 50,000 cycles), when

A Probabilistic Approach to Fatigue Design

damage condition.

1.E-08

introduces costs.

after 31,100/3 = 10,367 cycles.

expensive.

1.E-07

1.E-06

1.E-05

1.E-04

**Probability of failure**

1.E-03

1.E-02

1.E-01

1.E+00

of Aerospace Components by Using the Risk Assessment Evaluation 45

cycles the component should be substituted, without any consideration about its real

**Deterministic vs Probabilistic Approach in fatigue design of aerospace components**

Second inspection after (63000-51000) = 12000 cycles

Fig. 15. Example of PISA simulation and maintenance strategy for a lap-joint panel.

The probability of failure at 18,638 cycles is extremely low, so the component could still be used without any loss in safety. In this situation, the component replacement only

As far as the damage tolerance criterion is concerned, the first inspection (threshold) is fixed by evaluating the number of cycles necessary for a crack of assigned dimension (regulations state a 1.27 mm size) to grow till the final failure. The inspections cited below are planned considering the propagation period of a sure visible crack (depending on the selected inspection method, in this case 6.35 mm) till the final failure. A safety factor equal to 2 at the

For this component, by analytical calculation or by using the PISA code itself, it can be found that the period necessary for a crack to grow from 1.27 mm to the final failure is equal

Therefore, the first inspection will be carried out at 57,700/2=28,850 cycles and the next one

As it can be seen in Fig 15, the corresponding probability of failures is very low, and so the inspection plan, based on the deterministic damage tolerance approach, might be very

to 57,700 cycles, and from 6.35 mm to the final failure is equal to 31,100 cycles.

Without inspections

threshold and 3 at the following period are additionally applied.

First inspection at 51000 cycles Second inspection at 63000 cycles

40000 50000 60000 70000 80000 90000 100000 110000 120000 130000 **Cycles to failure**

First inspection = Threshold at 51000 cycles

Third inspection after (73000-63000) = 10000 cycles

Mean Life for Safe Life criterion

five cracks were detected and the panel repaired. The simulation went on till the following inspection, when four more cracks were detected and repaired. With this inspection plan the panel could reach the target life.

## **6.2 Applications of the PISA code for the probabilistic risk evaluation**

Before applying the Pisa code for the evaluation of the probability of failure of an aerospace component, an acceptable risk level must be identified. Indeed, one among the most debated items connected with the application of this methodology is the definition of the 'acceptable' risk level. Usually, 'risk' defines the probability of failure of some components within an assigned period.

Lincoln (Lincoln, 1998), says that for the USAF an acceptable global risk failure is 10-7 for flight, even if other authors suggest a safer *r*(t) 10-9 per hour (Lundberg, 1959).

We fixed *r*(t) 10-7. To reach a 10-7 probability of failure, at least 3x10+7 simulations must be run.

Starting from a lap-joint in Al 2024-T3, Fig. 7, loaded at C.A. with *S*max=120 MPa, *R*=0.1, our aims were the definition of a 'safe' maintenance plan, the comparison of the effects of the deterministic (safe-life and damage tolerance) and the probabilistic approaches, and the evaluation of their respective advantages and disadvantages, (Cavallini & R. Lazzeri, 2007).

We assumed the following distributions for the stochastic parameters:


At first, we simulated the fatigue behaviour of 3x10+7 lap-joint panels without any inspection actions. The number of cycles with probability of failure equal to 10-7 is 51,000 cycles, Fig. 15. So, this number of cycles can be fixed for the first inspection (threshold).

The second run was made after having fixed, for each panel, the first inspection at 51,000 cycles. In such a way, we obtained the new probability of failure curve and it was possible to fix the second inspection at 63,000 cycles, that is (63,000 - 51,000) = 12,000 cycles after the first one.

In Fig. 15 the probability of failures corresponding to the different deterministic approaches are also shown.

The safe life criterion requires the component replacement after a portion (as for example ¼) of its mean life. The mean life (probability of failure equal to 50%) corresponds to 74,550 cycles that, divided by four, gives 18,638 cycles. So, for the safe life criterion, after 18,638

five cracks were detected and the panel repaired. The simulation went on till the following inspection, when four more cracks were detected and repaired. With this inspection plan the

Before applying the Pisa code for the evaluation of the probability of failure of an aerospace component, an acceptable risk level must be identified. Indeed, one among the most debated items connected with the application of this methodology is the definition of the 'acceptable' risk level. Usually, 'risk' defines the probability of failure of some components within an

Lincoln (Lincoln, 1998), says that for the USAF an acceptable global risk failure is 10-7 for

We fixed *r*(t) 10-7. To reach a 10-7 probability of failure, at least 3x10+7 simulations must be

Starting from a lap-joint in Al 2024-T3, Fig. 7, loaded at C.A. with *S*max=120 MPa, *R*=0.1, our aims were the definition of a 'safe' maintenance plan, the comparison of the effects of the deterministic (safe-life and damage tolerance) and the probabilistic approaches, and the evaluation of their respective advantages and disadvantages, (Cavallini & R. Lazzeri, 2007).





At first, we simulated the fatigue behaviour of 3x10+7 lap-joint panels without any inspection actions. The number of cycles with probability of failure equal to 10-7 is 51,000 cycles, Fig. 15. So, this number of cycles can be fixed for the first inspection (threshold).

The second run was made after having fixed, for each panel, the first inspection at 51,000 cycles. In such a way, we obtained the new probability of failure curve and it was possible to fix the second inspection at 63,000 cycles, that is (63,000 - 51,000) = 12,000 cycles after the

In Fig. 15 the probability of failures corresponding to the different deterministic approaches

The safe life criterion requires the component replacement after a portion (as for example ¼) of its mean life. The mean life (probability of failure equal to 50%) corresponds to 74,550 cycles that, divided by four, gives 18,638 cycles. So, for the safe life criterion, after 18,638

flight, even if other authors suggest a safer *r*(t) 10-9 per hour (Lundberg, 1959).

We assumed the following distributions for the stochastic parameters:

[(*C*)]=0.036792. The corresponding *m* value is *m*=2.555.

[Log10(*c*0)]=0.28456, [*c*0] in mm.

(extrapolated from Ratwani, 1996).

**6.2 Applications of the PISA code for the probabilistic risk evaluation** 

panel could reach the target life.

assigned period.

1980),

first one.

are also shown.

run.

cycles the component should be substituted, without any consideration about its real damage condition.

Fig. 15. Example of PISA simulation and maintenance strategy for a lap-joint panel.

The probability of failure at 18,638 cycles is extremely low, so the component could still be used without any loss in safety. In this situation, the component replacement only introduces costs.

As far as the damage tolerance criterion is concerned, the first inspection (threshold) is fixed by evaluating the number of cycles necessary for a crack of assigned dimension (regulations state a 1.27 mm size) to grow till the final failure. The inspections cited below are planned considering the propagation period of a sure visible crack (depending on the selected inspection method, in this case 6.35 mm) till the final failure. A safety factor equal to 2 at the threshold and 3 at the following period are additionally applied.

For this component, by analytical calculation or by using the PISA code itself, it can be found that the period necessary for a crack to grow from 1.27 mm to the final failure is equal to 57,700 cycles, and from 6.35 mm to the final failure is equal to 31,100 cycles.

Therefore, the first inspection will be carried out at 57,700/2=28,850 cycles and the next one after 31,100/3 = 10,367 cycles.

As it can be seen in Fig 15, the corresponding probability of failures is very low, and so the inspection plan, based on the deterministic damage tolerance approach, might be very expensive.

A Probabilistic Approach to Fatigue Design

OH 45433-6553.

04-1.1/NLR.

Belgium, May 1998.

Martinus Nijhoff.

7923-7013-9, Dordrecht, NL.

1.

evaluation of structures.

Publ., ISBN 0-412-15870-1, New York.

Aerospace Engineering, University of Pisa.

American Society for Testing Materials.

Wright-Patterson Air Force Base, Dayton, Ohio.

ISBN 0-13-579475-7, Englewood Cliffs, NJ.

of Aerospace Components by Using the Risk Assessment Evaluation 47

Grooteman F. P. (2002). *WP4.4: Structural Reliability Solution Methods – Advanced Stochastic Method*, Admire Document N. ADMIRE-TR-4.4-03-3.1/NLR-CR-2002-544. Hammersley J. M. & Handscomb D. C. (1983). *Monte Carlo Methods,* Chapman and Hall

Hovey P. W., Berens A. P. & Skinn D. A. (1991). *Risk Analysis for Aging Aircraft Volume 1 –* 

Imparato G. & Santini L. (1997). *Prove sperimentali sul comportamento a fatica di strutture con* 

Joint Aviation Authorities (1994). Joint Airworthiness Requirements, JAR-25, Large

Johnston G. O. (1983). Statistical scatter in fracture toughness and fatigue crack growth rates,

Koolloons M. (2002). Details on Round Tobin Tests, ADMIRE Document, ADMIRE-TR-5.1-

Kuo, A., Yasgur, D. & Levy, M. (1986). Assessment of damage tolerance requirements and

Lewis W. H., Sproat W.H., Dodd B. D. & Hamilton J. M. (1978). *Reliability of nondestructive* 

Liao M. & Komorowski J. P. (2004). Corrosion risk assessment of aircraft structures. *Journal* 

Lincoln J.W. (1998). Role of nondestructive inspection airworthiness assurance, *RTO AVT* 

Lundberg, B. (1959). The Quantitative Statistical Approach to the Aircraft Fatigue Problem,

Manning, S.D., Yang, J.N. & Rudd, J.L. (1987). Durability of Aircraft Structures, In:

Ratwani M. M. (1996). Visual and non-destructive inspection technologies, In: *Aging Combat* 

Sampath S. & Broek D. (1991). Estimation of requirements of inspection intervals for panels

*Aircraft Fleets - Long Term Implications*, AGARD SMP LS-206.

*of ASTM International*, vol. 1, no. 8 (September 2004), pp. 183-198.

*Analysis*, Flight Dynamics Directorate, Wright Laboratory, Wright-Patterson AFB,

*danneggiamento multiplo*, Thesis in Aeronautical Engineering, Department of

Aeroplanes, Section 1, Subpart D, JAR 25.571, Damage-tolerance and fatigue

In: *Probabilistic fracture Mechanics and Fatigue Methods: Applications for structural design and maintenance*, ASTM STP 798, pp. 42-66, Bloom J.M. & Ekvall J.C.,

analyses - Task I report., *ICAF Doc. 1583*, AFVAL-TR-86-3003, vol. II, AFVAL

*inspection-final report*, San Antonio Air Logistic Center, Rep. SA-ALC/MME 76-6-38-

*Workshop on Airframe Inspection Reliability under field/depot conditions*, Brussels,

Full-Scale Fatigue Testing of Aircraft Structures, *Proceedings of the 1st ICAF Symposium*, Amsterdam, Netherlands, 1959, Pergamon Press, pp. 393-412 (1961). Madsen H. O., Krenk S. & Lind N. C. (1986). *Methods of Structural Safety*, Prentice Hall, Inc.,

*Probabilistic Fracture Mechanics and Reliability*, Provan J.W. (ed.), pp. 213-267,

susceptible to multiple site damage, In: *Structural Integrity of Aging Airplanes*, Atluri S.N., Sampath, S.G. & Tong, P., Editors, , pp*.* 339-389, Springer-Verlag, Berlin. Schijve J., (2001). *Fatigue of Structures and Materials*, Kluwer Academic Publishers, ISBN 0-

## **7. Conclusion**

In this Chapter, a new possible and useful approach to fatigue design of aerospace metallic components is explained, founded on probabilistic bases, together with a tool – the PISA code - and the experimental test results used for the validation of the tool, and of the approach.

The validation analysis provided good results and therefore the PISA code can be used for the risk assessment analysis and to compare the effect of the deterministic approaches (damage tolerance and safe-life) with those of the probabilistic approach in the fatigue design of a wide lap-joint panel.

The advantages appeared to be very significant:


The comparison between the different approaches, applied to a lap joint panel, shows that a more economic inspection plan can be applied if the probabilistic approach is used, without loss of safety.

Of course, this new methodology can be safely applied only if reliable models for the crack growth are available, and the parameter distributions have been carefully obtained.

## **8. References**


In this Chapter, a new possible and useful approach to fatigue design of aerospace metallic components is explained, founded on probabilistic bases, together with a tool – the PISA code - and the experimental test results used for the validation of the tool, and of the

The validation analysis provided good results and therefore the PISA code can be used for the risk assessment analysis and to compare the effect of the deterministic approaches (damage tolerance and safe-life) with those of the probabilistic approach in the fatigue





The comparison between the different approaches, applied to a lap joint panel, shows that a more economic inspection plan can be applied if the probabilistic approach is used, without

Of course, this new methodology can be safely applied only if reliable models for the crack

Besuner P.M. (1987). Probabilistic Fracture Mechanics, In: *Probabilistic fracture mechanics and* 

Cavallini G., Lanciotti A. & Lazzeri L. (1997). A Probabilistic Approach to Aircraft Structures

Cavallini G. & Lazzeri R. (2007). A probabilistic approach to fatigue risk assessment in

Federal Aviation Administration (1998). Federal Aviation Regulations – Part 25.

http://rgl.faa.gov/Regulatory\_and\_Guidance\_Library/rgFAR.nsf/Frameset?Ope

tolerance and fatigue evaluation of structures. Available from

*reliability*, Provan Ed., pp. 387-436, Martinus Nijhoff Publ., ISBN 90-247-3334-0,

Risk Assessment, Proceedings of the 19th ICAF Symposium, Edinburgh (UK), June

aerospace components. *Eng. Fracture Mech.*, vol. 74, issue 18, (Dec. 2007), pp. 2964-

Airworthiness Standards: Transport Category Airplanes, Section 571, Damage-

growth are available, and the parameter distributions have been carefully obtained.

avoiding too early inspections or the substitution of intact components.

so the risk level is well defined. In deterministic approaches, this important element is not known and the assumption of conservative values of the inputs can produce

;

**7. Conclusion** 

design of a wide lap-joint panel.

The advantages appeared to be very significant:

uneconomical designs without benefits;

to heavy and/or very expensive solutions;

in terms of probability of failure;

approach.

loss of safety.

**8. References** 

Dordrecht (NL).

1997, pp. 421-440.

2970.

nPage


**3** 

Juraj Belan

*Slovak Republic* 

**Study of Advanced Materials for Aircraft Jet** 

The aerospace industry is one of the biggest consumers of advanced materials because of its unique combination of mechanical and physical properties and chemical stability. Highly alloyed stainless steel, titanium alloys and nickel based superalloys are mostly used for aerospace applications. High alloyed stainless steel is used for the shafts of aero engine turbines, titanium alloys for compressor blades and finally nickel base superalloys are used for the most stressed parts of the jet engine – the turbine blades. Nickel base superalloys were used in various structural modifications: as cast polycrystalline, a directionally solidified, single crystal and in last year's materials which were produced by powder

So what exactly is a superalloy? Let us have a closer look to its definition. An interesting thing about it is that there is no standard definition of what constitutes a superalloy. The definitions which are provided in the various handbooks and reference books, although somewhat vague, are typically based on the service conditions in which superalloys are utilised. The most concise definition might be that provided by Sims et al. (1987): "...superalloys are alloys based on Group VIII-A base elements developed for elevatedtemperature service, which demonstrate combined mechanical strength and surface stability." Superalloys are typically used at service temperatures above 540 C° (1000 F°), and within a wide range of fields and applications, such as components in turbine engines, nuclear reactors, chemical processing equipment and biomedical devices; by volume, its predominant use is in aerospace applications. Superalloys are processed by a wide range of techniques, such as investment casting, forging and forming, and powder metallurgy.

The superalloys are often divided into three classes based on the major alloying constituent: iron-nickel-based, nickel-based and cobalt-based. The iron-nickel-based superalloys are considered to have developed as an extension of stainless steel technology. Superalloys are highly alloyed, and a wide range of alloying elements are used to enhance specific microstructural features (and - therefore - mechanical properties). Superalloys can be further divided into three additional groups based on their primary

**1. Introduction** 

metallurgy.

strengthening mechanism:

solid-solution strengthened;

**Engines Using Quantitative Metallography** 

*University of Žilina, Faculty of Mechanical Engineering,* 

*Department of Materials Engineering, Žilina* 


http://www.everyspec.com/USAF/USAF+(General)/JSSG-2006\_10206/.

Wang G. S. (1988). An Elastic-Plastic Solution for a Normally Loaded Center Hole in a finite Circular Body, *Int. J. Press-Ves & Piping*, vol. 33, pp. 269-284.

## **Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography**

## Juraj Belan

*University of Žilina, Faculty of Mechanical Engineering, Department of Materials Engineering, Žilina Slovak Republic* 

## **1. Introduction**

48 Recent Advances in Aircraft Technology

Schutz W. (1980). Treatment of scatter of fracture toughness data for design purpose, In: *Practical Applications of fracture Mechanics*, AGARD-AG-257, Liebowitz H. (ed). Spindel J.E. & Haibach E. (1979). The method of maximum likelihood applied to the

Tong Y. C. (2001). Literature Review on Aircraft Structural Risk and Reliability Analysis,

US Department of Defence (1998). Joint Service Specification Guide - Aircraft Structures,

Wang G. S. (1988). An Elastic-Plastic Solution for a Normally Loaded Center Hole in a finite

http://www.everyspec.com/USAF/USAF+(General)/JSSG-2006\_10206/.

Circular Body, *Int. J. Press-Ves & Piping*, vol. 33, pp. 269-284.

1979), pp. 81-88.

1110%20PR.pdf

JSSG-2006, Available from

statistical analysis of fatigue data. *International Journal of Fatigue*, vol. I, no. 2, (April

Department of Defence DSTO, Melbourne. Available from http://dspace.dsto.defence.gov.au/dspace/bitstream/1947/4289/1/DSTO-TR-

> The aerospace industry is one of the biggest consumers of advanced materials because of its unique combination of mechanical and physical properties and chemical stability. Highly alloyed stainless steel, titanium alloys and nickel based superalloys are mostly used for aerospace applications. High alloyed stainless steel is used for the shafts of aero engine turbines, titanium alloys for compressor blades and finally nickel base superalloys are used for the most stressed parts of the jet engine – the turbine blades. Nickel base superalloys were used in various structural modifications: as cast polycrystalline, a directionally solidified, single crystal and in last year's materials which were produced by powder metallurgy.

> So what exactly is a superalloy? Let us have a closer look to its definition. An interesting thing about it is that there is no standard definition of what constitutes a superalloy. The definitions which are provided in the various handbooks and reference books, although somewhat vague, are typically based on the service conditions in which superalloys are utilised. The most concise definition might be that provided by Sims et al. (1987): "...superalloys are alloys based on Group VIII-A base elements developed for elevatedtemperature service, which demonstrate combined mechanical strength and surface stability." Superalloys are typically used at service temperatures above 540 C° (1000 F°), and within a wide range of fields and applications, such as components in turbine engines, nuclear reactors, chemical processing equipment and biomedical devices; by volume, its predominant use is in aerospace applications. Superalloys are processed by a wide range of techniques, such as investment casting, forging and forming, and powder metallurgy.

> The superalloys are often divided into three classes based on the major alloying constituent: iron-nickel-based, nickel-based and cobalt-based. The iron-nickel-based superalloys are considered to have developed as an extension of stainless steel technology. Superalloys are highly alloyed, and a wide range of alloying elements are used to enhance specific microstructural features (and - therefore - mechanical properties). Superalloys can be further divided into three additional groups based on their primary strengthening mechanism:

solid-solution strengthened;

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 51

Fig. 2. Turbine blading in the (a) equiaxed-, (b) columnar- and (c) single–crystal forms.

during heat treatment. The fatigue life is then improved.

equiaxed forms is still practiced in many instances).

Once this development had occurred, it was quite natural to remove the grain boundaries completely such that monocrystalline (single-crystal) superalloys were produced. This allowed, in turn, the removal of grain boundary strengthening elements such as boron and carbon which had traditionally been added, thereby enabling better heat treatments to reduce microsegregation and induced eutectic content, whilst avoiding incipient melting

Nowadays, single–crystal superalloys are being used in increasing quantities in the gas turbine engine; if the very best creep properties are required, then the turbine engineers turn to them (although it should be recognised that the use of castings in the columnar and

In this chapter, a problem of polycrystalline (equiaxed) nickel base superalloy turbine blades

The structure of polycrystalline Ni–based superalloys - depending on heat–treatment consists of a solid solution of elements in Ni (-phase, an austenitic fcc matrix phase) and inter-metallic strengthening precipitate Ni3(Al, Ti) (΄-phase, which is an ordered coherent precipitate phase with a LI2 structure). A schematic showing representative microstructures of both a wrought and a cast nickel-base superalloy is shown in Figure 3. The precipitates in precipitate strengthened nickel-base superalloys remain coherent up to large precipitate sizes due to the small lattice mismatch between the matrix phase and the precipitates. The precipitates are usually present in volume fractions in the range of 20-60%, depending on the alloy (Sims et al. 1987), with typical shapes from the spherical at small sizes to cuboid at larger sizes, although more complex dendritic shapes are also observed in some cases (see Figure 4). The alignment of precipitates along the elastically soft (100) directions is frequently observed. Nickel based superalloys are precipitation hardened, with a typical


precipitate size of 0.25-0.5 μm for high temperature applications (Sims et al. 1987).


Solid-solution strengthening results from lattice distortions caused by solute atoms. These solute atoms produce a strain field which interacts with the strain field associated with the dislocations and acts to impede the dislocation motion. In precipitation strengthened alloys, coherent precipitates resist dislocation motion. At small precipitate sizes, strengthening occurs by the dislocation cutting of the precipitates, while at larger precipitate sizes strengthening occurs through Orowan looping. Oxide dispersion strengthened alloys are produced by mechanical alloying and contain fine incoherent oxide particles which are harder than the matrix phase and which inhibit dislocation motion by Orowan looping (MacSleyne 2008).

Figure 1. provides a representation of the alloy and process development which has occurred since the first superalloys began to appear in the 1940s; the data relates to the materials and processes used in turbine blading, such that the creep performance is a suitable measure for the progress which has been made. Various points emerge from a study of the figure. First, one can see that - for the blading application - cast rather than wrought materials are now preferred since the very best creep performance is then conferred. However, the first aerofoils were produced in wrought form. During this time, alloy development work – which saw the development of the first Nimonic alloys - enabled the performance of blading to be improved considerably; the vacuum introduction casting technologies which were introduced in the 1950s helped with this since the quality and cleanliness of the alloy were dramatically improved.

Fig. 1. Evolution of the high–temperature capability of superalloys over a 70 year period, since their emergence in the 1940s (Reed 2006).

Second, the introduction of improved casting methods and - later - the introduction of processing by directional solidification enabled significant improvements to be made; this was due to the columnar microstructures that were produced in which the transverse grain boundaries were absent (see Figure 2.)

Solid-solution strengthening results from lattice distortions caused by solute atoms. These solute atoms produce a strain field which interacts with the strain field associated with the dislocations and acts to impede the dislocation motion. In precipitation strengthened alloys, coherent precipitates resist dislocation motion. At small precipitate sizes, strengthening occurs by the dislocation cutting of the precipitates, while at larger precipitate sizes strengthening occurs through Orowan looping. Oxide dispersion strengthened alloys are produced by mechanical alloying and contain fine incoherent oxide particles which are harder than the matrix phase and which inhibit dislocation motion by Orowan looping

Figure 1. provides a representation of the alloy and process development which has occurred since the first superalloys began to appear in the 1940s; the data relates to the materials and processes used in turbine blading, such that the creep performance is a suitable measure for the progress which has been made. Various points emerge from a study of the figure. First, one can see that - for the blading application - cast rather than wrought materials are now preferred since the very best creep performance is then conferred. However, the first aerofoils were produced in wrought form. During this time, alloy development work – which saw the development of the first Nimonic alloys - enabled the performance of blading to be improved considerably; the vacuum introduction casting technologies which were introduced in the 1950s helped with this since the quality and

Fig. 1. Evolution of the high–temperature capability of superalloys over a 70 year period,

Second, the introduction of improved casting methods and - later - the introduction of processing by directional solidification enabled significant improvements to be made; this was due to the columnar microstructures that were produced in which the transverse grain

precipitation strengthened;

(MacSleyne 2008).

oxide dispersion strengthened (ODS) alloys.

cleanliness of the alloy were dramatically improved.

since their emergence in the 1940s (Reed 2006).

boundaries were absent (see Figure 2.)

Fig. 2. Turbine blading in the (a) equiaxed-, (b) columnar- and (c) single–crystal forms.

Once this development had occurred, it was quite natural to remove the grain boundaries completely such that monocrystalline (single-crystal) superalloys were produced. This allowed, in turn, the removal of grain boundary strengthening elements such as boron and carbon which had traditionally been added, thereby enabling better heat treatments to reduce microsegregation and induced eutectic content, whilst avoiding incipient melting during heat treatment. The fatigue life is then improved.

Nowadays, single–crystal superalloys are being used in increasing quantities in the gas turbine engine; if the very best creep properties are required, then the turbine engineers turn to them (although it should be recognised that the use of castings in the columnar and equiaxed forms is still practiced in many instances).

In this chapter, a problem of polycrystalline (equiaxed) nickel base superalloy turbine blades - such as the most stressed parts of the aero jet engine - will be discussed.

The structure of polycrystalline Ni–based superalloys - depending on heat–treatment consists of a solid solution of elements in Ni (-phase, an austenitic fcc matrix phase) and inter-metallic strengthening precipitate Ni3(Al, Ti) (΄-phase, which is an ordered coherent precipitate phase with a LI2 structure). A schematic showing representative microstructures of both a wrought and a cast nickel-base superalloy is shown in Figure 3. The precipitates in precipitate strengthened nickel-base superalloys remain coherent up to large precipitate sizes due to the small lattice mismatch between the matrix phase and the precipitates. The precipitates are usually present in volume fractions in the range of 20-60%, depending on the alloy (Sims et al. 1987), with typical shapes from the spherical at small sizes to cuboid at larger sizes, although more complex dendritic shapes are also observed in some cases (see Figure 4). The alignment of precipitates along the elastically soft (100) directions is frequently observed. Nickel based superalloys are precipitation hardened, with a typical precipitate size of 0.25-0.5 μm for high temperature applications (Sims et al. 1987).

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 53

Close-Packed) phases are also presented, such as σ-phase AxBy (Cr, Mo)x(Fe, Ni)y, μ–phase A7B6 (Co, Fe, Ni)7(Mo, W, Cr)6, Laves phases A2B (Fe, Cr, Mn, Si)2(Mo, Ti, Nb) and A3B phases ( Ni3(AlTa), Ni3Ti, Ni3Ta and (NiFeCo)3(NbTi)). The shape and size of these structural components have a significant influence on final the mechanical properties of

Although alloy-specific heat treatments are generally proprietary, the typical heat treatment of nickel-base superalloys consists of a solution treatment followed by an aging step (precipitation and coarsening). For additional details on alloy-specific heat treatments, see Sims et al. (1987) and M. J. Donachie & S. J. Donachie (2002). Nickel-base superalloys are highly-alloyed: because of the complexity which this adds, many experimental studies use binary or ternary alloys as model alloy systems. The nickel-rich region of the binary Ni-Al alloy system is frequently used as a model alloy system. The Al-Ni phase diagram is shown in Figure 5, and we will use it to consider the typical heat treatments of nickel-base

The solution treatment occurs above the solvus and is required for the ordered to go into the solution. The solvus separates the + and regions in Figure 5. This is usually followed by a quench (air, water or oil, depending upon the alloy) to room temperature. The

alloys and - mainly - on creep rupture life.

Fig. 5. Al-Ni phase diagram (ASM, 1992).

superalloys.

Fig. 3. Structure of a wrought and a cast nickel-base superalloy (M. J. Donachie & S. J. Donachie 2002).

Fig. 4. Schematic showing the evolution of morphology during continuous cooling. Sphere → cube → ogdoadically diced cubes → octodendrite → dendrite (Durand–Charre 1997).

In niobium-strengthened nickel-base superalloys - such as IN-718 - the principal strengthening phase is (Ni3Nb), which has a bct ordered DO22 structure. When precipitates are observed, they form as disk-shaped precipitates on {100} planes with a thickness of 5-9 nm and an average diameter of 60 nm (Durand–Charre 1997).

The next structural components are MC type primary carbides (created by such elements as Cr and Ti) and M23C6 type secondary carbides (created by such elements as Cr, Co, Mo and W). However, except of these structural components, "unwanted" TCP (Topologically

Fig. 3. Structure of a wrought and a cast nickel-base superalloy (M. J. Donachie & S. J.

Fig. 4. Schematic showing the evolution of morphology during continuous cooling. Sphere → cube → ogdoadically diced cubes → octodendrite → dendrite (Durand–Charre 1997).

In niobium-strengthened nickel-base superalloys - such as IN-718 - the principal strengthening phase is (Ni3Nb), which has a bct ordered DO22 structure. When precipitates are observed, they form as disk-shaped precipitates on {100} planes with a

The next structural components are MC type primary carbides (created by such elements as Cr and Ti) and M23C6 type secondary carbides (created by such elements as Cr, Co, Mo and W). However, except of these structural components, "unwanted" TCP (Topologically

thickness of 5-9 nm and an average diameter of 60 nm (Durand–Charre 1997).

Donachie 2002).

Close-Packed) phases are also presented, such as σ-phase AxBy (Cr, Mo)x(Fe, Ni)y, μ–phase A7B6 (Co, Fe, Ni)7(Mo, W, Cr)6, Laves phases A2B (Fe, Cr, Mn, Si)2(Mo, Ti, Nb) and A3B phases ( Ni3(AlTa), Ni3Ti, Ni3Ta and (NiFeCo)3(NbTi)). The shape and size of these structural components have a significant influence on final the mechanical properties of alloys and - mainly - on creep rupture life.

Although alloy-specific heat treatments are generally proprietary, the typical heat treatment of nickel-base superalloys consists of a solution treatment followed by an aging step (precipitation and coarsening). For additional details on alloy-specific heat treatments, see Sims et al. (1987) and M. J. Donachie & S. J. Donachie (2002). Nickel-base superalloys are highly-alloyed: because of the complexity which this adds, many experimental studies use binary or ternary alloys as model alloy systems. The nickel-rich region of the binary Ni-Al alloy system is frequently used as a model alloy system. The Al-Ni phase diagram is shown in Figure 5, and we will use it to consider the typical heat treatments of nickel-base superalloys.

Fig. 5. Al-Ni phase diagram (ASM, 1992).

The solution treatment occurs above the solvus and is required for the ordered to go into the solution. The solvus separates the + and regions in Figure 5. This is usually followed by a quench (air, water or oil, depending upon the alloy) to room temperature. The

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 55

For the number of -phase particles, a coherent testing grid with 9 square shape area

For the volume of -phase particles, a coherent testing grid with 50 dot probes made

Secondary dendrite arm spacing was evaluated according to Figure 6 and calculated with formula (1). The changing of the distance between the secondary dendrite arms "d" is an important characteristic because of base material, matrix , degradation via the equalising of

probes was used;

from backslash crossing was used.

chemical heterogeneity and also grain size growing.

Fig. 6. Scheme for the evaluation of secondary dendrite arm-spacing.

and as causing decreasing mechanical strength at higher temperatures.

and "z" is the magnification used.

<sup>1</sup> <sup>1000</sup> *<sup>L</sup> d m*


For the evaluation of the - and -phases the method of coherent testing grid was used, and the number of "N" was evaluated by a grid with 9 square-shaped area probes (Figure 7a) and the volume of "V" was evaluated by grid with 50 dot probes (Figure 7b). Afterwards, measurement of the values was calculated with formulas (2) and (3). For a detailed description of the methods used, see (Skočovský & Vaško 2007, Tillová & Panušková 2008, Tillová et al. 2011). The size of is also important from the point of view of creep rupture life. A precipitate with a size higher than 0.8 μm can be considered to be heavily degraded

(1)

*n z*

aging occurs at a temperature below the solvus temperature and allows for the homogeneous nucleation, growth and coarsening of , followed by air or furnace cooling to room temperature. Although heterogeneous nucleation is observed on grain boundaries and dislocations - for example - nucleation is primarily homogeneous. The temperature and duration of the aging treatment are selected so as to optimise the morphology, alignment and size distribution of precipitates. The resulting microstructure, in addition to its dependence on heat-treatment parameters, is also dependent on the physical properties of the alloy (and their isotropic or anisotropic nature) such as the lattice mismatch, the coherent interface energy, the volume fraction of and the elastic properties of the matrix and precipitate.

Polycrystalline turbine blades typically work within a temperature range from 705°C up to 800°C. As such, they must be protected from heat by various heat-proof layers; for example an alitise layer, MCrAlY coating or TBC (Thermal Barrier Coating). For this reason, dendrite arm-spacing, carbide size and distribution, morphology, the number and value of the phase and protective layer degradation are very important structural characteristics for the prediction of a blade's lifetime as well as the aero engine itself. In this chapter, the methods of quantitative metallography (Image Analyzer software NIS – Elements for carbide evaluation, the measurement of secondary dendrite arm-spacing and a coherent testing grid for -phase evaluation) are used for the evaluation of the structural characteristics mentioned above on experimental material – Ni base superalloy ŽS6K.

For instance, a precipitate size greater than 0.8 m significantly decreases the creep rupture life of superalloys and a carbide size greater than 5 m is not desirable because of the initiation of fatigue cracks (Cetel, A. D. & Duhl, D. N. 1988).

For this reason, the needs of new methods of the evaluation of non–conventional structure parameters were developed. Quantitative metallography, deep etching and colour-contrast belong to the basic methods. The analysis of quantitative metallography has a statistical nature. The elementary tasks of quantitative metallography are:


The application of quantitative metallography and colour contrast on the ŽS6K Ni–base superalloy are the main objectives discussed in this chapter.

## **2. Description of experimental methods and experimental material**

#### **2.1 Experimental methods**

For the evaluation of structural characteristics the following methods of quantitative metallography were used:


aging occurs at a temperature below the solvus temperature and allows for the homogeneous nucleation, growth and coarsening of , followed by air or furnace cooling to room temperature. Although heterogeneous nucleation is observed on grain boundaries and dislocations - for example - nucleation is primarily homogeneous. The temperature and duration of the aging treatment are selected so as to optimise the morphology, alignment and size distribution of precipitates. The resulting microstructure, in addition to its dependence on heat-treatment parameters, is also dependent on the physical properties of the alloy (and their isotropic or anisotropic nature) such as the lattice mismatch, the coherent interface energy, the volume fraction of and the elastic properties of the matrix

Polycrystalline turbine blades typically work within a temperature range from 705°C up to 800°C. As such, they must be protected from heat by various heat-proof layers; for example an alitise layer, MCrAlY coating or TBC (Thermal Barrier Coating). For this reason, dendrite arm-spacing, carbide size and distribution, morphology, the number and value of the phase and protective layer degradation are very important structural characteristics for the prediction of a blade's lifetime as well as the aero engine itself. In this chapter, the methods of quantitative metallography (Image Analyzer software NIS – Elements for carbide evaluation, the measurement of secondary dendrite arm-spacing and a coherent testing grid for -phase evaluation) are used for the evaluation of the structural characteristics

For instance, a precipitate size greater than 0.8 m significantly decreases the creep rupture life of superalloys and a carbide size greater than 5 m is not desirable because of

For this reason, the needs of new methods of the evaluation of non–conventional structure parameters were developed. Quantitative metallography, deep etching and colour-contrast belong to the basic methods. The analysis of quantitative metallography has a statistical

The application of quantitative metallography and colour contrast on the ŽS6K Ni–base

For the evaluation of structural characteristics the following methods of quantitative

Carbide distribution and average size was evaluated by the software NIS-Elements;

**2. Description of experimental methods and experimental material** 

mentioned above on experimental material – Ni base superalloy ŽS6K.

the initiation of fatigue cracks (Cetel, A. D. & Duhl, D. N. 1988).

nature. The elementary tasks of quantitative metallography are:

 Volume ratio of evaluated gamma prime phase; Number ratio of evaluated gamma prime phase;

Secondary dendrite arm-spacing measurement;

superalloy are the main objectives discussed in this chapter.

 Dendrite arm-spacing evaluation; Carbide size and distribution;

**2.1 Experimental methods** 

metallography were used:

 Size of evaluated gamma prime phase; Protective alitise layer degradation.

and precipitate.


Secondary dendrite arm spacing was evaluated according to Figure 6 and calculated with formula (1). The changing of the distance between the secondary dendrite arms "d" is an important characteristic because of base material, matrix , degradation via the equalising of chemical heterogeneity and also grain size growing.

Fig. 6. Scheme for the evaluation of secondary dendrite arm-spacing.

$$d = \frac{L}{n} \cdot \frac{1}{z} \cdot 1000 \quad \text{(\mu m)}\tag{1}$$


For the evaluation of the - and -phases the method of coherent testing grid was used, and the number of "N" was evaluated by a grid with 9 square-shaped area probes (Figure 7a) and the volume of "V" was evaluated by grid with 50 dot probes (Figure 7b). Afterwards, measurement of the values was calculated with formulas (2) and (3). For a detailed description of the methods used, see (Skočovský & Vaško 2007, Tillová & Panušková 2008, Tillová et al. 2011). The size of is also important from the point of view of creep rupture life. A precipitate with a size higher than 0.8 μm can be considered to be heavily degraded and as causing decreasing mechanical strength at higher temperatures.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 57

caused by chemical heterogeneity (Fig. 8a) and particles of primary MC and secondary M23C6 carbides (Fig. 8b). Primary carbides MC (where M is (Ti, Mo and W)) are presented as block-shaped particles, mainly inside grains. Secondary carbides are presented by "Chinese"

a) dendritic segregation b) MC and M23C6 carbides

However, the microstructure also contains a solid solution of elements in the nickel matrix – the so-called -phase (Ni (Cr, Co and Fe)) and strengthening-phase, which is a product of artificial age–hardening and has a significant influence on mechanical properties and creep rupture life – so-called -phase (gamma prime, Ni3 (Al and Ti)), Fig. 9a. Of course, both of

a) matrix and phases b) / eutectic

Fig. 9. Ni–base superalloy ŽS6K microstructure, as–cast.

Fig. 8. Microstructure of as–cast Ni–base superalloy ŽS6K, Beraha III.

these phases - (gamma) and (gamma prime) - create an eutectic /, Fig. 9b.

Table 1. Experimental alloy's chemical composition.

script-shaped particles on grain boundaries.

C **Ni Co Ti Cr Al W Mo Fe Mn**  0.13 ÷0.2 Bal. 4.0 ÷ 5.5 2.5 ÷ 3.2 9.5 ÷ 12 5.0 ÷ 6.0 4.5 ÷ 5.5 3.5 ÷ 4.8 2 0.4 **Adulterants P S Pb Bi** 0.015 0.015 0.001 0.0005

Fig. 7. Coherent testing grid for evaluation.

$$N = 1,1 \, 1 \cdot z^2 \cdot x\_{str} \cdot 10^{-9} \quad \left(\mu m^{-2}\right) \tag{2}$$


$$V = \frac{n\_s}{n} \cdot 100 \quad \text{(\%)}\tag{3}$$


#### **2.2 Experimental material**

The cast Ni–base superalloy ŽS6K was used as an experimental material. Alloy ŽS6K is a former USSR superalloy which was used in the DV–2 jet engine. It is used for turbine rotor blades and whole-cast small-sized rotors with a working temperature of up to 800 ÷ 1050°C. The alloy is made in vacuum furnaces. Parts are made by the method of precise casting. The temperature of the liquid at casting in a vacuum to form is 1500 ÷ 1600°C, depending on the part's shape and its quantity. The cast ability of this alloy is very good, with only 2 ÷ 2.5% of shrinkage. Blades made of this alloy are also protected against hot corrosion, with a protective heat-proof alitise layer, and so they are able to work at temperatures of up to 750°C for 500 flying hours.

This alloy was evaluated at the starting stage, the stage with normal heat treatment after 600, 1000, 1500 and 2000 hours of regular working (for these evaluations, real ŽS6K turbine blades with a protective alitise layer were used as an experimental material), and different samples made from the same experimental material ŽS6K after annealing at 800 °C/ 10 and 800 °C/15 hours. This was followed by cooling at various rates in water, oil and air. The chemical composition in wt % is presented in Table 1.

A typical microstructure of the ŽS6K Ni–base superalloy as cast is shown by Figures 8 and 9. The microstructure of the as–cast superalloy consists of significant dendritic segregation


Table 1. Experimental alloy's chemical composition.

56 Recent Advances in Aircraft Technology

<sup>2</sup> <sup>9</sup> <sup>2</sup> 11,1 10 *xzN str*


%100


The cast Ni–base superalloy ŽS6K was used as an experimental material. Alloy ŽS6K is a former USSR superalloy which was used in the DV–2 jet engine. It is used for turbine rotor blades and whole-cast small-sized rotors with a working temperature of up to 800 ÷ 1050°C. The alloy is made in vacuum furnaces. Parts are made by the method of precise casting. The temperature of the liquid at casting in a vacuum to form is 1500 ÷ 1600°C, depending on the part's shape and its quantity. The cast ability of this alloy is very good, with only 2 ÷ 2.5% of shrinkage. Blades made of this alloy are also protected against hot corrosion, with a protective heat-proof alitise layer, and so they are able to work at temperatures of up to

This alloy was evaluated at the starting stage, the stage with normal heat treatment after 600, 1000, 1500 and 2000 hours of regular working (for these evaluations, real ŽS6K turbine blades with a protective alitise layer were used as an experimental material), and different samples made from the same experimental material ŽS6K after annealing at 800 °C/ 10 and 800 °C/15 hours. This was followed by cooling at various rates in water, oil and air. The

A typical microstructure of the ŽS6K Ni–base superalloy as cast is shown by Figures 8 and 9. The microstructure of the as–cast superalloy consists of significant dendritic segregation

*n <sup>n</sup> <sup>V</sup> <sup>s</sup>* *<sup>m</sup>* (2)

(3)

a) number of particles b) volume of particles

Fig. 7. Coherent testing grid for evaluation.

medium value of -phase measurements.

**2.2 Experimental material** 

750°C for 500 flying hours.

probes, the equation become more simple: V=2ns).

chemical composition in wt % is presented in Table 1.

caused by chemical heterogeneity (Fig. 8a) and particles of primary MC and secondary M23C6 carbides (Fig. 8b). Primary carbides MC (where M is (Ti, Mo and W)) are presented as block-shaped particles, mainly inside grains. Secondary carbides are presented by "Chinese" script-shaped particles on grain boundaries.

Fig. 8. Microstructure of as–cast Ni–base superalloy ŽS6K, Beraha III.

However, the microstructure also contains a solid solution of elements in the nickel matrix – the so-called -phase (Ni (Cr, Co and Fe)) and strengthening-phase, which is a product of artificial age–hardening and has a significant influence on mechanical properties and creep rupture life – so-called -phase (gamma prime, Ni3 (Al and Ti)), Fig. 9a. Of course, both of these phases - (gamma) and (gamma prime) - create an eutectic /, Fig. 9b.

Fig. 9. Ni–base superalloy ŽS6K microstructure, as–cast.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 59

From the relations presented (Figure 11) it is obvious that the holding time on various temperatures for annealing and cooling in selected mediums does not have a significant influence on carbide particle size. More significant, the influence on the ratio of carbide particles has a cooling rate (Figure 10). With increasing speed of cooling and a longer

Generally, we can suppose that carbide particles are partially dissolved with the temperature of annealing and elements, which are consider as an carbide creators (in this case mainly Ti) have create a new particles of phase. This phenomenon has an influence on decreasing the segregated carbide percentage ratio. With an increase of the cooling rate (water, oil), an amount of the -phase decreases and the carbides percentage ratio is higher. At slow cooling and a longer time of holding is higher amount of segregate and, therefore,

*→ M23C6 +* 

The microstructures which are equivalent to these evaluations are in Figures 12 and 13. For

holding time on the annealing temperature, the carbide particles' ratio decreases.

*MC +* 

carbide evaluation, etching is not necessary. All of the micrographs are non-etched.

Fig. 12. Microstructure of ŽS6K, carbides ratio after 800°C annealing/10 hrs: a) water

the ratio of carbides decreases. It is all happen according to scheme:

a) b)

c)

cooling; b) oil cooling; c) air cooling.

## **3. Experimental results and discussion**

## **3.1 Carbide evaluation**

Polycrystalline and columnar grain alloys contain carbon additions to help improve grain– boundary strength and ductility. While the addition of carbon is beneficial to grain boundary ductility, the large carbides that form can adversely affect fatigue life. Both lowand high-cycle fatigue-cracking have been observed to initiate with the large (lengths greater than 0.005 mm) carbides presented in these alloys. When polycrystalline alloys were cast in a single crystal form, it was determined that carbides did not impart any beneficial strengthening effects in the absence of grain boundaries, and thus could be eliminated. Producing essentially carbon–free single crystal alloys led to significant improvements in fatigue life as large carbide colonies were no-longer present to initiate fatigue cracks (Cetel, A. D. & Duhl, D. N. 1988).

The first characteristic were carbide size and its distribution evaluated. Specimens made of the ŽS6K superalloy were compared at the starting stage (non-heat-treated, as-cast) after 800°C/10 hrs and 800°C/15 hrs. The cooling rate depends on the cooling medium; in our case these were air, oil and water. The results for the ratio of carbide particles in the observed area are in Figure 10 and the results on the average carbide size are in Figure 11.

Fig. 10. The ratio of carbide particles from the observed area.

**Starting stage 10 hrs. 15 hrs.**

Fig. 11. Average carbide size [μm].

Polycrystalline and columnar grain alloys contain carbon additions to help improve grain– boundary strength and ductility. While the addition of carbon is beneficial to grain boundary ductility, the large carbides that form can adversely affect fatigue life. Both lowand high-cycle fatigue-cracking have been observed to initiate with the large (lengths greater than 0.005 mm) carbides presented in these alloys. When polycrystalline alloys were cast in a single crystal form, it was determined that carbides did not impart any beneficial strengthening effects in the absence of grain boundaries, and thus could be eliminated. Producing essentially carbon–free single crystal alloys led to significant improvements in fatigue life as large carbide colonies were no-longer present to initiate fatigue cracks (Cetel,

The first characteristic were carbide size and its distribution evaluated. Specimens made of the ŽS6K superalloy were compared at the starting stage (non-heat-treated, as-cast) after 800°C/10 hrs and 800°C/15 hrs. The cooling rate depends on the cooling medium; in our case these were air, oil and water. The results for the ratio of carbide particles in the observed area are in Figure 10 and the results on the average carbide size are in Figure 11.

**3,55 3,55 3,55**

**water oil air Cooling medium**

**5,74 6,12 6,2**

**5,15**

**6,05 6,07**

**3. Experimental results and discussion** 

Fig. 10. The ratio of carbide particles from the observed area.

**Starting stage 10 hrs. 15 hrs.**

**3.1 Carbide evaluation** 

A. D. & Duhl, D. N. 1988).

Fig. 11. Average carbide size [μm].

**Carbide particles average size** 

**[µm]**

From the relations presented (Figure 11) it is obvious that the holding time on various temperatures for annealing and cooling in selected mediums does not have a significant influence on carbide particle size. More significant, the influence on the ratio of carbide particles has a cooling rate (Figure 10). With increasing speed of cooling and a longer holding time on the annealing temperature, the carbide particles' ratio decreases.

Generally, we can suppose that carbide particles are partially dissolved with the temperature of annealing and elements, which are consider as an carbide creators (in this case mainly Ti) have create a new particles of phase. This phenomenon has an influence on decreasing the segregated carbide percentage ratio. With an increase of the cooling rate (water, oil), an amount of the -phase decreases and the carbides percentage ratio is higher. At slow cooling and a longer time of holding is higher amount of segregate and, therefore, the ratio of carbides decreases. It is all happen according to scheme:

$$\text{MC} + \text{\$\gamma \to M\$}\_2\text{C}\_6 + \text{\$\gamma'} $$

The microstructures which are equivalent to these evaluations are in Figures 12 and 13. For carbide evaluation, etching is not necessary. All of the micrographs are non-etched.

Fig. 12. Microstructure of ŽS6K, carbides ratio after 800°C annealing/10 hrs: a) water cooling; b) oil cooling; c) air cooling.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 61

Fig. 14. Dendritic segregation of ŽS6K, starting stage, Marble etchant.

a)

b)

c)

cooling, Marble etchant.

Fig. 15. Dendritic segregation of ŽS6K, 800°C/10 hrs: a) water cooling; b) oil cooling,; c) air

Fig. 13. Microstructure of ŽS6K, carbides ratio after 800°C annealing/15 hrs: a) water cooling; b) oil cooling; c) air cooling.

#### **3.2 Evaluation of secondary dendrite arm-spacing**

The second characteristic which is evaluated is dendrite arm-spacing. In this evaluation, two different approaches were taken. For the first evaluation, non-heat treated ŽS6K specimens were used and compared with loading at 800°C/10(15) hrs. The results of these first evaluations can be seen in Table 2 and Figures 14, 15 and 16. The second evaluation was performed on ŽS6K turbine blades used in the DV-2 (LPT – Low Pressure Turbine and HPT – High Pressure Turbine) aero jet engine at the starting stage (basic heat treatment) and after an engine exposition (at real working temperatures) for 600, 1000, 1500 and 2000 hours. Again, the results are in Table 3 and the microstructures are in Figure 17.


Table 2. Results from secondary dendrite arm-spacing for a non-heat treated ŽS6K alloy

a) b)

c)

**3.2 Evaluation of secondary dendrite arm-spacing** 

cooling; b) oil cooling; c) air cooling.

Fig. 13. Microstructure of ŽS6K, carbides ratio after 800°C annealing/15 hrs: a) water

Again, the results are in Table 3 and the microstructures are in Figure 17.

The second characteristic which is evaluated is dendrite arm-spacing. In this evaluation, two different approaches were taken. For the first evaluation, non-heat treated ŽS6K specimens were used and compared with loading at 800°C/10(15) hrs. The results of these first evaluations can be seen in Table 2 and Figures 14, 15 and 16. The second evaluation was performed on ŽS6K turbine blades used in the DV-2 (LPT – Low Pressure Turbine and HPT – High Pressure Turbine) aero jet engine at the starting stage (basic heat treatment) and after an engine exposition (at real working temperatures) for 600, 1000, 1500 and 2000 hours.

> **Secondary dendrite arm spacing [μm] ŽS6K – starting stage** 185.19 **Cooling medium**

Table 2. Results from secondary dendrite arm-spacing for a non-heat treated ŽS6K alloy

**ŽS6K/10hrs.** 126.58 131.58 138.89 **ŽS6K/15hrs.** 113.64 131.58 156.25

**Water Oil Air** 

Fig. 14. Dendritic segregation of ŽS6K, starting stage, Marble etchant.

Fig. 15. Dendritic segregation of ŽS6K, 800°C/10 hrs: a) water cooling; b) oil cooling,; c) air cooling, Marble etchant.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 63

increases (the dendrites are growing). From the results mentioned above (Table 2), it is clear to see that with a higher cooling rate comes a slowing of the diffusion processes and the dendrite arm-spacing decreases in comparison with the starting stage (Figure 14). All of these changes are also obvious in Figures 15 and 16. The ŽS6K dendrite arm-spacing increases in relation to the annealing time, with an annealing temperature and cooling

The same phenomena can be observed with heat-treated turbine blades after various working times. Of course, the secondary dendrite arm-spacing is smaller, but again it has a tendency to growth. So, this confirms the results from Table 2: that a longer time of exposure

Since the advanced high-strength nickel–base alloys owe their exceptional high temperature properties to the high volume fraction of the ordered -phase that they contain, it should not be surprising that control of precipitate distribution and morphology can profoundly affect their properties. The post-casting processing of these alloys - especially solution heat

The high-strength alloys typically contain about 55 ÷ 75 % of the precipitates which, in the cast condition, are coarse (0.4 ÷ 1.0 μm) and irregularly-shaped cuboid particles (see Figure 9). The evaluation of the -phase is also divided into two parts, just as the dendrite evaluation was. Firstly, the -phase was evaluated on the cast stage, and secondly on turbine blades. The characteristics of γ΄-phase morphology were also measured using the coherent testing grid methods. As was mentioned above, the number and volume of the γ΄-phase have a significant influence on the mechanical properties of this alloy, especially on creep rupture life. The average satisfactory size of the γ΄-phase is about 0.35–0.45 m (Figure 18) and also the carbide size should not exceed a size of 5 m because of fatigue crack initiation (M. J. Donachie & S. J. Donachie 2002). Another risk in using high temperature loading (or annealing) is the creation of TCP phases - such σ-phase or Laves-phase - within the temperature range of 750 °C–800 °C. The results of first evaluation are in Table 4. The

Fig. 18. Influence of γ΄-phase size on the lifetime and mechanical properties of Ni superalloy.

medium of between 113.64 and 156.25 μm.

**3.3 Evaluation of morphology** 

has a significant influence on dendrite and grain size.

treatment - can radically affect microstructure.

microstructures related to this evaluation are in Figures 19 and 20.

Fig. 16. Dendritic segregation of ŽS6K, 800°C/15 hrs.: a) water cooling, b) oil cooling, c) air cooling, Marble etchant.


Table 3. Results from secondary dendrite arm spacing for real turbine blades, heat-treated ŽS6K alloy.

Fig. 17. Dendritic segregation of ŽS6K turbine blades: a) 1°LPT – starting stage; b) HPT – after 1500 hrs of work, Marble etchant.

The cast materials are characterised by dendritic segregation, which is caused by chemical heterogeneity. With the influence of holding at an annealing temperature, chemical heterogeneity decreases. This means that the distance between secondary dendrite arms increases (the dendrites are growing). From the results mentioned above (Table 2), it is clear to see that with a higher cooling rate comes a slowing of the diffusion processes and the dendrite arm-spacing decreases in comparison with the starting stage (Figure 14). All of these changes are also obvious in Figures 15 and 16. The ŽS6K dendrite arm-spacing increases in relation to the annealing time, with an annealing temperature and cooling medium of between 113.64 and 156.25 μm.

The same phenomena can be observed with heat-treated turbine blades after various working times. Of course, the secondary dendrite arm-spacing is smaller, but again it has a tendency to growth. So, this confirms the results from Table 2: that a longer time of exposure has a significant influence on dendrite and grain size.

#### **3.3 Evaluation of morphology**

62 Recent Advances in Aircraft Technology

(a) (b)

(c) Fig. 16. Dendritic segregation of ŽS6K, 800°C/15 hrs.: a) water cooling, b) oil cooling, c) air

Table 3. Results from secondary dendrite arm spacing for real turbine blades, heat-treated

Fig. 17. Dendritic segregation of ŽS6K turbine blades: a) 1°LPT – starting stage; b) HPT –

The cast materials are characterised by dendritic segregation, which is caused by chemical heterogeneity. With the influence of holding at an annealing temperature, chemical heterogeneity decreases. This means that the distance between secondary dendrite arms

**Blade of 1°LPT – starting stage** 24.38 **Blade of HPT - 600 hrs.** 24.78 **Blade of HPT - 1000 hrs.** 27.98 **Blade of HPT - 1500 hrs.** 48.73 **Blade of HPT - 2000 hrs.** 66.66

a) b)

after 1500 hrs of work, Marble etchant.

**Type of blade Secondary dendrite arm-spacing [μm]** 

cooling, Marble etchant.

ŽS6K alloy.

Since the advanced high-strength nickel–base alloys owe their exceptional high temperature properties to the high volume fraction of the ordered -phase that they contain, it should not be surprising that control of precipitate distribution and morphology can profoundly affect their properties. The post-casting processing of these alloys - especially solution heat treatment - can radically affect microstructure.

The high-strength alloys typically contain about 55 ÷ 75 % of the precipitates which, in the cast condition, are coarse (0.4 ÷ 1.0 μm) and irregularly-shaped cuboid particles (see Figure 9).

The evaluation of the -phase is also divided into two parts, just as the dendrite evaluation was. Firstly, the -phase was evaluated on the cast stage, and secondly on turbine blades. The characteristics of γ΄-phase morphology were also measured using the coherent testing grid methods. As was mentioned above, the number and volume of the γ΄-phase have a significant influence on the mechanical properties of this alloy, especially on creep rupture life. The average satisfactory size of the γ΄-phase is about 0.35–0.45 m (Figure 18) and also the carbide size should not exceed a size of 5 m because of fatigue crack initiation (M. J. Donachie & S. J. Donachie 2002). Another risk in using high temperature loading (or annealing) is the creation of TCP phases - such σ-phase or Laves-phase - within the temperature range of 750 °C–800 °C. The results of first evaluation are in Table 4. The microstructures related to this evaluation are in Figures 19 and 20.

Fig. 18. Influence of γ΄-phase size on the lifetime and mechanical properties of Ni superalloy.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 65

a) b)

c)

cooling, Marble etchant, SEM.

1988).

Fig. 20. Morphology of -phase after 800°C/15 hrs: a) air cooling; b) oil cooling; c) water

The highly alloyed nickel–base alloys solidify dendritically and, due to the effects of chemical segregation across the dendrites, a higher concentration of the -phase forms elements such as aluminium and titanium which are more present in the inter-dendritic areas than in the dendrite core. This results in the solvus (the temperature at which first precipitates upon cooling) being lower in the core region than the inter-dendritically region. Varying the cooling rate from the solution heat treatment temperature can significantly affect the particle size, as rapid rates do not allow sufficient time for the particles to coarsen as they precipitate upon cooling below the solvus temperature. Increasing the cooling rate of the solution heat treatment temperature from 30 to 120°C/minute results in an average particle size refinement of more than 30% (Figure 21) (Cetel, A. D. & Duhl, D. N.

By controlling both the solution heat treatment and the cooling rate, both the volume fraction of the fine particles as well as their size can be controlled. Heat treating an alloy close to its solvus temperature and completely dissolving its coarse particles can

produce consistently high-elevated temperature creep–rupture strength.


Table 4. Results from -phase evaluation at the cast stage at 800°C/10 (15) hrs.

With exposure for 10 hours at an annealing temperature, the volume of -phase was increased by about 16.8–33% when compared with the starting stage (Figure 19). The significant increase of the -phase was observed at a holding time of 15 hours (Figure 20), and cooling on air, where volume of -phase is 76.6 %.

Fig. 19. Morphology of -phase after 800°C/10 hrs: a) air cooling; b) oil cooling; c) water cooling, Marble etchant, SEM.

With exposure for 10 hours at an annealing temperature, the volume of -phase was increased by about 16.8–33% when compared with the starting stage (Figure 19). The significant increase of the -phase was observed at a holding time of 15 hours (Figure 20),

**Start. stage** 2.47 39.4 0.61 **10h water** 1.95 56.2 0.54 **10h oil** 1.60 63 0.63 **10h air** 1.50 72.4 0.69 **15h water** 1.90 66.8 0.59 **15h oil** 1.59 71.8 0.67 **15h air** 1.49 76.6 0.72 Table 4. Results from -phase evaluation at the cast stage at 800°C/10 (15) hrs.

**Volume of - phase V [%]** 

**Average size of - phase u [μm]** 

**Cooling medium**

**Number of - phase N [μm-2]** 

and cooling on air, where volume of -phase is 76.6 %.

a) b)

c)

cooling, Marble etchant, SEM.

Fig. 19. Morphology of -phase after 800°C/10 hrs: a) air cooling; b) oil cooling; c) water

Fig. 20. Morphology of -phase after 800°C/15 hrs: a) air cooling; b) oil cooling; c) water cooling, Marble etchant, SEM.

The highly alloyed nickel–base alloys solidify dendritically and, due to the effects of chemical segregation across the dendrites, a higher concentration of the -phase forms elements such as aluminium and titanium which are more present in the inter-dendritic areas than in the dendrite core. This results in the solvus (the temperature at which first precipitates upon cooling) being lower in the core region than the inter-dendritically region.

Varying the cooling rate from the solution heat treatment temperature can significantly affect the particle size, as rapid rates do not allow sufficient time for the particles to coarsen as they precipitate upon cooling below the solvus temperature. Increasing the cooling rate of the solution heat treatment temperature from 30 to 120°C/minute results in an average particle size refinement of more than 30% (Figure 21) (Cetel, A. D. & Duhl, D. N. 1988).

By controlling both the solution heat treatment and the cooling rate, both the volume fraction of the fine particles as well as their size can be controlled. Heat treating an alloy close to its solvus temperature and completely dissolving its coarse particles can produce consistently high-elevated temperature creep–rupture strength.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 67

0.98542 67.2 0.6819 1.1242 67.6 0.60131 1.1004 59 0.53615 0.81938 57.4 0.7005 0.6968 40.6 0.5826 Table 5. Results from the -phase evaluation on heat-treated turbine blades at various

Fig. 23. Morphology of the -phase heat-treated turbine blade, starting stage (0 hours), HCl

The morphology of the -phase at the starting stage is cuboid and distributed equally in the base matrix (Figure 23.). With an increase of the hours of work at a temperature of up to 750°C, the -phase morphology changes. The particles of the -phase gradually coarsen (time of work to 1000 hours, Figure 24 a, b), which confirms the results of a number of phase evaluations "N" (see Table 5). A decrease of this value at a longer duration of work (1500 and 2000 hours) is caused by reprecipitation of new, fine -phase particles in the area between the primal -phase (Figure 24 c, d). From the results in Table 5, it is obvious that the -phase of the ŽS6K alloy coarsen uniformly and increase its volume ratio in the structure after up to 1000 hours of exposition (regular work of a jet engine). However, after longer durations of work (1500 or 2000 hours) there occurs the reprecipitation of new, fine particles of the -phase in the free space of matrix and which has caused structural

In terms of structure degradation and the prediction of the life time of turbine blades - as well as the jet engine itself – and according to the results in Table 5, after up to 1000 hours of exposition the structure (with "N" = 1.1004, "V" = 59 and average size "u" = 0.53615) is at the "edge" of use because of its mechanical properties, as shown by Figure 18. However, the -phase size is not the only parameter influencing the life time. In addition, the number "N"

**Volume of -phase V [%]** 

**Average size of -phase u [μm2]** 

**Number of -phase N [μm-2]** 

**Time of work [hours]** 

working times.

+ H2O2 etchant.

heterogeneity.

Fig. 21. Optimum size achieved by rapid cooling of the solution temperature combined with post-solution heat treatment.

Work performed by (Nathal et al. 1987) indicates that the optimum -phase size for an alloy is dependent on the lattice mismatch between the - and -phases (Figure 22), which is composition dependent.

Fig. 22. The optimum -phase size to maximize creep strength is dependent on mismatch between the - and -phases (Nathal et al. 1987).

The second evaluation of the -phase was provided on heat-treated turbine blades of a DV-2 aero jet engine after various working times. The results obtained are shown by Table 5. For the evaluation a coherent testing grid was used - the same procedure as in the first evaluation. The microstructures related to this evaluation are shown in Figures 23 and 24.

Fig. 21. Optimum size achieved by rapid cooling of the solution temperature combined

Work performed by (Nathal et al. 1987) indicates that the optimum -phase size for an alloy is dependent on the lattice mismatch between the - and -phases (Figure 22), which is

Fig. 22. The optimum -phase size to maximize creep strength is dependent on mismatch

The second evaluation of the -phase was provided on heat-treated turbine blades of a DV-2 aero jet engine after various working times. The results obtained are shown by Table 5. For the evaluation a coherent testing grid was used - the same procedure as in the first evaluation. The microstructures related to this evaluation are shown in Figures 23 and 24.

with post-solution heat treatment.

between the - and -phases (Nathal et al. 1987).

composition dependent.


Table 5. Results from the -phase evaluation on heat-treated turbine blades at various working times.

Fig. 23. Morphology of the -phase heat-treated turbine blade, starting stage (0 hours), HCl + H2O2 etchant.

The morphology of the -phase at the starting stage is cuboid and distributed equally in the base matrix (Figure 23.). With an increase of the hours of work at a temperature of up to 750°C, the -phase morphology changes. The particles of the -phase gradually coarsen (time of work to 1000 hours, Figure 24 a, b), which confirms the results of a number of phase evaluations "N" (see Table 5). A decrease of this value at a longer duration of work (1500 and 2000 hours) is caused by reprecipitation of new, fine -phase particles in the area between the primal -phase (Figure 24 c, d). From the results in Table 5, it is obvious that the -phase of the ŽS6K alloy coarsen uniformly and increase its volume ratio in the structure after up to 1000 hours of exposition (regular work of a jet engine). However, after longer durations of work (1500 or 2000 hours) there occurs the reprecipitation of new, fine particles of the -phase in the free space of matrix and which has caused structural heterogeneity.

In terms of structure degradation and the prediction of the life time of turbine blades - as well as the jet engine itself – and according to the results in Table 5, after up to 1000 hours of exposition the structure (with "N" = 1.1004, "V" = 59 and average size "u" = 0.53615) is at the "edge" of use because of its mechanical properties, as shown by Figure 18. However, the -phase size is not the only parameter influencing the life time. In addition, the number "N"

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 69

a) starting stage b) 600 hours of exposition

c) 1500 hours of exposition d) 2000 hours of exposition

To improve the lifetime of turbine blades made from the ŽS6K alloy against a hot corrosion environment, an alitise Al–Si protective layer is used. What is important about this kind of layer is that it does not improve the high temperature properties of the base alloy but only

An alitise layer is used for the protection of HPT blades (Figure 26) and only 1° of LPT blades, which means that Al-Si suspension is applied on to the surface of the blades. Silicon is added due to its ability to increase resistance to corrosion in sulphide and sea environments. Generally, an alitise layer AS-2 type is used for corrosion protection of aero gas turbine parts which work at temperatures of up to 950 C; type AS-1 up to a temperature of 1100 C (DV–2–I–62: Company standard). The standard procedure of applying Al–Si

Fig. 25. Detail of the ŽS6K alloy's -phase showing the increasing distance between particles as an affect of working exposition - at normal working loading of a jet engine -

which has a significant influence on the dislocation hardening effect.

**3.4 Evaluation of the Al–Si protective layer** 

its hot corrosion resistance.

protective coating is in Table 6.

and volume "V" is important from the point of view of dislocation hardening. When "N" and "V" are smaller, this means that the distances between single particles are greater and that fact causes a decrease of the dislocation hardening effect. On the other hand, M23C6 carbides form a carbide net on the grain boundary which also decreases the creep rupture life by developing brittle grain boundaries. For a comparison of the increasing distance between -phase particles see Figure 25.

Fig. 24. Morphology of the -phase, heat treated turbine blade made of the ŽS6K alloy, after various exposition, HCl + H2O2 etchant.

and volume "V" is important from the point of view of dislocation hardening. When "N" and "V" are smaller, this means that the distances between single particles are greater and that fact causes a decrease of the dislocation hardening effect. On the other hand, M23C6 carbides form a carbide net on the grain boundary which also decreases the creep rupture life by developing brittle grain boundaries. For a comparison of the increasing distance

a) 600 hours of exposition b) 1000 hours of exposition

c) 1500 hours of exposition d) 2000 hours of exposition

Fig. 24. Morphology of the -phase, heat treated turbine blade made of the ŽS6K alloy, after

between -phase particles see Figure 25.

various exposition, HCl + H2O2 etchant.

Fig. 25. Detail of the ŽS6K alloy's -phase showing the increasing distance between particles as an affect of working exposition - at normal working loading of a jet engine which has a significant influence on the dislocation hardening effect.

#### **3.4 Evaluation of the Al–Si protective layer**

To improve the lifetime of turbine blades made from the ŽS6K alloy against a hot corrosion environment, an alitise Al–Si protective layer is used. What is important about this kind of layer is that it does not improve the high temperature properties of the base alloy but only its hot corrosion resistance.

An alitise layer is used for the protection of HPT blades (Figure 26) and only 1° of LPT blades, which means that Al-Si suspension is applied on to the surface of the blades. Silicon is added due to its ability to increase resistance to corrosion in sulphide and sea environments. Generally, an alitise layer AS-2 type is used for corrosion protection of aero gas turbine parts which work at temperatures of up to 950 C; type AS-1 up to a temperature of 1100 C (DV–2–I–62: Company standard). The standard procedure of applying Al–Si protective coating is in Table 6.

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 71

a) b)

c) d)

Starting stage, Fig. 27b

Overheating at 1000°C, Fig. 27d

Figure 27b, d.

Fig. 27. Alitise layer a, b) starting stage; c, d) after overheating at 1000°C, SEM.

Table 7. Spot analysis of selected particles. The marked spots (in wt%) correspond with

The alitise layer on the blades which have worked at regular conditions is also degraded, which is represented by the changing of the layer thickness and the surface relief. Changes in layer thickness are caused by heterogeneity of the temperature field along the blade and the abrasive and erosive effect of gases and exhaust gases. The level of layer degradation varies, depending on the area of blade. From a metallographic point of view, the highest degradation is in the flap pantile region close to the flow edge, in the case of blades after

Sample Marked spots AlK SiK MoK TiK CrK CoK NiK Wk

1 19.7 2.21 0.23 0.83 6.54 4.14 65.5 0.34 2 15.01 3.09 2.56 2.28 5.16 4.06 65 2.27 3 1.57 7.57 17.51 5.82 13.53 3.01 28.45 21.94

1 9.04 5.94 7.42 0.92 9.25 4.15 52.6 10.6 2 18.2 3.55 3.56 0.73 6.18 3.81 58.2 5.7 3 - 9.54 13.5 9.5 11.3 3.25 33.1 19.6


Table 6. Steps for protective Al–Si coating as applied on to a ŽS6K turbine blade.

Fig. 26. A high pressure turbine blade of a DV–2 aero jet engine, left-side, right-side and cross-section with cooling chambers.

The Al–Si layer consists of two layers at the starting stage. The upper layer (the aluminiumrich layer) is created by aluminides - a complex compound of Si, Cr and Mo - and by carbides. The lower part of layer (the silicon rich layer) is created mainly by silicon and titanium carbides and the matrix. The average thickness of the layer is 0.04 mm. According to the evaluation by metallography, the alitise layer is equally distributed across the whole blade surface at the starting stage (Figure 27 a, b).

In cases of overheating (here, at 1000°C - the normal working temperature is 705°C ÷ 750°C) the alitise layer is significantly degraded (Figure 27 c, d). The layer is non-homogeneous, with a rough surface and in place of the flow edge in the area of the flap pantile is a layer which is evenly broken (Figure 27d). Layer degradation is connected with the diffusion of elementary elements - such as Cr, Ti, Ni and Al - from the base material into the surface area (Table 7.). Where Cr and Ti creates carbides, Ni and Al form fine particles and Al as itself also creates NiAl (β–phase) and Al2O3 oxides on the surface of the layer. With decreasing of the layer's heat resistance, the base material is impoverished, which leads to the growth of particles and decreasing of its volume.

In vacuum, temperature 1225 °C, holding 4 hrs, cooling with argon to 900 °C per 10 min.

1. Spraying of AS2 layer (AS2 – koloxylin solution 350 ml, Al – powder 112 g, Si – powder 112 g) 2. Diffusion annealing temperature 1000 °C, 3 hrs, slowly cooled in retort

**Heat – treatment Conditions** 

Table 6. Steps for protective Al–Si coating as applied on to a ŽS6K turbine blade.

Fig. 26. A high pressure turbine blade of a DV–2 aero jet engine, left-side, right-side and

The Al–Si layer consists of two layers at the starting stage. The upper layer (the aluminiumrich layer) is created by aluminides - a complex compound of Si, Cr and Mo - and by carbides. The lower part of layer (the silicon rich layer) is created mainly by silicon and titanium carbides and the matrix. The average thickness of the layer is 0.04 mm. According to the evaluation by metallography, the alitise layer is equally distributed across the whole

In cases of overheating (here, at 1000°C - the normal working temperature is 705°C ÷ 750°C) the alitise layer is significantly degraded (Figure 27 c, d). The layer is non-homogeneous, with a rough surface and in place of the flow edge in the area of the flap pantile is a layer which is evenly broken (Figure 27d). Layer degradation is connected with the diffusion of elementary elements - such as Cr, Ti, Ni and Al - from the base material into the surface area (Table 7.). Where Cr and Ti creates carbides, Ni and Al form fine particles and Al as itself also creates NiAl (β–phase) and Al2O3 oxides on the surface of the layer. With decreasing of the layer's heat resistance, the base material is impoverished, which leads to the growth of

**Homogenization annealing** 

**Alitise AS2** 

cross-section with cooling chambers.

blade surface at the starting stage (Figure 27 a, b).

particles and decreasing of its volume.

Fig. 27. Alitise layer a, b) starting stage; c, d) after overheating at 1000°C, SEM.


Table 7. Spot analysis of selected particles. The marked spots (in wt%) correspond with Figure 27b, d.

The alitise layer on the blades which have worked at regular conditions is also degraded, which is represented by the changing of the layer thickness and the surface relief. Changes in layer thickness are caused by heterogeneity of the temperature field along the blade and the abrasive and erosive effect of gases and exhaust gases. The level of layer degradation varies, depending on the area of blade. From a metallographic point of view, the highest degradation is in the flap pantile region close to the flow edge, in the case of blades after

Study of Advanced Materials for Aircraft Jet Engines Using Quantitative Metallography 73

 The structure of the samples is characterised by dendritic segregation. In dendritic areas, fine -phase is segregated. In inter dendritic areas, eutectic cells / and carbides

 The holding time (10–15 hrs.) has a significant influence on the carbide particles' size. The size of the carbides is under a critical level for the initiation of fatigue crack only at the starting stage. An increase in the rate of cooling has a significant effect on the

 The chemical heterogeneity of the samples with a longer holding time decreases. This is a reason of the fact that there is sufficient time for the diffusion mechanism, which is

The volume of the -phase with a longer holding time increases and the -phase size

There was no evidence of the presence of TCP phase even at a high annealing

 Cooling rate also has an influence on the hardness. At a lower rate of cooling, the internal stresses are relaxed, which causes hardness to increase – a changing of the

The cooling rates, represented by various cooling mediums, have a significant influence on the diffusion processes which are operating within the structure. These diffusion processes are the main mechanisms for the formation and segregation of carbide particles, the equalising of chemical heterogeneity (represented by dendrite arm-spacing) and segregation

Air - as a cooling medium - provides sufficient time for the realisation of diffusion reactions and it leads to a decrease of chemical heterogeneity, which is presented by an increase of secondary dendrite arm-spacing. Also, this "slow" cooling rate has a positive effect on carbides' segregation and on the morphology, number and volume of the strength

Water is the most intensive cooling medium, which breaks diffusion processes and which leads to an increase of carbide particles in the observed area; the precipitate is smaller,

From a general point of view we can perform cooling in oil, which might be consider as a

For the turbine blades, which have been worked at normal loading and for various

The medium distance of secondary dendrite arms "d" grows in dependence on the time

 The gradual dissolving of primary carbides rests and the reprecipitation of secondary carbides on grain boundary. After longer durations of work (1000–2000 hours) it changes its chain morphology onto the carbide net, which has a significant influence on

 The inter-metallic phase- was evaluated with the methods of quantitative metallography; this evaluation shows gradual morphology changes of the -phase –

of the -phase; they are also responsible for structural degradation of such alloys.

precipitate (the precipitate has a greater diameter and its volume increases).

durations (600, 1000, 1500 and 2000 hours), the following results were achieved:

increasing the hardness and at least also increasing the strength.

of work, caused by changes of the grain size of the -matrix.

medium point between cooling in air and cooling in water.

lowering the mechanical properties of the alloy.

coarsening, spheroidisation and reprecipitation.

confirmed by the measurement results of secondary dendrite arm-spacing.

grows. With a higher rate of cooling the particles become finer.

are segregated.

temperature.

dislocation structure.

carbide particles' ratio.

1500 and 2000 hours of work (Figure 28 c, d). In region close to the Si sub-layer, needle particles (probably a special form of Cr base carbides) are created which grow depending upon the time of work (compare Figures 27 a, b and 28). These needle particles start to form after 600 hours of loading, which means that after 600 (Figure 28 a, b) hours of work and aero jet engine should to be taken in for overhauling and the old alitise layer replaced by a new one. However, when it comes to the local overheating of the turbine blades, all of the degradation processes are much faster.

Fig. 28. Creation of needle particles in the region under the Si sub-layer: a) 600 hours; b) 1000 hours; c) 1500 hours; d) 2000 hours of regular work, SEM, Marble etchant.

## **4. Conclusion**

As cast Ni–base ŽS6K superalloy was used as an experimental material, the structural characteristics were evaluated from the starting stage of the sample, after annealing at 800 °C/10 and 800 °C/15 hrs and after various working times in real jet engines with the use of the methods of quantitative metallography. The results are as follows:

1500 and 2000 hours of work (Figure 28 c, d). In region close to the Si sub-layer, needle particles (probably a special form of Cr base carbides) are created which grow depending upon the time of work (compare Figures 27 a, b and 28). These needle particles start to form after 600 hours of loading, which means that after 600 (Figure 28 a, b) hours of work and aero jet engine should to be taken in for overhauling and the old alitise layer replaced by a new one. However, when it comes to the local overheating of the turbine blades, all of the

degradation processes are much faster.

a) b)

c) d)

**4. Conclusion** 

Fig. 28. Creation of needle particles in the region under the Si sub-layer: a) 600 hours; b) 1000

As cast Ni–base ŽS6K superalloy was used as an experimental material, the structural characteristics were evaluated from the starting stage of the sample, after annealing at 800 °C/10 and 800 °C/15 hrs and after various working times in real jet engines with the use

hours; c) 1500 hours; d) 2000 hours of regular work, SEM, Marble etchant.

of the methods of quantitative metallography. The results are as follows:


The cooling rates, represented by various cooling mediums, have a significant influence on the diffusion processes which are operating within the structure. These diffusion processes are the main mechanisms for the formation and segregation of carbide particles, the equalising of chemical heterogeneity (represented by dendrite arm-spacing) and segregation of the -phase; they are also responsible for structural degradation of such alloys.

Air - as a cooling medium - provides sufficient time for the realisation of diffusion reactions and it leads to a decrease of chemical heterogeneity, which is presented by an increase of secondary dendrite arm-spacing. Also, this "slow" cooling rate has a positive effect on carbides' segregation and on the morphology, number and volume of the strength precipitate (the precipitate has a greater diameter and its volume increases).

Water is the most intensive cooling medium, which breaks diffusion processes and which leads to an increase of carbide particles in the observed area; the precipitate is smaller, increasing the hardness and at least also increasing the strength.

From a general point of view we can perform cooling in oil, which might be consider as a medium point between cooling in air and cooling in water.

For the turbine blades, which have been worked at normal loading and for various durations (600, 1000, 1500 and 2000 hours), the following results were achieved:


**4** 

**ALLVAC 718 Plus™ Superalloy** 

**for Aircraft Engine Applications** 

Innovations on the aerospace and aircraft industry have been throwing light upon building to future's engineering architecture at the today's globalization world where technology is the indispensable part of life. On the basis of aviation sector, the improvements of materials used in aircraft gas turbine engines which constitute 50 % of total aircraft weight must protect its actuality continuously. On the other hand utilization of super alloys in aerospace and defense industries can not be ignored because of excellent corrosion and oxidation

Materials that can be used at the homologous temperature of 0.6 Tm and still remain stable to withstand severe mechanical stresses and strains in oxidizing environments are so-called superalloys, usually based on Ni, Fe or Co (Sims et al, 1987). Nickel-based superalloys are the exceptional group of superalloys with superior materials properties. Their excellent properties range from high temperature mechanical strength, toughness to resistance to degradation in oxidizing and corrosive environment. Therefore they are not only used in aerospace and aircraft industry, but also in ship, locomotive, petro- chemistry and nuclear

Inconel 718 is Ni-based, precipitation- hardening superalloy with Nb as a major hardening element, used for high temperature aerospace applications very widely in recent years (Yaman & Kushan, 1998). However, the metastability of the primary strengthening (γ", gamma double prime) phase is typically unacceptable for applications above about 650°C. As a result, other more costly and difficult to process alloys, like Waspalloy, are used in such applications. Although Waspalloy is strengthened primarily by γ', it is still more susceptible to weld-related cracking than Inconel 718 (Otti et al, 2005). In these circumstances ALLVAC 718 Plus™ come to stage, which is strengthened with uniform cubic FCC inter metallic γ' phase, innovated by ATI ALLVAC Company very recently. In recent years it has been becoming widespread dramatically for using of disc material in aerospace gas turbine engine parts. The most important reason of this is the high yield and ultimate tensile strength and very good corrosion and oxidation resistance of material together with

resistance, high strength and long creep life at elevated temperatures.

**1. Introduction** 

reactor industries.

Melih Cemal Kushan1, Sinem Cevik Uzgur2,

Yagiz Uzunonat3 and Fehmi Diltemiz4

*1Eskisehir Osmangazi University 2Ondokuz Mayis University* 

*4Air Supply and Maintenance Base* 

*3Anadolu University* 

*Turkey* 

 The alitise layer degradation was expressed by a changing thickness and needle-like Cr carbide segregation at the sub-layer region, which has a negative influence on the layer's lifetime. There is strong recommendation for overhauling after every 500 hours of regular work.

### **5. Acknowledgment**

The authors acknowledge the financial support of the projects VEGA No. 1/0841/11 and No. 1/0460/11; KEGA No. 220-009ŽU-4/2010 and European Union - the Project "*Systematization of advanced technologies and knowledge transfer between industry and universities (ITMS 26110230004)*".

#### **6. References**


## **ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications**

Melih Cemal Kushan1, Sinem Cevik Uzgur2, Yagiz Uzunonat3 and Fehmi Diltemiz4 *1Eskisehir Osmangazi University 2Ondokuz Mayis University 3Anadolu University 4Air Supply and Maintenance Base Turkey* 

## **1. Introduction**

74 Recent Advances in Aircraft Technology

 The alitise layer degradation was expressed by a changing thickness and needle-like Cr carbide segregation at the sub-layer region, which has a negative influence on the layer's lifetime. There is strong recommendation for overhauling after every 500 hours

The authors acknowledge the financial support of the projects VEGA No. 1/0841/11 and No. 1/0460/11; KEGA No. 220-009ŽU-4/2010 and European Union - the Project "*Systematization of advanced technologies and knowledge transfer between industry and universities*

ASM. (1992). *ASM Handbook Volume 3: Alloy Phase Diagrams* (10th edition), ASM

Cetel, A. D. & Duhl, D. N. (1988). Microstructure – Property Relationships In Advanced

Donachie, M. J. & Donachie, S. J. (2002). *Superalloys – A technical Guide* (2nd edition)*,* ASM

Durand–Chare, M. (1997). *The Microstructure of Superalloys*, Gordon & Breach Science

DV–2–I–62: Company standard, Považské machine industry, Division of Aircraft Engine

MacSleyne, J. P. (2008). Moment invariants for two-dimensional and three-dimensional

<http://www.grin.com/en/doc/263761/moment-invariants-for-two-dimensional-

Reed, R. C. (2006). *The superalloys: fundamentals and applications*, Cambridge University Press,

Sims, Ch. T., Stoloff, N. S. & Hagel, W. C. (1987). *Superalloys II* (2nd edition), Wiley-

Skočovský, P. & Vaško, A. (2007). *The quantitative evaluation of cast iron structure* (1st edition),

Tillová, E. & Panuškova, M. (2008). Effect of Solution Treatment on Intermetallic Phase's

Tillová, E., Chalupová, M., Hurtalová, L., Bonek, M., & Dobrzanski, L. A. (2011). Structural

Morphology in AlSi9Cu3 Cast Alloy. *Mettalurgija/METABK*, No. 47, pp. 133-137, 1-

analysis of heat treated automotive cast alloy. *Journal of Achievements in Materials and Manufacturing Engineering/JAMME*, Vol. 47, No. 1, (July 2011), pp. 19-25, ISSN

characterization of the morphology of gamma-prime precipitates in nickel-base

Publishers, ISBN 90–5699–097–7, Amsterdam, Netherlands.

superalloys, In: *Doctoral Thesis / Dissertation*, n.d., Available from:

Nickel Base Superalloy Airfoil Castings, *2nd International SAMPE Metals Conference,* 

International, ISBN 0–871–70381–5, USA.

International, ISBN 0–87170–749–7, USA.

DV–2, Považská Bystrica, Slovakia, 1989.

and-three-dimensional-characterization> Nathal, M. V. (1987) *Met. Trans.*, Vol. 18 A, pp. 1961–1970.

ISBN 0–521–85904-2, New York, USA.

Interscience, ISBN 0–471–01147–9, USA.

4, ISSN 0543-5846.

1734-8412.

EDIS, ISBN 978-80-8070-748-4, Žilina, Slovak Republic.

pp. 37–48, USA, August 2–4, 1988.

of regular work.

**5. Acknowledgment** 

*(ITMS 26110230004)*".

**6. References** 

Innovations on the aerospace and aircraft industry have been throwing light upon building to future's engineering architecture at the today's globalization world where technology is the indispensable part of life. On the basis of aviation sector, the improvements of materials used in aircraft gas turbine engines which constitute 50 % of total aircraft weight must protect its actuality continuously. On the other hand utilization of super alloys in aerospace and defense industries can not be ignored because of excellent corrosion and oxidation resistance, high strength and long creep life at elevated temperatures.

Materials that can be used at the homologous temperature of 0.6 Tm and still remain stable to withstand severe mechanical stresses and strains in oxidizing environments are so-called superalloys, usually based on Ni, Fe or Co (Sims et al, 1987). Nickel-based superalloys are the exceptional group of superalloys with superior materials properties. Their excellent properties range from high temperature mechanical strength, toughness to resistance to degradation in oxidizing and corrosive environment. Therefore they are not only used in aerospace and aircraft industry, but also in ship, locomotive, petro- chemistry and nuclear reactor industries.

Inconel 718 is Ni-based, precipitation- hardening superalloy with Nb as a major hardening element, used for high temperature aerospace applications very widely in recent years (Yaman & Kushan, 1998). However, the metastability of the primary strengthening (γ", gamma double prime) phase is typically unacceptable for applications above about 650°C. As a result, other more costly and difficult to process alloys, like Waspalloy, are used in such applications. Although Waspalloy is strengthened primarily by γ', it is still more susceptible to weld-related cracking than Inconel 718 (Otti et al, 2005). In these circumstances ALLVAC 718 Plus™ come to stage, which is strengthened with uniform cubic FCC inter metallic γ' phase, innovated by ATI ALLVAC Company very recently. In recent years it has been becoming widespread dramatically for using of disc material in aerospace gas turbine engine parts. The most important reason of this is the high yield and ultimate tensile strength and very good corrosion and oxidation resistance of material together with

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 77

Leaving the combustor, the hot exhaust is passed through the turbine (shown in a (c) part of Fig. 2), in which the gases are partially expanded through alternate stator and rotor rows. Depending on the engine type, the turbine may consist of one or several stages. Like the compressor, the turbine is divided into low-pressure and high-pressure sections (shown in

Fig. 3. The temperature and pressure profile in gas turbine (Carlos & Estrada, 2007).

shaft, as well as the power for the fuel pump, generator, and other accessories. From

The turbine provides the power to turn the compressor, to which it is connected via a central

Fig. 2. Some basic sections of a gas turbine engine (Eliaz et al, 2002).

Fig. 3), the latter being closer to the combustor.

excellent creep resistance at elevated temperatures. Fig. 1 shows that where wrought alloy 718Plus can be used as a disc material for high pressure (HP) compressor as well as for high pressure (HP) turbine discs (Bond & Kennedy, 2005).

Fig. 1. Potential applications for alloy 718PlusTM in a future high pressure core section (Bond & Kennedy, 2005).

The newly innovated ALLVAC 718 Plus superalloy which is the last version of Inconel 718 has been proceeding in the way to become a material that aerospace and defense industries never replace of any other material with combining its good mechanical properties, easy machinability and low cost.

## **2. Gas turbine engines**

Gas turbine engines, also known as jet engines, power most modern civilian and military aircraft. Fig. 2 shows some sections of this kind of an engine. The inlet (intake) directs outside air into the engine. The compressor (shown in a (a) part of Fig. 2) is situated at the exit of the inlet. In order to produce thrust, it is essential to compress the air before fuel is added. In an axial-flow compressor, the air flows in the direction of the shaft axis through alternate rows of stationary and rotating blades, called stators and rotors, respectively. Modern axial-flow compressors can increase the pressure 24 times in 15 stages, with each set of stators and rotors making up a stage. The compressors in most modern engines are divided into low-pressure and high-pressure sections which run off two different shafts. In the combustor, or burner (shown in a (b) part of Fig. 2), the compressed air is mixed with fuel and burned. Fuel is introduced through an array of spray nozzles that atomize it. An electric igniter is used to begin combustion. The combustor adds heat energy to the air stream and increases its temperature (up to about 1930°C), a process which is accompanied by a slight decrease in pressure (~ 1-2%). For best performances, the combustion temperature should be the maximum obtainable from the complete combustion of the oxygen and the fuel. However, turbine inlet temperatures currently cannot exceed about 1100°C because of material limits. Hence, only part of the compressed air is burned in the combustor; the remainder is used to cool the turbine.

excellent creep resistance at elevated temperatures. Fig. 1 shows that where wrought alloy 718Plus can be used as a disc material for high pressure (HP) compressor as well as for high

Fig. 1. Potential applications for alloy 718PlusTM in a future high pressure core section

The newly innovated ALLVAC 718 Plus superalloy which is the last version of Inconel 718 has been proceeding in the way to become a material that aerospace and defense industries never replace of any other material with combining its good mechanical properties, easy

Gas turbine engines, also known as jet engines, power most modern civilian and military aircraft. Fig. 2 shows some sections of this kind of an engine. The inlet (intake) directs outside air into the engine. The compressor (shown in a (a) part of Fig. 2) is situated at the exit of the inlet. In order to produce thrust, it is essential to compress the air before fuel is added. In an axial-flow compressor, the air flows in the direction of the shaft axis through alternate rows of stationary and rotating blades, called stators and rotors, respectively. Modern axial-flow compressors can increase the pressure 24 times in 15 stages, with each set of stators and rotors making up a stage. The compressors in most modern engines are divided into low-pressure and high-pressure sections which run off two different shafts. In the combustor, or burner (shown in a (b) part of Fig. 2), the compressed air is mixed with fuel and burned. Fuel is introduced through an array of spray nozzles that atomize it. An electric igniter is used to begin combustion. The combustor adds heat energy to the air stream and increases its temperature (up to about 1930°C), a process which is accompanied by a slight decrease in pressure (~ 1-2%). For best performances, the combustion temperature should be the maximum obtainable from the complete combustion of the oxygen and the fuel. However, turbine inlet temperatures currently cannot exceed about 1100°C because of material limits. Hence, only part of the compressed air is burned in the

pressure (HP) turbine discs (Bond & Kennedy, 2005).

(Bond & Kennedy, 2005).

machinability and low cost.

**2. Gas turbine engines** 

combustor; the remainder is used to cool the turbine.

Fig. 2. Some basic sections of a gas turbine engine (Eliaz et al, 2002).

Leaving the combustor, the hot exhaust is passed through the turbine (shown in a (c) part of Fig. 2), in which the gases are partially expanded through alternate stator and rotor rows. Depending on the engine type, the turbine may consist of one or several stages. Like the compressor, the turbine is divided into low-pressure and high-pressure sections (shown in Fig. 3), the latter being closer to the combustor.

Fig. 3. The temperature and pressure profile in gas turbine (Carlos & Estrada, 2007).

The turbine provides the power to turn the compressor, to which it is connected via a central shaft, as well as the power for the fuel pump, generator, and other accessories. From

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 79

still used in industrial turbines, but were later replaced by *Waspaloy* and *Astroloy* as stress and temperature requirements increased. These alloys were turbine blade alloys with a suitably modified heat-treatment for discs. However, blade material is designed for creep, whereas disc material requires tensile strength coupled with low cycle fatigue life to cope with the stress changes in the flight cycle. To meet these requirements *Waspaloy* was thermomechanically processed (TMP) to give a fine-grain size and a 40% increase in tensile strength over the corresponding blade material, but at the expense of creep life. Similar improvements for discs have been produced in *Inco 901* by TMP. More highly-alloyed nickel-based discs suffer from excessive ingot segregation which makes grain size control difficult. Further development led to alloys produced by powder processing by gas atomization of a molten stream of metal in an inert argon atmosphere and consolidating the resultant powder by HIPing to near-net shape. Such products are limited in stress application because of inclusions in the powder and, hence, to realize the maximum advantage of this process it is necessary to produce 'superclean' material by electron beam

Improvements in turbine materials were initially developed by increasing the volume fraction of γ´ in changing *Nimonic 80A* up to *Nimonic 115*. Unfortunately increasing the (Ti +Al) content lowers the melting point, thereby narrowing the forging range makes processing more difficult. Improved high-temperature oxidation and hot corrosion performance has led to the introduction of aluminide and overlay coatings and subsequently the development of *IN 738* and *IN 939* with much improved hot-corrosion

Further improvements in superalloys have depended on alternative manufacturing routes, particularly using modern casting technology like Vacuum casting (Smallman & Bishop,

In these alloys *γ'* (Ni3Al) and *γ״*) Ni3Nb) are the principal strengtheners by chemical and coherency strain hardening. The ordered *γ'-*Ni3Al phase is an equilibrium second phase in both the binary Ni–Al and Ni–Cr–Al systems and a metastable phase in the Ni–Ti and Ni– Cr–Ti systems, with close matching of the *γ'* and the FCC matrix. The two phases have very similar lattice parameters and the coherency confers a very low coarsening rate on the precipitate so that the alloy overages extremely slowly even at 0.7*T*m. In alloys containing Nb, a metastable Ni3Nb phase occurs but, although ordered and coherent, it is less stable

In high-temperature service, the properties of the grain boundaries are as important as the strengthening by γ' within the grains. Grain boundary strengthening is produced mainly by precipitation of chromium and refractory metal carbides; small additions of Zr and B improve the morphology and stability of these carbides. Optimum properties are developed by multistage heat treatment; the intermediate stages produce the desired grain boundary microstructure of carbide particles enveloped in a film of *γ*' and the other stages produce two size ranges of *γ*' for the best combination of strength at both intermediate and high temperatures (Smallman & Ngan, 2007). Table 2 indicates the effect of the different alloying elements and Table 3 indicates the common ranges of main alloying additions and their

than γ' at high temperatures (Smallman & Ngan, 2007).

effects on superalloy properties.

or plasma melting.

resistance.

1999).

thermodynamics, the turbine work per mass airflow is equal to the change in the specific enthalpy of the flow from the entrance to the exit of the turbine. This change is related to the temperatures at these points. The temperature at the entrance to the turbine can be as high as 1650°C, considerably above the melting point of the material from which the blades are made.

The gases, leaving the turbine at an intermediate pressure, are finally accelerated through a nozzle to reach the desired high jet-exit velocity. Because the exit velocity is greater than the free stream velocity, thrust is produced. The amount of thrust generated depends on the rate of mass flow through the engine and the leaving jet velocity, according to Newton's Second Law. Thus, the gas is accelerated to the rear, and the engine (as well as the aircraft) is accelerated in the opposite direction according to Newton's Third Law (Eliaz et al, 2002).

Modern gas turbines have the most advanced and sophisticated technology in all aspects; construction materials are not the exception due their extreme operating conditions. The most difficult and challenging point is the one located at the turbine inlet, because, several difficulties associated to it; like extreme temperature, high pressure, high rotational speed, vibration, small circulation area, and so on. These rush characteristics produce effects on the gas turbine components that are shown on the Table 1 (Carlos & Estrada, 2007).


Table 1. Severity of the different surface related problems for gas turbine applications (Carlos & Estrada, 2007).

In order to overcome those barriers, gas turbine components are made using advanced materials and modern alloys (superalloys) that contains up to ten significant alloying elements.

## **3. Superalloys**

These alloys have been developed for high-temperature service and include iron-, cobaltand nickel based materials, although nowadays they are principally nickel based. These materials are widely used in aircraft and power-generation turbines, rocket engines, and other challenging environments, including nuclear power and chemical processing plants. The aero gas turbine was the impetus for the development of superalloys in the early 1940s, when conventional materials available at that time were insufficient for the demanding environment of the turbine. Therefore it can be said that "The development of superalloys made the modern gas turbine possible".

A major application of superalloys is in turbine materials, jet engines, both disc and blades. Initial disc alloys were *Inco 718* and *Inco 901* produced by conventional casting ingot, forged billet and forged disc route. These alloys were developed from austenitic steels, which are

The gases, leaving the turbine at an intermediate pressure, are finally accelerated through a nozzle to reach the desired high jet-exit velocity. Because the exit velocity is greater than the free stream velocity, thrust is produced. The amount of thrust generated depends on the rate of mass flow through the engine and the leaving jet velocity, according to Newton's Second Law. Thus, the gas is accelerated to the rear, and the engine (as well as the aircraft) is accelerated in the opposite direction according to Newton's Third Law (Eliaz et al, 2002).

Modern gas turbines have the most advanced and sophisticated technology in all aspects; construction materials are not the exception due their extreme operating conditions. The most difficult and challenging point is the one located at the turbine inlet, because, several difficulties associated to it; like extreme temperature, high pressure, high rotational speed, vibration, small circulation area, and so on. These rush characteristics produce effects on the

**Oxidation Hot corrosion Interdiffusion Thermal Fatigue** 

gas turbine components that are shown on the Table 1 (Carlos & Estrada, 2007).

**Aircraft** Severe Moderate Severe Severe

**Generator** Moderate Severe Moderate Light

**Marine Engines** Moderate Severe Light Moderate Table 1. Severity of the different surface related problems for gas turbine applications

In order to overcome those barriers, gas turbine components are made using advanced materials and modern alloys (superalloys) that contains up to ten significant alloying

These alloys have been developed for high-temperature service and include iron-, cobaltand nickel based materials, although nowadays they are principally nickel based. These materials are widely used in aircraft and power-generation turbines, rocket engines, and other challenging environments, including nuclear power and chemical processing plants. The aero gas turbine was the impetus for the development of superalloys in the early 1940s, when conventional materials available at that time were insufficient for the demanding environment of the turbine. Therefore it can be said that "The development of superalloys

A major application of superalloys is in turbine materials, jet engines, both disc and blades. Initial disc alloys were *Inco 718* and *Inco 901* produced by conventional casting ingot, forged billet and forged disc route. These alloys were developed from austenitic steels, which are

thermodynamics, the turbine work per mass airflow is equal to the change in the specific enthalpy of the flow from the entrance to the exit of the turbine. This change is related to the temperatures at these points. The temperature at the entrance to the turbine can be as high as 1650°C, considerably above the melting point of the material from which the blades are

made.

**Land-based Power** 

(Carlos & Estrada, 2007).

made the modern gas turbine possible".

elements.

**3. Superalloys** 

still used in industrial turbines, but were later replaced by *Waspaloy* and *Astroloy* as stress and temperature requirements increased. These alloys were turbine blade alloys with a suitably modified heat-treatment for discs. However, blade material is designed for creep, whereas disc material requires tensile strength coupled with low cycle fatigue life to cope with the stress changes in the flight cycle. To meet these requirements *Waspaloy* was thermomechanically processed (TMP) to give a fine-grain size and a 40% increase in tensile strength over the corresponding blade material, but at the expense of creep life. Similar improvements for discs have been produced in *Inco 901* by TMP. More highly-alloyed nickel-based discs suffer from excessive ingot segregation which makes grain size control difficult. Further development led to alloys produced by powder processing by gas atomization of a molten stream of metal in an inert argon atmosphere and consolidating the resultant powder by HIPing to near-net shape. Such products are limited in stress application because of inclusions in the powder and, hence, to realize the maximum advantage of this process it is necessary to produce 'superclean' material by electron beam or plasma melting.

Improvements in turbine materials were initially developed by increasing the volume fraction of γ´ in changing *Nimonic 80A* up to *Nimonic 115*. Unfortunately increasing the (Ti +Al) content lowers the melting point, thereby narrowing the forging range makes processing more difficult. Improved high-temperature oxidation and hot corrosion performance has led to the introduction of aluminide and overlay coatings and subsequently the development of *IN 738* and *IN 939* with much improved hot-corrosion resistance.

Further improvements in superalloys have depended on alternative manufacturing routes, particularly using modern casting technology like Vacuum casting (Smallman & Bishop, 1999).

In these alloys *γ'* (Ni3Al) and *γ״*) Ni3Nb) are the principal strengtheners by chemical and coherency strain hardening. The ordered *γ'-*Ni3Al phase is an equilibrium second phase in both the binary Ni–Al and Ni–Cr–Al systems and a metastable phase in the Ni–Ti and Ni– Cr–Ti systems, with close matching of the *γ'* and the FCC matrix. The two phases have very similar lattice parameters and the coherency confers a very low coarsening rate on the precipitate so that the alloy overages extremely slowly even at 0.7*T*m. In alloys containing Nb, a metastable Ni3Nb phase occurs but, although ordered and coherent, it is less stable than γ' at high temperatures (Smallman & Ngan, 2007).

In high-temperature service, the properties of the grain boundaries are as important as the strengthening by γ' within the grains. Grain boundary strengthening is produced mainly by precipitation of chromium and refractory metal carbides; small additions of Zr and B improve the morphology and stability of these carbides. Optimum properties are developed by multistage heat treatment; the intermediate stages produce the desired grain boundary microstructure of carbide particles enveloped in a film of *γ*' and the other stages produce two size ranges of *γ*' for the best combination of strength at both intermediate and high temperatures (Smallman & Ngan, 2007). Table 2 indicates the effect of the different alloying elements and Table 3 indicates the common ranges of main alloying additions and their effects on superalloy properties.

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 81

strengthening mechanisms, there are several groupings of iron-nickel superalloys

The most common precipitate is γ', typified by A-286, V-57 or Incoloy 901. Some alloys, typified by Inconel (IN)- 718, which precipitate γ״, were formerly classed as iron-nickel-base

The most common type of iron-nickel-base superalloys is INCONEL 718 which is a precipitation- hardening alloy used for high-temperature applications. In particular, the reputation of wrought Inconel 718 for being relatively easy to weld is generally attributed to the sluggish precipitation kinetics of the tetragonal γ״ strengthening phase. Inconel 718 is a relatively recent alloy as its industrial use started in 1965. It is a precipitation hardenable alloy, containing significant amounts of Fe, Nb and Mo. Minor contents of Al and Ti are also present. Inconel 718 combines good corrosion and high mechanical properties with and excellent weldability. It is employed in gas turbines, rocket engines, turbine blades, and in

Ni and Cr contribute to the corrosion resistance of this material. They crystallize as a γ phase (face centred cubic). Nb is added to form hardening precipitates γ״) a metastable inter metallic compound Ni3Nb, centred tetragonal crystal). Ti and Al are added to precipitate in the form of intermetallic γ' (Ni3(Ti,Al), simple cubic crystal). They have a lower hardening effect than particles. C is also added to precipitate in the form of MC carbides (M = Ti or Nb). In this case the C content must be low enough to allow Nb and Ti precipitation in the form of γ' and γ״ particles. Mo is also frequent in Inconel 718 in order to increase the mechanical resistance by solid solution hardening. Finally, a β phase (intermetallic Ni3Nb), (sometimes called δ phase) can also appear. It is an equilibrium particle with orthorhombic structure. All theses particles can precipitate along the grain boundaries of the γ matrix increasing the intergranular flow resistance of the present alloy. A typical precipitation time

temperature (PTT) diagram for this alloy is shown in Fig. 4 (Thomas et al, 2006).

Fig. 4. PTT diagram of different phases in Inconel 718 (Thomas et al, 2006).

(Campbell, 2008).

extrusion dies and containers.

superalloys but now are considered to be nickel-base.


Table 2. The effect of the different alloying elements (Smallman & Ngan, 2007).


Table 3. Common Ranges of Main Alloying Additions and Their Effects on Superalloys.

The wide range of applications for superalloys has expanded many other areas since they were developed and now includes aircraft and land-based gas turbines, rocket engines, chemical, and petroleum plants. The performance of an industrial gas turbine engines depends strongly on service conditions and the environment in which it operates.

#### **3.1 Iron-nickel-based superalloys**

Iron-nickel base superalloys evolved from austenitic stainless steels and are based on the principle of combining both solid-solution hardening and precipitate forming elements. As a class, the iron nickel superalloys have useful strengths to approximately 650°C (1200°C). The austenitic matrix based on nickel and iron, with at least 25 wt % Ni needed to stabilize the FCC phase. Other alloying elements, such as Chromium partition primarily to austenite to provide solid-solution hardening. Most alloys contain 25 to 45 wt % Nickel. Chromium in the range of 15 to 28 wt% is added for oxidation resistance at elevated temperature, while 1 to 6 wt% Mo provides solid solution strengthening. The main elements that facilitate precipitation hardening are titanium, aluminum and niobium.

The strengthening precipitates are primarily γ' (Ni3Al), η (Ni3Ti), and γ״) Ni3Nb). Elements that partition to grain boundaries, such as Boron and Zirconium, suppress grain boundary creep, resulting in significant increases in rupture life. Boron in quantities of 0.003 to 0.03 wt% and, less frequently, small additions of zirconium are added to improve stress-rupture properties and hot workability. Zirconium also forms the MC carbide ZrC. Another MC carbide (NbC) is found in alloys that contain niobium such as Inconel 706 and Inconel 718. Vanadium is also added in small quantities to iron-nickel superalloys to improve both notch ductility at service temperatures and hot workability. Based on their composition and

Table 2. The effect of the different alloying elements (Smallman & Ngan, 2007).

Table 3. Common Ranges of Main Alloying Additions and Their Effects on Superalloys.

depends strongly on service conditions and the environment in which it operates.

precipitation hardening are titanium, aluminum and niobium.

**3.1 Iron-nickel-based superalloys** 

The wide range of applications for superalloys has expanded many other areas since they were developed and now includes aircraft and land-based gas turbines, rocket engines, chemical, and petroleum plants. The performance of an industrial gas turbine engines

Iron-nickel base superalloys evolved from austenitic stainless steels and are based on the principle of combining both solid-solution hardening and precipitate forming elements. As a class, the iron nickel superalloys have useful strengths to approximately 650°C (1200°C). The austenitic matrix based on nickel and iron, with at least 25 wt % Ni needed to stabilize the FCC phase. Other alloying elements, such as Chromium partition primarily to austenite to provide solid-solution hardening. Most alloys contain 25 to 45 wt % Nickel. Chromium in the range of 15 to 28 wt% is added for oxidation resistance at elevated temperature, while 1 to 6 wt% Mo provides solid solution strengthening. The main elements that facilitate

The strengthening precipitates are primarily γ' (Ni3Al), η (Ni3Ti), and γ״) Ni3Nb). Elements that partition to grain boundaries, such as Boron and Zirconium, suppress grain boundary creep, resulting in significant increases in rupture life. Boron in quantities of 0.003 to 0.03 wt% and, less frequently, small additions of zirconium are added to improve stress-rupture properties and hot workability. Zirconium also forms the MC carbide ZrC. Another MC carbide (NbC) is found in alloys that contain niobium such as Inconel 706 and Inconel 718. Vanadium is also added in small quantities to iron-nickel superalloys to improve both notch ductility at service temperatures and hot workability. Based on their composition and strengthening mechanisms, there are several groupings of iron-nickel superalloys (Campbell, 2008).

The most common precipitate is γ', typified by A-286, V-57 or Incoloy 901. Some alloys, typified by Inconel (IN)- 718, which precipitate γ״, were formerly classed as iron-nickel-base superalloys but now are considered to be nickel-base.

The most common type of iron-nickel-base superalloys is INCONEL 718 which is a precipitation- hardening alloy used for high-temperature applications. In particular, the reputation of wrought Inconel 718 for being relatively easy to weld is generally attributed to the sluggish precipitation kinetics of the tetragonal γ״ strengthening phase. Inconel 718 is a relatively recent alloy as its industrial use started in 1965. It is a precipitation hardenable alloy, containing significant amounts of Fe, Nb and Mo. Minor contents of Al and Ti are also present. Inconel 718 combines good corrosion and high mechanical properties with and excellent weldability. It is employed in gas turbines, rocket engines, turbine blades, and in extrusion dies and containers.

Ni and Cr contribute to the corrosion resistance of this material. They crystallize as a γ phase (face centred cubic). Nb is added to form hardening precipitates γ״) a metastable inter metallic compound Ni3Nb, centred tetragonal crystal). Ti and Al are added to precipitate in the form of intermetallic γ' (Ni3(Ti,Al), simple cubic crystal). They have a lower hardening effect than particles. C is also added to precipitate in the form of MC carbides (M = Ti or Nb). In this case the C content must be low enough to allow Nb and Ti precipitation in the form of γ' and γ״ particles. Mo is also frequent in Inconel 718 in order to increase the mechanical resistance by solid solution hardening. Finally, a β phase (intermetallic Ni3Nb), (sometimes called δ phase) can also appear. It is an equilibrium particle with orthorhombic structure. All theses particles can precipitate along the grain boundaries of the γ matrix increasing the intergranular flow resistance of the present alloy. A typical precipitation time temperature (PTT) diagram for this alloy is shown in Fig. 4 (Thomas et al, 2006).

Fig. 4. PTT diagram of different phases in Inconel 718 (Thomas et al, 2006).

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 83

Table 4. The chemical compositions of several superalloys (wt.%) (Furrer & Fecht, 1999).

Table 5. The chemical composition of some Cobalt-based superalloys (Jovanović et al).

The cobalt-based superalloys (Table 5) are not as strong as nickel-based superalloys, but they retain their strength up to higher temperatures. They derive their strength largely from a distribution of refractory metal carbides (combinations of carbon and metals such as Mo and W), which tend to collect at grain boundaries (Fig. 6). This network of carbides strengthens grain boundaries and alloy becomes stable nearly up to the melting point. In addition to refractory metals and metal carbides, cobalt superalloys generally contain high levels of Cr to make them more resistant to corrosion that normally takes place in the presence of hot exhaust gases. The Cr atoms react with oxygen atoms to form a protective layer of Cr2O3 which protects the alloy from corrosive gases. Being not as hard as nickelbased superalloys, cobalt superalloys are not so sensitive to cracking under thermal shocks as other superalloys. Co-based superalloys are therefore more suitable for parts that need to be worked or welded, such as those in the intricate structures of the combustion chamber

**3.3 Cobalt-based Superalloys** 

(Jovanović et al).

## **3.2 Nickel-based Superalloys**

Nickel-based superalloys are an unusual class of metallic materials with an exceptional combination of high temperature strength, toughness, and resistance to degradation in corrosive or oxidizing environments. The nickel-based alloys show a wider range of application than any other class of alloys.

The austenitic stainless steels were developed and utilized early in the 1900s, whereas the development of the nickel-based alloys did not begin until about 1930. In aerospace applications nickel-based superalloys are used widely as components of jet engine turbines. Therefore important position of super-alloys in this area is manifested by the fact that they represent at present more than 50 % of mass of advanced aircraft engines. Extensive use of super-alloys in turbines, supported by the fact that thermo-dynamic efficiency of turbines increases with increasing temperatures at the turbine inlet, became partial reason of the effort aimed at increasing of the maximum service temperature of high-alloyed alloys (Jonsta et al, 2007). Therefore in gas turbine applications alloys with good stability and very low crack-growth rates that are readily inspectable by nondestructive means are desired. Fuel efficiency and emissions are also key commercial and environmental drivers impacting turbine-engine materials. To meet these demands, modern nickel-based alloys offer an efficient compromise between performance and economics. The chemistries of several common and advanced nickel-based superalloys are listed in Table 4 and the parts of gas turbine engine in which Nickel-based superalloys (marked red) commonly used are shown in Fig. 5.

Fig. 5. Commonly used materials in gas turbine engine components.

In the environmental series nickel is nobler than iron but more active than copper. Reducing environments, such as dilute sulfuric acid, find nickel more corrosion resistant than iron but not as resistant as copper or nickel-copper alloys. The nickel molybdenum alloys are more corrosion resistant to reducing environment than nickel or nickel- copper alloys (Philip & Schweitzer, 2003). Nickel-based superalloys are extremely prone to weld cracking.

High-temperature strength of Ni-base superalloys depends mainly, on the volume fraction and morphology of γ´ precipitates. Several basic factors contribute to the magnitude of hardening of the alloy (Sajjadi & Zebarjad, 2006).

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 83


Table 4. The chemical compositions of several superalloys (wt.%) (Furrer & Fecht, 1999).

#### **3.3 Cobalt-based Superalloys**

82 Recent Advances in Aircraft Technology

Nickel-based superalloys are an unusual class of metallic materials with an exceptional combination of high temperature strength, toughness, and resistance to degradation in corrosive or oxidizing environments. The nickel-based alloys show a wider range of

The austenitic stainless steels were developed and utilized early in the 1900s, whereas the development of the nickel-based alloys did not begin until about 1930. In aerospace applications nickel-based superalloys are used widely as components of jet engine turbines. Therefore important position of super-alloys in this area is manifested by the fact that they represent at present more than 50 % of mass of advanced aircraft engines. Extensive use of super-alloys in turbines, supported by the fact that thermo-dynamic efficiency of turbines increases with increasing temperatures at the turbine inlet, became partial reason of the effort aimed at increasing of the maximum service temperature of high-alloyed alloys (Jonsta et al, 2007). Therefore in gas turbine applications alloys with good stability and very low crack-growth rates that are readily inspectable by nondestructive means are desired. Fuel efficiency and emissions are also key commercial and environmental drivers impacting turbine-engine materials. To meet these demands, modern nickel-based alloys offer an efficient compromise between performance and economics. The chemistries of several common and advanced nickel-based superalloys are listed in Table 4 and the parts of gas turbine engine in which Nickel-based superalloys (marked red) commonly used are shown

Fig. 5. Commonly used materials in gas turbine engine components.

hardening of the alloy (Sajjadi & Zebarjad, 2006).

In the environmental series nickel is nobler than iron but more active than copper. Reducing environments, such as dilute sulfuric acid, find nickel more corrosion resistant than iron but not as resistant as copper or nickel-copper alloys. The nickel molybdenum alloys are more corrosion resistant to reducing environment than nickel or nickel- copper alloys (Philip &

High-temperature strength of Ni-base superalloys depends mainly, on the volume fraction and morphology of γ´ precipitates. Several basic factors contribute to the magnitude of

Schweitzer, 2003). Nickel-based superalloys are extremely prone to weld cracking.

**3.2 Nickel-based Superalloys** 

in Fig. 5.

application than any other class of alloys.

The cobalt-based superalloys (Table 5) are not as strong as nickel-based superalloys, but they retain their strength up to higher temperatures. They derive their strength largely from a distribution of refractory metal carbides (combinations of carbon and metals such as Mo and W), which tend to collect at grain boundaries (Fig. 6). This network of carbides strengthens grain boundaries and alloy becomes stable nearly up to the melting point. In addition to refractory metals and metal carbides, cobalt superalloys generally contain high levels of Cr to make them more resistant to corrosion that normally takes place in the presence of hot exhaust gases. The Cr atoms react with oxygen atoms to form a protective layer of Cr2O3 which protects the alloy from corrosive gases. Being not as hard as nickelbased superalloys, cobalt superalloys are not so sensitive to cracking under thermal shocks as other superalloys. Co-based superalloys are therefore more suitable for parts that need to be worked or welded, such as those in the intricate structures of the combustion chamber (Jovanović et al).


Table 5. The chemical composition of some Cobalt-based superalloys (Jovanović et al).

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 85

The use of 718Plus alloy in elevated temperature applications is of interest for military systems. In particular, the manufacturing difficulties associated with alloys such as Waspaloy provide a need for a material with similar component capabilities, but with better producibility. Initial characterization shows that the alloy exhibits many similarities to Alloy 718, including good workability, weldability and intermediate temperature strength

Fig. 7. Developments leading up to alloy 718 and subsequent efforts to improve capability

Since the advent of the first superalloys over 60 years ago, alloy developers have worked to promote strength and high temperature stability while balancing processability. Processing constraints for many alloy systems preclude their general use for cast and wrought forging applications. Instead these compositions are used in the cast form, or are producible only using powder metallurgy. The development and introduction of alloy 718 in the late 1950's offered a significant breakthrough in malleability and weldability relative to other high strength alloys available at that time including Waspaloy and René 41 which are primarily gamma prime strengthened. Since the introduction of alloy 718 a significant number of alloys have been examined, including cast as well as wrought alloys, with the primary intent to maintain or improve properties and provide increased thermal stability while maintaining favorable processability. Some of the alloys developed subsequently are shown

Minimal cost increase; intermediate to 718 and Waspaloy alloys

Good workability; better than Waspaloy alloy

capability (Bergstrom & Bayhan, 2005).

over 718 (Otti et al, 2005).

Fig. 6. Optical micrograph of Haynes-25. G mainly M6C carbides (Jovanović et al).

## **4. ALLVAC 718 plus™**

Inconel 718 is a nickel base superalloy that is used extensively in aerospace applications because of its unique high temperature mechanical properties. Since it was invented by Eiselstein, it has been used as a material of construction for aero-engine and land based turbine components. The reasons for alloy 718's popularity include excellent strength, good hot and cold workability, the best weldability of any of the superalloys and last, but not least, moderate cost. However, the application of the alloy has been limited to a temperature below 650 ◦C, as its properties deteriorate rapidly on exposure above this temperature due to the instability of the main strengthening phase of the alloy, γ״) Idowu & Ojo, 2007). With prolonged exposure at this temperature or higher, γ" rapidly overages and transforms to the equilibrium δ phase with an accompanying loss of strength and especially creep life (Kennedy, 2005).

Other wrought, commercial superalloys exist which have significantly greater temperature capability such as Waspaloy and René 41. These alloys are typically γ' hardened and are significantly more difficult to fabricate and weld. Because of this and because of their intrinsic raw material content, these alloys are significantly more expensive than alloy 718. There have been numerous attempts to develop an affordable, workable 718-type alloy with increased temperature capability. After a number of years of systematic work, including both computer modeling and experimental melting trials, ATI Allvac has developed a new alloy, Allvac® 718Plus™, which offers a full 55°C temperature advantage over alloy 718. The alloy maintains many of the desirable features of alloy 718, including good workability, weldability and moderate cost (Kennedy, 2005).

ATI Allvac has extensively investigated the 718Plus alloy billet properties, both as an internal program and as part of the Metals Affordability Initiative program entitled "Low-Cost, High Temperature Structural Material" for turbine engine ring-rolling applications. The objective of all these programs is to develop an alloy with the following characteristics:


Fig. 6. Optical micrograph of Haynes-25. G mainly M6C carbides (Jovanović et al).

Inconel 718 is a nickel base superalloy that is used extensively in aerospace applications because of its unique high temperature mechanical properties. Since it was invented by Eiselstein, it has been used as a material of construction for aero-engine and land based turbine components. The reasons for alloy 718's popularity include excellent strength, good hot and cold workability, the best weldability of any of the superalloys and last, but not least, moderate cost. However, the application of the alloy has been limited to a temperature below 650 ◦C, as its properties deteriorate rapidly on exposure above this temperature due to the instability of the main strengthening phase of the alloy, γ״) Idowu & Ojo, 2007). With prolonged exposure at this temperature or higher, γ" rapidly overages and transforms to the equilibrium δ phase with an accompanying loss of strength and especially creep life

Other wrought, commercial superalloys exist which have significantly greater temperature capability such as Waspaloy and René 41. These alloys are typically γ' hardened and are significantly more difficult to fabricate and weld. Because of this and because of their intrinsic raw material content, these alloys are significantly more expensive than alloy 718. There have been numerous attempts to develop an affordable, workable 718-type alloy with increased temperature capability. After a number of years of systematic work, including both computer modeling and experimental melting trials, ATI Allvac has developed a new alloy, Allvac® 718Plus™, which offers a full 55°C temperature advantage over alloy 718. The alloy maintains many of the desirable features of alloy 718, including good workability,

ATI Allvac has extensively investigated the 718Plus alloy billet properties, both as an internal program and as part of the Metals Affordability Initiative program entitled "Low-Cost, High Temperature Structural Material" for turbine engine ring-rolling applications. The objective of all these programs is to develop an alloy with the following characteristics: 55°C temperature advantage based on the Larson-Miller, time-temperature parameter

**4. ALLVAC 718 plus™** 

(Kennedy, 2005).

weldability and moderate cost (Kennedy, 2005).

Improved thermal stability; equal to Waspaloy at 704°C

Good weldability; at least intermediate to 718 and Waspaloy alloys

The use of 718Plus alloy in elevated temperature applications is of interest for military systems. In particular, the manufacturing difficulties associated with alloys such as Waspaloy provide a need for a material with similar component capabilities, but with better producibility. Initial characterization shows that the alloy exhibits many similarities to Alloy 718, including good workability, weldability and intermediate temperature strength capability (Bergstrom & Bayhan, 2005).

Fig. 7. Developments leading up to alloy 718 and subsequent efforts to improve capability over 718 (Otti et al, 2005).

Since the advent of the first superalloys over 60 years ago, alloy developers have worked to promote strength and high temperature stability while balancing processability. Processing constraints for many alloy systems preclude their general use for cast and wrought forging applications. Instead these compositions are used in the cast form, or are producible only using powder metallurgy. The development and introduction of alloy 718 in the late 1950's offered a significant breakthrough in malleability and weldability relative to other high strength alloys available at that time including Waspaloy and René 41 which are primarily gamma prime strengthened. Since the introduction of alloy 718 a significant number of alloys have been examined, including cast as well as wrought alloys, with the primary intent to maintain or improve properties and provide increased thermal stability while maintaining favorable processability. Some of the alloys developed subsequently are shown

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 87

As mentioned before, the primary strengthening phase is γ′ with a volume fraction ranging from 19.7-23.2 %, depending on the quantity of δ phase. Gamma prime strengthened alloys like Waspaloy and René 41 have much greater stability at higher temperature than γ′′ strengthened alloys like 718 since γ′′ grows rapidly and partially decomposes to equilibrium δ phase at temperatures in the 650–760°C range. Studies of the γ′ phase in 718Plus alloy show it to be high in Nb and Al, which is very different from the γ′ present in Waspaloy and René 41 and may account for its unique precipitation behavior and strengthening effects.

Like most superalloys there is a strong relationship between processing, structure and properties for alloy 718Plus. Optimum mechanical properties are achieved with a microstructure which has a small amount of rod shaped δ particles on the grain boundaries like that shown in Fig. 8 (a). Excessively high forging temperatures or high solution heat treating temperatures will result in structures with little or no δ phase precipitates that are prone to notch stress rupture failure. It is reported that no notch problems have been experienced using the 954°C solution temperature, probably because some δ phase can be precipitated at this temperature. However, excessively long heating times and possibly large amounts of stored, strain energy can result in large amounts of δ phase appearing on grain boundaries, twin lines and intragranularly, Fig. 8 (b). Such structures can lead to lower than

**4.2 Strengthening mechanisms** 

expected tensile and rupture strength (Kennedy, 2005).

(a) (b)

Excessive δ Phase (Kennedy, 2005).

Kennedy, 2006).

Fig. 8. SEM Micrographs of Alloy 718Plus™ with (a) Preferred δ Phase Morphology and (b)

Alloy 718Plus does contain δ phase which is beneficial for conferring stress rupture notch ductility and controlling microstructure during thermo-mechanical processing. However, the volume fraction of the delta phase is considerably less than is found in alloy 718 and tends to be more stable with a much slower growth rate at elevated temperatures. Some γ′′ may also be present in 718Plus alloy but in a much lower quantity, less than 7% (Jeniski &

along with 718, Waspaloy, and René 41 in the development timeline of Fig. 7. A key requirement beyond strength, toughness, fatigue, creep, crack growth resistance, and processability which has also driven composition development is weldability (Otti et al, 2005).

### **4.1 Chemistry**

There are lots of wrought alloys in use for gas turbine engine parts, such as Waspalloy, which have high temperature capability. But they are typically much more difficult to manufacture and fabricate into finished parts and also significantly more expensive than alloy 718 (Bond & Kennedy, 2005). Therefore when ALLVAC 718 plus is compared to Inconel 718 this newly modified super alloy has the higher content of Al+ Ti, the higher ratio of Al/ Ti and the addition of W and Co instead of Fe. As a result it provides increased temperature capacity up to 55°C and impressive thermal stability. Therefore it closes the gap between Inconel 718 and Waspalloy, as combining the good processability and weldability of Inconel 718 with the temperature capability of Waspalloy. (Schreiber et al, 2006). The chemical compositions of the ALLVAC 718 plus with Inconel 718 and Waspalloy are given in Table 6.


Table 6. Nominal chemistry comparison of the ALLVAC 718 plus, Inconel 718 and Waspalloy (Cao, 2005).

Alloy 718Plus has a much larger content of γ' and γ " than alloy 718 and a smaller amount of δ phase. Solvus temperatures for γ' and γ" are also higher in alloy 718Plus. All of these points likely contribute to improved high temperature properties. One of the major differences between alloy 718 and Waspaloy is the speed of the precipitation reaction. The γ" precipitation in alloy 718 is very sluggish and accounts in part for the good weldability and processing characteristics of the alloy.

In 718- type alloys primarily Fe, Co, Mo and W are the matrix elements. The effects of alloying elements on microstructure, mechanical properties, thermal stability and processing characteristics of alloy are important factors. Niobium is one of the major hardening elements and the other two is Al and Ti. The change in Al/ Ti ratio and the increase in Al+ Ti content converts the alloy into a predominantly γ' strengthening alloy and it gives the alloy an improved thermal stability. Furthermore the modification on the content of Al and Ti develop the optimum mechanical properties of the alloy. Another factor on the improvement of the mechanical properties and thermal stability is the addition of Co up to about 9 wt%. Still further improvement occurs with Fe content of 10 wt%, 2.8 wt% Mo and 1 wt% W (Cao & Kennedy, 2004). Very small additions of P and B further increases stress rupture and creep resistance.

#### **4.2 Strengthening mechanisms**

86 Recent Advances in Aircraft Technology

along with 718, Waspaloy, and René 41 in the development timeline of Fig. 7. A key requirement beyond strength, toughness, fatigue, creep, crack growth resistance, and processability which has also driven composition development is weldability (Otti et al,

There are lots of wrought alloys in use for gas turbine engine parts, such as Waspalloy, which have high temperature capability. But they are typically much more difficult to manufacture and fabricate into finished parts and also significantly more expensive than alloy 718 (Bond & Kennedy, 2005). Therefore when ALLVAC 718 plus is compared to Inconel 718 this newly modified super alloy has the higher content of Al+ Ti, the higher ratio of Al/ Ti and the addition of W and Co instead of Fe. As a result it provides increased temperature capacity up to 55°C and impressive thermal stability. Therefore it closes the gap between Inconel 718 and Waspalloy, as combining the good processability and weldability of Inconel 718 with the temperature capability of Waspalloy. (Schreiber et al, 2006). The chemical compositions of the ALLVAC 718 plus with Inconel 718 and Waspalloy are given

Table 6. Nominal chemistry comparison of the ALLVAC 718 plus, Inconel 718 and

Alloy 718Plus has a much larger content of γ' and γ " than alloy 718 and a smaller amount of δ phase. Solvus temperatures for γ' and γ" are also higher in alloy 718Plus. All of these points likely contribute to improved high temperature properties. One of the major differences between alloy 718 and Waspaloy is the speed of the precipitation reaction. The γ" precipitation in alloy 718 is very sluggish and accounts in part for the good weldability

In 718- type alloys primarily Fe, Co, Mo and W are the matrix elements. The effects of alloying elements on microstructure, mechanical properties, thermal stability and processing characteristics of alloy are important factors. Niobium is one of the major hardening elements and the other two is Al and Ti. The change in Al/ Ti ratio and the increase in Al+ Ti content converts the alloy into a predominantly γ' strengthening alloy and it gives the alloy an improved thermal stability. Furthermore the modification on the content of Al and Ti develop the optimum mechanical properties of the alloy. Another factor on the improvement of the mechanical properties and thermal stability is the addition of Co up to about 9 wt%. Still further improvement occurs with Fe content of 10 wt%, 2.8 wt% Mo and 1 wt% W (Cao & Kennedy, 2004). Very small additions of P and B further increases stress

2005).

**4.1 Chemistry** 

in Table 6.

Waspalloy (Cao, 2005).

and processing characteristics of the alloy.

rupture and creep resistance.

As mentioned before, the primary strengthening phase is γ′ with a volume fraction ranging from 19.7-23.2 %, depending on the quantity of δ phase. Gamma prime strengthened alloys like Waspaloy and René 41 have much greater stability at higher temperature than γ′′ strengthened alloys like 718 since γ′′ grows rapidly and partially decomposes to equilibrium δ phase at temperatures in the 650–760°C range. Studies of the γ′ phase in 718Plus alloy show it to be high in Nb and Al, which is very different from the γ′ present in Waspaloy and René 41 and may account for its unique precipitation behavior and strengthening effects.

Like most superalloys there is a strong relationship between processing, structure and properties for alloy 718Plus. Optimum mechanical properties are achieved with a microstructure which has a small amount of rod shaped δ particles on the grain boundaries like that shown in Fig. 8 (a). Excessively high forging temperatures or high solution heat treating temperatures will result in structures with little or no δ phase precipitates that are prone to notch stress rupture failure. It is reported that no notch problems have been experienced using the 954°C solution temperature, probably because some δ phase can be precipitated at this temperature. However, excessively long heating times and possibly large amounts of stored, strain energy can result in large amounts of δ phase appearing on grain boundaries, twin lines and intragranularly, Fig. 8 (b). Such structures can lead to lower than expected tensile and rupture strength (Kennedy, 2005).

Fig. 8. SEM Micrographs of Alloy 718Plus™ with (a) Preferred δ Phase Morphology and (b) Excessive δ Phase (Kennedy, 2005).

Alloy 718Plus does contain δ phase which is beneficial for conferring stress rupture notch ductility and controlling microstructure during thermo-mechanical processing. However, the volume fraction of the delta phase is considerably less than is found in alloy 718 and tends to be more stable with a much slower growth rate at elevated temperatures. Some γ′′ may also be present in 718Plus alloy but in a much lower quantity, less than 7% (Jeniski & Kennedy, 2006).

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 89

Fig. 11. The coarsening of strengthening phase γ´ in alloy 718 plus after 760°C Aging (Xie et

The microstructure of 718 plus in as received hot-rolled condition consists of FCC austenitic matrix with an average grain size of 50 μm. Fig. 12 shows the optical micrograph of the alloy. It can be seen that precipitates with round-to-blocky morphology are randomly

Fig. 12. Optical microstructure of as received 718 plus alloy (Vishwakarma et al, 2007).

SEM/EDS analysis of the precipitates shows them to be mainly Nb-rich MC type carbides containing Ti and C. As laves phase can be eliminated by high temperature homogenization and thermo–mechanical processing of wrought Inconel 718 and 718 Plus type of alloys, it is therefore not observed in the microstructure of the as received material. The delta phase, which is commonly observed in Inconel 718, is not observed in the as received

Heat treated at 950ºC for 1 hour microstructure of 718 plus alloy, grain size of 54 μm, which has normal B and P concentrations, has shown in Fig. 13. It can be seen that needle like δ

dispersed within the microstructure (Vishwakarma et al, 2007).

microstructure of 718 plus alloy (Vishwakarma et al, 2007).

al, 2005).

**4.3 Microstructure** 

When Inconel 718 is compared with the ALLVAC 718 plus it is reported that the size of strengthening phases increases in both alloys after long time thermal exposure (Fig. 9), but more significantly in alloy 718. In alloy 718, the average size of γ″+γ′ grows from about 15 nm at as heat-treated condition to almost 100 nm after 500 hrs long time aging at 760°C as indicated in Fig. 10 and the main strengthening phase γ″ grows to about 200nm in estimation (see Fig. 11). However, in alloy 718Plus, the main strengthening phase γ′ coarsens slowly and the average size of γ′ is still about 70 nm as indicated in Figure 9. These important quantitative phase analyses results convince us that alloy 718Plus has a superior stable microstructure in comparison with alloy 718 (Xie et al, 2005).

Fig. 9. The size of strengthening phases in Alloy 718 and Alloy 718 plus (Xie et al, 2005).

Fig. 10. The coarsening of strengthening phases γ´´ and γ´ in alloy 718 after 760°C (Xie et al, 2005).

Fig. 11. The coarsening of strengthening phase γ´ in alloy 718 plus after 760°C Aging (Xie et al, 2005).

## **4.3 Microstructure**

88 Recent Advances in Aircraft Technology

When Inconel 718 is compared with the ALLVAC 718 plus it is reported that the size of strengthening phases increases in both alloys after long time thermal exposure (Fig. 9), but more significantly in alloy 718. In alloy 718, the average size of γ″+γ′ grows from about 15 nm at as heat-treated condition to almost 100 nm after 500 hrs long time aging at 760°C as indicated in Fig. 10 and the main strengthening phase γ″ grows to about 200nm in estimation (see Fig. 11). However, in alloy 718Plus, the main strengthening phase γ′ coarsens slowly and the average size of γ′ is still about 70 nm as indicated in Figure 9. These important quantitative phase analyses results convince us that alloy 718Plus has a superior

Fig. 9. The size of strengthening phases in Alloy 718 and Alloy 718 plus (Xie et al, 2005).

Fig. 10. The coarsening of strengthening phases γ´´ and γ´ in alloy 718 after 760°C (Xie et al,

2005).

stable microstructure in comparison with alloy 718 (Xie et al, 2005).

The microstructure of 718 plus in as received hot-rolled condition consists of FCC austenitic matrix with an average grain size of 50 μm. Fig. 12 shows the optical micrograph of the alloy. It can be seen that precipitates with round-to-blocky morphology are randomly dispersed within the microstructure (Vishwakarma et al, 2007).

Fig. 12. Optical microstructure of as received 718 plus alloy (Vishwakarma et al, 2007).

SEM/EDS analysis of the precipitates shows them to be mainly Nb-rich MC type carbides containing Ti and C. As laves phase can be eliminated by high temperature homogenization and thermo–mechanical processing of wrought Inconel 718 and 718 Plus type of alloys, it is therefore not observed in the microstructure of the as received material. The delta phase, which is commonly observed in Inconel 718, is not observed in the as received microstructure of 718 plus alloy (Vishwakarma et al, 2007).

Heat treated at 950ºC for 1 hour microstructure of 718 plus alloy, grain size of 54 μm, which has normal B and P concentrations, has shown in Fig. 13. It can be seen that needle like δ

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 91

Fig. 14. Effect of test temperature on room temperature tensile ultimate tensile strength for

Fig. 15. The tensile strength of Alloy 718 plus and Waspaloy (Otti et al, 2005).

Direct aging can be effectively applied to alloy 718Plus to improve its mechanical properties, including strength and stress rupture life of alloy 718Plus. Considering the fine grain size and high strength resulting from direct aging, the low cycle fatigue resistance of this alloy should also be significantly improved although further experimental verification is necessary. DA processing of this alloy is also different from Waspaloy in that hot working at

several alloys (Bond & Kennedy, 2005).

phase is observed on the grain boundaries and occasionally intra-granularly on the twin boundaries. And also seen in the microstructure, round and blocky shaped MC type carbide particles are randomly distributed. Ti-rich carbo-nitride particles can be also observed (Vishwakarma & Chaturvedi, 2008). Intermetallic phases like FCC γ' and BCT γ" are expected to form in 718 plus alloys but γ' is the main strengthening phase in these superalloys (Cao & Kennedy, 2004).

#### **4.4 Mechanical properties**

Alloy 718Plus™ has a significant strength advantage over alloy 718 at temperatures above 650°C and over the entire temperature range compared to Waspaloy and A286. Elongation for alloy 718Plus™ over the entire temperature range remained high at 18% minimum. These data are consistent with comparisons of alloy 718, Waspaloy, A286 and alloy 718Plus™ in other product forms, including billet, rolled rings, forgings and sheet (Bond & Kennedy, 2005). In Fig. 14 shows the effect of temperature on room temperature ultimate tensile strength for several alloys. The tensile strength of ALLVAC 718 plus and Waspaloy is shown in Fig. 15.

It is reported by ATI ALLVAC Cooperation that extensive studies demonstrated that this alloy has shown superior tensile and stress rupture properties to alloy 718 and comparable properties to Waspaloy at the temperature up to 704°C. However, relatively speaking, the data on fatigue crack propagation (FCP) resistance of this alloy are still insufficient. Alloys 718Plus, 718 and Waspaloy have similar fatigue crack growth rates under 3 seconds triangle loading at 650°C with 718Plus being slightly better. Waspalloy shows the best resistance to fatigue crack growth under hold time fatigue condition while the resistance of 718Plus is better than that of Alloy 718 (Liu et al, 2005).

Examination of the fatigue fracture surfaces by scanning electronic microscope (SEM) revealed transgranular crack propagation with striations for 718Plus at room temperature. The fracture mode at 650ºC is the mixture of intergranular and transgranular modes (Liu et al, 2004).

phase is observed on the grain boundaries and occasionally intra-granularly on the twin boundaries. And also seen in the microstructure, round and blocky shaped MC type carbide particles are randomly distributed. Ti-rich carbo-nitride particles can be also observed (Vishwakarma & Chaturvedi, 2008). Intermetallic phases like FCC γ' and BCT γ" are expected to form in 718 plus alloys but γ' is the main strengthening phase in these

Fig. 13. Microstructure of 718 plus alloy heat treated at 950ºC for 1 h (Vishwakarma &

Alloy 718Plus™ has a significant strength advantage over alloy 718 at temperatures above 650°C and over the entire temperature range compared to Waspaloy and A286. Elongation for alloy 718Plus™ over the entire temperature range remained high at 18% minimum. These data are consistent with comparisons of alloy 718, Waspaloy, A286 and alloy 718Plus™ in other product forms, including billet, rolled rings, forgings and sheet (Bond & Kennedy, 2005). In Fig. 14 shows the effect of temperature on room temperature ultimate tensile strength for several alloys. The tensile strength of ALLVAC 718 plus and Waspaloy is

It is reported by ATI ALLVAC Cooperation that extensive studies demonstrated that this alloy has shown superior tensile and stress rupture properties to alloy 718 and comparable properties to Waspaloy at the temperature up to 704°C. However, relatively speaking, the data on fatigue crack propagation (FCP) resistance of this alloy are still insufficient. Alloys 718Plus, 718 and Waspaloy have similar fatigue crack growth rates under 3 seconds triangle loading at 650°C with 718Plus being slightly better. Waspalloy shows the best resistance to fatigue crack growth under hold time fatigue condition while the resistance of 718Plus is

Examination of the fatigue fracture surfaces by scanning electronic microscope (SEM) revealed transgranular crack propagation with striations for 718Plus at room temperature. The fracture mode at 650ºC is the mixture of intergranular and transgranular modes (Liu et

superalloys (Cao & Kennedy, 2004).

Chaturvedi, 2008).

shown in Fig. 15.

al, 2004).

**4.4 Mechanical properties** 

better than that of Alloy 718 (Liu et al, 2005).

Fig. 14. Effect of test temperature on room temperature tensile ultimate tensile strength for several alloys (Bond & Kennedy, 2005).

Fig. 15. The tensile strength of Alloy 718 plus and Waspaloy (Otti et al, 2005).

Direct aging can be effectively applied to alloy 718Plus to improve its mechanical properties, including strength and stress rupture life of alloy 718Plus. Considering the fine grain size and high strength resulting from direct aging, the low cycle fatigue resistance of this alloy should also be significantly improved although further experimental verification is necessary. DA processing of this alloy is also different from Waspaloy in that hot working at

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 93

It is reported that limited weldability testing has been conducted on 718Plus alloy but results have been encouraging. Weldability of alloy 718Plus is believed to be quite good, at least intermediate to alloys 718 and Waspaloy (Kennedy, 2005). Improved weldability over Waspaloy is one of the primary drivers for 718Plus alloy in engine applications. Figure 16 shows the weld cracking tendency for a number of well known commercial alloys and illustrates the good welding characteristics expected with 718Plus alloy based on its chemistry (Jeniski & Kennedy, 2006). Some micrographs of the Electron Beam Welded

ALLVAC 718 plus, Inconel 718 and Waspaloy rings are shown in Fig. 17.

Fig. 17. EB welding of 890 mm diameter rolled rings, (a) typical weld location for 718, Waspaloy, and 718Plus welds and typical welds for (b) Waspaloy, and (c) 718Plus (Otti et al,

rings, closed die forgings, bar, rod, wire, sheet, plate and castings.

limited due to elevated operating temperatures.

alloy 718 is used (Jeniski & Kennedy, 2006).

The first commitment to a production use of alloy 718Plus has been made for a high temperature tooling application, replacing Waspaloy as a hot shear knife. Other applications include aero and land-base turbine disks, forged compressor blades, fasteners, engine shafts and fabricated sheet/plate components. Product forms include rolled or flash butt welded

The alloy also can be used for flash-butt welded ring applications. Sheet form of the alloy is being considered for fabricated engine parts such as turbine exhaust cases and engine seals. Fasteners remain another potential application for 718Plus alloy. The property advantages for 718Plus alloy have also led to its being considered for rotating parts. Cao and Kennedy have shown that 718Plus alloy is capable of direct aging (DA), low temperature working followed by aging with no prior solution heat treatment. DA processing resulted in the production of very fine grain material with yield strength improvement at 704°C of 70-100 MPa. The alloy is also being considered for blading applications in areas where alloy 718 is

The alloy has also other applications outside of the jet and power turbine engines. Any application that currently uses alloys 718, Waspaloy, René 41 or other nickel-based superalloys can consider 718Plus alloy as a substitute for reasons of cost savings or increased temperature capability. Other markets where 718Plus alloy has potential are automotive turbo-chargers or industrial markets like chemical process or oil and gas where

2005).

**4.6 Cost and Applications** 

temperatures above the γ' solvus can achieve a good, direct age response (Cao & Kennedy, 2005).

#### **4.5 Weldability**

There are numerous types of superalloys with a difference in weldability among the types. The solid solution alloys are the easiest to weld because they don't undergo drastic metallurgical changes when heated and cooled. Because of their limited strength, however, they are only used in certain areas of a gas turbine, such as the combustor.

The precipitation-strengthened alloys are more demanding during welding and post welding because of the precipitation of the hardening phase that usually contains aluminum, titanium, or niobium. These elements oxidize very easily and, therefore, alloys that contain them need better gas protection during welding. A third type of superalloy is the mechanically alloyed materials that cannot be welded without suffering a drastic drop in strength. These alloys are usually joined by mechanical means or diffusion bonding. In addition to those elements that enable a superalloy to undergo precipitation hardening, such as aluminum, titanium, and niobium, other elements are added to enhance mechanical properties or corrosion resistance. These include boron and zirconium, which are often intentionally added to some alloys to improve high temperature performance but at a cost to weldability. There are numerous other elements that are not intentionally added but can be present in very small quantities that are harmful, such as lead and zinc. These are practically insoluble in superalloys and can cause hot cracking during solidification of the welds. Small quantities of these elements on the surface of a metal can cause localized weld cracking. Sulfur is considered detrimental if present in too large a quantity, but can cause low weld penetration problems if present in very low amounts (Donald & Tillack, 2007).

Fig. 16. Effect of chemistry on post-weld heat cracking (Jeniski & Kennedy, 2006).

temperatures above the γ' solvus can achieve a good, direct age response (Cao & Kennedy,

There are numerous types of superalloys with a difference in weldability among the types. The solid solution alloys are the easiest to weld because they don't undergo drastic metallurgical changes when heated and cooled. Because of their limited strength, however,

The precipitation-strengthened alloys are more demanding during welding and post welding because of the precipitation of the hardening phase that usually contains aluminum, titanium, or niobium. These elements oxidize very easily and, therefore, alloys that contain them need better gas protection during welding. A third type of superalloy is the mechanically alloyed materials that cannot be welded without suffering a drastic drop in strength. These alloys are usually joined by mechanical means or diffusion bonding. In addition to those elements that enable a superalloy to undergo precipitation hardening, such as aluminum, titanium, and niobium, other elements are added to enhance mechanical properties or corrosion resistance. These include boron and zirconium, which are often intentionally added to some alloys to improve high temperature performance but at a cost to weldability. There are numerous other elements that are not intentionally added but can be present in very small quantities that are harmful, such as lead and zinc. These are practically insoluble in superalloys and can cause hot cracking during solidification of the welds. Small quantities of these elements on the surface of a metal can cause localized weld cracking. Sulfur is considered detrimental if present in too large a quantity, but can cause low weld

they are only used in certain areas of a gas turbine, such as the combustor.

penetration problems if present in very low amounts (Donald & Tillack, 2007).

Fig. 16. Effect of chemistry on post-weld heat cracking (Jeniski & Kennedy, 2006).

2005).

**4.5 Weldability** 

It is reported that limited weldability testing has been conducted on 718Plus alloy but results have been encouraging. Weldability of alloy 718Plus is believed to be quite good, at least intermediate to alloys 718 and Waspaloy (Kennedy, 2005). Improved weldability over Waspaloy is one of the primary drivers for 718Plus alloy in engine applications. Figure 16 shows the weld cracking tendency for a number of well known commercial alloys and illustrates the good welding characteristics expected with 718Plus alloy based on its chemistry (Jeniski & Kennedy, 2006). Some micrographs of the Electron Beam Welded ALLVAC 718 plus, Inconel 718 and Waspaloy rings are shown in Fig. 17.

Fig. 17. EB welding of 890 mm diameter rolled rings, (a) typical weld location for 718, Waspaloy, and 718Plus welds and typical welds for (b) Waspaloy, and (c) 718Plus (Otti et al, 2005).

### **4.6 Cost and Applications**

The first commitment to a production use of alloy 718Plus has been made for a high temperature tooling application, replacing Waspaloy as a hot shear knife. Other applications include aero and land-base turbine disks, forged compressor blades, fasteners, engine shafts and fabricated sheet/plate components. Product forms include rolled or flash butt welded rings, closed die forgings, bar, rod, wire, sheet, plate and castings.

The alloy also can be used for flash-butt welded ring applications. Sheet form of the alloy is being considered for fabricated engine parts such as turbine exhaust cases and engine seals. Fasteners remain another potential application for 718Plus alloy. The property advantages for 718Plus alloy have also led to its being considered for rotating parts. Cao and Kennedy have shown that 718Plus alloy is capable of direct aging (DA), low temperature working followed by aging with no prior solution heat treatment. DA processing resulted in the production of very fine grain material with yield strength improvement at 704°C of 70-100 MPa. The alloy is also being considered for blading applications in areas where alloy 718 is limited due to elevated operating temperatures.

The alloy has also other applications outside of the jet and power turbine engines. Any application that currently uses alloys 718, Waspaloy, René 41 or other nickel-based superalloys can consider 718Plus alloy as a substitute for reasons of cost savings or increased temperature capability. Other markets where 718Plus alloy has potential are automotive turbo-chargers or industrial markets like chemical process or oil and gas where alloy 718 is used (Jeniski & Kennedy, 2006).

ALLVAC 718 Plus™ Superalloy for Aircraft Engine Applications 95

Cao. W., and Kennedy. R. L., (2005) Application Of Direct Aging To ALLVAC 718 Plus

Carlos A. E. M., (2007) New Technology Used In Gas Turbine Blade Materials, *Scientia et* 

Donald. T. J., (2007) Welding Superalloys For Aerospace Applications, *Welding Journal.* pp

Eliaz. N., Shemesh. G., Latanision. R.M., (2002) Hot Corrosion in Gas Turbine Components,

Idowu, O.A. , Ojo, O.A., Chaturvedi, M.C. (2007) Effect of heat input on heat affected zone

Jeniski. R. A., Jr. and Kennedy. R. L., (2006) Development of ATI Allvac 718Plus Alloy and

Jonšta Z., Jonšta P., Vodárek V., Mazanec K. (2007) Physical-Metallurgical Characteristics Of Nickel Super-Alloys Of Inconel Type*. Acta Metallurgica Slovaca*, 13, 4 (546 - 553). Jovanović T. M., Lukic. B., Miskovic. Z., Bobic. I., Ivana, Cvijovic. B. D., Processing And

Kennedy. R. L., (2005) Allvac® 718plus™ Superalloy For The Next Forty Years, *Superalloys* 

Liu. X., Xu. J., Deem. N., Chang. K., Barbero. E., Cao. W., Kennedy. R. L., Carneiro. T., (2005)

Liu. X., Rangararan. S., Barbero. E., Chang. K., Cao. W., Kennedy. R.L., and Carneiro. T.,

Otti. E.A., Grohi. J. And Sizek. H., (2005) Metals Affordability Initiative: Application of

Philip A. Schweitzer, P. E., (2003). *Metallic Materials: Physical, Mechanical and Corrosion* 

Sajjadi, S.A., Zebarjad, S.M. (2006) Study of fracture mechanisms of a Ni-Base superalloy at

Schreiber. K., Loehnert. K., Singer. R.F., (2006) Opportunities and Challenges for the New

Sims. C.T., Stoloff. N.S. and Hagel. W.C., (1987) Superalloys II- High Temperature Materials

Smallman R. E. Ngan, A. W. H., (2007). *Physical Metallurgy and Advanced Materials,* Elsevier

for Aerospace and Industrial Power, *John Wiley & Sons Inc.,* USA.

*Properties,* Marcel Dekker, Inc., ISBN: 0-8247-0878-4, USA.

cracking in laser welded ATI Allvac 718Plus superalloy. *Materials Science and* 

Applications*, II. Symposium on Recent Advantages of Nb-Containing Materials in* 

Some Applications Of Nickel, Cobalt And Titanium-Based Alloys, *Association of Metallurgical Engineers of Serbia Review paper,* MJoM *Metalurgija Journal Of* 

Effect Of Thermal-Mechanical Treatment On The Fatigue Crack Propagaton Behavior Of Newly Developed Allvac 718plus Alloy, *Superalloys 718, 625, 706 and* 

(2004) Fatigue Crack Propagaton Behavors Of New Developed Allvac 718plus

Allvac Alloy 718Plus for Aircraft Engine Static Structural Components, *Superalloys 718, 625, 706 and Derivatives Edited by E.A. Loria TMS (The Minerals, Metals &* 

different temperatures. *Journal of Achievements in Materials and Manufacturing* 

Nickel-Based Alloy 718 Plus, *II. Symposium on Recent Advantages of Nb-Containing* 

*E.A. Loria TMS (The Minerals, Metals & Materials Society).* 

Furrer. D., Fecht. H., (1999) Ni-Based Superalloys For Turbine Discs, *JOM*

*Technica Ano XIII, No:36, ISSN: 0122-1701* 

*Engineering.* Vol. A No. 454–455 pp.389–397.

*718, 625, 706 and Derivatives 2005, TMS.* 

Superalloy, *Superalloys 2004, TMS.*

*Engineering*. Vol. 18 Issue 1-2.

Ltd., ISBN: 978 0 7506 6906 1, UK.

*Engineering Failure Analysis* 9 31–43

28-32 January.

*Europe* 

*Metallurgy*

*Derivatives 2005, TMS*.

*Materials Society).*

*Materials in Europe.*

Alloy For Improved Performance, *Superalloys 718, 625, 706 and Derivatives Edited by* 

The cost of finished components of alloy 718 Plus is expected to be intermediate to alloys 718 and Waspaloy (Kennedy, 2005).

## **5. Acknowledgment**

The author would like to thank Eskisehir Osmangazi University for supporting the research study with Eskisehir Osmangazi University Research Funding Project, Project No: 2011/15020.

## **6. Conclusion**

A superalloy is a metallic alloy which is developed to resist most of all high temperatures, usually in cases until 70 % of the absolute melting temperature. All of these alloys have an excellent creep, corrosion and oxidation resistance as well as a good surface stability and fatigue life.

The main alloying elements are nickel, cobalt or nickel – iron, which can be found in the VIII. group of the periodic system of the elements. Fields of application are found particularly in the aerospace industry and in the nuclear industries, e.g. for engines and turbines.

The development of these advanced alloys allows a better exploitation of engines, which work at high temperatures, because the Turbine Inlet Temperature ( TIT ) depends on the temperature capability of the material which forms the turbine blades. Nickel-based superalloys can be strengthened through solid-solution and precipitation hardening.

Nickel-based superalloys can be used for a higher fraction of melting temperature and are therefore more favourable than cobal-based and iron-nickel-based superalloys at operating temperatures close to the melting temperature of the materials.

The newly innovated nickel based ALLVAC 718 Plus superalloy which is the last version of Inconel 718 has been proceeding in the way to become a material that aerospace and defense industries never replace of any other material with combining its good mechanical properties, easy machinability and low cost.

## **7. References**


The cost of finished components of alloy 718 Plus is expected to be intermediate to alloys

The author would like to thank Eskisehir Osmangazi University for supporting the research study with Eskisehir Osmangazi University Research Funding Project, Project No:

A superalloy is a metallic alloy which is developed to resist most of all high temperatures, usually in cases until 70 % of the absolute melting temperature. All of these alloys have an excellent creep, corrosion and oxidation resistance as well as a good surface stability and

The main alloying elements are nickel, cobalt or nickel – iron, which can be found in the VIII. group of the periodic system of the elements. Fields of application are found particularly in

The development of these advanced alloys allows a better exploitation of engines, which work at high temperatures, because the Turbine Inlet Temperature ( TIT ) depends on the temperature capability of the material which forms the turbine blades. Nickel-based

Nickel-based superalloys can be used for a higher fraction of melting temperature and are therefore more favourable than cobal-based and iron-nickel-based superalloys at operating

The newly innovated nickel based ALLVAC 718 Plus superalloy which is the last version of Inconel 718 has been proceeding in the way to become a material that aerospace and defense industries never replace of any other material with combining its good mechanical

Bergstrom D. S., and Bayhan. T. D., (2005) Properties and Microstructure Of ALLVAC 718

Bond. B.J. and Kennedy. R.L., (2005) Evaluation of ALLVAC 718 plus Alloy In the Cold

Campbell F. C. (2008). *Elements Of Metallurgy And Engineering Alloys,* ASTM International.,

Cao. W., (2005) Solidification and Solid State Phase Transformation of ALLVAC 718 Plus

Cao. W. and Kennedy. R., (2004) Role of Chemistry in 718-Type Alloys- Alloy 718 plus

Plus Alloy Rolled Sheet, *Superalloys 718, 625, 706 and Derivatives Edited by E.A. Loria* 

Worked and Heat Treated Condition, *Superalloys 718, 625, 706 and Derivatives Edited* 

Development, *Edited by K.A. Green, T.M. Pollock, H. Harada T.E. Howson, R.C. Reed, J.J. Schirra, and S, Walston, Superalloys, TMS (The Minerals, Metals & Materials Society)*

the aerospace industry and in the nuclear industries, e.g. for engines and turbines.

superalloys can be strengthened through solid-solution and precipitation hardening.

temperatures close to the melting temperature of the materials.

*TMS (The Minerals, Metals & Materials Society).* 

*by E.A. Loria TMS (The Minerals, Metals & Materials Society).* 

Alloy, *Superalloys 718, 625, 706 and Derivatives 2005, TMS.*

properties, easy machinability and low cost.

ISBN: 978-0-87170-867-0, USA.

718 and Waspaloy (Kennedy, 2005).

**5. Acknowledgment** 

2011/15020.

fatigue life.

**7. References** 

**6. Conclusion** 


**5** 

**Potential of MoSi2 and MoSi2-Si3N4 Composites** 

It has been expected that gas turbine engines in high temperature environments where aggressive mechanical stresses may occur and a good surface stability is needed should operate more efficiently. So the investigations about the materials which will be able to carry the aviation technology to the next level are beginning to accelerate in this direction. And also it expected that those new materials using in gas turbine engines as a high temperature structural material will exceed the superalloys' mechanical and physical limits. The intended development can only be achieved by providing the improvement of the essential properties of the structural materials such as thermal fatigue, oxidation resistance, strength/weigth ratio and fracture toughness. There are two different type of materials which are candidate to resist the operating conditions about 1200oC; first one is structural

ceramics such as SiC, Si3N4 and the second one is structural silicides such as MoSi2.

proper materials for high service applications (Vaseduvan & Petrovic, 1992).

**2. Superalloys and their limitations at elevated temperatures** 

After the propulsion systems with high strength/weight ratio, it observed that development of new materials with high strength and low density was necessary, thus the studies about the intermetallics began. The most important ones of these intermetallic compounds are silicides and aluminides. By the oxide layers in Al2O3, it can be used as a protective material in high temperature applications. Moreover, aluminides such as FeAl, TiAl, Ni3Al, can be suitable for some special applications in low and medium temperatures. In spite of these advantages, they remain inadequate above the temperatures 1200oC for their melting points with 1400-1600oC. Their low strength and creep resistance is not suitable for the temperatures above 1000oC. For this reason, it seems that silicides and aluminides are the

In aviation applications, advanced gas turbine elements are exposed to several mechanic, thermal and corrosive environments and intensive studies for the developing of these parts are still continuing. However, these alloys are needed to be cooled during the operation of

**1. Introduction** 

**for Aircraft Gas Turbine Engines** 

Melih Cemal Kushan1, Yagiz Uzunonat2, Sinem Cevik Uzgur3 and Fehmi Diltemiz4

*1Eskisehir Osmangazi University* 

 *41st Air Supply and Maintenance Base* 

*2 Anadolu University 3Ondokuz Mayis University* 

*Turkey* 


## **Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines**

Melih Cemal Kushan1, Yagiz Uzunonat2, Sinem Cevik Uzgur3 and Fehmi Diltemiz4 *1Eskisehir Osmangazi University 2 Anadolu University 3Ondokuz Mayis University 41st Air Supply and Maintenance Base Turkey* 

### **1. Introduction**

96 Recent Advances in Aircraft Technology

Smallman R. E. Bishop, R. J., (1999). *Modern Physical Metallurgy and Materials Engineering,*  Reed Educational and Professional Publishing Ltd., ISBN: 0 7506 4564 4, UK. Thomas. A., El-Wahabi. M., Cabrera. J.M., Prado. J. M., (2006) High Temperature Deformation of Inconel 718, *Journal of Materials Processing Technology* 177 469–472. Vishwakarma. K.R., Richards. N.L., Chaturvedi. M.C., (2007) Microstructural Analysis of

Vishwakarma. K.R. and Chaturvedi. M.C., (2007) A Study Of Haz Mcrofssurng In A

Yaman. Y.M., Kushan. M.C., (1998) Hot Cracking Susceptibilities In the Heat Affected Zone

Xie. X., Wang. G., Dong. J., Xu. C., Cao. W., Kennedy. R. L., (2005) Structure Stability Study

superalloy, *Materials Science and Engineering* A 480 517–528

Newly Developed Allvac® 718 Plus Tm Superalloy.

1234.

*Materials Society).* 

Fusion and Heat Affected Zones in Electron Beam Welded ALLVAC® 718PLUSTM

of Electron Beam Welded Inconel 718, *Journal of Materials Science Letters* 17, 1231-

On A Newly Developed Nickel-Base Superalloy- ALLVAC 718 Plus, *Superalloys 718, 625, 706 and Derivatives Edited by E.A. Loria TMS (The Minerals, Metals &* 

> It has been expected that gas turbine engines in high temperature environments where aggressive mechanical stresses may occur and a good surface stability is needed should operate more efficiently. So the investigations about the materials which will be able to carry the aviation technology to the next level are beginning to accelerate in this direction. And also it expected that those new materials using in gas turbine engines as a high temperature structural material will exceed the superalloys' mechanical and physical limits. The intended development can only be achieved by providing the improvement of the essential properties of the structural materials such as thermal fatigue, oxidation resistance, strength/weigth ratio and fracture toughness. There are two different type of materials which are candidate to resist the operating conditions about 1200oC; first one is structural ceramics such as SiC, Si3N4 and the second one is structural silicides such as MoSi2.

> After the propulsion systems with high strength/weight ratio, it observed that development of new materials with high strength and low density was necessary, thus the studies about the intermetallics began. The most important ones of these intermetallic compounds are silicides and aluminides. By the oxide layers in Al2O3, it can be used as a protective material in high temperature applications. Moreover, aluminides such as FeAl, TiAl, Ni3Al, can be suitable for some special applications in low and medium temperatures. In spite of these advantages, they remain inadequate above the temperatures 1200oC for their melting points with 1400-1600oC. Their low strength and creep resistance is not suitable for the temperatures above 1000oC. For this reason, it seems that silicides and aluminides are the proper materials for high service applications (Vaseduvan & Petrovic, 1992).

#### **2. Superalloys and their limitations at elevated temperatures**

In aviation applications, advanced gas turbine elements are exposed to several mechanic, thermal and corrosive environments and intensive studies for the developing of these parts are still continuing. However, these alloys are needed to be cooled during the operation of

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 99

excellent oxidation resistance, high thermal conductivity, and thermodynamic compatibility with many ceramic reinforcements. However, low fracture toughness at near-ambient temperatures, low strength at elevated temperatures in the monolithic form and tendency to pest degradation at ~500oC have seriously limited the development of MoSi2-based structural materials. Several recent studies have attempted to address these issues and have shown promising results. For example, pest resistant MoSi2-based materials have been

For general polycrystalline ductility five independent deformation modes are necessary. Changing the critical resolved shear stress of the slip systems through alloying may be a way to activate all three slip vectors, and obtain polycrystalline ductility. In fact, solid solution softening has been observed at room tempera ture in MoSi2 alloyed with Al and transition metals such as Nb, V and Ta. The mechanism of softening is not clearly understood, although first principles calculations indicate that solutes such as Al, Mg, V and Nb may change the Peierls stress so as to enhance relative to cleavage. Clearly, more work is

needed to understand how alloying may influence the mechanical behavior of MoSi2.

With regard to elevated temperature strengthening of MoSi2, both alloying with W to form C11b (Mo, W)Si2 alloys and composites with ceramic re inforcements such as SiC have been tried. A (Mo, W)Si2./20 vol.% SiC composite was shown to have significantly higher strength than Mar-M247 superalloy at temperatures above 1000oC. However, the strength of the (Mo, W)Si2./20 vol.% SiC composite dropped by almost an order of mag -nitude from 1200 to 1500oC; the yield strength at 1500oC was only ~75 MPa. A simpler and more e€ective way of strengthening MoSi2 at elevated temperatures is needed where the strength can be better retained with increasing temperature above 1200oC. Our preliminary studies using hot hardness experiments have shown that Re addition to MoSi2 caused signifcant hardening up to 1300oC. Further, it has been reported that alloying with Re, perhaps in synergism with carbon, increased the pesting resistance in the temperature range of 500 ± 800oC. In another preliminary study, polycrystalline (Mo, Re)Si2 alloys exhibited a minimum creep rate of ~5 x 10-6/s at 100 MPa applied stress at 1400o C as compared with the ~1 x 10- 4/s creep rate exhibited by MoSi2. No detailed mechanistic study has been performed to understand the effects of Re alloying on the elevated temperature mechanical behavior of MoSi2. In the present investigation, we have evaluated the mechanical properties, in compression, of arc-melted polycrystalline MoSi2 and (Mo, Re) Si2 alloys. We find that significant strengthening is achieved up to 1600oC by only small additions of Re. The mechanisms of elevated temperature solid solution strengthening are elucidated by considering the generation of constitutional Si vacancies that may pair with Re substitutionals to form tetragonally distorted point defect complexes. Characteristics of MoSi2 make it an interesting material as high temperature structural silicide. Not only it has a low density and a high melting point but also it can excellently resist the free oxygen of air in high temperature environments for a long time period. On the other hand, researchers noticed its potential as a structural material due to its electrical resistance increasing after every use and high modulus of elasticity at high temperatures. This makes MoSi2 a candidate material for structural high temperature applications particularly in gas turbine engines. MoSi2 and its composites offer a higher rate of resistance to oxidizing and aggressive environments during the combustion processes with their high melting points.

developed using silicon nitride reinforcement or alloying with Al.

the turbine engine and the practical temperature limits for metallic alloys remain below 1100oC. But in this situation, the elevation of turbine inlet temperature will be quite difficult and expansive. Because of these given limitations, there is not any important improvements on nickel based superalloys since 1985 (Soetching, 1995).

The basic facts that can directly effect the performance of superalloys in high temperatures are oxidation, hot corrosion and thermal fatigue. These effects cause the superalloy elements' surfaces may react with hot gases easier, and then their surface stability decreases (Bradley, 1988). Furthermore, during operation and stand-by period of turbine, there occurs a oscillation motion in the hot section elements respectively. This causes thermal fatigues on the superalloy parts.

## **2.1 Oxidation**

Oxidation is one of the most serious factors acting on the gas turbine's service life and can be determined as the reaction of materials with oxygen in 2-4 atm. partial pressure (Tein & Caulfield, 1989). Mostly the uniform oxidation is not accepted as a considerable problem in relatively low temperatures (870oC and below). But in temperatures about 1100oC, the aluminum content in the form of Al2O3 as a protective oxide can not provide the expected protection in long term periods. For this reason, it is necessary to use the silicide based structural composite materials or to make protective coatings with respect to the segment's location in gas turbine.

#### **2.2 Hot corrosion**

The process of hot corrosion contains a structural element and the reactions occurring in its surroundings. In operating conditions at high temperatures there is a possible accelerated oxidation for superalloys. Another name for this reaction is hot corrosion and it consists of two different mechanisms as low temperature (680-750oC) and high temperature (900- 1050oC) hot corrosion (Akkuş, 1999).

The basic principle to avoid from the hot corrosion in superalloys is using of the high content of chrome (≥ %20) during the manufacturing of material. But only a few types of nickel based superalloys have this rate for their high proportion of γ' and γ'' structure.

### **2.3 Thermal fatigue**

Heating with non-uniform distribution make interior stresses in the zones hotter than the average temperature of the turbine, and tension stresses in the colder zones. Superalloy turbine vanes are the good examples of elements exposed to thermal fatigue in aeroplane jet engines. During the acceleration, inlet and outlet edges of the turbine vane can heat and expand easier than the medium part under cooling. But in deceleration, inlet and outlet parts can quietly cool off than the medium parts. This case results as fatigue crack at the edges.

## **3. Physical and mechanical properties of MoSi2**

MoSi2 is a potential material for high temperature structural applications primarily due to its high melting point (2020oC), lower density (6.3 g/cm3) compared with superalloys,

the turbine engine and the practical temperature limits for metallic alloys remain below 1100oC. But in this situation, the elevation of turbine inlet temperature will be quite difficult and expansive. Because of these given limitations, there is not any important improvements

The basic facts that can directly effect the performance of superalloys in high temperatures are oxidation, hot corrosion and thermal fatigue. These effects cause the superalloy elements' surfaces may react with hot gases easier, and then their surface stability decreases (Bradley, 1988). Furthermore, during operation and stand-by period of turbine, there occurs a oscillation motion in the hot section elements respectively. This causes thermal fatigues on

Oxidation is one of the most serious factors acting on the gas turbine's service life and can be determined as the reaction of materials with oxygen in 2-4 atm. partial pressure (Tein & Caulfield, 1989). Mostly the uniform oxidation is not accepted as a considerable problem in relatively low temperatures (870oC and below). But in temperatures about 1100oC, the aluminum content in the form of Al2O3 as a protective oxide can not provide the expected protection in long term periods. For this reason, it is necessary to use the silicide based structural composite materials or to make protective coatings with respect to the segment's

The process of hot corrosion contains a structural element and the reactions occurring in its surroundings. In operating conditions at high temperatures there is a possible accelerated oxidation for superalloys. Another name for this reaction is hot corrosion and it consists of two different mechanisms as low temperature (680-750oC) and high temperature (900-

The basic principle to avoid from the hot corrosion in superalloys is using of the high content of chrome (≥ %20) during the manufacturing of material. But only a few types of

Heating with non-uniform distribution make interior stresses in the zones hotter than the average temperature of the turbine, and tension stresses in the colder zones. Superalloy turbine vanes are the good examples of elements exposed to thermal fatigue in aeroplane jet engines. During the acceleration, inlet and outlet edges of the turbine vane can heat and expand easier than the medium part under cooling. But in deceleration, inlet and outlet parts can quietly cool off than the medium parts. This case results as fatigue crack at the

MoSi2 is a potential material for high temperature structural applications primarily due to its high melting point (2020oC), lower density (6.3 g/cm3) compared with superalloys,

and γ'' structure.

nickel based superalloys have this rate for their high proportion of γ'

**3. Physical and mechanical properties of MoSi2** 

on nickel based superalloys since 1985 (Soetching, 1995).

the superalloy parts.

location in gas turbine.

1050oC) hot corrosion (Akkuş, 1999).

**2.2 Hot corrosion** 

**2.3 Thermal fatigue** 

edges.

**2.1 Oxidation** 

excellent oxidation resistance, high thermal conductivity, and thermodynamic compatibility with many ceramic reinforcements. However, low fracture toughness at near-ambient temperatures, low strength at elevated temperatures in the monolithic form and tendency to pest degradation at ~500oC have seriously limited the development of MoSi2-based structural materials. Several recent studies have attempted to address these issues and have shown promising results. For example, pest resistant MoSi2-based materials have been developed using silicon nitride reinforcement or alloying with Al.

For general polycrystalline ductility five independent deformation modes are necessary. Changing the critical resolved shear stress of the slip systems through alloying may be a way to activate all three slip vectors, and obtain polycrystalline ductility. In fact, solid solution softening has been observed at room tempera ture in MoSi2 alloyed with Al and transition metals such as Nb, V and Ta. The mechanism of softening is not clearly understood, although first principles calculations indicate that solutes such as Al, Mg, V and Nb may change the Peierls stress so as to enhance relative to cleavage. Clearly, more work is needed to understand how alloying may influence the mechanical behavior of MoSi2.

With regard to elevated temperature strengthening of MoSi2, both alloying with W to form C11b (Mo, W)Si2 alloys and composites with ceramic re inforcements such as SiC have been tried. A (Mo, W)Si2./20 vol.% SiC composite was shown to have significantly higher strength than Mar-M247 superalloy at temperatures above 1000oC. However, the strength of the (Mo, W)Si2./20 vol.% SiC composite dropped by almost an order of mag -nitude from 1200 to 1500oC; the yield strength at 1500oC was only ~75 MPa. A simpler and more e€ective way of strengthening MoSi2 at elevated temperatures is needed where the strength can be better retained with increasing temperature above 1200oC. Our preliminary studies using hot hardness experiments have shown that Re addition to MoSi2 caused signifcant hardening up to 1300oC. Further, it has been reported that alloying with Re, perhaps in synergism with carbon, increased the pesting resistance in the temperature range of 500 ± 800oC. In another preliminary study, polycrystalline (Mo, Re)Si2 alloys exhibited a minimum creep rate of ~5 x 10-6/s at 100 MPa applied stress at 1400o C as compared with the ~1 x 10- 4/s creep rate exhibited by MoSi2. No detailed mechanistic study has been performed to understand the effects of Re alloying on the elevated temperature mechanical behavior of MoSi2. In the present investigation, we have evaluated the mechanical properties, in compression, of arc-melted polycrystalline MoSi2 and (Mo, Re) Si2 alloys. We find that significant strengthening is achieved up to 1600oC by only small additions of Re. The mechanisms of elevated temperature solid solution strengthening are elucidated by considering the generation of constitutional Si vacancies that may pair with Re substitutionals to form tetragonally distorted point defect complexes. Characteristics of MoSi2 make it an interesting material as high temperature structural silicide. Not only it has a low density and a high melting point but also it can excellently resist the free oxygen of air in high temperature environments for a long time period. On the other hand, researchers noticed its potential as a structural material due to its electrical resistance increasing after every use and high modulus of elasticity at high temperatures. This makes MoSi2 a candidate material for structural high temperature applications particularly in gas turbine engines. MoSi2 and its composites offer a higher rate of resistance to oxidizing and aggressive environments during the combustion processes with their high melting points.

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 101

One of the most considerable limitations in MoSi2 applications is the structural disintegration during the low temperature oxidation which is known as pesting oxidation (Meschter, 1992). Previously, we noted that MoSi2 has an excellent oxidation resistance above the 1000oC, but at the temperatures about 500oC as it is presented Figure 3., the oxidation mechanism accelerates because of the volume expansion, MoO3 crystals, amorphshaped SiO2 bulks and MoSi2 particles residual from the reaction. If the material is porous and the surface accuracy is low, this state can be observed along the cracks or grain

 Fig. 3. Isothermal oxidation curves (a) at room temperature, (b) at the temperatures above

Fig. 4. Surfaces oxidized at 773K for 600-7200s in O2 (Chen et al., 1999).

During the oxidation reactions above 600oC, no pesting effect can be observed. MoSi2 based composites have considerably higher isothermal oxidation resistance than any other titanium, niobium or tantalum based composites, intermetallic compounds and nickel based superalloys, MoSi2 perfectly keeps this condition to 1600oC (Vaseduvan & Petrovic, 1992).

This fact was discovered in 1955 and predicted as the grain boundary fracture due to solution oxygen at the grain boundaries after the short-term cyclic diffusion, even though its complete nature is still a phenomena (Chou & Nieh, 1992, 1993). Methods for preventing from the pest effect are continuing. These methods are; making a protective SiO2 coating on the material and increasing the relative density of MoSi2 in the structure

boundaries, and granular oxide particles occur as a result.

1000o C (Liu et al., 2001).

(Wang et al., 2003)

Fig. 1. Unit cell of the body-centered tetragonal C11b structure of MoSi2. (Misra et al., 1999).

Fracture toughness of the material shows similarities with the other silicon based ceramics and yet it receives a brittle fracture resulted with low toughness. Table 1. shows the considerable characteristics of MoSi2.


Table 1. Basic characteristics of MoSi2.

The figure below shows the tetragonal lattice structure directions, red and blue points inidicate silicon and molybdenum atoms respectively.

Fig. 2. Tetragonal MoSi2 lattice structure.

Fig. 1. Unit cell of the body-centered tetragonal C11b structure of MoSi2. (Misra et al., 1999).

Fracture toughness of the material shows similarities with the other silicon based ceramics and yet it receives a brittle fracture resulted with low toughness. Table 1. shows the

The figure below shows the tetragonal lattice structure directions, red and blue points

 Metric English Density 6.23 g/cm3 0.225 lb/in3 Molecular Weigth 152.11 g/mol 152.11 g/mol Electrical Resistance (20C) 3.5x10-7 ohm-cm 3.5x10-7 ohm-cm Electrical Resistance(1700C) 4.0x10-6 ohm-cm 4.0x10-6 ohm-cm Thermal Capacity 0.437 J/g-C 0.104 BTU/lb-F Thermal Conductivity 66.2 W/m-K 459 BTU-in/hr-ft2-F

Melting Point 2030C 4046F Maximum Service Temp. 1600C 2912F Crystal Structure Tetragonal Tetragonal

considerable characteristics of MoSi2.

Table 1. Basic characteristics of MoSi2.

Fig. 2. Tetragonal MoSi2 lattice structure.

inidicate silicon and molybdenum atoms respectively.

One of the most considerable limitations in MoSi2 applications is the structural disintegration during the low temperature oxidation which is known as pesting oxidation (Meschter, 1992). Previously, we noted that MoSi2 has an excellent oxidation resistance above the 1000oC, but at the temperatures about 500oC as it is presented Figure 3., the oxidation mechanism accelerates because of the volume expansion, MoO3 crystals, amorphshaped SiO2 bulks and MoSi2 particles residual from the reaction. If the material is porous and the surface accuracy is low, this state can be observed along the cracks or grain boundaries, and granular oxide particles occur as a result.

Fig. 3. Isothermal oxidation curves (a) at room temperature, (b) at the temperatures above 1000o C (Liu et al., 2001).

This fact was discovered in 1955 and predicted as the grain boundary fracture due to solution oxygen at the grain boundaries after the short-term cyclic diffusion, even though its complete nature is still a phenomena (Chou & Nieh, 1992, 1993). Methods for preventing from the pest effect are continuing. These methods are; making a protective SiO2 coating on the material and increasing the relative density of MoSi2 in the structure (Wang et al., 2003)

Fig. 4. Surfaces oxidized at 773K for 600-7200s in O2 (Chen et al., 1999).

During the oxidation reactions above 600oC, no pesting effect can be observed. MoSi2 based composites have considerably higher isothermal oxidation resistance than any other titanium, niobium or tantalum based composites, intermetallic compounds and nickel based superalloys, MoSi2 perfectly keeps this condition to 1600oC (Vaseduvan & Petrovic, 1992).

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 103

value of 89968 Hv. Samples containing 2 at.% Al or 1 at.% Nb had average Vickers hardness values of 72928 Hv and 72950 Hv, respectively. 2.5 at.% Re containing samples had the highest hardness value of 103971 Hv. Samples containing 1 at.% Re+2 at.% Al had an average hardness value of 74230 Hv, slightly higher than Al containing samples but significantly lower than both MoSi2 and (Mo, 1 at.% Re)Si2 samples. Slip lines were observed around indentations in all samples except the unalloyed MoSi2 and (Mo, 2.5 at.% Re)Si2 samples. Samples containing 1 at.% Nb+2 at.% Al did not exhibit any improvements in the

mechanical properties and were excluded from further considerations.

Fig. 6. Vickers hardness values for polycrystalline samples of all composition under investigation. Hardness of polycrystalline Al containing samples is also compared to the values obtained on monocrystalline Al containing samples on (100) and (001) surfaces of the

Compression testing at room temperature and at 1600oC was used to determine the 0.2% offset yield strength for all polycrystalline samples. Unalloyed MoSi2 and 2.5 at.% Re containing samples could not be deformed plastically below 900 and 1200o C, respectively. Below these temperatures, the aforementioned samples would undergo brittle fracture during compression testing. The addition of 2.5 at.% Re increased the BDTT, in compression, of MoSi2 by about 300o C while increasing its yield strength from 14 MPa to 170 at 1600o C. Among alloying elements investigated here, 2.5 at.% Re was most effective in increasing strength at 1600o C. The addition of 2 at.% Al was effective in both increasing the high temperature strength to 55 MPa and lowering the BDTT to 425o C. Mo(Si, 2 at.% Al)2 samples exhibited the lowest room temperature yield strength of 415 MPa. Addition of 1 at.% Re+2 at.% Al combined the beneficial effects of both alloying elements and resulted in enhanced ambient temperature compressive plasticity and high temperature strength

However, the improvements in room temperature plasticity was less than that of samples alloyed with 2 at.% Al alone as evident from the value of the room temperature yield

crystal.

**4.2 Yield strength** 

compared to the unalloyed samples.

Fig. 5. Surfaces oxidized at 673-873*K* for 7200s in O2 (Chen et al., 1999).

Despite excellent oxidation resistance, high melting point, and low density, the potentials of molybdenum disilicide as a high temperature structural material have not been utilized due to its brittleness at low temperatures and low strength at high temperatures . For example, below 900o C, the fracture toughness of MoSi2 is in the range of 2–4 MPam1/2 , and the 0.2% offset yield strength of MoSi2 at 1600o C is about 20 MPa. Alloying or reinforcing with a second phase may lower the brittle to ductile transition temperature (BDTT) of MoSi2. However, ductile-phase toughening with metallic phases has limited applicability in MoSi2 due to the chemical reaction with silicon to form silicides, and reinforcing with ceramic second phases such as SiC and ZrO2 has only a modest effect on enhancing plastic flow and increasing toughness.

First principles calculations indicate that alloying of MoSi2 while maintaining its bodycentered tetragonal (C11b) structure may result in improved mechanical properties. For example, Al and Nb may enhance ductility and Re may increase strength. Improvements in both low and high temperature mechanical properties of MoSi2 have been reported by alloying MoSi2 with small amounts of Al, Nb, and Re (<2 at.%). During alloying, below the solubility limits of alloying elements in the C11b structure of MoSi2, Al substitutes for Si, whereas Re and Nb substitute for Mo. The solubility limits of Re, Nb and Al in MoSi2 have been reported as ~2.5, 1.3 and 2.7 at.%, respectively. Although improvements in the ambient temperature toughness have been reported by alloying of MoSi2 beyond the solubility limits with Nb and Al, the rates of improvement per fraction of solute are not as considerable as those observed in single phase alloys. Furthermore, the presence of secondary phases with a lower high temperature strength than the matrix alloy would degrade the mechanical properties at high temperature (>1500oC) for which applications MoSi2 is an excellent candidate. The aim of this investigation was to explore the possibility of obtaining concurrently enhanced room temperature ductility and high temperature strength in singlephase MoSi2 by combining the high temperature hardening and the low temperature softening effects of Re, Al, and Nb. Hardness testing at room temperature and compression testing at 1600o C are conducted on unalloyed and alloyed MoSi2 samples in order to study both low and high temperature effects of each alloying composition on the mechanical properties of MoSi2. (Sharif et al., 2001).

#### **4. Effects of alloying**

#### **4.1 Hardness**

Microhardness testing performed on stoichiometric samples obtained from melting (Mo, Re or Nb)(Si, Al)2.01 samples indicated that unalloyed MoSi2 had an average Vickers hardness value of 89968 Hv. Samples containing 2 at.% Al or 1 at.% Nb had average Vickers hardness values of 72928 Hv and 72950 Hv, respectively. 2.5 at.% Re containing samples had the highest hardness value of 103971 Hv. Samples containing 1 at.% Re+2 at.% Al had an average hardness value of 74230 Hv, slightly higher than Al containing samples but significantly lower than both MoSi2 and (Mo, 1 at.% Re)Si2 samples. Slip lines were observed around indentations in all samples except the unalloyed MoSi2 and (Mo, 2.5 at.% Re)Si2 samples. Samples containing 1 at.% Nb+2 at.% Al did not exhibit any improvements in the mechanical properties and were excluded from further considerations.

Fig. 6. Vickers hardness values for polycrystalline samples of all composition under investigation. Hardness of polycrystalline Al containing samples is also compared to the values obtained on monocrystalline Al containing samples on (100) and (001) surfaces of the crystal.

#### **4.2 Yield strength**

102 Recent Advances in Aircraft Technology

Despite excellent oxidation resistance, high melting point, and low density, the potentials of molybdenum disilicide as a high temperature structural material have not been utilized due to its brittleness at low temperatures and low strength at high temperatures . For example, below 900o C, the fracture toughness of MoSi2 is in the range of 2–4 MPam1/2 , and the 0.2% offset yield strength of MoSi2 at 1600o C is about 20 MPa. Alloying or reinforcing with a second phase may lower the brittle to ductile transition temperature (BDTT) of MoSi2. However, ductile-phase toughening with metallic phases has limited applicability in MoSi2 due to the chemical reaction with silicon to form silicides, and reinforcing with ceramic second phases such as SiC and ZrO2 has only a modest effect on enhancing plastic flow and

First principles calculations indicate that alloying of MoSi2 while maintaining its bodycentered tetragonal (C11b) structure may result in improved mechanical properties. For example, Al and Nb may enhance ductility and Re may increase strength. Improvements in both low and high temperature mechanical properties of MoSi2 have been reported by alloying MoSi2 with small amounts of Al, Nb, and Re (<2 at.%). During alloying, below the solubility limits of alloying elements in the C11b structure of MoSi2, Al substitutes for Si, whereas Re and Nb substitute for Mo. The solubility limits of Re, Nb and Al in MoSi2 have been reported as ~2.5, 1.3 and 2.7 at.%, respectively. Although improvements in the ambient temperature toughness have been reported by alloying of MoSi2 beyond the solubility limits with Nb and Al, the rates of improvement per fraction of solute are not as considerable as those observed in single phase alloys. Furthermore, the presence of secondary phases with a lower high temperature strength than the matrix alloy would degrade the mechanical properties at high temperature (>1500oC) for which applications MoSi2 is an excellent candidate. The aim of this investigation was to explore the possibility of obtaining concurrently enhanced room temperature ductility and high temperature strength in singlephase MoSi2 by combining the high temperature hardening and the low temperature softening effects of Re, Al, and Nb. Hardness testing at room temperature and compression testing at 1600o C are conducted on unalloyed and alloyed MoSi2 samples in order to study both low and high temperature effects of each alloying composition on the mechanical

Microhardness testing performed on stoichiometric samples obtained from melting (Mo, Re or Nb)(Si, Al)2.01 samples indicated that unalloyed MoSi2 had an average Vickers hardness

Fig. 5. Surfaces oxidized at 673-873*K* for 7200s in O2 (Chen et al., 1999).

increasing toughness.

properties of MoSi2. (Sharif et al., 2001).

**4. Effects of alloying** 

**4.1 Hardness** 

Compression testing at room temperature and at 1600oC was used to determine the 0.2% offset yield strength for all polycrystalline samples. Unalloyed MoSi2 and 2.5 at.% Re containing samples could not be deformed plastically below 900 and 1200o C, respectively. Below these temperatures, the aforementioned samples would undergo brittle fracture during compression testing. The addition of 2.5 at.% Re increased the BDTT, in compression, of MoSi2 by about 300o C while increasing its yield strength from 14 MPa to 170 at 1600o C. Among alloying elements investigated here, 2.5 at.% Re was most effective in increasing strength at 1600o C. The addition of 2 at.% Al was effective in both increasing the high temperature strength to 55 MPa and lowering the BDTT to 425o C. Mo(Si, 2 at.% Al)2 samples exhibited the lowest room temperature yield strength of 415 MPa. Addition of 1 at.% Re+2 at.% Al combined the beneficial effects of both alloying elements and resulted in enhanced ambient temperature compressive plasticity and high temperature strength compared to the unalloyed samples.

However, the improvements in room temperature plasticity was less than that of samples alloyed with 2 at.% Al alone as evident from the value of the room temperature yield

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 105

density functional calculations, offer an alternative route which can be used as a costeffective precursor to experiment. The need, in the case of MoSi2, is for an element or elements which can be introduced at microalloy levels (less than 5%) and which will perturb the brittle–ductile behavior in favor of ductility without adversely affecting the advantageous physical properties. While the method of choice would normally be an atomistic calculation, bonding in MoSi2 is known to have hybrid metallic and covalent character. Determination of the effects of alloying on such bonding requires accurate quantum mechanical treatment of the electrons, and generation of reliable interatomic potentials, which are an essential prerequisite to atomistic methods, is impractical1. Instead, use is made of recent advances in the theory of dislocation nucleation and mobility which provide approximate links between these properties and the generalized stacking fault energy surface, which can be calculated accurately using first principles quantum mechanical techniques. A similar approach has been used successfully by two of the present authors to investigate the DBT in silicon. Even with these gross approximations, the numerical work is intensive. The calculations are restricted to small supercells, with correspondingly large alloy content, and the effects of true microalloying must be estimated by interpolation. An overview of the experimental background will be presented in the next

Fig. 8. Crystal structure of MoSi2. (a) Unit cell for the body centered C11b structure; solid circles represent Mo atoms and open circles represent Si atoms. (b) (013) plane and the

The creep behavior of MoSi2-based materials has been extensively studied. It has been observed that the grain size has a large effect on creep resistance of monolithic MoSi2. Reinforcing with SiC also refined grain size that enhanced creep rates overshadowing any beneficial effects of reinforcement Increased creep resistance has been noted only when volume fractions of SiC are above 20%. Another important factor strongly affecting the creep strain rate of MoSi2 is the presence of silica particles (SiO2). During high temperature deformation, the SiO2 particles at the grain boundaries flow to form intergranular film, which slides or cracks. A high volume fraction of SiO2 and reduction in grain size, both

Burgers vector for {013}331 slip systems (Waghmare et al., 1999).

section.

**4.4 Creep** 

strength of (Mo, 1 at.% Re)(Si, 2 at.% Al)2 alloy, 670 MPa. Similarly, the enhancement in high temperature strength was less than that of only Re containing samples but greater than that of only Al containing samples. The effects of 1 at.% Nb as an alloying element by itself in lowering BDTT and enhancing high temperature strength of MoSi2 was more pronounced than the combined effects of 1 at.% Re and 2 at.% Al. The room temperature yield strength of (Mo, 1 at.% Nb)Si2, 500 MPa, was higher than that of 2 at.% Al containing samples and lower than that of (Mo, 1 at.% Re)(Si, 2 at.% Al)2 samples. The 0.2% offset strength of (Mo, 1 at.% Nb)Si2 samples at 1600oC, 143 MPa, was an order of magnitude greater than that of unalloyed MoSi2 (Sharif et al., 2001).

Fig. 7. Effects of alloying on the room temperature and high temperature (1600oC) strength of MoSi2.

#### **4.3 Ductility**

Molybdenum disilicide crystallizes in an ordered body centered tetragonal structure with a=0.320 nm and c=0.785 nm, formed by alternate stacking of single Mo and double Si (001) layers. With its high temperature ductility and exceptional resistance to corrosion and fatigue crack growth, MoSi2 combines the toughness of a metal with the strength of a ceramic and is a promising candidate to replace nickel alloys in the next generation of hightemperature gas turbines. Unfortunately, it undergoes a ductile–brittle transition (DBT) at 1200°C, with the fracture toughness dropping to 2–3 MPa m1/2, well below the minimum of 20 MPa m1/2 required for engine applications. This brittleness at low temperature means that MoSi2 must be formed by costly electro-discharge machining and places a severe limitation on its potential technological utility. However, there is a reasonable chance that the DBT in MoSi2 may be manipulated or even eliminated. Many of the slip systems in MoSi2 are ductile and it is only for a stress axis near [001] that a DBT is observed.

It is desirable therefore, to alter the properties of MoSi2 in very specific ways. This can be, and has in the past been, attempted by heuristically changing the composition or structure of the material and studying experimentally the effect of these changes. It will be argued in this paper that advances in the theory of bonding in solids, based on quantum mechanical density functional calculations, offer an alternative route which can be used as a costeffective precursor to experiment. The need, in the case of MoSi2, is for an element or elements which can be introduced at microalloy levels (less than 5%) and which will perturb the brittle–ductile behavior in favor of ductility without adversely affecting the advantageous physical properties. While the method of choice would normally be an atomistic calculation, bonding in MoSi2 is known to have hybrid metallic and covalent character. Determination of the effects of alloying on such bonding requires accurate quantum mechanical treatment of the electrons, and generation of reliable interatomic potentials, which are an essential prerequisite to atomistic methods, is impractical1. Instead, use is made of recent advances in the theory of dislocation nucleation and mobility which provide approximate links between these properties and the generalized stacking fault energy surface, which can be calculated accurately using first principles quantum mechanical techniques. A similar approach has been used successfully by two of the present authors to investigate the DBT in silicon. Even with these gross approximations, the numerical work is intensive. The calculations are restricted to small supercells, with correspondingly large alloy content, and the effects of true microalloying must be estimated by interpolation. An overview of the experimental background will be presented in the next section.

Fig. 8. Crystal structure of MoSi2. (a) Unit cell for the body centered C11b structure; solid circles represent Mo atoms and open circles represent Si atoms. (b) (013) plane and the Burgers vector for {013}331 slip systems (Waghmare et al., 1999).

#### **4.4 Creep**

104 Recent Advances in Aircraft Technology

strength of (Mo, 1 at.% Re)(Si, 2 at.% Al)2 alloy, 670 MPa. Similarly, the enhancement in high temperature strength was less than that of only Re containing samples but greater than that of only Al containing samples. The effects of 1 at.% Nb as an alloying element by itself in lowering BDTT and enhancing high temperature strength of MoSi2 was more pronounced than the combined effects of 1 at.% Re and 2 at.% Al. The room temperature yield strength of (Mo, 1 at.% Nb)Si2, 500 MPa, was higher than that of 2 at.% Al containing samples and lower than that of (Mo, 1 at.% Re)(Si, 2 at.% Al)2 samples. The 0.2% offset strength of (Mo, 1 at.% Nb)Si2 samples at 1600oC, 143 MPa, was an order of magnitude greater than that of

Fig. 7. Effects of alloying on the room temperature and high temperature (1600oC) strength

Molybdenum disilicide crystallizes in an ordered body centered tetragonal structure with a=0.320 nm and c=0.785 nm, formed by alternate stacking of single Mo and double Si (001) layers. With its high temperature ductility and exceptional resistance to corrosion and fatigue crack growth, MoSi2 combines the toughness of a metal with the strength of a ceramic and is a promising candidate to replace nickel alloys in the next generation of hightemperature gas turbines. Unfortunately, it undergoes a ductile–brittle transition (DBT) at 1200°C, with the fracture toughness dropping to 2–3 MPa m1/2, well below the minimum of 20 MPa m1/2 required for engine applications. This brittleness at low temperature means that MoSi2 must be formed by costly electro-discharge machining and places a severe limitation on its potential technological utility. However, there is a reasonable chance that the DBT in MoSi2 may be manipulated or even eliminated. Many of the slip systems in

MoSi2 are ductile and it is only for a stress axis near [001] that a DBT is observed.

It is desirable therefore, to alter the properties of MoSi2 in very specific ways. This can be, and has in the past been, attempted by heuristically changing the composition or structure of the material and studying experimentally the effect of these changes. It will be argued in this paper that advances in the theory of bonding in solids, based on quantum mechanical

unalloyed MoSi2 (Sharif et al., 2001).

of MoSi2.

**4.3 Ductility** 

The creep behavior of MoSi2-based materials has been extensively studied. It has been observed that the grain size has a large effect on creep resistance of monolithic MoSi2. Reinforcing with SiC also refined grain size that enhanced creep rates overshadowing any beneficial effects of reinforcement Increased creep resistance has been noted only when volume fractions of SiC are above 20%. Another important factor strongly affecting the creep strain rate of MoSi2 is the presence of silica particles (SiO2). During high temperature deformation, the SiO2 particles at the grain boundaries flow to form intergranular film, which slides or cracks. A high volume fraction of SiO2 and reduction in grain size, both

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 107

Monolithic MoSi2 exhibits only a modest value of fracture toughness at low temperatures and inadequate strength at high temperatures. Thus, many of recent studies on the development of MoSi2-based alloys have focused on improving these poor mechanical properties through forming composites with ceramics and with other silicides. These properties have recently been reported to be significantly improved in composites formed with Si3N4 and SiC. However, the volume fraction of Si3N4 and SiC ceramic reinforcements in these MoSi2-composites generally exceeds 50%. Further improvements in mechanical properties of these composites will be achieved if those of the MoSi2 matrix phase are improved. The present study was undertaken to achieve this by alloying additions to MoSi2. Transition-metal atoms that form disilicides with tetragonal C11b, hexagonal C40 and orthorhombic C54 structures are considered as alloying elements to MoSi2. These three structures commonly possess (pseudo-) hexagonally arranged TMSi2 layers and differ from each other only in the stacking sequence of these TMSi2 layers; the C11b, C40 and C54 structures are based on the AB, ABC and ADBC stacking of these layers, respectively. W and Re have been known to form a C11b disilicide with Si and they are believed to form a complete C11b solid-solution with MoSi2, although recent studies have indicated that the disilicide formed with Re is an off-stoichiometric (defective) one for-mulated to be ReSi1.75 having a monoclinic crystal structure. The details of our crystal structure assessment for ReSi1.75 as well as phase equilibria in the MoSi2±ReSi1.75 pseudobinary system will be published elsewhere. Large amounts of alloying additions are possible for these alloying elements, and high temperature strength is expected to be improved through a solid solution hardening mechanism since the hardness of both WSi2 and ReSi1.75 is reported to be larger than that of MoSi2. The yield strength of MoSi2 powder compacts is greatly increased when WSi2 is alloyed with MoSi2 by more than 50 vol%. In addition, our previous study on single crystals of MoSi2±WSi2 solid solutions has indicated that the compression yield stres above 1200o C greatly increases when the WSi2 content in the solutions exceeds 50 vol.%. However, low temperature deformability may be declined upon alloying with these elements because of the increased strength. Indeed, the room temperature hardness of both MoSi2± WSi2 and MoSi2± ReSi1.75 solid solutions is reported to monotonically increase with

V, Cr, Nb and Ta have been known to form a C40 disilicide with Si. Al is also known to transform MoSi2 from the C11b to the C40 structures by substituting it for Si. Of the five slip systems identified to be operative in MoSi2, slip on {110}<111> is operative from 500oC. 1/2<111> dislocations of this slip system are reported to dissociate into two identical 1/4<111> partials separated by a stacking fault. The stacking across the fault is ABC and resembles the stacking of (0001) in the C40 structure. Hence, the addition of elements that form a C40 disilicide may cause the energy difference between C11b and C40 structures to decrease so that the energy of the stacking fault would also be decreased, although the solid solubility of these alloying elements in MoSi2 has been reported to be rather limited to the level of a few atomic %. From this point of view, we may expect that the deformability of MoSi2 at low temperatures increases upon alloying with elements that form a C40 disilicide. This is consistent that V and Nb may enhance the ductility of MoSi2. Indeed, room temperature hardness of MoSi2 polycrystals decreases upon alloying with Cr, Nb, Ta and Al and similar observations were made for Al-bearing MoSi2 polycrystals. Compression deformation experiments made so far on ternary MoSi2 single crystals containing these

**4.5 Plastic deformation** 

the increase in either WSi2 or ReSi1.75 content.

enhance creep rates; but it is of interest to examine how the two are interrelated. Alloying of polycrystalline MoSi2 with Al and C converts SiO2 to Al2O3 and SiC, respectively, which leads to the enhancement in creep resistance. When C is added, oxygen is got rid off in the form of CO or CO2, which may leave behind fine pores, which are difficult to close. When Al is added, the reaction between Al and SiO2 forms Al2O3 and Si. The Si may remain in elemental form or react with Mo5Si3 particles that are present in small volume fractions. These particles form due to partial oxidation of MoSi2 during hot pressing, particularly when vacuum is low. The probability of the reaction between free Si and residual Mo5Si3 in the present study is high, as free Si has been observed in the microstructure only very rarely. The figure below presents the effect of alloying of single and polycrystalline MoSi2 with Al, on the creep rates at 1300oC. The single crystals of Mo(Si0.97 Al0.03)2, with hexagonal C40 or hP9 (Pearson's symbol) structure, have shown higher creep rates, compared to those of single crystals of MoSi2 along the [0 15 1] orientation of stress axis. However, the trend reverses with change of stress-axis to [001] direction. On the other hand, polycrystalline MoSi2–5.5Al alloy has shown improvement in creep resistance, compared to polycrystalline MoSi2 at 1300oC. Unlike Mo(Si0.97 Al0.03)2, the matrix phase of MoSi2–5.5Al has tetragonal, C11b structure.

Fig. 9. Comparison of steady state creep rates, measured at 1300oC on MoSi2 and MoSi2– 5.5Al alloy, as well as single crystals of MoSi2 and Mo(Si0.97 Al0.03)2 tested with [0 15 1] and [001] orientations. Single crystals are marked as X (Mitra et al., 2004).

As expected, the creep rates of polycrystalline MoSi2 and MoSi2–5.5Al alloy are higher compared to those of single crystals at 1300oC, because of the role of grain boundaries at 0.68 Tm: In the present investigation, samples of MoSi2 with varying grain sizes and SiO2 contents, as well as those of MoSi2–20 vol% SiC composite and MoSi2–Al alloys have been creep tested at 1200oC and their behaviors analyzed. The values of activation volume and threshold stress have been calculated. These provide an insight into the ratecontrolling and strengthening mechanisms. The creep behavior of the above materials has also been compared with deformation behavior under constant strain rate tests.

**4.5 Plastic deformation** 

106 Recent Advances in Aircraft Technology

enhance creep rates; but it is of interest to examine how the two are interrelated. Alloying of polycrystalline MoSi2 with Al and C converts SiO2 to Al2O3 and SiC, respectively, which leads to the enhancement in creep resistance. When C is added, oxygen is got rid off in the form of CO or CO2, which may leave behind fine pores, which are difficult to close. When Al is added, the reaction between Al and SiO2 forms Al2O3 and Si. The Si may remain in elemental form or react with Mo5Si3 particles that are present in small volume fractions. These particles form due to partial oxidation of MoSi2 during hot pressing, particularly when vacuum is low. The probability of the reaction between free Si and residual Mo5Si3 in the present study is high, as free Si has been observed in the microstructure only very rarely. The figure below presents the effect of alloying of single and polycrystalline MoSi2 with Al, on the creep rates at 1300oC. The single crystals of Mo(Si0.97 Al0.03)2, with hexagonal C40 or hP9 (Pearson's symbol) structure, have shown higher creep rates, compared to those of single crystals of MoSi2 along the [0 15 1] orientation of stress axis. However, the trend reverses with change of stress-axis to [001] direction. On the other hand, polycrystalline MoSi2–5.5Al alloy has shown improvement in creep resistance, compared to polycrystalline MoSi2 at 1300oC. Unlike Mo(Si0.97 Al0.03)2, the matrix phase of MoSi2–5.5Al has tetragonal,

Fig. 9. Comparison of steady state creep rates, measured at 1300oC on MoSi2 and MoSi2– 5.5Al alloy, as well as single crystals of MoSi2 and Mo(Si0.97 Al0.03)2 tested with [0 15 1] and

As expected, the creep rates of polycrystalline MoSi2 and MoSi2–5.5Al alloy are higher compared to those of single crystals at 1300oC, because of the role of grain boundaries at 0.68 Tm: In the present investigation, samples of MoSi2 with varying grain sizes and SiO2 contents, as well as those of MoSi2–20 vol% SiC composite and MoSi2–Al alloys have been creep tested at 1200oC and their behaviors analyzed. The values of activation volume and threshold stress have been calculated. These provide an insight into the ratecontrolling and strengthening mechanisms. The creep behavior of the above materials has also been

[001] orientations. Single crystals are marked as X (Mitra et al., 2004).

compared with deformation behavior under constant strain rate tests.

C11b structure.

Monolithic MoSi2 exhibits only a modest value of fracture toughness at low temperatures and inadequate strength at high temperatures. Thus, many of recent studies on the development of MoSi2-based alloys have focused on improving these poor mechanical properties through forming composites with ceramics and with other silicides. These properties have recently been reported to be significantly improved in composites formed with Si3N4 and SiC. However, the volume fraction of Si3N4 and SiC ceramic reinforcements in these MoSi2-composites generally exceeds 50%. Further improvements in mechanical properties of these composites will be achieved if those of the MoSi2 matrix phase are improved. The present study was undertaken to achieve this by alloying additions to MoSi2. Transition-metal atoms that form disilicides with tetragonal C11b, hexagonal C40 and orthorhombic C54 structures are considered as alloying elements to MoSi2. These three structures commonly possess (pseudo-) hexagonally arranged TMSi2 layers and differ from each other only in the stacking sequence of these TMSi2 layers; the C11b, C40 and C54 structures are based on the AB, ABC and ADBC stacking of these layers, respectively. W and Re have been known to form a C11b disilicide with Si and they are believed to form a complete C11b solid-solution with MoSi2, although recent studies have indicated that the disilicide formed with Re is an off-stoichiometric (defective) one for-mulated to be ReSi1.75 having a monoclinic crystal structure. The details of our crystal structure assessment for ReSi1.75 as well as phase equilibria in the MoSi2±ReSi1.75 pseudobinary system will be published elsewhere. Large amounts of alloying additions are possible for these alloying elements, and high temperature strength is expected to be improved through a solid solution hardening mechanism since the hardness of both WSi2 and ReSi1.75 is reported to be larger than that of MoSi2. The yield strength of MoSi2 powder compacts is greatly increased when WSi2 is alloyed with MoSi2 by more than 50 vol%. In addition, our previous study on single crystals of MoSi2±WSi2 solid solutions has indicated that the compression yield stres above 1200o C greatly increases when the WSi2 content in the solutions exceeds 50 vol.%. However, low temperature deformability may be declined upon alloying with these elements because of the increased strength. Indeed, the room temperature hardness of both MoSi2± WSi2 and MoSi2± ReSi1.75 solid solutions is reported to monotonically increase with the increase in either WSi2 or ReSi1.75 content.

V, Cr, Nb and Ta have been known to form a C40 disilicide with Si. Al is also known to transform MoSi2 from the C11b to the C40 structures by substituting it for Si. Of the five slip systems identified to be operative in MoSi2, slip on {110}<111> is operative from 500oC. 1/2<111> dislocations of this slip system are reported to dissociate into two identical 1/4<111> partials separated by a stacking fault. The stacking across the fault is ABC and resembles the stacking of (0001) in the C40 structure. Hence, the addition of elements that form a C40 disilicide may cause the energy difference between C11b and C40 structures to decrease so that the energy of the stacking fault would also be decreased, although the solid solubility of these alloying elements in MoSi2 has been reported to be rather limited to the level of a few atomic %. From this point of view, we may expect that the deformability of MoSi2 at low temperatures increases upon alloying with elements that form a C40 disilicide. This is consistent that V and Nb may enhance the ductility of MoSi2. Indeed, room temperature hardness of MoSi2 polycrystals decreases upon alloying with Cr, Nb, Ta and Al and similar observations were made for Al-bearing MoSi2 polycrystals. Compression deformation experiments made so far on ternary MoSi2 single crystals containing these

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 109

acceptably compliant. Unfortunately, most of the incremental toughening and attended fiber pullout occurs at engineering strains that exceed the strain which occurs at the ultimate strength of the material, and the toughening benefit is, therefore, not useful for design purposes. Thus, the degree of incremental toughening that occurs during the inelastic portion of the stress-strain curve up to the ultimate strength will have to be improved before

It has also been shown that MoSi2 offers the potential of combining the effects of second phase reinforcements with metallurgical alloying to improve mechanical properties without degrading oxidation resistance. For example, high temperature (1200°C) creep was reduced by a factor of 10 by alloying with WSi2 and by another factor of 10–15 with the addition of SiC whiskers. Additions of carbon and zirconia have also proven to be beneficial. Carbon reacts with the oxygen impurities in MoSi2 to improve the toughness at high temperatures by removing SiO2, and leaves behind a compatible SiC phase. Zirconia, which is thermochemically stable with MoSi2, can also be used to increase fracture toughness. a 20 v/o loading of particulate ZrO2 increased the low temperature fracture toughness of MoSi2 by a factor of four. The toughening transformation occurs above the ductile–brittle transition and therefore enhances the low temperature properties. In an attempt to reinforce MoSi2 with SCS-6 silicon fibers, it was discovered that the large thermal expansion difference between the matrix and fiber introduced matrix cracking upon cooling from the densification temperature. This problem was solved by adding Si3N4 to form a two phase composite matrix with a coefficient of expansion that more closely matches the fiber. There is no reaction between Si3N4 and MoSi2, even at fabrication temperatures as high as 1750°C. No gross cracking occurs on cool down, although some microcracking has been observed and is a function of grain size. The critical particle size below which microcracking will not occur was calculated to be 3 mm. Coarse phase MoSi2– Si3N4 composites also exhibit higher room temperature toughness than fine phase material, reaching values of 8 MPam1/2 . Fracture toughness also increases with temperature and the trend is quite significant above

However, the fine phase materials are stronger than the coarse phase materials with bend strengths reaching 1000 MPa. The MoSi2– Si3N4 composites have also been shown to exhibit R-curve behavior, and crack deflection and particle pullout have been observed. Molybdenum disilicide does not have good creep resistance at high temperatures above its brittle-to-ductile transition. When high volume fractions of Si3N4 are added, creep is improved significantly and the activation energy is comparable to monolithic silicon

Additions of carbon can also improve creep resistance as well as toughness. In situ processing with carbon additions have produced material with creep resistance comparable to Ni-base superalloys. The silicon-base composite systems of current interest typically utilize carbon or silicon carbide fibers and silicon nitride or silicon carbide matrices. A popular designation is to display the fiber first followed by the matrix phase, e.g. C/SIC, SiC/SiC, and SiC/Si3N4. Mixtures composed of MoSi2 and Si3N4 form two phase composites that are also candidates as matrices in C or SiC fiber reinforced composite systems. The combination of a fiber reinforced composite with a composite matrix becomes a little

increased use of FRCMCs can be realized.

800°C with toughness values exceeding 10 MPam1/2.

confusing, but can be represented by SiC/ MoSi2– Si3N4.

nitride.

elements have focused attention to the high temperature deformation behavior. The yield stress of MoSi2 increases upon allying with Cr above 1100oC. A similar observation was made for Nb-bearing MoSi2 at 1400oC. However, since these compression experiments were made only at high temperatures above 1100oC, almost nothing is know about the low temperature strength and deformability of these ternary MoSi2 single crystals. V, Cr, Nb and Al that form a disilicide with the C40 structure and W and Re that form a disilicide with the C11b structure as alloying elements to MoSi2, and investigated the deformation behavior of single crystals of MoSi2 containing these elements in a wide temperature range from room temperature to 1500oC. The crystal orientations investigated were the [0 15 1] orientation, in which slip on {110}<111> is operative, and the [001] orientation, in which the highest strength is obtained at high temperatures for binary MoSi2.

Fig. 10. Atomic arrangement on TMSi2 layers corresponding to {110}, (0001) and (001) planes in the C11b, C40 and C54 structures, respectively. The stacking positions of A±D and crystallographic directions with respect to these three structures are indicated (Inui et al., 2000)

## **5. Development of MoSi2 – Si3N4 composites**

Interest in fiber reinforced ceramic matrix composites (FRCMCs) has increased steadily over the past 15 years, and several refined silicon-base composite systems are now being produced commercially. These composites offer very good structural stiffness, high specific strength to weight, and good high temperature environmental resistance. Industrial applications include, hot gas filters, shrouds, and combuster liners. In addition, silicon-base ceramic composites are being considered for gas turbine hot gas flow path components, e.g. combusters transition pieces, and nozzles. The manufacturers of liquid rocket engines are also looking to ceramic composites in hopes of obtaining better efficiency in the next generation of designs. Applications include inlet nozzles, fuel turbopump rotors, injectors, combustion chambers, nozzle throats, and nozzle extensions. In order to maximize properties, materials developers have now begun to pay more attention to engineered interfaces between the matrix and the fiber reinforcement. If the interfacial debonding energy and sliding resistance is low, the fibers can pull away or out of the matrix and form bridges behind the advancing crack front which renders these otherwise brittle materials

elements have focused attention to the high temperature deformation behavior. The yield stress of MoSi2 increases upon allying with Cr above 1100oC. A similar observation was made for Nb-bearing MoSi2 at 1400oC. However, since these compression experiments were made only at high temperatures above 1100oC, almost nothing is know about the low temperature strength and deformability of these ternary MoSi2 single crystals. V, Cr, Nb and Al that form a disilicide with the C40 structure and W and Re that form a disilicide with the C11b structure as alloying elements to MoSi2, and investigated the deformation behavior of single crystals of MoSi2 containing these elements in a wide temperature range from room temperature to 1500oC. The crystal orientations investigated were the [0 15 1] orientation, in which slip on {110}<111> is operative, and the [001] orientation, in which the highest

Fig. 10. Atomic arrangement on TMSi2 layers corresponding to {110}, (0001) and (001) planes in the C11b, C40 and C54 structures, respectively. The stacking positions of A±D and crystallographic directions with respect to these three structures are indicated (Inui et al.,

Interest in fiber reinforced ceramic matrix composites (FRCMCs) has increased steadily over the past 15 years, and several refined silicon-base composite systems are now being produced commercially. These composites offer very good structural stiffness, high specific strength to weight, and good high temperature environmental resistance. Industrial applications include, hot gas filters, shrouds, and combuster liners. In addition, silicon-base ceramic composites are being considered for gas turbine hot gas flow path components, e.g. combusters transition pieces, and nozzles. The manufacturers of liquid rocket engines are also looking to ceramic composites in hopes of obtaining better efficiency in the next generation of designs. Applications include inlet nozzles, fuel turbopump rotors, injectors, combustion chambers, nozzle throats, and nozzle extensions. In order to maximize properties, materials developers have now begun to pay more attention to engineered interfaces between the matrix and the fiber reinforcement. If the interfacial debonding energy and sliding resistance is low, the fibers can pull away or out of the matrix and form bridges behind the advancing crack front which renders these otherwise brittle materials

strength is obtained at high temperatures for binary MoSi2.

**5. Development of MoSi2 – Si3N4 composites** 

2000)

acceptably compliant. Unfortunately, most of the incremental toughening and attended fiber pullout occurs at engineering strains that exceed the strain which occurs at the ultimate strength of the material, and the toughening benefit is, therefore, not useful for design purposes. Thus, the degree of incremental toughening that occurs during the inelastic portion of the stress-strain curve up to the ultimate strength will have to be improved before increased use of FRCMCs can be realized.

It has also been shown that MoSi2 offers the potential of combining the effects of second phase reinforcements with metallurgical alloying to improve mechanical properties without degrading oxidation resistance. For example, high temperature (1200°C) creep was reduced by a factor of 10 by alloying with WSi2 and by another factor of 10–15 with the addition of SiC whiskers. Additions of carbon and zirconia have also proven to be beneficial. Carbon reacts with the oxygen impurities in MoSi2 to improve the toughness at high temperatures by removing SiO2, and leaves behind a compatible SiC phase. Zirconia, which is thermochemically stable with MoSi2, can also be used to increase fracture toughness. a 20 v/o loading of particulate ZrO2 increased the low temperature fracture toughness of MoSi2 by a factor of four. The toughening transformation occurs above the ductile–brittle transition and therefore enhances the low temperature properties. In an attempt to reinforce MoSi2 with SCS-6 silicon fibers, it was discovered that the large thermal expansion difference between the matrix and fiber introduced matrix cracking upon cooling from the densification temperature. This problem was solved by adding Si3N4 to form a two phase composite matrix with a coefficient of expansion that more closely matches the fiber. There is no reaction between Si3N4 and MoSi2, even at fabrication temperatures as high as 1750°C. No gross cracking occurs on cool down, although some microcracking has been observed and is a function of grain size. The critical particle size below which microcracking will not occur was calculated to be 3 mm. Coarse phase MoSi2– Si3N4 composites also exhibit higher room temperature toughness than fine phase material, reaching values of 8 MPam1/2 . Fracture toughness also increases with temperature and the trend is quite significant above 800°C with toughness values exceeding 10 MPam1/2.

However, the fine phase materials are stronger than the coarse phase materials with bend strengths reaching 1000 MPa. The MoSi2– Si3N4 composites have also been shown to exhibit R-curve behavior, and crack deflection and particle pullout have been observed. Molybdenum disilicide does not have good creep resistance at high temperatures above its brittle-to-ductile transition. When high volume fractions of Si3N4 are added, creep is improved significantly and the activation energy is comparable to monolithic silicon nitride.

Additions of carbon can also improve creep resistance as well as toughness. In situ processing with carbon additions have produced material with creep resistance comparable to Ni-base superalloys. The silicon-base composite systems of current interest typically utilize carbon or silicon carbide fibers and silicon nitride or silicon carbide matrices. A popular designation is to display the fiber first followed by the matrix phase, e.g. C/SIC, SiC/SiC, and SiC/Si3N4. Mixtures composed of MoSi2 and Si3N4 form two phase composites that are also candidates as matrices in C or SiC fiber reinforced composite systems. The combination of a fiber reinforced composite with a composite matrix becomes a little confusing, but can be represented by SiC/ MoSi2– Si3N4.

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 111

Upon cooling from the consolidation temperature, the fiber can theoretically contract and debond from the matrix unless there is enough surface roughness for asperity contact. Upon reloading, the matrix will not efficiently transfer stress to the fiber unless it is in intimate contact. However, the high elastic modulus, as indicated by the stress– strain curve, suggests that good load transfer is occurring in this system. Inelastic behavior starts at about 350 MPa, but the ultimate is reached at about 450 MPa which is well below the expected fiber bundle failure. The ultimate strength for the Silcomp matrix composite is on the order of 650 MPa and is in reasonable agreement with bundle fiber failure, as is the MoSi2–

Fig. 12. Tensile stress–strain curves for uniaxial SCS-6 reinforced silicon-base composite

A number of composite approaches have been developed to toughen brittle high temperature structural ceramic materials. Many of these approaches have also been applied

The MoSi2–Si3N4 composite system is an interesting and important one. Si3N4 is considered to be the most important structural ceramic, due to its high strength, good thermal shock resistance, and relatively high (for a structural ceramic) room temperature fracture toughness. Si3N4 and MoSi2 are thermodynamically stable species at elevated temperatures.

**Property MoSi2 Si3N4** Density(g/cm3) 6.2 3.2 Thermal expansion coefficent (10-6/oC) 7.2 3.8 Thermal conductivity (W/mK) 65 37 Melting point(oC) 2030 2100 Creep resistance (0C) 1200 1400 Toughness Low Low Oxidation resistance Good Excellent Structural stability Good Good Intricate machinability Good Difficult Cost Low High Table 2. Some physical and thermal properties of MoSi2 and Si3N4 (Nathesan & Devi, 2000).

0.5Si3N4 matrix composite.

systems (Courtright, 1999).

to high temperature structural silicides.

The matrix properties for several silicon-base composite systems and their fiber properties are presented below. The high coefficient of thermal expansion of SCS-6, i.e. 4.8 x 1-6 °C-1, indicates that this fiber will have a larger expansion coefficient than the matrix phase for SiC/SiC and SiC/ Si3N4 fiber reinforced composite systems. It is generally more difficult to weave and fabricate structural components from large diameter fibers. The size can also influence properties. For example, toughness scales directly with fiber radius while the matrix cracking strength is inversely proportional to the radius. The properties for MoSi2– Si3N4 compare quite favorably with both SiC and Si3N4. The fracture toughness is slightly above the middle range for SiC and comparable to Si3N4. The highest matrix toughness values are on the order of 10 MPam1/2 for in situ toughened silicon nitride. The toughness for all the candidate matrix materials will depend upon processing conditions and microstructure. It is anticipated that in situ toughening of the silicon nitride phase in the two phase MoSi2–Si3N4 composites should yield further improvements for this matrix candidate.

The onset of nonlinear behavior s often found in tension tests marked by a distinct load drop, indicating the initiation of matrix cracking, whereas in flexure, this important feature may go undetected. It is the region between matrix cracking and fiber bundle failure at maximum load where matrix enhancement can make the greatest contribution. Many FRCMC composites exhibit much of their toughness beyond this point, because as the fibers pull away from the matrix, they bridge cracks and impose traction forces that retard crack growth. Pullout toughening extends life after failure, and adds a margin of safety from the catastrophic nature of failure often found in brittle materials, but this phenomenon is not useful as a design property. Three of the composites exhibit matrix cracking stresses in the range of 150–175 MPa. Of the four systems, the SiC/Si3N4 has the highest matrix cracking stress, which is near 350 MPa. This composite also has the largest coefficient of thermal expansion mismatch, CTE, with the fiber having the larger value.

Fig. 11. Stress–strain behaviour for ceramic fiber reinforced: ceramic matrix composites.

The matrix properties for several silicon-base composite systems and their fiber properties are presented below. The high coefficient of thermal expansion of SCS-6, i.e. 4.8 x 1-6 °C-1, indicates that this fiber will have a larger expansion coefficient than the matrix phase for SiC/SiC and SiC/ Si3N4 fiber reinforced composite systems. It is generally more difficult to weave and fabricate structural components from large diameter fibers. The size can also influence properties. For example, toughness scales directly with fiber radius while the matrix cracking strength is inversely proportional to the radius. The properties for MoSi2– Si3N4 compare quite favorably with both SiC and Si3N4. The fracture toughness is slightly above the middle range for SiC and comparable to Si3N4. The highest matrix toughness values are on the order of 10 MPam1/2 for in situ toughened silicon nitride. The toughness for all the candidate matrix materials will depend upon processing conditions and microstructure. It is anticipated that in situ toughening of the silicon nitride phase in the two phase MoSi2–Si3N4 composites should yield further improvements for this matrix candidate. The onset of nonlinear behavior s often found in tension tests marked by a distinct load drop, indicating the initiation of matrix cracking, whereas in flexure, this important feature may go undetected. It is the region between matrix cracking and fiber bundle failure at maximum load where matrix enhancement can make the greatest contribution. Many FRCMC composites exhibit much of their toughness beyond this point, because as the fibers pull away from the matrix, they bridge cracks and impose traction forces that retard crack growth. Pullout toughening extends life after failure, and adds a margin of safety from the catastrophic nature of failure often found in brittle materials, but this phenomenon is not useful as a design property. Three of the composites exhibit matrix cracking stresses in the range of 150–175 MPa. Of the four systems, the SiC/Si3N4 has the highest matrix cracking stress, which is near 350 MPa. This composite also has the largest coefficient of thermal

expansion mismatch, CTE, with the fiber having the larger value.

Fig. 11. Stress–strain behaviour for ceramic fiber reinforced: ceramic matrix composites.

Upon cooling from the consolidation temperature, the fiber can theoretically contract and debond from the matrix unless there is enough surface roughness for asperity contact. Upon reloading, the matrix will not efficiently transfer stress to the fiber unless it is in intimate contact. However, the high elastic modulus, as indicated by the stress– strain curve, suggests that good load transfer is occurring in this system. Inelastic behavior starts at about 350 MPa, but the ultimate is reached at about 450 MPa which is well below the expected fiber bundle failure. The ultimate strength for the Silcomp matrix composite is on the order of 650 MPa and is in reasonable agreement with bundle fiber failure, as is the MoSi2– 0.5Si3N4 matrix composite.

Fig. 12. Tensile stress–strain curves for uniaxial SCS-6 reinforced silicon-base composite systems (Courtright, 1999).

A number of composite approaches have been developed to toughen brittle high temperature structural ceramic materials. Many of these approaches have also been applied to high temperature structural silicides.

The MoSi2–Si3N4 composite system is an interesting and important one. Si3N4 is considered to be the most important structural ceramic, due to its high strength, good thermal shock resistance, and relatively high (for a structural ceramic) room temperature fracture toughness. Si3N4 and MoSi2 are thermodynamically stable species at elevated temperatures.


Table 2. Some physical and thermal properties of MoSi2 and Si3N4 (Nathesan & Devi, 2000).

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 113

MoSi2–Si3N4–SiC hybrid discontinuous particle-continuous fiber composites have been developed with excellent room temperature fracture toughness, thermal shock resistance, and thermo-mechanical impact behavior. These hybrid composites consist of MoSi2–Si3N4 particulate composites which form the matrix for SiC continuous fibers. The MoSi2–Si3N4 portion of the hybrid composites has two functions. First, additions of 30-50.% Si3N4 to the MoSi2 completely eliminates the oxidation pest behavior at the intermediate 500°C temperature. Second, the Si3N4 addition aids to match the thermal expansion coefficient of the matrix to that of the SiC fibers. This prevents thermal expansion coefficient mismatch

Figure 14.(a) shows a SEM back scattered image of a fully dense MoSi2-βSi3N4 composite (MS-70). During processing, the original α-Si3N4 powder particles are transformed into randomly oriented whiskers of β-Si3N4. These long whiskers are well dispersed throughout the material and appear to be quite stable, with very little or no reaction with the MoSi2, even at 1900oC. In some isolated areas, the Mo5Si3 phase is visible. Figure 14.(b) shows a back scattered image of MoSi2-βSi3N4 (MS-80) with the β-Si3N4 exhibiting a blocky

 Fig. 14.(a) randomly oriented in-situ grown long whiskers of β-Si3N4 and large MoSi2

Density of (MS-70) is 4.57±0.01g/cm3 and Vickers microhardness is 10.7±0.6GPa. Figure 15. shows the coefficient of thermal expansion as a function of temperature for (MS-70). From this data the average coefficient for expansion of this composite material is about

particle size, (b) Si3N4 has a blocky particulate structure (Hebsur et al., 2001).

Fig. 15. The coefficient of linear expansion for MoSi2-βSi3N4 (Hebsur et al., 2001).

cracking in the hybrid composite matrix.

aggregate-type morphology.

4.0ppm/oC.


When composites were synthesized with elongated Si3N4 grains toughness can reach to 15 MPa m1/2 (Nathal & Hebsur, 1997).

Fig. 3. Microstructures of different MoSi2– Si3N4 composites.

However, a drawback of transformation toughening is that toughness decreases with increasing temperature, due to the thermodynamics of the phase transformation. Discontinuously reinforced ceramic composites have typically employed ceramic whiskers or particles as the reinforcing phases. An example is SiC whisker reinforced Si3N4. Toughening mechanisms here are crack deflection and crack bridging. Discontinuous ceramic composites can reach toughness levels of 10 MPam1/2. One important variant of this approach is the in-situ toughening of Si3N4 due to the presence of elongated Si3N4 grains. By way of comparison to structural ceramics, the room temperature fracture toughness of polycrystalline MoSi2 is approximately 3 MPam1/2, while the room temperature fracture toughness of equiaxed polycrystalline Si3N4 which is densified without densification aids is also 3 MPam1/2 (Petrovic, 2000). For comparison, two monolithic ceramics SiC and Si3N4 are also included in the figure. Further improvement in room temperature fracture can be achieved by microalloying MoSi2 with elements like Nb, Al an Mg or by randomly oriented long whisker type β-Si3N4 grains (Hebsur, 1999).

Fig. 13. Temperature dependence of fracture toughness of MoSi2-based materials compared with ceramic matrices (Hebsur, 1999).

When composites were synthesized with elongated Si3N4 grains toughness can reach to 15

However, a drawback of transformation toughening is that toughness decreases with increasing temperature, due to the thermodynamics of the phase transformation. Discontinuously reinforced ceramic composites have typically employed ceramic whiskers or particles as the reinforcing phases. An example is SiC whisker reinforced Si3N4. Toughening mechanisms here are crack deflection and crack bridging. Discontinuous ceramic composites can reach toughness levels of 10 MPam1/2. One important variant of this approach is the in-situ toughening of Si3N4 due to the presence of elongated Si3N4 grains. By way of comparison to structural ceramics, the room temperature fracture toughness of polycrystalline MoSi2 is approximately 3 MPam1/2, while the room temperature fracture toughness of equiaxed polycrystalline Si3N4 which is densified without densification aids is also 3 MPam1/2 (Petrovic, 2000). For comparison, two monolithic ceramics SiC and Si3N4 are also included in the figure. Further improvement in room temperature fracture can be achieved by microalloying MoSi2 with elements like Nb, Al an Mg or by randomly oriented

Fig. 13. Temperature dependence of fracture toughness of MoSi2-based materials compared

MS-60 Fully dense β-Si3 N4, with long whisker-type morphology MS-70 Fully dense β- Si3 N4, with long whisker-type morphology

MS-40 Not fully dense α- Si3 N4, fine grained MoSi2 and blocky α- Si3 N4

MS-80 Not fully dense β- Si3 N4, with blocky morphology MS-50 Fully dense α- Si3 N4, with blocky morphology

Fig. 3. Microstructures of different MoSi2– Si3N4 composites.

long whisker type β-Si3N4 grains (Hebsur, 1999).

with ceramic matrices (Hebsur, 1999).

MPa m1/2 (Nathal & Hebsur, 1997).

Designation Microstructure

MoSi2–Si3N4–SiC hybrid discontinuous particle-continuous fiber composites have been developed with excellent room temperature fracture toughness, thermal shock resistance, and thermo-mechanical impact behavior. These hybrid composites consist of MoSi2–Si3N4 particulate composites which form the matrix for SiC continuous fibers. The MoSi2–Si3N4 portion of the hybrid composites has two functions. First, additions of 30-50.% Si3N4 to the MoSi2 completely eliminates the oxidation pest behavior at the intermediate 500°C temperature. Second, the Si3N4 addition aids to match the thermal expansion coefficient of the matrix to that of the SiC fibers. This prevents thermal expansion coefficient mismatch cracking in the hybrid composite matrix.

Figure 14.(a) shows a SEM back scattered image of a fully dense MoSi2-βSi3N4 composite (MS-70). During processing, the original α-Si3N4 powder particles are transformed into randomly oriented whiskers of β-Si3N4. These long whiskers are well dispersed throughout the material and appear to be quite stable, with very little or no reaction with the MoSi2, even at 1900oC. In some isolated areas, the Mo5Si3 phase is visible. Figure 14.(b) shows a back scattered image of MoSi2-βSi3N4 (MS-80) with the β-Si3N4 exhibiting a blocky aggregate-type morphology.

Fig. 14.(a) randomly oriented in-situ grown long whiskers of β-Si3N4 and large MoSi2 particle size, (b) Si3N4 has a blocky particulate structure (Hebsur et al., 2001).

Density of (MS-70) is 4.57±0.01g/cm3 and Vickers microhardness is 10.7±0.6GPa. Figure 15. shows the coefficient of thermal expansion as a function of temperature for (MS-70). From this data the average coefficient for expansion of this composite material is about 4.0ppm/oC.

Fig. 15. The coefficient of linear expansion for MoSi2-βSi3N4 (Hebsur et al., 2001).

Potential of MoSi2 and MoSi2-Si3N4 Composites for Aircraft Gas Turbine Engines 115

Soetching, F.O. (1995). A Design Perspective on Thermal Barrier Coatings. *Proceedings of a* 

Misra, A.; Sharif, A.A. & Petrovic, J.J. (2000). Rapid Solution Hardening at Elevated

Akkus, I. (1999). The Aluminide Coating of Superalloys with Pack Cementation Method. *Journal of Institution of Science Osmangazi University*, Vol. 18, pp. 27-28 Meschter, P.J. (1992). Low Temperature Oxidation of Molybdenum Disilicide, *Metallurg.* 

Liu, Q.; Shao G. & Tsakiropoulos, P. (2001). On the Oxidation Behaviour of MoSi2.

Chou, T.C. & Nieh, T.G. (1992). New Observation of MoSi2 Pest at 500°C. *Script. Metallurg.* 

Chou, T.C. & Nieh, T.G. (1993). Pesting of the High Temperature Intermetallic MoSi2. *Journal* 

Wang, G.; Jiang, W. & Bai, G. (2003). Effect of Addition of Oxides on Low-Temperature

Chen, J.; Li, C.; Fu, Z.; Tu, X.; Sundberg, M. & Pompe, R. (1999). Low Temperature Oxidation

Sharif, A.A.; Misra, A. & Petrovic, J.J. (2001). Rapid Solution Hardening at Elevated

Waghmare, U.V.; Bulatov, V. & Kasiras, E. (1999). Microalloying for Ductility in

Mitra, R.; Sadananda, K. & Feng, C.R. (2004). Effect of microstructural parameters and Al

Inui, H.; Ishikawa, K. & Yamaguchi, M. (2000). Effects of Alloying Elements on Plastic Deformation of Single Crystals of MoSi2. *Intermetallics*, Vol. 8, pp. 1131-1145 Courtright, E.R. (1999). A Comparison of MoSi2 Matrix Composites with Other Silicon-Base Composite Systems. *Materials Science and Engineering*, Vol. A261, pp. 53-63 Natesan, K. & Deevi, S.C. (2000). Oxidation Behaviour of Molybdenum Disilicides and Their

Nathal, M.V. & Hebsur, M.G. (1997). Strong, Tough, and Pest-Resistant MoSi2-Base Hybrid

Petrovic, J.J. (2000). Toughening Strategies for MoSi2-Based High Temperature Structural

Hebsur, M.G. (1999). Development and Characterization of SiC(f)/MoSi2-Si3N4(p) Hybrid

Hebsur, M.G.; Choi, S.R.; Whittenberger, J.D.; Salem, J.A. & Noebe, R.D. (2001).

Composites. *Material Science and Engineering*, Vol. A261, pp. 24-37

Composite for Structural Applications. *Structural Intermetallics*, Warrendale (USA):

Development of Tough, Strong, and Pest-Resistant MoSi2-βSi3N4 Composites for

Molybdenum Disilicides. *Intermetallics*, Vol. 12, pp. 827-836

Composites. *Intermetallics*, Vol. 8, pp. 1147-1158

Silicides, *Intermetallics*, Vol. 8, pp. 1175-1182

Oxidation of Molybdenum Disilicide. *Journal of American Ceramic Society*, Vol. 86,

Behaviour of MoSi2Bbased Material. *Materials Science and Engineering*, Vol. A26, pp.

Temperatures by Substitional Re Alloying in MoSi2, *Acta Mater*, Vol. 48, pp. 925-

Molybdenum Disilicide, *Materials Science and Engineering*, Vol. A261, pp. 147-

Alloying on Creep Behavior, Threshold Stress and Activation Volumes of

Temperatures by Substitional Re Alloying in MoSi2, *Acta Mater*, Vol.48, pp. 925-

*Conference at NASA Lewis Research Center*, September 1994

Bradley, E.F. (1988). *Superalloys a Technical Guide*, Metals Park, Ohio

*Trans. A*, Vol. 23A, pp. 1763-1772

*Intermetallics*, Vol. 8, pp. 1147-1158

*Mater.*, Vol. 26, pp. 1637-1642

*of Materials*, Vol. 30, pp. 15-22

pp. 731-734

239-244

932

157

TMS, pp. 949-953

932

The oxidation behaviour of a MoSiB alloy is also included for comparison. 500oC is the temperature for maximum accelerated oxidation and pest for MoSi2-base alloys. There is interest in this alloy, over MoSi2, for structural aerospace applications due to its attractive high temperature oxidation resistance (Bose, 1992; Berczik, 1997).

Fig. 16. Specific weight gain versus number of cycles of (MS-70) at 500oC (Hebsur, 2001).

However, the (MS-70) shows very little weight gain compared to binary MoSi2 and the MoSiB alloy, indicating the absence of accelerated oxidation. In contrast the binary MoSi2 and MoSiB alloys exhibits accelerated oxidation followed by pesting.

## **6. Conclusion**

Based on the cyclic oxidation properties at 900oC, the family of MoSi2-Si3N4 composites show promise for aircraft applications. The composites do not exhibit the phenomena of pesting, and the weight gain after 500h is negligible and superior to base line hybrid composites.

A wide spectrum of mechanical and environmental properties have been measured in order to establish feasibility of an MoSi2 composite with Si3N4 particulate. The high impact resistance of the composite is of particular note, as it was a key property of interest for engine applications. Processing issues have also been addressed in order to lower cost and improve shape making capability. These results indicate that this composite system remains competitive with other ceramics as potential replacement for superalloys.

## **7. References**

Vaseduvan, A.K. & Petrovic, J.J. (1992). A Comparative Overview of Molybdenum Disilicide Composites. *Materials Science and Engineering*, A155, pp. 1-17

Tein, J. K. & Caulfield, T. (1989). *Superalloys, Supercomposites, and Superceramics*, Boston Academic Press, New York

The oxidation behaviour of a MoSiB alloy is also included for comparison. 500oC is the temperature for maximum accelerated oxidation and pest for MoSi2-base alloys. There is interest in this alloy, over MoSi2, for structural aerospace applications due to its attractive

Fig. 16. Specific weight gain versus number of cycles of (MS-70) at 500oC (Hebsur, 2001).

and MoSiB alloys exhibits accelerated oxidation followed by pesting.

competitive with other ceramics as potential replacement for superalloys.

Composites. *Materials Science and Engineering*, A155, pp. 1-17

**6. Conclusion** 

composites.

**7. References** 

Academic Press, New York

However, the (MS-70) shows very little weight gain compared to binary MoSi2 and the MoSiB alloy, indicating the absence of accelerated oxidation. In contrast the binary MoSi2

Based on the cyclic oxidation properties at 900oC, the family of MoSi2-Si3N4 composites show promise for aircraft applications. The composites do not exhibit the phenomena of pesting, and the weight gain after 500h is negligible and superior to base line hybrid

A wide spectrum of mechanical and environmental properties have been measured in order to establish feasibility of an MoSi2 composite with Si3N4 particulate. The high impact resistance of the composite is of particular note, as it was a key property of interest for engine applications. Processing issues have also been addressed in order to lower cost and improve shape making capability. These results indicate that this composite system remains

Vaseduvan, A.K. & Petrovic, J.J. (1992). A Comparative Overview of Molybdenum Disilicide

Tein, J. K. & Caulfield, T. (1989). *Superalloys, Supercomposites, and Superceramics*, Boston

high temperature oxidation resistance (Bose, 1992; Berczik, 1997).


**Part 2** 

**Aircraft Control Systems** 

High Temperature Structural Applications, *International Symposium on Structural Intermetallics*, NASA, 2001


**Part 2** 

**Aircraft Control Systems** 

116 Recent Advances in Aircraft Technology

Berczik, D.M. (1997). Oxidation Resistant Molybdenum Alloy, U.S. Patent, No. 5, 696,

*Intermetallics*, NASA, 2001

150

Bose, S. (1992). *High Temperature Silicides*, North-Holland, NY

High Temperature Structural Applications, *International Symposium on Structural* 

**1. Introduction**

of aerodynamics.

special flight conditions are realized,

(5) testing whether the aircraft specification is compliant.

Development of efficient parameter identification methods for the model of a dynamic system based on real-time measurements of some components of its state vector should be taken as one of the most important problems of applied statistics and computational mathematics. Calculating the motion of the system given the initial conditions and its mathematical model is conventionally called the direct problem of dynamics. Then, the inverse problem of dynamics would be the problem of identifying the system model parameters based on measurements of certain components of the state vector provided that the general structural scheme of the model is known from physical considerations. Such an inverse problem corresponds to identification problem for the dynamic system representing an aircraft. In this case, the general structural scheme of the model (motion equations) follows from the fundamental laws

**An Algorithm for Parameters Identification of** 

*State Institute of Aviation Systems, Moskow Physical Technical Institute* 

**an Aircraft's Dynamics\***

I. A. Boguslavsky

*Russia* 

**6**

In many cases, modern computational methods and wind tunnel experiments can provide sufficient data on nominal parameters of the mathematical model - nominal aerodynamic characteristics of the aircraft. Nevertheless, there exist problems [1] that require correcting

(1) verifying and interpreting theoretical predictions and results of wind tunnel experiments

(2) obtaining more exact and complete mathematical models of the aircraft dynamics to be

(3) designing flight simulators that require more accurate dynamic aircraft profile in all flight modes (many motions of aircrafts and flight conditions can be neither reconstructed in the

(4) extending the range of flight modes for new aircrafts, which can include quantitative determination of stability and impact of control when the configuration is changed or when

nominal parameters based on measurements taken in real flights. These imply

applied in designing stability enhancement methods and flight control systems,

wind tunnel nor calculated analytically up to sufficient accuracy or efficiency),

\*This work has been supported by the Russian Foundation for Basic Research.

(flight data can also be used to improve ground prediction methods),

## **An Algorithm for Parameters Identification of an Aircraft's Dynamics\***

I. A. Boguslavsky *State Institute of Aviation Systems, Moskow Physical Technical Institute Russia* 

#### **1. Introduction**

Development of efficient parameter identification methods for the model of a dynamic system based on real-time measurements of some components of its state vector should be taken as one of the most important problems of applied statistics and computational mathematics. Calculating the motion of the system given the initial conditions and its mathematical model is conventionally called the direct problem of dynamics. Then, the inverse problem of dynamics would be the problem of identifying the system model parameters based on measurements of certain components of the state vector provided that the general structural scheme of the model is known from physical considerations. Such an inverse problem corresponds to identification problem for the dynamic system representing an aircraft. In this case, the general structural scheme of the model (motion equations) follows from the fundamental laws of aerodynamics.

In many cases, modern computational methods and wind tunnel experiments can provide sufficient data on nominal parameters of the mathematical model - nominal aerodynamic characteristics of the aircraft. Nevertheless, there exist problems [1] that require correcting nominal parameters based on measurements taken in real flights. These imply

(1) verifying and interpreting theoretical predictions and results of wind tunnel experiments (flight data can also be used to improve ground prediction methods),

(2) obtaining more exact and complete mathematical models of the aircraft dynamics to be applied in designing stability enhancement methods and flight control systems,

(3) designing flight simulators that require more accurate dynamic aircraft profile in all flight modes (many motions of aircrafts and flight conditions can be neither reconstructed in the wind tunnel nor calculated analytically up to sufficient accuracy or efficiency),

(4) extending the range of flight modes for new aircrafts, which can include quantitative determination of stability and impact of control when the configuration is changed or when special flight conditions are realized,

(5) testing whether the aircraft specification is compliant.

<sup>\*</sup>This work has been supported by the Russian Foundation for Basic Research.

The regression method given in [1] solves this problem under the following limitations

(2) at the measurement instants *tk*, the algorithm constructs the estimate of the vector of

An Algorithm for Parameters Identification of an Aircraft's Dynamics 121

These fundamental limitations of the regression method duplicate features of the identification algorithm from [5]. The substantial drawback of the algorithm [5] and the algorithm of the regression method is that they do not allow using the mathematical model to analyze theoretically (without applying the Monte-Carlo method) observability conditions of components of the vector of parameters to be identified for the preliminary given control law for the test flight of the aircraft and information on random errors of its sensors. Note that this is the drawback of all known numerical methods that solve nonlinear identification problems. Relations (0.1) and (0.2) show that when conditions (1)-(3) are met and *N* is sufficiently big, the estimation vector satisfies the overdetermind system of linear algebraic equations, with methods to solve it being well known. The given conditions seem to be rather rigid and may be hard-to-implement. For instance, it is arguable whether one can construct the vector of derivatives dx/dt sufficiently accurately given the real turbulent atmosphere conditions, which imply that the outputs of the angle of attack and sideslip sensors inevitably include

All this justifies the development of new identification algorithms that can be applied to dynamic systems of a rather general class and do not possess drawbacks of NASA algorithms. The proposed multipolynomial approximation algorithm (MPA algorithm) serves as such a

The general scheme for identifying aerodynamic characteristics of the aircraft by the test flight data is as follows [1]. Motion equations of the aircraft (0.1) and system (0.2) of measurements of motion characteristics of the aircraft are given. The vector *ϑ* is the vector of nominal aerodynamic parameters determined in the wind tunnel experiment. Calculated by the results

When the aircraft flies, its computer fixes the digital array of initial conditions and time functions, viz. current control surface angles and measurements of some motion parameters of the aircraft (some components of the vector x(t) of the state of the aircraft) received from its sensors. Note that selecting the criterion for optimal or, at least, rational mode to control the test flight is a separate problem and lies beyond our further consideration. The current motion characteristics measured as the time function such as angles of attack and sideslip and components of the vector of angular velocity and g-load obtained by the inertial system of the aircraft are registered for real (not known for sure) aerodynamic parameters of the aircraft (parameters *ϑ* + *η*) and can be called measured characteristics of the perturbed motion.

**2. Statement of the problem and basic scheme of the proposed identification**

(1) all components of the state vector can be measured : *yk* = *x*(*tk*) + *ξk*,

(3) the vector function *f*(*x*, *ϑ* + *η*, *u*) linearly depends on the vector *η*.

random and unpredictable frequency components.

of real (test) flight, the vector *η* is used to correct the vector *ϑ*.

new identification algorithm.

**algorithm**

derivatives *dx*/*dt*,

Furthermore, dimensionless numbers at the nodes of one-or two-dimensional tables found in wind tunnel experiments serve as nominal values in the aerodynamic parameter identification problem of the aircraft. This causes the vector that corrects these parameters determined by the algorithm processing digital data flows received from the aircraft sensors to have a significant dimension of the order about several tens or hundreds.

It is worth noting that the USA (NASA) is doing extensive work on theoretical and practical aircraft identification by test flights. In 2006 alone, in addition to many journal publications, American Institute of Aeronatics and Astronatics (AIAA) published three fundamental monographs [1-3] on the subject. An implementation of multiple NASA recommended algorithms for identification problems, SIDPAS (Systems Identification Programs for Aircraft) software package written in MATLAB M-files language is available on the Internet as an appendix to [1]. Various existing identification methods published in monographs on statistics and computational mathematics are widely reviewed in [1].

For the most general identification method, one should take the known nonlinear least squares method [4] that forms the sum of errors squared - differences between the real measurements and their calculated analogues obtained by numerical integration of motion equations of the system for some realization of the vector of unknown parameters.

Successful identification yields the vector of parameters that delivers the global minimum to the above mentioned sum of errors squared. Still, this criterion is statistically valid only for linear identification problems, in which measurements are linear with respect to the unknown vector of parameters.

Implementing the nonlinear least squares method to correct nominal parameters of the aircraft based on its test flight data involves computational challenges. These arise when the dimension of the correction vector is big and the sum of errors squared as the function of the correction vector has multiple relative minimums or when variations of the Newton's method are applied, with the sequence of local linearizations performed to find stationary points of this function. In [1], the regression method supported by *lesq.m, smoo.m, derive.m, and xstep.m* files in SIDPAS is recommended for practical applications.

Suppose the motion equations of the system and the sequence of measurements have the form

$$d\mathbf{x}/dt = f(\mathbf{x}, \theta + \eta, \mu)\_{\prime} ... (0.1)$$

$$y\_k = H\_k(\mathfrak{x}(t\_k)) + \mathfrak{f}\_{k\prime} \dots (0.2)$$

where *x*(*tk*) is the *n* × 1-dimensional vector of the system states at the current instant *t* and at the given instants *tk*, *k* = 1, ..., *N*, *ϑ* is the *r* × 1-vector of nominal (known) parameters of the system, *η* is the vector of unknown parameters that serves as the correction vector for the nominal vector *ϑ* after the results of measurements are stochastically processed, *u* is the control vector of the system, *f*(...) is the given vector-function, *yk* is the sequence of vectors-results of measurements, *Hk*(...) is the given vector function, and *ξk*, *k* = 1, ..., *N* is the sequence of random vectors-errors of measurements with the given random generator for the mathematical simulation.

We can state the identification problem for the vector *η* as follows. Find the estimate as the function of the vector *YN* formed of the results of all measurements *y*1, ...*yN*.

2 Will-be-set-by-IN-TECH

Furthermore, dimensionless numbers at the nodes of one-or two-dimensional tables found in wind tunnel experiments serve as nominal values in the aerodynamic parameter identification problem of the aircraft. This causes the vector that corrects these parameters determined by the algorithm processing digital data flows received from the aircraft sensors to have a

It is worth noting that the USA (NASA) is doing extensive work on theoretical and practical aircraft identification by test flights. In 2006 alone, in addition to many journal publications, American Institute of Aeronatics and Astronatics (AIAA) published three fundamental monographs [1-3] on the subject. An implementation of multiple NASA recommended algorithms for identification problems, SIDPAS (Systems Identification Programs for Aircraft) software package written in MATLAB M-files language is available on the Internet as an appendix to [1]. Various existing identification methods published in monographs on statistics

For the most general identification method, one should take the known nonlinear least squares method [4] that forms the sum of errors squared - differences between the real measurements and their calculated analogues obtained by numerical integration of motion equations of the

Successful identification yields the vector of parameters that delivers the global minimum to the above mentioned sum of errors squared. Still, this criterion is statistically valid only for linear identification problems, in which measurements are linear with respect to the unknown

Implementing the nonlinear least squares method to correct nominal parameters of the aircraft based on its test flight data involves computational challenges. These arise when the dimension of the correction vector is big and the sum of errors squared as the function of the correction vector has multiple relative minimums or when variations of the Newton's method are applied, with the sequence of local linearizations performed to find stationary points of this function. In [1], the regression method supported by *lesq.m, smoo.m, derive.m, and xstep.m*

Suppose the motion equations of the system and the sequence of measurements have the form

*dx*/*dt* = *f*(*x*, *ϑ* + *η*, *u*), ...(0.1)

*yk* = *Hk*(*x*(*tk*)) + *ξk*, ...(0.2) where *x*(*tk*) is the *n* × 1-dimensional vector of the system states at the current instant *t* and at the given instants *tk*, *k* = 1, ..., *N*, *ϑ* is the *r* × 1-vector of nominal (known) parameters of the system, *η* is the vector of unknown parameters that serves as the correction vector for the nominal vector *ϑ* after the results of measurements are stochastically processed, *u* is the control vector of the system, *f*(...) is the given vector-function, *yk* is the sequence of vectors-results of measurements, *Hk*(...) is the given vector function, and *ξk*, *k* = 1, ..., *N* is the sequence of random vectors-errors of measurements with the given random generator for the

We can state the identification problem for the vector *η* as follows. Find the estimate as the

function of the vector *YN* formed of the results of all measurements *y*1, ...*yN*.

significant dimension of the order about several tens or hundreds.

and computational mathematics are widely reviewed in [1].

files in SIDPAS is recommended for practical applications.

vector of parameters.

mathematical simulation.

system for some realization of the vector of unknown parameters.

The regression method given in [1] solves this problem under the following limitations

(1) all components of the state vector can be measured : *yk* = *x*(*tk*) + *ξk*,

(2) at the measurement instants *tk*, the algorithm constructs the estimate of the vector of derivatives *dx*/*dt*,

(3) the vector function *f*(*x*, *ϑ* + *η*, *u*) linearly depends on the vector *η*.

These fundamental limitations of the regression method duplicate features of the identification algorithm from [5]. The substantial drawback of the algorithm [5] and the algorithm of the regression method is that they do not allow using the mathematical model to analyze theoretically (without applying the Monte-Carlo method) observability conditions of components of the vector of parameters to be identified for the preliminary given control law for the test flight of the aircraft and information on random errors of its sensors. Note that this is the drawback of all known numerical methods that solve nonlinear identification problems.

Relations (0.1) and (0.2) show that when conditions (1)-(3) are met and *N* is sufficiently big, the estimation vector satisfies the overdetermind system of linear algebraic equations, with methods to solve it being well known. The given conditions seem to be rather rigid and may be hard-to-implement. For instance, it is arguable whether one can construct the vector of derivatives dx/dt sufficiently accurately given the real turbulent atmosphere conditions, which imply that the outputs of the angle of attack and sideslip sensors inevitably include random and unpredictable frequency components.

All this justifies the development of new identification algorithms that can be applied to dynamic systems of a rather general class and do not possess drawbacks of NASA algorithms. The proposed multipolynomial approximation algorithm (MPA algorithm) serves as such a new identification algorithm.

## **2. Statement of the problem and basic scheme of the proposed identification algorithm**

The general scheme for identifying aerodynamic characteristics of the aircraft by the test flight data is as follows [1]. Motion equations of the aircraft (0.1) and system (0.2) of measurements of motion characteristics of the aircraft are given. The vector *ϑ* is the vector of nominal aerodynamic parameters determined in the wind tunnel experiment. Calculated by the results of real (test) flight, the vector *η* is used to correct the vector *ϑ*.

When the aircraft flies, its computer fixes the digital array of initial conditions and time functions, viz. current control surface angles and measurements of some motion parameters of the aircraft (some components of the vector x(t) of the state of the aircraft) received from its sensors. Note that selecting the criterion for optimal or, at least, rational mode to control the test flight is a separate problem and lies beyond our further consideration. The current motion characteristics measured as the time function such as angles of attack and sideslip and components of the vector of angular velocity and g-load obtained by the inertial system of the aircraft are registered for real (not known for sure) aerodynamic parameters of the aircraft (parameters *ϑ* + *η*) and can be called measured characteristics of the perturbed motion.

induction.

values.

�*ηWN*(*d*)�.

which are all possible values *ya*<sup>1</sup>

realizations of random vectors *η* and *xik*.

of components of the vector *WN*<sup>1</sup> (*d*)

by the vector *<sup>η</sup>*(*WN*(*d*)).

*CV*(*d*, *<sup>N</sup>*) = *<sup>E</sup>*((*V*(*d*, *<sup>N</sup>*) <sup>−</sup> *<sup>V</sup>*¯(*d*, *<sup>N</sup>*))(*V*(*d*, *<sup>N</sup>*) <sup>−</sup> *<sup>V</sup>*¯(*d*, *<sup>N</sup>*))*T*) .

*<sup>η</sup>*(*WN*(*d*)) = <sup>∑</sup> *<sup>a</sup>*1+...+*aN* <sup>≤</sup>*<sup>d</sup>*

particular problem described by Eqs. (1) and (2).

vectors *η* and *ξ<sup>k</sup>* allowed by the a priori conditions.

*m*(*d*, *N*) = *m*(*d* − 1, *N*)+(*N* + *d* − 1) ··· *N*/*d*!, *m*(1, *N*) = *N*. We obtain the vector *WN*(*d*) of dimension *m*(*d*, *N*) × 1, the components *w*1, ..., *wm*(*d*, *N*) of

An Algorithm for Parameters Identification of an Aircraft's Dynamics 123

Then, we construct the base vector *V*(*d*, *N*) of dimension (*r* + *m*(*d*, *N*)) × 1, *V*(*d*, *N*) =

Step 2. We use a known statistical generator of random vectors *η* and *ξ<sup>k</sup>* to solve repeatedly the Cauchy problem for *Eq*.(1) for given initial conditions *x*(0), a control law *u*(*t*) and various

We apply the Monte-Carlo method to find the prior first and second statistical moments of the vector *V*(*d*, *N*), i.e., the mathematical expectation *V*¯(*d*, *N*), and the covariance matrix

Implementation of step 2 is a learning process for the algorithm, adjusting it to solve the

Step 3. For given *<sup>d</sup>* and *<sup>N</sup>* and a fixed vector *YN*, we assign the vector *<sup>η</sup>*(*WN*(*d*)) to be the solution to the estimation problem. This vector gives an approximate estimate of the vector *E*(*η*|*YN*) that is optimal in the root-mean-square sense on the set of vector linear combinations

The vector *V*¯(*d*, *N*) and the matrix *CV*(*d*, *N*) are the initial conditions for the process of recurrent calculations that realizes the principle of observation decomposition [6] and consists of *m*(*d*, *N*) steps. Once the final step is performed, we obtain vector coefficients *λ*(*a*1, ..., *aN*) for (1.1). Moreover, we determine the matrix *C*(*d*, *N*), which is the covariance matrix of the estimation errors for the vector *E*(*ηN*|*YN*) of conditional mathematical expectation estimated

Calculating the elements of the matrix *C*(*d*, *N*), we have the method of preliminary (prior to the actual flight) analysis of observability of identified parameters for the given control law, structure of measurements and their expected random errors. Recurrent calculations do not require matrix inversion and indicate the situations when the next component of the vector *WN*(*d*) is close to linear combination of its previous components. To implement the recursion, we process the components of the vector *WN*(*d*) one after another. However, the adjustment of the algorithm performed by applying the Monte-Carlo method to find the vector *V*¯(*d*, *N*) and the matrix *CV*(*d*, *N*) takes into account a priori ideas on stochastic structure of components of the whole set of possible vectors *WN*(*d*) that can appear in any realizations of the random

This adjustment is the price we have to pay if we want the MPA algorithm to solve nonlinear identification problems efficiently. This is what makes the MPA algorithm differ fundamentally from, for instance, the standard Kalman filter designed to solve linear

*<sup>λ</sup>*(*a*1, ..., *aN*)*ya*<sup>1</sup>

<sup>1</sup> ··· *<sup>y</sup>aN*

*<sup>N</sup>* . (1.1)

*<sup>N</sup>* of the form that represent the powers of measurable

<sup>1</sup> ...*yan*

Once the flight under the mentioned (given) initial conditions and time functions (control surface angles) is completed, nominal motion equations (equations of form (1) for *η* = 0) are integrated numerically for the nominal aerodynamic parameters of the aircraft. For the calculated characteristics of the nominal motion of the aircraft one should take the obtained data - components of the state vector of the aircraft as the function of discrete time. Differences between measurable characteristics of the perturbed motion and calculated characteristics of the nominal motion serve as carriers of data on the unknown vector *η* that shows the difference between real and nominal aerodynamic parameters.

The input of the MPA identification algorithm receives the vector of initial conditions and control surface angles as functions of time and arrays of characteristics of nominal and perturbed motions.

The output of the algorithm is *η*ˆ(*YN*) , which is the correction vector for nominal aerodynamic parameters.

The identification algorithm is efficient if the motion equations integrated numerically with the corrected aerodynamic parameters yield such motion characteristics *ϑ* + *η*ˆ(*YN*) (*corrected* characteristics, in what follows) that are close to real (measurable) characteristics.

In this work, we consider the technology of applying the Bayes MPA algorithm [6, 7] to solve identification problems on the example of the aircraft, for which nominal aerodynamic parameters of the pitching motion are the nominal parameters of one of an "pseudo" F-16 aircraft.

We replace real flights by mathematical simulation, with characteristics of the perturbed motion obtained by integrating the motion equations of the aircraft numerically. In these equations, nominal aerodynamic parameters at the nodes of the corresponding tables are changed to random values that do not exceed in modulus the given 25 ÷ 50 percents of nominal values at these nodes.

Fundamentally, the MPA algorithm assumes that the vector of unknown parameters *η* is random on the set of possible flights. We assume that the a priori statistical-generator for computer generated random vectors *η* and *ξ<sup>k</sup>* is given. This generator makes the algorithm estimating components of the vector *η* (the identification algorithm) Bayesian. Further, for particular calculations, we assume that random components of the mentioned vectors are distributed uniformly and can be called by the standard Random program in Turbo Pascal.

The MPA algorithm provides the approximation method we implement with the multidimensional power series of the vector *E*(*η*|*YN*) of the conditional mathematical expectation of the vector *η* if the vector of measurements *YN* is fixed and a priori statistical data on random vectors *η* and *ξ<sup>k</sup>* are given.

The vector *E*(*η*|*YN*) is known to be optimal, in root-mean-square sense, estimate of the random vector *η*.

We describe the steps of operation of the MPA algorithm when it identifies the vector *η*[6, 7].

Step 1. Suppose *d* is a given positive integer number and the set of integer numbers *a*1, ..., *aN* consists of all nonnegative solutions of the integer inequality *a*<sup>1</sup> + ... + *aN* ≤ *d*, the number of which we denote by *m*(*d*, *N*). The value *m*(*d*, *N*) is given by the recurrent formula proved by induction.

4 Will-be-set-by-IN-TECH

Once the flight under the mentioned (given) initial conditions and time functions (control surface angles) is completed, nominal motion equations (equations of form (1) for *η* = 0) are integrated numerically for the nominal aerodynamic parameters of the aircraft. For the calculated characteristics of the nominal motion of the aircraft one should take the obtained data - components of the state vector of the aircraft as the function of discrete time. Differences between measurable characteristics of the perturbed motion and calculated characteristics of the nominal motion serve as carriers of data on the unknown vector *η* that shows the

The input of the MPA identification algorithm receives the vector of initial conditions and control surface angles as functions of time and arrays of characteristics of nominal and

The output of the algorithm is *η*ˆ(*YN*) , which is the correction vector for nominal aerodynamic

The identification algorithm is efficient if the motion equations integrated numerically with the corrected aerodynamic parameters yield such motion characteristics *ϑ* + *η*ˆ(*YN*) (*corrected*

In this work, we consider the technology of applying the Bayes MPA algorithm [6, 7] to solve identification problems on the example of the aircraft, for which nominal aerodynamic parameters of the pitching motion are the nominal parameters of one of an "pseudo" F-16

We replace real flights by mathematical simulation, with characteristics of the perturbed motion obtained by integrating the motion equations of the aircraft numerically. In these equations, nominal aerodynamic parameters at the nodes of the corresponding tables are changed to random values that do not exceed in modulus the given 25 ÷ 50 percents of

Fundamentally, the MPA algorithm assumes that the vector of unknown parameters *η* is random on the set of possible flights. We assume that the a priori statistical-generator for computer generated random vectors *η* and *ξ<sup>k</sup>* is given. This generator makes the algorithm estimating components of the vector *η* (the identification algorithm) Bayesian. Further, for particular calculations, we assume that random components of the mentioned vectors are distributed uniformly and can be called by the standard Random program in Turbo Pascal. The MPA algorithm provides the approximation method we implement with the multidimensional power series of the vector *E*(*η*|*YN*) of the conditional mathematical expectation of the vector *η* if the vector of measurements *YN* is fixed and a priori statistical

The vector *E*(*η*|*YN*) is known to be optimal, in root-mean-square sense, estimate of the random

We describe the steps of operation of the MPA algorithm when it identifies the vector *η*[6, 7]. Step 1. Suppose *d* is a given positive integer number and the set of integer numbers *a*1, ..., *aN* consists of all nonnegative solutions of the integer inequality *a*<sup>1</sup> + ... + *aN* ≤ *d*, the number of which we denote by *m*(*d*, *N*). The value *m*(*d*, *N*) is given by the recurrent formula proved by

characteristics, in what follows) that are close to real (measurable) characteristics.

difference between real and nominal aerodynamic parameters.

perturbed motions.

parameters.

aircraft.

vector *η*.

nominal values at these nodes.

data on random vectors *η* and *ξ<sup>k</sup>* are given.

$$m(d,N) = m(d-1,N) + (N+d-1) \cdot \cdots \cdot N/d!, m(1,N) = N.$$

We obtain the vector *WN*(*d*) of dimension *m*(*d*, *N*) × 1, the components *w*1, ..., *wm*(*d*, *N*) of which are all possible values *ya*<sup>1</sup> <sup>1</sup> ...*yan <sup>N</sup>* of the form that represent the powers of measurable values.

Then, we construct the base vector *V*(*d*, *N*) of dimension (*r* + *m*(*d*, *N*)) × 1, *V*(*d*, *N*) = �*ηWN*(*d*)�.

Step 2. We use a known statistical generator of random vectors *η* and *ξ<sup>k</sup>* to solve repeatedly the Cauchy problem for *Eq*.(1) for given initial conditions *x*(0), a control law *u*(*t*) and various realizations of random vectors *η* and *xik*.

We apply the Monte-Carlo method to find the prior first and second statistical moments of the vector *V*(*d*, *N*), i.e., the mathematical expectation *V*¯(*d*, *N*), and the covariance matrix *CV*(*d*, *<sup>N</sup>*) = *<sup>E</sup>*((*V*(*d*, *<sup>N</sup>*) <sup>−</sup> *<sup>V</sup>*¯(*d*, *<sup>N</sup>*))(*V*(*d*, *<sup>N</sup>*) <sup>−</sup> *<sup>V</sup>*¯(*d*, *<sup>N</sup>*))*T*) .

Implementation of step 2 is a learning process for the algorithm, adjusting it to solve the particular problem described by Eqs. (1) and (2).

Step 3. For given *<sup>d</sup>* and *<sup>N</sup>* and a fixed vector *YN*, we assign the vector *<sup>η</sup>*(*WN*(*d*)) to be the solution to the estimation problem. This vector gives an approximate estimate of the vector *E*(*η*|*YN*) that is optimal in the root-mean-square sense on the set of vector linear combinations of components of the vector *WN*<sup>1</sup> (*d*)

$$\hat{\eta}(\mathcal{W}\_N(d)) = \sum\_{a\_1 + \ldots + a\_N \le d} \lambda(a\_1, \ldots, a\_N) y\_1^{a\_1} \cdots y\_N^{a\_N} . \quad (1.1)$$

The vector *V*¯(*d*, *N*) and the matrix *CV*(*d*, *N*) are the initial conditions for the process of recurrent calculations that realizes the principle of observation decomposition [6] and consists of *m*(*d*, *N*) steps. Once the final step is performed, we obtain vector coefficients *λ*(*a*1, ..., *aN*) for (1.1). Moreover, we determine the matrix *C*(*d*, *N*), which is the covariance matrix of the estimation errors for the vector *E*(*ηN*|*YN*) of conditional mathematical expectation estimated by the vector *<sup>η</sup>*(*WN*(*d*)).

Calculating the elements of the matrix *C*(*d*, *N*), we have the method of preliminary (prior to the actual flight) analysis of observability of identified parameters for the given control law, structure of measurements and their expected random errors. Recurrent calculations do not require matrix inversion and indicate the situations when the next component of the vector *WN*(*d*) is close to linear combination of its previous components. To implement the recursion, we process the components of the vector *WN*(*d*) one after another. However, the adjustment of the algorithm performed by applying the Monte-Carlo method to find the vector *V*¯(*d*, *N*) and the matrix *CV*(*d*, *N*) takes into account a priori ideas on stochastic structure of components of the whole set of possible vectors *WN*(*d*) that can appear in any realizations of the random vectors *η* and *ξ<sup>k</sup>* allowed by the a priori conditions.

This adjustment is the price we have to pay if we want the MPA algorithm to solve nonlinear identification problems efficiently. This is what makes the MPA algorithm differ fundamentally from, for instance, the standard Kalman filter designed to solve linear

ii. the current covariance matrix of estimation errors (we emphasize that known numerical methods of constructing approximations of the vector of nonlinear estimates cannot calculate

An Algorithm for Parameters Identification of an Aircraft's Dynamics 125

Implementation of items 2 and 2.1 makes the MPA algorithm more efficient than any known

ii. does not apply variants of the Newton method to solve systems of nonlinear algebraic

iii. forms the estimation vector that tends uniformly to the vector of conditional mathematical

It is worth emphasizing that in this work we just develop the fundamental ground of computational technique for solving the complex problem of aircraft parameter identification.

We consider the boundary inverse problem for the attractor whose equations are presented in [8 ]. The three parameters are the initial conditions: *X*1[0] = *η*1, *X*2[0] = *η*2, *X*3[0] = *η*3. The six parameters *η*3+*i*, *i* = 1, ..., 6 correspond to combinations of the inductance, the resistances

*X*1[*k* − 1] < −*η*3+<sup>6</sup> : *f* = *η*3+5;

*X*1[*k* − 1] > *η*3+<sup>6</sup> : *f* = −*η*3+5; *X*1[*k*] = *X*1[*k* − 1] + *δX*2[*k* − 1]; *X*2[*k*] = *X*2[*k* − 1] + *δ*(−*X*1[*k* − 1] − *η*3+1*X*2[*k* − 1] + *X*3[*k* − 1]); *X*3[*k*] = *X*3[*k* − 1] + *δ*(*θ*3+2(*η*3+<sup>3</sup> *f* − *X*3[*k* − 1]) − *η*3+4*X*2[*k* − 1]);

*η<sup>i</sup>* ∈ 1 + (*ε<sup>i</sup>* − 0.5).

*η*3+<sup>1</sup> ∈ 0.5(1 + (*ε*<sup>1</sup> − 0.5)); *η*3+<sup>2</sup> = 0.3(1 + (*ε*<sup>2</sup> − 0.5)); *η*3+<sup>3</sup> = 15(1 + (*ε*<sup>3</sup> − 0.5));

*η*3+<sup>4</sup> ∈ 1.5(1 + (*ε*<sup>4</sup> − 0.5)); *η*3+<sup>5</sup> = 0.5(1 + (*ε*<sup>5</sup> − 0.5)); *η*3+<sup>6</sup> = 1.2(1 + (*ε*<sup>6</sup> − 0.5)); *yk* = *X*1(*tk*)) + *ζ<sup>k</sup>*

*<sup>k</sup>*=<sup>1</sup> *yk*, *<sup>z</sup>*<sup>2</sup> <sup>=</sup> <sup>∑</sup>*k*=2×*N*/*<sup>T</sup>*

*<sup>k</sup>*=1+*N*/*<sup>T</sup> yk*,...

2);

−*η*3+<sup>6</sup> < *X*1[*k* − 1] < *η*3+<sup>6</sup> : *f* = *X*1[*k* − 1](1 − *X*1[*k* − 1]

where *X*1[*k*] corresponds to a voltage, *X*2[*k*] to a current and *X*3[*k*] to another voltage.

**3. Testing the MPA algorithm: Problem reconstruction (identification of parameters) for the attractor from units of an electrical chain**

The equations of the mathematical model of the circuit take the following form [8]:

current covariance matrices of estimation errors).

iiii. obtains the covariance matrix of estimation errors.

linear identification algorithm since it

expectation for the growing integer *d*,

and the two capacitances of a circuit.

We suppose that by *i* = 1, 2, 3

*<sup>δ</sup>* = 0.01, *<sup>k</sup>* = 1, ..., *<sup>N</sup>* = 1200, *<sup>z</sup>*<sup>1</sup> = <sup>∑</sup>*k*=*N*/*<sup>T</sup>*

We also suppose [ 8] that

i. does not involve linearization,

equations,

identification problems only or from multiple variations of algorithms resulted from attempts to extend the Kalman filter to nonlinear filtration problems.

In [6], a multidimensional analogue of the K. Weierstrass theorem (the corollary of the M. Stone theorem [9]) is used to prove that when the integer *d* is increases then the error estimates of the vector *<sup>E</sup>*(*η*|*YN*) the vector <sup>|</sup>*<sup>η</sup>*(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*)<sup>|</sup> tend to zero uniformly on some region. Formulas of the recurrent algorithm are given and justified in [6, 7] and in the Appendix.

This scheme for the MPA algorithm operation shows that it can be applied to identify parameters of almost any dynamic system provided that the structures of the motion equations and measurements of form (0.1) and (0.2) and prior statistical generators of random unknown parameters and errors of measurements are given. The MPA algorithm is devoid of the above listed limitations and drawbacks, which gives it substantial advantages over NASA identification algorithms. Apart from errors of computations, the algorithm does not add any other errors (such as errors due to linearization of nonlinear functions) into the identified parameters. Therefore, one should expect that the priori spread of identifiable parameters to be always greater than the posterior spread. This is why we can use iterations.

Let us compare the sequential steps of the standard discrete Kalman filter and the MPA algorithm.

(1) The Kalman filter identifies the vector *η*, which can be represented by part of components of the state vector of the linear dynamic system for the observations that linearly depend on state vectors. The a priori data are the first and second moments of components of random initial state vectors, uncorrelated random vectors of perturbations and observation errors. We need these data for sequential (recurrent) construction of the estimation vector that is root-mean-square optimal. Usually assigned, a priori data can be also determined by the Monte-Carlo method if the complex mechanism of their appearance is given.

(2) To find an asymptotic solution to the nonlinear identification problem, the MPA algorithm, unlike the Kalman filter, requires a priori statistical data on both the initial and all hypothesized future state vectors of the dynamic system and observations. These a priori data are represented by the first and second statistical moments for the random vector V(d, N): the vector *V*(*d*, *N*)) and the matrix *CV*(*d*, *N*). These moments are calculated using the Monte-Carlo method. However, there are cases when they can be obtained by numerical multidimensional region integration.

(1.1) Once conditions from (1) are met, the Kalman filter constructs the recurrent process, at every step of which the current estimation vector optimal in the root-mean-square sense and the covariance matrix of errors of the estimate are calculated.

(2.1) Based on (2), the MPA algorithm implements the recurrent computational process that do not require matrix inversion. At each step of the process, we construct

i. the current estimation vector *η*ˆ(*WN*(*d*)) linear with respect to components of the vector *WN*(*d*) and optimal in the root-mean-square sense on the set of linear combinations of components of this vector; moreover, the uniform convergence *η*ˆ(*WN*(*d*)) → *E*(*η*|*YN*), *d* → ∞. is attained on some region,

ii. the current covariance matrix of estimation errors (we emphasize that known numerical methods of constructing approximations of the vector of nonlinear estimates cannot calculate current covariance matrices of estimation errors).

Implementation of items 2 and 2.1 makes the MPA algorithm more efficient than any known linear identification algorithm since it

i. does not involve linearization,

6 Will-be-set-by-IN-TECH

identification problems only or from multiple variations of algorithms resulted from attempts

In [6], a multidimensional analogue of the K. Weierstrass theorem (the corollary of the M. Stone theorem [9]) is used to prove that when the integer *d* is increases then the error estimates of the vector *<sup>E</sup>*(*η*|*YN*) the vector <sup>|</sup>*<sup>η</sup>*(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*)<sup>|</sup> tend to zero uniformly on some region. Formulas of the recurrent algorithm are given and justified in [6, 7] and in the Appendix.

This scheme for the MPA algorithm operation shows that it can be applied to identify parameters of almost any dynamic system provided that the structures of the motion equations and measurements of form (0.1) and (0.2) and prior statistical generators of random unknown parameters and errors of measurements are given. The MPA algorithm is devoid of the above listed limitations and drawbacks, which gives it substantial advantages over NASA identification algorithms. Apart from errors of computations, the algorithm does not add any other errors (such as errors due to linearization of nonlinear functions) into the identified parameters. Therefore, one should expect that the priori spread of identifiable parameters to

Let us compare the sequential steps of the standard discrete Kalman filter and the MPA

(1) The Kalman filter identifies the vector *η*, which can be represented by part of components of the state vector of the linear dynamic system for the observations that linearly depend on state vectors. The a priori data are the first and second moments of components of random initial state vectors, uncorrelated random vectors of perturbations and observation errors. We need these data for sequential (recurrent) construction of the estimation vector that is root-mean-square optimal. Usually assigned, a priori data can be also determined by the

(2) To find an asymptotic solution to the nonlinear identification problem, the MPA algorithm, unlike the Kalman filter, requires a priori statistical data on both the initial and all hypothesized future state vectors of the dynamic system and observations. These a priori data are represented by the first and second statistical moments for the random vector V(d, N): the vector *V*(*d*, *N*)) and the matrix *CV*(*d*, *N*). These moments are calculated using the Monte-Carlo method. However, there are cases when they can be obtained by numerical

(1.1) Once conditions from (1) are met, the Kalman filter constructs the recurrent process, at every step of which the current estimation vector optimal in the root-mean-square sense and

(2.1) Based on (2), the MPA algorithm implements the recurrent computational process that

i. the current estimation vector *η*ˆ(*WN*(*d*)) linear with respect to components of the vector *WN*(*d*) and optimal in the root-mean-square sense on the set of linear combinations of components of this vector; moreover, the uniform convergence *η*ˆ(*WN*(*d*)) → *E*(*η*|*YN*), *d* → ∞.

be always greater than the posterior spread. This is why we can use iterations.

Monte-Carlo method if the complex mechanism of their appearance is given.

to extend the Kalman filter to nonlinear filtration problems.

algorithm.

multidimensional region integration.

is attained on some region,

the covariance matrix of errors of the estimate are calculated.

do not require matrix inversion. At each step of the process, we construct

ii. does not apply variants of the Newton method to solve systems of nonlinear algebraic equations,

iii. forms the estimation vector that tends uniformly to the vector of conditional mathematical expectation for the growing integer *d*,

iiii. obtains the covariance matrix of estimation errors.

It is worth emphasizing that in this work we just develop the fundamental ground of computational technique for solving the complex problem of aircraft parameter identification.

#### **3. Testing the MPA algorithm: Problem reconstruction (identification of parameters) for the attractor from units of an electrical chain**

We consider the boundary inverse problem for the attractor whose equations are presented in [8 ]. The three parameters are the initial conditions: *X*1[0] = *η*1, *X*2[0] = *η*2, *X*3[0] = *η*3. The six parameters *η*3+*i*, *i* = 1, ..., 6 correspond to combinations of the inductance, the resistances and the two capacitances of a circuit.

The equations of the mathematical model of the circuit take the following form [8]:

$$X\_1[k-1] < -\eta\_{3+6} : f = \eta\_{3+5};$$

$$-\eta\_{3+6} < X\_1[k-1] < \eta\_{3+6} : f = X\_1[k-1](1 - X\_1[k-1]^2);$$

$$X\_1[k-1] > \eta\_{3+6} : f = -\eta\_{3+5};$$

$$X\_1[k] = X\_1[k-1] + \delta X\_2[k-1];$$

$$X\_2[k] = X\_2[k-1] + \delta(-X\_1[k-1] - \eta\_{3+1}X\_2[k-1] + X\_3[k-1]);$$

$$X\_3[k] = X\_3[k-1] + \delta(\theta\_{3+2}(\eta\_{3+3}f - X\_3[k-1]) - \eta\_{3+4}X\_2[k-1]);$$

where *X*1[*k*] corresponds to a voltage, *X*2[*k*] to a current and *X*3[*k*] to another voltage. We suppose that by *i* = 1, 2, 3

$$
\eta\_i \in 1 + (\varepsilon\_i - 0.5).
$$

We also suppose [ 8] that

$$\eta\_{3+1} \in 0.5(1 + (\varepsilon\_1 - 0.5)); \eta\_{3+2} = 0.3(1 + (\varepsilon\_2 - 0.5)); \eta\_{3+3} = 15(1 + (\varepsilon\_3 - 0.5));$$

$$\eta\_{3+4} \in 1.5(1 + (\varepsilon\_4 - 0.5)); \eta\_{3+5} = 0.5(1 + (\varepsilon\_5 - 0.5)); \eta\_{3+6} = 1.2(1 + (\varepsilon\_6 - 0.5));$$

$$y\_k = \mathcal{X}\_1(t\_k)) + \zeta\_k$$

$$\delta = 0.01, k = 1, \dots, N = 1200, z\_1 = \sum\_{k=1}^{k=N/T} y\_k, z\_2 = \sum\_{k=1+N/T}^{k=2\times N/T} y\_k.$$

number *α<sup>i</sup> CZ*<sup>0</sup> (*αi*) *Cm*<sup>0</sup> (*αi*) *CZq* (*αi*) *Cmq* (*αi*) 1 0.7700 −0.1740 −8.8000 −7.2100 2 0.2410 −0.1450 −25.8000 −5.4000 −0.1000 −0.1210 −28.9000 −5.2300 −0.4160 −0.1270 −31.4000 −5.2600 −0.7310 −0.1290 −31.2000 −6.1100 −1.0530 −0.1020 −30.7000 −6.6400 −1.3660 −0.0970 −27.7000 −5.6900 −1.6460 −0.1130 −28.2000 −6.0000 −1.9170 −0.0870 −29.0000 −6.2000 −2.1200 −0.0840 −29.8000 −6.4000 −2.2480 −0.0690 −38.3000 −6.6000 −2.2290 −0.0060 −35.3000 −6.0000

An Algorithm for Parameters Identification of an Aircraft's Dynamics 127

We use the rectangular coordinate system XYZ adopted in NASA. Then for the unperturbed

*dα*/*dt* = *ω<sup>Y</sup>* + (*g*/*V*)(*NZ* + *cos*(*θ* − *α*)),

where *V*=300 ft/sec,*H*=20000 ft, *α* is the angle of attack, *NZ* is the g-load, which is the vector of aerodynamic forces projected onto the axis *Z* and divided by the weight of the aircraft, *MY* is the vector of the moment of aerodynamic forces projected onto the axis *Y*, *ω* is the vector of the angular velocity of the aircraft projected onto the axis *Y*,*θ* is the angle between the the axis *X* and the horizontal plane, *q* is the value of the dynamic pressure, *G* is the weight, *JY* is the moment of inertia with respect to the axis *Y*, *S* is the area of the surface generating aerodynamic forces, *b* is the mean aerodynamic of the wing, *CZ*(*α*, *δ*) and *Cm*(*α*, *δ*) are dimensionless coefficients of the aerodynamic force and moment,*δ<sup>s</sup>* is the angle of the

> *CZ*(*α*, *δs*) = *CZ*<sup>0</sup> (*α*) − 0.19(*δs*/25) + *CZq* (*α*)(*b*/(2*V*))*ω*, *Cm*(*α*, *δs*) = *Cm*<sup>0</sup> (*α*)*δ<sup>s</sup>* + *Cmq* (*α*)(*b*/(2*V*))*ω* + 0.1*CZ*.

Nominal values of 4 functions of the angle of attack *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*) are given with the argument step (55 − 1)/12 degree at 12 nodes (Table 1) in range −10◦ ≤ *α* ≤ 45◦ . To determine values of functions between the nodes, we use linear interpolation. Having analyzed Table 1, we can see that functions *CZ*<sup>0</sup> (*αi*), *Cm*<sup>0</sup> (*αi*), *CZq* (*αi*), *Cmq* (*αi*) are essentially

atmosphere and conditions *V* = *const*, pitching motion equations have the form [1]:

*NZ* = *CZ*(*α*, *δs*)*qS*/*G*, *MY* = *Cm*(*α*, *δs*)*qSb*,

Table 1. Nominal values of the functions *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*)

*dωY*/*dt* = *MY*/*JY*, *dθ*/*dt* = *ωY*,

**4.1 Pitching motion equations**

stabilator devlection measured in degrees.

Functions *CZ*(*α*, *δs*) and *Cm*(*α*, *δs*) are given by the relations [1],:

**4.2 Parametric model aerodynamic forces and moments**

The algorithm uses approximations of parameters by means of linear combinations of the constructed values *zi*(*d* = 1). Values *z*1, *z*<sup>2</sup> - are the sums of values of flowing observations serve as inputs of MPA algorithm

The relative errors of the boundary problem are

*i* 123 *T* = 24 Δ*<sup>i</sup>* 0.025 0.264 0.272 *T* = 48 Δ*<sup>i</sup>* 0.0007 −0.003 0.046 *T* = 120 Δ*<sup>i</sup>* 0.00005 −0.00264 0.01687 *T* = 240 Δ*<sup>i</sup>* 0.00001 −0.00049 0.02686 .

The relative errors of the inverse problem are

$$\begin{array}{ccccccc} i & 1 & 2 & 3 & 4 & 5 & 6\\ T = 24 & & & & & & \\ \Delta\_{3+i} & -0.347 \; 0.198 & -0.250 & 0.097 & 0.095 \; 0.136\\ T = 48 & & & & & & \\ \Delta\_{3+i} & -0.140 \; 0.234 & -0.222 & 0.104 & 0.143 \; 0.133 & \\ T = 120 & & & & & \\ \Delta\_{3+i} & -0.169 \; 0.205 & -0.167 & 0.094 & 0.179 \; 0.097 & \\ T = 240 & & & & & \\ \Delta\_{3+i} & -0.042 \; 0.129 & -0.031 & -0.0001 \; 0.151 & 0.146 & \\ \end{array}$$

The resulted tables show, that corresponding adjustment the MPA algorithm - a corresponding selection of value *T* allows to make small relative errors of an estimation of parameters of the non-linear dynamic system.

#### **4. Identification of aerodynamic coefficients of the pitching motion for an pseudo f-16 aircraft**

We illustrate efficiency of offered MPA algorithm on an example of identification of 48 dimensionless aerodynamic coefficients for the aircraft of near F-16. The aircraft we shall conditionally name " pseudo F-16 ". The term "near" is justified by that, what is the coefficients are taken from SIDPAS [1], but are perturbed by addition of some random numbers.

The tables resulted below, show, that errors of identification are small also modules of their relative values do not surpass several hundredth. The considered problem corresponds to minimization of object function of 48 variables, which it is made of the sum of squares of differences of actual and computational angles of attack, g-load, pitch angles, observable with frequency 10 hertz during 25 sec. flight of the aircraft maneuvering in a vertical plane.


Table 1. Nominal values of the functions *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*)

#### **4.1 Pitching motion equations**

8 Will-be-set-by-IN-TECH

The algorithm uses approximations of parameters by means of linear combinations of the constructed values *zi*(*d* = 1). Values *z*1, *z*<sup>2</sup> - are the sums of values of flowing observations -

*i* 123

Δ*<sup>i</sup>* 0.025 0.264 0.272

Δ*<sup>i</sup>* 0.0007 −0.003 0.046

.

.

Δ*<sup>i</sup>* 0.00005 −0.00264 0.01687

Δ*<sup>i</sup>* 0.00001 −0.00049 0.02686

*i* 1 2 3 4 56

Δ3+*<sup>i</sup>* −0.347 0.198 −0.250 0.097 0.095 0.136

Δ3+*<sup>i</sup>* −0.140 0.234 −0.222 0.104 0.143 0.133

Δ3+*<sup>i</sup>* −0.169 0.205 −0.167 0.094 0.179 0.097

Δ3+*<sup>i</sup>* −0.042 0.129 −0.031 −0.0001 0.151 0.146

The resulted tables show, that corresponding adjustment the MPA algorithm - a corresponding selection of value *T* allows to make small relative errors of an estimation of parameters of the

**4. Identification of aerodynamic coefficients of the pitching motion for an pseudo**

We illustrate efficiency of offered MPA algorithm on an example of identification of 48 dimensionless aerodynamic coefficients for the aircraft of near F-16. The aircraft we shall conditionally name " pseudo F-16 ". The term "near" is justified by that, what is the coefficients

The tables resulted below, show, that errors of identification are small also modules of their relative values do not surpass several hundredth. The considered problem corresponds to minimization of object function of 48 variables, which it is made of the sum of squares of differences of actual and computational angles of attack, g-load, pitch angles, observable with frequency 10 hertz during 25 sec. flight of the aircraft maneuvering in a vertical plane.

are taken from SIDPAS [1], but are perturbed by addition of some random numbers.

serve as inputs of MPA algorithm

The relative errors of the boundary problem are

The relative errors of the inverse problem are

*T* = 24

*T* = 48

*T* = 120

*T* = 240

non-linear dynamic system.

**f-16 aircraft**

*T* = 24

*T* = 48

*T* = 120

*T* = 240

We use the rectangular coordinate system XYZ adopted in NASA. Then for the unperturbed atmosphere and conditions *V* = *const*, pitching motion equations have the form [1]:

$$\begin{aligned} d\alpha/dt &= \omega\_Y + (\operatorname{g}/V)(N\_Z + \cos(\theta - \mathfrak{a}))\_\prime \\ d\omega\_Y/dt &= M\_Y/f\_{Y\prime} \\ d\theta/dt &= \omega\_{Y\prime} \\ N\_Z &= C\_Z(\mathfrak{a}\_\prime \delta\_\mathrm{s}) qS/G\_\prime \\ M\_Y &= C\_m(\mathfrak{a}\_\prime \delta\_\mathrm{s}) qSb\_\prime \end{aligned}$$

where *V*=300 ft/sec,*H*=20000 ft, *α* is the angle of attack, *NZ* is the g-load, which is the vector of aerodynamic forces projected onto the axis *Z* and divided by the weight of the aircraft, *MY* is the vector of the moment of aerodynamic forces projected onto the axis *Y*, *ω* is the vector of the angular velocity of the aircraft projected onto the axis *Y*,*θ* is the angle between the the axis *X* and the horizontal plane, *q* is the value of the dynamic pressure, *G* is the weight, *JY* is the moment of inertia with respect to the axis *Y*, *S* is the area of the surface generating aerodynamic forces, *b* is the mean aerodynamic of the wing, *CZ*(*α*, *δ*) and *Cm*(*α*, *δ*) are dimensionless coefficients of the aerodynamic force and moment,*δ<sup>s</sup>* is the angle of the stabilator devlection measured in degrees.

Functions *CZ*(*α*, *δs*) and *Cm*(*α*, *δs*) are given by the relations [1],:

$$\begin{aligned} \mathsf{C}\_{\mathsf{Z}}(\mathsf{a},\delta\_{\mathsf{s}}) &= \mathsf{C}\_{\mathsf{Z}\_{\mathsf{0}}}(\mathsf{a}) - 0.19(\delta\_{\mathsf{s}}/25) + \mathsf{C}\_{\mathsf{Z}\_{\mathsf{q}}}(\mathsf{a})(\mathsf{b}/(2V))\omega, \\ \mathsf{C}\_{\mathsf{m}}(\mathsf{a},\delta\_{\mathsf{s}}) &= \mathsf{C}\_{\mathsf{m}\_{\mathsf{0}}}(\mathsf{a})\delta\_{\mathsf{s}} + \mathsf{C}\_{\mathsf{m}\_{\mathsf{q}}}(\mathsf{a})(\mathsf{b}/(2V))\omega + 0.1\mathsf{C}\_{\mathsf{Z}}. \end{aligned}$$

#### **4.2 Parametric model aerodynamic forces and moments**

Nominal values of 4 functions of the angle of attack *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*) are given with the argument step (55 − 1)/12 degree at 12 nodes (Table 1) in range −10◦ ≤ *α* ≤ 45◦ .

To determine values of functions between the nodes, we use linear interpolation. Having analyzed Table 1, we can see that functions *CZ*<sup>0</sup> (*αi*), *Cm*<sup>0</sup> (*αi*), *CZq* (*αi*), *Cmq* (*αi*) are essentially

number. obs. *k δs*(*k*) *α*(*k*) *NZ*(*k*) *θ*(*k*)

An Algorithm for Parameters Identification of an Aircraft's Dynamics 129

 −0.0200 3.6820 0.1021 0.0132 −0.0600 5.1462 −0.3525 0.1388 −0.1000 6.0119 −0.2956 0.1689 −0.1400 6.7707 −0.2493 0.0300 −0.1800 7.5061 −0.2085 −0.1851 −0.2200 8.2964 −0.1685 −0.3945 −0.2600 9.2186 −0.1253 −0.5227 −0.3000 10.2083 −0.6016 −0.5119 −0.3400 10.6145 −0.5691 −0.6187 −0.3800 10.8889 −0.5477 −0.8891 −0.4200 11.0977 −0.5334 −1.2461 −0.4600 11.2993 −0.5223 −1.6252 −0.5000 11.5494 −0.5114 −1.9688 −0.5400 11.9047 −0.4974 −2.2222 −0.5800 12.4277 −0.4774 −2.3286 −0.6200 13.1919 −0.4477 −2.2247 −0.6600 14.2870 −0.4043 −1.8352 −0.7000 15.4810 −0.8822 −1.1481 −0.7400 16.1493 −0.8343 −0.6582 −0.7800 16.6530 −0.7993 −0.3828 −0.8200 17.0629 −0.7728 −0.2382 −0.8600 17.4401 −0.7511 −0.1552 −0.9000 17.8400 −0.7310 −0.0747 −0.9400 18.3168 −0.7095 0.0571 −0.9800 18.9260 −0.6838 0.2926 −1.0200 19.7287 −0.6512 0.6855 −1.0600 20.1833 −1.1389 1.1343 −1.1000 20.1954 −1.1266 1.3075 −1.1400 19.9812 −1.1273 1.2486 −1.1800 19.9888 −1.1293 1.1572 −1.2200 19.9789 −1.1308 1.1146 −1.2600 20.0083 −1.1316 1.1129 −1.3000 20.0049 −1.1325 1.1437 −1.3400 20.0371 −1.1330 1.2113 −1.3800 20.0328 −1.1338 1.3073 −1.4200 20.0598 −1.1344 1.4359 −1.4600 20.0636 −1.1346 1.6053 −1.5000 20.0993 −1.1344 1.8069 −1.5400 20.1945 −1.1326 2.0671 −1.5800 20.3760 −1.1278 2.4096 −1.6200 20.6696 −1.1186 2.8558 −1.6600 21.1005 −1.1037 3.4247 −1.7000 21.6926 −1.0820 4.1331 −1.7400 22.4690 −1.0523 4.9948 −1.7800 23.4509 −1.0133 6.0212


Table 2. Nominal values of increment Δ(*CZ*<sup>0</sup> (*αi*)), Δ(*Cm*<sup>0</sup> (*αi*)), Δ(*CZq* (*αi*)), Δ(*Cmq* (*αi*))

nonlinear. Table 2 confirms this visual impression. In it increments are presented 4 functions on each step of Table 1. Apparently, increments noticeably vary.

We study the identification problem for the perturbed analogues of the functions *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*). The number of nominal coefficients that determine these functions is 12+12+12+12 = 48. Let us single out the problem which is the most complex for the MPA algorithm, when the actual coefficients differs from the nominal coefficients by the unknown bounded by the prior limits value *η<sup>i</sup>* at each point of the table. Then, for accumulated results of measurements of parameters of the perturbed motion, the MPA algorithm is to estimate 48 components of the vector of random estimates, - the vector of differences between actual and nominal coefficients.

Suppose *ϑ<sup>i</sup>* and *Bi* are the i-th components of the nominal and actual (perturbed) vectors of aerodynamic coefficients , *i* = 1, ..., 48, i.e. the number of actual coefficients to be identified is 48 in this case. We assume that the parametric model

$$B\_i = \vartheta\_i + \eta\_i.$$

holds. The vector *η* serves as the vector of perturbations of nominal data errors of aerodynamic parameters, and identification yields the estimates of its components. We give the structure of these components by the formula *η<sup>i</sup>* = *ϑiρiεi*, 0 < *ρ<sup>i</sup>* < 1, −1 < *ε<sup>i</sup>* < 1. The positive number *ρ<sup>i</sup>* gives the maximum value that, by identification conditions, can be attained by the ratio of the absolute values of the random value of perturbations *η<sup>i</sup>* and nominal coefficients *ϑ<sup>i</sup>* .

#### **4.3 Transient processes of characteristics of nominal motions**

We wish to identify-estimate - during one test flight the 48 unknown aerodynamic coefficients for 12 nodes-12 the set angles of attack *αi*, *i* = 1..., 12. For a testing maneuver the characteristics *α*(*t*), *NZ*(*t*), *θ*(*t*) of Transient Processes are carrier of information of the the identified coefficients. Therefore during flight the aircraft should "visit" vicinities of angles of attack −10◦ ≤ *α* ≤ 45◦

Will-be-set-by-IN-TECH

number *α<sup>i</sup>* Δ(*CZ*<sup>0</sup> (*αi*)) Δ(*Cm*<sup>0</sup> (*αi*)) Δ(*CZq* (*αi*)) Δ(*Cmq* (*αi*)) 1 0.7700 −0.1740 −8.8000 −7.2100 −0.5290 0.0290 −17.0000 1.8100 −0.3410 0.0240 −3.1000 0.1700 −0.3160 −0.0060 −2.5000 −0.0300 −0.3150 −0.0020 0.2000 −0.8500 −0.3220 0.0270 0.5000 −0.5300 −0.3130 0.0050 3.0000 0.9500 −0.2800 −0.0160 −0.5000 −0.3100 −0.2710 0.0260 −0.8000 −0.2000 −0.2030 0.0030 −0.8000 −0.2000 −0.1280 0.0150 −8.5000 −0.2000 12 0.0190 0.0630 3.0000 0.6000

Table 2. Nominal values of increment Δ(*CZ*<sup>0</sup> (*αi*)), Δ(*Cm*<sup>0</sup> (*αi*)), Δ(*CZq* (*αi*)), Δ(*Cmq* (*αi*))

on each step of Table 1. Apparently, increments noticeably vary.

differences between actual and nominal coefficients.

48 in this case. We assume that the parametric model

**4.3 Transient processes of characteristics of nominal motions**

nominal coefficients *ϑ<sup>i</sup>* .

of attack −10◦ ≤ *α* ≤ 45◦

nonlinear. Table 2 confirms this visual impression. In it increments are presented 4 functions

We study the identification problem for the perturbed analogues of the functions *CZ*<sup>0</sup> (*α*), *Cm*<sup>0</sup> (*α*), *CZq* (*α*), *Cmq* (*α*). The number of nominal coefficients that determine these functions is 12+12+12+12 = 48. Let us single out the problem which is the most complex for the MPA algorithm, when the actual coefficients differs from the nominal coefficients by the unknown bounded by the prior limits value *η<sup>i</sup>* at each point of the table. Then, for accumulated results of measurements of parameters of the perturbed motion, the MPA algorithm is to estimate 48 components of the vector of random estimates, - the vector of

Suppose *ϑ<sup>i</sup>* and *Bi* are the i-th components of the nominal and actual (perturbed) vectors of aerodynamic coefficients , *i* = 1, ..., 48, i.e. the number of actual coefficients to be identified is

*Bi* = *ϑ<sup>i</sup>* + *ηi*.

holds. The vector *η* serves as the vector of perturbations of nominal data errors of aerodynamic parameters, and identification yields the estimates of its components. We give the structure of these components by the formula *η<sup>i</sup>* = *ϑiρiεi*, 0 < *ρ<sup>i</sup>* < 1, −1 < *ε<sup>i</sup>* < 1. The positive number *ρ<sup>i</sup>* gives the maximum value that, by identification conditions, can be attained by the ratio of the absolute values of the random value of perturbations *η<sup>i</sup>* and

We wish to identify-estimate - during one test flight the 48 unknown aerodynamic coefficients for 12 nodes-12 the set angles of attack *αi*, *i* = 1..., 12. For a testing maneuver the characteristics *α*(*t*), *NZ*(*t*), *θ*(*t*) of Transient Processes are carrier of information of the the identified coefficients. Therefore during flight the aircraft should "visit" vicinities of angles


 −0.7800 27.4286 −1.2336 39.7559 −0.7400 26.0585 −1.2787 38.9816 −0.7000 24.4358 −0.8717 37.7788 −0.6600 22.7903 −0.9362 36.2669 −0.6200 20.7622 −1.0163 34.4409 −0.5800 18.7935 −0.5743 32.3636 −0.5400 17.0475 −0.6702 30.3402 −0.5000 15.1676 −0.7674 28.2787 −0.4600 13.9500 −0.3141 26.3775 −0.4200 13.1154 −0.3750 24.8646 −0.3800 12.4703 −0.4193 23.5913 −0.3400 11.9129 −0.4537 22.4424 −0.3000 11.3566 −0.4837 21.3242 −0.2600 10.7223 −0.5143 20.1561 −0.2200 9.9324 −0.5497 18.8636 −0.1800 9.8365 −0.0427 17.6449 −0.1400 9.9588 −0.0463 16.6494 −0.1000 9.9853 −0.0452 15.7456 −0.0600 9.9853 −0.0437 14.8087 −0.0200 10.0132 −0.0415 13.8278 221 0.0200 9.9999 −0.0398 12.7981 223 0.0600 9.9409 −0.5639 11.7156 225 0.1000 9.9557 −0.0371 10.5715 227 0.1400 9.9311 −0.0359 9.3815 229 0.1800 9.8312 −0.0369 8.1116 231 0.2200 9.6174 −0.0423 6.7279 233 0.2600 9.2466 −0.0540 5.1941 235 0.3000 8.6685 −0.0746 3.4700 237 0.3400 7.8225 −0.1068 1.5084 239 0.3800 6.6340 −0.1539 −0.7475 241 0.4200 5.0095 −0.2203 −3.3684 243 0.4600 3.6383 0.2178 −6.2070 245 0.5000 2.2053 0.1484 −9.0796 247 0.5400 0.5330 0.0717 −12.0920 249 0.5800 −0.9453 0.5373 −15.2658 Table 3. The characteristics *α*(*t*), *NZ*(*t*), *θ*(*t*) of the nominal motions for the chosen control

An Algorithm for Parameters Identification of an Aircraft's Dynamics 131

**4.4 Estimating identification accuracy of 48 errors of aerodynamic parameters of the aircraft** Primary task of MPA algorithm consists in identification - estimation-48 increments of 4 functions. If entry conditions and increments are determined, values of the unknown

To estimate the accuracy, we assume that the current values of *α*, *NY*, *θ* are measured every 0.1 sec. during 25 seconds .We assume that random errors of measurement represent the discrete white noise bounded by the true measurable value multiplied by the given value *�*.

coefficients follow from obvious recurrent formulas.

An amount of the primary observations equal 3\*250=750.

law *δs*(*t*).


Will-be-set-by-IN-TECH

 −1.8200 24.6576 −0.9641 7.2201 −1.8600 25.7086 −1.3912 8.6103 −1.9000 26.9187 −1.3506 10.2913 −1.9400 28.6178 −1.2942 12.4085 −1.9800 30.7696 −1.6768 15.0754 −2.0200 32.4354 −1.6031 17.5706 −2.0600 33.8743 −1.5431 19.7731 −2.1000 35.1432 −1.8357 21.7958 −2.1400 35.7317 −1.8014 23.4808 −2.1800 35.9655 −1.7815 24.7918 −2.1800 35.9159 −1.7715 25.8128 −2.1400 35.6010 −1.7680 26.5691 −2.1000 34.9990 −1.7695 27.0436 −2.0600 34.7166 −1.4423 27.4338 −2.0200 34.4743 −1.4525 27.8824 −1.9800 34.2309 −1.4610 28.3457 −1.9400 33.9488 −1.4692 28.7851 −1.9000 33.5921 −1.4785 29.1649 −1.8600 33.1250 −1.4900 29.4506 −1.8200 32.5103 −1.5050 29.6075 −1.7800 31.7080 −1.5248 29.5992 −1.7400 30.6740 −1.5505 29.3864 −1.7000 29.6897 −1.1403 29.0369 −1.6600 29.5906 −1.1751 29.2924 −1.6200 30.0407 −1.6327 30.1792 −1.5800 30.0324 −1.1643 30.9954 −1.5400 30.0007 −1.1623 31.6784 −1.5000 29.9971 −1.1623 32.3405 −1.4600 29.9834 −1.6165 32.9999 −1.4200 29.9916 −1.1620 33.6324 −1.3800 29.9805 −1.1621 34.2532 −1.3400 30.0190 −1.1626 34.8756 −1.3000 29.9687 −1.6164 35.4719 −1.2600 30.0181 −1.1623 36.0635 −1.2200 29.9808 −1.1614 36.6311 −1.1800 29.9772 −1.1614 37.1835 −1.1400 29.9490 −1.6153 37.7184 −1.1000 29.9574 −1.1606 38.2407 −1.0600 29.9417 −1.6141 38.7453 −1.0200 29.9178 −1.1614 39.1922 −0.9800 29.8635 −1.1625 39.6161 −0.9400 29.7346 −1.1650 39.9729 −0.9000 29.4842 −1.1711 40.2178 −0.8600 29.0596 −1.1829 40.3020 −0.8200 28.3990 −1.2028 40.1699


Table 3. The characteristics *α*(*t*), *NZ*(*t*), *θ*(*t*) of the nominal motions for the chosen control law *δs*(*t*).

#### **4.4 Estimating identification accuracy of 48 errors of aerodynamic parameters of the aircraft**

Primary task of MPA algorithm consists in identification - estimation-48 increments of 4 functions. If entry conditions and increments are determined, values of the unknown coefficients follow from obvious recurrent formulas.

To estimate the accuracy, we assume that the current values of *α*, *NY*, *θ* are measured every 0.1 sec. during 25 seconds .We assume that random errors of measurement represent the discrete white noise bounded by the true measurable value multiplied by the given value *�*. An amount of the primary observations equal 3\*250=750.

number *α<sup>i</sup>* nom.koef. *CZq* (*αi*) perturb.koef. *CZq* (*αi*) *δ*(*CZq* (*αi*)) −9.9636 −8.8984 0.10691 −25.2235 −26.1655 −0.03735 −28.4644 −29.2857 −0.02885 −31.4821 −31.8270 −0.01096 −31.3125 −31.6274 −0.01006 −30.8417 −31.1249 −0.00918 −27.5461 −28.0921 −0.01982 −28.1388 −28.6036 −0.01652 −28.9682 −29.4069 −0.01515 −29.7908 −30.2114 −0.01412 −38.6789 −38.7933 −0.00296 −35.7355 −35.8053 −0.00195

An Algorithm for Parameters Identification of an Aircraft's Dynamics 133

number *α<sup>i</sup>* nom.koef. *Cmq* (*αi*) perturb.koef. *Cmq* (*αi*) *δ*(*Cmq* (*αi*)) −5.5807 −6.1771 −0.10686 −4.1294 −4.3066 −0.04291 −3.9913 −4.1368 −0.03645 −4.0250 −4.1662 −0.03510 −4.9363 −5.0012 −0.01315 −5.5024 −5.5314 −0.00527 −4.5272 −4.5870 −0.01320 −4.8711 −4.8936 −0.00462 −5.0970 −5.0915 0.00108 −5.3245 −5.2912 0.00626 −5.5637 −5.4908 0.01310 −4.8726 −4.8937 −0.00434

number *α<sup>i</sup>* nom.koef. *CZ*<sup>0</sup> (*αi*) perturb.koef. *CZ*<sup>0</sup> (*αi*) *δ*(*CZ*<sup>0</sup> (*αi*)) 0.5324 0.4255 0.20083 −0.1999 −0.2092 −0.04637 −0.6556 −0.5969 0.08959 −1.0629 −0.9303 0.12481 −1.3911 −1.2772 0.08188 −1.7546 −1.6697 0.04839 −2.1699 −2.0331 0.06304 −2.4704 −2.3417 0.05209 −2.6379 −2.6342 0.00138 −2.7936 −2.8450 −0.01839 −2.8799 −2.9723 −0.03208 −2.8518 −2.9530 −0.03548

Table 6. The Relative errors of the identifications of *CZq* (*αi*) by *ρ* = 0.25

Table 7. The Relative errors of the identifications of *Cmq* (*αi*) by *ρ* = 0.25

Table 8. The Relative errors of the identifications of *CZ*<sup>0</sup> (*αi*) by *ρ* = 0.50


Table 4. The Relative errors of the identifications of *CZ*<sup>0</sup> (*αi*) by *ρ* = 0.25


Table 5. The Relative errors of the identifications of *Cm*<sup>0</sup> (*αi*) by *ρ* = 0.25

We compress primary observations for a smoothing the high-frequency errors and reduction of a dimension of matrixes covariance . The file of the primary observations is divided into 12 groups and as an input of the algorithm of the identification the vector of the dimension × 1 serves. Components of this vector are the sums of elements of each of 12 groups.

To characterize the accuracy of identification of the random parameter *η<sup>i</sup>* the degree of perturbation of the aerodynamic coefficients *ϑ* , we determine the relative errors of estimation (*η<sup>i</sup>* − *η*ˆ*i*)/*η<sup>i</sup>* for every component the identifiable functions . The relative errors designate *δ*(*CZ*<sup>0</sup> (*αi*)), *δ*(*Cm*<sup>0</sup> (*αi*)), *δ*(*CZq* (*αi*)), *δ*(*Cmq* (*αi*)), *i* = 1, ..., 12.

Apparently, relative errors of identification are small and do not surpass several hundredth at *ρ* = 0.25

Will-be-set-by-IN-TECH

number *α<sup>i</sup>* nom.koef. *CZ*<sup>0</sup> (*αi*) perturb.koef. *CZ*<sup>0</sup> (*αi*) *δ*(*CZ*<sup>0</sup> (*αi*)) 0.6512 0.6326 0.02854 0.0205 0.0260 −0.26410 −0.3778 −0.3646 0.03491 −0.7395 −0.7213 0.02456 −1.0610 −1.0657 −0.00443 −1.4038 −1.4016 0.00159 −1.7679 −1.7424 0.01444 −2.0582 −2.0453 0.00627 −2.2774 −2.3388 −0.02693 −2.4568 −2.5459 −0.03625 −2.5639 −2.6698 −0.04130 −2.5404 −2.6505 −0.04334

number *α<sup>i</sup>* nom.koef. *Cm*<sup>0</sup> (*αi*) perturb.koef. *Cm*<sup>0</sup> (*αi*) *δ*(*Cm*<sup>0</sup> (*αi*)) −0.2130 −0.2054 0.03582 −0.1816 −0.1783 0.01851 −0.1567 −0.1550 0.01061 −0.1618 −0.1611 0.00439 −0.1634 −0.1631 0.00209 −0.1427 −0.1388 0.02754 −0.1372 −0.1338 0.02439 −0.1495 −0.1502 −0.00467 −0.1175 −0.1220 −0.03771 −0.1139 −0.1190 −0.04484 −0.0957 −0.1043 −0.08937 −0.0399 −0.0394 0.01236

We compress primary observations for a smoothing the high-frequency errors and reduction of a dimension of matrixes covariance . The file of the primary observations is divided into 12 groups and as an input of the algorithm of the identification the vector of the dimension × 1 serves. Components of this vector are the sums of elements of each of 12 groups.

To characterize the accuracy of identification of the random parameter *η<sup>i</sup>* the degree of perturbation of the aerodynamic coefficients *ϑ* , we determine the relative errors of estimation (*η<sup>i</sup>* − *η*ˆ*i*)/*η<sup>i</sup>* for every component the identifiable functions . The relative errors designate

Apparently, relative errors of identification are small and do not surpass several hundredth at

Table 4. The Relative errors of the identifications of *CZ*<sup>0</sup> (*αi*) by *ρ* = 0.25

Table 5. The Relative errors of the identifications of *Cm*<sup>0</sup> (*αi*) by *ρ* = 0.25

*δ*(*CZ*<sup>0</sup> (*αi*)), *δ*(*Cm*<sup>0</sup> (*αi*)), *δ*(*CZq* (*αi*)), *δ*(*Cmq* (*αi*)), *i* = 1, ..., 12.

*ρ* = 0.25


Table 6. The Relative errors of the identifications of *CZq* (*αi*) by *ρ* = 0.25


Table 7. The Relative errors of the identifications of *Cmq* (*αi*) by *ρ* = 0.25


Table 8. The Relative errors of the identifications of *CZ*<sup>0</sup> (*αi*) by *ρ* = 0.50

**5. Conclusions**

forces and moments.

estimate.

**A.1. An algorithm fundamental (AF)**

The presented data show that the multipolynomial approximation algorithm can provide a computational ground for developing an efficient parameter identification technique for the nonlinear dynamic system, including identification of aerodynamic parameters of an aircraft. We emphasize that tables characterizing a sufficiently high accuracy of aerodynamic parameter identification are obtained when there are no iterations and d = 1, which corresponds to the case when the estimation vector <sup>ˆ</sup> (*<sup>ϑ</sup>* + *<sup>η</sup>*)(*WN*(*d*)) is represented by the vector linear combination of measured data that is optimal on the family of linear operators over the vector of measurements. This is due to good (in terms of the identification problem) properties of the parametric system of equations of the pitching motion of the "pseudo F-16 " aircraft. It can become much more complicated when it comes to the identification problem of the parametric system of equations of complete (spatial) motion of the aircraft. In such case, we may need to use polynomials of the power d > 1 and increase requirements on the computer performance and RAM. This was the case for identification attempts made for some parameters of F-16 complete motion equations. We emphasize that the inputs of the MPA algorithm we considered were not real (were not the results of operation of real sensors of the aircraft during its test flight); they were determined by mathematical simulation - by means the numerically integrations motion equations for perturbed parameters of aerodynamic

An Algorithm for Parameters Identification of an Aircraft's Dynamics 135

**6. Appendix A: An estimate of the vector of the conditional mathematical**

We consider the algorithm fundamental (AF) for solving the problem of finding the estimate of the vector *E*(*η*|*YN*) that is optimal in the root-mean-square sense. This vector is known to be the estimate optimal in the root-mean-square sense of the vector *η* once the vector *YN* is fixed. Therefore, it is justified that it is the vector of conditional expectation that AF tends to

We construct AF that ensures polynomial approximation of the vector *E*(*η*|*YN*). To do this, we find the approximate estimate of the vector *E*(*η*|*YN*), which is linear with respect to components of the vector *WN*(*d*) and optimal in the root-mean-square sense. We denote the vector of this estimate by *<sup>η</sup>*(*WN*(*d*)) . To obtain the explicit expression for the estimation vector, we calculate elements of the vector *V*(*d*, *N*) and the covariance matrix *CV*(*d*, *N*) that are the first and second (centered) statistical moments for the vector *V*(*d*, *N*). These vector

*E*(*E*(*η*|*YN*)) = *E*(*η*);

*E*(*E*(*W*(*d*, *N*)|*YN*)) = *E*(*W*(*d*, *N*)); <sup>=</sup> *<sup>E</sup>*((*E*(*η*|*YN*) <sup>−</sup> *<sup>E</sup>*(*η*))(*E*(*η*|*YN*) <sup>−</sup> *<sup>E</sup>*(*η*))*T*) = *<sup>E</sup>*((*<sup>η</sup>* <sup>−</sup> *<sup>E</sup>*(*η*))(*<sup>η</sup>* <sup>−</sup> *<sup>E</sup>*(*η*))*T*). *LN*(*d*) = *<sup>E</sup>*((*E*(*η*|*YN*) <sup>−</sup> *<sup>E</sup>*(*E*(*η*|*YN*(*d*)))(*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*)))*T*) = *<sup>E</sup>*(*η*)*WN*(*d*)*T*) <sup>−</sup> *<sup>E</sup>*(*η*)*E*(*WN*(*d*))*T*,

**expectation that is optimal in the root-mean-square sense**

and matrix can be divided into blocks of the following structure


number *α<sup>i</sup>* nom.koef. *Cm*<sup>0</sup> (*αi*) perturb.koef. *Cm*<sup>0</sup> (*αi*) *δ*(*Cm*<sup>0</sup> (*αi*))

Table 9. The Relative errors of the identifications of *Cm*<sup>0</sup> (*αi*) by *ρ* = 0.50


Table 10. The Relative errors of the identifications of *CZq* (*αi*) by *ρ* = 0.25


Table 11. The Relative errors of the identifications of *Cmq* (*αi*) by *ρ* = 0.50

#### **5. Conclusions**

16 Will-be-set-by-IN-TECH

number *α<sup>i</sup>* nom.koef. *Cm*<sup>0</sup> (*αi*) perturb.koef. *Cm*<sup>0</sup> (*αi*) *δ*(*Cm*<sup>0</sup> (*αi*)) −0.2520 −0.2441 0.03123 −0.2183 −0.2166 0.00781 −0.1924 −0.1934 −0.00523 −0.1966 −0.1994 −0.01457 −0.1979 −0.2014 −0.01792 −0.1834 −0.1747 0.04747 −0.1773 −0.1697 0.04301 −0.1860 −0.1858 0.00145 −0.1481 −0.1599 −0.08004 −0.1438 −0.1569 −0.09149 −0.1225 −0.1420 −0.15942 −0.0738 −0.0811 −0.09934

number *α<sup>i</sup>* nom.koef. *CZq* (*αi*) perturb.koef. *CZq* (*αi*) *δ*(*CZq* (*αi*)) −11.1272 −8.6840 0.21957 −24.6470 −25.6672 −0.04139 −28.0288 −28.8049 −0.02769 −31.5642 −31.3356 0.00724 −31.4249 −31.1306 0.00937 −30.9833 −30.6296 0.01142 −27.3921 −27.6113 −0.00800 −28.0776 −28.1104 −0.00117 −28.9364 −28.9144 0.00076 −29.7817 −29.7303 0.00172 −39.0577 −38.2130 0.02163 −36.1709 −35.1346 0.02865

number *α<sup>i</sup>* nom.koef. *Cmq* (*αi*) perturb.koef. *Cmq* (*αi*) *δ*(*Cmq* (*αi*)) −3.9514 −6.9359 −0.75528 −2.8588 −5.1596 −0.80480 −2.7526 −4.9893 −0.81258 −2.7899 −5.0189 −0.79894 −3.7625 −5.8672 −0.55939 −4.3649 −6.3936 −0.46477 −3.3644 −5.4530 −0.62079 −3.7422 −5.7627 −0.53993 −3.9940 −5.9631 −0.49299 −4.2490 −6.1601 −0.44976 −4.5274 −6.3594 −0.40464 −3.7451 −5.7631 −0.53882

Table 9. The Relative errors of the identifications of *Cm*<sup>0</sup> (*αi*) by *ρ* = 0.50

Table 10. The Relative errors of the identifications of *CZq* (*αi*) by *ρ* = 0.25

Table 11. The Relative errors of the identifications of *Cmq* (*αi*) by *ρ* = 0.50

The presented data show that the multipolynomial approximation algorithm can provide a computational ground for developing an efficient parameter identification technique for the nonlinear dynamic system, including identification of aerodynamic parameters of an aircraft. We emphasize that tables characterizing a sufficiently high accuracy of aerodynamic parameter identification are obtained when there are no iterations and d = 1, which corresponds to the case when the estimation vector <sup>ˆ</sup> (*<sup>ϑ</sup>* + *<sup>η</sup>*)(*WN*(*d*)) is represented by the vector linear combination of measured data that is optimal on the family of linear operators over the vector of measurements. This is due to good (in terms of the identification problem) properties of the parametric system of equations of the pitching motion of the "pseudo F-16 " aircraft. It can become much more complicated when it comes to the identification problem of the parametric system of equations of complete (spatial) motion of the aircraft. In such case, we may need to use polynomials of the power d > 1 and increase requirements on the computer performance and RAM. This was the case for identification attempts made for some parameters of F-16 complete motion equations. We emphasize that the inputs of the MPA algorithm we considered were not real (were not the results of operation of real sensors of the aircraft during its test flight); they were determined by mathematical simulation - by means the numerically integrations motion equations for perturbed parameters of aerodynamic forces and moments.

#### **6. Appendix A: An estimate of the vector of the conditional mathematical expectation that is optimal in the root-mean-square sense**

#### **A.1. An algorithm fundamental (AF)**

We consider the algorithm fundamental (AF) for solving the problem of finding the estimate of the vector *E*(*η*|*YN*) that is optimal in the root-mean-square sense. This vector is known to be the estimate optimal in the root-mean-square sense of the vector *η* once the vector *YN* is fixed. Therefore, it is justified that it is the vector of conditional expectation that AF tends to estimate.

We construct AF that ensures polynomial approximation of the vector *E*(*η*|*YN*). To do this, we find the approximate estimate of the vector *E*(*η*|*YN*), which is linear with respect to components of the vector *WN*(*d*) and optimal in the root-mean-square sense. We denote the vector of this estimate by *<sup>η</sup>*(*WN*(*d*)) . To obtain the explicit expression for the estimation vector, we calculate elements of the vector *V*(*d*, *N*) and the covariance matrix *CV*(*d*, *N*) that are the first and second (centered) statistical moments for the vector *V*(*d*, *N*). These vector and matrix can be divided into blocks of the following structure

$$E(E(\eta|Y\_N)) = E(\eta);$$

$$\begin{split} E(E(W(d,N)|Y\_N)) &= E(W(d,N)); \\ &= E((E(\eta|Y\_N) - E(\eta))(E(\eta|Y\_N) - E(\eta))^T) = \\ E((\eta - E(\eta))(\eta - E(\eta))^T). \end{split}$$

$$L\_N(d) = E((E(\eta|Y\_N) - E(E(\eta|Y\_N(d)))(W\_N(d) - E(W\_N(d))))^T) = \\ E(\eta)W\_N(d)^T) - E(\eta)E(W\_N(d))^T.$$

*WN*(*d*) and the right-hand side of (A.1) and equate them to the right-hand side of formula

An Algorithm for Parameters Identification of an Aircraft's Dynamics 137

We consider asymptotic estimation errors when we use (A.1). Suppose the vector *YN* is fixed. We assume that the vector *E*(*η*|*YN* is given by the function of *YN* on some a priori region that is a compact; the function is continuous on this region. Then, the following theorem holds.

*Sup YN* <sup>∈</sup>Ω*YN* <sup>|</sup>*<sup>η</sup>*(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*)| ⇒ 0, *<sup>d</sup>* <sup>⇒</sup> <sup>∞</sup>. (*A*.6) Proof. The multidimensional analogue of the K. Weierstrass theorem, which is the corollary of the M. Stone theorem [9], states that for any number *ε* > 0 there exists a multidimensional

*Sup YN* <sup>∈</sup>Ω*YN* |*P*(*WN*(*dε*)) − *<sup>E</sup>*(*η*|*YN*)| < *<sup>ε</sup>*.

*Sup YN* <sup>∈</sup>Ω*YN* |*P*(*WN*(*d*)) − *<sup>E</sup>*(*η*|*YN*)| ⇒ 0, *<sup>d</sup>* ⇒ <sup>∞</sup>. (*A*.7)

*<sup>C</sup>* <sup>=</sup> *<sup>E</sup>*((*P*(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*))(*P*(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*))*T*.

*C* ⇒ 0*n*, *d* ⇒ ∞ (*A*.8). By construction, the vector *<sup>η</sup>*(*WN*(*d*)) provides the estimate of the vector *<sup>η</sup>* that is linear with respect to components *WN*(*d*) and optimal in the root-mean-square sense. However, it follows from the lemma that for any other non-optimal linear estimate, including estimates of the form

*C*(*d*, *N*) ⇒ 0*n*, *d* ⇒ ∞. (*A*.9)

*p*(*η*,*YN*)*dηdYN*,

where *p*(*η*,*YN*) is the joint probability density of the random vectors *η* and *YN*. The theorem

Thus, by (A.1), AF determines the vector series that, with the increasing number *m*(*d*, *N*) of its terms, approximates the vector of conditional mathematical expectation of the vector *θ* of

To use formula (A.1), we need to find the matrix inverse to the matrix *QN*(*d*). When the dimension *m*(*d*, *N*) × *m*(*d*, *N*) of the matrix *QN*(*d*) is high and *QN*(*d*) is close to the singular

the estimated parameters with an arbitrary uniformly small root-mean-square error.

(*ZE*(*η*|*YN* )(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*))(*ZE*(*η*|*YN* )(*WN*(*d*)) <sup>−</sup> *<sup>E</sup>*(*η*|*YN*))*<sup>T</sup>*

We assume that C is the covariance matrix of the random vector *P*(*WN*(*d*) − *E*(*θ*|*YN*) :

*P*(*WN*(*d*), the relation *C* ≥ *C*(*d*, *N*) holds. Hence, taking into account (A.8), we obtain

Proposition (A.9) is equivalent to (A.6) if we recall that

(1.1).

Theorem.

polynomial *P*(*WN*(*dε*)) such that

We can rewrite this relation as

It follows from (A.7) that

*<sup>C</sup>*(*d*, *<sup>N</sup>*) =

**A.2. Recurrent (Realizable) MPA algorithm**

is proved.

$$Q\_N(d) = E((W\_N(d) - E(W\_N(d)))(W\_N(d) - E(W\_N(d)))^T);$$

The right-hand sides of these blocks are the first and second (centered) statistical moments calculated by the Monte-Carlo method. However, their left-hand sides also serve as the first and second (centered) statistical moments of components of the vector of conditional mathematical expectations. Hence, we can use mathematical models of form (0.1) and (0.2) to find these statistical moments experimentally for vectors of conditional expectations as well. This obvious proposition gives us the basis for practical implementation of the computational procedure of estimating the vector of the conditional expectation.

We introduce

$$\hat{\eta}(\mathcal{W}\_{\mathcal{N}}(d)) = E(\eta) + \Lambda\_{\mathcal{N}}(d)(\mathcal{W}\_{\mathcal{N}}(d) - E(\mathcal{W}\_{\mathcal{N}}(d))), \quad (A.1)$$

where the matrix Λ*N*(*d*),*r* × *m*(*d*, *N*) satisfies the equation

$$
\Lambda\_N(d) Q\_N(d) = L\_N(d) \,.
$$

We also introduce

$$\tilde{\eta}(\mathcal{W}\_N(d)) = z + \bar{\Lambda}\_N(d)(\mathcal{W}\_N(d) - E(\mathcal{W}\_N(d))), \quad (A.2)$$

where *z* and Λ *<sup>N</sup>*(*d*) are the arbitrary vector and matrix of dimensions *r* × 1 and *r* × *m*(*d*, *N*). Suppose *C*(*d*, *N*) and *C*(*d*, *N*) are the covariance matrices of estimation errors for the vector *<sup>E</sup>*(*η*|*YN*) generated by the estimates *<sup>η</sup>*(*WN*(*d*)) and *<sup>η</sup>*(*WN*(*d*)).

Lemma. The matrix *C*(*d*, *N*) − *C*(*d*, *N*) is a nonnegative definite matrix : *C*(*d*, *N*) ≤ *C*(*d*, *N*). The lemma follows from the identity

$$\bar{\mathcal{C}}(d,N) = \mathcal{C}(d,N) + (\Lambda\_N(d) - \tilde{\Lambda}\_N(d))(\Lambda\_N(d) - \tilde{\Lambda}\_N(d))^T +$$

$$(\Lambda\_N(d)Q\_N(d) - L\_N(d))(\tilde{\Lambda}\_N(d) - \Lambda\_N(d))^T +$$

$$((\Lambda\_N(d)Q\_N(d) - L\_N(d))(\tilde{\Lambda}\_N(d) - \Lambda\_N(d))^T)^T + (z - E(\eta)(z - E\eta)^T. \tag{A.3}$$

Corollary of the lemma. For the vector *<sup>E</sup>*(*η*|*YN*, the vector *<sup>η</sup>*(*WN*(*d*)) is the estimate optimal in the root-mean-square sense among the set of estimates linear with respect to components of the vector *WN*(*d*). If *QN*(*d*) > 0, the estimation vector is unique and

$$\hat{\eta}(\mathcal{W}\_N(d)) = E(\eta) + L\_N(d)Q\_N(d)^{-1}(\mathcal{W}\_N(d) - E(\mathcal{W}\_N(d))). \quad (A.4)$$

The covariance matrix *C*(*d*, *N*) of estimation errors of the vector *E*(*η*|*YN*) is given by the formula

$$\mathcal{C}(d, N) = \mathbb{C}\_{\eta} - \Lambda\_N(d) L\_N(d). (A.5)$$

If *QN*(*d*) ≥ 0, the vectors that provide linear and optimal in the root-mean-square sense estimate are not unique; however, the variances of components of the difference between these vectors are zeros.

Formula (A.1) gives explicit expressions for the vector coefficients of the form *λ*(*a*1, ..., *aN*) in (1.1). To find these relations, we open the explicit expressions for components of the vector *WN*(*d*) and the right-hand side of (A.1) and equate them to the right-hand side of formula (1.1).

We consider asymptotic estimation errors when we use (A.1). Suppose the vector *YN* is fixed. We assume that the vector *E*(*η*|*YN* is given by the function of *YN* on some a priori region that is a compact; the function is continuous on this region. Then, the following theorem holds. Theorem.

$$\operatorname{Supp}\_{\ Y\_N \in \Omega\_{Y\_N}} |\widehat{\eta}(\mathcal{W}\_N(d)) - E(\eta | Y\_N)| \Rightarrow 0, d \Rightarrow \infty. \quad (A.6)$$

Proof. The multidimensional analogue of the K. Weierstrass theorem, which is the corollary of the M. Stone theorem [9], states that for any number *ε* > 0 there exists a multidimensional polynomial *P*(*WN*(*dε*)) such that

$$\sup\_{Y\_N \in \Omega\_{Y\_N}} |P(W\_N(d\_\varepsilon)) - E(\eta | Y\_N)| < \varepsilon.$$

We can rewrite this relation as

18 Will-be-set-by-IN-TECH

*QN*(*d*) = *<sup>E</sup>*((*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*)))(*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*)))*T*);

The right-hand sides of these blocks are the first and second (centered) statistical moments calculated by the Monte-Carlo method. However, their left-hand sides also serve as the first and second (centered) statistical moments of components of the vector of conditional mathematical expectations. Hence, we can use mathematical models of form (0.1) and (0.2) to find these statistical moments experimentally for vectors of conditional expectations as well. This obvious proposition gives us the basis for practical implementation of the computational

*<sup>η</sup>*(*WN*(*d*)) = *<sup>E</sup>*(*η*) + <sup>Λ</sup>*N*(*d*)(*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*))), (*A*.1)

Λ*N*(*d*)*QN*(*d*) = *LN*(*d*).

*<sup>η</sup>*(*WN*(*d*)) = *<sup>z</sup>* <sup>+</sup> <sup>Λ</sup> *<sup>N</sup>*(*d*)(*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*))), (*A*.2)

where *z* and Λ *<sup>N</sup>*(*d*) are the arbitrary vector and matrix of dimensions *r* × 1 and *r* × *m*(*d*, *N*). Suppose *C*(*d*, *N*) and *C*(*d*, *N*) are the covariance matrices of estimation errors for the vector

Lemma. The matrix *C*(*d*, *N*) − *C*(*d*, *N*) is a nonnegative definite matrix : *C*(*d*, *N*) ≤ *C*(*d*, *N*).

*<sup>C</sup>*(*d*, *<sup>N</sup>*) = *<sup>C</sup>*(*d*, *<sup>N</sup>*)+(Λ*N*(*d*) <sup>−</sup> <sup>Λ</sup> *<sup>N</sup>*(*d*))(Λ*N*(*d*) <sup>−</sup> <sup>Λ</sup> *<sup>N</sup>*(*d*))*T*<sup>+</sup>

(Λ*N*(*d*)*QN*(*d*) <sup>−</sup> *LN*(*d*))(<sup>Λ</sup> *<sup>N</sup>*(*d*) <sup>−</sup> <sup>Λ</sup>*N*(*d*))*T*<sup>+</sup> ((Λ*N*(*d*)*QN*(*d*) <sup>−</sup> *LN*(*d*))(<sup>Λ</sup> *<sup>N</sup>*(*d*) <sup>−</sup> <sup>Λ</sup>*N*(*d*))*T*)*<sup>T</sup>* + (*<sup>z</sup>* <sup>−</sup> *<sup>E</sup>*(*η*)(*<sup>z</sup>* <sup>−</sup> *<sup>E</sup>η*)*T*. (*A*.3) Corollary of the lemma. For the vector *<sup>E</sup>*(*η*|*YN*, the vector *<sup>η</sup>*(*WN*(*d*)) is the estimate optimal in the root-mean-square sense among the set of estimates linear with respect to components

*<sup>η</sup>*(*WN*(*d*)) = *<sup>E</sup>*(*η*) + *LN*(*d*)*QN*(*d*)−1(*WN*(*d*) <sup>−</sup> *<sup>E</sup>*(*WN*(*d*))). (*A*.4)

The covariance matrix *C*(*d*, *N*) of estimation errors of the vector *E*(*η*|*YN*) is given by the

*C*(*d*, *N*) = *C<sup>η</sup>* − Λ*N*(*d*)*LN*(*d*).(*A*.5) If *QN*(*d*) ≥ 0, the vectors that provide linear and optimal in the root-mean-square sense estimate are not unique; however, the variances of components of the difference between

Formula (A.1) gives explicit expressions for the vector coefficients of the form *λ*(*a*1, ..., *aN*) in (1.1). To find these relations, we open the explicit expressions for components of the vector

procedure of estimating the vector of the conditional expectation.

where the matrix Λ*N*(*d*),*r* × *m*(*d*, *N*) satisfies the equation

*<sup>E</sup>*(*η*|*YN*) generated by the estimates *<sup>η</sup>*(*WN*(*d*)) and *<sup>η</sup>*(*WN*(*d*)).

of the vector *WN*(*d*). If *QN*(*d*) > 0, the estimation vector is unique and

The lemma follows from the identity

We introduce

We also introduce

formula

these vectors are zeros.

$$\sup\_{Y\_N \in \Omega\_{Y\_N}} |P(\mathcal{W}\_N(d)) - E(\eta | Y\_N)| \Rightarrow 0, \quad d \Rightarrow \infty. \quad (A.7)$$

We assume that C is the covariance matrix of the random vector *P*(*WN*(*d*) − *E*(*θ*|*YN*) :

$$\mathcal{C} = E(\left(P(\mathcal{W}\_N(d)) - E(\eta|\mathcal{Y}\_N)\right) \left(P(\mathcal{W}\_N(d)) - E(\eta|\mathcal{Y}\_N)\right)^T.$$

It follows from (A.7) that

$$
\mathbb{C} \Rightarrow \mathbf{0\_{n\nu}} \quad d \Rightarrow \infty \quad (A.8).
$$

By construction, the vector *<sup>η</sup>*(*WN*(*d*)) provides the estimate of the vector *<sup>η</sup>* that is linear with respect to components *WN*(*d*) and optimal in the root-mean-square sense. However, it follows from the lemma that for any other non-optimal linear estimate, including estimates of the form *P*(*WN*(*d*), the relation *C* ≥ *C*(*d*, *N*) holds. Hence, taking into account (A.8), we obtain

$$C(d, N) \Rightarrow 0\_{n\prime}d \Rightarrow \infty. \quad (A.9)$$

Proposition (A.9) is equivalent to (A.6) if we recall that

$$\mathcal{C}(d,\mathcal{N}) = \int (Z\_{E(\eta|Y\_N)}(\mathcal{W}\_{\mathcal{N}}(d)) - E(\eta|Y\_{\mathcal{N}})) (Z\_{E(\eta|Y\_{\mathcal{N}})}(\mathcal{W}\_{\mathcal{N}}(d)) - E(\eta|Y\_{\mathcal{N}}))^T$$

$$p(\eta, Y\_{\mathcal{N}}) d\eta dY\_{\mathcal{N}}$$

where *p*(*η*,*YN*) is the joint probability density of the random vectors *η* and *YN*. The theorem is proved.

Thus, by (A.1), AF determines the vector series that, with the increasing number *m*(*d*, *N*) of its terms, approximates the vector of conditional mathematical expectation of the vector *θ* of the estimated parameters with an arbitrary uniformly small root-mean-square error.

#### **A.2. Recurrent (Realizable) MPA algorithm**

To use formula (A.1), we need to find the matrix inverse to the matrix *QN*(*d*). When the dimension *m*(*d*, *N*) × *m*(*d*, *N*) of the matrix *QN*(*d*) is high and *QN*(*d*) is close to the singular

where the scalar *zwi*<sup>+</sup><sup>1</sup> is the (*r* + 1)-th component of the vector *V*¯

has processed the components *w*1, ..., *wi*, *V*¯ <sup>−</sup><sup>1</sup>

matrix *C*(*Vi*) with its (*r* + 1)-th component deleted.

*E*(*η*|*wi*, ..., *wi*) after the algorithm processed the vector *Wi*(*d*).

observable components of the vector *WN*(*d*) has the form

*<sup>C</sup>*(*d*, *<sup>i</sup>*) = *<sup>C</sup>η*(0) <sup>−</sup> *<sup>q</sup>*−<sup>1</sup>

problem.

**7. Acknowledgments**

10-08-00415a).

**8. References**

and optimal in the root-mean-square sense estimate of the component after the algorithm

An Algorithm for Parameters Identification of an Aircraft's Dynamics 139

eliminating its component *zwi*<sup>+</sup><sup>1</sup> , the scalar *qi* is the (*r* + 1)-th diagonal element of the matrix *C*(*Vi*), which is the variance of the estimation error of the component *wi*<sup>+</sup><sup>1</sup> after components *w*1, ..., *wi* were processed, *C*(*Vi*)<sup>1</sup> is the matrix formed of *C*(*Vi*) after the (*r* + 1)-th row vector and (*r* + 1)-th column vector were excluded, and *bi* is the (*r* + 1)-th column vector of the

If the scalar *qi* turned out to be close to zero, the component *wi*<sup>+</sup><sup>1</sup> corresponds to a linear combination of components *w*1, ..., *wi*. Then, *wi*<sup>+</sup><sup>1</sup> do not give any new information on *θ* and should be excluded from the computational process. Note that the sequence of random variables like (*wi*<sup>+</sup><sup>1</sup> − *zwi*<sup>+</sup><sup>1</sup> ) forms an updating sequence. The upper left block of the (*r* × *r*)-matrix *C*(*Vi*) includes the covariance matrix *C*(*d*, *i*) of estimation errors of the vector

We assume that *l*(*i*) is the vector composed of r first components of the vector *bi*. The formula representing the evolution of the covariance matrix *C*(*d*, *i*) in the function of the number *i* of

<sup>1</sup> *<sup>l</sup>*(1)*l*(1)*<sup>T</sup>* <sup>−</sup> ... <sup>−</sup> *<sup>q</sup>*−<sup>1</sup>

To test this MPA algorithm, we solved numerically several problems of estimating the components of the state vector for essentially nonlinear dynamic systems. The estimated components are unknown random constant parameters *η*1, ..., *η<sup>r</sup>* of the dynamic system.

As for particular applied problems, we considered smoothing problems and the filtration

In the above examples, we applied the Monte-Carlo method for the number of random realizations lying within 5000 - 10000. This number does not affect the estimation errors provided by the MPA algorithm significantly. The estimated random parameters are assumed to be statistically independent and are a priori uniformly distributed. The value of the root-mean- square deviation *σ*(*i*, *theo*) is determined theoretically by calculating variances :the diagonal elements of the covariance matrix *C*(*d*, *N*). The value of the root-mean-square deviation *σ*(*i*,*exp*) is obtained experimentally by applying the Monte-Carlo method for 5000 realizations. Experimental and theoretical root-mean-square deviations almost coincide,

This study was financially supported by the Russian Foundation for Basic Research (grant no.

[1] V. Klein and A. G. Morelli, Aircraft System Identification: Theory and Methods

which proves that the above given formulas of the MPA algorithm are correct.

(American Institute of Aeronautics and Astronautics, Reston 2006).

*<sup>i</sup>*, the scalar is the linear

*<sup>i</sup>* by

*<sup>i</sup>* is the vector obtained from the vector *<sup>V</sup>*¯

*<sup>i</sup> <sup>l</sup>*(*i*)*l*(*i*)*T*. (*A*.12)

matrix, it is difficult to calculate elements of the inverse matrix. Below, we give the recurrent computational process based on the principle of decomposing observations, described in [6, 7]. Above, we specified the vector *WN*(*d*) of dimension *m*(*d*, *N*) × 1 with the components *w*1, ..., *wm*(*d*,*N*). The computational process consists of *m*(*d*, *N*) successive steps. At each step, we use new updated prior data to find the new estimate of the vector *θ* and perform the prediction, which provides estimates for the rest part of the observation vector. At the same time, we determine the covariance matrix of the estimation errors attained at this step. There is no prediction at the last *m*(*d*, *N*)− th step, and the vector *θ* is refined for the last time.

Let us construct the recurrent algorithm (the MPA algorithm) that does not calculate inverse matrices and consists of *m*(*d*, *N*) steps of calculating the first and second statistical moments for the sequence of special vectors *<sup>V</sup>*1, ..., *Vi*, ..., *Vm*(*d*,*N*) performed after prior moments *<sup>V</sup>*¯(*d*, *<sup>N</sup>*) and *CV*(*d*, *N*) are found for the basic vector *V*(*d*, *N*). We assume that *V*<sup>1</sup> is composed of *r* + *m*(*d*, *N*) − 1 components of the basic vector *V*(*d*, *N*) left after the component *w*<sup>1</sup> was excluded, *<sup>w</sup>*1, ...; *Vi* composed of components of the vector *Vi*−<sup>1</sup> left after the component *wi* was excluded, etc. The component *wm*(*d*,*N*) is the last component of the vector *WN*(*d*), and once we exclude it, the resulting vector *Vm*(*d*,*N*) turns out to equal the estimation vector *η*ˆ(*WN*(*d*)).

At step 1, we use the particular case of formulas of form (A.1) and (A.5) to calculate the vector *V*¯ <sup>1</sup> that estimates the vector *V*<sup>1</sup> and is optimal in the root-mean-square sense and linear with respect to *w*1, and the covariance matrix of the estimation errors *C*(*V*1).

The estimation vector is formed of the estimate of the vector of conditional mathematical expectation *E*(*η*|*YN*) and the vector of dimension *m*(*d*, *N*) − 1) × 1. Once we fix the value *w*1, the latter becomes the vector of statistical prediction of the mean values of "future" values *w*2, ..., *wm*(*d*,*N*). We emphasize that calculations performed at step 1 are based on the preliminary found prior , *V*¯(*d*, *N*), *CV*(*d*, *N*).

Suppose steps 1, ..., *i* of the computational process yielded the vector *V*¯(*d*, *N*) and the matrix *C*(*Vi*) after the values *w*1, ...*wi*, wi were fixed. At step *i* + 1, we have from the particular case of formulas (A.1) and (A.5) the vector *V*¯ *<sup>i</sup>*+<sup>1</sup> that estimates the vector *Vi*<sup>+</sup><sup>1</sup> and is optimal in the root-mean-square sense and linear with respect to *w*1, ..., *wi*+1, and the covariance matrix *C*(*Vi*<sup>+</sup>1) of estimation errors. The vector *V*¯ *<sup>i</sup>*+<sup>1</sup> is still formed of the estimate of the vector of conditional mathematical expectation *<sup>E</sup>*(*η*|*w*1, ..., *wi*<sup>+</sup><sup>1</sup> (first r components of the vector *<sup>V</sup>*¯ *<sup>i</sup>*<sup>+</sup>1) and the vector of statistical prediction of mean values of "future" - h values *wi*+2, ..., *wm*(*d*,*N*) after *<sup>w</sup>*1, ..., *wi*<sup>+</sup><sup>1</sup> (the rest *<sup>m</sup>*(*d*, *<sup>N</sup>*) <sup>−</sup> (*<sup>i</sup>* <sup>+</sup> <sup>1</sup>) components of the vector *<sup>V</sup>*¯ *<sup>i</sup>*+<sup>1</sup> ) are fixed. We emphasize that calculations at step *i* + 1 are based on the preliminary found *V*¯ *<sup>i</sup>*and *C*(*Vi*), which can be naturally called the first and second statistical moments for "future" random values *wi*+1, ..., *wm*(*d*,*N*). These vectors and matrices represent a priori data on statistical moments of components of the vector *Vi*<sup>+</sup><sup>1</sup> before the algorithm receives the value *wi*<sup>+</sup><sup>1</sup> at its input.

Recurrent formulas that corresponds exactly to the above given qualitative description of the computational process have the form

$$
\overline{V}\_{i+1} = \overline{V}\_i^1 + q\_i^{-1} b\_i (w\_{i+1} - z\_{w\_{i+1}})\_\prime \quad (A.10)
$$

$$
\mathbb{C}(V\_{i+1}) = \mathbb{C}(V\_i)^1 - q\_i^{-1} b\_i b\_i^T \quad (A.11)
$$

20 Will-be-set-by-IN-TECH

matrix, it is difficult to calculate elements of the inverse matrix. Below, we give the recurrent computational process based on the principle of decomposing observations, described in [6, 7]. Above, we specified the vector *WN*(*d*) of dimension *m*(*d*, *N*) × 1 with the components *w*1, ..., *wm*(*d*,*N*). The computational process consists of *m*(*d*, *N*) successive steps. At each step, we use new updated prior data to find the new estimate of the vector *θ* and perform the prediction, which provides estimates for the rest part of the observation vector. At the same time, we determine the covariance matrix of the estimation errors attained at this step. There is no prediction at the last *m*(*d*, *N*)− th step, and the vector *θ* is refined for the last time.

Let us construct the recurrent algorithm (the MPA algorithm) that does not calculate inverse matrices and consists of *m*(*d*, *N*) steps of calculating the first and second statistical moments for the sequence of special vectors *<sup>V</sup>*1, ..., *Vi*, ..., *Vm*(*d*,*N*) performed after prior moments *<sup>V</sup>*¯(*d*, *<sup>N</sup>*) and *CV*(*d*, *N*) are found for the basic vector *V*(*d*, *N*). We assume that *V*<sup>1</sup> is composed of *r* + *m*(*d*, *N*) − 1 components of the basic vector *V*(*d*, *N*) left after the component *w*<sup>1</sup> was excluded, *<sup>w</sup>*1, ...; *Vi* composed of components of the vector *Vi*−<sup>1</sup> left after the component *wi* was excluded, etc. The component *wm*(*d*,*N*) is the last component of the vector *WN*(*d*), and once we exclude

At step 1, we use the particular case of formulas of form (A.1) and (A.5) to calculate the vector

The estimation vector is formed of the estimate of the vector of conditional mathematical expectation *E*(*η*|*YN*) and the vector of dimension *m*(*d*, *N*) − 1) × 1. Once we fix the value *w*1, the latter becomes the vector of statistical prediction of the mean values of "future" values *w*2, ..., *wm*(*d*,*N*). We emphasize that calculations performed at step 1 are based on the

Suppose steps 1, ..., *i* of the computational process yielded the vector *V*¯(*d*, *N*) and the matrix *C*(*Vi*) after the values *w*1, ...*wi*, wi were fixed. At step *i* + 1, we have from the particular case

the root-mean-square sense and linear with respect to *w*1, ..., *wi*+1, and the covariance matrix

and the vector of statistical prediction of mean values of "future" - h values *wi*+2, ..., *wm*(*d*,*N*)

which can be naturally called the first and second statistical moments for "future" random values *wi*+1, ..., *wm*(*d*,*N*). These vectors and matrices represent a priori data on statistical moments of components of the vector *Vi*<sup>+</sup><sup>1</sup> before the algorithm receives the value *wi*<sup>+</sup><sup>1</sup> at

Recurrent formulas that corresponds exactly to the above given qualitative description of the

*<sup>i</sup> bi*(*wi*<sup>+</sup><sup>1</sup> − *zwi*<sup>+</sup><sup>1</sup> ), (*A*.10)

*<sup>i</sup>* , (*A*.11)

*<sup>i</sup> bib<sup>T</sup>*

conditional mathematical expectation *<sup>E</sup>*(*η*|*w*1, ..., *wi*<sup>+</sup><sup>1</sup> (first r components of the vector *<sup>V</sup>*¯

after *<sup>w</sup>*1, ..., *wi*<sup>+</sup><sup>1</sup> (the rest *<sup>m</sup>*(*d*, *<sup>N</sup>*) <sup>−</sup> (*<sup>i</sup>* <sup>+</sup> <sup>1</sup>) components of the vector *<sup>V</sup>*¯

*<sup>i</sup>* + *<sup>q</sup>*−<sup>1</sup>

*<sup>C</sup>*(*Vi*<sup>+</sup>1) = *<sup>C</sup>*(*Vi*)<sup>1</sup> <sup>−</sup> *<sup>q</sup>*−<sup>1</sup>

*Vi*<sup>+</sup><sup>1</sup> <sup>=</sup> *<sup>V</sup>*<sup>1</sup>

emphasize that calculations at step *i* + 1 are based on the preliminary found *V*¯

*<sup>i</sup>*+<sup>1</sup> that estimates the vector *Vi*<sup>+</sup><sup>1</sup> and is optimal in

*<sup>i</sup>*+<sup>1</sup> is still formed of the estimate of the vector of

*<sup>i</sup>*<sup>+</sup>1)

*<sup>i</sup>*+<sup>1</sup> ) are fixed. We

*<sup>i</sup>*and *C*(*Vi*),

<sup>1</sup> that estimates the vector *V*<sup>1</sup> and is optimal in the root-mean-square sense and linear with

it, the resulting vector *Vm*(*d*,*N*) turns out to equal the estimation vector *η*ˆ(*WN*(*d*)).

respect to *w*1, and the covariance matrix of the estimation errors *C*(*V*1).

preliminary found prior , *V*¯(*d*, *N*), *CV*(*d*, *N*).

of formulas (A.1) and (A.5) the vector *V*¯

*C*(*Vi*<sup>+</sup>1) of estimation errors. The vector *V*¯

computational process have the form

*V*¯

its input.

where the scalar *zwi*<sup>+</sup><sup>1</sup> is the (*r* + 1)-th component of the vector *V*¯ *<sup>i</sup>*, the scalar is the linear and optimal in the root-mean-square sense estimate of the component after the algorithm has processed the components *w*1, ..., *wi*, *V*¯ <sup>−</sup><sup>1</sup> *<sup>i</sup>* is the vector obtained from the vector *<sup>V</sup>*¯ *<sup>i</sup>* by eliminating its component *zwi*<sup>+</sup><sup>1</sup> , the scalar *qi* is the (*r* + 1)-th diagonal element of the matrix *C*(*Vi*), which is the variance of the estimation error of the component *wi*<sup>+</sup><sup>1</sup> after components *w*1, ..., *wi* were processed, *C*(*Vi*)<sup>1</sup> is the matrix formed of *C*(*Vi*) after the (*r* + 1)-th row vector and (*r* + 1)-th column vector were excluded, and *bi* is the (*r* + 1)-th column vector of the matrix *C*(*Vi*) with its (*r* + 1)-th component deleted.

If the scalar *qi* turned out to be close to zero, the component *wi*<sup>+</sup><sup>1</sup> corresponds to a linear combination of components *w*1, ..., *wi*. Then, *wi*<sup>+</sup><sup>1</sup> do not give any new information on *θ* and should be excluded from the computational process. Note that the sequence of random variables like (*wi*<sup>+</sup><sup>1</sup> − *zwi*<sup>+</sup><sup>1</sup> ) forms an updating sequence. The upper left block of the (*r* × *r*)-matrix *C*(*Vi*) includes the covariance matrix *C*(*d*, *i*) of estimation errors of the vector *E*(*η*|*wi*, ..., *wi*) after the algorithm processed the vector *Wi*(*d*).

We assume that *l*(*i*) is the vector composed of r first components of the vector *bi*. The formula representing the evolution of the covariance matrix *C*(*d*, *i*) in the function of the number *i* of observable components of the vector *WN*(*d*) has the form

$$\mathcal{C}(d, i) = \mathcal{C}\_{\eta}(0) - q\_1^{-1}l(1)l(1)^T - \dots - q\_i^{-1}l(i)l(i)^T. \quad (A.12)$$

To test this MPA algorithm, we solved numerically several problems of estimating the components of the state vector for essentially nonlinear dynamic systems. The estimated components are unknown random constant parameters *η*1, ..., *η<sup>r</sup>* of the dynamic system.

As for particular applied problems, we considered smoothing problems and the filtration problem.

In the above examples, we applied the Monte-Carlo method for the number of random realizations lying within 5000 - 10000. This number does not affect the estimation errors provided by the MPA algorithm significantly. The estimated random parameters are assumed to be statistically independent and are a priori uniformly distributed. The value of the root-mean- square deviation *σ*(*i*, *theo*) is determined theoretically by calculating variances :the diagonal elements of the covariance matrix *C*(*d*, *N*). The value of the root-mean-square deviation *σ*(*i*,*exp*) is obtained experimentally by applying the Monte-Carlo method for 5000 realizations. Experimental and theoretical root-mean-square deviations almost coincide, which proves that the above given formulas of the MPA algorithm are correct.

#### **7. Acknowledgments**

This study was financially supported by the Russian Foundation for Basic Research (grant no. 10-08-00415a).

#### **8. References**

[1] V. Klein and A. G. Morelli, Aircraft System Identification: Theory and Methods (American Institute of Aeronautics and Astronautics, Reston 2006).

**1. Introduction**

The focus of this chapter is an aircraft propelled with four rotors, called the quadrotor. Quadrotor was among the first rotorcrafts ever built. The first successful quadrotor flight was recorded in 1921, when De Bothezat Quadrotor remained airborne for two minutes and 45 seconds. Later he perfected his design, which was then powered by 180-horse power engine and was capable of carrying 3 passengers on limited altitudes. Quadrotor rotorcrafts actually preceded the more common helicopters, but were later replaced by them because of very sophisticated control requirements Gessow & Myers (1952). At the moment, quadrotors are mostly designed as small or micro aircrafts capable of carrying only surveillance equipment. In the future, however, some designs, like Bell Boeing Quad TiltRotor, are being planned for

**Influence of Forward and Descent Flight** 

*Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb* 

*LARICS-Laboratory for Robotics and Intelligent Control Systems* 

**on Quadrotor Dynamics** 

*Department of Control and Computer Engineering,* 

Matko Orsag and Stjepan Bogdan

*Croatia* 

**7**

In the last couple of years, quadrotor aircrafts have been a subject of extensive research in the field of autonomous control systems. This is mostly because of their small size, which prevents them to carry any passengers. Various control algorithms, both for stabilization and control, have been proposed. The authors in Bouabdallah et al. (2004) synthesized and compared PID and LQ controllers used for stabilization of a similar aircraft. They have concluded that classical PID controllers achieve more robust results. In Adigbli et al. (2007); Bouabdallah & Siegwart (2005) "Backstepping" and "Sliding-mode" control techniques are compared. The research presented in Adigbli et al. (2007) shows how PID controllers cannot be used as effective set point tracking controller. Fuzzy based controller is presented in Varga & Bogdan (2009). This controller exhibits good tracking results for simple, predefined trajectories. Each of these control algorithms proved to be successful and energy efficient for a single flying

This chapter examines the behaviour of a quadrotor propulsion system focusing on its limitations (i.e. saturation and dynamic capabilities) and influence that the forward and descent flights have on this propulsion system. A lot of previous research failed to address this practical problem. However, in case of demanding flight trajectories, such as fast forward and descent flight manoeuvres, as well as in the presence of the In Ground Effect, these aerodynamic phenomena could significantly influence quadrotor's dynamics. Authors in Hoffmann et al. (2007) show how control performance can be diminished if aerodynamic effects are not considered. In these situations control signals could drive the propulsion

heavy lift operations Anderson (1981); Warwick (2007).

manoeuvre (hovering, liftoff, horizontal flight, etc.).


## **Influence of Forward and Descent Flight on Quadrotor Dynamics**

Matko Orsag and Stjepan Bogdan

*LARICS-Laboratory for Robotics and Intelligent Control Systems Department of Control and Computer Engineering, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb Croatia* 

### **1. Introduction**

22 Will-be-set-by-IN-TECH

140 Recent Advances in Aircraft Technology

[2] M. B. Tischle and R. K. Remple, Aircraft and Rotorcraft System Identification: Engineering Methods with Flight Test Examples (American Institute of Aeronautics and

[3] R. Jategaonkar, Flying Vehicle System Identification: A Time Domain Methodology

[5] B. K. Poplavskii and G. N. Sirotkin, Integrated Approach to Analysis of Processes of Identification of Model Parameters of a Spacecraft, in Transactions No. 429 of p/ya

[6] J. A. Boguslavskiy, Bayes Estimators of Nonlinear Regression and Allied Issues, Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 4, 14.24 (1996) [Comp. Syst. Sci. 35 (4), 511.521

[7] J. A. Boguslavskiy, A Polynomial Approximation for Nonlinear Problems of Estimation

[8] J. Timmer, H.Rust, W.Horbelt, H.U. Voss, Parametric, nonparametric and parametric modelling of a chaotic circuit time series, Physics Letters A 274 (2000) 123 - 134 [9] A. F. Timan, The Theory of Approximation of Functions of a Real Variable (Nauka,

(American Institute of Aeronautics and Astronautics, Reston, 2006). [4] L. Ljung, System Identification: Theory for the USER (Prentice Hall, 1987).

and Control (Fizmatlit,Moscow, 2006) [in Russian].

Astronautics, Reston, 2006).

Moscow, 1960) [in Russian].

V-8759 [in Russian].

(1996)].

The focus of this chapter is an aircraft propelled with four rotors, called the quadrotor. Quadrotor was among the first rotorcrafts ever built. The first successful quadrotor flight was recorded in 1921, when De Bothezat Quadrotor remained airborne for two minutes and 45 seconds. Later he perfected his design, which was then powered by 180-horse power engine and was capable of carrying 3 passengers on limited altitudes. Quadrotor rotorcrafts actually preceded the more common helicopters, but were later replaced by them because of very sophisticated control requirements Gessow & Myers (1952). At the moment, quadrotors are mostly designed as small or micro aircrafts capable of carrying only surveillance equipment. In the future, however, some designs, like Bell Boeing Quad TiltRotor, are being planned for heavy lift operations Anderson (1981); Warwick (2007).

In the last couple of years, quadrotor aircrafts have been a subject of extensive research in the field of autonomous control systems. This is mostly because of their small size, which prevents them to carry any passengers. Various control algorithms, both for stabilization and control, have been proposed. The authors in Bouabdallah et al. (2004) synthesized and compared PID and LQ controllers used for stabilization of a similar aircraft. They have concluded that classical PID controllers achieve more robust results. In Adigbli et al. (2007); Bouabdallah & Siegwart (2005) "Backstepping" and "Sliding-mode" control techniques are compared. The research presented in Adigbli et al. (2007) shows how PID controllers cannot be used as effective set point tracking controller. Fuzzy based controller is presented in Varga & Bogdan (2009). This controller exhibits good tracking results for simple, predefined trajectories. Each of these control algorithms proved to be successful and energy efficient for a single flying manoeuvre (hovering, liftoff, horizontal flight, etc.).

This chapter examines the behaviour of a quadrotor propulsion system focusing on its limitations (i.e. saturation and dynamic capabilities) and influence that the forward and descent flights have on this propulsion system. A lot of previous research failed to address this practical problem. However, in case of demanding flight trajectories, such as fast forward and descent flight manoeuvres, as well as in the presence of the In Ground Effect, these aerodynamic phenomena could significantly influence quadrotor's dynamics. Authors in Hoffmann et al. (2007) show how control performance can be diminished if aerodynamic effects are not considered. In these situations control signals could drive the propulsion

Fig. 1. Transformation from the body frame to the world frame coordinate system

−→**<sup>F</sup>** <sup>=</sup> *<sup>∂</sup>*

rotor inertia tensor - **Ir**. Angular motion equations can be derived as follows:

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> −→<sup>ω</sup> <sup>×</sup> −→**<sup>M</sup>** <sup>=</sup> *<sup>∂</sup>*−→<sup>ω</sup>

−→**<sup>F</sup>** <sup>=</sup> *<sup>∂</sup>*−→**<sup>v</sup>** *∂t*

not change over time, resulting in a simple equation 3.

−→**<sup>T</sup>** <sup>=</sup> *<sup>∂</sup>*

−→**M**

−→**L** *<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup>*−→**<sup>v</sup>** *∂t*

*mq* = *mq*

where *mq* represents quadrotor mass and −→**v** its velocity vector. Due to the fact that most unmanned quadrotors are electrically driven, it is safe to assume that quadrotor mass does

Same analysis can be applied to angular momentum, having in mind, the angular momentum is produced from the quadrotor motion as well as from the rotors spinning to produce the desired thrust. There are four important variables concerning angular momentum: quadrotor angular speed vector - −→<sup>ω</sup> , rotor angular speed vector - −→**Ω**, quadrotor inertia tensor - **Iq** and

*∂t*

Quadrotors are normally constructed to be completely symmetric. Therefore, their tensor of inertia is a diagonal matrix 5. The same rule applies for rotors as well(otherwise they would be misbalanced and completely useless). Furthermore, rotors spin in one direction only, so

**Iq** <sup>+</sup> −→<sup>ω</sup> <sup>×</sup> **Iq**

Forces and torques, produced from the propulsion system and the surroundings, move and turn the quadrotor. In this paragraph, the quadrotor is viewed as a rigid body with linear and circular momentum, −→*L* and −→*M* respectively. According to the 2nd Newtons law, the force applied to the body equals the change of linear momentum. Using the principal of the change of momentum used in Jazar (2010), the following equation maps the change of quadrotor's

Influence of Forward and Descent Flight on Quadrotor Dynamics 143

*mq* +

 *∂*<sup>2</sup> *x ∂t*<sup>2</sup> *∂*2*y ∂t*<sup>2</sup> *∂*2*z ∂t*<sup>2</sup> *T*

*∂mq ∂t* −→**v**

−→**<sup>M</sup>** <sup>=</sup> −→<sup>ω</sup> <sup>+</sup> **Ir**

−→<sup>ω</sup> <sup>+</sup> −→<sup>ω</sup> <sup>×</sup> **Ir**

−→**Ω**

−→**Ω**

(3)

(4)

**2.2 Dynamic motion equations**

position with respect to the applied force:

system well within the region of saturation, thus causing undesired or unstable quadrotor behaviour. This effect is especially important in situations where the aircraft is operating at its limits (i.e. carrying heavy load, single engine breakdown, etc.).

The proposed analysis of propulsion system is based on the thin airfoil (blade element) theory combined with the momentum theory Bramwell et al. (2001). The analysis takes into account the important aerodynamic effects, specific to quadrotor construction. As a result, the chapter presents analytical expressions showing how thrust, produced by a small propeller used in quadrotor propulsion system, can be significantly influenced by airflow induced from certain manoeuvres.

#### **2. Basic dynamic model**

This section introduces the basic quadrotor dynamic modeling, which includes rigid body dynamics (i.e. Euler equations), kinematics and static nonlinear rotor thrust equation. This model, based on the first order approximation, has been successfully utilized in various quadrotor control designs so far. Nevertheless, recent shift in Unmanned Aerial Vehicle research community towards more payload oriented missions (i.e. pick and place or mobile manipulation missions) emphasized the need for a more complete dynamic model.

#### **2.1 Kinematics**

Quadrotor kinematics problem is, actually, a rigid-body attitude representation problem. Rigid-body attitude can be accurately described with a set of 3-by-3 orthogonal matrices. Additionally, the determinant of these matrices has to be one Chaturvedi et al. (2011). Since matrix representation cannot give a clear insight into the exact rigid body pose, attitude is often studied using parameterizations Shuster (1993). Regardless of the choice, every parameterization at some point fails to fully represent rigid body pose. Due to the gimbal lock, Euler angles cannot globally represent rigid body pose, whereas quaternions cannot define it uniquely.Chaturvedi et al. (2011)

Although researchers proved the effectiveness of using quaternions in quadrotor control Stingu & Lewis (2009), Euler angles are still the most common way of representing rigid body pose. To uniquely describe quadrotor pose using Euler angles, a composition of 3 elemental rotations is chosen. Following *X* − *Y* − *Z* convention, a world reference coordinate system is first rotated Ψ degrees around *X* axis. After this, a Θ degree rotation around an intermediate *Y* axis is applied. Finally, a Φ degree rotation around a newly formed *Z* axis is applied to yield a transformation matrix from the world coordinate system W to the body frame B, as shown in figure 1. Equations 1 and 2 formalize this procedure:

$$\text{Rot}\left(\Phi/\Theta,\Psi\right) = \text{Rot}\left(z^w,\Phi\right)\text{Rot}\left(y^w,\Theta\right)\text{Rot}\left(\mathbf{x}^w,\Psi\right) \tag{1}$$

$$\operatorname{Rot}\left(\Phi,\Theta,\Psi\right) = \begin{bmatrix} c\phi c\theta \ c\phi s\theta s\psi - s\phi c\psi \ c\phi s\theta c\psi + s\phi s\psi\\ s\phi c\theta \ s\phi s\theta s\psi + c\phi c\psi \ s\phi s\theta c\psi - c\phi s\psi\\ -s\theta & c\theta s\psi & c\theta c\psi \end{bmatrix} \tag{2}$$

where *cφ* and *sφ* stand for *cos*(*φ*) and *sin*(*φ*), respectively. The same abbreviations are applied to other angles as well.

Fig. 1. Transformation from the body frame to the world frame coordinate system

#### **2.2 Dynamic motion equations**

2 Will-be-set-by-IN-TECH

system well within the region of saturation, thus causing undesired or unstable quadrotor behaviour. This effect is especially important in situations where the aircraft is operating at its

The proposed analysis of propulsion system is based on the thin airfoil (blade element) theory combined with the momentum theory Bramwell et al. (2001). The analysis takes into account the important aerodynamic effects, specific to quadrotor construction. As a result, the chapter presents analytical expressions showing how thrust, produced by a small propeller used in quadrotor propulsion system, can be significantly influenced by airflow induced from certain

This section introduces the basic quadrotor dynamic modeling, which includes rigid body dynamics (i.e. Euler equations), kinematics and static nonlinear rotor thrust equation. This model, based on the first order approximation, has been successfully utilized in various quadrotor control designs so far. Nevertheless, recent shift in Unmanned Aerial Vehicle research community towards more payload oriented missions (i.e. pick and place or mobile

Quadrotor kinematics problem is, actually, a rigid-body attitude representation problem. Rigid-body attitude can be accurately described with a set of 3-by-3 orthogonal matrices. Additionally, the determinant of these matrices has to be one Chaturvedi et al. (2011). Since matrix representation cannot give a clear insight into the exact rigid body pose, attitude is often studied using parameterizations Shuster (1993). Regardless of the choice, every parameterization at some point fails to fully represent rigid body pose. Due to the gimbal lock, Euler angles cannot globally represent rigid body pose, whereas quaternions cannot

Although researchers proved the effectiveness of using quaternions in quadrotor control Stingu & Lewis (2009), Euler angles are still the most common way of representing rigid body pose. To uniquely describe quadrotor pose using Euler angles, a composition of 3 elemental rotations is chosen. Following *X* − *Y* − *Z* convention, a world reference coordinate system is first rotated Ψ degrees around *X* axis. After this, a Θ degree rotation around an intermediate *Y* axis is applied. Finally, a Φ degree rotation around a newly formed *Z* axis is applied to yield a transformation matrix from the world coordinate system W to the body frame B, as shown

where *cφ* and *sφ* stand for *cos*(*φ*) and *sin*(*φ*), respectively. The same abbreviations are applied

*Rot*(Φ, Θ, Ψ) = *Rot*(*zw*, Φ) *Rot*(*yw*, Θ) *Rot*(*xw*, Ψ) (1)

⎤

⎦ (2)

*cφcθ cφsθsψ* − *sφcψ cφsθcψ* + *sφsψ sφcθ sφsθsψ* + *cφcψ sφsθcψ* − *cφsψ* −*sθ cθsψ cθcψ*

manipulation missions) emphasized the need for a more complete dynamic model.

limits (i.e. carrying heavy load, single engine breakdown, etc.).

manoeuvres.

**2.1 Kinematics**

**2. Basic dynamic model**

define it uniquely.Chaturvedi et al. (2011)

in figure 1. Equations 1 and 2 formalize this procedure:

*Rot*(Φ, Θ, Ψ) =

to other angles as well.

⎡ ⎣ Forces and torques, produced from the propulsion system and the surroundings, move and turn the quadrotor. In this paragraph, the quadrotor is viewed as a rigid body with linear and circular momentum, −→*L* and −→*M* respectively. According to the 2nd Newtons law, the force applied to the body equals the change of linear momentum. Using the principal of the change of momentum used in Jazar (2010), the following equation maps the change of quadrotor's position with respect to the applied force:

$$\begin{aligned} \overrightarrow{\mathbf{F}} &= \frac{\partial \overrightarrow{\mathbf{L}}}{\partial t} = \frac{\partial \overrightarrow{\mathbf{v}}}{\partial t} m\_q + \frac{\partial m\_q}{\partial t} \overrightarrow{\mathbf{v}} \\ \overrightarrow{\mathbf{F}} &= \frac{\partial \overrightarrow{\mathbf{v}}}{\partial t} m\_q = m\_q \begin{bmatrix} \frac{\partial^2 x}{\partial t^2} & \frac{\partial^2 y}{\partial t^2} & \frac{\partial^2 z}{\partial t^2} \end{bmatrix}^T \end{aligned} \tag{3}$$

where *mq* represents quadrotor mass and −→**v** its velocity vector. Due to the fact that most unmanned quadrotors are electrically driven, it is safe to assume that quadrotor mass does not change over time, resulting in a simple equation 3.

Same analysis can be applied to angular momentum, having in mind, the angular momentum is produced from the quadrotor motion as well as from the rotors spinning to produce the desired thrust. There are four important variables concerning angular momentum: quadrotor angular speed vector - −→<sup>ω</sup> , rotor angular speed vector - −→**Ω**, quadrotor inertia tensor - **Iq** and rotor inertia tensor - **Ir**. Angular motion equations can be derived as follows:

$$\begin{aligned} \overrightarrow{\mathbf{M}} &= \overrightarrow{\omega} + \mathbf{I\_r} \overrightarrow{\mathbf{M}} \\ \overrightarrow{\mathbf{T}} = \frac{\partial \overrightarrow{\mathbf{M}}}{\partial t} + \overrightarrow{\omega} \times \overrightarrow{\mathbf{M}} &= \frac{\partial \overrightarrow{\omega}}{\partial t} \mathbf{I\_q} + \overrightarrow{\omega} \times \mathbf{I\_q} \overrightarrow{\omega} + \overrightarrow{\omega} \times \mathbf{I\_r} \overrightarrow{\mathbf{Q}} \end{aligned} \tag{4}$$

Quadrotors are normally constructed to be completely symmetric. Therefore, their tensor of inertia is a diagonal matrix 5. The same rule applies for rotors as well(otherwise they would be misbalanced and completely useless). Furthermore, rotors spin in one direction only, so

Fig. 2. A side by side image of X and Plus quadrotor configurations

Influence of Forward and Descent Flight on Quadrotor Dynamics 145

Fig. 3. Plus configuration control inputs for rotation, lift and forward motion. Arrow

payload and operates near the point of saturation, it is wiser to use the cross configuration. Changing the speed of each blade for a small amount, as opposed to changing only two blades but doubling the amount of speed change, will keep the engines safe from saturation point. Basic control sequences of cross configuration are shown in figure 3. First approximation of rotor dynamics implies that rotors produce only the vertical thrust force. As the rotors are displaced from the axis of rotation (i.e. *x* and *y* axis) they produce corresponding torques,

thickness stands for higher speed.

the rotor angular speed vector −→**<sup>Ω</sup>** has only one component <sup>Ω</sup>*z*. Evaluating 3 yields a circular motion equation 6.

$$\mathbf{I\_{q}} = \begin{bmatrix} I\_{xx} & 0 & 0 \\ 0 & I\_{yy} & 0 \\ 0 & 0 & I\_{zz} \end{bmatrix} \tag{5}$$

$$M\_{\rm X} = I\_{\rm xx} \frac{\mathbf{d}\omega\_{\rm x}}{\mathbf{d}t} - \left(I\_{\rm yy} - I\_{zz}\right)\omega\_{\rm y}\omega\_{\rm z} + I\_{\rm r}\omega\_{\rm y}\Omega\_{\rm z}$$

$$M\_{\rm Y} = I\_{\rm yy}\frac{\mathbf{d}\omega\_{\rm y}}{\mathbf{d}t} - \left(I\_{zz} - I\_{\rm xx}\right)\omega\_{\rm x}\omega\_{\rm z} - I\_{\rm r}\omega\_{\rm x}\Omega\_{\rm z} \tag{6}$$

$$M\_{\rm z} = I\_{\rm zz}\frac{\mathbf{d}\omega\_{\rm z}}{\mathbf{d}t} - \left(I\_{\rm xx} - I\_{\rm yy}\right)\omega\_{\rm x}\omega\_{\rm y}$$

Equation 6 calculates rotation speeds in the body frame coordinate system. To transform these body frame angular velocities into world frame rotations, one needs a transformation matrix 8. This matrix is derived from successive elemental transformations 7 similarly as kinematics equation 2. Infinitesimal changes in Euler angles, affect the rotation vector in a way that the first Euler angle Ψ undergoes two additional rotations, the second angle Θ only one additional rotation, and the final Euler angle Φ no additional rotations Jazar (2010):

$$
\begin{bmatrix} \omega\_x\\ \omega\_y\\ \omega\_z \end{bmatrix}^{\mathfrak{B}} = \begin{bmatrix} 0\\ 0\\ \dot{\Phi} \end{bmatrix}^{\mathfrak{W}} + \operatorname{Rot} \left( \Phi, z^{\mathfrak{W}} \right)^{\mathrm{T}} \begin{bmatrix} 0\\ \dot{\Theta}\\ 0 \end{bmatrix}^{\mathfrak{W}} + \operatorname{Rot} \left( \Phi, z^{\mathfrak{W}} \right)^{\mathrm{T}} \operatorname{Rot} \left( \Theta, y^{\mathfrak{W}} \right)^{\mathrm{T}} \begin{bmatrix} \dot{\Psi}\\ 0\\ 0 \end{bmatrix}^{\mathfrak{W}} \tag{7}
$$

$$\mathbf{J} = \begin{bmatrix} \cos(\Psi) / \cos(\Theta) \sin(\Psi) / \cos(\Theta) \ 0 \\ -\sin(\Psi) & \cos(\Psi) \ 0 \\ \cos(\Psi) \tan(\Theta) & \sin(\Psi) \tan(\Theta) & 1 \end{bmatrix} \tag{8}$$

#### **2.3 Rotor forces and torques**

Four quadrotor blades are placed in a square shaped form. Blades that are next to each other spin in opposite directions, thus maintaining inherent stability of the aircraft. The same four blades that make the quadrotor hover enable it to move in the desired direction. Therefore, in order for quadrotor to move, it has to be pitched and rolled in the desired direction. To pitch and roll the quadrotor, some blades need to spin faster, while others spin slower. This produces the desired torques, which in term affect aircraft attitude and position Orsag et al. (2010).

Depending on the orientation of the blades, relative to the body coordinate system, there are two basic types of quadrotor configurations: cross and plus configuration shown in figure 2. In the plus configuration, a pair of blades spinning in the same direction, are placed on *x* and *y* coordinates of the body frame coordinate system. With this configuration it is easier to control the aircraft, because each move (i.e. *x* or *y* direction) requires a controller to disbalance only the speeds of two blades placed on the desired direction.

The cross configuration, on the other hand, requires that the blades are placed in each quadrant of the body frame coordinate system. In such a configuration each move requires all four blades to vary their rotation speed. Although the control system seems to be more complex, there is one big advantage to the cross construction. Keeping in mind that the amount of torque needed to rotate the aircraft is very similar for both configurations, it takes less change per blade if all four blades change their speeds. Therefore, when the aircraft carries 4 Will-be-set-by-IN-TECH

the rotor angular speed vector −→**<sup>Ω</sup>** has only one component <sup>Ω</sup>*z*. Evaluating 3 yields a circular

*Ixx* 0 0 0 *Iyy* 0 0 0 *Izz*

*Iyy* − *Izz*

d*ω<sup>z</sup>* <sup>d</sup>*<sup>t</sup>* <sup>−</sup> �

Equation 6 calculates rotation speeds in the body frame coordinate system. To transform these body frame angular velocities into world frame rotations, one needs a transformation matrix 8. This matrix is derived from successive elemental transformations 7 similarly as kinematics equation 2. Infinitesimal changes in Euler angles, affect the rotation vector in a way that the first Euler angle Ψ undergoes two additional rotations, the second angle Θ only one additional

�

<sup>d</sup>*<sup>t</sup>* <sup>−</sup> (*Izz* <sup>−</sup> *Ixx*) *<sup>ω</sup>xω<sup>z</sup>* <sup>−</sup> *Irωx*Ω*<sup>z</sup>*

⎤

*ωyω<sup>z</sup>* + *Irωy*Ω*<sup>z</sup>*

� *ωxωy*

<sup>Φ</sup>, *<sup>z</sup>*W�*<sup>T</sup>*

*Rot* �

⎤

<sup>Θ</sup>, *<sup>y</sup>*W�*<sup>T</sup>*

⎡ ⎣ Ψ˙ 0 0

⎦ (8)

⎤ ⎦ W

(7)

*Ixx* − *Iyy*

⎦ (5)

(6)

⎡ ⎣

**Iq** =

*Mz* = *Izz*

d*ω<sup>x</sup>* <sup>d</sup>*<sup>t</sup>* <sup>−</sup> �

d*ω<sup>y</sup>*

rotation, and the final Euler angle Φ no additional rotations Jazar (2010):

<sup>Φ</sup>, *<sup>z</sup>*W�*<sup>T</sup>*

⎡ ⎣ 0 Θ˙ 0

⎤ ⎦

W

*cos*(Ψ)/*cos*(Θ) *sin*(Ψ)/*cos*(Θ) 0 −*sin*(Ψ) *cos*(Ψ) 0 *cos*(Ψ)*tan*(Θ) *sin*(Ψ)*tan*(Θ) 1

Four quadrotor blades are placed in a square shaped form. Blades that are next to each other spin in opposite directions, thus maintaining inherent stability of the aircraft. The same four blades that make the quadrotor hover enable it to move in the desired direction. Therefore, in order for quadrotor to move, it has to be pitched and rolled in the desired direction. To pitch and roll the quadrotor, some blades need to spin faster, while others spin slower. This produces the desired torques, which in term affect aircraft attitude and position Orsag et al.

Depending on the orientation of the blades, relative to the body coordinate system, there are two basic types of quadrotor configurations: cross and plus configuration shown in figure 2. In the plus configuration, a pair of blades spinning in the same direction, are placed on *x* and *y* coordinates of the body frame coordinate system. With this configuration it is easier to control the aircraft, because each move (i.e. *x* or *y* direction) requires a controller to disbalance only

The cross configuration, on the other hand, requires that the blades are placed in each quadrant of the body frame coordinate system. In such a configuration each move requires all four blades to vary their rotation speed. Although the control system seems to be more complex, there is one big advantage to the cross construction. Keeping in mind that the amount of torque needed to rotate the aircraft is very similar for both configurations, it takes less change per blade if all four blades change their speeds. Therefore, when the aircraft carries

+ *Rot* �

*Mx* = *Ixx*

*My* = *Iyy*

motion equation 6.

⎡ ⎣ *ωx ωy ωz*

(2010).

⎤ ⎦

B = ⎡ ⎣ 0 0 Φ˙

**2.3 Rotor forces and torques**

⎤ ⎦

W

+ *Rot* �

> **J** = ⎡ ⎣

the speeds of two blades placed on the desired direction.

Fig. 2. A side by side image of X and Plus quadrotor configurations

Fig. 3. Plus configuration control inputs for rotation, lift and forward motion. Arrow thickness stands for higher speed.

payload and operates near the point of saturation, it is wiser to use the cross configuration. Changing the speed of each blade for a small amount, as opposed to changing only two blades but doubling the amount of speed change, will keep the engines safe from saturation point. Basic control sequences of cross configuration are shown in figure 3. First approximation of rotor dynamics implies that rotors produce only the vertical thrust force. As the rotors are displaced from the axis of rotation (i.e. *x* and *y* axis) they produce corresponding torques,

Fig. 4. Momentum theory - horizontal motion, vertical motion and induced speed total

Influence of Forward and Descent Flight on Quadrotor Dynamics 147

Blade element theory observes a small rotor blade element Δ*r* 5. Figure 5 shows this infinitesimal part of quadrotor's blade together with elemental lift and drag forces it produces Bramwell et al. (2001). For better clarity angles are drawn larger than they actually are:

where *CL* and *CD* are lift and drag coefficients, *S* is the surface of the element and *Vstr* the airflow around the blade element. The airflow is mostly produced from the rotor spin Ω*R* and therefore depends on the distance of each blade element to the center of blade rotation. Adding to this airflow is the total air stream coming from quadrotor's vertical and horizontal movement, *VS* = *Vxy* + *Vz*. Finally, blade rotation produces additional induced speed *vi*. The ideal airfoil lift coefficient *CL* can be calculated using equation 12 Gessow & Myers (1952).

where *a* is an aerodynamic coefficient, ideally equal to 2*π*. The effective angle of attack *αe f* , is the angle between the airflow and the blade. Its value changes with the change of airflow

Standard rotor blades are twisted because the dominant airflow coming from blade rotation increases linearly towards the end of the blade. According to equation 11 this causes the increase of lift and drag forces. The difference in forces produced near and far from the center of rotation would cause the blade to twist, and ultimately brake. To avoid that, a linear twist,

*<sup>α</sup>m*(*r*) = <sup>Θ</sup><sup>0</sup> <sup>−</sup> *<sup>r</sup>*

*ρVstrCLS*

*ρVstrCDS*

*<sup>z</sup>* of a free-falling plate with a drag coefficient

*CL* = *aαe f* = 2*παe f* (12)

*<sup>R</sup> Qtw* (13)

(11)

2*CDρR*2*πV*<sup>2</sup>

Δ*L* <sup>Δ</sup>*<sup>R</sup>* <sup>=</sup> <sup>1</sup> 2

Δ*D* <sup>Δ</sup>*<sup>R</sup>* <sup>=</sup> <sup>1</sup> 2

airflow vector sum

*CD* = 1.

equal to the drag equation *D* = <sup>1</sup>

direction and due to the blade twist.

**3.1.2 Blade element theory**

−→**Mx** <sup>=</sup> −→**<sup>F</sup>** <sup>×</sup> −→**rx** and −→**My** <sup>=</sup> −→**<sup>F</sup>** <sup>×</sup> −→**ry** respectively. Torque −→**Mz** comes from the spinning of each rotor blade **Ir** −→**Ω**. Adding the corresponding thrust forces and torques yields the following equation:

$$\begin{aligned} \overrightarrow{\mathbf{F\_{tot}}} &= \overrightarrow{\mathbf{T\_1}} + \overrightarrow{\mathbf{T\_2}} + \overrightarrow{\mathbf{T\_3}} + \overrightarrow{\mathbf{T\_4}} \\ M\_x^{tot} &= M\_x^2 + M\_x^3 - M\_x^1 - M\_x^4 \\ M\_y^{tot} &= M\_y^3 + M\_y^4 - M\_y^1 - M\_y^2 \\ M\_z^{tot} &= M\_z^2 + M\_z^4 - M\_z^1 - M\_z^3 \end{aligned} \tag{9}$$

#### **3. Aerodynamics**

As the quadrotor research shifts to new research areas (i.e. Mobile manipulation, Aerobatic moves, etc.) Korpela et al. (2011); Mellinger et al. (2010), the need for an elaborate mathematical model arises. The model needs to incorporate a full spectrum of aerodynamic effects that act on the quadrotor during climb, descent and forward flight. To derive a more complete mathematical model of a quadrotor, one needs to start with basic concepts of momentum theory and blade elemental theory.

#### **3.1 Combining momentum and blade elemental theory**

The momentum theory of a rotor, also known as classical actuator disk theory, combines rotor thrust, induced velocity (i.e. airspeed produced in rotor) and aircraft speed into a single equation. On the other hand, blade elemental theory is used to calculate forces and torques acting on the rotor by studying a small rotor blade element modeled as an airplane wing so that the airfoil theory can be applied.Bramwell et al. (2001) A combination of these two views, macroscopic and microscopic, yields a base ground for a good approximative mathematical model.

#### **3.1.1 Momentum theory**

Basic momentum theory offers two solutions, one for each of the two operational states in which the defined rotor slipstream exists. The solutions refer to rotorcraft climb and descent, the so called helicopter and the windmill states. Quadrotor in a combined lateral and vertical move is shown in figure 4. The figure shows the most important airflows viewed in Momentum theory: *Vz* and *Vxy* that are induced by quadrotor's movement, together with the induced speed *vi* that is produced by the rotors.

Unfortunately, classic momentum theory implies no steady state transition between the helicopter and the windmill states. Experimental results, however, show that this transition exists. In order for momentum theory to comply with experimental results, the augmented momentum theory equation 10 is proposed Gessow & Myers (1952),

$$T = 2\rho R^2 \pi v\_i \sqrt{(v\_i + V\_z)^2 + V\_{xy}^2 + \frac{V\_z^2}{7.67}}\tag{10}$$

where *<sup>V</sup>*<sup>2</sup> *z* 7.67 term is introduced to assure that the augmented momentum theory equation complies with experimental results, *R* stands for rotor radius and *ρ* is the air density. It is easy to show that in case of autorotation with no forward speed, thrust in equation 10 becomes

Fig. 4. Momentum theory - horizontal motion, vertical motion and induced speed total airflow vector sum

equal to the drag equation *D* = <sup>1</sup> 2*CDρR*2*πV*<sup>2</sup> *<sup>z</sup>* of a free-falling plate with a drag coefficient *CD* = 1.

#### **3.1.2 Blade element theory**

6 Will-be-set-by-IN-TECH

−→**Mx** <sup>=</sup> −→**<sup>F</sup>** <sup>×</sup> −→**rx** and −→**My** <sup>=</sup> −→**<sup>F</sup>** <sup>×</sup> −→**ry** respectively. Torque −→**Mz** comes from the spinning of each

*<sup>x</sup>* + *<sup>M</sup>*<sup>3</sup>

*<sup>y</sup>* + *<sup>M</sup>*<sup>4</sup>

*<sup>z</sup>* + *<sup>M</sup>*<sup>4</sup>

As the quadrotor research shifts to new research areas (i.e. Mobile manipulation, Aerobatic moves, etc.) Korpela et al. (2011); Mellinger et al. (2010), the need for an elaborate mathematical model arises. The model needs to incorporate a full spectrum of aerodynamic effects that act on the quadrotor during climb, descent and forward flight. To derive a more complete mathematical model of a quadrotor, one needs to start with basic concepts

The momentum theory of a rotor, also known as classical actuator disk theory, combines rotor thrust, induced velocity (i.e. airspeed produced in rotor) and aircraft speed into a single equation. On the other hand, blade elemental theory is used to calculate forces and torques acting on the rotor by studying a small rotor blade element modeled as an airplane wing so that the airfoil theory can be applied.Bramwell et al. (2001) A combination of these two views, macroscopic and microscopic, yields a base ground for a good approximative mathematical

Basic momentum theory offers two solutions, one for each of the two operational states in which the defined rotor slipstream exists. The solutions refer to rotorcraft climb and descent, the so called helicopter and the windmill states. Quadrotor in a combined lateral and vertical move is shown in figure 4. The figure shows the most important airflows viewed in Momentum theory: *Vz* and *Vxy* that are induced by quadrotor's movement, together with

Unfortunately, classic momentum theory implies no steady state transition between the helicopter and the windmill states. Experimental results, however, show that this transition exists. In order for momentum theory to comply with experimental results, the augmented

(*vi* + *Vz*)

7.67 term is introduced to assure that the augmented momentum theory equation complies with experimental results, *R* stands for rotor radius and *ρ* is the air density. It is easy to show that in case of autorotation with no forward speed, thrust in equation 10 becomes

<sup>2</sup> + *Vxy*

<sup>2</sup> + *Vz* 2

7.67 (10)

*Mtot <sup>x</sup>* = *<sup>M</sup>*<sup>2</sup>

*Mtot <sup>y</sup>* = *<sup>M</sup>*<sup>3</sup>

*Mtot <sup>z</sup>* = *<sup>M</sup>*<sup>2</sup>

of momentum theory and blade elemental theory.

**3.1 Combining momentum and blade elemental theory**

the induced speed *vi* that is produced by the rotors.

momentum theory equation 10 is proposed Gessow & Myers (1952),

*T* = 2*ρR*2*πvi*

−→**Ftot** <sup>=</sup> −→**T1** <sup>+</sup> −→**T2** <sup>+</sup> −→**T3** <sup>+</sup> −→**T4**

*<sup>x</sup>* <sup>−</sup> *<sup>M</sup>*<sup>1</sup>

*<sup>y</sup>* <sup>−</sup> *<sup>M</sup>*<sup>1</sup>

*<sup>z</sup>* <sup>−</sup> *<sup>M</sup>*<sup>1</sup>

*<sup>x</sup>* <sup>−</sup> *<sup>M</sup>*<sup>4</sup> *x*

(9)

*<sup>y</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> *y*

*<sup>z</sup>* <sup>−</sup> *<sup>M</sup>*<sup>3</sup> *z*

−→**Ω**. Adding the corresponding thrust forces and torques yields the following

rotor blade **Ir**

**3. Aerodynamics**

equation:

model.

where *<sup>V</sup>*<sup>2</sup> *z*

**3.1.1 Momentum theory**

Blade element theory observes a small rotor blade element Δ*r* 5. Figure 5 shows this infinitesimal part of quadrotor's blade together with elemental lift and drag forces it produces Bramwell et al. (2001). For better clarity angles are drawn larger than they actually are:

$$\begin{aligned} \frac{\Delta L}{\Delta R} &= \frac{1}{2} \rho V\_{str} \mathbf{C}\_{LS} \\ \frac{\Delta D}{\Delta R} &= \frac{1}{2} \rho V\_{str} \mathbf{C}\_{D} S \end{aligned} \tag{11}$$

where *CL* and *CD* are lift and drag coefficients, *S* is the surface of the element and *Vstr* the airflow around the blade element. The airflow is mostly produced from the rotor spin Ω*R* and therefore depends on the distance of each blade element to the center of blade rotation. Adding to this airflow is the total air stream coming from quadrotor's vertical and horizontal movement, *VS* = *Vxy* + *Vz*. Finally, blade rotation produces additional induced speed *vi*. The ideal airfoil lift coefficient *CL* can be calculated using equation 12 Gessow & Myers (1952).

$$\mathbb{C}\_{L} = a \mathfrak{a}\_{ef} = 2 \,\pi \mathfrak{a}\_{ef} \tag{12}$$

where *a* is an aerodynamic coefficient, ideally equal to 2*π*. The effective angle of attack *αe f* , is the angle between the airflow and the blade. Its value changes with the change of airflow direction and due to the blade twist.

Standard rotor blades are twisted because the dominant airflow coming from blade rotation increases linearly towards the end of the blade. According to equation 11 this causes the increase of lift and drag forces. The difference in forces produced near and far from the center of rotation would cause the blade to twist, and ultimately brake. To avoid that, a linear twist,

$$
\mathfrak{a}\_{m}(r) = \Theta\_{0} - \frac{r}{R} \mathcal{Q}\_{tw} \tag{13}
$$

Fig. 6. Blade element in quadrotor coordinate system

*FV* <sup>=</sup> *<sup>N</sup>ρacR*¯ <sup>3</sup>Ω<sup>2</sup> 4

*CT* =

Variables *<sup>μ</sup>*,*λ<sup>i</sup>* and *<sup>λ</sup><sup>c</sup>* are speed coefficients *Vxy*

2 <sup>3</sup> <sup>+</sup> *<sup>μ</sup>*<sup>2</sup>

average cord length of the blade element shown in figure 5.

*CHx* = cos (*α*) *μ*

*CHy* = sin (*α*) *μ*

separately in 18.

2

<sup>3</sup> <sup>+</sup> *<sup>μ</sup>*<sup>2</sup>

 Θ<sup>0</sup> − 1 + *μ*<sup>2</sup>

y components, coming both from the drag and lift of the rotor, given in 19.

 *CD a*

 *CD a*

where *ψ* is the blade angle due to rotation, taken at a certain sample time. Solving integral equation 16 yields the expression for rotor thrust (i.e. vertical force) Orsag & Bogdan (2009):

Influence of Forward and Descent Flight on Quadrotor Dynamics 149

The term inside the brackets of equation 17 is known as a thrust coefficient, and is given

*<sup>R</sup>*<sup>Ω</sup> , *Vz*

The same approach can be applied for the calculation of horizontal forces and torques produced within the quadrotor Orsag & Bogdan (2009). Calculated lateral force has x and

+ (*λ<sup>i</sup>* + *λc*)

+ (*λ<sup>i</sup>* + *λc*)

In case of torque equations the angles between the forces and directions are easily derived from basic geometric relations shown in figure 6, resulting in the elemental torque equations

Θ*tw*

Θ*tw*

<sup>Θ</sup><sup>0</sup> <sup>−</sup> <sup>Θ</sup>*tw* 2

<sup>Θ</sup><sup>0</sup> <sup>−</sup> <sup>Θ</sup>*tw* 2

*<sup>R</sup>*<sup>Ω</sup> and *vi*

<sup>2</sup> <sup>−</sup> *<sup>λ</sup><sup>i</sup>* <sup>−</sup> *<sup>λ</sup><sup>c</sup>*

<sup>2</sup> <sup>−</sup> *<sup>λ</sup><sup>i</sup>* <sup>−</sup> *<sup>λ</sup><sup>c</sup>* (18)

*<sup>R</sup>*<sup>Ω</sup> respectively. New constant *c*¯ is the

(19)

(17)

 Θ<sup>0</sup> − 1 + *μ*<sup>2</sup>

Fig. 5. Infitesimal rotor blade element Δ*r* in surrounding airflow Orsag & Bogdan (2009)

is introduced to the blade design.

The effect of varying airflow can be calculated separating the vertical components *Vz* + *vi* and horizontal ones *Vxy* + Ω*r*. The airflow direction angle Φ can be easily calculated from the equation

$$\Phi = \arctan\left(\frac{V\_z + v\_i}{V\_x y + \Omega r}\right) \approx \arctan\left(\frac{V\_z + v\_i}{\Omega r}\right) \tag{14}$$

As lift and drag forces are not aligned with body frame of reference, horizontal and vertical projection forces need to be derived. Keeping in mind that Ω*r* � {*Vz*, *vi*, *Vxy*} small angle approximations cos(Φ) ≈ 1 and sin(Φ) ≈ Φ can be used. Moreover, in a well balanced rotor blade, drag force should be negligible compared to the lift Gessow & Myers (1952). Applying this considerations to 11 and keeping in mind the relations from figure 5 enables the derivation of horizontal and vertical force equations 15.

$$\begin{aligned} \frac{\text{d}F\_V}{\text{d}r} &= \frac{\text{d}L}{\text{d}r} \cos(\Phi) + \frac{\text{d}D}{\text{d}r} \sin(\Phi) \approx \frac{\text{d}L}{\text{d}r} = \rho V\_{\text{tot}}^2 c \pi \alpha\_{\text{ef}}\\ \frac{\text{d}F\_H}{\text{d}r} &= \frac{\text{d}L}{\text{d}r} \sin(\Phi) + \frac{\text{d}D}{\text{d}r} \cos(\Phi) \approx \frac{1}{2}\rho V\_{\text{tot}}^2 \mathbb{C}\_D S + \frac{1}{2}\rho V\_{\text{tot}}^2 \mathbb{C}\_L S \Phi \end{aligned} \tag{15}$$

#### **3.1.3 Applying blade element theory to quadrotor construction**

This section continues with the observation of a small rotor blade element Δ*r* from the previous section, placing it in real surroundings shown in figure 6. Since the blades rotate, the forces produced by blade elements tend to change both in size and direction. This is the reason why an average elemental thrust of all blade elements should be calculated.

Figure 6 shows the relative position of one rotor as it is seen from quadrotor's body frame. This rotor is displaced from the body frame origin and forms an angle of 45◦ with quadrotor's body frame *x* axis. Similar relations can be shown for other rotors. Accounting for the number of rotor blades *N*, the following equation for rotor vertical thrust force calculation is proposed Orsag & Bogdan (2009):

$$T = F\_V = \frac{1}{2\pi} \int\_0^{2\pi} \int\_0^R N \frac{\Delta F\_V}{\Delta R} dr d\psi \tag{16}$$

8 Will-be-set-by-IN-TECH

Fig. 5. Infitesimal rotor blade element Δ*r* in surrounding airflow Orsag & Bogdan (2009)

 *Vz* + *vi Vxy* + Ω*r*

The effect of varying airflow can be calculated separating the vertical components *Vz* + *vi* and horizontal ones *Vxy* + Ω*r*. The airflow direction angle Φ can be easily calculated from the

As lift and drag forces are not aligned with body frame of reference, horizontal and vertical projection forces need to be derived. Keeping in mind that Ω*r* � {*Vz*, *vi*, *Vxy*} small angle approximations cos(Φ) ≈ 1 and sin(Φ) ≈ Φ can be used. Moreover, in a well balanced rotor blade, drag force should be negligible compared to the lift Gessow & Myers (1952). Applying this considerations to 11 and keeping in mind the relations from figure 5 enables the derivation

cos(Φ) + <sup>d</sup>*<sup>D</sup>*

d*r*

2 *ρV*<sup>2</sup>

cos(Φ) <sup>≈</sup> <sup>1</sup>

This section continues with the observation of a small rotor blade element Δ*r* from the previous section, placing it in real surroundings shown in figure 6. Since the blades rotate, the forces produced by blade elements tend to change both in size and direction. This is the

Figure 6 shows the relative position of one rotor as it is seen from quadrotor's body frame. This rotor is displaced from the body frame origin and forms an angle of 45◦ with quadrotor's body frame *x* axis. Similar relations can be shown for other rotors. Accounting for the number of rotor blades *N*, the following equation for rotor vertical thrust force calculation is proposed

> 2*π* 0

 *R* 0

*<sup>N</sup>* <sup>Δ</sup>*FV*

≈ arctan

sin(Φ) <sup>≈</sup> <sup>d</sup>*<sup>L</sup>*

*totCDS* +

 *Vz* + *vi* Ω*r*

<sup>d</sup>*<sup>r</sup>* <sup>=</sup> *<sup>ρ</sup>V*<sup>2</sup>

1 2 *ρV*<sup>2</sup> *totCLS*Φ

*totcπαe f*

<sup>Δ</sup>*<sup>R</sup> drd<sup>ψ</sup>* (16)

(14)

(15)

is introduced to the blade design.

Φ = arctan

of horizontal and vertical force equations 15.

d*FH* <sup>d</sup>*<sup>r</sup>* <sup>=</sup> <sup>d</sup>*<sup>L</sup>* d*r*

Orsag & Bogdan (2009):

d*FV* <sup>d</sup>*<sup>r</sup>* <sup>=</sup> <sup>d</sup>*<sup>L</sup>* d*r*

sin(Φ) + <sup>d</sup>*<sup>D</sup>*

**3.1.3 Applying blade element theory to quadrotor construction**

d*r*

reason why an average elemental thrust of all blade elements should be calculated.

2*π*

*<sup>T</sup>* <sup>=</sup> *FV* <sup>=</sup> <sup>1</sup>

equation

Fig. 6. Blade element in quadrotor coordinate system

where *ψ* is the blade angle due to rotation, taken at a certain sample time. Solving integral equation 16 yields the expression for rotor thrust (i.e. vertical force) Orsag & Bogdan (2009):

$$F\_V = \frac{N\rho a\bar{c}R^3\Omega^2}{4} \left[ \left(\frac{2}{3} + \mu^2\right)\Theta\_0 - \left(1 + \mu^2\right)\frac{\Theta\_{tw}}{2} - \lambda\_i - \lambda\_c \right] \tag{17}$$

The term inside the brackets of equation 17 is known as a thrust coefficient, and is given separately in 18.

$$\mathcal{C}\_{T} = \left(\frac{2}{3} + \mu^2\right) \Theta\_0 - \left(1 + \mu^2\right) \frac{\Theta\_{\rm tw}}{2} - \lambda\_i - \lambda\_c \tag{18}$$

Variables *<sup>μ</sup>*,*λ<sup>i</sup>* and *<sup>λ</sup><sup>c</sup>* are speed coefficients *Vxy <sup>R</sup>*<sup>Ω</sup> , *Vz <sup>R</sup>*<sup>Ω</sup> and *vi <sup>R</sup>*<sup>Ω</sup> respectively. New constant *c*¯ is the average cord length of the blade element shown in figure 5.

The same approach can be applied for the calculation of horizontal forces and torques produced within the quadrotor Orsag & Bogdan (2009). Calculated lateral force has x and y components, coming both from the drag and lift of the rotor, given in 19.

$$\begin{aligned} \mathsf{C}\_{Hx} &= \cos\left(\mathfrak{a}\right)\mu \left[ \frac{\mathsf{C}\_{D}}{a} + \left(\lambda\_{i} + \lambda\_{\mathfrak{c}}\right) \left(\Theta\_{0} - \frac{\Theta\_{tw}}{2}\right) \right] \\ \mathsf{C}\_{Hy} &= \sin\left(\mathfrak{a}\right)\mu \left[ \frac{\mathsf{C}\_{D}}{a} + \left(\lambda\_{i} + \lambda\_{\mathfrak{c}}\right) \left(\Theta\_{0} - \frac{\Theta\_{tw}}{2}\right) \right] \end{aligned} \tag{19}$$

In case of torque equations the angles between the forces and directions are easily derived from basic geometric relations shown in figure 6, resulting in the elemental torque equations

Voltage [V] Rotation speedΩ [rpm] Induced speed *vi* [m/s] Thrust [N] 4.04 194.465 1.5 0.16 5.01 241.17 2 0.29 5.99 284.105 2.45 0.44 6.99 328.82 2.7 0.58 8.00 367.357 3.2 0.72 8.98 403.171 3.5 0.94 10.02 433.540 3.8 1.16 10.99 464.223 4.05 1.34 12.05 490.088 4.3 1.42

Influence of Forward and Descent Flight on Quadrotor Dynamics 151

In order to use thrust equation 18, certain coefficients need to be known. Some of them like rotor radius *R* and cord length *c* can be measured. Others, like the mechanical angle Θ<sup>0</sup> have to be calculated. Solving thrust equation 18 for *μ* = 0 and *λ<sup>c</sup>* = 0 (i.e. static conditions) yields:

> Θ<sup>3</sup> 4 − *λ<sup>i</sup>*

*<sup>ρ</sup>ac*Ω2*R*<sup>3</sup> <sup>+</sup> *<sup>λ</sup><sup>i</sup>*

<sup>4</sup> of the blade length *R* 13. Θ*tw* can later be assessed

4

4

*CT* (0, 0, 0) *<sup>T</sup>* (0, 0, 0) (25)

(23)

(24)

. For given set of

= 11.6291*o*, which is

*ρacω*2*R*<sup>3</sup>

from the blade construction. Rearranging equation 23 yields an equation for solving the

2*FV*

<sup>Ω</sup>*<sup>R</sup>* = 0.0766. Therefore the mechanical angle Θ<sup>3</sup>

Obtained data is piecewise linearized, in order to clearly demonstrate the differences between various voltage ranges. From Fig. 7 it can be seen how thrust declines near the point of saturation. This is important to notice, when deriving valid algorithms for quadrotor

Voltage [V] Linear gain [N/V]

To apply aerodynamic coefficient 18 to the static thrust experimental results, one needs to multiply experimental results with dynamic-to-static aerodynamic coefficient ratio 25.

*<sup>T</sup>* (*μ*, *<sup>λ</sup>c*, *<sup>λ</sup>i*) <sup>=</sup> *CT* (*μ*, *<sup>λ</sup>c*, *<sup>λ</sup>i*)

[0 − 3] 0 [3 − 8] 0.1433 [8 − 11] 0.2070 [11 − 12] 0.08

*FV* <sup>=</sup> <sup>1</sup> 2

Θ<sup>3</sup> 4 <sup>=</sup> <sup>3</sup> 2

stabilization and control. Linearizaton coefficients are given in table 2.

Using experimental data from table 1 it is easy to calculate rotor angle Θ<sup>3</sup>

Table 1. Data collected from the experiments

is a mechanical angle at the <sup>3</sup>

where Θ<sup>3</sup>

4

mechanical angle problem 24.

data the average *<sup>λ</sup><sup>i</sup>* = *vi*

well between the expected boundaries.

Table 2. Piecewise linearization coefficients

**3.2.2 Applying aerodynamics to rotor dynamic model**

Orsag & Bogdan (2009):

$$\frac{\Delta M\_z}{\Delta r} = -\frac{\Delta F\_H}{\Delta r} \left( D \cos \left( \Psi - \frac{pii}{4} \right) - r \right) \tag{20}$$

$$\frac{\Delta M\_{xy}}{\Delta r} = -\frac{\sqrt{2} \Delta F\_V}{2 \Delta r} \left( D - r \cos \left( \Psi - \frac{\pi}{4} \right) \pm r \sin \left( \Psi - \frac{pii}{4} \right) \right)$$

Using the same methods which were used for force calculation, the following momentum coefficients were calculated:

$$\mathbf{C}\_{Mz} = R\left[\frac{1+\mu^2}{2a}\mathbf{C}\_D - \mathbf{C}\_T \left(\mu, \lambda, \lambda\_i\right)|\_{\mu=0}\right] \pm D\mu \cos\left(\frac{\pi}{4} + \phi \frac{\mathbf{C}\_{Hx}}{\cos(\phi)\mu}\right)$$

$$\mathbf{C}\_{Mx} = D\frac{\sqrt{2}}{2}\mathbf{C}\_T \pm R\mu \sin(\phi)\left[\frac{2}{3}\Theta\_0 - \frac{1}{2}\left(\Theta\_{tw} + \lambda\right)\right] \tag{21}$$

$$\mathbf{C}\_{My} = D\frac{\sqrt{2}}{2}\mathbf{C}\_T \pm R\mu \cos(\phi)\left[\frac{2}{3}\Theta\_0 - \frac{1}{2}\left(\Theta\_{tw} + \lambda\right)\right]$$

It is important to notice that equations 20 have two solutions, since the rotors spin in different directions, as seen in figure 3. Different rotational directions have the opposite effect on torques. This is why the ± sign is used in torque equations. These differences, induced from the specific quadrotor construction, along with the augmented momentum equation provide an improved insight to quadrotor aerodynamics. Regardless of the flying state of the quadrotor, by using these equations one can effectively model its behavior.

#### **3.2 Building a more realistic rotor model**

Building a more realistic rotor model begins with redefining its widely accepted static thrust equation 22 with real experimental results. No matter how precise, static equation is valid only when quadrotor remains stationary (i.e. hover mode). In order for the equation to be valid during quadrotor maneuvers, aerodynamic effects from 3.1 need to be incorporated into the equation.

$$T \sim k\_T \Omega^2 \tag{22}$$

#### **3.2.1 Experimental results**

This section presents the experimental results of a static thrust equation for an example quadrotor. Most of researched quadrotors use DC motors to drive the rotors. Although new designs use brushless DC motors (BLDC), brushed motors are still used due to their lower cost. Some advantages of brushless over brushed DC motors include more torque per weight, more torque per watt (increased efficiency) and increased reliability Sanchez et al. (2011); Solomon & Famouri (2006); Y. (2003).

Quadrotor used in described experiments is equipped with a standard brushed DC motor. Experimental results show that quadratic relationship between rotor speed (applied voltage) and resulting thrust is valid for certain range of voltages. Moving close to saturation point (i.e. 11V-12V), the quadratic relation of thrust and rotor speed deteriorates. Experimental results are shown in figure 7 and in the table 1.

10 Will-be-set-by-IN-TECH

<sup>Δ</sup>*<sup>r</sup>* <sup>=</sup> <sup>−</sup>Δ*FH*

Δ*r*

<sup>2</sup> *CT* <sup>±</sup> *<sup>R</sup><sup>μ</sup>* sin(*φ*)

<sup>2</sup> *CT* <sup>±</sup> *<sup>R</sup><sup>μ</sup>* cos(*φ*)

It is important to notice that equations 20 have two solutions, since the rotors spin in different directions, as seen in figure 3. Different rotational directions have the opposite effect on torques. This is why the ± sign is used in torque equations. These differences, induced from the specific quadrotor construction, along with the augmented momentum equation provide an improved insight to quadrotor aerodynamics. Regardless of the flying state of

Building a more realistic rotor model begins with redefining its widely accepted static thrust equation 22 with real experimental results. No matter how precise, static equation is valid only when quadrotor remains stationary (i.e. hover mode). In order for the equation to be valid during quadrotor maneuvers, aerodynamic effects from 3.1 need to be incorporated into

This section presents the experimental results of a static thrust equation for an example quadrotor. Most of researched quadrotors use DC motors to drive the rotors. Although new designs use brushless DC motors (BLDC), brushed motors are still used due to their lower cost. Some advantages of brushless over brushed DC motors include more torque per weight, more torque per watt (increased efficiency) and increased reliability Sanchez et al.

Quadrotor used in described experiments is equipped with a standard brushed DC motor. Experimental results show that quadratic relationship between rotor speed (applied voltage) and resulting thrust is valid for certain range of voltages. Moving close to saturation point (i.e. 11V-12V), the quadratic relation of thrust and rotor speed deteriorates. Experimental results

± *Dμ* cos

 2 3

 2 3

 <sup>Ψ</sup> <sup>−</sup> *<sup>π</sup>* 4 ± *r* sin

Using the same methods which were used for force calculation, the following momentum

 *D* cos <sup>Ψ</sup> <sup>−</sup> *pi* 4 − *r* 

> <sup>Ψ</sup> <sup>−</sup> *pi* 4

 *π* <sup>4</sup> <sup>+</sup> *<sup>φ</sup>*

<sup>Θ</sup><sup>0</sup> <sup>−</sup> <sup>1</sup>

<sup>Θ</sup><sup>0</sup> <sup>−</sup> <sup>1</sup>

*<sup>T</sup>* <sup>∼</sup> *kT*Ω<sup>2</sup> (22)

*CHx* cos(*φ*)*μ*

<sup>2</sup> (Θ*tw* <sup>+</sup> *<sup>λ</sup>*)

<sup>2</sup> (Θ*tw* <sup>+</sup> *<sup>λ</sup>*)

(21)

(20)

Δ*Mz*

*D* − *r* cos

*CD* − *CT* (*μ*, *λ*, *λi*)|*μ*=<sup>0</sup>

√2

√2

the quadrotor, by using these equations one can effectively model its behavior.

*CMx* = *D*

*CMy* = *D*

<sup>√</sup>2Δ*FV* 2Δ*r*

Orsag & Bogdan (2009):

coefficients were calculated:

*CMz* = *R*

Δ*Mxy* <sup>Δ</sup>*<sup>r</sup>* <sup>=</sup> <sup>−</sup>

> 1 + *μ*<sup>2</sup> 2*a*

**3.2 Building a more realistic rotor model**

(2011); Solomon & Famouri (2006); Y. (2003).

are shown in figure 7 and in the table 1.

the equation.

**3.2.1 Experimental results**



Table 1. Data collected from the experiments

In order to use thrust equation 18, certain coefficients need to be known. Some of them like rotor radius *R* and cord length *c* can be measured. Others, like the mechanical angle Θ<sup>0</sup> have to be calculated. Solving thrust equation 18 for *μ* = 0 and *λ<sup>c</sup>* = 0 (i.e. static conditions) yields:

$$F\_V = \frac{1}{2} \rho a c \omega^2 \mathbb{R}^3 \left(\Theta\_{\frac{3}{4}} - \lambda\_i\right) \tag{23}$$

where Θ<sup>3</sup> 4 is a mechanical angle at the <sup>3</sup> <sup>4</sup> of the blade length *R* 13. Θ*tw* can later be assessed from the blade construction. Rearranging equation 23 yields an equation for solving the mechanical angle problem 24.

$$\Theta\_{\frac{3}{4}} = \frac{3}{2} \left( \frac{2F\_V}{\rho ac \Omega^2 R^3} + \lambda\_i \right) \tag{24}$$

Using experimental data from table 1 it is easy to calculate rotor angle Θ<sup>3</sup> 4 . For given set of data the average *<sup>λ</sup><sup>i</sup>* = *vi* <sup>Ω</sup>*<sup>R</sup>* = 0.0766. Therefore the mechanical angle Θ<sup>3</sup> 4 = 11.6291*o*, which is well between the expected boundaries.

Obtained data is piecewise linearized, in order to clearly demonstrate the differences between various voltage ranges. From Fig. 7 it can be seen how thrust declines near the point of saturation. This is important to notice, when deriving valid algorithms for quadrotor stabilization and control. Linearizaton coefficients are given in table 2.


Table 2. Piecewise linearization coefficients

#### **3.2.2 Applying aerodynamics to rotor dynamic model**

To apply aerodynamic coefficient 18 to the static thrust experimental results, one needs to multiply experimental results with dynamic-to-static aerodynamic coefficient ratio 25.

$$T\left(\mu,\lambda\_{\rm c},\lambda\_{\rm i}\right) = \frac{\mathbb{C}\_{T}\left(\mu,\lambda\_{\rm c},\lambda\_{\rm i}\right)}{\mathbb{C}\_{T}\left(0,0,0\right)}T\left(0,0,0\right) \tag{25}$$

Fig. 8. 3D representation of *λ<sup>i</sup>* change during horizontal and vertical movement

3D representation of final results is shown in figure 9.

increased and enables more aggressive maneuvers.

**3.2.3 Quadrotor model**

in figure 7.

The results of solving this quadrinome can be shown in a 3D graph 8. Although equations 27 look straightforward to solve, it still requires a substantial amount of processor capacity. This is why an offline calculation is proposed. This way, the calculated data can be used during simulation without the need for online computation. By using calculated values of the induced velocity, it is easy to calculate the dynamic thrust coefficient from equation 18. The

Influence of Forward and Descent Flight on Quadrotor Dynamics 153

Due to an increase of airflow produced by quadrotor movement, the induced velocity decreases. This can be seen in figure 8. Although both movements tend to increase induced velocity, only the vertical movement decreases the thrust coefficient. As a result, during takeoff the quadrotor looses rotor thrust, but during horizontal movement that same thrust is

A complete quadrotor model, incorporating previously mentioned effects is shown in figure 10. A control input block feeds the voltage signals to calculate statics thrust, which is easily interpolated from the available experimental data, using an interpolation function as shown

Static rotor thrust is applied to equation 25 along with aerodynamic coefficient *CT*(*μ*, *λc*, *λi*). Induced velocity and aerodynamic coefficient are calculated using inputs from the current flight data (i.e. *λc*, *μ*). This data is supplied from the Quadrotor Dynamics block. The calculation can be done offline, so that a set of data points from figure 9 can be used to

Fig. 7. Static rotor thrust experimental results with interpolation function and piecewise linear approximations

For the calculation of the aerodynamic coefficient *CT* it is crucial to know three airspeed coefficients *μ*, *λ<sup>c</sup>* and *λi*. Two of them, *μ*, *λc*, can easily be obtained from the available motion data *Vxy*, *Vz* and Ω*R*. *λ<sup>i</sup>* however, is very hard to know, because it is impossible to measure the induced velocity *vi*.

One way to solve this problem is to calculate the induced velocity coefficient *λ<sup>i</sup>* from the two aerodynamic principals, momentum and blade element theories. The macroscopic momentum equation 10 and the microscopic blade element equation 17 provide the same rotor thrust using different physical approach:

$$T = \frac{1}{4}\rho aR^3\Omega^2c\left[\frac{2}{3}a\_{m\text{hel}}\left(1 + \frac{3}{2}\mu^2\right) - \lambda\_i - \lambda\_c\right] = 2\rho R^2\pi\lambda\_i\sqrt{\left(\lambda\_c + \lambda\_i\right)^2 + \mu^2 + \frac{\lambda\_c^2}{7,67}}\tag{26}$$

When squared, equation 26 can be easily solved as a quadrinome:

$$\begin{aligned} \lambda\_i^4 + p\_3 \lambda\_i^3 + p\_2 \lambda\_i^2 + p\_1 \lambda\_i + p\_0 &= 0\\ p\_0 &= -c\_1 c\_2^2\\ p\_1 &= 2c\_1 c\_2 \end{aligned} $$

$$p\_2 = \left( 1 + \frac{1}{7.67} \right) \lambda\_c^2 + \mu^2 - c\_1 \tag{27}$$

$$\begin{aligned} p\_3 &= 2\lambda\_c\\ c\_1 &= \frac{a^2 s^2}{64} \\ c\_2 &= 2\Theta\_0 \left( 1 + 1.5\mu^2 \right) / 3 - \lambda\_c \end{aligned} \tag{28}$$

12 Will-be-set-by-IN-TECH

Fig. 7. Static rotor thrust experimental results with interpolation function and piecewise

For the calculation of the aerodynamic coefficient *CT* it is crucial to know three airspeed coefficients *μ*, *λ<sup>c</sup>* and *λi*. Two of them, *μ*, *λc*, can easily be obtained from the available motion data *Vxy*, *Vz* and Ω*R*. *λ<sup>i</sup>* however, is very hard to know, because it is impossible to measure

One way to solve this problem is to calculate the induced velocity coefficient *λ<sup>i</sup>* from the two aerodynamic principals, momentum and blade element theories. The macroscopic momentum equation 10 and the microscopic blade element equation 17 provide the same

= 2*ρR*2*πλ<sup>i</sup>*

*<sup>i</sup>* + *p*1*λ<sup>i</sup>* + *p*<sup>0</sup> = 0

*<sup>p</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>*c*1*c*<sup>2</sup>

*p*<sup>1</sup> = 2*c*1*c*<sup>2</sup>

*<sup>c</sup>* <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*<sup>1</sup>

*p*<sup>3</sup> = 2*λ<sup>c</sup> <sup>c</sup>*<sup>1</sup> <sup>=</sup> *<sup>a</sup>*2*s*<sup>2</sup> 64

/3 − *λ<sup>c</sup>*

2

(*λ<sup>c</sup>* + *λi*)

<sup>2</sup> + *μ*<sup>2</sup> +

*λ*2 *c* 7, 67 (26)

(27)

− *λ<sup>i</sup>* − *λ<sup>c</sup>*

*<sup>i</sup>* <sup>+</sup> *<sup>p</sup>*2*λ*<sup>2</sup>

When squared, equation 26 can be easily solved as a quadrinome:

*p*<sup>2</sup> = 1 + 1 7.67 *λ*2

*c*<sup>2</sup> = 2Θ<sup>0</sup>

1 + 1.5*μ*<sup>2</sup>

*λ*4 *<sup>i</sup>* <sup>+</sup> *<sup>p</sup>*3*λ*<sup>3</sup>

linear approximations

the induced velocity *vi*.

*ρaR*3Ω2*c*

*<sup>T</sup>* <sup>=</sup> <sup>1</sup> 4

rotor thrust using different physical approach:

 2 3 *αmeh* 1 + 3 2 *μ*2 

Fig. 8. 3D representation of *λ<sup>i</sup>* change during horizontal and vertical movement

The results of solving this quadrinome can be shown in a 3D graph 8. Although equations 27 look straightforward to solve, it still requires a substantial amount of processor capacity. This is why an offline calculation is proposed. This way, the calculated data can be used during simulation without the need for online computation. By using calculated values of the induced velocity, it is easy to calculate the dynamic thrust coefficient from equation 18. The 3D representation of final results is shown in figure 9.

Due to an increase of airflow produced by quadrotor movement, the induced velocity decreases. This can be seen in figure 8. Although both movements tend to increase induced velocity, only the vertical movement decreases the thrust coefficient. As a result, during takeoff the quadrotor looses rotor thrust, but during horizontal movement that same thrust is increased and enables more aggressive maneuvers.

#### **3.2.3 Quadrotor model**

A complete quadrotor model, incorporating previously mentioned effects is shown in figure 10. A control input block feeds the voltage signals to calculate statics thrust, which is easily interpolated from the available experimental data, using an interpolation function as shown in figure 7.

Static rotor thrust is applied to equation 25 along with aerodynamic coefficient *CT*(*μ*, *λc*, *λi*). Induced velocity and aerodynamic coefficient are calculated using inputs from the current flight data (i.e. *λc*, *μ*). This data is supplied from the Quadrotor Dynamics block. The calculation can be done offline, so that a set of data points from figure 9 can be used to

interpolate true aerodynamic coefficient. This speeds up the simulation, as opposed to solving

Influence of Forward and Descent Flight on Quadrotor Dynamics 155

A combination of the results provided from these two blocks using equation 25 gives the true aerodynamic rotor thrust. The same procedure is used to calculate the induced speed from the data shown in figure 8. Once the exact induced speed is known it can be applied to horizontal coefficients 19 and torque coefficients 21. In this way, quadrotor dynamics block can calculate

Dynamics data is finally fed into the kinematics block, that calculates quadrotor motion in

As the unmanned aerial research community shifts its efforts towards more and more aggressive flying maneuvers as well as mobile manipulation, the need for a more complete

The chapter introduces a nonlinear mathematical model that incorporates aerodynamic effects of forward and vertical flights. A clear insight on how to incorporate these effects to a basic quadrotor model is given. Experimental results of widely used brushed DC motors are presented. The results show negative saturation effects observed when using this type of

The proposed model incorporates aerodynamic effects using offline precalculated data, that can easily be added to existing basic quadrotor model. Furthermore, the model described in

Adigbli, P., Grand, C., Mouret, J.-B. & Doncieux, S. (2007). Nonlinear attitude and position

Anderson, S. B. (1981). Historical overview of v/stol aircraft technology, *NASA Technical*

Bouabdallah, S., Noth, A. & Siegwart, R. (2004). Pid vs lq control techniques applied to an

Bouabdallah, S. & Siegwart, R. (2005). Backstepping and sliding-mode techniques applied to

Bramwell, A., Done, G. & Balmford, D. (2001). *Bramwell's helicopter dynamics*, American

Chaturvedi, N., Sanyal, A. & McClamroch, N. (2011). Rigid-body attitude control, *Control*

Hoffmann, G. M., Huang, H., Wasl, S. L. & Tomlin, E. C. J. (2007). Quadrotor helicopter

Gessow, A. & Myers, G. (1952). *Aerodynamics of the helicopter*, F. Ungar Pub. Co.

control of a micro quadrotor using sliding mode and backstepping techniques, *7th*

indoor micro quadrotor, *Proc. of The IEEE International Conference on Intelligent Robots*

an indoor micro quadrotor, *Proc. of The IEEE International Conference on Robotics and*

flight dynamics and control: Theory and experiment, *In Proc. of the AIAA Guidance,*

DC motors, as well as the phenomenon of thrust variations during quadrotor's flight.

the paper can incorporate additional aerodynamic effects like the In Ground Effect.

*European Micro Air Vehicle Conference (MAV07)*, Toulouse.

aerodynamic quadrotor model, such as the one presented in this chapter arises.

quadrotors angular and linear dynamics using equations 6 and 3.

world coordinate system using transformation matrices 2 and 7.

the quadrinome problem online.

**4. Conclusion**

**5. References**

*Memorandum* .

*and Systems (IROS)*.

*Automation (ICRA)*.

*systems magazine* .

Institute of Aeronautics and Astronautics.

*Navigation, and Control Conference*.

Fig. 9. 3D representation of *<sup>T</sup>*(*λi*,*λ<sup>c</sup>* ,*μ*) *<sup>T</sup>*(0,0,0) ratio during horizontal and vertical movement

Fig. 10. Quadrotor model

interpolate true aerodynamic coefficient. This speeds up the simulation, as opposed to solving the quadrinome problem online.

A combination of the results provided from these two blocks using equation 25 gives the true aerodynamic rotor thrust. The same procedure is used to calculate the induced speed from the data shown in figure 8. Once the exact induced speed is known it can be applied to horizontal coefficients 19 and torque coefficients 21. In this way, quadrotor dynamics block can calculate quadrotors angular and linear dynamics using equations 6 and 3.

Dynamics data is finally fed into the kinematics block, that calculates quadrotor motion in world coordinate system using transformation matrices 2 and 7.

## **4. Conclusion**

14 Will-be-set-by-IN-TECH

*<sup>T</sup>*(0,0,0) ratio during horizontal and vertical movement

Fig. 9. 3D representation of *<sup>T</sup>*(*λi*,*λ<sup>c</sup>* ,*μ*)

Fig. 10. Quadrotor model

As the unmanned aerial research community shifts its efforts towards more and more aggressive flying maneuvers as well as mobile manipulation, the need for a more complete aerodynamic quadrotor model, such as the one presented in this chapter arises.

The chapter introduces a nonlinear mathematical model that incorporates aerodynamic effects of forward and vertical flights. A clear insight on how to incorporate these effects to a basic quadrotor model is given. Experimental results of widely used brushed DC motors are presented. The results show negative saturation effects observed when using this type of DC motors, as well as the phenomenon of thrust variations during quadrotor's flight.

The proposed model incorporates aerodynamic effects using offline precalculated data, that can easily be added to existing basic quadrotor model. Furthermore, the model described in the paper can incorporate additional aerodynamic effects like the In Ground Effect.

#### **5. References**


**8** 

*Italy* 

**Advanced Graph Search Algorithms** 

**for Path Planning of Flight Vehicles** 

Path planning is one of the most important tasks for mission definition and management of manned flight vehicles and it is crucial for Unmanned Aerial Vehicles (UAVs) that have autonomous flight capabilities. This task involves mission constraints, vehicle's characteristics and mission environment that must be combined in order to comply with the mission requirements. Nevertheless, to implement an effective path planning strategy, a deep analysis of various contributing elements is needed. Mission tasks, required payload and surveillance systems drive the aircraft selection, but its characteristics strongly influence the path. As an example, quad-rotors have hovering capabilities. This feature permits to relax turning constraints on the path (which represents a crucial problem for fixed-wing vehicles). The type of mission defines the environment for planning actions, the path constraints (mountains, hills, valleys, …) and the required optimization process. The need for off-line or real-time re-planning may also substantially revise the path planning strategy for the selected type of missions. Finally, the computational performances of the Remote Control Station (RCS), where the mission management system is generally running, can influence the algorithm selection and design, as time constraints can be a serious operational

Describe the most important algorithms developed for path planning of flying vehicles

Focus on graph search algorithms in order to define their main characteristics and

 Present a new graph search algorithm (called Kinematic A\*) that has been developed on the base of the well-known A\* algorithm and aims to fill the relation gap between the path planned with classical graph search solutions and the aircraft kinematic

 Introduction to first approaches to path planning: manual path planning and Dubins curves. Also some simple applications developed by this research group

in order to compare them and depict their merits and drawbacks.

General description of the most important path planning algorithms:

provide a complete overview of the most important methods developed.

**1. Introduction** 

issue.

constraints.

The chapter is structured as follow:

are presented.

This chapter aims to cover three main topics:

Luca De Filippis and Giorgio Guglieri

*Politecnico di Torino* 


URL: *http://dx.doi.org/10.1007/s10846-010-9470-3*


## **Advanced Graph Search Algorithms for Path Planning of Flight Vehicles**

Luca De Filippis and Giorgio Guglieri *Politecnico di Torino Italy* 

## **1. Introduction**

16 Will-be-set-by-IN-TECH

156 Recent Advances in Aircraft Technology

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URL: *http://dx.doi.org/10.1007/s10846-010-9470-3*

Path planning is one of the most important tasks for mission definition and management of manned flight vehicles and it is crucial for Unmanned Aerial Vehicles (UAVs) that have autonomous flight capabilities. This task involves mission constraints, vehicle's characteristics and mission environment that must be combined in order to comply with the mission requirements. Nevertheless, to implement an effective path planning strategy, a deep analysis of various contributing elements is needed. Mission tasks, required payload and surveillance systems drive the aircraft selection, but its characteristics strongly influence the path. As an example, quad-rotors have hovering capabilities. This feature permits to relax turning constraints on the path (which represents a crucial problem for fixed-wing vehicles). The type of mission defines the environment for planning actions, the path constraints (mountains, hills, valleys, …) and the required optimization process. The need for off-line or real-time re-planning may also substantially revise the path planning strategy for the selected type of missions. Finally, the computational performances of the Remote Control Station (RCS), where the mission management system is generally running, can influence the algorithm selection and design, as time constraints can be a serious operational issue.

This chapter aims to cover three main topics:


The chapter is structured as follow:

	- Introduction to first approaches to path planning: manual path planning and Dubins curves. Also some simple applications developed by this research group are presented.

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 159

First studies on path planning of unmanned aircrafts evidenced task complexities, strict safety requirements and reduced technological capabilities that imposed as unique solution manual approaches for path planning of UASs. The waypoint sequences where based on the environment map and on the mission tasks, taking into account some basic kinematic constraints. The flight programs were then loaded on the aircraft flight control system (FCS) and the path tracking were monitored in real time. These approaches were overtaken researching on this problem, but some of them are still used for industrial applications where the plan complexity requires a human supervision at all stages. In these cases computer tools driving the waypoints allocation, the path feasibility verification and the waypoints-sequence conversion in formats compatible with the aircraft FCS assist the

The above-mentioned path-planning procedures were investigated (De Filippis et al., 2009) and some simple tools were developed in Matlab/Simulink and integrated into a single software package. This software (named PCube) handles geotiff and Digital Elevation Models to generate waypoint sequences compatible with the programming scripts of Micropilot commercial autopilots. The tool has a basic Graphical User Interface (GUI) used

generate automatic grid type waypoint sequences (grid patterns for photogrammetric

If manual planning is the first and basic approach to path planning, it motivated research of more accurate solutions. In this direction optimization of the path with respect to some performance parameters was the challenge. Many approaches from optimal theory were studied and adapted to path planning and the Dubins curves are one of the most used and

In a bi-dimensional space a couple of points each one associated to a unitary vector is given such that a vehicle is supposed to pass from these points with its trajectory tangent to the vector in that point. Dubins considered a non-holonomic vehicle moving at constant speed with limited turning capabilities and tried to find the shortest path between the two points under such constraints. He demonstrated that assuming constant turning radiuses this path exists and analysing each possible case a set of geodesic curves can be defined (Dubins, 1957). The same work was moved forward through successive studies on holonomic

Dubins curves are used in PCube to take into account the UAVs turn performances and average flight speed, reallocating waypoints violating the constraints. For grid type patterns, the path generation is optimized for optical type payloads, specifying image overlaps and focal length. The package also allows the manipulation of maps and flight paths (i.e. sizing, scaling and rotation of mapped patterns). 3D surface and contour level plots are available for enhancing the visualization of the flight path. Coordinates and map formats can also be

to manage the map and the path planning sequence. This tool can be used to:

choose predefined path shapes (square, rectangular and butterfly shapes),

attractive solutions for their conceptual and implementation simplicity.

converted in different standards according to user specifications.

**2.1 Manual path planning and Dubins curves** 

generate point and click waypoint sequences,

human agent.

use).

vehicles (Reeds & Shepp, 1990).

	- General description of commonality and differences between methods composing this family. Basic algorithm structure identification and introduction to the general features of these methods.
	- First graph search solutions focusing on the A\* algorithm.
	- Introduction to dynamic-graph search and to the principal developed methods.
	- "Any heading" algorithms description focusing on Theta\*.
	- Brief comparison between Theta\* and A\* on paths planned with the tools developed by this research group, focusing on the main improvements introduced with Theta\*.
	- State space definition: in order to implement Kinematic A\*, redefinition of the state space is needed.
	- Kinematic model description: the system of differential equation modelling the aircraft kinematic behaviour.
	- Introduction of wind in the kinematic model in order to take into account this disturbance on the path.
	- Formulation of the optimization problem solved with the graph search approach.
	- Constraints definition identifying the set of states evaluated to find the optimal path.
	- Algorithm description.
	- Algorithm test on a square map collecting four obstacles placed close to the four corners. A\* path comparison with the Kinematic A\* one planned with and without wind.
	- Algorithm test on a square map with one obstacle obstructing the path. This test is made to verify the algorithm search performances.

## **2. The path planning task**

Generally, path planning aims to generate a real-time trajectory to a target, avoiding obstacles or collisions (assuming reference flight-conditions and providing maps of the environment), but also optimizing a given functional under kinematic and/or dynamic constraints. Several solutions were developed matching different planning requirements: performances optimization, collision avoidance, real-time planning or risk minimization, etc. Several algorithms were designed for robotic systems and ground vehicles. They took hints from research fields like physics for potential field algorithms, mathematics for probabilistic approaches, or computer science for graph search algorithms. Each family of algorithms has been tailored for path planning of UAVs, and future work will enforce the development of new strategies.

#### **2.1 Manual path planning and Dubins curves**

158 Recent Advances in Aircraft Technology

 General description of commonality and differences between methods composing this family. Basic algorithm structure identification and introduction to the general

Introduction to dynamic-graph search and to the principal developed methods.

 Brief comparison between Theta\* and A\* on paths planned with the tools developed by this research group, focusing on the main improvements introduced

State space definition: in order to implement Kinematic A\*, redefinition of the state

Kinematic model description: the system of differential equation modelling the

Introduction of wind in the kinematic model in order to take into account this

Formulation of the optimization problem solved with the graph search

Constraints definition identifying the set of states evaluated to find the optimal

 Algorithm test on a square map collecting four obstacles placed close to the four corners. A\* path comparison with the Kinematic A\* one planned with and without

Algorithm test on a square map with one obstacle obstructing the path. This test is

Generally, path planning aims to generate a real-time trajectory to a target, avoiding obstacles or collisions (assuming reference flight-conditions and providing maps of the environment), but also optimizing a given functional under kinematic and/or dynamic constraints. Several solutions were developed matching different planning requirements: performances optimization, collision avoidance, real-time planning or risk minimization, etc. Several algorithms were designed for robotic systems and ground vehicles. They took hints from research fields like physics for potential field algorithms, mathematics for probabilistic approaches, or computer science for graph search algorithms. Each family of algorithms has been tailored for path planning of UAVs, and future work will enforce the

Results presentation in order to identify new algorithm merits and drawbacks:

made to verify the algorithm search performances.

 General description of probabilistic and graph search algorithms. General description of potential field and model predictive algorithms.

Introduction to some generic optimization algorithms.

First graph search solutions focusing on the A\* algorithm.

"Any heading" algorithms description focusing on Theta\*.

Study on graph search algorithms:

with Theta\*.

approach.

path.

wind.

**2. The path planning task** 

development of new strategies.

space is needed.

aircraft kinematic behaviour.

disturbance on the path.

Algorithm description.

Conclusion and future work description.

Kinematic A\*:

features of these methods.

First studies on path planning of unmanned aircrafts evidenced task complexities, strict safety requirements and reduced technological capabilities that imposed as unique solution manual approaches for path planning of UASs. The waypoint sequences where based on the environment map and on the mission tasks, taking into account some basic kinematic constraints. The flight programs were then loaded on the aircraft flight control system (FCS) and the path tracking were monitored in real time. These approaches were overtaken researching on this problem, but some of them are still used for industrial applications where the plan complexity requires a human supervision at all stages. In these cases computer tools driving the waypoints allocation, the path feasibility verification and the waypoints-sequence conversion in formats compatible with the aircraft FCS assist the human agent.

The above-mentioned path-planning procedures were investigated (De Filippis et al., 2009) and some simple tools were developed in Matlab/Simulink and integrated into a single software package. This software (named PCube) handles geotiff and Digital Elevation Models to generate waypoint sequences compatible with the programming scripts of Micropilot commercial autopilots. The tool has a basic Graphical User Interface (GUI) used to manage the map and the path planning sequence. This tool can be used to:


If manual planning is the first and basic approach to path planning, it motivated research of more accurate solutions. In this direction optimization of the path with respect to some performance parameters was the challenge. Many approaches from optimal theory were studied and adapted to path planning and the Dubins curves are one of the most used and attractive solutions for their conceptual and implementation simplicity.

In a bi-dimensional space a couple of points each one associated to a unitary vector is given such that a vehicle is supposed to pass from these points with its trajectory tangent to the vector in that point. Dubins considered a non-holonomic vehicle moving at constant speed with limited turning capabilities and tried to find the shortest path between the two points under such constraints. He demonstrated that assuming constant turning radiuses this path exists and analysing each possible case a set of geodesic curves can be defined (Dubins, 1957). The same work was moved forward through successive studies on holonomic vehicles (Reeds & Shepp, 1990).

Dubins curves are used in PCube to take into account the UAVs turn performances and average flight speed, reallocating waypoints violating the constraints. For grid type patterns, the path generation is optimized for optical type payloads, specifying image overlaps and focal length. The package also allows the manipulation of maps and flight paths (i.e. sizing, scaling and rotation of mapped patterns). 3D surface and contour level plots are available for enhancing the visualization of the flight path. Coordinates and map formats can also be converted in different standards according to user specifications.

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 161

the risk distribution of some kind of threat on a map (obstacles, forbidden areas, wind, etc.) and the algorithm creates probabilistic maps (Bertuccelli & How 2005) or look-up tables (that can be updated in real-time) modelling this distribution with various theories and logics (Pfeiffer, B et al., 2008). Markov processes are commonly used to introduce probabilistic uncertainties on the problem of path planning and Markov decision processes (MDPs) are defined all the approaches connecting this uncertainty with the taken action. These techniques are useful for all the cases where the optimization parameters are uncertain and can change in time and space, like conditions of flight, environment, and

Graph search algorithms are then interesting techniques coming from computer science. They were developed to find optimal plans to drive data exchanges on computer networks. These algorithms are commonly defined "greedy" as they generate a local optimal solution that can be quite far from the global optimal one. These algorithms are widely used in different fields thanks to their simplicity and small computational load and in the last five decades they evolved from basic approaches as Djikstra and Bellman-Ford algorithms to more complex solutions as D\* Lite and Theta\*. All of them differ in some aspects related to arc-weights definition and cost-function, but they are very similar in the implementation

The main drawback of probabilistic and graph search algorithms resides in the lack of correlation between the aircraft kinematics and the planned path. Commonly, after the path between nodes of the graph has been generated with the minimum path algorithms, it has to be smoothed in order to be adapted to the vehicle flight performances. Indeed greedy algorithms provide a path constituted by line segments connected with edges that can't be followed by any type of flight vehicle. In order to obtain a more feasible and realistic path, refinement algorithms have to be used. This kind of algorithms can be very different in nature, starting from geometric curve definition algorithms also line flow smoothing logic can be used, but in any case at the end of this process a more realistic path is obtained,

which better matches with autopilot control characteristics and flight performances.

performance constrains has been investigated.

**2.3 Potential field and model predictive algorithms** 

Successive research on path planning algorithms brought to development of potential field based solutions. First potential field implementations came out to solve obstacles avoidance and formation flight problems, but in the last few years trajectory optimization under some

Potential field algorithms come from robotic science and have been adapted to UASs simply modifying the kinematic models and the obstacles models. The environment is modelled to generate attractive forces toward the goal and repulsive ones around the obstacles (Dogan, 2003). The potential field model can be magnetic or electric (Horner & Healey, 2004), but the methods derived from aerodynamics provide the best choice in generation of trajectories for flight (Waydo & Murray, 2003). The vehicle motion is forced to follow the energy minimum respecting some dynamic constraints connected with its characteristics (Ford & Fulkerson, 1962). Two important aerodynamic field methods can be mentioned here: one obtained modelling path through propagation of pressure waves and another based on streamline modelling the motion field. The first method has been implemented supposing the fluid

mission tasks.

philosophy.

An example of manual planning using the point and click technique on a highland area is shown in **Figure 1**. Where the waypoint sequence defined by the user has been modified exploiting the Dubins curves. Manual path planning can generate paths with very simple logics when the optimization constraints do not affect the task and more complex solutions were developed and implemented.

Fig. 1. Manula path planning with PCube Graphical User Interface (GUI).

#### **2.2 Probabilistic and graph search algorithms**

The problem of path planning is just an optimization problem made complex by the concurring parameters to be optimized on the same path. These parameters sometimes jar each other and they have to be balanced with respect to the mission tasks. All the more advanced algorithms developed for path planning try to identify the object of the optimization and reformulate the problem to cope with the prominent task, finding different approaches to optimize the parameter connected with this task. Many of them were developed for other applications and were modified to match with the problem of path planning. It's the case of the probability algorithms.

These algorithms generate a probability distribution connected with the parameter to be optimized and they implement statistic techniques to find the most probable path that optimizes this parameter (Jun & D'Andrea, 2002). Many implementations are related with

An example of manual planning using the point and click technique on a highland area is shown in **Figure 1**. Where the waypoint sequence defined by the user has been modified exploiting the Dubins curves. Manual path planning can generate paths with very simple logics when the optimization constraints do not affect the task and more complex solutions

Fig. 1. Manula path planning with PCube Graphical User Interface (GUI).

The problem of path planning is just an optimization problem made complex by the concurring parameters to be optimized on the same path. These parameters sometimes jar each other and they have to be balanced with respect to the mission tasks. All the more advanced algorithms developed for path planning try to identify the object of the optimization and reformulate the problem to cope with the prominent task, finding different approaches to optimize the parameter connected with this task. Many of them were developed for other applications and were modified to match with the problem of path

These algorithms generate a probability distribution connected with the parameter to be optimized and they implement statistic techniques to find the most probable path that optimizes this parameter (Jun & D'Andrea, 2002). Many implementations are related with

**2.2 Probabilistic and graph search algorithms** 

planning. It's the case of the probability algorithms.

were developed and implemented.

the risk distribution of some kind of threat on a map (obstacles, forbidden areas, wind, etc.) and the algorithm creates probabilistic maps (Bertuccelli & How 2005) or look-up tables (that can be updated in real-time) modelling this distribution with various theories and logics (Pfeiffer, B et al., 2008). Markov processes are commonly used to introduce probabilistic uncertainties on the problem of path planning and Markov decision processes (MDPs) are defined all the approaches connecting this uncertainty with the taken action. These techniques are useful for all the cases where the optimization parameters are uncertain and can change in time and space, like conditions of flight, environment, and mission tasks.

Graph search algorithms are then interesting techniques coming from computer science. They were developed to find optimal plans to drive data exchanges on computer networks. These algorithms are commonly defined "greedy" as they generate a local optimal solution that can be quite far from the global optimal one. These algorithms are widely used in different fields thanks to their simplicity and small computational load and in the last five decades they evolved from basic approaches as Djikstra and Bellman-Ford algorithms to more complex solutions as D\* Lite and Theta\*. All of them differ in some aspects related to arc-weights definition and cost-function, but they are very similar in the implementation philosophy.

The main drawback of probabilistic and graph search algorithms resides in the lack of correlation between the aircraft kinematics and the planned path. Commonly, after the path between nodes of the graph has been generated with the minimum path algorithms, it has to be smoothed in order to be adapted to the vehicle flight performances. Indeed greedy algorithms provide a path constituted by line segments connected with edges that can't be followed by any type of flight vehicle. In order to obtain a more feasible and realistic path, refinement algorithms have to be used. This kind of algorithms can be very different in nature, starting from geometric curve definition algorithms also line flow smoothing logic can be used, but in any case at the end of this process a more realistic path is obtained, which better matches with autopilot control characteristics and flight performances.

Successive research on path planning algorithms brought to development of potential field based solutions. First potential field implementations came out to solve obstacles avoidance and formation flight problems, but in the last few years trajectory optimization under some performance constrains has been investigated.

## **2.3 Potential field and model predictive algorithms**

Potential field algorithms come from robotic science and have been adapted to UASs simply modifying the kinematic models and the obstacles models. The environment is modelled to generate attractive forces toward the goal and repulsive ones around the obstacles (Dogan, 2003). The potential field model can be magnetic or electric (Horner & Healey, 2004), but the methods derived from aerodynamics provide the best choice in generation of trajectories for flight (Waydo & Murray, 2003). The vehicle motion is forced to follow the energy minimum respecting some dynamic constraints connected with its characteristics (Ford & Fulkerson, 1962). Two important aerodynamic field methods can be mentioned here: one obtained modelling path through propagation of pressure waves and another based on streamline modelling the motion field. The first method has been implemented supposing the fluid

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 163

Mathematical methods to solve optimization problems, known as indirect methods, are the most important and referenced techniques in this field. Algorithms based on Pontryagin minimum principle and Lagrange multipliers have been widely used to reduce optimization problems to a boundary condition one (Chitsaz & LaValle, 2007). Sequential Gradient Restoration Algorithm (SGRA) represents an indirect method used for several problems like space trajectories optimization (Miele & Pritchard, 1969, Miele, 1970). These techniques are elegant and reliable thanks to decades of research and application to thousands of different problems. They require a complex problem formulation and simplification in order to reach the required mathematical structure that ensures convergence. In some cases where complex and non-linear problems need to be solved these methods can result impracticable and other

Genetic algorithms are nowadays the most attractive solution in problems where constraints and optimization variables are the issue (Carroll,1996). They are based on the concept of natural selection, modelling the solutions like a population of individuals and evaluating evolution of this population over an environment represented by the problem itself. Using Splines or random threes to model the trajectory, these algorithms can reallocate the waypoint sequence to generate optimum solutions under constraints on complex environments (Nikolos et al., 2003). Being interesting and flexible, the evolutionary algorithms are spreading on different planning problems, but their solving complexity is

Finally, more advanced optimization techniques inspired to biological behaviours must be mentioned. These techniques recall biological behaviours to find the optimal solution to the problem. The key aspect of these solutions is the observation of biological phenomena and the adaptation to path planning problems. These algorithms permit to improve the system flexibility to changes in mission constraints and environmental conditions and with respect to genetic approaches these algorithms optimize the solution through a cooperative search.

Graph search algorithms were developed for computer science to find the shortest path between two nodes of connected graphs. They were designed for computer networks to develop routing protocols and were applied to path planning through decomposition of the path in waypoint sequences. The optimization logics behind these algorithms attain the minimization of the distance covered by the vehicle, but none of its performances or

a finite or countably infinite state space that collects all the possible states or nodes of

an actions space that collects for each state the set of action that can be taken to move

**2.4 Generic optimization algorithms** 

paid with a heavy computational effort.

optimization techniques are needed (Sussmann & Tang, 1991).

**3. The graph search algorithms for path planning** 

kinematic characteristics is involved in the path search.

Basic elements common to each graph search method are (LaValle, 2006):

**3.1 General overview** 

the graph (*X*),

from a state to the next (*U*), a state transition function:

expanding from the target position through the starting one and modelling objects in the environment as obstacles. The second method instead models the environment like an aerodynamic field where obstacles are represented with singularities characterized by outgoing flow direction and target position like attracting singularities. The trajectory is chosen between all the streamlines defined in the field, to minimize the potential field gradient.

As a matter of fact these algorithms give smoothed and flyable paths, avoiding static and dynamic obstacles according with the field complexity. In the last years they have been widely investigated and interesting applications have been published. Even tough they are a promising solution for path planning and collision avoidance their application to some problems seemed hard due to their tendency to local minima on complex potential models.

The last and more advanced family of methods presented here, is based on technique coming from control science and applied to path planning and collision avoidance in the last decades. These algorithms apply model predictive control techniques to path planning problems linking a simplified model of the vehicle to some optimization parameters.

These algorithms solve in open loop an optimization problem constrained with a set of differential equations over a finite time horizon. The fundamental idea is to generate a control input that respects vehicle dynamics, environment characteristics and optimization constrains inside the defined time step and to repeat this process each step up to reach the goal. Sensors data can be integrated to update the model states so that these algorithms are used for collision avoidance in presence of active obstacles and particular harsh environments.

The big merit of model predictive solutions is the inclusion inside the optimization problem of the vehicle kinematics and dynamics in order to generate flyable trajectories. Model Predictive Control (MPC) or Receding Horizon Control (RHC) are the first techniques developed for industrial processes control that have been adapted to path planning (Ma & Castanon, 2006). Relation between control theory and path planning underlines another important characteristic of these methods. Indeed using the same logic for control and path planning opens the possibility to generate an integrated system that provides trajectories and control signals. On the other hand because complex sets of differential equations solved iteratively to generate the path are used in these methods, computation speed has been a real issue for these algorithms to spread. Also, as more as the problem complexity increases, as more the optimization space becomes complex and convergence to the optimal solution becomes an issue. Though, successive evolution of the model predictive technique is the Mixed-Integer Linear Programming (MILP). This algorithm applies the same logics of the model predictive one but allows inclusion of integer variables and discrete logics in a continuous linear optimization problem. Variables are used to model obstacles and to generate collision avoidance rules, while dynamics can be modelled with continuous constrains.

As it was stated previously, path planning is an optimization process then classical optimization techniques must be described to give a complete overview of the main tools developed to cope with this problem.

## **2.4 Generic optimization algorithms**

162 Recent Advances in Aircraft Technology

expanding from the target position through the starting one and modelling objects in the environment as obstacles. The second method instead models the environment like an aerodynamic field where obstacles are represented with singularities characterized by outgoing flow direction and target position like attracting singularities. The trajectory is chosen between all the streamlines defined in the field, to minimize the potential field

As a matter of fact these algorithms give smoothed and flyable paths, avoiding static and dynamic obstacles according with the field complexity. In the last years they have been widely investigated and interesting applications have been published. Even tough they are a promising solution for path planning and collision avoidance their application to some problems seemed hard due to their tendency to local minima on complex potential

The last and more advanced family of methods presented here, is based on technique coming from control science and applied to path planning and collision avoidance in the last decades. These algorithms apply model predictive control techniques to path planning

These algorithms solve in open loop an optimization problem constrained with a set of differential equations over a finite time horizon. The fundamental idea is to generate a control input that respects vehicle dynamics, environment characteristics and optimization constrains inside the defined time step and to repeat this process each step up to reach the goal. Sensors data can be integrated to update the model states so that these algorithms are used for collision avoidance in presence of active obstacles and particular harsh

The big merit of model predictive solutions is the inclusion inside the optimization problem of the vehicle kinematics and dynamics in order to generate flyable trajectories. Model Predictive Control (MPC) or Receding Horizon Control (RHC) are the first techniques developed for industrial processes control that have been adapted to path planning (Ma & Castanon, 2006). Relation between control theory and path planning underlines another important characteristic of these methods. Indeed using the same logic for control and path planning opens the possibility to generate an integrated system that provides trajectories and control signals. On the other hand because complex sets of differential equations solved iteratively to generate the path are used in these methods, computation speed has been a real issue for these algorithms to spread. Also, as more as the problem complexity increases, as more the optimization space becomes complex and convergence to the optimal solution becomes an issue. Though, successive evolution of the model predictive technique is the Mixed-Integer Linear Programming (MILP). This algorithm applies the same logics of the model predictive one but allows inclusion of integer variables and discrete logics in a continuous linear optimization problem. Variables are used to model obstacles and to generate collision avoidance rules, while dynamics can be modelled with continuous

As it was stated previously, path planning is an optimization process then classical optimization techniques must be described to give a complete overview of the main tools

problems linking a simplified model of the vehicle to some optimization parameters.

gradient.

models.

environments.

constrains.

developed to cope with this problem.

Mathematical methods to solve optimization problems, known as indirect methods, are the most important and referenced techniques in this field. Algorithms based on Pontryagin minimum principle and Lagrange multipliers have been widely used to reduce optimization problems to a boundary condition one (Chitsaz & LaValle, 2007). Sequential Gradient Restoration Algorithm (SGRA) represents an indirect method used for several problems like space trajectories optimization (Miele & Pritchard, 1969, Miele, 1970). These techniques are elegant and reliable thanks to decades of research and application to thousands of different problems. They require a complex problem formulation and simplification in order to reach the required mathematical structure that ensures convergence. In some cases where complex and non-linear problems need to be solved these methods can result impracticable and other optimization techniques are needed (Sussmann & Tang, 1991).

Genetic algorithms are nowadays the most attractive solution in problems where constraints and optimization variables are the issue (Carroll,1996). They are based on the concept of natural selection, modelling the solutions like a population of individuals and evaluating evolution of this population over an environment represented by the problem itself. Using Splines or random threes to model the trajectory, these algorithms can reallocate the waypoint sequence to generate optimum solutions under constraints on complex environments (Nikolos et al., 2003). Being interesting and flexible, the evolutionary algorithms are spreading on different planning problems, but their solving complexity is paid with a heavy computational effort.

Finally, more advanced optimization techniques inspired to biological behaviours must be mentioned. These techniques recall biological behaviours to find the optimal solution to the problem. The key aspect of these solutions is the observation of biological phenomena and the adaptation to path planning problems. These algorithms permit to improve the system flexibility to changes in mission constraints and environmental conditions and with respect to genetic approaches these algorithms optimize the solution through a cooperative search.

## **3. The graph search algorithms for path planning**

Graph search algorithms were developed for computer science to find the shortest path between two nodes of connected graphs. They were designed for computer networks to develop routing protocols and were applied to path planning through decomposition of the path in waypoint sequences. The optimization logics behind these algorithms attain the minimization of the distance covered by the vehicle, but none of its performances or kinematic characteristics is involved in the path search.

### **3.1 General overview**

Basic elements common to each graph search method are (LaValle, 2006):


Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 165

Cell expansion cycle (this cycle breaks when each new cell has been evaluated)

 Check if the new cost is lower then the previous one: True: substitute the new cost and the cell parent

The algorithm expands systematically the cells up to reach the goal and the different solutions composing this family of algorithms differ each other because of the logics driving the expansion. However the algorithm breaks when the goal cell is reached without providing any guaranty of global optimality on the solution. More advanced algorithms include more complex cost functions driving the expansion in such a way to provide some guarantees of local optimality of the solution, but the "greedy" optimization logics

From late 50s wide research activity was performed on graph-search algorithms within computer science, trying to support the design of computer networks. Soon after, the possibility of their application in robotics resulted evident and new solutions were developed to implement algorithms tailored for autonomous agents. As a consequence, research on graph-search methods brought new solutions and still continues nowadays. Therefore, an accurate analysis is required to understand advantages and drawbacks of each

The Dijkstra algorithm (Dijkstra, 1959) is one of the first and most important algorithms for graph search and permits to find the minimum path between two nodes of a graph with positive arc costs (Chandler et al, 2000). The structure of this algorithm is the one reported in the previous section and it represents the basic code for all the successive developments. An evolution of the Dijkstra algorithm is the Bellman-Ford (Bellman, 1958) algorithm; this method finds the minimum path on oriented graphs with positive, but also negative costs (Papaefthymiou & Rodriguez, 1991). Another method arose by the previous two is the Floyd-Warshall algorithm (Floyd, 1962, Warshall, 1962), that finds the shortest path on a

 Use the transition function to find a new cell Check inclusion of the new cell in the closed list

Check inclusion of the new cell in the open list

Evaluate the cost to come

 Evaluate the cost to come Add the cell to the open list

False: jump the state

characterizing these path planning techniques has in this one of its drawbacks.

proposed approach, in order to find possible improvements.

True: jump the state

False: go on

True:

False:

End of the searching cycle.

**3.2 From Dijkstra to A\*** 

End of the new state evaluation cycle

 True: go on False: cycle break Add this cell to the closed list Cancel this cell from the open list

: ''),( *XxxuxfUuandXxf* (1)


Classical graph search algorithms applied to path planning tasks then have other common elements:


Classical graph search algorithms treat each cell as a graph node and they search the shortest path with "greedy" logics. The algorithm applies the transition function to the current cell to move to the next one and it analyses systematically the state space from the starting cell trying to reach the goal one. Each analysed cell can be:

	- Alive: a cell that the algorithm could reach from another neighbouring cell. A cell alive is yet in the open list. The algorithm computes the new cost to come and substitutes the new cost associated to the cell whether it is lower then the previous one.
	- Dead: a cell that the algorithm already reached and its cost to come cannot be reduced further. These cells are stored in a list called *closed list.*

For each cell together with its coordinates and the cost to come, the algorithm stores in the lists also the parent coordinates. The parent is the cell left to reach a current one (i.e. *x0* used in *f(x0,u0)* is the *x* parent assuming that *x* is the current cell and *x' = f(x,u)* is the *x* neighbour).

Main structure of any classical graph search algorithms is:

	- Check that the open list is not empty
		- True: go on
		- False: cycle break
	- Sort the open list with respect to the cost to come
	- Take the cell with the lower cost
	- Check that this cell is not the target one

True: go on

164 Recent Advances in Aircraft Technology

Classical graph search algorithms applied to path planning tasks then have other common

the state space is the set of cells obtained meshing the environment in discrete fractions,

 the transition function checks the neighbours of a given cell to determine whether motion is possible (i.e. for an eight connected mesh the transition function checks the

the cost function evaluates the cost to move from a given cell to one of its neighbours.

Classical graph search algorithms treat each cell as a graph node and they search the shortest path with "greedy" logics. The algorithm applies the transition function to the current cell to move to the next one and it analyses systematically the state space from the

 Unexpanded: a cell that the algorithm has not been reached yet. When the algorithm reaches an unexpanded cell the cost to come to that cell is computed and the cell is

 Alive: a cell that the algorithm could reach from another neighbouring cell. A cell alive is yet in the open list. The algorithm computes the new cost to come and substitutes the new cost associated to the cell whether it is lower then the previous

Dead: a cell that the algorithm already reached and its cost to come cannot be

For each cell together with its coordinates and the cost to come, the algorithm stores in the lists also the parent coordinates. The parent is the cell left to reach a current one (i.e. *x0* used in *f(x0,u0)* is the *x* parent assuming that *x* is the current cell and *x' = f(x,u)* is the *x* neighbour).

Searching cycle (this cycle breaks when the goal cell is reached or the open list is

reduced further. These cells are stored in a list called *closed list.*

the action space is the set of cells reachable from a given cell,

starting cell trying to reach the goal one. Each analysed cell can be:

Main structure of any classical graph search algorithms is:

Sort the open list with respect to the cost to come

Check that the open list is not empty

 Take the cell with the lower cost Check that this cell is not the target one

 the initial state is the starting cell where the aircraft is supposed to be, the goal state is the goal cell where the aircraft is supposed to arrive.

 an initial state *XxI* , a goal state *<sup>G</sup> Xx* ,

eight neighbours of a given cell),

stored in a list called *open list.* Expanded (a cell already reached):

Insert the starting cell in open list

 True: go on False: cycle break

one.

empty):

elements:

: ''),( *XxxuxfUuandXxf* (1)

	- Use the transition function to find a new cell
	- Check inclusion of the new cell in the closed list
		- True: jump the state
		- False: go on
	- Check inclusion of the new cell in the open list
		- True:
			- Evaluate the cost to come
			- Check if the new cost is lower then the previous one:
				- True: substitute the new cost and the cell parent
				- False: jump the state
		- False:
			- Evaluate the cost to come
			- Add the cell to the open list

The algorithm expands systematically the cells up to reach the goal and the different solutions composing this family of algorithms differ each other because of the logics driving the expansion. However the algorithm breaks when the goal cell is reached without providing any guaranty of global optimality on the solution. More advanced algorithms include more complex cost functions driving the expansion in such a way to provide some guarantees of local optimality of the solution, but the "greedy" optimization logics characterizing these path planning techniques has in this one of its drawbacks.

From late 50s wide research activity was performed on graph-search algorithms within computer science, trying to support the design of computer networks. Soon after, the possibility of their application in robotics resulted evident and new solutions were developed to implement algorithms tailored for autonomous agents. As a consequence, research on graph-search methods brought new solutions and still continues nowadays. Therefore, an accurate analysis is required to understand advantages and drawbacks of each proposed approach, in order to find possible improvements.

## **3.2 From Dijkstra to A\***

The Dijkstra algorithm (Dijkstra, 1959) is one of the first and most important algorithms for graph search and permits to find the minimum path between two nodes of a graph with positive arc costs (Chandler et al, 2000). The structure of this algorithm is the one reported in the previous section and it represents the basic code for all the successive developments. An evolution of the Dijkstra algorithm is the Bellman-Ford (Bellman, 1958) algorithm; this method finds the minimum path on oriented graphs with positive, but also negative costs (Papaefthymiou & Rodriguez, 1991). Another method arose by the previous two is the Floyd-Warshall algorithm (Floyd, 1962, Warshall, 1962), that finds the shortest path on a

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 167

of D\* and D\* focused, but they recall the heuristic cost component of A\* to drive the cell expansion process. They are very similar and can be described together. LPA\* and D\* Lite exploit an incremental search method to update modified nodes, recalculating only the start distances (i.e. distance from the start cell) that have changed or have not been calculated before. These algorithms exploit the change of *consistency* of the path to replan. When obstacles move, graph cells are updated and their cost to come changes. The algorithm records the cell cost to come before modifications and compares the new cost with the old one to verify consistency. The change in consistency of the path drives the algorithm search.

Dynamic algorithms allowed new applications of graph search methods to path planning of robotic systems. More recently, other drawbacks and possible improvements were discovered. Particularly, one of the most important drawbacks of A\* and the entire dynamic algorithms resides on the heading constraints connected with the graph structure. The graph obtained from a surface map is a mesh of eight-connected cells with undirected edges. Moving from a given cell to the next means to move along the graph edge. The edges of these graphs are the straight lines connecting the centre of the current cell with the one of the neighbour. As a matter of fact the edges between cells of an eight connected graph can

80 *Nnnna*

Then the paths obtained with A\* and its successors is made of steps with heading defined in equation [3]. This limit is demonstrated prevents these algorithms to find the real shortest path between goal and start cells in many cases (it is easy to imagine a straight line connecting the start with goal cell having heading different from the ones of equation [3]). A\* and dynamic algorithms generate strongly suboptimal solutions because of this limit, that comes out in any application to path planning. Suboptimal solutions are paths with continuous heading changes and useless vehicle steering (increasing control losses) that require some kind of post processing to become feasible. Different approaches were developed to cope with this problem, based on post-processing algorithms or on improvements of the graph-search algorithm itself. Very important examples are Field D\* and Theta\*. These algorithms refined the graph search obtaining generalized paths with

To exploit Field D\*, the map must be meshed with cells of given geometry and the algorithm propagates information along the edges of the cells (Ferguson & Stentz, 2006). Field D\* evaluates neighbours of the current cell like D\*, but it also considers any path from the cell to any point along the perimeter of the neighbouring cell. A functional defines the point on the perimeter characterising the shortest path. With this method a wider range of headings

Theta\* represents the cutting edge algorithm on graph search, solving with a simple and effective method the heading constraint issue (Nash et al., 2007). It evaluates the distance from the parent to one of the neighbours for the current cell so that the shortest path is obtained. When the algorithm expands a cell, it evaluates two types of paths: from the

(3)

4 

**3.4 Any heading graph search** 

have slope *a* such that:

almost "any" heading.

can be achieved and shorter paths are obtained.

weighted graph with positive and negative weights, but it reduces the number of evaluated nodes compared with Dijkstra.

The A\* algorithm is one of the most important solvers developed between 50s and 70s, explicitly oriented to motion-robotics (Hart et al., 1968). A\* improved the logic of graph search adding a heuristic component to the cost function. Together with the evaluation of the cost to come (i.e. the distance between the current node and a neighbour), it also considers the cost to go (i.e. an heuristic evaluation of the distance between a neighbour and the goal cell). Indeed the cost function (F) exploited by the A\* algorithm is obtained summing up two terms:


The G-value is 0 for the starting cell and it increases while the algorithm expands successive cells. The H-value is used to drive the cells expansion toward the goal, reducing this way the amount of expanded cells and improving the convergence. Because in many cases is hard to determine the exact cost to go for a given cell, the H-function is an heuristic evaluation of this cost that has to be monotone or consistent. In other words, at each step the H-value of a cell has not to overestimate the cost to go and H has to vary along the path in such a way that:

$$H(\mathbf{x}', \mathbf{x}\_G) \le H(\mathbf{x}, \mathbf{x}\_G) + G(\mathbf{x}, \mathbf{x}') \tag{2}$$

#### **3.3 Dynamic graph search**

The graph-search algorithms developed between 60s and 80s were widely used in many fields, from robotics to video games, assuming fixed and known positions of the obstacles on the map. This is a logic assumption for many planning problems, but represents a limit when robots move in unknown environments. This problem excited research on algorithms able to face with map modifications during the path execution. Particularly, results on sensing robots, able to detect obstacles along the path, induced research on algorithms used to re-plan the trajectory with a more effective strategy than static solvers were able to implement.

Dynamic re-planning with graph search algorithms was introduced. D\* (Dynamic A\*) was published in 1993 (Stentz, 1993) and it represents the evolution of A\* for re-planning . When changes occur on the obstacle distribution some of the cell costs to come changes. Dynamic algorithms update the cost for these cells and replan only the portion of path around them keeping the remaining path unchanged. This way D\* expands less cells than A\* because it has not to re-plan the whole path through the end. D\* focused was the evolution of D\*, published by the same authors and developed to improve its characteristics (Stentz, 1995). This algorithm improved the expansion, reducing the amount of analysed nodes and the computational time.

Then, research on dynamic re-planning brought to the development of Lifelong Planning A\* (LPA\*) and D\* Lite (Koenig & Likhachev, 2001, 2002). They are based on the same principles of D\* and D\* focused, but they recall the heuristic cost component of A\* to drive the cell expansion process. They are very similar and can be described together. LPA\* and D\* Lite exploit an incremental search method to update modified nodes, recalculating only the start distances (i.e. distance from the start cell) that have changed or have not been calculated before. These algorithms exploit the change of *consistency* of the path to replan. When obstacles move, graph cells are updated and their cost to come changes. The algorithm records the cell cost to come before modifications and compares the new cost with the old one to verify consistency. The change in consistency of the path drives the algorithm search.

#### **3.4 Any heading graph search**

166 Recent Advances in Aircraft Technology

weighted graph with positive and negative weights, but it reduces the number of evaluated

The A\* algorithm is one of the most important solvers developed between 50s and 70s, explicitly oriented to motion-robotics (Hart et al., 1968). A\* improved the logic of graph search adding a heuristic component to the cost function. Together with the evaluation of the cost to come (i.e. the distance between the current node and a neighbour), it also considers the cost to go (i.e. an heuristic evaluation of the distance between a neighbour and the goal cell). Indeed the cost function (F) exploited by the A\* algorithm is obtained

The cost to go H: a heuristic estimation of the distance from the neighbouring cell *x'* to

The cost to come G: the distance between the expanded cell *x* and the neighbouring one

The G-value is 0 for the starting cell and it increases while the algorithm expands successive cells. The H-value is used to drive the cells expansion toward the goal, reducing this way the amount of expanded cells and improving the convergence. Because in many cases is hard to determine the exact cost to go for a given cell, the H-function is an heuristic evaluation of this cost that has to be monotone or consistent. In other words, at each step the H-value of a cell has not to overestimate the cost to go and H has to vary along the path in such a way

The graph-search algorithms developed between 60s and 80s were widely used in many fields, from robotics to video games, assuming fixed and known positions of the obstacles on the map. This is a logic assumption for many planning problems, but represents a limit when robots move in unknown environments. This problem excited research on algorithms able to face with map modifications during the path execution. Particularly, results on sensing robots, able to detect obstacles along the path, induced research on algorithms used to re-plan the trajectory with a more effective strategy than static solvers were able to

Dynamic re-planning with graph search algorithms was introduced. D\* (Dynamic A\*) was published in 1993 (Stentz, 1993) and it represents the evolution of A\* for re-planning . When changes occur on the obstacle distribution some of the cell costs to come changes. Dynamic algorithms update the cost for these cells and replan only the portion of path around them keeping the remaining path unchanged. This way D\* expands less cells than A\* because it has not to re-plan the whole path through the end. D\* focused was the evolution of D\*, published by the same authors and developed to improve its characteristics (Stentz, 1995). This algorithm improved the expansion, reducing the amount of analysed nodes and the

Then, research on dynamic re-planning brought to the development of Lifelong Planning A\* (LPA\*) and D\* Lite (Koenig & Likhachev, 2001, 2002). They are based on the same principles

*<sup>G</sup> <sup>G</sup> xxGxxHxxH* )',(),(),'( (2)

nodes compared with Dijkstra.

summing up two terms:

**3.3 Dynamic graph search** 

the goal *xG*.

*x'*.

that:

implement.

computational time.

Dynamic algorithms allowed new applications of graph search methods to path planning of robotic systems. More recently, other drawbacks and possible improvements were discovered. Particularly, one of the most important drawbacks of A\* and the entire dynamic algorithms resides on the heading constraints connected with the graph structure. The graph obtained from a surface map is a mesh of eight-connected cells with undirected edges. Moving from a given cell to the next means to move along the graph edge. The edges of these graphs are the straight lines connecting the centre of the current cell with the one of the neighbour. As a matter of fact the edges between cells of an eight connected graph can have slope *a* such that:

$$a = n \cdot \frac{\pi}{4} \quad 0 \le n \le 8 \quad n \in N \tag{3}$$

Then the paths obtained with A\* and its successors is made of steps with heading defined in equation [3]. This limit is demonstrated prevents these algorithms to find the real shortest path between goal and start cells in many cases (it is easy to imagine a straight line connecting the start with goal cell having heading different from the ones of equation [3]). A\* and dynamic algorithms generate strongly suboptimal solutions because of this limit, that comes out in any application to path planning. Suboptimal solutions are paths with continuous heading changes and useless vehicle steering (increasing control losses) that require some kind of post processing to become feasible. Different approaches were developed to cope with this problem, based on post-processing algorithms or on improvements of the graph-search algorithm itself. Very important examples are Field D\* and Theta\*. These algorithms refined the graph search obtaining generalized paths with almost "any" heading.

To exploit Field D\*, the map must be meshed with cells of given geometry and the algorithm propagates information along the edges of the cells (Ferguson & Stentz, 2006). Field D\* evaluates neighbours of the current cell like D\*, but it also considers any path from the cell to any point along the perimeter of the neighbouring cell. A functional defines the point on the perimeter characterising the shortest path. With this method a wider range of headings can be achieved and shorter paths are obtained.

Theta\* represents the cutting edge algorithm on graph search, solving with a simple and effective method the heading constraint issue (Nash et al., 2007). It evaluates the distance from the parent to one of the neighbours for the current cell so that the shortest path is obtained. When the algorithm expands a cell, it evaluates two types of paths: from the

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 169

Longitude [m]

Fig. 3. Comparison between Theta\* and A\* (Longitude-Latitude plane).

50 100 150 200 250 300

0

5

10

15

20

25

30

Theta star A star

Fig. 2. Comparison between Theta\* and A\* (3D view).

Latitude [m]

50

100

150

200

250

300

current cell to the neighbour (like in A\*) and from the current-cell parent to the neighbour. As a conclusion, paths obtained by the Theta\* solver are smoother and shorter than those generated by A\*.

Apparently, Theta\* is the most promising solution for path planning. As a matter of fact, some other graph search algorithms were not considered here, as this chapter would provide a general overview on the main concepts converging in development of these pathplanning methods. By the way all the algorithms described have the common drawback of missing any kind of vehicle kinematic constraints in the path generation. The algorithm presented in the following chapter (Kinematic A\*) has been developed to bridge this gap and open investigations in this direction.

## **3.5 Tridimensional path planning with A\* and Theta\***

The application of A\* and Theta\* to 3D path planning for mini and micro UAVs was extensively investigated (De Filippis et al., 2010, 2011). The A\*-basic algorithm was improved and applied to tri-dimensional path planning on highlands and urban environments. Then this algorithm has been compared with Theta\* for the same applications in order to investigate merits and drawbacks of these solutions.

Here is reported the comparison between a path planned with A\* with the same one planned with Theta\* in order to show the improvements introduced adopting the last algorithm. **Figure 2** is the tri-dimensional view of the two paths implemented for this example.

Map characteristics:



Table 1. Example parameters.

current cell to the neighbour (like in A\*) and from the current-cell parent to the neighbour. As a conclusion, paths obtained by the Theta\* solver are smoother and shorter than those

Apparently, Theta\* is the most promising solution for path planning. As a matter of fact, some other graph search algorithms were not considered here, as this chapter would provide a general overview on the main concepts converging in development of these pathplanning methods. By the way all the algorithms described have the common drawback of missing any kind of vehicle kinematic constraints in the path generation. The algorithm presented in the following chapter (Kinematic A\*) has been developed to bridge this gap

The application of A\* and Theta\* to 3D path planning for mini and micro UAVs was extensively investigated (De Filippis et al., 2010, 2011). The A\*-basic algorithm was improved and applied to tri-dimensional path planning on highlands and urban environments. Then this algorithm has been compared with Theta\* for the same

Here is reported the comparison between a path planned with A\* with the same one planned with Theta\* in order to show the improvements introduced adopting the last algorithm. **Figure 2** is the tri-dimensional view of the two paths implemented for this

applications in order to investigate merits and drawbacks of these solutions.

Environment matrix dimensions: 300 x 300 x 111 (lat x long x Z).

 A\* Theta\* Path length 386.5 m 372.4 m Computation time 3.1 s 3.6 s Number of heading changes 327 6 Number of altitude changes 0 0 Number of waypoints 327 6

generated by A\*.

example.

Path 1

Map characteristics:

 Δlat: 1 m. Δlong: 1 m. ΔZ: 1 m.

Cells number: 9990000.

Table 1. Example parameters.

and open investigations in this direction.

**3.5 Tridimensional path planning with A\* and Theta\*** 

Fig. 2. Comparison between Theta\* and A\* (3D view).

Fig. 3. Comparison between Theta\* and A\* (Longitude-Latitude plane).

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 171

The model is a set of four differential equations describing the aircraft motion in Ground reference frame (G frame). This is not the typical Nort-East-Down (NED) frame used to write navigation equations in aeronautics. The Ground frame is typical of ground robotic applications that inspired this work. The G-frame origin is placed in the aircraft center of mass. The X and Y axes are aligned with the longitude and latitude directions respectively.

In the G frame distances are measured in meters and two control angles (χ and γ) act as

χ is the angle between the X axis and the projection of the speed vector (V) on the X-Y

γ is the angle between the speed vector and its projection on the X-Z plane (see Figure

The model is obtained considering the aircraft flying at constant speed and the Body frame (B frame) aligned with the Wind frame (W frame). The rate of turn is assumed bounded

The speed vector is constant and aligned with the XB axis. Using the Euler transformation matrix from the body to the ground frame the speed components in G frame are obtained. Combining these differential equations with the turning-rate the aircraft model becomes:

*u* 1

*<sup>w</sup>* <sup>1</sup> (4)

*X V* cos(

*Y V* sin(

*Z V* sin(

 *<sup>V</sup> R*min *u*

 

 

X,Y,Z = aircraft positions vector P on the ground frame [m].

where:

V = aircraft speed [m/s].

)cos(max *w*)

 )cos(max *w*)

max *w*)

with the minimum turn radius and the rate of climb with the maximum climb angle.

plane, the variation of this angle is connected with the rate of turn.

Then the Z axis points up completing the frame.

4), this angle controls the rate of climb.

Fig. 4. The Ground Reference frame (G frame).

gains on rate of turn and rate of climb along the path:

**Table 1** collects the map parameters and the algorithm performances while **Figure 3** is the longitude-latitude path view. The paths are planned without altitude changes so the last picture is sufficient to depict differences between them. The path obtained with Theta\* is slightly shorter then the A\* one, but huge difference is in the number of waypoints composing the path and in the amount of heading changes. These parameters testify the previous statements identifying in Theta\* an interesting algorithm among the classical graph search solutions here mentioned.

## **4. Kinematic A\***

The main drawback of applying classical graph search algorithms to path planning problems resides in the lack of correlation between the path and the vehicle kinematic constraints. In this section, a new path-planning algorithm (Kinematic A\*) is presented, implementing the graph search logics to generate feasible paths and introducing basic vehicle characteristics to drive the search.

Kinematic A\* (KA\*) includes a simple kinematic model of the vehicle to evaluate the moving cost between the waypoints of the path in a tridimensional environment. Movements are constrained with the minimum turning radius and the maximum rate of climb. Furthermore, separation from obstacles is imposed, defining a volume along the path free from obstacles (tube-type boundaries), as inside these limits the navigation of the vehicle is assumed to be safe.

The main structure of the algorithm will be presented in this section, together with the most important subroutines composing the path planner.

## **4.1 From cells to state variables**

Classical graph search algorithms solve a discrete optimization problem linking the cost function evaluation to the distance between cells. These cells discretize the motion space representing the discrete state of the system. The states space is finite and discrete containing the positions of the cell centres. The optimization problem requires finding the sequence of states minimizing the total covered distance between the starting and the target cell.

Kinematic A\* introduces a vehicle model to generate the states and evaluate the cost function. Each state is made of the model variables and is discrete because the command space is made of discrete variables. So the optimization problem is transformed in finding the discrete sequence of optimal commands generating the minimum path between the starting and the target state.

In the following sections then the concept of cells or nodes of the graph, representing the discrete set of states defining the optimization problem is substituted with the concept of states of the vehicle model and the optimization problem is reformulated.

#### **4.2 The kinematic model**

In the following description S is the state of the aircraft at the current position. S is the vector of the model state variables. This simple model is used to generate the possible movements from a given state to the next, i.e. the evolution of S from the current condition to the next.

The model is a set of four differential equations describing the aircraft motion in Ground reference frame (G frame). This is not the typical Nort-East-Down (NED) frame used to write navigation equations in aeronautics. The Ground frame is typical of ground robotic applications that inspired this work. The G-frame origin is placed in the aircraft center of mass. The X and Y axes are aligned with the longitude and latitude directions respectively. Then the Z axis points up completing the frame.

In the G frame distances are measured in meters and two control angles (χ and γ) act as gains on rate of turn and rate of climb along the path:


The model is obtained considering the aircraft flying at constant speed and the Body frame (B frame) aligned with the Wind frame (W frame). The rate of turn is assumed bounded with the minimum turn radius and the rate of climb with the maximum climb angle.

Fig. 4. The Ground Reference frame (G frame).

The speed vector is constant and aligned with the XB axis. Using the Euler transformation matrix from the body to the ground frame the speed components in G frame are obtained. Combining these differential equations with the turning-rate the aircraft model becomes:

$$\begin{cases} \dot{X} = V \cos(\boldsymbol{\chi}) \cos(\boldsymbol{\chi}\_{\text{max}} \cdot \boldsymbol{w}) \\ \dot{Y} = V \sin(\boldsymbol{\chi}) \cos(\boldsymbol{\chi}\_{\text{max}} \cdot \boldsymbol{w}) \\ \dot{Z} = V \sin(\boldsymbol{\chi}\_{\text{max}} \cdot \boldsymbol{w}) \\ \dot{Z} = \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u} \end{cases} \quad \left| \boldsymbol{u} \right| \le 1 \tag{4}$$

where:

170 Recent Advances in Aircraft Technology

**Table 1** collects the map parameters and the algorithm performances while **Figure 3** is the longitude-latitude path view. The paths are planned without altitude changes so the last picture is sufficient to depict differences between them. The path obtained with Theta\* is slightly shorter then the A\* one, but huge difference is in the number of waypoints composing the path and in the amount of heading changes. These parameters testify the previous statements identifying in Theta\* an interesting algorithm among the classical graph

The main drawback of applying classical graph search algorithms to path planning problems resides in the lack of correlation between the path and the vehicle kinematic constraints. In this section, a new path-planning algorithm (Kinematic A\*) is presented, implementing the graph search logics to generate feasible paths and introducing basic

Kinematic A\* (KA\*) includes a simple kinematic model of the vehicle to evaluate the moving cost between the waypoints of the path in a tridimensional environment. Movements are constrained with the minimum turning radius and the maximum rate of climb. Furthermore, separation from obstacles is imposed, defining a volume along the path free from obstacles (tube-type boundaries), as inside these limits the navigation of the vehicle is

The main structure of the algorithm will be presented in this section, together with the most

Classical graph search algorithms solve a discrete optimization problem linking the cost function evaluation to the distance between cells. These cells discretize the motion space representing the discrete state of the system. The states space is finite and discrete containing the positions of the cell centres. The optimization problem requires finding the sequence of states minimizing the total covered distance between the starting and the target

Kinematic A\* introduces a vehicle model to generate the states and evaluate the cost function. Each state is made of the model variables and is discrete because the command space is made of discrete variables. So the optimization problem is transformed in finding the discrete sequence of optimal commands generating the minimum path between the

In the following sections then the concept of cells or nodes of the graph, representing the discrete set of states defining the optimization problem is substituted with the concept of

In the following description S is the state of the aircraft at the current position. S is the vector of the model state variables. This simple model is used to generate the possible movements from a given state to the next, i.e. the evolution of S from the current condition to the next.

states of the vehicle model and the optimization problem is reformulated.

search solutions here mentioned.

vehicle characteristics to drive the search.

**4.1 From cells to state variables** 

starting and the target state.

**4.2 The kinematic model** 

important subroutines composing the path planner.

**4. Kinematic A\*** 

assumed to be safe.

cell.

X,Y,Z = aircraft positions vector P on the ground frame [m].

V = aircraft speed [m/s].

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 173

The kinematic model can be improved taking into account the wind effect on the states evolution. Summing to the aircraft speed the wind components in G frame and assuming

*R V*

*si <sup>i</sup> xs <sup>i</sup>*

cos( sin) )sin(

*R V*

cos( cos) )cos(

 

*i s i ys*

 

 

*w*

min max min

*w*

min max min

 

 

In Figure 6 and Figure 7 a state evolution with wind is compared with the same without

 

(7)

 

*tu tW*

*tu tW*

 

Fig. 5. Sequences of states for a time horizon of 8 seconds.

these components constant on Δt the system of equations [6] become:

**4.3 The kinematic model with wind** 

min

*u*

*i*

*R V*

*si i*

 

 

*si*

*<sup>R</sup> XX*

*<sup>R</sup> YY*

*si*

 

 *VZZ tWtw*

sin( )

max

*si zi*

*tu*

[Wx Wy Wz] = wind speed components in G frame [m/s].

wind. The state and parameters used as an example are:

 

*u*

 

 

 

Ps = [0 0 0 0], V = 25 [m/s], Rmin = 120 [m], γmin = 4 [deg], Δt = 8 [s].

where:

Rmin =maximum turning radius [m].

χ = turning angle.

γmax = maximum climbing angle.

u,w = command parameters.

To generate the set of possible movements discrete command values (ui and wi) are chosen and the system of equations [4] is integrated in time with the initial conditions given by the current state S:

$$\begin{cases} X(0) = X\_s \\ Y(0) = Y\_s \\ Z(0) = Z\_s \\ X(0) = Z\_s \end{cases} \quad \mu\_i = \begin{bmatrix} -1 & -0.5 & 0 & 0.5 & 1 \end{bmatrix} \tag{5}$$
 
$$\begin{cases} Z(0) = Z\_s \\ X(0) = Z\_s \end{cases} \quad \text{or} \begin{bmatrix} -1 & -0.5 & 0 & 0.5 & 1 \end{bmatrix} \tag{6}$$

If the command values are constant along the integration time (Δt), the equations in [4] become:

$$\begin{cases} X\_i = X\_s + \left(\frac{R\_{\text{min}}}{u\_i}\right) \cdot \cos(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \left[\sin\left(\mathcal{Z}\_s + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t\right) - \sin(\mathcal{Z}\_s)\right] \\ Y\_i = Y\_s - \left(\frac{R\_{\text{min}}}{u\_i}\right) \cdot \cos(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \left[\cos\left(\mathcal{Z}\_s + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t\right) - \cos(\mathcal{Z}\_s)\right] \\ Z\_i = Z\_s + V \cdot \sin(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \Delta t \\ \mathcal{X}\_i = \mathcal{X}\_s + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t \end{cases} \tag{6}$$

providing the evolution of S for each controls space. On Figure 5 25 trajectories are represented. They are obtained combining the two vectors *u* and *w* presented in [5] and substituting each command couple (5 *ui* values x 5 *wi* values) in [6]. For each couple the system of equation is integrated over the time step with initial conditions and parameters equal to:

Ps = [0 0 0 0], V = 25 [m/s], Rmin = 120 [m], γmax = 4 [deg], Δt = 8 [s].

Once Δt, aircraft speed, minimum turning radius and maximum climbing angle are chosen according with the aircraft kinematic constraints the equations in [6] can be solved at each cycle for the current state and the algorithm can generate the set of possible movements looking for the optimal path.

Fig. 5. Sequences of states for a time horizon of 8 seconds.

#### **4.3 The kinematic model with wind**

The kinematic model can be improved taking into account the wind effect on the states evolution. Summing to the aircraft speed the wind components in G frame and assuming these components constant on Δt the system of equations [6] become:

$$\begin{cases} X\_i = X\_S + \left(\frac{R\_{\text{min}}}{u\_i}\right) \cdot \cos(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \left[\sin\left(\mathcal{Z}\_S + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t\right) - \sin(\mathcal{Z}\_S)\right] + W\_X \cdot \Delta t \\\ Y\_i = Y\_S - \left(\frac{R\_{\text{min}}}{u\_i}\right) \cdot \cos(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \left[\cos\left(\mathcal{Z}\_S + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t\right) - \cos(\mathcal{Z}\_S)\right] + W\_Y \cdot \Delta t \\\ Z\_i = Z\_S + V \cdot \sin(\mathcal{Y}\_{\text{max}} \cdot \boldsymbol{w}\_i) \cdot \Delta t + W\_Z \cdot \Delta t \\\ \mathcal{Z}\_i = \mathcal{Z}\_S + \frac{V}{R\_{\text{min}}} \cdot \boldsymbol{u}\_i \cdot \Delta t \end{cases} (7)$$

where:

172 Recent Advances in Aircraft Technology

To generate the set of possible movements discrete command values (ui and wi) are chosen and the system of equations [4] is integrated in time with the initial conditions given by the

If the command values are constant along the integration time (Δt), the equations in [4]

) sin

 

) cos

providing the evolution of S for each controls space. On Figure 5 25 trajectories are represented. They are obtained combining the two vectors *u* and *w* presented in [5] and substituting each command couple (5 *ui* values x 5 *wi* values) in [6]. For each couple the system of equation is integrated over the time step with initial conditions and parameters

Once Δt, aircraft speed, minimum turning radius and maximum climbing angle are chosen according with the aircraft kinematic constraints the equations in [6] can be solved at each cycle for the current state and the algorithm can generate the set of possible movements

 

) *t*

*<sup>s</sup>*

 

  *<sup>s</sup>*

*V R*min

*V R*min *ui t*

*ui t*

 

 

(6)

 sin( *s* )

 cos(*s* )

*ui* [ 1 0.5 0 0.5 1]

*wi* [ <sup>1</sup> 0.5 0 0.5 1] (5)

Rmin =maximum turning radius [m].

*X*(0) *Xs Y*(0) *Ys Z*(0) *Zs*

(0) *s*

*R*min *ui*

*ui*

*V R*min  cos(max *wi*

 cos(max *wi*

max *wi*

*ui t*

 

 

*Zi Zs V* sin(

 

 

*Xi Xs*

 

 

 *<sup>i</sup> <sup>s</sup>*

looking for the optimal path.

*Yi Ys <sup>R</sup>*min

γmax = maximum climbing angle. u,w = command parameters.

χ = turning angle.

current state S:

become:

equal to:

Ps = [0 0 0 0], V = 25 [m/s], Rmin = 120 [m], γmax = 4 [deg], Δt = 8 [s].

[Wx Wy Wz] = wind speed components in G frame [m/s].

In Figure 6 and Figure 7 a state evolution with wind is compared with the same without wind. The state and parameters used as an example are:

Ps = [0 0 0 0], V = 25 [m/s], Rmin = 120 [m], γmin = 4 [deg], Δt = 8 [s].

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 175

These pictures show how wind affects the evolution of a given state and in turn the effect on

The functional J minimized through the optimization process is made up of the costs Fij of each state composing the path. The minimum of J is found summing up the smaller cost Fij of each state. Fij is made of two terms related respectively with the states and the commands. At each step the algorithm generates the set of movements from the current state (shown in Figure 5). Then it evaluates Fij for each new state and chooses the one with the smaller value.

*J Fij GGHH*

The H and G vectors take into account respectively the error on the states and the amount of command due to reach a new state. The matrices α and β are diagonal matrices of gains on

*T ji*

> 

 

 

 

2

0

2

The H vector is the distance between the new state [Xi Yi Zi]' and the target one [Xt Yt Zt]'. On the other hand the G vector evaluates the amount of command needed to reach this new state from the current one. Then choosing the smaller value of F the algorithm selects a new state that reduces the distance from the target minimizing the commands. The gain matrices are used to weight the state variables and the commands in order to tune their importance

To complete the problem formulation, the states in J must be included in the state space respecting the differential equations given in [4] and the initial conditions given in [5]. Then the commands must be chosen in the command space given in [5] in order to minimize the functional J. The state space is constrained by the map limits, the obstacles and the

00 00 00

1

0

1

*i i i*

*Z Y X*

 

3

*ij*

(9)

(10)

) (8)

*T ijij*

*S*

 

*i i*

 

separation requirements and will be described in the following section.

 

 

*w u*

 

*ZZ YY XX*

*it it it*

 

*i*

*H*

*i*

*G*

the set of possible movements from the current state to the next.

*<sup>t</sup> St*

min()min()min(

0 0

*S*

*S*

**4.4 The problem formulation** 

the states and on the commands:

in F.

The global optimization problem is finding:

Fig. 6. Comparison of states evolution (3D view) with and without wind (Wx=5 m/s, Wy=5 m/s, Wz=0).

Fig. 7. Comparison of states evolution (2D views) with and without wind (Wx=5 m/s, Wy=5 m/s, Wz=0).

These pictures show how wind affects the evolution of a given state and in turn the effect on the set of possible movements from the current state to the next.

#### **4.4 The problem formulation**

174 Recent Advances in Aircraft Technology

Fig. 6. Comparison of states evolution (3D view) with and without wind (Wx=5 m/s, Wy=5

Fig. 7. Comparison of states evolution (2D views) with and without wind (Wx=5 m/s, Wy=5

Wx = 5 [m/s] Wy = 5 [m/s] Wz = 0 [m/s]

m/s, Wz=0).

m/s, Wz=0).

The functional J minimized through the optimization process is made up of the costs Fij of each state composing the path. The minimum of J is found summing up the smaller cost Fij of each state. Fij is made of two terms related respectively with the states and the commands. At each step the algorithm generates the set of movements from the current state (shown in Figure 5). Then it evaluates Fij for each new state and chooses the one with the smaller value. The global optimization problem is finding:

$$\min(J) = \sum\_{S\_0}^{S\_t} \min(F\_{ij}) = \sum\_{S\_0}^{S\_t} \min(\overline{H}\_{ji}^T \cdot \overline{\alpha} \cdot \overline{H}\_{ij} + \overline{G}\_{ij}^T \cdot \overline{\beta} \cdot \overline{G}\_{ij}) \tag{8}$$

The H and G vectors take into account respectively the error on the states and the amount of command due to reach a new state. The matrices α and β are diagonal matrices of gains on the states and on the commands:

$$
\overline{H}\_i = \begin{bmatrix} X\_t - X\_i \\ Y\_t - Y\_i \\ Z\_t - Z\_i \end{bmatrix} = \begin{bmatrix} \Delta X\_i \\ \Delta Y\_i \\ \Delta Z\_i \end{bmatrix} \tag{9}
$$

$$
\overline{G}\_i = \begin{bmatrix} u\_i \\ u\_i \\ w\_i \end{bmatrix}
$$

$$
\overline{\alpha} = \begin{bmatrix} \alpha\_1 & 0 & 0 \\ 0 & \alpha\_2 & 0 \\ 0 & 0 & \alpha\_3 \end{bmatrix} \tag{10}
$$

$$
\overline{\beta} = \begin{bmatrix} \beta\_1 & 0 \\ 0 & \beta\_2 \end{bmatrix}
$$

The H vector is the distance between the new state [Xi Yi Zi]' and the target one [Xt Yt Zt]'. On the other hand the G vector evaluates the amount of command needed to reach this new state from the current one. Then choosing the smaller value of F the algorithm selects a new state that reduces the distance from the target minimizing the commands. The gain matrices are used to weight the state variables and the commands in order to tune their importance in F.

To complete the problem formulation, the states in J must be included in the state space respecting the differential equations given in [4] and the initial conditions given in [5]. Then the commands must be chosen in the command space given in [5] in order to minimize the functional J. The state space is constrained by the map limits, the obstacles and the separation requirements and will be described in the following section.

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 177

New states evaluation cycle (this cycle breaks when each new state has been

Check if the new cost is lower then the previous one:

True: substitute this new cost and the new current state to

Open list search for the state with the lower F value

 New states generation through the model equations Check inclusion of the new state inside the state space

Check inclusion of this state in the closed list

Check inclusion of this state in the open list

Evaluate the F cost for the state

the open list False: jump the state

 Evaluate the F cost for the state Add the state to the open list

limitations of this new technique in order to stimulate future developments.

The following chapter collects two tests with the obstacles placed on the map in such a way to force KA\* toward its limits. The new algorithm is compared with A\* in order to show the improvements introduced. The paths are generated with and without wind to show its effects on the path and to compare the results. These tests then give the opportunity to show

The first path is on a map with four obstacles symmetrically placed. They are close to the four corners of a square area and the aircraft is forced to slalom between them. The starting and target points are placed respectively at the bottom-left and top-right corner with different altitudes in order to force the algorithm to plan a descent meeting the four obstacles along the flight. Finding the path for this test is easier then finding it for the next one. The algorithm is able to follow the minimum of the cost function without analyzing too

The map of the second path has just one wide obstacle placed in the middle. The obstacle is placed slightly closer to the right border of the map in order to obstruct the path to the aircraft that is supposed to move from the bottom-left to the top-right corner. The path

End of the new state evaluation cycle

Check that this state is not the target one

True: go on

False: go on

True:

False:

End of the searching cycle.

many states and it converges rapidly.

**5. Results** 

False: jump the state

True: jump the state

 True: go on False: cycle break Add this state to the closed list Cancel this state from the open list

evaluated)

#### **4.5 The state space**

If the command space of the problem solved with KA\* is bounded by the kinematic constraints and is discretized according with the optimization requirements, the states are bounded only on the X and Y sets (longitude and latitude coordinates) because of constraints on the Z state and on the X and Y states themselves:


Figure 8 shows the horizontal (HZ1 and HZ2) and vertical (VZ1) separation constraints imposed on the path from the current state S to the next state I. These constraints guarantee the flight safety along the path because possible tracking errors of the guidance system are acceptable and safe inside the boundaries imposed by the separation constraints.

Fig. 8. Horizontal and Vertical separation from an obstacle.

### **4.6 Algorithm description**

	- Check that the open list is not empty
		- True: go on
		- False: cycle break
	- True: go on

If the command space of the problem solved with KA\* is bounded by the kinematic constraints and is discretized according with the optimization requirements, the states are bounded only on the X and Y sets (longitude and latitude coordinates) because of

The map bounds: these bounds affect the X-Y sets because points outside the map limits

 The ground obstacles: these constraints bound the X-Y sets if the Z component of the new state is lower then the ground altitude at the same X-Y coordinates, so the relative

 The separation constraints: the new state not only has to have a Z component higher then the ground one, but has also to respect the horizontal and vertical separations from the obstacles. The X-Y sets are bounded because states too close to the obstacles must be

Figure 8 shows the horizontal (HZ1 and HZ2) and vertical (VZ1) separation constraints imposed on the path from the current state S to the next state I. These constraints guarantee the flight safety along the path because possible tracking errors of the guidance system are

Searching cycle (this cycle breaks when the target state is reached or the open list is

acceptable and safe inside the boundaries imposed by the separation constraints.

constraints on the Z state and on the X and Y states themselves:

Fig. 8. Horizontal and Vertical separation from an obstacle.

Initialize the control variables and parameters.

Check that the open list is not empty

Evaluate the F cost for the initial state

Add this state to the open list.

 True: go on False: cycle break

**4.6 Algorithm description** 

empty):

can not be accepted as new states.

new state must be rejected.

**4.5 The state space** 

rejected.

	- New states generation through the model equations
	- Check inclusion of the new state inside the state space
		- True: go on
		- False: jump the state
	- Check inclusion of this state in the closed list
		- True: jump the state
		- False: go on
	- Check inclusion of this state in the open list
		- True:
			- Evaluate the F cost for the state
			- Check if the new cost is lower then the previous one:
				- True: substitute this new cost and the new current state to the open list
				- False: jump the state
		- False:
			- Evaluate the F cost for the state
			- Add the state to the open list
	- End of the new state evaluation cycle

## **5. Results**

The following chapter collects two tests with the obstacles placed on the map in such a way to force KA\* toward its limits. The new algorithm is compared with A\* in order to show the improvements introduced. The paths are generated with and without wind to show its effects on the path and to compare the results. These tests then give the opportunity to show limitations of this new technique in order to stimulate future developments.

The first path is on a map with four obstacles symmetrically placed. They are close to the four corners of a square area and the aircraft is forced to slalom between them. The starting and target points are placed respectively at the bottom-left and top-right corner with different altitudes in order to force the algorithm to plan a descent meeting the four obstacles along the flight. Finding the path for this test is easier then finding it for the next one. The algorithm is able to follow the minimum of the cost function without analyzing too many states and it converges rapidly.

The map of the second path has just one wide obstacle placed in the middle. The obstacle is placed slightly closer to the right border of the map in order to obstruct the path to the aircraft that is supposed to move from the bottom-left to the top-right corner. The path

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 179

**Figure 9** shows the obstacles position on the map and the three paths (KA\* with wind, KA\*

**Table 2** collects the numerical data used to implement this test. The time step between two states is set to 2 seconds in order to have a sufficiently discretized path without increasing too much the computation time. Horizontal and vertical obstacles separations are set to 15 m and 10 m respectively. This should guarantee sufficient safety without limiting the aircraft

The constant wind along the *Y* ground direction has 5 m/s intensity. This value is sufficiently high and it affects deeply the path as the computation time, the number of waypoints and the path shape testify. The wind pushes the aircraft toward the target,

In order to compare the KA\* performances with the A\* one, it can be noticed that the computation time for the two algorithms is almost the same for the path with wind, but it is increased without wind. This is due to the reduced speed of the aircraft without wind and to the higher number of possible movements from one state to the next. With wind the feasible movements between states are strongly reduced because of the wind disturbance. Flying at lower speed the algorithm is forced to analyze much more states and for each position much more possible movements are feasible. Then the optimization process takes more time. Finally shall be noticed how KA\* generates a path with a really small number of waypoints

reducing the computation time with respect to the case without wind.

path length,

**5.1 Four obstacles** 

number of waypoints.

without wind, A\*) in tridimensional view.

Fig. 9. Four obstacles test (3D view).

agility between the obstacles.

search for this test is harder then the previous one because the algorithm has to analyze many states to find the optimum path. Following a monotonic decrease of the cost function along the path search is impossible for this case. The obstacle in the middle forces the aircraft far from the target point. The aircraft has to go around the obstacle to reach the target; this induces a cost increase to move from a state to the next that makes the optimization harder.

The component of wind introduced to implement the following tests is considered constant in time and space on the whole map. This approach clearly does not mean to solve the problems due to wind disturbances in the path optimization. This is a complex and hard problem due to wind model complexity, effects of the wind on the aircraft performances and dynamics, turbulent components effects, etc. Face properly this problem requires specific studies and techniques, but it is useful to introduce this simple study at this level in order to show potential developments of this path planning technique for future applications.

Then the aircraft parameters chosen to implement the tests must be motivated. The small area of the map, induced to chose accordingly the aircraft parameters needed for the model. The reference vehicle is a mini UAV with reduced cruise speed, turning radius and climbing performances but agile enough to perform the required paths. Particularly speed is chosen so that the trajectories needed to avoid the obstacles would be feasible and the turning radius is calculated considering coordinated turns:

$$R\_{\min} = \sqrt{\frac{V^4}{g^2 \cdot \left(\left(\frac{1}{\cos(\varphi\_{\max})}\right)^2 - 1\right)}}\tag{11}$$

where:

*Rmin =* minimum turning radius.


Finally for each test a table collecting all the data is reported. All the reported paths are obtained with the MATLAB version 7.11.0 (R2010b), running on MacBook Pro with Intel Core 2 Duo (2 X 2.53 GHz), 4 Gb RAM and MAC OS X 10.5.8. The table contains:


path length,

178 Recent Advances in Aircraft Technology

search for this test is harder then the previous one because the algorithm has to analyze many states to find the optimum path. Following a monotonic decrease of the cost function along the path search is impossible for this case. The obstacle in the middle forces the aircraft far from the target point. The aircraft has to go around the obstacle to reach the target; this induces a cost increase to move from a state to the next that makes the

The component of wind introduced to implement the following tests is considered constant in time and space on the whole map. This approach clearly does not mean to solve the problems due to wind disturbances in the path optimization. This is a complex and hard problem due to wind model complexity, effects of the wind on the aircraft performances and dynamics, turbulent components effects, etc. Face properly this problem requires specific studies and techniques, but it is useful to introduce this simple study at this level in order to show potential developments of this path planning technique for future

Then the aircraft parameters chosen to implement the tests must be motivated. The small area of the map, induced to chose accordingly the aircraft parameters needed for the model. The reference vehicle is a mini UAV with reduced cruise speed, turning radius and climbing performances but agile enough to perform the required paths. Particularly speed is chosen so that the trajectories needed to avoid the obstacles would be feasible and the turning

> 

 

*<sup>V</sup> <sup>R</sup>*

Finally for each test a table collecting all the data is reported. All the reported paths are obtained with the MATLAB version 7.11.0 (R2010b), running on MacBook Pro with Intel

)cos( <sup>1</sup> <sup>2</sup> max

4

2

*g*

Core 2 Duo (2 X 2.53 GHz), 4 Gb RAM and MAC OS X 10.5.8. The table contains:

min

 

1

 

(11)

optimization harder.

applications.

where:

*Rmin =* minimum turning radius. *V =* aircraft cruise speed. *g =* gravitational acceleration. *φmax =* maximum bank angle.

 map dimensions, obstacles dimensions, obstacles center position,

 starting point target point, aircraft parameters, optimization parameters, obstacles separation parameters,

 wind speed, computation time,

radius is calculated considering coordinated turns:

number of waypoints.

### **5.1 Four obstacles**

**Figure 9** shows the obstacles position on the map and the three paths (KA\* with wind, KA\* without wind, A\*) in tridimensional view.

Fig. 9. Four obstacles test (3D view).

**Table 2** collects the numerical data used to implement this test. The time step between two states is set to 2 seconds in order to have a sufficiently discretized path without increasing too much the computation time. Horizontal and vertical obstacles separations are set to 15 m and 10 m respectively. This should guarantee sufficient safety without limiting the aircraft agility between the obstacles.

The constant wind along the *Y* ground direction has 5 m/s intensity. This value is sufficiently high and it affects deeply the path as the computation time, the number of waypoints and the path shape testify. The wind pushes the aircraft toward the target, reducing the computation time with respect to the case without wind.

In order to compare the KA\* performances with the A\* one, it can be noticed that the computation time for the two algorithms is almost the same for the path with wind, but it is increased without wind. This is due to the reduced speed of the aircraft without wind and to the higher number of possible movements from one state to the next. With wind the feasible movements between states are strongly reduced because of the wind disturbance. Flying at lower speed the algorithm is forced to analyze much more states and for each position much more possible movements are feasible. Then the optimization process takes more time. Finally shall be noticed how KA\* generates a path with a really small number of waypoints

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 181

α 10 β 1

Time step 2 [s]

Horizontal 15 [m] Vertical 10 [m]

KA\* with Wind 2.1827 [s] KA\* without Wind 9.9657 [s] A\* 2.3674 [s]

KA\* with Wind 772 [m] KA\* without Wind 769 [m] A\* 740 [m]

X 0 [m/s] Y 5 [m/s] Z 0 [m/s]

KA\* with Wind 34 KA\* without Wind 40 A\* 591

In **Figure 10** the paths on the Longitude-Latitude plane are presented. The path obtained with A\* pass over the bottom-left obstacle and very close to the top-right one. Planning sharp heading changes to reach the target. This is typical of classical graph search algorithms that do not take into account the vehicle kinematic constraints. The path obtained with KA\* on the other hand is smooth and obstacles separation constraints is evident. Comparing the path with wind with the one without wind between the obstacles on the left is evident the disturbance induced by the wind that pushes the path closer to the

In **Figure 11** on the *X-*axis is plotted the distance covered from start to target point and on the *Y-*axis the aircraft altitude. Again A\* plans sharp altitude changes and particularly sharp descends to reach the target. These changes are unfeasible with real aircrafts. As a matter of fact in general the A\* path requires deep post processing and waypoints reallocation to make the path flyable. Analyzing in detail though the algorithm plans a path passing over the bottom-left obstacle and then descending close to the top-right one. Being this descent unfeasible for the aircraft a complete re-planning is needed to reallocate the waypoints sequence. This is one of many cases evidencing that classical graph search algorithms used for tridimensional path planning can generate unfeasible paths because of the strong

**Optimization parameters** 

**Obstacles separation** 

**Wind Speed** 

**Path length** 

**WayPoints** 

top-left obstacle.

Table 2. Four-obstacles test parameters.

**Computation time** 


with respect to A\*. This permits to obtain more handy waypoints lists without need of post processing, ready to be loaded on the flight control system.

with respect to A\*. This permits to obtain more handy waypoints lists without need of post

ΔXY 1 [m] ΔXZ 1 [m]

X 250 [m] Y 125 [m] Z 50 [m]

X 125 [m] Y 125 [m]

X 125 [m] Y 375 [m]

X 375 [m] Y 375 [m]

X 375 [m] Y 125 [m]

X 20 [m] Y 20 [m] Z 60 [m]

X 480 [m] Y 480 [m] Z 30 [m]

Speed 10 [m/s] Min turning radius 25 [m] Max climbing angle 4 [deg]

 X 500 [m] Y 500 [m] Z 80 [m]

processing, ready to be loaded on the flight control system.

**Map dimension** 

**Obstacles dimension** 

**Obstacles-center position** 

**Obstacles-center position** 

**Obstacles-center position** 

**Obstacles-center position** 

**(1)** 

**(2)** 

**(3)** 

**(4)** 

**Starting point** 

**Target point** 

**Aircraft parameters** 


Table 2. Four-obstacles test parameters.

In **Figure 10** the paths on the Longitude-Latitude plane are presented. The path obtained with A\* pass over the bottom-left obstacle and very close to the top-right one. Planning sharp heading changes to reach the target. This is typical of classical graph search algorithms that do not take into account the vehicle kinematic constraints. The path obtained with KA\* on the other hand is smooth and obstacles separation constraints is evident. Comparing the path with wind with the one without wind between the obstacles on the left is evident the disturbance induced by the wind that pushes the path closer to the top-left obstacle.

In **Figure 11** on the *X-*axis is plotted the distance covered from start to target point and on the *Y-*axis the aircraft altitude. Again A\* plans sharp altitude changes and particularly sharp descends to reach the target. These changes are unfeasible with real aircrafts. As a matter of fact in general the A\* path requires deep post processing and waypoints reallocation to make the path flyable. Analyzing in detail though the algorithm plans a path passing over the bottom-left obstacle and then descending close to the top-right one. Being this descent unfeasible for the aircraft a complete re-planning is needed to reallocate the waypoints sequence. This is one of many cases evidencing that classical graph search algorithms used for tridimensional path planning can generate unfeasible paths because of the strong

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 183

Fig. 11. Distance-Altitude view (four obstacles test).

Fig. 12. Turning rate (four obstacles test).

Fig. 13. Climbing angle (four obstacles test).

longitudinal constraints of aircrafts and they need high intrusive post processing algorithms to modify the waypoint sequence.

In **Figure 12** the time history of the turning rate (connected with the *u* command) is plotted. The comparison between the command sequence with and without wind puts in evidence that the path needs more aggressive commands to compensate disturbances introduced from wind, but the average value remains limited thanks to the *G* value in the cost function that takes care of the amount of command needed to perform the path. On the other hand in **Figure 13** the climbing angle (due to the *w* command) is plotted. Also in this case the average amount of command is limited. Limiting turning rates and climbing angles required to follow the path is important. The main path-planning task is to generate a trajectory driving the aircraft from start to target in safe conditions. If tracking the path planned requires aggressive maneuvers, the aircraft performances will be completely absorbed by this task. However in many cases tracking the path is just a low-level task prerogative to accomplish with the high-level mission task (i.e in a save and rescue mission tracking the path could be one of the tasks together with many others. As an example it could be required also to avoid collision with dynamic obstacles along the flight, to deploy the payload and collect data, to interact with other aircrafts involved in the mission). If the aircraft must exploit its best performances to track the path it will not be able to accomplish also with the other mission tasks and this is not acceptable.

Fig. 10. Longitude-Latitude view (four obstacles test).

longitudinal constraints of aircrafts and they need high intrusive post processing algorithms

In **Figure 12** the time history of the turning rate (connected with the *u* command) is plotted. The comparison between the command sequence with and without wind puts in evidence that the path needs more aggressive commands to compensate disturbances introduced from wind, but the average value remains limited thanks to the *G* value in the cost function that takes care of the amount of command needed to perform the path. On the other hand in **Figure 13** the climbing angle (due to the *w* command) is plotted. Also in this case the average amount of command is limited. Limiting turning rates and climbing angles required to follow the path is important. The main path-planning task is to generate a trajectory driving the aircraft from start to target in safe conditions. If tracking the path planned requires aggressive maneuvers, the aircraft performances will be completely absorbed by this task. However in many cases tracking the path is just a low-level task prerogative to accomplish with the high-level mission task (i.e in a save and rescue mission tracking the path could be one of the tasks together with many others. As an example it could be required also to avoid collision with dynamic obstacles along the flight, to deploy the payload and collect data, to interact with other aircrafts involved in the mission). If the aircraft must exploit its best performances to track the path it will not be able to accomplish

to modify the waypoint sequence.

also with the other mission tasks and this is not acceptable.

Fig. 10. Longitude-Latitude view (four obstacles test).

Fig. 11. Distance-Altitude view (four obstacles test).

Fig. 12. Turning rate (four obstacles test).

Fig. 13. Climbing angle (four obstacles test).

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 185

ΔXY 1 [m] ΔXZ 1 [m]

X 300 [m] Y 125 [m] Z 50 [m]

X 300 [m] Y 250 [m]

X 20 [m] Y 20 [m] Z 40 [m]

X 450 [m] Y 450 [m] Z 50 [m]

Speed 10 [m/s]

Min turning radius 25 [m] Max climbing angle 4 [deg]

Time step 2 [s]

Horizontal 15 [m] Vertical 10 [m]

X -2 [m/s] Y 2 [m/s]

α 10 β 1

 X 500 [m] Y 500 [m] Z80 [m]

**Map dimension** 

**Obstacles dimension** 

**Obstacles-center** 

**Starting point** 

**Target point** 

**Aircraft parameters** 

**Obstacles separation** 

**Optimization parameters** 

**Wind** 

**position** 

#### **5.2 One obstacle**

**Figure 14** is the tridimensional view of the three paths (KA\* with wind, KA\* without wind, A\*) generated with this test. The picture shows that the obstacle obstructs almost completely the path to the aircraft on the right, leaving just a small aisle to reach the target.

Fig. 14. One obstacle test (3D view).

All the data collected in **Table 3** are almost the same of the previous test. The environment, the aircraft, starting and target point, obstacles separation and optimization parameters do not change. Just the number and distribution of the obstacles is changed, together with the wind speed. The last parameter is changed to investigate the effects of diagonal wind on the path. The wind intensity is reduced to avoid reaching conditions too harsh for the flight. Two [m/s] of wind along the *X* and *Y* ground axes are introduced and the effects on the path are evidenced by the search performances.

The computation time between the path with and without wind is strongly different. As in the previous case wind forces the aircraft to move faster with respect to the ground, but the big difference between the paths now is due to the different path followed to reach the target. As shown in **Figure 14** the path with wind goes to the left of the obstacle and reaches the target directly. This is due to the negative wind speed along the *X*-axis that opposes the tendency of the aircraft to go straight from start to target (as the first part of the path without wind shows). The aircraft is pushed to the left forcing it to find a different path to reach the goal. In this way the computation time is strongly reduced because crossed the obstacle the aircraft can go straight to the target.

On the other hand KA\* with wind and A\* look for a way to reach the goal crossing the obstacle to the right. This is due to the *H* component in the cost function that drives the

**Figure 14** is the tridimensional view of the three paths (KA\* with wind, KA\* without wind, A\*) generated with this test. The picture shows that the obstacle obstructs almost completely

All the data collected in **Table 3** are almost the same of the previous test. The environment, the aircraft, starting and target point, obstacles separation and optimization parameters do not change. Just the number and distribution of the obstacles is changed, together with the wind speed. The last parameter is changed to investigate the effects of diagonal wind on the path. The wind intensity is reduced to avoid reaching conditions too harsh for the flight. Two [m/s] of wind along the *X* and *Y* ground axes are introduced and the effects on the

The computation time between the path with and without wind is strongly different. As in the previous case wind forces the aircraft to move faster with respect to the ground, but the big difference between the paths now is due to the different path followed to reach the target. As shown in **Figure 14** the path with wind goes to the left of the obstacle and reaches the target directly. This is due to the negative wind speed along the *X*-axis that opposes the tendency of the aircraft to go straight from start to target (as the first part of the path without wind shows). The aircraft is pushed to the left forcing it to find a different path to reach the goal. In this way the computation time is strongly reduced because crossed the

On the other hand KA\* with wind and A\* look for a way to reach the goal crossing the obstacle to the right. This is due to the *H* component in the cost function that drives the

the path to the aircraft on the right, leaving just a small aisle to reach the target.

**5.2 One obstacle** 

Fig. 14. One obstacle test (3D view).

path are evidenced by the search performances.

obstacle the aircraft can go straight to the target.


Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 187

integration time must be increased. The integration time provides to the algorithm the capability to forecast the possible movements from the current state to the next. Then, if the aircraft has to climb to avoid an obstacle flying above it, a longer integration horizon is needed in order to plan in time the climbing maneuver reducing the computation time.

In **Figure 17** the turn rate (related to the *u* command) is plotted. In this case the two command sequences cannot be compared because of the different paths followed. Anyway it is possible to see the strong turning rate imposed to the aircraft reaching the bottom-right corner of the obstacle. Reaching that corner the aircraft has to turn in order to go toward the target respecting the obstacle separation constrains, than strong heading changes are needed. In order to limit turning radiuses and climbing angles along the path and generate smooth and flyable trajectories for the aircraft the *u* and *w* command vectors provided to the model to generate the possible movements are limited to half of the maximum turning rate and climbing angle. Finally **Figure 18** shows the climb angle (related to the *w* command) time history as for the previous test. Here the climb angle is always small for both the paths

and the aircraft climbs slowly to the target altitude.

Fig. 15. Longitude-Latitude view (one obstacle test).


Table 3. One-obstacle test parameters.

search along the diagonal between the start and the target point. In this way the algorithms look for the optimal path following the diagonal up to meeting the obstacle. Then the search continues choosing to turn on the right because in that direction the *F*-value is decreasing. Because of this process the computation time is higher and also the covered distance is more then the one with wind.

Analyzing this behavior an important limit of this optimization technique comes out: greedy algorithms become slow when the optimum search does not provide a continuing monotonic decrease of the cost function. Because of this tendency this first version of KA\* must be improved in order to accelerate the convergence to the optimal solution in cases where the continuing descent to the minim is not guaranteed.

**Figure 15** shows on the Longitude-Latitude plane the different paths planned in this test. In this case, the post processing phase for the A\* path would be less intrusive because of the slight altitude variation and of the few heading changes, but the 90 degrees heading change on the right of the obstacle is clearly unfeasible. This is evident comparing the turning radius planned by KA\* with the sharp angle planned with A\*. Some of these sharp heading changes can be easily corrected with a smoothing algorithm in post processing, but some of them can require a complete waypoints reallocation (as shown in the previous example). The important advantage of KA\* is to generate a feasible path respecting the basic aircraft kinematic constraints with low computation workload.

**Figure 16** again has on the *X-*axis the distance covered from start to target and on the *Y-*axis the aircraft altitude. For this test the altitude changes are smother then the one in the previous test, but here also it is possible to see the stepwise approach to climb of the A\* path compared with the smooth climbing maneuver planned with KA\*. About altitude variations the relation with the integration time step must be mentioned. Because of the slower behavior of an aircraft to altitude variations with respect to heading changes, when KA\* is used on environments requiring strong altitude variations to avoid the obstacles, the

KA\* with Wind 37 KA\* without Wind 41

search along the diagonal between the start and the target point. In this way the algorithms look for the optimal path following the diagonal up to meeting the obstacle. Then the search continues choosing to turn on the right because in that direction the *F*-value is decreasing. Because of this process the computation time is higher and also the covered distance is more

Analyzing this behavior an important limit of this optimization technique comes out: greedy algorithms become slow when the optimum search does not provide a continuing monotonic decrease of the cost function. Because of this tendency this first version of KA\* must be improved in order to accelerate the convergence to the optimal solution in cases

**Figure 15** shows on the Longitude-Latitude plane the different paths planned in this test. In this case, the post processing phase for the A\* path would be less intrusive because of the slight altitude variation and of the few heading changes, but the 90 degrees heading change on the right of the obstacle is clearly unfeasible. This is evident comparing the turning radius planned by KA\* with the sharp angle planned with A\*. Some of these sharp heading changes can be easily corrected with a smoothing algorithm in post processing, but some of them can require a complete waypoints reallocation (as shown in the previous example). The important advantage of KA\* is to generate a feasible path respecting the basic aircraft

**Figure 16** again has on the *X-*axis the distance covered from start to target and on the *Y-*axis the aircraft altitude. For this test the altitude changes are smother then the one in the previous test, but here also it is possible to see the stepwise approach to climb of the A\* path compared with the smooth climbing maneuver planned with KA\*. About altitude variations the relation with the integration time step must be mentioned. Because of the slower behavior of an aircraft to altitude variations with respect to heading changes, when KA\* is used on environments requiring strong altitude variations to avoid the obstacles, the

where the continuing descent to the minim is not guaranteed.

kinematic constraints with low computation workload.

A\* 698

**Computation time** 

**Path length** 

**WayPoints** 

Table 3. One-obstacle test parameters.

then the one with wind.

Z 0 [m/s]

A\* 6.5682 [s]

A\* 776 [m]

KA\* with Wind 0.4854 [s] KA\* without Wind 32.476 [s]

KA\* with Wind 689 [m] KA\* without Wind 796 [m] integration time must be increased. The integration time provides to the algorithm the capability to forecast the possible movements from the current state to the next. Then, if the aircraft has to climb to avoid an obstacle flying above it, a longer integration horizon is needed in order to plan in time the climbing maneuver reducing the computation time.

In **Figure 17** the turn rate (related to the *u* command) is plotted. In this case the two command sequences cannot be compared because of the different paths followed. Anyway it is possible to see the strong turning rate imposed to the aircraft reaching the bottom-right corner of the obstacle. Reaching that corner the aircraft has to turn in order to go toward the target respecting the obstacle separation constrains, than strong heading changes are needed. In order to limit turning radiuses and climbing angles along the path and generate smooth and flyable trajectories for the aircraft the *u* and *w* command vectors provided to the model to generate the possible movements are limited to half of the maximum turning rate and climbing angle. Finally **Figure 18** shows the climb angle (related to the *w* command) time history as for the previous test. Here the climb angle is always small for both the paths and the aircraft climbs slowly to the target altitude.

Fig. 15. Longitude-Latitude view (one obstacle test).

Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 189

Kinematic A\* has been developed to fill the relation gap between the aircraft kinematic constraints and the logics used in graph search approaches to find the optimal path. The simple aircraft model generates the state transitions in the state space and drives the search toward feasible directions. This approach has been tested on several cases. The algorithm generates feasible paths respecting limits on vehicle turning rates and climbing angles but the tests evidence also some issues that have to be investigated to improve the

 The goal to obtain paths respecting the basic aircraft kinematic constraints has been reached. KA\* generates smooth and safe paths respecting the imposed constraints. No sharp heading changes or strong altitude variations typical of classical graph search paths are shown. Tests put in evidence that some paths obtained with classical graph search algorithms cannot be adapted in post processing with waypoints reallocation, to reach the aircraft kinematic constraints. Some heading changes or altitude variations affect deeply the whole path and a complete re-planning is needed when these unfeasible trajectories are included in the full path. This point addresses the development of an algorithm that like KA\* may be able to match the graph search logics

 Obstacles separation represents an important improvement with respect to other graph search solutions. In classic graph search formulations obstacles separation was implemented just imposing to skip a given amount of cells around the obstacles whether the algorithm should try to expand them. In KA\* the obstacle separation is more elegant; the algorithm skips the states with positions too close to the obstacles

 Introducing the model to expand the states and perform the graph search is the fundamental novelty introduced with KA\*, but is paid with an algorithm increased complexity. A longer computation time was expected but several tests demonstrated just slight time increases to obtain the solution. This is important to save the merit of

 Tests show another important KA\* merit: the lower waypoints number on the path generated by KA\* with respect to A\*. KA\* algorithm naturally generates just the amount of waypoints needed to reach the goal and because of this the waypoints

 In spite of the simplified wind model, the effects of this disturbance are hardly relevant also for the preliminary tests reported in this paper. Wind modifies the state space and forces the algorithm to obtain solutions very different from the one without wind. Because of this further and deeper investigations are required to better understand this

 Analyzing heading changes and altitude variations needed to follow the KA\* paths it is evident the strong effect that wind has on the path following performances. Paths with wind require harder heading and altitude variations pushing the aircraft to reach its limits in order to follow the path. This aspect shall be taken into account in the following work in order to study deeper the problem and find modifications on the cost

low computation effort characteristic of graph search algorithms.

filtering and reallocation needed for A\* can be skipped.

problem and improve accordingly the implementation.

function that can improve the algorithm performances.

**6. Conclusions and future work** 

As a matter of fact the following conclusion can be taken:

with the aircraft kinematic constraints.

modifying accordingly the full planned path.

algorithm.

Fig. 16. Distance-Altitude view (one obstacle test).

Fig. 17. Turn rate (one obstacle test).

Fig. 18. Climb angle (one obstacle test).

## **6. Conclusions and future work**

188 Recent Advances in Aircraft Technology

Fig. 16. Distance-Altitude view (one obstacle test).

Fig. 17. Turn rate (one obstacle test).

Fig. 18. Climb angle (one obstacle test).

Kinematic A\* has been developed to fill the relation gap between the aircraft kinematic constraints and the logics used in graph search approaches to find the optimal path. The simple aircraft model generates the state transitions in the state space and drives the search toward feasible directions. This approach has been tested on several cases. The algorithm generates feasible paths respecting limits on vehicle turning rates and climbing angles but the tests evidence also some issues that have to be investigated to improve the algorithm.

As a matter of fact the following conclusion can be taken:


Advanced Graph Search Algorithms for Path Planning of Flight Vehicles 191

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#### **7. References**


 The second test puts in evidence an issue still present in KA\*: the optimality of the paths planned. The wind effect on the model drives the states expansion along directions that are not taken into account otherwise. This way a shorter and computationally lighter solution is found. This means that the algorithm search must be improved modifying the cost function in order to investigate possible optimality proofs for the generated path. However this is a hard task because the graph search logics in it self makes optimality proof possible just for limited and simple problem formulations. Another issue outlined by the tests that must be analyzed in the following work is the exponential increase of computational time when the algorithm cannot follow a monotonic cost function decrease along the states expansion. Part of the problem is due to the graph structure and needs a deep state space analysis to be improved. On the other side, the cost function then needs to be modified, investigating solutions able to

 Finally tests show that the different aircraft behavior on longitudinal with respect to lateral-directional plane affects largely the model time step selection. Particularly, longer time steps are needed when strong altitude variations are required to cross the obstacles. As a consequence, the time step must be tuned in order to improve the

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algorithm performances according with the test cases.


**0**

**9**

<sup>1</sup>*Germany* <sup>2</sup>*Australia* <sup>3</sup>*The Netherlands*

**Determination**

<sup>1</sup> *Technische Universität München* <sup>2</sup> *Curtin University of Technology* <sup>3</sup> *Delft University of Technology*

**GNSS Carrier Phase-Based Attitude**

The GNSS (Global Navigation Satellite Systems) are a valid aid in support of the aeronautic science. GNSS technology has been successfully implemented in aircraft design, in order to provide accurate position, velocity and heading estimations. Although it does not yet comply with aviation integrity requirements, GNSS-based aircraft navigation is one of the alternative means to traditional dead-reckoning systems. It can provide fast, accurate, and driftless positioning solutions. Additionally, ground-based GNSS receivers may be employed

One of the main issues in airborne navigation is the determination of the aircraft attitude, i.e., the orientation of the aircraft with respect to a defined reference system. Many sensors and technologies are available to estimate the attitude of a aircraft, but there is a growing interest in GNSS-based attitude determination (AD), often integrated at various levels of tightness to other types of sensors, typically Inertial Measurements Units (IMU). Although the accuracy of a stand-alone GNSS attitude system might not be comparable with the one obtainable with other modern attitude sensors, a GNSS-based system presents several advantages. It is inherently driftless, a GNSS receiver has low power consumption, it requires minor maintenance, and it is not as expensive as other high-precision systems, such as laser

GNSS-based AD employs a number of antennas rigidly mounted on the aircraft's structure, as depicted in Figure 1. The orientation of each of the baselines formed between the antennas is determined by computing their relative positions. The use of GNSS carrier phase signals enables very precise range measurements, which can then be related to angular estimations. However, carrier phase measurements are affected by unknown integer ambiguities, since only their fractional part is measured by the receiver. The process of reconstructing the number of whole cycles from a set of measurements affected by errors goes under the name of ambiguity resolution (AR). Only after these ambiguities are correctly resolved to their correct integer values, will reliable baseline measurements and attitude estimations become available. This chapter focuses on novel AR and AD methods. Recent advances in GNSS-based attitude

to aid navigation in critical applications, such as precision approaches and landings.

**1. Introduction**

gyroscopes.

Gabriele Giorgi1 and Peter J. G. Teunissen2,3


## **GNSS Carrier Phase-Based Attitude Determination**

Gabriele Giorgi1 and Peter J. G. Teunissen2,3

 *Technische Universität München Curtin University of Technology Delft University of Technology* <sup>1</sup>*Germany* <sup>2</sup>*Australia The Netherlands*

#### **1. Introduction**

192 Recent Advances in Aircraft Technology

Sussmann, H.J. & Tang, W. (1991). Shortest Paths for the Reeds-Shepp Car: a Worked Out

Warshall, S. (1962). A theorem on Boolean matrices. *Journal of the ACM*, 9(1), pp. 11–12. Waydo, S. & Murray, R.M. (2003). Vehicle Motion Planning Using Stream Functions.

*SYCON-91-10*, Rutgers University.

September.

Example of the Use of Geometric Technique in Nonlinear Optimal Control. *Report* 

P*roceedings of 2003 IEEE International Conference on Robotics and Automation*,

The GNSS (Global Navigation Satellite Systems) are a valid aid in support of the aeronautic science. GNSS technology has been successfully implemented in aircraft design, in order to provide accurate position, velocity and heading estimations. Although it does not yet comply with aviation integrity requirements, GNSS-based aircraft navigation is one of the alternative means to traditional dead-reckoning systems. It can provide fast, accurate, and driftless positioning solutions. Additionally, ground-based GNSS receivers may be employed to aid navigation in critical applications, such as precision approaches and landings.

One of the main issues in airborne navigation is the determination of the aircraft attitude, i.e., the orientation of the aircraft with respect to a defined reference system. Many sensors and technologies are available to estimate the attitude of a aircraft, but there is a growing interest in GNSS-based attitude determination (AD), often integrated at various levels of tightness to other types of sensors, typically Inertial Measurements Units (IMU). Although the accuracy of a stand-alone GNSS attitude system might not be comparable with the one obtainable with other modern attitude sensors, a GNSS-based system presents several advantages. It is inherently driftless, a GNSS receiver has low power consumption, it requires minor maintenance, and it is not as expensive as other high-precision systems, such as laser gyroscopes.

GNSS-based AD employs a number of antennas rigidly mounted on the aircraft's structure, as depicted in Figure 1. The orientation of each of the baselines formed between the antennas is determined by computing their relative positions. The use of GNSS carrier phase signals enables very precise range measurements, which can then be related to angular estimations. However, carrier phase measurements are affected by unknown integer ambiguities, since only their fractional part is measured by the receiver. The process of reconstructing the number of whole cycles from a set of measurements affected by errors goes under the name of ambiguity resolution (AR). Only after these ambiguities are correctly resolved to their correct integer values, will reliable baseline measurements and attitude estimations become available. This chapter focuses on novel AR and AD methods. Recent advances in GNSS-based attitude

**2. The GNSS-based attitude model**

observable is therefore modeled as *Ps r*, *f*

*P* code observation [m] *τ* signal travel time [s]

*dm* multipath error [m]

*d* instrumental delays [s]

the tracked signal, modeled as

*<sup>r</sup>*, *<sup>f</sup>*(*t*) = *<sup>ρ</sup><sup>s</sup>*

+ *c* 

Φ*s*

*dt* clock errors [s]

*c* speed of light : 299 792 458 [ <sup>m</sup>

(*t*) = *ρ<sup>s</sup>*

+ *c* 

*<sup>r</sup>*(*t*, *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup>*

and the frequency, respectively. The different terms are:

*I* , *T* ionospheric and tropospheric delays [m]

*ε<sup>P</sup>* remaining unmodeled code errors [m]

*<sup>r</sup>*(*t*, *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup>*

*<sup>r</sup>* ) <sup>−</sup> *<sup>I</sup><sup>s</sup>*

*δr*, *<sup>f</sup>*(*t*) + *δ<sup>s</sup>*

*<sup>r</sup>*, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup><sup>s</sup>*

*<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup> r* ) 

*<sup>r</sup>* ) + *I<sup>s</sup>*

*dr*, *<sup>f</sup>*(*t*) + *d<sup>s</sup>*

*ρ* geometrical distance between receiver and satellite [m]

*<sup>r</sup>*, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup><sup>s</sup>*

*<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup> r* ) + *ε<sup>s</sup> P*,*r*,*f*

s ]

*<sup>r</sup>* + *dm<sup>s</sup>*

where the superscript *s* indicates the satellite and the subscripts *r* and *f* indicate the receiver

The magnitude of errors involved in these observations - decimeter or meter level - would not allow high-precision applications, such AD, which require cm- or mm-level accuracy in the final positioning product. Therefore, another set of observations is considered: the phase of

*<sup>r</sup>* + *δm<sup>s</sup>*

with *ϕ* the phase of the generated carrier signal (original or replica) in cycles, *t*<sup>0</sup> the time of reference for phase synchronization, and *λ<sup>f</sup>* the wavelength of frequency *f* . The phase reading is characterized by different atmospheric delays (the ionosphere causes an anticipation of phase instead of a delay), different instrumental biases (indicated with *δ*), different multipath and an additional bias which is represented by the unknown number of whole cycles that cannot be detected by the tracking loop, since only the fractional part is measured. These are the integer ambiguities *z*. In case of GNSS, the precision of the phase measurements

+ *λ<sup>f</sup>* [*ϕ<sup>s</sup> r*, *f*

*<sup>r</sup>*, *<sup>f</sup>* <sup>+</sup> *<sup>c</sup>* [*dtr*(*t*) <sup>−</sup> *dts*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup>*

(*t*0) <sup>−</sup> *<sup>ϕ</sup><sup>s</sup>*

*<sup>r</sup>* )]

*<sup>r</sup>*, *<sup>f</sup>* <sup>+</sup> *<sup>ε</sup><sup>s</sup>* Φ,*r*,*f*

*<sup>f</sup>*(*t*0)] + *<sup>λ</sup><sup>f</sup> <sup>z</sup><sup>s</sup>*

*<sup>r</sup>*, *<sup>f</sup>* <sup>+</sup> *<sup>c</sup>* [*dtr*(*t*) <sup>−</sup> *dts*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>s</sup>*

*<sup>r</sup>* )]

(1)

(2)

A GNSS receiver works by tracking satellites in view and storing the data received. Each GNSS satellite broadcasts a coded message with information about its orbit, the time of transmission, and few other parameters necessary for the correct processing at receiver side (Misra & Enge, 2001). By collecting signals from three or more satellites a GNSS receiver determines its own position with a triangulation procedure, exploiting the knowledge about both the satellites positions and the slant distance (range) by each satellite in view. The range measurements are obtained by detecting the time of arrival of the signal, from which the range can be inferred. This measurement is affected by several error sources: the satellite and receiver clocks are not perfectly synchronized; the signal travels through the atmosphere, which causes delays; the direct signal may be affected by unwanted reflections (multipath) that cannot be perfectly eliminated by careful antenna design. If not properly modeled, each of these effects will limit the achievable GNSS accuracy. The observed pseudorange or code

GNSS Carrier Phase-Based Attitude Determination 195

determination have demonstrated that the two problems can be formulated in an integrated manner, i.e., aircraft attitude and the phase ambiguities can be considered as the unknown parameters of a common ambiguity-attitude estimation method. In this integrated approach, the AR and AD problems are solved together by means of the theory of Constrained Integer Least-Squares (C-ILS). This theory extends the well-known least-squares theory (LS), by having geometrical constraints as well as integer constraints imposed on parameter subsets. The novel AR-AD estimation problem is discussed and its various properties are analyzed. The method's complexity is addressed by presenting new numerical algorithms that largely reduce the required processing load. The main objective of this chapter is to provide evidence that:


The structure of this contribution is as follows. Section 2 gives the observation and stochastic model which cast the set of GNSS observations, with special focus on the derivation of the GNSS-based attitude model. Section 3 reviews the most common attitude parameterization and estimation methods, mainly focusing on those widely used in aviation applications. Section 4 introduces a new ambiguity-attitude estimation method, which enhances the existing approach for attitude determination using GNSS signals. Section 5 presents flight-test results, which provide practical evidence of the novel method's performance. Finally, section 6 draws several conclusions.

Fig. 1. GNSS data collected on multiple antennas installed on the fuselage and wings allow the estimation of an aircraft's orientation (attitude).

#### **2. The GNSS-based attitude model**

2 Will-be-set-by-IN-TECH

determination have demonstrated that the two problems can be formulated in an integrated manner, i.e., aircraft attitude and the phase ambiguities can be considered as the unknown parameters of a common ambiguity-attitude estimation method. In this integrated approach, the AR and AD problems are solved together by means of the theory of Constrained Integer Least-Squares (C-ILS). This theory extends the well-known least-squares theory (LS), by having geometrical constraints as well as integer constraints imposed on parameter subsets. The novel AR-AD estimation problem is discussed and its various properties are analyzed. The method's complexity is addressed by presenting new numerical algorithms that largely reduce the required processing load. The main objective of this chapter is to provide evidence

• GNSS carrier-phase based attitude determination is a viable alternative to existing attitude

• Employing the new ambiguity-attitude estimation method enhances ambiguity resolution

The structure of this contribution is as follows. Section 2 gives the observation and stochastic model which cast the set of GNSS observations, with special focus on the derivation of the GNSS-based attitude model. Section 3 reviews the most common attitude parameterization and estimation methods, mainly focusing on those widely used in aviation applications. Section 4 introduces a new ambiguity-attitude estimation method, which enhances the existing approach for attitude determination using GNSS signals. Section 5 presents flight-test results, which provide practical evidence of the novel method's performance. Finally, section

Fig. 1. GNSS data collected on multiple antennas installed on the fuselage and wings allow

the estimation of an aircraft's orientation (attitude).

• The new method can be implemented such that it is suitable for real-time applications

that:

sensors

performance

6 draws several conclusions.

A GNSS receiver works by tracking satellites in view and storing the data received. Each GNSS satellite broadcasts a coded message with information about its orbit, the time of transmission, and few other parameters necessary for the correct processing at receiver side (Misra & Enge, 2001). By collecting signals from three or more satellites a GNSS receiver determines its own position with a triangulation procedure, exploiting the knowledge about both the satellites positions and the slant distance (range) by each satellite in view. The range measurements are obtained by detecting the time of arrival of the signal, from which the range can be inferred. This measurement is affected by several error sources: the satellite and receiver clocks are not perfectly synchronized; the signal travels through the atmosphere, which causes delays; the direct signal may be affected by unwanted reflections (multipath) that cannot be perfectly eliminated by careful antenna design. If not properly modeled, each of these effects will limit the achievable GNSS accuracy. The observed pseudorange or code observable is therefore modeled as

$$\begin{aligned} P\_{r,f}^{s}(t) &= \rho\_r^s(t, t - \tau\_r^s) + I\_{r,f}^s + T\_r^s + dm\_{r,f}^s + c \left[ dt\_{r}(t) - dt^s(t - \tau\_r^s) \right] \\ &+ c \left[ d\_{r,f}(t) + d\_f^s(t - \tau\_r^s) \right] + \varepsilon\_{r,f}^s \end{aligned} \tag{1}$$

where the superscript *s* indicates the satellite and the subscripts *r* and *f* indicate the receiver and the frequency, respectively. The different terms are:


The magnitude of errors involved in these observations - decimeter or meter level - would not allow high-precision applications, such AD, which require cm- or mm-level accuracy in the final positioning product. Therefore, another set of observations is considered: the phase of the tracked signal, modeled as

$$\begin{aligned} \Phi\_{r,f}^{s}(t) &= \rho\_r^s(t, t - \tau\_r^s) - I\_{r,f}^s + T\_r^s + \delta m\_{r,f}^s + c \left[ dt\_I(t) - dt^s(t - \tau\_r^s) \right] \\ &+ c \left[ \delta\_{r,f}(t) + \delta\_f^s(t - \tau\_r^s) \right] + \lambda\_f \left[ \boldsymbol{\varrho}\_{r,f}^s(t\_0) - \boldsymbol{\varrho}\_f^s(t\_0) \right] + \lambda\_f \boldsymbol{\varepsilon}\_{r,f}^s + \boldsymbol{\varepsilon}\_{\boldsymbol{\Phi},t}^s \end{aligned} \tag{2}$$

with *ϕ* the phase of the generated carrier signal (original or replica) in cycles, *t*<sup>0</sup> the time of reference for phase synchronization, and *λ<sup>f</sup>* the wavelength of frequency *f* . The phase reading is characterized by different atmospheric delays (the ionosphere causes an anticipation of phase instead of a delay), different instrumental biases (indicated with *δ*), different multipath and an additional bias which is represented by the unknown number of whole cycles that cannot be detected by the tracking loop, since only the fractional part is measured. These are the integer ambiguities *z*. In case of GNSS, the precision of the phase measurements

with *r<sup>s</sup>* and *rr* the satellite and receiver antenna position vectors, respectively. By assuming the atmospheric delays negligible and applying the Taylor expansion to expression (4) one

GNSS Carrier Phase-Based Attitude Determination 197

*<sup>r</sup>* )*T*�*rr*<sup>12</sup> <sup>+</sup> *<sup>ε</sup>*

where the observables are now 'observed minus computed' terms, and the unknowns are expressed as increments with respect to a computed approximate value. �*rr*<sup>12</sup> is the baseline

is the difference between unit line-of-sight vectors of different satellites. Also note that the

Consider now two antennas simultaneously tracking the same *m* + 1 satellites at *N* frequencies. The vector of DD observations of type (6) are cast in the linear(ized) functional

with *y* the 2*mN*-vector of code and carrier phase observations, *z* the unknown integer-valued ambiguities and *b* the vector of real-valued baseline coordinates. *A* and *G* are the design

with Λ the diagonal matrix of *N* carrier wavelengths and *U* the *m* × 3 matrix of DD unit

Model (7) describes the linear relationship between GNSS observables and the parameters of the two antennas. However, a single baseline is generally not sufficient to estimate the full orientation of an aircraft with respect to a given reference frame. At least three non-aligned antennas are necessary to guarantee that each rotation of the aircraft can be tracked unambiguously. It is straightforward to generalize the model formulation (7) to cast

This formulation is obtained by casting the observations at each baseline in the columns of the 2*mN* × *n* matrix *Y*. Consequently, *Z* = [*z*1,..., *zn*] is the matrix whose *n* columns are the integer ambiguity *mN*-vectors, and *B* = [*b*1,..., *bn*] is the 3 × *n* matrix that contains the *n* real-valued baseline vectors. We exploited here once again the short baseline hypothesis: the

Besides describing the functional relationship between observables and unknowns, a proper modeling should also capture the observation noise, i.e., the measurement error. The error is relative to the receiver, to the satellite, to the frequency and to the type of observations (code or phase). The variance-covariance (v-c) matrix of a vector of DD observations *y* collected at baseline *b* will be denoted as *D*(*y*) = *Qyy*, with *D*(·) the dispersion operator. For the multibaseline model (9), the description of measurement errors requires a further step: the

line-of-sight vectors. Symbol ⊗ denotes the Kronecker product (Van Loan, 2000).

*n* DD baseline observations, obtained with *n* + 1 GNSS antennas (Teunissen, 2007a):

⊗ *Im G* = *eN* ⊗

*<sup>r</sup>* )*T*�*rr*<sup>12</sup> <sup>+</sup> *<sup>λ</sup><sup>f</sup> <sup>z</sup>s*<sup>12</sup>

*s*<sup>12</sup> *P*,*r*12,*f*

*<sup>r</sup>*12, *<sup>f</sup>* + *<sup>ε</sup>*

*<sup>y</sup>* <sup>=</sup> *Az* <sup>+</sup> *Gb* <sup>+</sup> *<sup>ε</sup>* ; *<sup>z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN* , *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> (7)

 *U U* 

*<sup>Y</sup>* <sup>=</sup> *AZ* <sup>+</sup> *GB* <sup>+</sup> <sup>Ξ</sup> ; *<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* , *<sup>B</sup>* <sup>∈</sup> **<sup>R</sup>**3×*<sup>n</sup>* (9)

*s*<sup>12</sup> Φ,*r*12,*f* (6)

*<sup>r</sup>* <sup>−</sup> *<sup>u</sup>s*<sup>1</sup> *r*

(8)

*<sup>r</sup>* = *<sup>u</sup>s*<sup>2</sup>

*s*<sup>12</sup> <sup>Φ</sup>,*r*12,*<sup>f</sup>* .

*s*<sup>12</sup> *<sup>P</sup>*,*r*12,*<sup>f</sup>* and *ε*

obtains the linearized relations

model (Teunissen & Kleusberg, 1998)

matrices

�*Ps*<sup>12</sup>

�Φ*s*<sup>12</sup>

*<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> <sup>−</sup>(*us*<sup>12</sup>

*<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> <sup>−</sup>(*us*<sup>12</sup>

vector - the difference between the absolute antennas positions - whereas *us*<sup>12</sup>

multipath terms have been lumped into the remaining unmodeled errors *ε*

*A* = 0 Λ 

same matrix of line-of-sight vectors *U* is used for all baselines.

far exceeds the one of code observations: typically the phase observable is two orders of magnitude more accurate than the code measurement.

The many sources of error in (1) and (2) can be mitigated in relative positioning models. First, we form the so-called single difference (SD) code and carrier phase observations by taking the differences between observations simultaneously collected at two antennas tracking the same satellite:

$$\begin{aligned} P\_{r\_{2f}}^{s}(t) - P\_{r\_{1f}}^{s}(t) &= P\_{r\_{12f}}^{s} = \quad \rho\_{r\_{12f}}^{s} + I\_{r\_{12f}}^{s} + T\_{r\_{12}}^{s} + dm\_{r\_{12f}}^{s} + cdt\_{r\_{12}} + cd\_{r\_{12}f} + \varepsilon\_{r\_{12f}}^{s} \\\\ \Phi\_{r\_{2f}f}^{s}(t) - \Phi\_{r\_{1f}f}^{s}(t) &= \Phi\_{r\_{12f}}^{s} - I\_{r\_{12f}}^{s} + T\_{r\_{12}f}^{s} + \delta m\_{r\_{12f}}^{s} + cdt\_{r\_{12}} + c\delta\_{r\_{12}f} + \lambda\_{f}\rho\_{r\_{12f}}^{s}(t) \end{aligned}$$

$$\begin{aligned} \Phi\_{r\_{2f}f}^{s}(t) - \Phi\_{r\_{1f}f}^{s}(t) = \Phi\_{r\_{12}f}^{s} = \rho\_{r\_{12}f}^{s} - I\_{r\_{12}f}^{s} + T\_{r\_{12}}^{s} + \delta n\_{r\_{12}f}^{s} + cdt\_{r\_{12}} + c\delta\_{r\_{12}f} + \lambda\_{f}q\_{r\_{12}f}^{s}(t\_{0}) \\ + \lambda\_{f}z\_{r\_{12}f}^{s} + \epsilon\_{\Phi\_{r\_{12}f}}^{s} \end{aligned} \tag{3}$$

where subscript *r*<sup>12</sup> indicates the difference between two antennas: *<sup>r</sup>*<sup>12</sup> = *<sup>r</sup>*<sup>2</sup> − *<sup>r</sup>*<sup>1</sup> . The phase value *ϕ<sup>s</sup> <sup>f</sup>*(*t*0), relative to the common satellite, is eliminated. The instrumental delays and clock errors of the satellite are usually considered constant over short time spans, since the travel time difference with respect to any two points on the Earth surface is small (Teunissen & Kleusberg, 1998).

The terms *cdtr*<sup>12</sup> , *cdr*12, *<sup>f</sup>* and *δr*12, *<sup>f</sup>* refer to the relative clock errors and relative instrumental delays between the two receivers. A perfect synchronization between receivers implies the cancellation of the clock biases, and a correct calibration would reduce the impact of instrumental delays. In the case of a single receiver connected to two antennas, these two sources of relative error could cancel out with a proper calibration.

The receiver clock errors and hardware delays in the single difference equations (3) are common for all the satellites tracked at the same frequency. Therefore these terms can be eliminated by forming a double difference (DD) combination, obtained by subtracting two SD measurements from two different satellites:

$$\begin{aligned} P\_{r\_{12}f}^{\\$s\_{12}} &= \rho\_{r\_{12}f}^{\\$s\_{12}} + I\_{r\_{12}f}^{\\$s\_{12}} + T\_{r\_{12}}^{\\$s\_{12}} + dm\_{r\_{12}f}^{\\$s\_{12}} + \varepsilon\_{r\_{12}f}^{\\$s\_{12}} \\\\ \Phi\_{r\_{12}f}^{\\$s\_{12}} &= \rho\_{r\_{12}f}^{\\$s\_{12}} - I\_{r\_{12}f}^{\\$s\_{12}} + T\_{r\_{12}}^{\\$s\_{12}} + \delta m\_{r\_{12}f}^{\\$s\_{12}} + \lambda\_f \varepsilon\_{r\_{12}f}^{\\$s\_{12}} + \varepsilon\_{\Phi\_{\mathcal{I}r\_{12}f}}^{\\$s\_{12}} \end{aligned} \tag{4}$$

It has been assumed that the real-valued initial phase of the receiver replica does not vary for different tracked GNSS satellites.

The differential atmospheric delays depend on the distance between antennas. For sufficiently short baselines - typically shorter than a kilometer - the signals received by the antennas have traveled approximately the same path, thus the atmospheric delays becomes highly correlated. The differencing operation makes these errors negligible with respect to the measurement white noise for the baselines typically employed in AD applications, which rarely exceeds a few hundred meters.

Note that the relation between observations and baseline coordinates is nonlinear, since these are contained in the range term

$$\rho\_r^s = \left\| r^s(t\_r - \tau\_r^s) - r\_r(t\_r) \right\| \tag{5}$$

4 Will-be-set-by-IN-TECH

far exceeds the one of code observations: typically the phase observable is two orders of

The many sources of error in (1) and (2) can be mitigated in relative positioning models. First, we form the so-called single difference (SD) code and carrier phase observations by taking the differences between observations simultaneously collected at two antennas tracking the same

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup><sup>s</sup>*

Φ,*r*12,*f*

where subscript *r*<sup>12</sup> indicates the difference between two antennas: *<sup>r</sup>*<sup>12</sup> = *<sup>r</sup>*<sup>2</sup> − *<sup>r</sup>*<sup>1</sup> . The

and clock errors of the satellite are usually considered constant over short time spans, since the travel time difference with respect to any two points on the Earth surface is small (Teunissen

The terms *cdtr*<sup>12</sup> , *cdr*12, *<sup>f</sup>* and *δr*12, *<sup>f</sup>* refer to the relative clock errors and relative instrumental delays between the two receivers. A perfect synchronization between receivers implies the cancellation of the clock biases, and a correct calibration would reduce the impact of instrumental delays. In the case of a single receiver connected to two antennas, these two

The receiver clock errors and hardware delays in the single difference equations (3) are common for all the satellites tracked at the same frequency. Therefore these terms can be eliminated by forming a double difference (DD) combination, obtained by subtracting two SD

*<sup>r</sup>*<sup>12</sup> <sup>+</sup> *dms*<sup>12</sup>

*<sup>r</sup>*<sup>12</sup> <sup>+</sup> *<sup>δ</sup>ms*<sup>12</sup>

It has been assumed that the real-valued initial phase of the receiver replica does not vary for

The differential atmospheric delays depend on the distance between antennas. For sufficiently short baselines - typically shorter than a kilometer - the signals received by the antennas have traveled approximately the same path, thus the atmospheric delays becomes highly correlated. The differencing operation makes these errors negligible with respect to the measurement white noise for the baselines typically employed in AD applications, which

Note that the relation between observations and baseline coordinates is nonlinear, since these

(*tr* <sup>−</sup> *<sup>τ</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* + *<sup>ε</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* + *<sup>λ</sup><sup>f</sup> <sup>z</sup>*

*s*<sup>12</sup> *P*,*r*12,*f*

> *s*<sup>12</sup> *<sup>r</sup>*12, *<sup>f</sup>* + *<sup>ε</sup>*

*s*<sup>12</sup> Φ,*r*12,*f*

*<sup>r</sup>* ) − *rr*(*tr*)� (5)

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>ε</sup><sup>s</sup>*

*<sup>r</sup>*<sup>12</sup> + *dm<sup>s</sup>*

*<sup>r</sup>*<sup>12</sup> + *<sup>δ</sup>m<sup>s</sup>*

*<sup>f</sup>*(*t*0), relative to the common satellite, is eliminated. The instrumental delays

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *cdtr*<sup>12</sup> <sup>+</sup> *cdr*12, *<sup>f</sup>* <sup>+</sup> *<sup>ε</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *cdtr*<sup>12</sup> <sup>+</sup> *<sup>c</sup>δr*12, *<sup>f</sup>* <sup>+</sup> *<sup>λ</sup><sup>f</sup> <sup>ϕ</sup><sup>s</sup>*

*P*,*r*12,*f*

*<sup>r</sup>*12, *<sup>f</sup>*(*t*0)

(3)

(4)

magnitude more accurate than the code measurement.

*<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> *<sup>ρ</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> *<sup>ρ</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>I</sup><sup>s</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* <sup>−</sup> *<sup>I</sup><sup>s</sup>*

+*λ<sup>f</sup> z<sup>s</sup>*

sources of relative error could cancel out with a proper calibration.

*<sup>r</sup>*12, *<sup>f</sup>* + *<sup>I</sup>*

*<sup>r</sup>*12, *<sup>f</sup>* − *I*

*ρs <sup>r</sup>* <sup>=</sup> �*r<sup>s</sup>*

*s*<sup>12</sup> *<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup>s*<sup>12</sup>

*s*<sup>12</sup> *<sup>r</sup>*12, *<sup>f</sup>* <sup>+</sup> *<sup>T</sup>s*<sup>12</sup>

satellite:

*Ps r*2, *f*

Φ*s*

(*t*) <sup>−</sup> *<sup>P</sup><sup>s</sup> r*1, *f*

*<sup>r</sup>*2, *<sup>f</sup>*(*t*) <sup>−</sup> <sup>Φ</sup>*<sup>s</sup>*

phase value *ϕ<sup>s</sup>*

& Kleusberg, 1998).

*r*1, *f*

(*t*) = *P<sup>s</sup>*

(*t*) = Φ*<sup>s</sup>*

measurements from two different satellites: *Ps*<sup>12</sup> *<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> *<sup>ρ</sup>s*<sup>12</sup>

Φ*s*<sup>12</sup>

different tracked GNSS satellites.

rarely exceeds a few hundred meters.

are contained in the range term

*<sup>r</sup>*12, *<sup>f</sup>* <sup>=</sup> *<sup>ρ</sup>s*<sup>12</sup>

with *r<sup>s</sup>* and *rr* the satellite and receiver antenna position vectors, respectively. By assuming the atmospheric delays negligible and applying the Taylor expansion to expression (4) one obtains the linearized relations

$$\begin{aligned} \triangle P\_{r\_{12}f}^{s\_{12}} &= - (u\_r^{s\_{12}})^T \triangle r\_{r\_{12}} + \varepsilon\_{r\_{12}f}^{s\_{12}} \\ \triangle \Phi\_{r\_{12}f}^{s\_{12}} &= - (u\_r^{s\_{12}})^T \triangle r\_{r\_{12}} + \lambda\_f \varepsilon\_{r\_{12}f}^{s\_{12}} + \varepsilon\_{\Phi\_{r\_{12}f}}^{s\_{12}} \end{aligned} \tag{6}$$

where the observables are now 'observed minus computed' terms, and the unknowns are expressed as increments with respect to a computed approximate value. �*rr*<sup>12</sup> is the baseline vector - the difference between the absolute antennas positions - whereas *us*<sup>12</sup> *<sup>r</sup>* = *<sup>u</sup>s*<sup>2</sup> *<sup>r</sup>* <sup>−</sup> *<sup>u</sup>s*<sup>1</sup> *r* is the difference between unit line-of-sight vectors of different satellites. Also note that the multipath terms have been lumped into the remaining unmodeled errors *ε s*<sup>12</sup> *<sup>P</sup>*,*r*12,*<sup>f</sup>* and *ε s*<sup>12</sup> <sup>Φ</sup>,*r*12,*<sup>f</sup>* .

Consider now two antennas simultaneously tracking the same *m* + 1 satellites at *N* frequencies. The vector of DD observations of type (6) are cast in the linear(ized) functional model (Teunissen & Kleusberg, 1998)

$$y = Az + Gb + \varepsilon \quad ; \quad z \in \mathbb{Z}^{mN}, \quad b \in \mathbb{R}^3 \tag{7}$$

with *y* the 2*mN*-vector of code and carrier phase observations, *z* the unknown integer-valued ambiguities and *b* the vector of real-valued baseline coordinates. *A* and *G* are the design matrices

$$A = \begin{bmatrix} 0 \\ \Lambda \end{bmatrix} \otimes I\_{\mathfrak{m}} \qquad G = \mathfrak{e}\_{N} \otimes \begin{bmatrix} \mathcal{U} \\ \mathcal{U} \end{bmatrix} \tag{8}$$

with Λ the diagonal matrix of *N* carrier wavelengths and *U* the *m* × 3 matrix of DD unit line-of-sight vectors. Symbol ⊗ denotes the Kronecker product (Van Loan, 2000).

Model (7) describes the linear relationship between GNSS observables and the parameters of the two antennas. However, a single baseline is generally not sufficient to estimate the full orientation of an aircraft with respect to a given reference frame. At least three non-aligned antennas are necessary to guarantee that each rotation of the aircraft can be tracked unambiguously. It is straightforward to generalize the model formulation (7) to cast *n* DD baseline observations, obtained with *n* + 1 GNSS antennas (Teunissen, 2007a):

$$\mathbf{Y} = A\mathbf{Z} + \mathbf{G}\mathbf{B} + \boldsymbol{\Xi} \quad ; \quad \mathbf{Z} \in \mathbb{Z}^{mN \times n} , \quad \mathbf{B} \in \mathbb{R}^{3 \times n} \tag{9}$$

This formulation is obtained by casting the observations at each baseline in the columns of the 2*mN* × *n* matrix *Y*. Consequently, *Z* = [*z*1,..., *zn*] is the matrix whose *n* columns are the integer ambiguity *mN*-vectors, and *B* = [*b*1,..., *bn*] is the 3 × *n* matrix that contains the *n* real-valued baseline vectors. We exploited here once again the short baseline hypothesis: the same matrix of line-of-sight vectors *U* is used for all baselines.

Besides describing the functional relationship between observables and unknowns, a proper modeling should also capture the observation noise, i.e., the measurement error. The error is relative to the receiver, to the satellite, to the frequency and to the type of observations (code or phase). The variance-covariance (v-c) matrix of a vector of DD observations *y* collected at baseline *b* will be denoted as *D*(*y*) = *Qyy*, with *D*(·) the dispersion operator. For the multibaseline model (9), the description of measurement errors requires a further step: the

These conditions are fulfilled for orthonormal rotation matrices with determinant equal to

GNSS Carrier Phase-Based Attitude Determination 199

Model (9) can then be reformulated by means of the linear transformation *B* = *RF*, where *F* is used to cast the set of known local baseline coordinates and *R* is the orthonormal (*RTR* = *Iq*) matrix that rotates *B* into *F*. The complete GNSS attitude model reads then (Teunissen, 2007a;

*Y* = *AZ* + *GRF* + Ξ;

(16)

*<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* , *<sup>R</sup>* <sup>∈</sup> **<sup>O</sup>**3×*<sup>q</sup>*

*D*(*vec*(*Y*)) = *QYY*

Parameter *q* is introduced in order to make model (16) of general applicability. The *n* baselines may be aligned or coplanar, impeding the estimation of a full 3 × 3 matrix *R*. Therefore, *q* defines the span of matrix *F*. For baseline sets formed by aligning *n* + 1 antennas we set *q* = 1, whereas configurations of coplanar antennas are defined by *q* = 2. With four or more

The GNSS attitude model (16) is a nonlinear model. Although the relation between observables and unknowns remain linear, the orthonormal constraint is of a nonlinear nature, and profoundly affects the estimation process. This is investigated in section 4. First, the following section gives an overview of common attitude parameterization and estimation

matrix *R* can then be parameterized with a properly chosen set of variables, whose number can be as little as two (if *q* = 1) or three (if *q* ≤ 2). To this purpose, several representations

From a set of code and phase observations cast as in (16), the problem of extracting the components of the attitude representation involves, as shown in section 4, a nonlinear least squares problem. Its formulation and solution are the second topic discussed in this section.

Several attitude parameterizations are available in the literature, see e.g., Shuster (1993) and references therein. The most common parameterizations are briefly reviewed in the following.

<sup>3</sup>} = *R*{*u*1, *u*2, *u*3} =⇒ *u*�

*<sup>i</sup>* = 3 ∑ *j*=1

*rijuj* (17)

<sup>2</sup> constraints on its components *rij*. The full

one.

2011)

methods.

non-coplanar antennas, *q* = 3.

**3.1 Attitude parameterization**

**3.1.1 Direction cosine matrix**

{*u*� <sup>1</sup>, *u*� <sup>2</sup>, *u*�

**3. Attitude parameterization and estimation**

The orthonormality of *<sup>R</sup>* (*RTR* = *Iq*) imposes *<sup>q</sup>*(*q*+1)

may be used, and few are briefly reviewed in the following.

The transformation between two basis of orthonormal frames reads

observations are cast into a 2*mNn* vector by applying the *vec* operator, which stacks the columns of a matrix. The v-c matrix *QYY* that characterizes the error statistic of *vec*(*Y*) is

$$D(vec(\mathcal{Y})) = \mathcal{Q}\_{\mathcal{Y}} \tag{10}$$

A simple expression for *QYY* is obtained by assuming that each of the baselines is characterized by the same v-c matrix *Qyy*:

$$D(vec(Y)) = Q\_{Y^\gamma} = P\_\hbar \otimes Q\_{yy} \tag{11}$$

with *Pn* the *n* × *n* matrix that takes care of the correlation which is introduced by having a common antenna:

$$P\_{\rm ll} = \frac{1}{2} \left[ I\_{\rm ll} + e\_{\rm l} e\_{\rm l}^{\rm r} \right] = \begin{bmatrix} 1 & 0.5 & \cdots & 0.5 \\ 0.5 & 1 & & \vdots \\ \vdots & & \ddots & 0.5 \\ 0.5 & \ldots & 0.5 & 1 \end{bmatrix} \tag{12}$$

Expressions (9) and (11) define the *GNSS multibaseline model* that we use in this contribution as the foundation of our GNSS-based attitude estimation theory.

With the available code and phase observations it is possible to estimate the set of baseline coordinates. These can then be used to provide the aircraft attitude, but *only* when a further condition is realized: the positions of the antennas installed aboard the given aircraft are known, rigid and do not change over time (or, if change occurs, it is perfectly known and predictable). This is so because it is necessary to have a one-to-one relationship between aircraft attitude and baselines attitude. As an example, consider two antennas mounted on the two extremities of a flexible mast: it is not possible to separate the rotations of the mast from its deformations by only observing the variations of the mutual position between the two antennas.

The rigidity assumption is formalized in the following way. Consider two orthonormal frames, defined by the basis {*u*1, *u*2, *u*3} and {*u*� <sup>1</sup>, *u*� <sup>2</sup>, *u*� <sup>3</sup>}. Let us assume that the second frame is integrally fixed with the aircraft. An arbitrary vector *x* can be equivalently described by using either reference system:

$$\begin{array}{l} \mathbf{x}' = \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{1}\right)\boldsymbol{u}\_{1} + \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{2}\right)\boldsymbol{u}\_{2} + \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{3}\right)\boldsymbol{u}\_{3} \\ \mathbf{x}' = \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{1}'\right)\boldsymbol{u}\_{1}' + \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{2}'\right)\boldsymbol{u}\_{2}' + \left(\mathbf{x}^{\iota}\boldsymbol{u}\_{3}'\right)\boldsymbol{u}\_{3}' \end{array} \tag{13}$$

The relation between the components of vectors *x* and *x*� is completely defined by the mutual orientation of the two reference systems. The linear transformation *x* = *Rx*� allows for a one-to-one relationship. Matrix *R*, hereafter referred to as *rotation matrix* or *attitude matrix*, belongs to the class of orthonormal matrices **O**: its column vectors *ri* are normal and their product null: *rT <sup>i</sup> rj* = *δi*,*j*, with *δi*,*<sup>j</sup>* the Kronecker's delta (*δi*,*<sup>j</sup>* = 1 if *i* = *j*, 0 otherwise). These constraints are necessary for the admissibility of transformation *x* = *Rx*� . In absence of deformations, the scalar product between any two vectors should be invariant with respect to the transformation:

$$\mathbf{x}'^{\tau}y' = \mathbf{x}^{\tau}R^{\tau}Ry = \mathbf{x}^{\tau}y \tag{14}$$

whereas the vectorial product is invariant under rotations about the axis defined by *x*� × *y*� :

$$\mathbf{x}' \times \mathbf{y}' = (\mathbf{R}\mathbf{x}) \times (\mathbf{R}\mathbf{y}) = |\mathbf{R}| \, \mathbf{R} \, (\mathbf{x} \times \mathbf{y}) \tag{15}$$

6 Will-be-set-by-IN-TECH

observations are cast into a 2*mNn* vector by applying the *vec* operator, which stacks the columns of a matrix. The v-c matrix *QYY* that characterizes the error statistic of *vec*(*Y*) is

A simple expression for *QYY* is obtained by assuming that each of the baselines is characterized

with *Pn* the *n* × *n* matrix that takes care of the correlation which is introduced by having a

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

. .

*<sup>n</sup>*] =

Expressions (9) and (11) define the *GNSS multibaseline model* that we use in this contribution

With the available code and phase observations it is possible to estimate the set of baseline coordinates. These can then be used to provide the aircraft attitude, but *only* when a further condition is realized: the positions of the antennas installed aboard the given aircraft are known, rigid and do not change over time (or, if change occurs, it is perfectly known and predictable). This is so because it is necessary to have a one-to-one relationship between aircraft attitude and baselines attitude. As an example, consider two antennas mounted on the two extremities of a flexible mast: it is not possible to separate the rotations of the mast from its deformations by only observing the variations of the mutual position between the

The rigidity assumption is formalized in the following way. Consider two orthonormal

is integrally fixed with the aircraft. An arbitrary vector *x* can be equivalently described by

*x* = (*xT u*1) *u*<sup>1</sup> + (*xTu*2) *u*<sup>2</sup> + (*xTu*3) *u*<sup>3</sup>

The relation between the components of vectors *x* and *x*� is completely defined by the mutual orientation of the two reference systems. The linear transformation *x* = *Rx*� allows for a one-to-one relationship. Matrix *R*, hereafter referred to as *rotation matrix* or *attitude matrix*, belongs to the class of orthonormal matrices **O**: its column vectors *ri* are normal and their

deformations, the scalar product between any two vectors should be invariant with respect to

whereas the vectorial product is invariant under rotations about the axis defined by *x*� × *y*�

*<sup>i</sup> rj* = *δi*,*j*, with *δi*,*<sup>j</sup>* the Kronecker's delta (*δi*,*<sup>j</sup>* = 1 if *i* = *j*, 0 otherwise).

*y*� = *xTRTRy* = *xTy* (14)

*x*� × *y*� = (*Rx*) × (*Ry*) = |*R*| *R* (*x* × *y*) (15)

<sup>1</sup>, *u*� <sup>2</sup>, *u*�

*Pn* <sup>=</sup> <sup>1</sup>

as the foundation of our GNSS-based attitude estimation theory.

frames, defined by the basis {*u*1, *u*2, *u*3} and {*u*�

*x*� = �

*xT u*� 1 � *u*� <sup>1</sup> <sup>+</sup> � *xTu*� 2 � *u*� <sup>2</sup> <sup>+</sup> � *xTu*� 3 � *u*� 3

These constraints are necessary for the admissibility of transformation *x* = *Rx*�

*x*�*<sup>T</sup>*

using either reference system:

<sup>2</sup> [*In* <sup>+</sup> *eneT*

by the same v-c matrix *Qyy*:

common antenna:

two antennas.

product null: *rT*

the transformation:

*D*(*vec*(*Y*)) = *QYY* (10)

*D*(*vec*(*Y*)) = *QYY* = *Pn* ⊗ *Qyy* (11)

. . ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

<sup>3</sup>}. Let us assume that the second frame

(12)

(13)

:

. In absence of

1 0.5 ··· 0.5 0.5 1 .

. ... 0.5 0.5 . . . 0.5 1

These conditions are fulfilled for orthonormal rotation matrices with determinant equal to one.

Model (9) can then be reformulated by means of the linear transformation *B* = *RF*, where *F* is used to cast the set of known local baseline coordinates and *R* is the orthonormal (*RTR* = *Iq*) matrix that rotates *B* into *F*. The complete GNSS attitude model reads then (Teunissen, 2007a; 2011)

$$\begin{aligned} Y &= AZ + GRF + \Xi; \\ Z &\in \mathbb{Z}^{mN \times n}, \\ R &\in \mathbb{O}^{3 \times q} \\ D(vec(Y)) &= Q\_{Y} \end{aligned} \tag{16}$$

Parameter *q* is introduced in order to make model (16) of general applicability. The *n* baselines may be aligned or coplanar, impeding the estimation of a full 3 × 3 matrix *R*. Therefore, *q* defines the span of matrix *F*. For baseline sets formed by aligning *n* + 1 antennas we set *q* = 1, whereas configurations of coplanar antennas are defined by *q* = 2. With four or more non-coplanar antennas, *q* = 3.

The GNSS attitude model (16) is a nonlinear model. Although the relation between observables and unknowns remain linear, the orthonormal constraint is of a nonlinear nature, and profoundly affects the estimation process. This is investigated in section 4. First, the following section gives an overview of common attitude parameterization and estimation methods.

#### **3. Attitude parameterization and estimation**

The orthonormality of *<sup>R</sup>* (*RTR* = *Iq*) imposes *<sup>q</sup>*(*q*+1) <sup>2</sup> constraints on its components *rij*. The full matrix *R* can then be parameterized with a properly chosen set of variables, whose number can be as little as two (if *q* = 1) or three (if *q* ≤ 2). To this purpose, several representations may be used, and few are briefly reviewed in the following.

From a set of code and phase observations cast as in (16), the problem of extracting the components of the attitude representation involves, as shown in section 4, a nonlinear least squares problem. Its formulation and solution are the second topic discussed in this section.

#### **3.1 Attitude parameterization**

Several attitude parameterizations are available in the literature, see e.g., Shuster (1993) and references therein. The most common parameterizations are briefly reviewed in the following.

#### **3.1.1 Direction cosine matrix**

The transformation between two basis of orthonormal frames reads

$$\{u\_1', u\_2', u\_3'\} = \mathcal{R}\{u\_1, u\_2, u\_3\} \quad \Longrightarrow \quad u\_i' = \sum\_{j=1}^3 r\_{ij} u\_j \tag{17}$$

Any arbitrary rotation can always be decomposed as a combination of three consecutive rotations about the main axis *u*1, *u*<sup>2</sup> or *u*3, represented by one of the relations in (20). Figure 3 shows the example of a 321 rotation: the first rotation is about the third main axis *u*<sup>3</sup> with

GNSS Carrier Phase-Based Attitude Determination 201

*θ* =⇒ *R*(*u*2,*θ*)

Therefore, *R*<sup>321</sup> (*ψ*, *θ*, *φ*) = *R* (*u*1, *φ*) *R* (*u*2, *θ*) *R* (*u*3, *ψ*). Twelve combinations of rotations are possible, whose choice depends on the application. As an example, the sequence 321 is commonly used to describe the orientation of an aircraft, where the angles *ψ*, *θ*, *φ* are named

It is easy to see that the Euler angles representation is not unique: e.g, the combination 321 is equivalently expressed as *R*<sup>321</sup> (*ψ*, *θ*, *φ*) or *R*<sup>321</sup> (*ψ* + *π*, *π* − *θ*, *φ* + *π*). This ambiguity is usually avoided by imposing −90◦ < *θ* ≤ 90◦. The main advantage of the Euler angles representation is its straightforward physical interpretation, of importance for human-machine interfaces. The disadvantage lies in fact that the construction of the attitude matrix requires the evaluation of trigonometric functions, of higher computational load than other parameterizations. Also, the derivatives of the components of the rotation matrix are

3 *u*cc

3 *u*c

1 *u*c

Fig. 3. The three consecutive rotations that rotate the frame {*u*1, *u*2, *u*3} into the frame

<sup>2</sup> and magnitude *θ* and the third is about the main axis *u*��

A quaternion is an order-4 vector whose components can be used to define the mutual rotation

*q*¯ = (*q*1, *q*2, *q*3, *q*4)

imaginary (or vectorial) part. The components of a quaternion must respect the constraint *q*¯*Tq*¯ = 1. Physically, the four components of *q*¯ define the magnitude and axis of the rotation necessary to rotate one reference system into the other, see Figure 4. The attitude matrix *R* is

*q*<sup>4</sup> is named the scalar (real) component of the quaternion, whereas (*q*1, *q*2, *q*3)

1 *u*cc

2 2 *u u* c cc {


<sup>3</sup> }. The first one is about the main axis *u*<sup>3</sup> and magnitude *ψ*, the second is about

{*u*�� <sup>1</sup> , *u*�� <sup>2</sup> , *u*�� 3 } <sup>2</sup> with magnitude *θ*, the last

<sup>3</sup> } (21)

<sup>1</sup> with magnitude *φ*. The rotation matrix that defines the

*φ* =⇒ *R*(*u*1,*φ*)

3 *u*cc

3 *u*ccc

I

1 1 *u u* cc ccc {

2 *u*cc

*<sup>T</sup>* (22)

2 *u*ccc

<sup>1</sup> and magnitude *φ*.

*<sup>T</sup>* forms the

<sup>3</sup> } is built as

{*u*��� <sup>1</sup> , *u*��� <sup>2</sup> , *u*���

<sup>1</sup> , *u*��� <sup>2</sup> , *u*���

magnitude *ψ*, the second is about the (new) second main axis *u*�

{*u*� <sup>1</sup>, *u*� <sup>2</sup>, *u*� 3}

transformation between the frames {*u*1, *u*2, *u*3} and {*u*���

*ψ* =⇒ *R*(*u*3,*ψ*)

nonlinear (trigonometric), and affected by singularities.

\

2 *u*c

2 *u*

about the (new) first main axis *u*��

{*u*1, *u*2, *u*3}

heading, elevation and bank, respectively.

1 *u*c

3 3 *u u* { c

1 *u*

between reference systems:

{*u*��� <sup>1</sup> , *u*��� <sup>2</sup> , *u*���

the main axis *u*�

**3.1.3 Quaternions**

with *rij* the entries of *R*. The scalar product between any two unit vectors of the two frames is

$$\boldsymbol{u}\_{i}^{\prime \tau} \boldsymbol{u}\_{j} = \sum\_{k=1}^{3} r\_{ik} \left( \boldsymbol{u}\_{k}^{\tau} \boldsymbol{u}\_{j} \right) = r\_{i\bar{j}} = \cos \left( \widehat{\boldsymbol{u}\_{i}^{\prime} \boldsymbol{u}\_{j}} \right) \tag{18}$$

Hence, the attitude matrix can be expressed by nine direction cosines, i.e., the nine cosines of the angles formed by the three unit vectors of the first frame and the three unit vectors of the second frame:

$$R = \begin{bmatrix} u\_1'^\tau u\_1 \ u\_1'^\tau u\_2 \ u\_1'^\tau u\_3 \\ u\_2'^\tau u\_1 \ u\_2'^\tau u\_2 \ u\_2'^\tau u\_3 \\ u\_3'^\tau u\_1 \ u\_3'^\tau u\_2 \ u\_3'^\tau u\_3 \end{bmatrix} \tag{19}$$

This representation fully defines the mutual orientation of the two frames, by using a set of nine parameters (see Figure 2). Each configuration can be described without incurring any singularity, at the cost of having a larger number of parameters than other representations.

Fig. 2. The main axis *u*� <sup>1</sup> is completely defined by the knowledge of the three direction cosines *uT* 1*u*� <sup>1</sup>, *uT* 2*u*� <sup>1</sup> and *uT* 3*u*� 1.

#### **3.1.2 Euler angles**

Consider counterclockwise rotations about one of the main axis of a frame {*u*1, *u*2, *u*3}. Then the rotation matrix *R* is obtained through one of the following expressions:

$$\begin{aligned} R\left(\mu\_{1\prime}\phi\right) &= \begin{bmatrix} 1 & 0 & 0\\ 0 & \mathcal{C}\_{\phi} & \mathcal{S}\_{\phi} \\ 0 & -\mathcal{S}\_{\phi} & \mathcal{C}\_{\phi} \end{bmatrix} \\ R\left(\mu\_{2\prime}\phi\right) &= \begin{bmatrix} \mathcal{C}\_{\phi} & 0 & -\mathcal{S}\_{\phi} \\ 0 & 1 & 0 \\ \mathcal{S}\_{\phi} & 0 & \mathcal{C}\_{\phi} \end{bmatrix} \\ R\left(\mu\_{3\prime}\phi\right) &= \begin{bmatrix} \mathcal{C}\_{\phi} & \mathcal{S}\_{\phi} & 0 \\ -\mathcal{S}\_{\phi} & \mathcal{C}\_{\phi} & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{20}$$

8 Will-be-set-by-IN-TECH

with *rij* the entries of *R*. The scalar product between any two unit vectors of the two frames is

Hence, the attitude matrix can be expressed by nine direction cosines, i.e., the nine cosines of the angles formed by the three unit vectors of the first frame and the three unit vectors of the

This representation fully defines the mutual orientation of the two frames, by using a set of nine parameters (see Figure 2). Each configuration can be described without incurring any singularity, at the cost of having a larger number of parameters than other representations.

3 *u*

 3 1 acos c *<sup>T</sup> u u*

= *rij* = cos

<sup>1</sup> *u*<sup>2</sup> *u*�*<sup>T</sup>* <sup>1</sup> *u*<sup>3</sup>

<sup>2</sup> *u*<sup>2</sup> *u*�*<sup>T</sup>* <sup>2</sup> *u*<sup>3</sup>

<sup>3</sup> *u*<sup>2</sup> *u*�*<sup>T</sup>* <sup>3</sup> *u*<sup>3</sup>

 2 1 acos c *<sup>T</sup> u u*

<sup>1</sup> is completely defined by the knowledge of the three direction cosines

⎤ ⎦

⎤ ⎦

⎤ ⎦ 2 *u*

2 *u*c

1 *u*c

Consider counterclockwise rotations about one of the main axis of a frame {*u*1, *u*2, *u*3}. Then

⎡ ⎣

⎡ ⎣

⎡ ⎣ 10 0 0 *C<sup>φ</sup> S<sup>φ</sup>* 0 −*S<sup>φ</sup> C<sup>φ</sup>*

*C<sup>φ</sup>* 0 −*S<sup>φ</sup>* 01 0 *S<sup>φ</sup>* 0 *C<sup>φ</sup>*

*C<sup>φ</sup> S<sup>φ</sup>* 0 −*S<sup>φ</sup> C<sup>φ</sup>* 0 0 01 � *u* �� *i uj* �

⎤

⎦ (19)

(18)

(20)

*u*�*<sup>T</sup> <sup>i</sup> uj* =

second frame:

Fig. 2. The main axis *u*�

**3.1.2 Euler angles**

<sup>1</sup> and *uT* 3*u*� 1.

*uT* 1*u*� <sup>1</sup>, *uT* 2*u*�

3 ∑ *k*=1 *rik* � *uT kuj* �

*R* =

3 *u*c

1 *u*

⎡ ⎣ *u*�*<sup>T</sup>* <sup>1</sup> *u*<sup>1</sup> *u*�*<sup>T</sup>*

*u*�*<sup>T</sup>* <sup>2</sup> *u*<sup>1</sup> *u*�*<sup>T</sup>*

*u*�*<sup>T</sup>* <sup>3</sup> *u*<sup>1</sup> *u*�*<sup>T</sup>*

 1 1 acos c *<sup>T</sup> u u*

the rotation matrix *R* is obtained through one of the following expressions:

*R* (*u*1, *φ*) =

*R* (*u*2, *φ*) =

*R* (*u*3, *φ*) =

Any arbitrary rotation can always be decomposed as a combination of three consecutive rotations about the main axis *u*1, *u*<sup>2</sup> or *u*3, represented by one of the relations in (20). Figure 3 shows the example of a 321 rotation: the first rotation is about the third main axis *u*<sup>3</sup> with magnitude *ψ*, the second is about the (new) second main axis *u*� <sup>2</sup> with magnitude *θ*, the last about the (new) first main axis *u*�� <sup>1</sup> with magnitude *φ*. The rotation matrix that defines the transformation between the frames {*u*1, *u*2, *u*3} and {*u*��� <sup>1</sup> , *u*��� <sup>2</sup> , *u*��� <sup>3</sup> } is built as

$$\{\{u\_1, u\_2, u\_3\}\} \stackrel{\Psi}{\Longrightarrow} \{u'\_1, u'\_2, u'\_3\} \stackrel{\theta}{\Longrightarrow} \{u''\_1, u''\_2, u''\_3\} \stackrel{\phi}{\Longrightarrow} \{u''\_1, u''\_2, u''\_3\} \tag{21}$$

Therefore, *R*<sup>321</sup> (*ψ*, *θ*, *φ*) = *R* (*u*1, *φ*) *R* (*u*2, *θ*) *R* (*u*3, *ψ*). Twelve combinations of rotations are possible, whose choice depends on the application. As an example, the sequence 321 is commonly used to describe the orientation of an aircraft, where the angles *ψ*, *θ*, *φ* are named heading, elevation and bank, respectively.

It is easy to see that the Euler angles representation is not unique: e.g, the combination 321 is equivalently expressed as *R*<sup>321</sup> (*ψ*, *θ*, *φ*) or *R*<sup>321</sup> (*ψ* + *π*, *π* − *θ*, *φ* + *π*). This ambiguity is usually avoided by imposing −90◦ < *θ* ≤ 90◦. The main advantage of the Euler angles representation is its straightforward physical interpretation, of importance for human-machine interfaces. The disadvantage lies in fact that the construction of the attitude matrix requires the evaluation of trigonometric functions, of higher computational load than other parameterizations. Also, the derivatives of the components of the rotation matrix are nonlinear (trigonometric), and affected by singularities.

Fig. 3. The three consecutive rotations that rotate the frame {*u*1, *u*2, *u*3} into the frame {*u*��� <sup>1</sup> , *u*��� <sup>2</sup> , *u*��� <sup>3</sup> }. The first one is about the main axis *u*<sup>3</sup> and magnitude *ψ*, the second is about the main axis *u*� <sup>2</sup> and magnitude *θ* and the third is about the main axis *u*�� <sup>1</sup> and magnitude *φ*.

#### **3.1.3 Quaternions**

A quaternion is an order-4 vector whose components can be used to define the mutual rotation between reference systems:

$$\vec{q} = (q\_{1\prime}q\_{2\prime}q\_{3\prime}q\_4)^{\dagger} \tag{22}$$

*q*<sup>4</sup> is named the scalar (real) component of the quaternion, whereas (*q*1, *q*2, *q*3) *<sup>T</sup>* forms the imaginary (or vectorial) part. The components of a quaternion must respect the constraint *q*¯*Tq*¯ = 1. Physically, the four components of *q*¯ define the magnitude and axis of the rotation necessary to rotate one reference system into the other, see Figure 4. The attitude matrix *R* is

(SVD) or the EIGenvalues decomposition (EIG), such as the QUaternion ESTimator (QUEST) (Shuster, 1978; Shuster & Oh, 1981), the Fast Optimal Attitude Matrix (FOAM) (Markley & Landis, 1993), the EStimator of the Optimal Quaternion (ESOQ) (Mortari, 1997) or the Second ESOQ (ESOQ2) (Mortari, 2000) algorithms, which have been extensively compared in Markley

GNSS Carrier Phase-Based Attitude Determination 203

For nondiagonal matrices *Q*, the extraction of the orthonormal attitude matrix *R*ˇ has to be performed through nonlinear estimation techniques. A first numerical scheme for the solution of (25) is derived by applying the Lagrangian multipliers method. The Lagrangian function is

> *<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>* � <sup>−</sup> tr � [*λ*]*<sup>q</sup>* �

> > ; [*λ*]<sup>3</sup> =

*vec* (*R*) <sup>−</sup> *<sup>Q</sup>*−1*vec* �

⎡ ⎣ *λ*1 1 <sup>2</sup>*λ*<sup>4</sup> 1 <sup>2</sup>*λ*<sup>5</sup>

1 <sup>2</sup>*λ*<sup>4</sup> *λ*<sup>2</sup>

1 <sup>2</sup>*λ*<sup>5</sup> 1 <sup>2</sup>*λ*<sup>6</sup> *λ*<sup>3</sup>

<sup>2</sup> constraining functions that follows from the

*R*ˆ � = 0

, only its upper (or lower) triangular part has to be

. Following the reparameterization, matrix *R* (*μ*)

. The nonlinear least-squares problem (29) is solved

*<sup>I</sup>* (29)

�

orthonormality of *R*: *q* constraints are given by the normality (unit length) of the columns

�

considered in (28). The Newton-Raphson method can then be applied to iteratively converge

This method is computationally heavier than other iterative schemes, since it requires the explicit computation of larger-sized matrices than other methods given in the following.

A second viable solution scheme is obtained by re-parameterizing the attitude matrix

by applying iterative methods, e.g., the Newton method. This approach (Euler angles parameterization) works with a minimal set of unknowns - the Euler angles - and it can quickly converge to the sought minimizer if an accurate initial guess is used. The disadvantage is that trigonometric functions have to be evaluated, increasing the

�*h*(*μ*)�<sup>2</sup>

*T*

*μ*ˇ = arg min *μ*∈**R**<sup>3</sup>

<sup>2</sup> constraints are given by the orthogonality of the columns of *R*.

*RTR* − *Iq*

��

⎤

<sup>2</sup> constraining functions,

⎦ (27)

1 <sup>2</sup>*λ*<sup>6</sup> (26)

(28)

*Q*−1*vec* �

& Mortari (1999; 2000) and Cheng & Shuster (2007).

*L*(*R*) = *vec* �

with [*λ*]*<sup>q</sup>* the *q* by *q* matrix of Lagrangian multipliers:

[*λ*]<sup>1</sup> = *λ* ; [*λ*]

The last term of (26) gives the *<sup>q</sup>*(*q*+1)

defines the nonlinear system to be solved:

*vec* �

to the sought orthonormal matrix of rotations.

with the vector of Euler angles *μ* = (*ψ*, *θ*, *φ*)

<sup>2</sup> *vec* �

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*μ*)

<sup>2</sup> <sup>∇</sup>*L*(*R*) = �

*RTR* − *Iq*

� 1

Due to the symmetry of matrix �

of *R*, whereas *<sup>q</sup>*(*q*−1)

with *h*(*μ*) = *Q*<sup>−</sup> <sup>1</sup>

computational load.

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>* �*T*

<sup>2</sup> =

The gradient of the Lagrangian function (26), together with the *<sup>q</sup>*(*q*+1)

� = 0

� *λ*<sup>1</sup> 1 <sup>2</sup>*λ*<sup>3</sup>

1 <sup>2</sup>*λ*<sup>3</sup> *λ*<sup>2</sup>

*<sup>Q</sup>*−<sup>1</sup> <sup>−</sup> [*λ*]*<sup>q</sup>* <sup>⊗</sup> *<sup>I</sup>*<sup>3</sup>

�

*RTR* − *Iq*

implicitly fulfills the constraint *RTR* = *Iq*, and problem (25) is rewritten as

�

parameterized in terms of quaternions as

$$\begin{aligned} R\left(\overrightarrow{q}\right) &= R\left(q\_{\prime}q\_{4}\right) = \left(q\_{4}^{2} - \|q\|^{2}\right)I\_{3} + 2qq^{\top} + 2q\_{4}\left[q^{+}\right] \\ &= \begin{bmatrix} q\_{1}^{2} - q\_{2}^{2} - q\_{3}^{2} + q\_{4}^{2} & 2(q\_{1}q\_{2} + q\_{3}q\_{4}) & 2(q\_{1}q\_{3} - q\_{2}q\_{4}) \\ 2(q\_{1}q\_{2} - q\_{3}q\_{4}) & -q\_{1}^{2} + q\_{2}^{2} - q\_{3}^{2} + q\_{4}^{2} & 2(q\_{2}q\_{3} + q\_{1}q\_{4}) \\ 2(q\_{1}q\_{3} + q\_{2}q\_{4}) & 2(q\_{2}q\_{3} - q\_{1}q\_{4}) & -q\_{1}^{2} - q\_{2}^{2} + q\_{3}^{2} + q\_{4}^{2} \end{bmatrix} \end{aligned} \tag{23}$$

with

$$
\begin{bmatrix} q^+ \end{bmatrix} = \begin{bmatrix} 0 & q\_3 & -q\_2 \\ -q\_3 & 0 & q\_1 \\ q\_2 & -q\_1 & 0 \end{bmatrix} \tag{24}
$$

This parameterization is non ambiguous and it does not involve any trigonometric function, so that the computational burden is lower than with other representations. The quaternion representation is of common use in attitude estimation and control applications, since it guarantees high numerical robustness.

Fig. 4. The frame {*u*� <sup>1</sup>, *u*� <sup>2</sup>, *u*� <sup>3</sup>} can be rotated to equal the orientation of frame {*u*1, *u*2, *u*3} by means of a single rotation of magnitude *φ* about axis *n*ˆ. The four components of a quaternion are proportional to the entries of the normal vector *n*ˆ and to the magnitude *φ*.

#### **3.2 Attitude estimation**

As it will be shown in the next sections, the least-squares solution of model (16), requires the solution of a constrained least-squares problem of the type:

$$\check{\mathcal{R}} = \arg\min\_{\mathcal{R}\in\mathbf{O}^{3\times q}} \left\lVert vec\left(\hat{\mathcal{R}} - \mathcal{R}\right)\right\rVert\_{\mathcal{Q}}^2\tag{25}$$

with �·�<sup>2</sup> *<sup>Q</sup>* = (·) *<sup>T</sup> <sup>Q</sup>*−<sup>1</sup> (·). The shape of *<sup>Q</sup>* drives the choice of the solution technique to be adopted for solving (25).

If *Q* is a diagonal matrix, problem (25) becomes an Orthogonal Procustes Problem (OPP), see Schonemann (1966). This class of constrained least-squares problem have been thoroughly analyzed, and fast algorithms have been devised to quickly extract the minimizer *R*ˇ, see (Davenport, 1968; Shuster & Oh, 1981). Various fast methods for the solution of an OPP have been introduced - and widely used in practice - based on the Singular Value Decomposition 10 Will-be-set-by-IN-TECH

*I*<sup>3</sup> + 2*qqT* + 2*q*<sup>4</sup>

<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

0 *q*<sup>3</sup> −*q*<sup>2</sup> −*q*<sup>3</sup> 0 *q*<sup>1</sup> *q*<sup>2</sup> −*q*<sup>1</sup> 0

1 *u*′

means of a single rotation of magnitude *φ* about axis *n*ˆ. The four components of a quaternion

As it will be shown in the next sections, the least-squares solution of model (16), requires the

� �*vec* �

If *Q* is a diagonal matrix, problem (25) becomes an Orthogonal Procustes Problem (OPP), see Schonemann (1966). This class of constrained least-squares problem have been thoroughly analyzed, and fast algorithms have been devised to quickly extract the minimizer *R*ˇ, see (Davenport, 1968; Shuster & Oh, 1981). Various fast methods for the solution of an OPP have been introduced - and widely used in practice - based on the Singular Value Decomposition

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>* �� � 2

*<sup>T</sup> <sup>Q</sup>*−<sup>1</sup> (·). The shape of *<sup>Q</sup>* drives the choice of the solution technique to be

are proportional to the entries of the normal vector *n*ˆ and to the magnitude *φ*.

*<sup>R</sup>*<sup>ˇ</sup> <sup>=</sup> arg min *<sup>R</sup>*∈**O**3×*<sup>q</sup>*

φ*n*ˆ � *q*+�

<sup>4</sup> 2(*q*2*q*<sup>3</sup> + *q*1*q*4)

<sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

<sup>3</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup> 4

⎦ (24)

*<sup>Q</sup>* (25)

⎤ ⎦ (23)

<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

2 *u*

<sup>3</sup>} can be rotated to equal the orientation of frame {*u*1, *u*2, *u*3} by

<sup>4</sup> 2(*q*1*q*<sup>2</sup> + *q*3*q*4) 2(*q*1*q*<sup>3</sup> − *q*2*q*4)

⎤

<sup>3</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

2 *u*′

�

<sup>2</sup>(*q*1*q*<sup>3</sup> <sup>+</sup> *<sup>q</sup>*2*q*4) <sup>2</sup>(*q*2*q*<sup>3</sup> <sup>−</sup> *<sup>q</sup>*1*q*4) <sup>−</sup>*q*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

This parameterization is non ambiguous and it does not involve any trigonometric function, so that the computational burden is lower than with other representations. The quaternion representation is of common use in attitude estimation and control applications, since it

3 *u*

parameterized in terms of quaternions as

= ⎡ ⎣ *q*2 <sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

guarantees high numerical robustness.

Fig. 4. The frame {*u*�

**3.2 Attitude estimation**

*<sup>Q</sup>* = (·)

adopted for solving (25).

with �·�<sup>2</sup>

with

*R* (*q*¯) = *R* (*q*, *q*4) =

� *q*2 <sup>4</sup> <sup>−</sup> �*q*�<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

<sup>3</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

<sup>2</sup>(*q*1*q*<sup>2</sup> <sup>−</sup> *<sup>q</sup>*3*q*4) <sup>−</sup>*q*<sup>2</sup>

� *q*+� = ⎡ ⎣

3 *u*′

1 *u*

solution of a constrained least-squares problem of the type:

<sup>1</sup>, *u*� <sup>2</sup>, *u*� (SVD) or the EIGenvalues decomposition (EIG), such as the QUaternion ESTimator (QUEST) (Shuster, 1978; Shuster & Oh, 1981), the Fast Optimal Attitude Matrix (FOAM) (Markley & Landis, 1993), the EStimator of the Optimal Quaternion (ESOQ) (Mortari, 1997) or the Second ESOQ (ESOQ2) (Mortari, 2000) algorithms, which have been extensively compared in Markley & Mortari (1999; 2000) and Cheng & Shuster (2007).

For nondiagonal matrices *Q*, the extraction of the orthonormal attitude matrix *R*ˇ has to be performed through nonlinear estimation techniques. A first numerical scheme for the solution of (25) is derived by applying the Lagrangian multipliers method. The Lagrangian function is

$$L(\mathbb{R}) = \text{vec}\left(\mathbb{R} - \mathbb{R}\right)^{\mathsf{T}} Q^{-1} \text{vec}\left(\mathbb{R} - \mathbb{R}\right) - \text{tr}\left[\left[\lambda\right]\_q \left[\mathbb{R}^{\mathsf{T}} \mathbb{R} - I\_q\right]\right] \tag{26}$$

with [*λ*]*<sup>q</sup>* the *q* by *q* matrix of Lagrangian multipliers:

$$[\lambda]\_1 = \lambda \quad ; \quad [\lambda]\_2 = \begin{bmatrix} \lambda\_1 & \frac{1}{2}\lambda\_3\\ \frac{1}{2}\lambda\_3 & \lambda\_2 \end{bmatrix} \quad ; \quad [\lambda]\_3 = \begin{bmatrix} \frac{\lambda\_1}{2}\frac{1}{2}\lambda\_4 & \frac{1}{2}\lambda\_5\\ \frac{1}{2}\lambda\_4 & \lambda\_2 & \frac{1}{2}\lambda\_6\\ \frac{1}{2}\lambda\_5 & \frac{1}{2}\lambda\_6 & \lambda\_3 \end{bmatrix} \tag{27}$$

The last term of (26) gives the *<sup>q</sup>*(*q*+1) <sup>2</sup> constraining functions that follows from the orthonormality of *R*: *q* constraints are given by the normality (unit length) of the columns of *R*, whereas *<sup>q</sup>*(*q*−1) <sup>2</sup> constraints are given by the orthogonality of the columns of *R*.

The gradient of the Lagrangian function (26), together with the *<sup>q</sup>*(*q*+1) <sup>2</sup> constraining functions, defines the nonlinear system to be solved:

$$\begin{cases} \frac{1}{2}\nabla L(\mathbf{R}) = \left[Q^{-1} - [\boldsymbol{\lambda}]\_q \otimes I\_3\right] \text{vec}\left(\mathbf{R}\right) - Q^{-1} \text{vec}\left(\hat{\mathbf{R}}\right) = \mathbf{0} \\ \text{vec}\left(\mathbf{R}^T \mathbf{R} - I\_q\right) = 0 \end{cases} \tag{28}$$

Due to the symmetry of matrix � *RTR* − *Iq* � , only its upper (or lower) triangular part has to be considered in (28). The Newton-Raphson method can then be applied to iteratively converge to the sought orthonormal matrix of rotations.

This method is computationally heavier than other iterative schemes, since it requires the explicit computation of larger-sized matrices than other methods given in the following.

A second viable solution scheme is obtained by re-parameterizing the attitude matrix with the vector of Euler angles *μ* = (*ψ*, *θ*, *φ*) *T* . Following the reparameterization, matrix *R* (*μ*) implicitly fulfills the constraint *RTR* = *Iq*, and problem (25) is rewritten as

$$\sharp = \arg\min\_{\mu \in \mathbb{R}^3} \|h(\mu)\|\_I^2 \tag{29}$$

with *h*(*μ*) = *Q*<sup>−</sup> <sup>1</sup> <sup>2</sup> *vec* � *<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*μ*) � . The nonlinear least-squares problem (29) is solved by applying iterative methods, e.g., the Newton method. This approach (Euler angles parameterization) works with a minimal set of unknowns - the Euler angles - and it can quickly converge to the sought minimizer if an accurate initial guess is used. The disadvantage is that trigonometric functions have to be evaluated, increasing the computational load.

(a) Floating point operations.

GNSS Carrier Phase-Based Attitude Determination 205

(b) Computational times.

*FP*−<sup>1</sup> *<sup>n</sup>* <sup>⊗</sup> *GTQ*−<sup>1</sup>

*<sup>P</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>⊗</sup> *ATQ*−<sup>1</sup>

*yy*

*yy G FP*−<sup>1</sup> *<sup>n</sup>* <sup>⊗</sup> *GTQ*−<sup>1</sup>

*yy G P*−<sup>1</sup> *<sup>n</sup>* <sup>⊗</sup> *ATQ*−<sup>1</sup>

*yy*

*vec*(*Y*)

*yy A*

(34)

*yy A*

Fig. 5. Mean, maximum and minimum numbers of floating-point operations (left) and computational times (right) per number of baseline, for each of the attitude estimation

This float solution follows from solving the system of normal equations

 = 

*FP*−<sup>1</sup> *<sup>n</sup> FT* <sup>⊗</sup> *GTQ*−<sup>1</sup>

*<sup>P</sup>*−<sup>1</sup> *<sup>n</sup> FT* <sup>⊗</sup> *ATQ*−<sup>1</sup>

 *vec*(*R*ˆ) *vec*(*Z*ˆ)

*M*

*M* = 

method analyzed.

A third viable approach is devised by employing the quaternions parameterization of *R* and to solve for (25):

$$\mathfrak{h} = \arg\min\_{\substack{\vec{q} \in \mathbb{R}^4, \|\vec{q}\| = 1}} \left\| vec\left(\mathring{\mathcal{R}} - \mathcal{R}(\vec{q})\right) \right\|\_{\mathcal{Q}}^2 \tag{30}$$

The orthonormality of *R* is guaranteed by the normality of the quaternion: this introduces a single constraint in the minimization problem (30). A Lagrangian function is formed as

$$L'(\overline{q}) = \text{vec}\left(\hat{\mathbb{R}} - \mathcal{R}(\overline{q})\right)' Q^{-1} \text{vec}\left(\hat{\mathbb{R}} - \mathcal{R}(\overline{q})\right) - \lambda \left(\overline{q}^{\overline{r}}\overline{q} - 1\right) \tag{31}$$

and the (nonlinear) system to be solved is

$$\begin{cases} \frac{1}{2} \nabla L'(R(\vec{q})) = J\_{\mathbb{R}(\vec{q})}^{\mathrm{r}} Q^{-1} \mathrm{vec} \left( \hat{\mathsf{R}} - R(\vec{q}) \right) - \lambda \vec{q} = 0\\ \vec{q}^{\mathrm{r}} \vec{q} - 1 = 0 \end{cases} \tag{32}$$

with *JR*(*q*¯) the Jacobian of *vec*(*R*(*q*¯)).

The three iterative solutions given above rigorously solve for problem (25), but are generally slower than the methods available for diagonal *Q* matrices (SVD, EIG, QUEST, FOAM, ESOQ, and ESOQ2). Figure 5a illustrates the mean number of floating-point operations for different attitude estimation methods, per number of baselines employed. 10<sup>4</sup> samples *R*ˆ have been generated via Monte Carlo simulations for a given fully-populated *Q* matrix. The gray bars span between the maximum and minimum numbers obtained for each algorithm. The off-diagonal elements of *Q* are disregarded when applying the SVD, EIG, QUEST, FOAM, ESOQ, and ESOQ2 methods. These techniques outperform each iterative method: the number of required floating-point operations is generally two to three orders of magnitude lower. Among the iterative methods, the Lagrangian multiplier technique generally requires the highest number of operations, making it the least efficient method, while the Euler angle method and the Quaternion parameterization provide better overall results. Figure 5b shows the corresponding mean, maximum and minimum computational times marked during the simulations. The Lagrangian parameterization method generally takes the longest time to converge, whereas the quaternion and Euler angle methods show better results. Note that higher number of floating operations does not directly translate into longer computational times, because modern processor architectures efficiently operate by means of multi-threading and parallel processing.

#### **4. Reliable attitude-ambiguity estimation methods**

This section reviews the solution of the GNSS attitude model (16). This can be presented by addressing two consecutive steps: float estimation and ambiguity resolution.

#### **4.1 Float ambiguity-attitude solution**

We indicate with *float* the solution of (16) obtained by disregarding the whole set of constraints, i.e., the integerness of *Z* and the orthonormality of *R*:

$$\left\{\hat{Z},\hat{\mathcal{R}}\right\} = \arg\min\_{Z\in\mathbb{R}^{m\times n},\,R\in\mathbb{R}^{3\times q}} \left\| \text{vec}\left(Y - AZ - \text{GRF}\right) \right\|\_{Q\_{YY}}^2\tag{33}$$

12 Will-be-set-by-IN-TECH

A third viable approach is devised by employing the quaternions parameterization of *R* and

 *vec*

The orthonormality of *R* is guaranteed by the normality of the quaternion: this introduces a single constraint in the minimization problem (30). A Lagrangian function is formed as

*Q*−1*vec*

*<sup>R</sup>*(*q*¯)*Q*−1*vec*

The three iterative solutions given above rigorously solve for problem (25), but are generally slower than the methods available for diagonal *Q* matrices (SVD, EIG, QUEST, FOAM, ESOQ, and ESOQ2). Figure 5a illustrates the mean number of floating-point operations for different attitude estimation methods, per number of baselines employed. 10<sup>4</sup> samples *R*ˆ have been generated via Monte Carlo simulations for a given fully-populated *Q* matrix. The gray bars span between the maximum and minimum numbers obtained for each algorithm. The off-diagonal elements of *Q* are disregarded when applying the SVD, EIG, QUEST, FOAM, ESOQ, and ESOQ2 methods. These techniques outperform each iterative method: the number of required floating-point operations is generally two to three orders of magnitude lower. Among the iterative methods, the Lagrangian multiplier technique generally requires the highest number of operations, making it the least efficient method, while the Euler angle method and the Quaternion parameterization provide better overall results. Figure 5b shows the corresponding mean, maximum and minimum computational times marked during the simulations. The Lagrangian parameterization method generally takes the longest time to converge, whereas the quaternion and Euler angle methods show better results. Note that higher number of floating operations does not directly translate into longer computational times, because modern processor architectures efficiently operate by means of multi-threading

This section reviews the solution of the GNSS attitude model (16). This can be presented by

We indicate with *float* the solution of (16) obtained by disregarding the whole set of constraints,

�*vec* (*<sup>Y</sup>* <sup>−</sup> *AZ* <sup>−</sup> *GRF*)�<sup>2</sup>

*QYY* (33)

addressing two consecutive steps: float estimation and ambiguity resolution.

<sup>=</sup> arg min *<sup>Z</sup>*∈**R***mN*×*<sup>n</sup>* , *<sup>R</sup>*∈**R**3×*<sup>q</sup>*

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*q*¯)

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*q*¯)

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*q*¯)

 2

 − *λ* (*q*¯ *T*

*<sup>q</sup>*¯*Tq*¯ <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>0</sup> (32)

− *λq*¯ = 0

*<sup>Q</sup>* (30)

*q*¯ − 1) (31)

*q*ˇ¯ = arg min

*<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>R</sup>*(*q*¯)

(*R*(*q*¯)) = *JT*

*q*¯∈**R**4,�*q*¯�=1

*T*

to solve for (25):

*L*�

with *JR*(*q*¯) the Jacobian of *vec*(*R*(*q*¯)).

and parallel processing.

**4.1 Float ambiguity-attitude solution**

 *Z*ˆ, *R*ˆ

**4. Reliable attitude-ambiguity estimation methods**

i.e., the integerness of *Z* and the orthonormality of *R*:

and the (nonlinear) system to be solved is 1 <sup>2</sup> ∇*L*�

(*q*¯) = *vec*

(b) Computational times.

Fig. 5. Mean, maximum and minimum numbers of floating-point operations (left) and computational times (right) per number of baseline, for each of the attitude estimation method analyzed.

This float solution follows from solving the system of normal equations

$$\begin{aligned} M\begin{pmatrix} vec(\hat{\mathcal{R}})\\ vec(\hat{\mathcal{Z}}) \end{pmatrix} &= \begin{bmatrix} FP\_n^{-1} \otimes G^\top Q\_{yy}^{-1} \\ P\_n^{-1} \otimes A^\top Q\_{yy}^{-1} \end{bmatrix} vec(Y) \\\\ M &= \begin{bmatrix} FP\_n^{-1}F^\top \otimes G^\top Q\_{yy}^{-1}G \ F P\_n^{-1} \otimes G^\top Q\_{yy}^{-1}A \\ P\_n^{-1}F^\top \otimes A^\top Q\_{yy}^{-1}G & P\_n^{-1} \otimes A^\top Q\_{yy}^{-1}A \end{bmatrix} \end{aligned} \tag{34}$$

The ratio between the entries of matrix *QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> and *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*) is then proportional to the ratio

GNSS Carrier Phase-Based Attitude Determination 207

The second step consists of the resolution of the carrier phase integer ambiguities. The solution of model (16) is obtained through the following C-ILS minimization problem:

Both sets of constraints are now imposed: the matrix of ambiguities *Z*ˇ is integer valued and the matrix *<sup>R</sup>*<sup>ˇ</sup> belongs to the class of 3 <sup>×</sup> *<sup>q</sup>* orthonormal matrices **<sup>O</sup>**3×*q*. The C-ILS solution *<sup>Z</sup>*<sup>ˇ</sup>

> 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> <sup>+</sup>

The cost function *C*(*Z*) is the sum of two terms. The first weighs the distance between a candidate integer matrix *<sup>Z</sup>* and the float solution *<sup>Z</sup>*ˆ, weighted by the v-c matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . The second weighs the distance between the conditional (on the candidate *Z*) attitude matrix *R*ˆ(*Z*) and the orthonormal matrix *R*ˇ(*Z*) that follows from the solution of (45). Therefore, the computation of cost function *C*(*Z*) also involves a term that weighs the distance of the conditional attitude matrix from its orthogonal projection. This second term greatly aids the search for the correct ambiguities: integer candidates *Z* that produce matrices *R*ˆ(*Z*) too far from their orthonormal projection contribute to a much higher value of the cost function.

Since the minimization problem (44) is not solvable analytically due to the integer nature of the parameter involved, an extensive search in a subset of the space of integer matrices **Z***mN*×*<sup>n</sup>* has to be performed. The definition of an efficient and fast solution scheme for problem (44) is not a trivial task. In order to highlight the intricacies of such formulation, we first give an

Consider first the integer minimization problem (44) without the orthonormality constraint on *R*. Then the second term of *C*(*Z*) reduces to zero and the integer minimization problem

This is the usual approach of doing GNSS integer ambiguity resolution. Due to the absence of the orthonormality constraint on *R* one may expect lower success rates, i.e., lower probability

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*)

 2

*QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> (46)

�*vec* (*<sup>Y</sup>* <sup>−</sup> *AZ* <sup>−</sup> *GRF*)�<sup>2</sup>

*vec*(*R*ˆ(*Z*) <sup>−</sup> *<sup>R</sup>*ˇ(*Z*))

 2 *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*)

 *C*(*Z*)

*vec*(*R*ˆ(*Z*) <sup>−</sup> *<sup>R</sup>*)

 2 *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*)

*QYY* (43)

(44)

(45)

<sup>=</sup> arg min *<sup>Z</sup>*∈**Z***mN*×*<sup>n</sup>* , *<sup>R</sup>*∈**O**3×*<sup>q</sup>*

can be computed from the float solutions as (Teunissen & Kleusberg, 1998):

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*)

*<sup>R</sup>*ˇ(*Z*) = arg min *<sup>R</sup>*∈**O**3×*<sup>q</sup>*

approximate solution, obtained by neglecting the orthonormal constraint.

*<sup>Z</sup>*<sup>ˇ</sup> *<sup>U</sup>* <sup>=</sup> arg min *<sup>Z</sup>*∈**Z***mN*×*<sup>n</sup>*

. In GNSS applications, this phase-code variance ratio is in the order of 10−4. This clearly demonstrates the importance of ambiguity resolution: if we can integer-estimate *Z* with sufficiently high probability, then the attitude matrix *R* can be estimated with a precision that

*σ*2 Φ *σ*2 *P*

with

is comparable with the high precision of *R*ˆ(*Z*).

 *Z*ˇ, *R*ˇ

*<sup>Z</sup>*<sup>ˇ</sup> <sup>=</sup> arg min *<sup>Z</sup>*∈**Z***mN*×*<sup>n</sup>*

**4.2.1 The LAMBDA method**

becomes

**4.2 Ambiguity resolution**

where the v-c matrix *QYY* is written as in (11). Inversion of the normal matrix *M* gives the v-c matrix of the float estimators *R*ˆ and *Z*ˆ:

$$
\begin{bmatrix} Q\_{\dot{x}\dot{\kappa}} \ Q\_{\dot{x}\dot{\tau}} \\ Q\_{\dot{x}\dot{\kappa}} \ Q\_{\dot{x}\dot{\tau}} \end{bmatrix} = M^{-1} \tag{35}
$$

The float estimators are explicitly derived as

$$\begin{aligned} \hat{\mathcal{R}} &= \left[\overline{\mathbf{G}}^{\tau} \mathbf{Q}\_{yy}^{-1} \overline{\mathbf{G}}\right]^{-1} \overline{\mathbf{G}}^{\tau} \mathbf{Q}\_{yy}^{-1} Y \mathbf{P}\_n^{-1} F^{\tau} \left[ F \mathbf{P}\_n^{-1} F^{\tau} \right]^{-1} \\ \hat{\mathcal{Q}} &= \left[A^{\tau} \mathbf{Q}\_{yy}^{-1} A\right]^{-1} A^{\tau} \mathbf{Q}\_{yy}^{-1} \left[Y - \mathbf{G} \hat{\mathcal{R}} F\right] \end{aligned} \tag{36}$$

with *G* = *I* − *A ATQ*−<sup>1</sup> *yy A* −<sup>1</sup> *ATQ*−<sup>1</sup> *yy G*. Next to the above float solution, we can also define the following *conditional* float solution for the attitude matrix:

$$\hat{\mathcal{R}}(Z) = \arg\min\_{\mathcal{R}\in\mathbb{R}^{3\times q}} \left\| \mathsf{vec}\left(Y - AZ - GRF\right) \right\|\_{Q\_{YY}}^2 \tag{37}$$

In this case the ambiguity matrix is assumed completely known. The solution *R*ˆ(*Z*) can be computed form the float solutions *R*ˆ and *Z*ˆ as:

$$\text{vec}(\hat{\mathbb{R}}(\mathbf{Z})) = \text{vec}(\hat{\mathbb{R}}) - \mathbb{Q}\_{\mathbb{R}\hat{\mathbb{Z}}} \mathbb{Q}\_{\hat{\mathbb{Z}}\hat{\mathbb{Z}}}^{-1} \text{vec}(\hat{\mathbb{Z}} - \mathbf{Z}) \tag{38}$$

Application of the variance propagation law gives

$$Q\_{\hat{R}(Z)\hat{R}(Z)} = Q\_{\hat{R}\hat{R}} - Q\_{\hat{R}\hat{Z}}Q\_{2\hat{Z}}^{-1}Q\_{\hat{Z}\hat{R}} = \left[F P\_{\eta}^{-1}F^{\tau}\right]^{-1} \otimes \left[G^{\tau}Q\_{yy}^{-1}G\right]^{-1} \tag{39}$$

There is a very large difference in the precision of the float solution *R*ˆ and the precision of the conditional float solution *R*ˆ(*Z*). This can be demonstrated by comparing expression (39) with *QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> in (35), whose relation with the design matrices can be made explicit as

$$Q\_{\mathfrak{R}\mathfrak{R}} = \left[ F P\_n^{-1} F^\top \right]^{-1} \otimes \left[ \overline{\mathbf{G}}^\tau Q\_{yy}^{-1} \overline{\mathbf{G}} \right]^{-1} \tag{40}$$

Matrix *GTQ*−<sup>1</sup> *yy G* −<sup>1</sup> is characterized by much smaller entries than *GT Q*−<sup>1</sup> *yy G* −<sup>1</sup> . This is demonstrated as follows. Matrices *A*, *G* and *Qyy* may be partitioned as

$$A = \begin{bmatrix} 0 \\ \Lambda \end{bmatrix} \otimes I\_{\mathfrak{M}} \qquad G = \mathfrak{e}\_{\aleph} \otimes \begin{bmatrix} \mathcal{U} \\ \mathcal{U} \end{bmatrix} \qquad Q\_{\mathcal{Y}\mathcal{Y}} = I\_{\mathcal{N}} \otimes \begin{bmatrix} \sigma\_{\mathcal{F}}^2 \mathcal{Q} & 0 \\ 0 & \sigma\_{\bullet}^2 \mathcal{Q} \end{bmatrix} \tag{41}$$

where we assumed, for simplicity, the same code and phase standard deviations for each observation, independent from the combination of satellites, receivers and frequency. Λ is the diagonal matrix of carrier wavelengths, whereas *Q* is the matrix that introduces correlation due to the DD operation.

It follows that

$$\begin{aligned} \left[\overline{\mathbf{G}}^{\tau}\mathbf{Q}\_{yy}^{-1}\overline{\mathbf{G}}\right]^{-1} &= \frac{\sigma\_p^2}{N} \left[\mathbf{U}^{\tau}\mathbf{Q}^{-1}\mathbf{U}\right]^{-1} \\ \left[\mathbf{G}^{\tau}\mathbf{Q}\_{yy}^{-1}\mathbf{G}\right]^{-1} &= \frac{1}{N} \frac{\sigma\_\Phi^2}{\frac{\sigma\_\Phi^2}{\sigma\_p^2} + 1} \left[\mathbf{U}^{\tau}\mathbf{Q}^{-1}\mathbf{U}\right]^{-1} \approx \frac{\sigma\_\Phi^2}{N} \left[\mathbf{U}^{\tau}\mathbf{Q}^{-1}\mathbf{U}\right]^{-1} \end{aligned} \tag{42}$$

The ratio between the entries of matrix *QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> and *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*) is then proportional to the ratio *σ*2 Φ *σ*2 *P* . In GNSS applications, this phase-code variance ratio is in the order of 10−4. This clearly demonstrates the importance of ambiguity resolution: if we can integer-estimate *Z* with sufficiently high probability, then the attitude matrix *R* can be estimated with a precision that is comparable with the high precision of *R*ˆ(*Z*).

#### **4.2 Ambiguity resolution**

The second step consists of the resolution of the carrier phase integer ambiguities. The solution of model (16) is obtained through the following C-ILS minimization problem:

$$\{\check{Z}, \check{R}\} = \arg\min\_{Z \in \mathbb{Z}^{n \times n}, R \in \mathbb{O}^{3 \times q}} \left\| \vec{rec} \left( Y - AZ - GRF \right) \right\|\_{Q\_{YY}}^2 \tag{43}$$

Both sets of constraints are now imposed: the matrix of ambiguities *Z*ˇ is integer valued and the matrix *<sup>R</sup>*<sup>ˇ</sup> belongs to the class of 3 <sup>×</sup> *<sup>q</sup>* orthonormal matrices **<sup>O</sup>**3×*q*. The C-ILS solution *<sup>Z</sup>*<sup>ˇ</sup> can be computed from the float solutions as (Teunissen & Kleusberg, 1998):

$$\check{Z} = \arg\min\_{Z \in \mathbb{Z}^{m \times n}} \underbrace{\left( \left\| vec(\hat{Z} - Z) \right\|\_{Q\_{22}}^2 + \left\| vec(\hat{\mathbb{R}}(Z) - \check{\mathbb{R}}(Z)) \right\|\_{Q\_{\mathbb{R}(Z)\mathbb{R}(Z)}}^2 \right)}\_{\mathsf{C}(Z)} \tag{44}$$

with

14 Will-be-set-by-IN-TECH

where the v-c matrix *QYY* is written as in (11). Inversion of the normal matrix *M* gives the v-c

*ATQ*−<sup>1</sup> *yy* 

In this case the ambiguity matrix is assumed completely known. The solution *R*ˆ(*Z*) can be

*<sup>Z</sup>*ˆ*Z*<sup>ˆ</sup> *QZ*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> =

There is a very large difference in the precision of the float solution *R*ˆ and the precision of the conditional float solution *R*ˆ(*Z*). This can be demonstrated by comparing expression (39) with

> −<sup>1</sup> ⊗ *GT Q*−<sup>1</sup> *yy G* −<sup>1</sup>

is characterized by much smaller entries than

 *U U* 

where we assumed, for simplicity, the same code and phase standard deviations for each observation, independent from the combination of satellites, receivers and frequency. Λ is the diagonal matrix of carrier wavelengths, whereas *Q* is the matrix that introduces correlation

*UTQ*−1*U*−<sup>1</sup>

*UTQ*−1*U*−<sup>1</sup> <sup>≈</sup> *<sup>σ</sup>*<sup>2</sup>

 *FP*−<sup>1</sup> *<sup>n</sup> FT*

*vec*(*R*ˆ(*Z*)) = *vec*(*R*ˆ) <sup>−</sup> *QR*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> *<sup>Q</sup>*−<sup>1</sup>

*QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> in (35), whose relation with the design matrices can be made explicit as

 *FP*−<sup>1</sup> *<sup>n</sup> FT*

demonstrated as follows. Matrices *A*, *G* and *Qyy* may be partitioned as

⊗ *Im G* = *eN* ⊗

<sup>=</sup> *<sup>σ</sup>*<sup>2</sup> *P N* 

*QR*<sup>ˆ</sup>*R*<sup>ˆ</sup> =

*yy YP*−<sup>1</sup> *<sup>n</sup> FT*

*<sup>Y</sup>* <sup>−</sup> *GRF*<sup>ˆ</sup>

�*vec* (*<sup>Y</sup>* <sup>−</sup> *AZ* <sup>−</sup> *GRF*)�<sup>2</sup>

*FP*−<sup>1</sup> *<sup>n</sup> FT*

−<sup>1</sup> ⊗ *GTQ*−<sup>1</sup> *yy G* −<sup>1</sup>

*Qyy* = *IN* ⊗

Φ *N* 

= *M*−<sup>1</sup> (35)

(36)

(39)

(40)

(41)

. This is

−<sup>1</sup>

*G*. Next to the above float solution, we can also define

*QYY* (37)

*<sup>Z</sup>*ˆ*Z*<sup>ˆ</sup> *vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*) (38)

 *GT Q*−<sup>1</sup> *yy G* −<sup>1</sup>

*UTQ*−1*U*−<sup>1</sup> (42)

 *σ*2 *<sup>P</sup> Q* 0 0 *σ*<sup>2</sup> <sup>Φ</sup> *Q* 

*QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> *QR*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> *QZ*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup>

*ATQ*−<sup>1</sup> *yy* 

the following *conditional* float solution for the attitude matrix:

*<sup>R</sup>*ˆ(*Z*) = arg min *<sup>R</sup>*∈**R**3×*<sup>q</sup>*

matrix of the float estimators *R*ˆ and *Z*ˆ:

with *G* =

Matrix

 *GTQ*−<sup>1</sup> *yy G* −<sup>1</sup>

> *A* = 0 Λ

> > *GT Q*−<sup>1</sup> *yy G* −<sup>1</sup>

 *GTQ*−<sup>1</sup> *yy G* −<sup>1</sup> = <sup>1</sup> *N σ*2 Φ *σ*2 Φ *σ*2 *P* +1 

due to the DD operation.

It follows that

 *I* − *A ATQ*−<sup>1</sup> *yy A* −<sup>1</sup>

The float estimators are explicitly derived as

*R*ˆ = *GT Q*−<sup>1</sup> *yy G* −<sup>1</sup> *GT Q*−<sup>1</sup>

*Z*ˆ = *ATQ*−<sup>1</sup> *yy A* −<sup>1</sup>

computed form the float solutions *R*ˆ and *Z*ˆ as:

Application of the variance propagation law gives

*QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*) <sup>=</sup> *QR*<sup>ˆ</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>−</sup> *QR*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> *<sup>Q</sup>*−<sup>1</sup>

$$\check{R}(Z) = \arg\min\_{\mathcal{R}\in\mathbf{O}^{3\times q}} \left\| vec(\hat{\mathcal{R}}(Z) - \mathcal{R}) \right\|\_{Q\_{\hat{\mathcal{R}}(Z)\hat{\mathcal{R}}(Z)}}^2 \tag{45}$$

The cost function *C*(*Z*) is the sum of two terms. The first weighs the distance between a candidate integer matrix *<sup>Z</sup>* and the float solution *<sup>Z</sup>*ˆ, weighted by the v-c matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . The second weighs the distance between the conditional (on the candidate *Z*) attitude matrix *R*ˆ(*Z*) and the orthonormal matrix *R*ˇ(*Z*) that follows from the solution of (45). Therefore, the computation of cost function *C*(*Z*) also involves a term that weighs the distance of the conditional attitude matrix from its orthogonal projection. This second term greatly aids the search for the correct ambiguities: integer candidates *Z* that produce matrices *R*ˆ(*Z*) too far from their orthonormal projection contribute to a much higher value of the cost function.

Since the minimization problem (44) is not solvable analytically due to the integer nature of the parameter involved, an extensive search in a subset of the space of integer matrices **Z***mN*×*<sup>n</sup>* has to be performed. The definition of an efficient and fast solution scheme for problem (44) is not a trivial task. In order to highlight the intricacies of such formulation, we first give an approximate solution, obtained by neglecting the orthonormal constraint.

#### **4.2.1 The LAMBDA method**

Consider first the integer minimization problem (44) without the orthonormality constraint on *R*. Then the second term of *C*(*Z*) reduces to zero and the integer minimization problem becomes

$$\mathcal{Z}^{\rm u} = \arg\min\_{\mathbf{Z}\in\mathbb{Z}^{mN\times u}} \left\| \operatorname{vec}(\mathbf{Z} - \mathbf{Z}) \right\|\_{\mathcal{Q}\_{2\mathbf{Z}}}^2 \tag{46}$$

This is the usual approach of doing GNSS integer ambiguity resolution. Due to the absence of the orthonormality constraint on *R* one may expect lower success rates, i.e., lower probability

*QZZ*ˆ ˆ

2 *z*

by *TkQZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> *TT*

ˆ ˆ *T k k ZZ TQ T*

2 *z*

*mNn* ∑ *i*=1

*<sup>i</sup>*|*<sup>I</sup>* are the conditional float ambiguity estimator and its

<sup>=</sup> {*<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* <sup>|</sup> *<sup>C</sup>*(*Z*) <sup>≤</sup> *<sup>χ</sup>*2} (51)

*<sup>Z</sup>*ˆ*Z*<sup>ˆ</sup> (Giorgi et al., 2011; Teunissen, 2007a). For this reason

, whose inverse has entries two orders of

 *z*ˆ� *<sup>i</sup>*|*<sup>I</sup>* <sup>−</sup> *<sup>z</sup>*� *i* 2

> *σ*2 *i*|*I*

1*z*

(c) Final decorrelated ellipse, defined by *TQZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> *TT*.

ˆ ˆ *T ZZ TQ T*

<sup>≤</sup> *<sup>χ</sup>*<sup>2</sup> (50)

1*z*

*k* .

Fig. 6. Initial, intermediate and decorrelated search space defined by the (transformed) v-c

*vec*(*Z*ˆ� <sup>−</sup> *<sup>Z</sup>*�

) 2 *LDLT* =

corresponding conditional variance, respectively. These are conditioned to the previous *I* = 1, . . . , *i* − 1 values, and directly follow from the entries of matrices *L* and *D*. More details on the way the search is actually performed can be found in de Jonge & Tiberius (1996).

Due to the decorrelation step, the extensive search for the integer minimizer *Z*ˇ *<sup>U</sup>* is performed quickly and efficiently, making the LAMBDA method perfectly suitable for real-time

The MC-LAMBDA method is an extension of the LAMBDA method that applies to the geometrically-constrained problem (44). The MC-LAMBDA method shares the same working principle of the LAMBDA method: first the search space is decorrelated, then the search for the integer minimizer is performed. However, an extensive search within a (decorrelated) search space is generally not efficient as it is with the LAMBDA method, as explained in the

The cost function *C*(*Z*) takes, for the same candidate *Z*, much larger values than the first

it is not trivial to set a proper value of *χ*2, since the cost function *C*(*Z*) is highly sensitive to the choice of *Z* (Giorgi, 2011; Giorgi et al., 2011). This problem becomes more marked for weaker

(b) Intermediate decorrelated ellipse, defined

GNSS Carrier Phase-Based Attitude Determination 209

1*z*

matrix of the ambiguities.

**4.2.2 The MC-LAMBDA method**

The search space is now defined as

magnitude larger than the entries of *Q*−<sup>1</sup>

where the scalars *z*ˆ�

*vec*(*Z*ˆ� <sup>−</sup> *<sup>Z</sup>*�

) 2 *QZ*<sup>ˆ</sup>�*Z*ˆ� <sup>=</sup>

*<sup>i</sup>*|*<sup>I</sup>* and *<sup>σ</sup>*<sup>2</sup>

Ω*<sup>C</sup> χ*2 

quadratic term in (44), due to the matrix *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*)

(a) Original ellipse, defined

2 *z*

by *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> .

summation:

applications.

following.

of identifying the correct ambiguity matrix *Z*. However, the ILS problem (46) is of lower complexity than (44) and a very fast implementation of it is available: the LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) (Teunissen, 1995) method, see, e.g., Boon & Ambrosius (1997); Cox & Brading (2000); Huang et al. (2009); Ji et al. (2007); Kroes et al. (2005). It consists of two steps, namely decorrelation and search.

The integer minimizer has to be extensively searched within a subset of the whole space of integers:

$$\Omega^{\mu}\left(\chi^{2}\right) = \left\{ Z \in \mathbb{Z}^{m \times n} \mid \left\| \text{vec}(\mathcal{Z} - Z) \right\|\_{\mathcal{Q} \, \text{z}}^{2} \le \chi^{2} \right\} \tag{47}$$

Ω*<sup>U</sup>* is the so-called search space, a region of the space of integer matrices that contains only those candidates *Z* for which the squared norm (46) is bounded by the value *χ*2. This can be set by choosing an integer matrix *Zc* and taking *χ*<sup>2</sup> = *vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *Zc*) 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> . Rounding the float solution, *Zc* = [*Z*ˆ], is an option, as well as bootstrapping an integer matrix, as in Teunissen (2000; 2007b).

Searching for the integer minimizer in Ω*<sup>U</sup>* proves inefficient due to the weight matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . Geometrically, the search space defines a hyperellipsoid centered in *Z*ˆ and whose shape and orientation are driven by the entries of matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . The difficulty of the search lies in the fact that the search space is highly elongated, as detailed in Teunissen & Kleusberg (1998). The reason is that the ambiguities are highly correlated. While the set wherein the independent ambiguities (e.g., three ambiguities for a single baseline scenario) can be chosen is rather large, the set of admissible values for the remaining ambiguities is very small. This causes major halting problems during the search, since many times the selected subset of independent ambiguities does not yield admissible integer matrix candidates. This issue is tackled and solved in the LAMBDA method with a decorrelation step. The decorrelation of matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> is achieved by an admissible transformation matrix *T*. In order to preserve the integerness, such matrix has to fulfill the following two conditions: *T* as well as its inverse *T*−<sup>1</sup> need to have integer entries. The matrix of transformed ambiguities *Z*� and corresponding v-c matrix are then obtained as

$$\mathbf{Z}' = \mathbf{T}\mathbf{Z} \quad ; \quad \mathbf{Q}\_{\hat{\mathbf{Z}}'\hat{\mathbf{Z}}'} = \mathbf{T}\mathbf{Q}\_{\hat{\mathbf{Z}}\hat{\mathbf{Z}}}\mathbf{T}^{\tau} \tag{48}$$

The decorrelation procedure is described in Teunissen & Kleusberg (1998). The v-c matrix is iteratively decorrelated by a sequence of admissible transformations *Ti*, until matrix

$$Q\_{\mathcal{Z}'\mathcal{Z}'} = \left(\prod\_i T\_i\right) Q\_{\mathcal{Z}\mathcal{Z}} \left(\prod\_i T\_i\right)^r = T Q\_{\mathcal{Z}\mathcal{Z}} T^r \tag{49}$$

cannot be further decorrelated. Note that due to the integer conditions on *T*, a full decorrelation cannot generally be achieved. Figure 6 shows three steps of the decorrelation process for a two-dimensional example. Figure 6a shows the original (elongated) ellipse associated to *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> , Figure 6b shows an intermediate decorrelation step, and Figure 6c shows the final decorrelated search space.

After the decorrelation step, the actual search is performed by operating the *LDLT* factorization of matrix *QZ*<sup>ˆ</sup> � *<sup>Z</sup>*ˆ� , so that the quadratic form in (46) can be written as a

Fig. 6. Initial, intermediate and decorrelated search space defined by the (transformed) v-c matrix of the ambiguities.

summation:

16 Will-be-set-by-IN-TECH

of identifying the correct ambiguity matrix *Z*. However, the ILS problem (46) is of lower complexity than (44) and a very fast implementation of it is available: the LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) (Teunissen, 1995) method, see, e.g., Boon & Ambrosius (1997); Cox & Brading (2000); Huang et al. (2009); Ji et al. (2007); Kroes

The integer minimizer has to be extensively searched within a subset of the whole space of

Ω*<sup>U</sup>* is the so-called search space, a region of the space of integer matrices that contains only those candidates *Z* for which the squared norm (46) is bounded by the value *χ*2. This can be

solution, *Zc* = [*Z*ˆ], is an option, as well as bootstrapping an integer matrix, as in Teunissen

Searching for the integer minimizer in Ω*<sup>U</sup>* proves inefficient due to the weight matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . Geometrically, the search space defines a hyperellipsoid centered in *Z*ˆ and whose shape and orientation are driven by the entries of matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> . The difficulty of the search lies in the fact that the search space is highly elongated, as detailed in Teunissen & Kleusberg (1998). The reason is that the ambiguities are highly correlated. While the set wherein the independent ambiguities (e.g., three ambiguities for a single baseline scenario) can be chosen is rather large, the set of admissible values for the remaining ambiguities is very small. This causes major halting problems during the search, since many times the selected subset of independent ambiguities does not yield admissible integer matrix candidates. This issue is tackled and solved in the LAMBDA method with a decorrelation step. The decorrelation of matrix *QZ*<sup>ˆ</sup> *<sup>Z</sup>*<sup>ˆ</sup> is achieved by an admissible transformation matrix *T*. In order to preserve the integerness, such matrix has to fulfill the following two conditions: *T* as well as its inverse *T*−<sup>1</sup> need to have integer entries. The matrix of transformed ambiguities *Z*� and corresponding v-c matrix are

The decorrelation procedure is described in Teunissen & Kleusberg (1998). The v-c matrix is

cannot be further decorrelated. Note that due to the integer conditions on *T*, a full decorrelation cannot generally be achieved. Figure 6 shows three steps of the decorrelation process for a two-dimensional example. Figure 6a shows the original (elongated) ellipse associated to *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> , Figure 6b shows an intermediate decorrelation step, and Figure 6c shows

After the decorrelation step, the actual search is performed by operating the *LDLT*

iteratively decorrelated by a sequence of admissible transformations *Ti*, until matrix

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*)

 2

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *Zc*)

*Z*� = *TZ* ; *QZ*<sup>ˆ</sup> �*Z*<sup>ˆ</sup> � = *TQZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> *TT* (48)

*<sup>Z</sup>*ˆ� , so that the quadratic form in (46) can be written as a

= *TQZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> *TT* (49)

 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup>

*QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> <sup>≤</sup> *<sup>χ</sup>*2} (47)

. Rounding the float

et al. (2005). It consists of two steps, namely decorrelation and search.

<sup>=</sup> {*<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* <sup>|</sup>

Ω*<sup>U</sup> χ*2 

set by choosing an integer matrix *Zc* and taking *χ*<sup>2</sup> =

*QZ*<sup>ˆ</sup> � *<sup>Z</sup>*ˆ� =

the final decorrelated search space.

factorization of matrix *QZ*<sup>ˆ</sup> �

 ∏ *i Ti QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> ∏ *i Ti <sup>T</sup>*

integers:

(2000; 2007b).

then obtained as

$$\left\| \left| \text{vec}(\hat{\mathbf{Z}}' - \mathbf{Z}') \right| \right\|\_{\mathbf{Q}\_{\hat{\mathbf{Z}}'\hat{\mathbf{Z}}}}^2 = \left\| \text{vec}(\hat{\mathbf{Z}}' - \mathbf{Z}') \right\|\_{L\text{DL}^T}^2 = \sum\_{i=1}^{m\text{Nn}} \frac{\left( \hat{\mathbf{z}}'\_{i|I} - \mathbf{z}'\_i \right)^2}{\sigma\_{i|I}^2} \le \chi^2 \tag{50}$$

where the scalars *z*ˆ� *<sup>i</sup>*|*<sup>I</sup>* and *<sup>σ</sup>*<sup>2</sup> *<sup>i</sup>*|*<sup>I</sup>* are the conditional float ambiguity estimator and its corresponding conditional variance, respectively. These are conditioned to the previous *I* = 1, . . . , *i* − 1 values, and directly follow from the entries of matrices *L* and *D*. More details on the way the search is actually performed can be found in de Jonge & Tiberius (1996).

Due to the decorrelation step, the extensive search for the integer minimizer *Z*ˇ *<sup>U</sup>* is performed quickly and efficiently, making the LAMBDA method perfectly suitable for real-time applications.

#### **4.2.2 The MC-LAMBDA method**

The MC-LAMBDA method is an extension of the LAMBDA method that applies to the geometrically-constrained problem (44). The MC-LAMBDA method shares the same working principle of the LAMBDA method: first the search space is decorrelated, then the search for the integer minimizer is performed. However, an extensive search within a (decorrelated) search space is generally not efficient as it is with the LAMBDA method, as explained in the following.

The search space is now defined as

$$\Omega^{\mathbb{C}}\left(\chi^{2}\right) = \{Z \in \mathbb{Z}^{\mathbb{m}\mathbb{N} \times n} \mid \mathbb{C}(Z) \le \chi^{2}\}\tag{51}$$

The cost function *C*(*Z*) takes, for the same candidate *Z*, much larger values than the first quadratic term in (44), due to the matrix *QR*<sup>ˆ</sup>(*Z*)*R*ˆ(*Z*) , whose inverse has entries two orders of magnitude larger than the entries of *Q*−<sup>1</sup> *<sup>Z</sup>*ˆ*Z*<sup>ˆ</sup> (Giorgi et al., 2011; Teunissen, 2007a). For this reason it is not trivial to set a proper value of *χ*2, since the cost function *C*(*Z*) is highly sensitive to the choice of *Z* (Giorgi, 2011; Giorgi et al., 2011). This problem becomes more marked for weaker

candidate (if any, since set Ω<sup>1</sup>

Ω*<sup>C</sup> χ*<sup>2</sup>

that Ω*<sup>C</sup>*

space is

Figure 8.

applications.

**5. Flight test results**

*χ*<sup>2</sup> is non-empty. If the search space Ω*<sup>C</sup>*

*χ*<sup>2</sup>

 *χ*<sup>2</sup>

scalar *χ*<sup>2</sup> is set such that it guarantees the non-emptiness of Ω<sup>2</sup>

non-empty either. This can be done by choosing *χ*<sup>2</sup> = *C*2(*Z*�

size of the search space is iteratively 'expanded'.

Ω<sup>2</sup> *χ*2 

which is contained in the set Ω*<sup>C</sup>*

for an integer candidate in the set Ω<sup>2</sup>

the value *C*2(*Z*2) = *χ*<sup>2</sup>

value for the upper bound *C*2(*Z*1) = *χ*<sup>2</sup>

<sup>2</sup> <sup>&</sup>lt; *<sup>χ</sup>*<sup>2</sup>

actual data collected during two different flights tests.

*χ*<sup>2</sup> may also turn out empty), in order to also evaluate

*χ*<sup>2</sup> is empty, the size of Ω<sup>1</sup>

 *χ*2 

, aiming to find a matrix *Z*<sup>1</sup> that provides a smaller

, and therefore Ω*<sup>C</sup>*

) for an integer matrix *Z*�

. Consider the following iterative procedure. First, the

 *χ*2 

<sup>1</sup> <sup>&</sup>lt; *<sup>χ</sup>*2. When it is found, the set is shrunk to <sup>Ω</sup><sup>2</sup>

<sup>1</sup>. This process is repeated until the minimizer of *<sup>C</sup>*2(*Z*), say *<sup>Z</sup>*ˇ2,

 *χ*<sup>2</sup> is

(55)

*χ*<sup>2</sup> is

, which

 *χ*2 1 

*χ*<sup>2</sup> , with

. If this set turns out non-empty, then one has simply to extract the minimizer *Z*ˇ by

sorting the integer matrices according to the values of *C*(*Z*). However, there is no guarantee

GNSS Carrier Phase-Based Attitude Determination 211

increased and the process repeated iteratively until the minimizer *Z*ˇ is found. This search scheme, illustrated with the flow chart in Figure 7, is named *Expansion* approach, since the

An alternative approach is devised by considering the upper bound *C*2(*Z*). Its search

can be the rounded float solution, a bootstrapped solution, or an integer matrix obtained by other means (see for further options Giorgi et al. (2008)). Then, the search proceeds by looking

and the search continues by looking for another integer candidate *Z*<sup>2</sup> capable of reducing

*χ*<sup>2</sup> = *C*2(*Z*ˇ <sup>2</sup>), is evaluated and the sought-for integer minimizer *Z*ˇ extracted. This iterative search scheme is named *Search and Shrink* approach, and it is detailed in the flow chart of

Both the *Expansion* and the *Search and Shrink* approaches implement the search for integer minimizer (44) in a fast and efficient way, such that the algorithm can be used for real-time

The MC-LAMBDA method achieves very high success rates. The success rate is defined as the probability of providing the correct set of integer ambiguities. The inclusion of geometrical constraints, which follow from the a priori knowledge of the antennas relative positions aboard the aircraft, largely aids the ambiguity resolution process, allowing for higher success rates in weaker models, such as with the single-frequency and/or high measurement noise scenarios. These performance improvements associated to the MC-LAMBDA method with respect to classical methods (such as the LAMBDA) are analyzed in the following section with

The performance of the MC-LAMBDA method is analyzed with data collected on two flight-tests performed with a Cessna Citation jet aircraft. The aircraft attitude is extracted from unaided, single-epoch, single-frequency (*N* = 1) GNSS observations, in order to demonstrate the method capabilities in the most challenging scenario, i.e., stand-alone, high observation noise and low measurements redundancy. Also, single-epoch performance is extremely important for dynamic platforms, where a quick recovery from changes of tracked satellites, cycle slips and losses of lock is necessary to avoid undesired loss of guidance. The

is found. Since this may differ from the minimizer of *C*(*Z*), the search space Ω*<sup>C</sup>*

<sup>=</sup> {*<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* <sup>|</sup> *<sup>C</sup>*2(*Z*) <sup>≤</sup> *<sup>χ</sup>*2} ⊂ <sup>Ω</sup>*<sup>C</sup>*

models (single frequency, low number of satellites tracked, high noise levels). Obviously, larger values of *χ*<sup>2</sup> imply longer computational times due to the larger number of candidates to be evaluated. Also, the constrained least-squares problem (45) has to be solved for each of the integer candidates in Ω*<sup>C</sup> χ*<sup>2</sup> , thereby further increasing the computational load.

The aforementioned issues are solved with a novel numerical efficient search scheme for the solution of (44). This is achieved by employing easier-to-evaluate bounding functions and introducing new search algorithms.

First, consider two functions, *C*1(*Z*) and *C*2(*Z*), that satisfy the following inequalities:

$$\mathbb{C}\_1(\mathbf{Z}) \le \mathbb{C}(\mathbf{Z}) \le \mathbb{C}\_2(\mathbf{Z}) \tag{52}$$

These functions provide a lower and an upper bound for the cost function *C*(*Z*). The choice for these bounding functions is driven by two requirements: their evaluation should be less time consuming than the evaluation of *C*(*Z*), and each bound should be sufficiently tight. Several alternatives have been studied in (Giorgi, 2011; Giorgi et al., 2012; Nadarajah et al., 2011; Teunissen, 2007a;c), based on


For example, the first method listed exploits the inequalities *<sup>ξ</sup><sup>m</sup>* �·�<sup>2</sup> *<sup>I</sup>* <sup>≤</sup> �·�<sup>2</sup> *<sup>Q</sup>* <sup>≤</sup> *<sup>ξ</sup> <sup>M</sup>* �·�<sup>2</sup> *I* , with *ξ<sup>m</sup>* and *ξ <sup>M</sup>* the smallest and largest eigenvalues of *Q*−<sup>1</sup> *R*ˆ(*Z*)*R*ˆ(*Z*) , respectively. After some manipulation, the two bounding functions read

$$\begin{aligned} \mathbf{C}\_1(\mathbf{Z}) &= \left\| \operatorname{vec}(\hat{\mathbf{Z}} - \mathbf{Z}) \right\|\_{Q\_{22}}^2 + \xi\_m \sum\_{i=1}^q \left( \left\| \mathbb{H}\_i(\mathbf{Z}) \right\| - 1 \right)^2 \\\\ \mathbf{C}\_2(\mathbf{Z}) &= \left\| \operatorname{vec}(\hat{\mathbf{Z}} - \mathbf{Z}) \right\|\_{Q\_{2\hat{\mathbf{Z}}}}^2 + \xi\_M \sum\_{i=1}^q \left( \left\| \mathbb{H}\_i(\mathbf{Z}) \right\| + 1 \right)^2 \end{aligned} \tag{53}$$

where *r*ˆ*i*(*Z*) are the column vectors of *R*ˆ(*Z*).

Two efficient search methods have been developed to reduce the computational burden associated to an extensive search. Independently from the bounding functions used, these novel search schemes allow for a quick minimization of *C*(*Z*).

Consider first the lower bound *C*1(*Z*). The search space associated to *C*1(*Z*) is

$$\Omega\_1\left(\chi^2\right) = \{Z \in \mathbb{Z}^{m \text{N} \times n} \mid \mathbb{C}\_1(Z) \le \chi^2\} \supset \Omega^c\left(\chi^2\right) \tag{54}$$

Obviously, the search space Ω*<sup>C</sup> χ*2 is contained within <sup>Ω</sup><sup>1</sup> *χ*<sup>2</sup> . One may proceed, for example, by choosing *χ*<sup>2</sup> = *vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*� ) 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> with *<sup>Z</sup>*� a given integer matrix (both rounding the float solution and bootstrapping an integer matrix are viable choices). Then, we can enumerate all the integers matrices contained in Ω<sup>1</sup> *χ*<sup>2</sup> and compute *C*(*Z*) for each 18 Will-be-set-by-IN-TECH

models (single frequency, low number of satellites tracked, high noise levels). Obviously, larger values of *χ*<sup>2</sup> imply longer computational times due to the larger number of candidates to be evaluated. Also, the constrained least-squares problem (45) has to be solved for each of

The aforementioned issues are solved with a novel numerical efficient search scheme for the solution of (44). This is achieved by employing easier-to-evaluate bounding functions and

These functions provide a lower and an upper bound for the cost function *C*(*Z*). The choice for these bounding functions is driven by two requirements: their evaluation should be less time consuming than the evaluation of *C*(*Z*), and each bound should be sufficiently tight. Several alternatives have been studied in (Giorgi, 2011; Giorgi et al., 2012; Nadarajah et al.,

First, consider two functions, *C*1(*Z*) and *C*2(*Z*), that satisfy the following inequalities:

, thereby further increasing the computational load.

*C*1(*Z*) ≤ *C*(*Z*) ≤ *C*2(*Z*) (52)

*R*ˆ(*Z*)*R*ˆ(*Z*)

(�*r*ˆ*i*(*Z*)� − 1)

(�*r*ˆ*i*(*Z*)� + 1)

*q* ∑ *i*=1

*q* ∑ *i*=1 *<sup>I</sup>* <sup>≤</sup> �·�<sup>2</sup>

2

2

 *χ*2 

*QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> with *<sup>Z</sup>*� a given integer matrix (both rounding

 *χ*2 

, respectively. After some

. One may proceed, for

*χ*<sup>2</sup> and compute *C*(*Z*) for each

*<sup>Q</sup>* <sup>≤</sup> *<sup>ξ</sup> <sup>M</sup>* �·�<sup>2</sup>

*I* ,

(53)

(54)

the integer candidates in Ω*<sup>C</sup>*

introducing new search algorithms.

2011; Teunissen, 2007a;c), based on - the eigenvalues of matrix *Q*−<sup>1</sup>

*χ*<sup>2</sup>

*R*ˆ(*Z*)*R*ˆ(*Z*)

For example, the first method listed exploits the inequalities *<sup>ξ</sup><sup>m</sup>* �·�<sup>2</sup>

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*)

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*)

Consider first the lower bound *C*1(*Z*). The search space associated to *C*1(*Z*) is

) 2

*χ*2

*vec*(*Z*<sup>ˆ</sup> <sup>−</sup> *<sup>Z</sup>*�

enumerate all the integers matrices contained in Ω<sup>1</sup>

 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> <sup>+</sup> *<sup>ξ</sup><sup>m</sup>*

 2 *QZ*<sup>ˆ</sup>*Z*<sup>ˆ</sup> <sup>+</sup> *<sup>ξ</sup> <sup>M</sup>*

Two efficient search methods have been developed to reduce the computational burden associated to an extensive search. Independently from the bounding functions used, these

<sup>=</sup> {*<sup>Z</sup>* <sup>∈</sup> **<sup>Z</sup>***mN*×*<sup>n</sup>* <sup>|</sup> *<sup>C</sup>*1(*Z*) <sup>≤</sup> *<sup>χ</sup>*2} ⊃ <sup>Ω</sup>*<sup>C</sup>*

the float solution and bootstrapping an integer matrix are viable choices). Then, we can

is contained within <sup>Ω</sup><sup>1</sup>



with *ξ<sup>m</sup>* and *ξ <sup>M</sup>* the smallest and largest eigenvalues of *Q*−<sup>1</sup>

novel search schemes allow for a quick minimization of *C*(*Z*).


manipulation, the two bounding functions read

*<sup>C</sup>*1(*Z*) =

*<sup>C</sup>*2(*Z*) =

where *r*ˆ*i*(*Z*) are the column vectors of *R*ˆ(*Z*).

Ω<sup>1</sup> *χ*2 

Obviously, the search space Ω*<sup>C</sup>*

example, by choosing *χ*<sup>2</sup> =

candidate (if any, since set Ω<sup>1</sup> *χ*<sup>2</sup> may also turn out empty), in order to also evaluate Ω*<sup>C</sup> χ*<sup>2</sup> . If this set turns out non-empty, then one has simply to extract the minimizer *Z*ˇ by sorting the integer matrices according to the values of *C*(*Z*). However, there is no guarantee that Ω*<sup>C</sup> χ*<sup>2</sup> is non-empty. If the search space Ω*<sup>C</sup> χ*<sup>2</sup> is empty, the size of Ω<sup>1</sup> *χ*<sup>2</sup> is increased and the process repeated iteratively until the minimizer *Z*ˇ is found. This search scheme, illustrated with the flow chart in Figure 7, is named *Expansion* approach, since the size of the search space is iteratively 'expanded'.

An alternative approach is devised by considering the upper bound *C*2(*Z*). Its search space is

$$\Omega\_2\left(\chi^2\right) = \{Z \in \mathbb{Z}^{\mathrm{m} \mathrm{N} \times \mathrm{n}} \mid \mathbb{C}\_2(Z) \le \chi^2\} \subset \Omega^{\mathbb{C}}\left(\chi^2\right) \tag{55}$$

which is contained in the set Ω*<sup>C</sup> χ*2 . Consider the following iterative procedure. First, the scalar *χ*<sup>2</sup> is set such that it guarantees the non-emptiness of Ω<sup>2</sup> *χ*<sup>2</sup> , and therefore Ω*<sup>C</sup> χ*2 is non-empty either. This can be done by choosing *χ*<sup>2</sup> = *C*2(*Z*� ) for an integer matrix *Z*� , which can be the rounded float solution, a bootstrapped solution, or an integer matrix obtained by other means (see for further options Giorgi et al. (2008)). Then, the search proceeds by looking for an integer candidate in the set Ω<sup>2</sup> *χ*<sup>2</sup> , aiming to find a matrix *Z*<sup>1</sup> that provides a smaller value for the upper bound *C*2(*Z*1) = *χ*<sup>2</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>χ</sup>*2. When it is found, the set is shrunk to <sup>Ω</sup><sup>2</sup> *χ*2 1 and the search continues by looking for another integer candidate *Z*<sup>2</sup> capable of reducing the value *C*2(*Z*2) = *χ*<sup>2</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>χ</sup>*<sup>2</sup> <sup>1</sup>. This process is repeated until the minimizer of *<sup>C</sup>*2(*Z*), say *<sup>Z</sup>*ˇ2, is found. Since this may differ from the minimizer of *C*(*Z*), the search space Ω*<sup>C</sup> χ*<sup>2</sup> , with *χ*<sup>2</sup> = *C*2(*Z*ˇ <sup>2</sup>), is evaluated and the sought-for integer minimizer *Z*ˇ extracted. This iterative search scheme is named *Search and Shrink* approach, and it is detailed in the flow chart of Figure 8.

Both the *Expansion* and the *Search and Shrink* approaches implement the search for integer minimizer (44) in a fast and efficient way, such that the algorithm can be used for real-time applications.

The MC-LAMBDA method achieves very high success rates. The success rate is defined as the probability of providing the correct set of integer ambiguities. The inclusion of geometrical constraints, which follow from the a priori knowledge of the antennas relative positions aboard the aircraft, largely aids the ambiguity resolution process, allowing for higher success rates in weaker models, such as with the single-frequency and/or high measurement noise scenarios. These performance improvements associated to the MC-LAMBDA method with respect to classical methods (such as the LAMBDA) are analyzed in the following section with actual data collected during two different flights tests.

#### **5. Flight test results**

The performance of the MC-LAMBDA method is analyzed with data collected on two flight-tests performed with a Cessna Citation jet aircraft. The aircraft attitude is extracted from unaided, single-epoch, single-frequency (*N* = 1) GNSS observations, in order to demonstrate the method capabilities in the most challenging scenario, i.e., stand-alone, high observation noise and low measurements redundancy. Also, single-epoch performance is extremely important for dynamic platforms, where a quick recovery from changes of tracked satellites, cycle slips and losses of lock is necessary to avoid undesired loss of guidance. The

Enumerate all the integer matrices in Ω1

*Z CZ* <sup>×</sup> Ω =∈ ≤

{ } 2 2 1 1 ( ) | () *mN n*

 χ

Evaluate the minimizer of C(Z) among the enumerated matrices in Ω1

() ()

Fig. 8. The *Search and Shrink* approach: flow chart.

*ZR* 

*R R <sup>R</sup> <sup>Q</sup> R Z RZ R* <sup>×</sup> <sup>∈</sup> = − o

( )( ) <sup>3</sup> ˆ ˆ <sup>2</sup> <sup>ˆ</sup> arg min*<sup>p</sup>*

*RZRZ*

No

*ZR* 

test analyzed (*T* − *I*) the nose antenna was placed on the extremity of a boom, whereas in second test (*T* − *I I*) it was directly placed on the aircraft body. The two tests largely differ by the flight dynamic. Test *T* − *I* was conducted with aggressive maneuvering and few zero-gravity parabolas, whereas *T* − *I I* was performed as part of a gravimetry campaign, with very few smooth maneuvers, as shown in Figure 10. During test *T* − *I I*, the aircraft

Yes

*k k* = +1

Shrink the space to 2

χ*k k* = *C Z*

GNSS Carrier Phase-Based Attitude Determination 213

<sup>2</sup> ( )

SHRINK

ENUMERATE

MINIMIZE

Yes
