**Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator**

G. M'boungui, A.A. Jimoh, B. Semail and F. Giraud

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57569

#### **1. Introduction**

Research works have focused on friction over more than 500 years. It is indeed a complex phenomenon arising at the contact of the surfaces that is encountered in a wide variety of engineering disciplines including contact mechanics, system dynamics and controls, aerome‐ chanics, geomechanics, fracture and fatigue, structural dynamics, and many others [1]. Recently, the action of friction generated by a surface under the finger has been exploited in continuous structure tactile (sensorial touch) interfaces.

Indeed, in daily life, various tasks may be more speedily and efficiently completed if kinaes‐ thetic feedback is exploited [2]. However, human beings are only using visual and audio feedback when interacting with numerous interfaces such as computers, mobile phones, etc. In such a context, the utilization of haptic devices to get back forces corresponding to feelings from virtual objects manipulation enhances the realism of the global experience. This is even true in the instance of forces and pressure applied on fingers and hand in tele-robotics applications. As a matter of fact, as soon as one has to grasp, touch, and feel objects it becomes necessary to involve haptic devices. Thus, for many researchers, such interfaces represent a new human-machine communication medium, which is a growing topic of interest and prime importance to the research community. Developments in the field have been observed over the last years. Besides tele-operation, haptic feedback can have so many over applications either in engineering, CAD, electronic games, education, learning, etc. Force feedback devices are commercialised. Phantom from Sensable® or Virtuose from Haption®, are examples of haptic feedback pen [3]. Some other solutions have been proposed including « exoskeleton » devices that directly apply forces on the hand or the finger. Typically, these haptic peripherals have an essential place in design, simulation and virtual assembly for instance in car produc‐ tion in the automobile industry. So many additional illustrations in terms of utilisation are

© 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

available and helping a user in identifying limits and virtual shape changes may be achieved by such interfaces.

we present the design, modelling and implementation of our proposal which is a passive 2Dof (two Degree of freedom) device able to provide different resistant feelings when a

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

http://dx.doi.org/10.5772/57569

279

In this section, piezoelectric materials such as lead zirconium titanate (PZT) utilised elsewhere in our design are briefly discussed. We focus on the benefits that transducers technology in

Piezoelectric materials produce an electric charge when subjected to mechanical loads (direct effect) and/or vibrations. Those materials deform when subjected to a magnetic field (inverse effect) [8]. The piezoelectric effect is expressed in materials such as single crystals, ceramics,

Of the existing piezoelectric materials polymers and ceramics have been widely explored as transducers materials [8]. Ceramic piezoelectric materials comprise barium titanate (BaTiO3), lead zirconate – lead titanate of general formula Pb(Zr-Ti)O3-PZT and lead titanate (PbTiO3, PCT). Those materials have been extensively studied because most of them have in common a perovskite (ABO3) structure [9][8], a material in which the application of an intense electric field aimed at aligning polarization of elementary ferroelectric microcrystal (polarization operation) enables introduction of necessary anisotropy to piezoelectric existence. The dielectric and piezoelectric constants of barium titanate vary with temperature, stoichiometry, microstructure and doping [10] whereas the piezoelectric constants for PZT are not strongly dependant upon temperature but rather on material composition. However, piezoelectric constants for certain stoichiometry compositions of PZT are far more sensitive to temperature dependencies than others [11]. In [12] it can be seen that the characteristics of BaTiO3 single crystal favoured their usage in certain applications such as electromechanical transducers for operation at high frequencies. However, nowadays, PZT have been the material of reference in the field of piezoelectric motors. Regarding, lead titanate material, Samarium-modified PCT was investigated for application in infrared sensors, electro-optic devices and ferroelectric

Polymer piezoelectric material such as polyvinylidene difluoride (PVDF) or PVF2 demonstrate the piezoelectric effect when they are stretched or formed during fabrication [10]. However, it is among composite piezoelectric materials made of a piezoelectric and a polymer that composites comprised of PZT rods embedded within a polymer matrix are predicted to be one

Relaxor-type ferroelectric materials differ from traditional ferroelectric material because they have a broad phase transition from paraelectric to ferroelectric state, dielectric relaxation and weak remnant polarization. Indeed those materials have the ability to become polarized when subjected to applied electric field. Unlike in traditional ferroelectric materials, this can happen even if there is no permanent electric dipole that exists in the material. The

of the most promising structures for transducers and acoustic applications [10].

**2. Piezoelectric materials contribution to transducers technology**

polymers, composites, thin films and relaxor-type ferroelectric materials.

user moves it.

memory devices [13].

general can draw from those materials.

Nevertheless, it is remarkable that force feedback devices are mostly based on electromagnetic technologies. Consequently, numerous mechanical links are often involved in movement transformation. This results in systems integration and dynamic problems, etc. For this reason piezoelectric actuators present a reliable alternative especially since high forces, rapid response and compactness are seen to be added advantages. It is indeed shown that piezoelectric actuators make variable friction phenomena available [4] and exploitable for the purpose of haptic feedback. added advantages. It is indeed shown that piezoelectric actuators make variable

The actuator to be discussed follows this trend: it is based on electro mechanical conversion principle specific to piezoelectric systems. friction phenomena available [4] and exploitable for the purpose of haptic feedback. The actuator to be discussed follows this trend: it is based on electro mechanical conversion principle specific to piezoelectric systems.

Indeed, exciting a plate in certain conditions of vibratory amplitude and frequency the texture represented in Figure 1 e.g. can be simulated [5][6]. For that it suffices to correlate the vibratory amplitude with the level of friction reduction between an exploring finger and the textured plate. To complete the process, the control of the finger position over the plate enables textures feeling. In fact, during the touching of the excited surface, that is vibrating, the finger perceives smoothness. If the vibrations source is switched off a roughness is perceived. To reform a given texture, we can thus measure the finger displacement and control the vibratory amplitude as a function of the explored feeling (smoothness or roughness) and the finger position. This example is chosen to illustrate that controlled friction is an interesting alternative in terms of tactile feelings generation. The challenge is however in modelling the friction and to deduce appropriate control laws. Indeed, exciting a plate in certain conditions of vibratory amplitude and frequency the texture represented in Figure 1 e.g. can be simulated [5][6]. For that it suffices to correlate the vibratory amplitude with the level of friction reduction between an exploring finger and the textured plate. To complete the process, the control of the finger position over the plate enables textures feeling. In fact, during the touching of the excited surface, that is vibrating, the finger perceives smoothness. If the vibrations source is switched off a roughness is perceived. To reform a given texture, we can thus measure the finger displacement and control the vibratory amplitude as a function of the explored feeling (smoothness or roughness) and the finger position. This example is chosen to illustrate that controlled friction is an interesting alternative in terms of tactile feelings generation. The challenge is however in modelling the friction and to deduce appropriate control laws.

**Figure 1.** Image of the texture

ferroelectric materials.

A feature of stick-slip vibration is a saw tooth displacement time evolution with stick and slip phases clearly defined in which the two surfaces in contact stick respectively slip over each other [7]. From the later definition a relative similarity may be perceived between the modelling of stick-slip vibration and the texture in figure 1. In the chapter is presented the approach we propose using Matlab-Simulink-Stateflow® to deal with our structure as a finite state structure. Thus an overview of piezoelectric materials and the impact they may have on A feature of stick-slip vibration is a saw tooth displacement time evolution with stick and slip phases clearly defined in which the two surfaces in contact stick respectively slip over each other [7]. From the later definition a relative similarity may be perceived between the model‐ ling of stick-slip vibration and the texture in figure 1. In the chapter is presented the approach we propose using Matlab-Simulink-Stateflow® to deal with our structure as a finite state structure.

**Figure 1 : Image of the texture** 

resulting piezoelectric systems, in particular on transducers technology is given. This is followed by a brief discussion on tactile feedback designs using actuation other than piezoelectric and the drawbacks and advantages of piezoelectric actuators versus other types. From now on we present the design, modelling and implementation of Thus an overview of piezoelectric materials and the impact they may have on resulting piezoelectric systems, in particular on transducers technology is given. This is followed by a brief discussion on tactile feedback designs using actuation other than piezoelectric and the drawbacks and advantages of piezoelectric actuators versus other types. From now on

**2. Piezoelectric materials contribution to transducers technology** 

transducers technology in general can draw from those materials.

different resistant feelings when a user moves it.

our proposal which is a passive 2Dof (two Degree of freedom) device able to provide

In this section, piezoelectric materials such as lead zirconium titanate (PZT) utilised elsewhere in our design are briefly discussed. We focus on the benefits that

Piezoelectric materials produce an electric charge when subjected to mechanical loads (direct effect) and/or vibrations. Those materials deform when subjected to a magnetic field (inverse effect) [8]. The piezoelectric effect is expressed in materials such as single crystals, ceramics, polymers, composites, thin films and relaxor-type we present the design, modelling and implementation of our proposal which is a passive 2Dof (two Degree of freedom) device able to provide different resistant feelings when a user moves it.

### **2. Piezoelectric materials contribution to transducers technology**

available and helping a user in identifying limits and virtual shape changes may be achieved

Nevertheless, it is remarkable that force feedback devices are mostly based on electromagnetic technologies. Consequently, numerous mechanical links are often involved in movement transformation. This results in systems integration and dynamic problems, etc. For this reason piezoelectric actuators present a reliable alternative especially since high forces, rapid response and compactness are seen to be added advantages. It is indeed shown that piezoelectric actuators make variable friction phenomena available [4] and exploitable for the purpose of

The actuator to be discussed follows this trend: it is based on electro mechanical conversion

added advantages. It is indeed shown that piezoelectric actuators make variable friction phenomena available [4] and exploitable for the purpose of haptic feedback. The actuator to be discussed follows this trend: it is based on electro mechanical

Indeed, exciting a plate in certain conditions of vibratory amplitude and frequency the texture represented in Figure 1 e.g. can be simulated [5][6]. For that it suffices to correlate the vibratory amplitude with the level of friction reduction between an exploring finger and the textured plate. To complete the process, the control of the finger position over the plate enables textures feeling. In fact, during the touching of the excited surface, that is vibrating, the finger perceives smoothness. If the vibrations source is switched off a roughness is perceived. To reform a given texture, we can thus measure the finger displacement and control the vibratory amplitude as a function of the explored feeling (smoothness or roughness) and the finger position. This example is chosen to illustrate that controlled friction is an interesting alternative in terms of tactile feelings generation. The challenge is however in modelling the

**Figure 1 : Image of the texture** 

Direction of displacement

Relative roughness magnitude

A feature of stick-slip vibration is a saw tooth displacement time evolution with stick and slip phases clearly defined in which the two surfaces in contact stick respectively slip over each other [7]. From the later definition a relative similarity may be perceived between the modelling of stick-slip vibration and the texture in figure 1. In the chapter is presented the approach we propose using Matlab-Simulink-Stateflow®

A feature of stick-slip vibration is a saw tooth displacement time evolution with stick and slip phases clearly defined in which the two surfaces in contact stick respectively slip over each other [7]. From the later definition a relative similarity may be perceived between the model‐ ling of stick-slip vibration and the texture in figure 1. In the chapter is presented the approach we propose using Matlab-Simulink-Stateflow® to deal with our structure as a finite state

Thus an overview of piezoelectric materials and the impact they may have on resulting piezoelectric systems, in particular on transducers technology is given. This is followed by a brief discussion on tactile feedback designs using actuation other than piezoelectric and the drawbacks and advantages of piezoelectric actuators versus other types. From now on we present the design, modelling and implementation of our proposal which is a passive 2Dof (two Degree of freedom) device able to provide

Thus an overview of piezoelectric materials and the impact they may have on resulting piezoelectric systems, in particular on transducers technology is given. This is followed by a brief discussion on tactile feedback designs using actuation other than piezoelectric and the drawbacks and advantages of piezoelectric actuators versus other types. From now on

In this section, piezoelectric materials such as lead zirconium titanate (PZT) utilised elsewhere in our design are briefly discussed. We focus on the benefits that

Piezoelectric materials produce an electric charge when subjected to mechanical loads (direct effect) and/or vibrations. Those materials deform when subjected to a magnetic field (inverse effect) [8]. The piezoelectric effect is expressed in materials such as single crystals, ceramics, polymers, composites, thin films and relaxor-type

Indeed, exciting a plate in certain conditions of vibratory amplitude and frequency the texture represented in Figure 1 e.g. can be simulated [5][6]. For that it suffices to correlate the vibratory amplitude with the level of friction reduction between an exploring finger and the textured plate. To complete the process, the control of the finger position over the plate enables textures feeling. In fact, during the touching of the excited surface, that is vibrating, the finger perceives smoothness. If the vibrations source is switched off a roughness is perceived. To reform a given texture, we can thus measure the finger displacement and control the vibratory amplitude as a function of the explored feeling (smoothness or roughness) and the finger position. This example is chosen to illustrate that controlled friction is an interesting alternative in terms of tactile feelings generation. The challenge is however in modelling the friction and to deduce

by such interfaces.

278 MATLAB Applications for the Practical Engineer

haptic feedback.

appropriate control laws.

Rough

Smooth

**Figure 1.** Image of the texture

structure.

principle specific to piezoelectric systems.

conversion principle specific to piezoelectric systems.

friction and to deduce appropriate control laws.

to deal with our structure as a finite state structure.

different resistant feelings when a user moves it.

ferroelectric materials.

**2. Piezoelectric materials contribution to transducers technology** 

transducers technology in general can draw from those materials.

In this section, piezoelectric materials such as lead zirconium titanate (PZT) utilised elsewhere in our design are briefly discussed. We focus on the benefits that transducers technology in general can draw from those materials.

Piezoelectric materials produce an electric charge when subjected to mechanical loads (direct effect) and/or vibrations. Those materials deform when subjected to a magnetic field (inverse effect) [8]. The piezoelectric effect is expressed in materials such as single crystals, ceramics, polymers, composites, thin films and relaxor-type ferroelectric materials.

Of the existing piezoelectric materials polymers and ceramics have been widely explored as transducers materials [8]. Ceramic piezoelectric materials comprise barium titanate (BaTiO3), lead zirconate – lead titanate of general formula Pb(Zr-Ti)O3-PZT and lead titanate (PbTiO3, PCT). Those materials have been extensively studied because most of them have in common a perovskite (ABO3) structure [9][8], a material in which the application of an intense electric field aimed at aligning polarization of elementary ferroelectric microcrystal (polarization operation) enables introduction of necessary anisotropy to piezoelectric existence. The dielectric and piezoelectric constants of barium titanate vary with temperature, stoichiometry, microstructure and doping [10] whereas the piezoelectric constants for PZT are not strongly dependant upon temperature but rather on material composition. However, piezoelectric constants for certain stoichiometry compositions of PZT are far more sensitive to temperature dependencies than others [11]. In [12] it can be seen that the characteristics of BaTiO3 single crystal favoured their usage in certain applications such as electromechanical transducers for operation at high frequencies. However, nowadays, PZT have been the material of reference in the field of piezoelectric motors. Regarding, lead titanate material, Samarium-modified PCT was investigated for application in infrared sensors, electro-optic devices and ferroelectric memory devices [13].

Polymer piezoelectric material such as polyvinylidene difluoride (PVDF) or PVF2 demonstrate the piezoelectric effect when they are stretched or formed during fabrication [10]. However, it is among composite piezoelectric materials made of a piezoelectric and a polymer that composites comprised of PZT rods embedded within a polymer matrix are predicted to be one of the most promising structures for transducers and acoustic applications [10].

Relaxor-type ferroelectric materials differ from traditional ferroelectric material because they have a broad phase transition from paraelectric to ferroelectric state, dielectric relaxation and weak remnant polarization. Indeed those materials have the ability to become polarized when subjected to applied electric field. Unlike in traditional ferroelectric materials, this can happen even if there is no permanent electric dipole that exists in the material. The result of removing the field is polarisation in the material returning to zero [14]. Single crystals of Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Zn1/3Nb2/3)O3 (PZN), PMN-PT and PZN-PT are currently under investigation for transducer technology because of their large coupling coefficients, large piezoelectric constants and high strain levels so far higher than other piezoelectric materials [15].

stimulation area [17][18]. From a technological point of view, tactile stimulation can be realised from different ways. Therefore, various actuation principles are discussed. So, in shape tactile displays (quasi static) actuators are of: electromagnetic, Shape Memory Alloy, pneumatic technology, etc. In a second group, vibrotactile matrices systems use piezoelectric and electromagnetic actuators. Finally, the third subset is made of vibrating systems with continuous structure and friction reduction generally

Electromagnetic (EM) actuators used in shape stimuli display are generally dc rotary actuators or electromagnets. They are generally bulky because of mechanisms involved and their miniaturisation is challenging. Consequently very few electromagnetic micro actuators are available in the market. Two examples are given

The FEELEX from University of Tsukuba is an interface actuated by rotary motors controlled in position and deforming a plane surface of 3 mm thick thanks to a rods

The FEELEX2 [19] allows a maximum rods displacement and force of respectively 18 mm and 1.1 kgf; such displacements and high forces are the main advantages of

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

http://dx.doi.org/10.5772/57569

281

(a) (b)

Figure 2 : Feelex2 a) principle of movement transformation b) interface [19]

A device similar to the one in this study was proposed by Schneider et al [20] who developed a common computer mouse to move on a steel mat, to which force feedback function is added by including in an electromagnet. In function of cursor position on the computer screen and effort needed a reference voltage is applied inducing therefore a magnetic field and then a continuous friction force depending upon the voltage value. The modified mouse is a Hewlett-Packard 5187-1556 (Figure 3). The maximum friction force obtained is 2.0 N which is still relatively high. However, a notable problem came from the localized magnetic force at the back of the mouse, causing a rotation around the magnet; a rotation which needed to be

A device similar to the one in this study was proposed by Schneider et al [20] who developed a common computer mouse to move on a steel mat, to which force feedback function is added by including in an electromagnet. In function of cursor position on the computer screen and effort needed a reference voltage is applied inducing therefore a magnetic field and then a continuous friction force depending upon the voltage value. The modified mouse is a Hewlett-Packard 5187-1556 (Figure 3). The maximum friction force obtained is 2.0 N which is still relatively high. However, a notable problem came from the localized magnetic force at the back of the mouse, causing a rotation around the

magnet; a rotation which needed to be counterbalanced.

**Figure 3.** Hewlett-Packard 5187-1556 modified mouse [20]

counterbalanced.

**Figure 2.** Feelex2 a) principle of movement transformation b) interface [19]

matrix. Each rod is actuated by a DC motor which the movement is transformed.

actuated using piezoelectric technology.

with a graphic illustration.

EM actuation.

3.1. Shape tactile displays (Quasi-static)

3.1.1. Electromagnetic actuators

#### **3. Tactile feedback interfaces overview**

As we have indicated in the introduction, piezoelectric technology is promising in haptics, restricted in this chapter to tactile feedback, but other alternatives have been exploited. Therefore several methods used for tactile or coetaneous feedback and several forms of technological proposals will be described. We chose to sort those proposals by highlighting related function and technology.

Until now, two methods of tactile feedback have mainly been proposed: shape tactile display and vibrotactile stimuli application. Shape tactile display appears to designers of tactile stimulators to reforming a material state of surface [16]. Conversely, tactile stimulation, as it is perceived actually deals with direct stimulation through vibrators or skin mechanoreceptors stimulation. Each stimulator has its own domain of validity and tactile devices are divided according to discrimination and space covered by stimulation area [17][18]. From a techno‐ logical point of view, tactile stimulation can be realised from different ways. Therefore, various actuation principles are discussed. So, in shape tactile displays (quasi static) actuators are of: electromagnetic, Shape Memory Alloy, pneumatic technology, etc. In a second group, vibrotactile matrices systems use piezoelectric and electromagnetic actuators. Finally, the third subset is made of vibrating systems with continuous structure and friction reduction generally actuated using piezoelectric technology.

#### **3.1. Shape tactile displays (Quasi-static)**

#### *3.1.1. Electromagnetic actuators*

Electromagnetic (EM) actuators used in shape stimuli display are generally DC rotary actuators or electromagnets. They are generally bulky because of mechanisms involved and their miniaturisation is challenging. Consequently very few electromagnetic micro actuators are available in the market. Two examples are given with a graphic illustration.

The FEELEX from University of Tsukuba is an interface actuated by rotary motors controlled in position and deforming a plane surface of 3 mm thick thanks to a rods matrix. Each rod is actuated by a DC motor which the movement is transformed.

The FEELEX2 [19] allows a maximum rods displacement and force of respectively 18 mm and 1.1 kgf; such displacements and high forces are the main advantages of EM actuation.

stimulation area [17][18]. From a technological point of view, tactile stimulation can be realised from different ways. Therefore, various actuation principles are discussed. So, in shape tactile displays (quasi static) actuators are of: electromagnetic, Shape Memory Alloy, pneumatic technology, etc. In a second group, vibrotactile matrices systems use piezoelectric and electromagnetic actuators. Finally, the third subset is made of vibrating systems with continuous structure and friction reduction generally

Electromagnetic (EM) actuators used in shape stimuli display are generally dc rotary actuators or electromagnets. They are generally bulky because of mechanisms involved and their miniaturisation is challenging. Consequently very few electromagnetic micro actuators are available in the market. Two examples are given

The FEELEX from University of Tsukuba is an interface actuated by rotary motors controlled in position and deforming a plane surface of 3 mm thick thanks to a rods

matrix. Each rod is actuated by a DC motor which the movement is transformed.

**Figure 2.** Feelex2 a) principle of movement transformation b) interface [19]

actuated using piezoelectric technology.

with a graphic illustration.

EM actuation.

3.1. Shape tactile displays (Quasi-static)

3.1.1. Electromagnetic actuators

result of removing the field is polarisation in the material returning to zero [14]. Single crystals of Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Zn1/3Nb2/3)O3 (PZN), PMN-PT and PZN-PT are currently under investigation for transducer technology because of their large coupling coefficients, large piezoelectric constants and high strain levels so far higher than other

As we have indicated in the introduction, piezoelectric technology is promising in haptics, restricted in this chapter to tactile feedback, but other alternatives have been exploited. Therefore several methods used for tactile or coetaneous feedback and several forms of technological proposals will be described. We chose to sort those proposals by highlighting

Until now, two methods of tactile feedback have mainly been proposed: shape tactile display and vibrotactile stimuli application. Shape tactile display appears to designers of tactile stimulators to reforming a material state of surface [16]. Conversely, tactile stimulation, as it is perceived actually deals with direct stimulation through vibrators or skin mechanoreceptors stimulation. Each stimulator has its own domain of validity and tactile devices are divided according to discrimination and space covered by stimulation area [17][18]. From a techno‐ logical point of view, tactile stimulation can be realised from different ways. Therefore, various actuation principles are discussed. So, in shape tactile displays (quasi static) actuators are of: electromagnetic, Shape Memory Alloy, pneumatic technology, etc. In a second group, vibrotactile matrices systems use piezoelectric and electromagnetic actuators. Finally, the third subset is made of vibrating systems with continuous structure and friction reduction generally

Electromagnetic (EM) actuators used in shape stimuli display are generally DC rotary actuators or electromagnets. They are generally bulky because of mechanisms involved and their miniaturisation is challenging. Consequently very few electromagnetic micro actuators

The FEELEX from University of Tsukuba is an interface actuated by rotary motors controlled in position and deforming a plane surface of 3 mm thick thanks to a rods matrix. Each rod is

The FEELEX2 [19] allows a maximum rods displacement and force of respectively 18 mm and

1.1 kgf; such displacements and high forces are the main advantages of EM actuation.

are available in the market. Two examples are given with a graphic illustration.

actuated by a DC motor which the movement is transformed.

piezoelectric materials [15].

280 MATLAB Applications for the Practical Engineer

related function and technology.

actuated using piezoelectric technology.

**3.1. Shape tactile displays (Quasi-static)**

*3.1.1. Electromagnetic actuators*

**3. Tactile feedback interfaces overview**

A device similar to the one in this study was proposed by Schneider et al [20] who developed a common computer mouse to move on a steel mat, to which force feedback function is added by including in an electromagnet. In function of cursor position on the computer screen and effort needed a reference voltage is applied inducing therefore a magnetic field and then a continuous friction force depending upon the voltage value. The modified mouse is a Hewlett-Packard 5187-1556 (Figure 3). The maximum friction force obtained is 2.0 N which is still relatively high. However, a notable problem came from the localized magnetic force at the back of the mouse, causing a rotation around the A device similar to the one in this study was proposed by Schneider et al [20] who developed a common computer mouse to move on a steel mat, to which force feedback function is added by including in an electromagnet. In function of cursor position on the computer screen and effort needed a reference voltage is applied inducing therefore a magnetic field and then a continuous friction force depending upon the voltage value. The modified mouse is a Hewlett-Packard 5187-1556 (Figure 3). The maximum friction force obtained is 2.0 N which is still relatively high. However, a notable problem came from the localized magnetic force at the back of the mouse, causing a rotation around the magnet; a rotation which needed to be counterbalanced.

Figure 2 : Feelex2 a) principle of movement transformation b) interface [19]

**Figure 3.** Hewlett-Packard 5187-1556 modified mouse [20]

#### *3.1.2. Shape Memory Alloys (SMA)*

SMA are alloys of tremendous properties among metal materials wherein the capability to "keep in memory" an initial shape and to get back even after a deformation. Usually SMA follows a plastic deformation at relatively low temperatures (Martensite) and gets back their original shape (Austenite) if they are heated at high temperature.

ent skin mechanoreceptor populations. Conceptually, electromagnetic and piezoelectric

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283

The devices belonging to the family to be discussed in next section are based on electro mechanical conversion principle specific to piezoelectric systems. That principle will be used

In order to create shape displays or vibro tactile feelings, the technologies described above generally have the particularity to communicate the explored feeling to the user finger through a rods matrix. The advantage of that rods matrix structure is to allow refinement in control (each rod being independently controllable) but the structure is limited in integration. Conversely, some other devices present a continuous structure producing a pulse or a

Using this principle, the impulse display proposed by Poupyrev et al [27] was able to be

The action of friction generated by a surface under the finger is a second alternative exploited in continuous structure tactile interfaces. Watanabee [28] pioneered friction coefficient adjustment. Let's describe Watanabee experience briefly for the slot it opened in the design of number of structures in this category. Watanabee used a steel beam which one end was attached to a Langevin transducer [29]. The Langevin transducer excited at 77 kHz commu‐ nicates its maximum 2 μm vibrations to the beam. As a result, the feeling procured to a finger that explored the surface of the non excited beam was different from the feeling obtained with the vibrating beam: in the latter case the surface is very slippery and smooth. Watanabee also observed that as long as the vibratory frequency is grater than 20 kHz it has no influence on

In the follow up T. Nara [30] proposed a tapered plate and as in the introduction the idea

From the description of the later devices it can be seen that controlled friction is an interesting alternative in terms of tactile feelings production. Controlled friction is generated by contin‐ uous surfaces, limited in size and hence fully inerrable. In addition the control of these devices is rather global since it is the entire structure that is excited not a matrix rods. The accurate knowledge of the finger position is necessary and texture to be explored have to be processed

in terms of friction coefficient in order to provide reference inputs to the effecter.

inspired among others the designs presented in [2] and [3].

incorporated on the sensitive screen of a PDA (Personal Digital Assistant).

In this category the Vital [17] is an example of electromagnetic system.

**3.3. Continuous structure based vibro tactile systems**

technologies are resorted to.

in the rest of this work.

controllable friction under the finger.

*3.4.1. Variable friction creation principle*

**3.4. Variable friction devices**

the perceived feeling.

For teleoperation and virtual reality applications, researchers from Harvard University have developed a tactile prototype [21] made of one line of rods.

Reduced space between rods, significant force and roughness developed, and displacements amplitude make up SMA technologically adapted to tactile feedback. However, SMA are not so often used because of their relatively long response time and their integration remains challenging.

#### *3.1.3. Pneumatic technology*

Apart from air bladders inserted in the gloves of TeleTact Glove and filled up by a compressor, other devices exist. That is the case for pistons actuated by motors [22] allowing the feeling of a variable roughness membrane, the case as well of devices using a pump to expel [23] or aspire some air by means of binary electromagnet micro valves [24]. Although those devices from pneumatic technology can generate high forces, they are not always comfortable and are relatively heavy.

#### *3.1.4. Electro-Rheologic Fluids (ERF)*

A matrix of bladders full of ERF can allow space distribution of normal forces on the finger pulp that pre constrains the device then. Indeed, a local change of fluid viscosity by electric field application yields a variable roughness under the finger pulp. It turns out that some progress in terms of precision is necessary [25].

#### *3.1.5. Other technologies*

In this field of application, research groups have investigated other solutions such as MEMS (Micro-Electro Mechanical Systems) [26] and active polymers but developments are still marginal.

#### **3.2. Vibro tactile matrices systems**

The chosen approach in this sub section differs from the latter one because this time it is the vibro tactile stimuli application that generates the prospected feeling. Previously the prospected effect was obtained by means of normal indentation of the skin according to shape reconstitution usually related to discrete representation of the state of surface or 3D explored asperity. High amplitude and low frequency of rods displacement feature those structures whereas in vibro tactile stimulators rods have a high frequency (around 200 Hz) and very low amplitude (around 10 μm) displacements. "Reproducing" surface asperities is no longer the aim but conversely, producing a sort of appropriate excitation on differ‐ ent skin mechanoreceptor populations. Conceptually, electromagnetic and piezoelectric technologies are resorted to.

In this category the Vital [17] is an example of electromagnetic system.

The devices belonging to the family to be discussed in next section are based on electro mechanical conversion principle specific to piezoelectric systems. That principle will be used in the rest of this work.

#### **3.3. Continuous structure based vibro tactile systems**

In order to create shape displays or vibro tactile feelings, the technologies described above generally have the particularity to communicate the explored feeling to the user finger through a rods matrix. The advantage of that rods matrix structure is to allow refinement in control (each rod being independently controllable) but the structure is limited in integration. Conversely, some other devices present a continuous structure producing a pulse or a controllable friction under the finger.

Using this principle, the impulse display proposed by Poupyrev et al [27] was able to be incorporated on the sensitive screen of a PDA (Personal Digital Assistant).

#### **3.4. Variable friction devices**

*3.1.2. Shape Memory Alloys (SMA)*

282 MATLAB Applications for the Practical Engineer

challenging.

relatively heavy.

*3.1.5. Other technologies*

marginal.

*3.1.3. Pneumatic technology*

*3.1.4. Electro-Rheologic Fluids (ERF)*

**3.2. Vibro tactile matrices systems**

progress in terms of precision is necessary [25].

SMA are alloys of tremendous properties among metal materials wherein the capability to "keep in memory" an initial shape and to get back even after a deformation. Usually SMA follows a plastic deformation at relatively low temperatures (Martensite) and gets back their

For teleoperation and virtual reality applications, researchers from Harvard University have

Reduced space between rods, significant force and roughness developed, and displacements amplitude make up SMA technologically adapted to tactile feedback. However, SMA are not so often used because of their relatively long response time and their integration remains

Apart from air bladders inserted in the gloves of TeleTact Glove and filled up by a compressor, other devices exist. That is the case for pistons actuated by motors [22] allowing the feeling of a variable roughness membrane, the case as well of devices using a pump to expel [23] or aspire some air by means of binary electromagnet micro valves [24]. Although those devices from pneumatic technology can generate high forces, they are not always comfortable and are

A matrix of bladders full of ERF can allow space distribution of normal forces on the finger pulp that pre constrains the device then. Indeed, a local change of fluid viscosity by electric field application yields a variable roughness under the finger pulp. It turns out that some

In this field of application, research groups have investigated other solutions such as MEMS (Micro-Electro Mechanical Systems) [26] and active polymers but developments are still

The chosen approach in this sub section differs from the latter one because this time it is the vibro tactile stimuli application that generates the prospected feeling. Previously the prospected effect was obtained by means of normal indentation of the skin according to shape reconstitution usually related to discrete representation of the state of surface or 3D explored asperity. High amplitude and low frequency of rods displacement feature those structures whereas in vibro tactile stimulators rods have a high frequency (around 200 Hz) and very low amplitude (around 10 μm) displacements. "Reproducing" surface asperities is no longer the aim but conversely, producing a sort of appropriate excitation on differ‐

original shape (Austenite) if they are heated at high temperature.

developed a tactile prototype [21] made of one line of rods.

#### *3.4.1. Variable friction creation principle*

The action of friction generated by a surface under the finger is a second alternative exploited in continuous structure tactile interfaces. Watanabee [28] pioneered friction coefficient adjustment. Let's describe Watanabee experience briefly for the slot it opened in the design of number of structures in this category. Watanabee used a steel beam which one end was attached to a Langevin transducer [29]. The Langevin transducer excited at 77 kHz commu‐ nicates its maximum 2 μm vibrations to the beam. As a result, the feeling procured to a finger that explored the surface of the non excited beam was different from the feeling obtained with the vibrating beam: in the latter case the surface is very slippery and smooth. Watanabee also observed that as long as the vibratory frequency is grater than 20 kHz it has no influence on the perceived feeling.

In the follow up T. Nara [30] proposed a tapered plate and as in the introduction the idea inspired among others the designs presented in [2] and [3].

From the description of the later devices it can be seen that controlled friction is an interesting alternative in terms of tactile feelings production. Controlled friction is generated by contin‐ uous surfaces, limited in size and hence fully inerrable. In addition the control of these devices is rather global since it is the entire structure that is excited not a matrix rods. The accurate knowledge of the finger position is necessary and texture to be explored have to be processed in terms of friction coefficient in order to provide reference inputs to the effecter.

#### **4. The system**

#### **4.1. Description**

Figure 4 shows the proposed structure; a resonant physical device that in this specific case is a piezoelectric transducer converting electrical energy into mechanical energy. Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezoceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to move by itself along the tangential direction. Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezo-ceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezo-ceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to

In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the

feet

Figure 4 : Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location

The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt in Figure 7 a) acting at the interface feet -

When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to amplitude vibration. As a matter of fact, from a given wave amplitude, an intermittent contact may occur at the interface. Consequently, at the feet base, transitions between

wavelength/4

antinode

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285

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

x

node

z

substrate

Figure 5 : Standing wave and foot trajectory

Many researchers studied stick-slip vibrations with switch models which Leine et al. [7] claim the solution they proposed is an improved version. The model proposed by Leine treats the system as three different sets of ordinary differential equations (ODE): one for the slip phase, a second for the stick phase and a third for the transition from stick to slip. Restricted to its normal movement component, our system can be seen as the "stick-slip" defined above with the "same" (contact, separation and transition from contact to separation) three states. The problem is solved using Simulink-Stateflow®, a convenient tool for finite state machine simulation and control: here is all the interest that is to see how it provides with a "switch block" to deal with the transition phase source of some difficulties including dealing with equation for stick to slip transition in the pseudo code used by Leine et al., sensitivity analysis,

The structure in study involves contact and separation sequences between the feet of a body and the floor or substrate: contact and separation are induced by the foot elasticity. The movement so defined on the component normal to the plane of contact is broken down into compression – relaxation – separation sequences. This approach identifies the feet in contact

The particular surface topology required by the device, that is a substrate highly rigid and a minimal surface roughness (*Ra* ≤0.6*μm*), the low clearances of the effecter tip in the range of a

foot

move by itself along the tangential direction.

structure voltage supply that is kept unbroken.

a) b)

4.2. Working principle

stick or slip conditions are created.

**Figure 5.** Standing wave and foot trajectory

**4.3. Modelling**

etc, as developed in [7].

*4.3.1. Feet-plan Contact model*

with the floor to a mass-spring system.

few micrometers make the approximation a priori acceptable.

The model will enable characterizing the contact intermittence.

substrate.

V


V

"+"

In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the structure voltage supply that is kept unbroken. move by itself along the tangential direction. In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the

structure voltage supply that is kept unbroken.

Figure 4 : Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location **Figure 4.** Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location

#### 4.2. Working principle **4.2. Working principle**

The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt in Figure 7 a) acting at the interface feet substrate. The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt in Figure 7 a) acting at the interface feet-substrate.

When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to amplitude vibration. As a matter of fact, from a given wave amplitude, an intermittent contact may occur at the interface. Consequently, at the feet base, transitions between stick or slip conditions are created. When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to amplitude vibration. As a matter of fact, from a given wave amplitude, an intermittent contact may occur at the interface. Consequently, at the feet base, transitions between stick or slip conditions are created.

antinode

x

node

z

substrate

Figure 5 : Standing wave and foot trajectory

foot

wavelength/4

Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezo-ceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to

In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the

feet

Figure 4 : Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location

The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt in Figure 7 a) acting at the interface feet -

When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to

move by itself along the tangential direction.

structure voltage supply that is kept unbroken.

a) b)

4.2. Working principle

stick or slip conditions are created.

substrate.

V


V

"+"

Figure 5 : Standing wave and foot trajectory **Figure 5.** Standing wave and foot trajectory

#### **4.3. Modelling**

**4. The system**

284 MATLAB Applications for the Practical Engineer

**4.1. Description**

itself along the tangential direction.

4.2. Working principle

stick or slip conditions are created.

in Figure 7 a) acting at the interface feet-substrate.

substrate.

move by itself along the tangential direction.

structure voltage supply that is kept unbroken.

a) b)

supply that is kept unbroken.

V

**4.2. Working principle**


V

Figure 4 shows the proposed structure; a resonant physical device that in this specific case is a piezoelectric transducer converting electrical energy into mechanical energy. Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezoceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to move by

Driven in a simple bending mode of vibration, the structure consists of a set of PZT polarised piezo-ceramics of 12x12x1mm glued on the upside of a cupper-beryllium substrate whose size is 64x38x3 mm. On the opposite side (Figure 1), four built-in feet support the plate. Considering this polarisation, piezo-ceramic electrodes are conveniently supplied by a sinusoidal voltage of some ten Volts to create a standing wave using the piezoelectricity inverse effect with 40.7 kHz driving frequency (resonant frequency). Earlier some studies have been carried out using a system closed to this structure [31][32]. The main difference is that this plate will only move normally. Indeed, as we will see later, this particular structure is not supposed to

In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the structure voltage

"+"

In Figure 4 b), alongside the four feet appears also a measurement ceramic glued on the plate. That flat round ceramic is acting as a vibratory sensor without altering the

feet

Figure 4 : Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location

**Figure 4.** Up and down view of the actuator: a) electrical connection b) four feet and measurement ceramic location

The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt

When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to amplitude vibration. As a matter of fact, from a given wave amplitude, an intermittent contact may occur at the interface. Consequently, at the feet base, transitions between stick or slip conditions are created.

The feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force (Rt in Figure 7 a) acting at the interface feet -

When voltage is applied to the ceramic electrodes, a standing wave is generated and the friction between feet and substrate is decreased: this happens according to amplitude vibration. As a matter of fact, from a given wave amplitude, an intermittent contact may occur at the interface. Consequently, at the feet base, transitions between

wavelength/4

antinode

x

node

z

substrate

Figure 5 : Standing wave and foot trajectory

foot

Many researchers studied stick-slip vibrations with switch models which Leine et al. [7] claim the solution they proposed is an improved version. The model proposed by Leine treats the system as three different sets of ordinary differential equations (ODE): one for the slip phase, a second for the stick phase and a third for the transition from stick to slip. Restricted to its normal movement component, our system can be seen as the "stick-slip" defined above with the "same" (contact, separation and transition from contact to separation) three states. The problem is solved using Simulink-Stateflow®, a convenient tool for finite state machine simulation and control: here is all the interest that is to see how it provides with a "switch block" to deal with the transition phase source of some difficulties including dealing with equation for stick to slip transition in the pseudo code used by Leine et al., sensitivity analysis, etc, as developed in [7].

#### *4.3.1. Feet-plan Contact model*

The structure in study involves contact and separation sequences between the feet of a body and the floor or substrate: contact and separation are induced by the foot elasticity. The movement so defined on the component normal to the plane of contact is broken down into compression – relaxation – separation sequences. This approach identifies the feet in contact with the floor to a mass-spring system.

The particular surface topology required by the device, that is a substrate highly rigid and a minimal surface roughness (*Ra* ≤0.6*μm*), the low clearances of the effecter tip in the range of a few micrometers make the approximation a priori acceptable.

The model will enable characterizing the contact intermittence.

#### *4.3.1.1. Foot mass – spring model*

From the plate which the kinetic is briefly reminded here, a refined analysis of the bond up effecter can be carried out. For sake of convenience, figure 6 shows the plate kinetic diagram in an *Oxyz* coordinates system. For a plate constrained in pure bending mode as in figure 6, more precisely a plate in vibratory mode (0,6) as in figure 5 with feet located at λ/4, it can be shown [33] [34][35] that:

$$\text{tr}\_{\text{A}}(\mathbf{u}, \mathbf{v}, t) = \mathbf{w}(t) = \mathcal{W}(t)\sin(\alpha t) \tag{1}$$

Fn due to Mext (*Fn=9.81Mext*) is constant. The mass of the vibrating plate is denoted m and the number of feet, n. The foot mass mf is too low to be considered and its normal stiffness is kn. Finally the displacement wA(t) is imposed by the plate vibrations whereas Rn denotes the

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

A

z

Figure 7: a) equivalent mechanical scheme and forces acting on the foot b) system

Describing the behaviour of actuator along the normal axis Oz amounts to writing the foot tip equation of movement characterizing the induced separation and contact periods along that Oz axis. The separation phase is also called flight. Equation obtained by applying the general dynamic laws to the system assuming the substrate is ideally rigid and therefore only the foot is storage element of potential energy,

Describing the behaviour of actuator along the normal axis *Oz* amounts to writing the foot tip equation of movement characterizing the induced separation and contact periods along that *Oz* axis. The separation phase is also called flight. Equation obtained by applying the general dynamic laws to the system assuming the substrate is ideally rigid and therefore only the foot

)( .. ..

n F

*mm m F*

*n n nn*

*n*

w n m

.. ..

z n m

*mm m F*

*n n nn*

*z w g Rt*

.. ..

is storage element of potential energy, which yields:

*4.3.1.2. Actuator behaviour along the normal axis*

n m

<sup>A</sup> <sup>A</sup> ( ) *<sup>n</sup>*

<sup>A</sup> <sup>A</sup> ( ) durin , g the contact phase *<sup>n</sup>*

*z w g Rt*

g R t

.

= -- + (3)

( ) 0, during the separation a ph se *R t <sup>n</sup>* = (4)

n

*n*

<sup>n</sup> <sup>A</sup> <sup>A</sup> = − − + (2)

= -- + (2)

x

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287

)( ( ))( <sup>n</sup> <sup>n</sup> <sup>A</sup> <sup>n</sup> <sup>G</sup> R t = k h − z t − d z , during the contact phase (3)

<sup>R</sup> <sup>t</sup>)( <sup>=</sup> <sup>0</sup> <sup>n</sup> , during the separation phase (4)

4.3.1.2. Actuator behaviour along the normal axis

x

That yields the equivalent mechanical diagram used to define the system dynamic

Each foot can now on be described as a mass-spring system and we represented it in Figure 7 a. In Figure 7 a, Mext depicts the load applied on the top of the device to assume pre-stress. This load lies on an elastic element whose stiffness km is low enough to assume that the force Fn due to Mext (Fn=9.81Mext) is constant. The mass of the vibrating plate is denoted m and the number of feet, n. The foot mass mf is too low to be considered and its normal stiffness is kn. Finally the displacement wA(t) is imposed by the plate vibrations whereas Rn denotes the normal reaction force at the

It is recalled in the preceding that only the displacement w is considered. Also, the feet location which the tip is at antinode (λ/4) is assumed constant because of the pure bending mode assumption. In addition, from the similar feet positioning with respect to the wave, the mechanical study may be restricted to that of one foot subjected to a pre stress Fn equally distributed upon n feet. Moreover, external loads are reported to the plate partial centre of gravity G. Conversely, the plate has four feet, making the problem hyperstatic because of many number of unknown contact variables. Elsewhere, planarity of the contact surface may contribute to invalidate the load equal distribution upon the feet. Nevertheless, considering the global contact approach, the

normal reaction force at the foot tip.

Mext

 

Rt Oo

Rn

m

W

z

a) b)

**Figure 7.** a) equivalent mechanical scheme and forces acting on the foot b) system

mf

kn

G u WA(t)

km

A

load equal repartition is retained.

relations.

foot tip.

which yields:

with

with

A is the foot end located at intersection of the foot and the plate.

**Figure 6.** Kinematic of plate deformation

W(t) is the dynamic vibration amplitude at the actuator centre and ω the vibratory mode pulsation.

It is recalled in the preceding that only the displacement w is considered. Also, the feet location which the tip is at antinode (λ/4) is assumed constant because of the pure bending mode assumption. In addition, from the similar feet positioning with respect to the wave, the mechanical study may be restricted to that of one foot subjected to a pre stress Fn equally distributed upon n feet. Moreover, external loads are reported to the plate partial centre of gravity G. Conversely, the plate has four feet, making the problem hyperstatic because of many number of unknown contact variables. Elsewhere, planarity of the contact surface may contribute to invalidate the load equal distribution upon the feet. Nevertheless, considering the global contact approach, the load equal repartition is retained.

That yields the equivalent mechanical diagram used to define the system dynamic relations.

Each foot can now on be described as a mass-spring system and we represented it in Figure 7 a. In Figure 7 a, Mext depicts the load applied on the top of the device to assume pre-stress. This load lies on an elastic element whose stiffness km is low enough to assume that the force

Fn due to Mext (*Fn=9.81Mext*) is constant. The mass of the vibrating plate is denoted m and the number of feet, n. The foot mass mf is too low to be considered and its normal stiffness is kn. Finally the displacement wA(t) is imposed by the plate vibrations whereas Rn denotes the normal reaction force at the foot tip. enough to assume that the force Fn due to Mext (Fn=9.81Mext) is constant. The mass of the vibrating plate is denoted m and the number of feet, n. The foot mass mf is too low to be considered and its normal stiffness is kn. Finally the displacement wA(t) is imposed by the plate vibrations whereas Rn denotes the normal reaction force at the foot tip.

That yields the equivalent mechanical diagram used to define the system dynamic

Each foot can now on be described as a mass-spring system and we represented it in

It is recalled in the preceding that only the displacement w is considered. Also, the feet location which the tip is at antinode (λ/4) is assumed constant because of the pure bending mode assumption. In addition, from the similar feet positioning with respect to the wave, the mechanical study may be restricted to that of one foot subjected to a pre stress Fn equally distributed upon n feet. Moreover, external loads are reported to the plate partial centre of gravity G. Conversely, the plate has four feet, making the problem hyperstatic because of many number of unknown contact variables. Elsewhere, planarity of the contact surface may contribute to invalidate the load equal distribution upon the feet. Nevertheless, considering the global contact approach, the

Figure 7: a) equivalent mechanical scheme and forces acting on the foot b) system **Figure 7.** a) equivalent mechanical scheme and forces acting on the foot b) system

#### 4.3.1.2. Actuator behaviour along the normal axis *4.3.1.2. Actuator behaviour along the normal axis*

z n m

load equal repartition is retained.

relations.

Describing the behaviour of actuator along the normal axis Oz amounts to writing the foot tip equation of movement characterizing the induced separation and contact periods along that Oz axis. The separation phase is also called flight. Equation obtained by applying the general dynamic laws to the system assuming the substrate is ideally rigid and therefore only the foot is storage element of potential energy, Describing the behaviour of actuator along the normal axis *Oz* amounts to writing the foot tip equation of movement characterizing the induced separation and contact periods along that *Oz* axis. The separation phase is also called flight. Equation obtained by applying the general dynamic laws to the system assuming the substrate is ideally rigid and therefore only the foot is storage element of potential energy, which yields:

$$\frac{m}{m}\overline{z}\_{\text{A}} = \frac{m}{n}\overline{w}\_{\text{A}} - \frac{F\_n}{n} - \frac{m}{n}\mathbf{g} + R\_n(t) \tag{2}$$

)( ( ))( <sup>n</sup> <sup>n</sup> <sup>A</sup> <sup>n</sup> <sup>G</sup> R t = k h − z t − d z , during the contact phase (3)

with

which yields:

with

*4.3.1.1. Foot mass – spring model*

286 MATLAB Applications for the Practical Engineer

shown [33] [34][35] that:

**Figure 6.** Kinematic of plate deformation

pulsation.

From the plate which the kinetic is briefly reminded here, a refined analysis of the bond up effecter can be carried out. For sake of convenience, figure 6 shows the plate kinetic diagram in an *Oxyz* coordinates system. For a plate constrained in pure bending mode as in figure 6, more precisely a plate in vibratory mode (0,6) as in figure 5 with feet located at λ/4, it can be

<sup>A</sup> *w u t wt Wt t* ( ,v, ) ( ) ( )sin( ) = =

W(t) is the dynamic vibration amplitude at the actuator centre and ω the vibratory mode

It is recalled in the preceding that only the displacement w is considered. Also, the feet location which the tip is at antinode (λ/4) is assumed constant because of the pure bending mode assumption. In addition, from the similar feet positioning with respect to the wave, the mechanical study may be restricted to that of one foot subjected to a pre stress Fn equally distributed upon n feet. Moreover, external loads are reported to the plate partial centre of gravity G. Conversely, the plate has four feet, making the problem hyperstatic because of many number of unknown contact variables. Elsewhere, planarity of the contact surface may contribute to invalidate the load equal distribution upon the feet. Nevertheless, considering

That yields the equivalent mechanical diagram used to define the system dynamic relations.

Each foot can now on be described as a mass-spring system and we represented it in Figure 7 a. In Figure 7 a, Mext depicts the load applied on the top of the device to assume pre-stress. This load lies on an elastic element whose stiffness km is low enough to assume that the force

A is the foot end located at intersection of the foot and the plate.

the global contact approach, the load equal repartition is retained.

w

(1)

$$\frac{m}{m}\overset{\cdot}{\text{m}}\_{\text{A}} = \frac{m}{n}\overset{\cdot}{\text{w}}\_{\text{A}} - \frac{F\_{\text{n}}}{n} - \frac{m}{n}\text{g} + R\_{\text{n}}(\text{t}), \qquad \text{during the contact phase} \tag{3}$$

.

$$R\_n(t) = 0,\qquad\text{during the separation phase}\tag{4}$$

Where kn is the elasticity of the foot, h the height of the foot when no pressure is exerted on it and dn a damping coefficient on the main compression of the foot induced by variation of zG.

Prior to solving equation (2) let us examine the detail of transition conditions from contact to separation phase. At rest, under external mass Mext, the plate plane is located from the ground at a distance zG lesser than h, the height of relaxed/released foot: the foot is compressed. Vibrating, the plate imposes a sinusoidal normal displacement of the foot end zA. In the instance of a displacement large enough for the ordinate zA to be greater than the height h, there is separation. It is underlined that for kn a priori high, eventual longitudinal vibrations of the foot are neglected during the separation. The following transition condition results:

$$z\_{\mathbf{A}} > h\_{\prime} \qquad \text{foot} \newline \text{in separation} \tag{5}$$

The set of these two equations may mathematically describe the system. The solutions of the piecewise ODE are of course function of the initial conditions and the focus is only on the steady state in presence of an intermittent contact; without the intermittence we are in the instance of a spring which one end is fixed while to the other end is attached a mass in sinusoidal motion. It is assumed that the first phase is a contact phase governed by equation (11) and the final conditions of the contact phase (*zA* >*h* ) are the initial conditions of the

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289

The structure requires taking into account vibratory phenomena alternating transient-steadytransient states. For an arbitrary value of vibratory amplitude high enough to induce inter‐ mittent contact, that behaviour is graphically illustrated in Figure 8 where tci denotes the start up of the ith phase arising after a transient state of the periodic phenomenon described by Equation 11 of period T. tc(i+1) represents the beginning of the (i+1)th phenomenon described by Equation 11 of period T. t contact phase. c(i+1) represents the beginning of

Figure 8 : Contact-separation-contact sequence and post tsi detail

Bearing in mind that the aim is to compute the normal reaction Rn, it is worth the while to look at the set of ODE implementation. Figure 9 shows the Simulink implementation of equation (10). The subsystem that is a self defined Simulink system uses the Constant, Sum, Integrator, outport and Gain templates taken from the linear block library of Simulink. The first summation generates the quantity in the right hand side of equation (10). The later sum is divided by the gain "1/n" and then integrated twice to obtain zm from which zG is derived. The reason for integrators chain (two integrators) is because we are dealing with a second order equation.

Bearing in mind that the aim is to compute the normal reaction Rn, it is worth the while to look at the set of ODE implementation. Figure 9 shows the Simulink implementation of equation (10). The subsystem that is a self defined Simulink system uses the Constant, Sum, Integrator, Outport and Gain templates taken from the linear block library of Simulink. The first summa‐ tion generates the quantity in the right hand side of equation (10). The later sum is divided by the gain "1/n" and then integrated twice to obtain zm from which zG is derived. The reason for integrators chain (two integrators) is because we are dealing with a second order equation.

Of course, the integrators must be initialized to correspond to initial values; thus in this specific case, the initial value of first integrator is set to 0 according to the system initial conditions (null foot speed). The initial value of the second integrator is set to h

Of course, the integrators must be initialized to correspond to initial values; thus in this specific case, the initial value of first integrator is set to 0 according to the system initial conditions (null foot speed). The initial value of the second integrator is set to h according to the initial

Figure 9 : Implementation of equation (10)

To proceed further, we use the Simulink system shown in Figure 10. The distinctive feature of this system is that it contains an algebraic loop and a Stateflow® chart.

z twice to get z.

twice to get z.

separation phase (*zA* ≤*h* ) governed by equation (12) and so on.

the (i+1)th contact phase.

Therefore we integrate ..

Therefore we integrate *z*

conditions.

according to the initial conditions.

A second sum is used to get zA as in equation (7).

A second sum is used to get zA as in equation (7).

**Figure 8.** Contact-separation-contact sequence and post tsi detail

..

$$z\_{\mathbf{A}} \le h\_{\prime} \qquad \text{foot in contact} \tag{6}$$

Taking

$$
\omega\_{\mathbf{A}} = \mathbf{z}\_{\mathbf{G}} - \mathbf{w}\_{\mathbf{A}'} \tag{7}
$$

yields

$$(z\_{\mathbb{G}} + w\_{\mathbb{A}}) \le h : \quad \text{contact} \tag{8}$$

$$(z\_{\mathcal{G}} + w\_{\mathcal{A}}) > h \colon \quad \text{separation} \tag{9}$$

Taking into account equation (7), equation (2) may be rewritten:

$$\frac{m}{m}\frac{m}{n} = -\frac{F\_n}{n} - \frac{m}{n}g + R\_n(t) \tag{10}$$

or after rewriting equation (2):

$$\frac{m}{n}\ddot{z}\_{\text{G}} + d\_{n}\dot{z}\_{\text{G}} + k\_{n}z\_{\text{A}} = -\frac{F\_{n}}{n} - \frac{m}{n}g + k\_{n}h \quad \text{If there is contact} \tag{11}$$

$$\frac{m}{n}\frac{m}{n}\overline{z}\_G = -\frac{F\_n}{n} - \frac{m}{n}G.\qquad\text{If there is separation}\tag{12}$$

The set of these two equations may mathematically describe the system. The solutions of the piecewise ODE are of course function of the initial conditions and the focus is only on the steady state in presence of an intermittent contact; without the intermittence we are in the instance of a spring which one end is fixed while to the other end is attached a mass in sinusoidal motion. It is assumed that the first phase is a contact phase governed by equation (11) and the final conditions of the contact phase (*zA* >*h* ) are the initial conditions of the separation phase (*zA* ≤*h* ) governed by equation (12) and so on.

The structure requires taking into account vibratory phenomena alternating transient-steadytransient states. For an arbitrary value of vibratory amplitude high enough to induce inter‐ mittent contact, that behaviour is graphically illustrated in Figure 8 where tci denotes the start up of the ith phase arising after a transient state of the periodic phenomenon described by Equation 11 of period T. tc(i+1) represents the beginning of the (i+1)th phenomenon described by Equation 11 of period T. t contact phase. c(i+1) represents the beginning of

**Figure 8.** Contact-separation-contact sequence and post tsi detail

Bearing in mind that the aim is to compute the normal reaction Rn, it is worth the while to look at the set of ODE implementation. Figure 9 shows the Simulink implementation of equation (10). The subsystem that is a self defined Simulink system uses the Constant, Sum, Integrator, outport and Gain templates taken from the linear block library of Simulink. The first summation generates the quantity in the right hand side of equation (10). The later sum is divided by the gain "1/n" and then integrated twice to obtain zm from which zG is derived. The reason for integrators chain (two integrators) is because we are dealing with a second order equation. Bearing in mind that the aim is to compute the normal reaction Rn, it is worth the while to look at the set of ODE implementation. Figure 9 shows the Simulink implementation of equation (10). The subsystem that is a self defined Simulink system uses the Constant, Sum, Integrator, Outport and Gain templates taken from the linear block library of Simulink. The first summa‐ tion generates the quantity in the right hand side of equation (10). The later sum is divided by the gain "1/n" and then integrated twice to obtain zm from which zG is derived. The reason for integrators chain (two integrators) is because we are dealing with a second order equation.

Figure 8 : Contact-separation-contact sequence and post tsi detail

Therefore we integrate .. z twice to get z. Therefore we integrate *z* .. twice to get z.

the (i+1)th contact phase.

Where kn is the elasticity of the foot, h the height of the foot when no pressure is exerted on it and dn a damping coefficient on the main compression of the foot induced by variation of zG.

Prior to solving equation (2) let us examine the detail of transition conditions from contact to separation phase. At rest, under external mass Mext, the plate plane is located from the ground at a distance zG lesser than h, the height of relaxed/released foot: the foot is compressed. Vibrating, the plate imposes a sinusoidal normal displacement of the foot end zA. In the instance of a displacement large enough for the ordinate zA to be greater than the height h, there is separation. It is underlined that for kn a priori high, eventual longitudinal vibrations of the foot are neglected during the separation. The following transition condition results:

Taking

yields

<sup>A</sup>*z h* > , foot in separation (5)

<sup>A</sup>*z h* £ , foot in contact (6)

A A , *<sup>G</sup> z zw* = - (7)

<sup>A</sup> ( ): t contac *Gzw h* + £ (8)

<sup>A</sup> ( ) : on separati *Gzw h* + > (9)

=- - + (10)

Taking into account equation (7), equation (2) may be rewritten:

or after rewriting equation (2):

288 MATLAB Applications for the Practical Engineer

.. .

..

*G*

*m m <sup>F</sup> z G*

..

*G nG n n m m <sup>F</sup> z d z kz g kh n n n*

*m m F*

*n nn*

( ) *<sup>n</sup> G n*

<sup>A</sup> If there is contact *<sup>n</sup>*

. If there is separati no *<sup>n</sup>*

+ + =- - + (11)

*n nn* =- - (12)

*z g Rt*

Of course, the integrators must be initialized to correspond to initial values; thus in this specific case, the initial value of first integrator is set to 0 according to the system initial conditions (null foot speed). The initial value of the second integrator is set to h according to the initial conditions. Of course, the integrators must be initialized to correspond to initial values; thus in this specific case, the initial value of first integrator is set to 0 according to the system initial conditions (null foot speed). The initial value of the second integrator is set to h according to the initial conditions.

Figure 9 : Implementation of equation (10)

To proceed further, we use the Simulink system shown in Figure 10. The distinctive feature of this system is that it contains an algebraic loop and a Stateflow® chart.

A second sum is used to get zA as in equation (7). A second sum is used to get zA as in equation (7).

In Stateflow® representations states and transitions form the basic building blocks of the

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291

Stateflow® block is used within the Simulink model here to dynamically simulate the system state changes. The chart block where the system is modelled is open in Figure 11. The device has two states: contact and separation that are represented by a rectangular block and named accordingly. Transition lines indicate the next state that the system in the current state can transit to. According to our algorithm these lines are from contact to fly and vice versa. Also, a default transition is assigned to a default or very first state, the state machine has to be in when it starts, chosen to be the "contact" state. In this case, the first state will be "contact" when

Actions to be taken when entering each state are defined. In this case, this is fly ( *flight* =1) or its logic complement, not fly ( *flight* =0) for contact and flight state respectively. For the Simulink model it is the machine output (0 or 1) that is present at the Stateflow® chart output

Conversely, let us notice the chart block input port set to control the Stateflow® chart that "sees" (*h* − *zA*) component from Simulink model. This is the "flight\_cond" standing for flight

Next, at its input 2, from Stateflow® chart, a switch block compares the machine output (0 or 1) to a threshold chosen to have the arbitrary value 0.5 as criterion for the switch to pass either

.

the 0 value. One or the other later amount will ultimately output the switch yielding the Rn(t) (see equation (11) and (12)) of the subsystem output. In every iteration step, Simulink will try

to bring the algebraic loop mentioned earlier which involves Rn "into balance".

is present or through switch input 3 holding

port. Entry command will be executed when entering the state.

system.

the execution begins.

condition associated to transition lines.

through switch input 1 where *kn*(*h* − *zA*(*t*))−*dnzG*

**Figure 11.** Open chart block named "motion state" in figure 10

This block enables watching state changes when simulating.

**Figure 9.** Implementation of equation (10)

To proceed further, we use the Simulink system shown in Figure 10. The distinctive feature of this system is that it contains an algebraic loop and a Stateflow® chart.

As can be seen in Figure 10, Rn is delivered to the input of FPDnorm broke down above.

On the other hand, this input depends directly on the output function given variables as expressed in equation (10). But, Rn is conditional, which suggests resorting to Stateflow® through "motion state" chart.

**Figure 10.** Normal reactive force Rn computation

**Stateflow**® is a Simulink toolbox convenient for modelling and simulating finite state machines. A finite state machine being a representation of an event-driven system that is a system making a transition from one state (mode) to another prescribed state, provided that the condition defining the change is true.

In Stateflow® representations states and transitions form the basic building blocks of the system.

Stateflow® block is used within the Simulink model here to dynamically simulate the system state changes. The chart block where the system is modelled is open in Figure 11. The device has two states: contact and separation that are represented by a rectangular block and named accordingly. Transition lines indicate the next state that the system in the current state can transit to. According to our algorithm these lines are from contact to fly and vice versa. Also, a default transition is assigned to a default or very first state, the state machine has to be in when it starts, chosen to be the "contact" state. In this case, the first state will be "contact" when the execution begins.

Actions to be taken when entering each state are defined. In this case, this is fly ( *flight* =1) or its logic complement, not fly ( *flight* =0) for contact and flight state respectively. For the Simulink model it is the machine output (0 or 1) that is present at the Stateflow® chart output port. Entry command will be executed when entering the state.

Conversely, let us notice the chart block input port set to control the Stateflow® chart that "sees" (*h* − *zA*) component from Simulink model. This is the "flight\_cond" standing for flight condition associated to transition lines.

Next, at its input 2, from Stateflow® chart, a switch block compares the machine output (0 or 1) to a threshold chosen to have the arbitrary value 0.5 as criterion for the switch to pass either

through switch input 1 where *kn*(*h* − *zA*(*t*))−*dnzG* . is present or through switch input 3 holding the 0 value. One or the other later amount will ultimately output the switch yielding the Rn(t) (see equation (11) and (12)) of the subsystem output. In every iteration step, Simulink will try to bring the algebraic loop mentioned earlier which involves Rn "into balance".

This block enables watching state changes when simulating.

**Figure 9.** Implementation of equation (10)

290 MATLAB Applications for the Practical Engineer

through "motion state" chart.

**Figure 10.** Normal reactive force Rn computation

the condition defining the change is true.

To proceed further, we use the Simulink system shown in Figure 10. The distinctive feature of

On the other hand, this input depends directly on the output function given variables as expressed in equation (10). But, Rn is conditional, which suggests resorting to Stateflow®

**Stateflow**® is a Simulink toolbox convenient for modelling and simulating finite state machines. A finite state machine being a representation of an event-driven system that is a system making a transition from one state (mode) to another prescribed state, provided that

As can be seen in Figure 10, Rn is delivered to the input of FPDnorm broke down above.

this system is that it contains an algebraic loop and a Stateflow® chart.

**Figure 11.** Open chart block named "motion state" in figure 10

Regarding the diagrams in Figure 9 and 10 the interested reader may connect outport carrying Rn to corresponding inport, the same for inports Wa, replaces outports dzm, zG and zA with scopes e.g., set up a sinewave for Wa and use the given values to run the simulation. The solution may be calculated using the ode23 (Bogacki-Shampine) procedure with step size control activated (parameters: Initial Step Size=Auto, Max Step Size=1e-7, Min Step Size=Auto, relative and absolute tolerance=Auto), over the time interval [0, 0.5]. Here is an example of numerical values set: *kn=9.7* MN/m, *Fn=10* N, *m=0.0723* kg, *dn=200* N/ms-1, *h=0.004* m. A MATLAB *function*, similar to the one in Appendix may be used to calculate the quantities of the example above, to be supplied to Simulink interfaces. Regarding the diagrams in Figure 9 and 10 the interested reader may connect outport carrying Rn to corresponding inport, the same for inports Wa, replaces outports dzm, zG and zA with scopes e.g., set up a sinewave for Wa and use the given values to run the simulation. The solution may be calculated using the ode23 (Bogacki-Shampine) procedure with step size control activated (parameters: Initial Step Size = Auto, Max Step Size = 1e-7, Min Step Size = Auto, relative and absolute tolerance = Auto), over the time interval [0, 0.5]. Here is an example of numerical values set: kn = 9.7 MN/m, Fn = 10 N, m = 0.0723 kg, dn = 200 N/ms-1, h = 0.004 m. A MATLAB function, similar to the one in Appendix may be used to calculate the quantities of the example above, to be supplied to Simulink interfaces.

switch to pass either through switch input 1 where .

mentioned earlier which involves Rn "into balance".

This block enables watching state changes when simulating.

through switch input 3 holding the 0 value. One or the other later amount will ultimately output the switch yielding the Rn(t) (see equation (11) and (12)) of the subsystem output. In every iteration step, Simulink will try to bring the algebraic loop

( ))( <sup>n</sup> <sup>A</sup> Gn k h − z t − d z is present or

tc: instant of contact debut.

with

and

actuator.

λ/8 b) feet at λ/4.

wave in steady state.

vibratory amplitude.

this point.

been used up to this point.

Flight ratio (%)

Figure 13 shows results for *β* that are compared to those obtained by [37] for feet located at λ/ 8. Since our simulation is somehow validated by this result, we show in Figure 9 b) the flight

a) b)

Figure 13 : Flight ratio as a function of vibratory amplitude a) for kn equals 10 MN/m ; comparison with [LJF00]; feet at λ/8 b) feet at λ/4.

**Figure 13.** Flight ratio as a function of vibratory amplitude a) for kn equals 10 MN/m ; comparison with [LJF00]; feet at

In steady state, for different preload values, for a stiffness of 10 MN/m and feet located at λ/8, results in Figure 13 a) show the expected flight ratio variations that

In steady state, for different preload values, for a stiffness of 10 MN/m and feet located at λ/8, results in Figure 13 a) show the expected flight ratio variations that increase with the vibratory

Experimental tests performed by [36] to measure the flight ratio allowed the verification of the model relevance observing foot contact intermittence with the floor. Also, the phenomenon remains periodic and of same period with the vibratory

Experimental tests performed by [36] to measure the flight ratio allowed the verification of the model relevance observing foot contact intermittence with the floor. Also, the phenomenon

Regarding the simulation results in conditions of feet located at the wave crest, they show the same trend with a general curves shift toward the left that is a flight ratio

Regarding the simulation results in conditions of feet located at the wave crest, they show the same trend with a general curves shift toward the left that is a flight ratio greater for given

The "similarity" introduced earlier on when the system was restricted to its normal movement component and the "stick-slip" vibrations as defined by Leine et al. [7] has

The "similarity" introduced earlier on when the system was restricted to its normal movement component and the "stick-slip" vibrations as defined by Leine et al. [7] has been used up to

To calculate the tangential force variation, a classical approach would have consisted of applying at the contact surface Coulomb friction modelling. It turns out that if the device is manipulated by a user, non-zero relative tangential speed always exists between the foot and the substrate. Coulomb law suggests therefore that *Rt* =*μRn* during contact phase and *Rt* =0

increase with the vibratory amplitude and decrease with the preload.

remains periodic and of same period with the vibratory wave in steady state.

otherwise. Since in flight event *Rn* =0, it is possible to generalize that *Rt* =*μRn*.

Fn=5.5 N Fn=10 N Fn=20 N Fn=5.5 N [LJF00] Fn=10N [LJF00] Fn=20N [LJF00]

Figure 13 shows results for β that are compared to those obtained by [37] for feet located at λ/8. Since our simulation is somehow validated by this result, we show in Figure 9 b) the flight rate variations as a function of vibratory amplitude for our

Figure 12 shows an annulment of Rn and thus an intermittent contact. The impact of the switch in transitions contact-separation-contact is remarkable. For a purpose, in steady state the exciting frequency is taken equal to 35 kHz in this simulation.

For the purpose of verification this simulation was compared to the one performed in [36]. To that end, attention was given to a parameter capable to learn on the flight duration over a vibratory period. The parameter is named flight ratio and defined as:

<sup>t</sup> <sup>t</sup> <sup>s</sup> <sup>c</sup> <sup>β</sup> (13)

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

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293

rate variations as a function of vibratory amplitude for our actuator.

<sup>×</sup><sup>100</sup> <sup>−</sup> <sup>=</sup> <sup>T</sup>

ts: instant of separation debut

tc: instant of contact debut.

0 0.5 1 1.5 2 Vibra tory amplitude (10-6 m)

greater for given vibratory amplitude.

*4.3.1.3. Actuator behaviour along the tangential axis*

amplitude and decrease with the preload.

In Figure 12 are shown the time evolution of normal reaction Rn: for low vibratory amplitude, the normal reaction does not get null but is modulated at the excitation frequency and according to the vibratory amplitude. For higher vibratory amplitudes In Figure 12 are shown the time evolution of normal reaction Rn: for low vibratory amplitude, the normal reaction does not get null but is modulated at the excitation frequency and according to the vibratory amplitude. For higher vibratory amplitudes

Figure 12 : Continuous-time variation of normal reaction Rn a) low b) high vibratory amplitude **Figure 12.** Continuous-time variation of normal reaction Rn a) low b) high vibratory amplitude

Figure 12 shows an annulment of Rn and thus an intermittent contact. The impact of the switch in transitions contact-separation-contact is remarkable. For a purpose, in steady state the exciting frequency is taken equal to 35 kHz in this simulation.

For the purpose of verification this simulation was compared to the one performed in [36]. To that end, attention was given to a parameter capable to learn on the flight duration over a vibratory period. The parameter is named flight ratio and defined as:

$$
\beta = \frac{t\_s - t\_c}{T} \times 100\tag{13}
$$

with

ts: instant of separation debut

and

<sup>t</sup> <sup>t</sup> <sup>s</sup> <sup>c</sup> <sup>β</sup> (13)

#### tc: instant of contact debut. tc: instant of contact debut.

with

and

Regarding the diagrams in Figure 9 and 10 the interested reader may connect outport carrying Rn to corresponding inport, the same for inports Wa, replaces outports dzm, zG and zA with scopes e.g., set up a sinewave for Wa and use the given values to run the simulation. The solution may be calculated using the ode23 (Bogacki-Shampine) procedure with step size control activated (parameters: Initial Step Size=Auto, Max Step Size=1e-7, Min Step Size=Auto, relative and absolute tolerance=Auto), over the time interval [0, 0.5]. Here is an example of numerical values set: *kn=9.7* MN/m, *Fn=10* N, *m=0.0723* kg, *dn=200* N/ms-1, *h=0.004* m. A MATLAB *function*, similar to the one in Appendix may be used to calculate the quantities of

Regarding the diagrams in Figure 9 and 10 the interested reader may connect outport carrying Rn to corresponding inport, the same for inports Wa, replaces outports dzm, zG and zA with scopes e.g., set up a sinewave for Wa and use the given values to run the simulation. The solution may be calculated using the ode23 (Bogacki-Shampine) procedure with step size control activated (parameters: Initial Step Size = Auto, Max Step Size = 1e-7, Min Step Size = Auto, relative and absolute tolerance = Auto), over the time interval [0, 0.5]. Here is an example of numerical values set: kn = 9.7 MN/m, Fn = 10 N, m = 0.0723 kg, dn = 200 N/ms-1, h = 0.004 m. A MATLAB function, similar to the one in Appendix may be used to calculate the quantities of the example

Figure 11 : Open chart block named "motion state" in figure 10

switch to pass either through switch input 1 where .

mentioned earlier which involves Rn "into balance".

This block enables watching state changes when simulating.

through switch input 3 holding the 0 value. One or the other later amount will ultimately output the switch yielding the Rn(t) (see equation (11) and (12)) of the subsystem output. In every iteration step, Simulink will try to bring the algebraic loop

( ))( <sup>n</sup> <sup>A</sup> Gn k h − z t − d z is present or

In Figure 12 are shown the time evolution of normal reaction Rn: for low vibratory amplitude, the normal reaction does not get null but is modulated at the excitation frequency and

In Figure 12 are shown the time evolution of normal reaction Rn: for low vibratory amplitude, the normal reaction does not get null but is modulated at the excitation frequency and according to the vibratory amplitude. For higher vibratory amplitudes

a) b)

Figure 12 : Continuous-time variation of normal reaction Rn a) low b) high vibratory amplitude **Figure 12.** Continuous-time variation of normal reaction Rn a) low b) high vibratory amplitude

Figure 12 shows an annulment of Rn and thus an intermittent contact. The impact of the switch in transitions contact-separation-contact is remarkable. For a purpose, in steady state the

For the purpose of verification this simulation was compared to the one performed in [36]. To that end, attention was given to a parameter capable to learn on the flight duration over a

> <sup>100</sup> *s c t t T*


the example above, to be supplied to Simulink interfaces.

above, to be supplied to Simulink interfaces.

292 MATLAB Applications for the Practical Engineer

according to the vibratory amplitude. For higher vibratory amplitudes

exciting frequency is taken equal to 35 kHz in this simulation.

with

and

ts: instant of separation debut

vibratory period. The parameter is named flight ratio and defined as:

b

Figure 13 shows results for *β* that are compared to those obtained by [37] for feet located at λ/ 8. Since our simulation is somehow validated by this result, we show in Figure 9 b) the flight rate variations as a function of vibratory amplitude for our actuator. Figure 13 shows results for β that are compared to those obtained by [37] for feet located at λ/8. Since our simulation is somehow validated by this result, we show in Figure 9 b) the flight rate variations as a function of vibratory amplitude for our actuator.

Figure 12 shows an annulment of Rn and thus an intermittent contact. The impact of the switch in transitions contact-separation-contact is remarkable. For a purpose, in steady state the exciting frequency is taken equal to 35 kHz in this simulation.

For the purpose of verification this simulation was compared to the one performed in [36]. To that end, attention was given to a parameter capable to learn on the flight duration over a vibratory period. The parameter is named flight ratio and defined as:

<sup>×</sup><sup>100</sup> <sup>−</sup> <sup>=</sup> <sup>T</sup>

Figure 13 : Flight ratio as a function of vibratory amplitude a) for kn equals 10 MN/m ; comparison with [LJF00]; feet at λ/8 b) feet at λ/4. **Figure 13.** Flight ratio as a function of vibratory amplitude a) for kn equals 10 MN/m ; comparison with [LJF00]; feet at λ/8 b) feet at λ/4.

In steady state, for different preload values, for a stiffness of 10 MN/m and feet located at λ/8, results in Figure 13 a) show the expected flight ratio variations that increase with the vibratory amplitude and decrease with the preload. Experimental tests performed by [36] to measure the flight ratio allowed the In steady state, for different preload values, for a stiffness of 10 MN/m and feet located at λ/8, results in Figure 13 a) show the expected flight ratio variations that increase with the vibratory amplitude and decrease with the preload.

verification of the model relevance observing foot contact intermittence with the floor. Also, the phenomenon remains periodic and of same period with the vibratory wave in steady state. Experimental tests performed by [36] to measure the flight ratio allowed the verification of the model relevance observing foot contact intermittence with the floor. Also, the phenomenon remains periodic and of same period with the vibratory wave in steady state.

Regarding the simulation results in conditions of feet located at the wave crest, they show the same trend with a general curves shift toward the left that is a flight ratio greater for given vibratory amplitude. Regarding the simulation results in conditions of feet located at the wave crest, they show the same trend with a general curves shift toward the left that is a flight ratio greater for given vibratory amplitude.

The "similarity" introduced earlier on when the system was restricted to its normal movement component and the "stick-slip" vibrations as defined by Leine et al. [7] has been used up to this point. The "similarity" introduced earlier on when the system was restricted to its normal movement component and the "stick-slip" vibrations as defined by Leine et al. [7] has been used up to this point.

#### *4.3.1.3. Actuator behaviour along the tangential axis*

To calculate the tangential force variation, a classical approach would have consisted of applying at the contact surface Coulomb friction modelling. It turns out that if the device is manipulated by a user, non-zero relative tangential speed always exists between the foot and the substrate. Coulomb law suggests therefore that *Rt* =*μRn* during contact phase and *Rt* =0 otherwise. Since in flight event *Rn* =0, it is possible to generalize that *Rt* =*μRn*.

Calculating *Rt* average value over a vibratory period, if a constant friction coefficient *μ* is considered, <*Rt* > =*μ* <*Rn* > (<f> denotes average value of f). But, Equation 10 shows that in steady state <*Rn* > = *Fn n* + *m <sup>n</sup> <sup>g</sup>*. It follows hence that regardless the flight ratio, <*Rt* > is constant for constant *μ*. That is not what was experimentally observed, justifying thus the consideration of a time variable friction coefficient. to generalize that = µRR nt . Calculating Rt average value over a vibratory period, if a constant friction coefficient µ is considered, < <sup>t</sup> >= µ < RR <sup>n</sup> > (<f> denotes average value of f). But, Equation 10 shows that in steady state g n m n <sup>F</sup> <sup>R</sup> <sup>n</sup> <sup>n</sup> +>=< . It follows hence that regardless the flight ratio, < R<sup>t</sup> > is constant for constant µ . That is not what was experimentally

4.3.1.3.Actuator behaviour along the tangential axis

To calculate the tangential force variation, a classical approach would have consisted of applying at the contact surface Coulomb friction modelling. It turns out that if the

during contact phase and = 0 Rt otherwise. Since in flight event = 0 R<sup>n</sup> , it is possible

As a consequence, we are able to obtain μ(t) during the foot/substrate contact time, limited by tc and ts which are respectively the contact and separation instants during one period of our

*crit*

> is a key point for this study since it is the reactive force sensed by

û ë (15)

*t V t dt* <sup>=</sup> ò (17)

m

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

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d

 mm> û ë (16)

m dd

<sup>0</sup> if , and then () ( ) *c s*

<sup>0</sup> if , an thend *<sup>s</sup> cc t tt* Î = ù é dd

> 0 () () *t t*

is the relative sliding speed between the two surfaces in contact.

the device user and, equations (11) and (12) show that it is a function of the pre-load, displace‐ ment speed and wave amplitude. To this end, we will have to control the wave amplitude, the two other variables being not suitable for control: the tangential speed will be imposed

The control of the vibratory amplitude may be achieved following different approaches. In [39], the wave amplitude control is done thanks to the phase control of the standing wave according to the voltage signal supply. The advantage of this method is its high robustness against resonance frequency variations. One drawback is a lower dynamic behaviour due to

Another way to control the wave amplitude is to tune the supply frequency around the resonance value. This approach comes from the characteristic frequency – vibratory amplitude which shows that beyond the resonant frequency, the wave amplitude W decreases quasi – linearly, making possible its control [40]. The method presents the advantage of being easy to implement and the loop dynamic is fast. Conversely, it has the disadvantage that changes in temperature displace the resonant frequency and lead to discrepancies in the control. Rigor‐ ously, to avoid that inconvenience, an algorithm to track the resonant frequency should be implemented to anticipate the preload influence on the resonant frequency. Nevertheless we

The displacement δ(t) is computed from the tangential speed integration.

d

externally by the user, and the normal pre-load is set at a fixed value.

**5. Control of the vibration amplitude**

the response time imposed by a phase locked loop (PLL).

have chosen this approach, also easier to implement.

 Î = ù é <sup>&</sup>lt; d

*<sup>c</sup> t tt t t*

vibrating device. We can then write:

where Vt

The determination of <*Rt*

The approach consists of considering in a more refined way the sliding triggering phenomena occurring over every vibratory period. Indeed it has been seen that feet – substrate contact can be intermittent. In such an instance, at every resuming contact, while the actuator is tangen‐ tially moving due to the user action, the feet are first in adhesion on the substrate, then speedily in partial slip and finally in total slip before flying again: the consideration of partial slip phase is source of tangential force average variation. That phase may be characterized by an elastic behaviour of the foot, characterized by its tangential stiffness kt . This corresponds to the definition of a time varying friction coefficient μ obeying Coulomb – Orowan law [38]. observed, justifying thus the consideration of a time variable friction coefficient. The approach consists of considering in a more refined way the sliding triggering phenomena occurring over every vibratory period. Indeed it has been seen that feet – substrate contact can be intermittent. In such an instance, at every resuming contact, while the actuator is tangentially moving due to the user action, the feet are first in adhesion on the substrate, then speedily in partial slip and finally in total slip before flying again: the consideration of partial slip phase is source of tangential force average variation. That phase may be characterized by an elastic behaviour of the foot, characterized by its tangential stiffness kt. This corresponds to the definition of a

Figure 14 depicts friction coefficient as a function of δ and the plot is divided approximately in two parts. The first part, relative to tangential stiffness of the contact, is linear and describes the partial slip. Then, from a critical displacement (δcrit) corresponding to total slip, μ is constant. time varying friction coefficient µ obeying Coulomb – Orowan law [38]. Figure 14 depicts friction coefficient as a function of δ and the plot is divided approximately in two parts. The first part, relative to tangential stiffness of the contact, is linear and describes the partial slip. Then, from a critical displacement (δcrit) corresponding to total slip, µ is constant.

**Figure 14.** Coulomb-Orowan model

vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force acting at the interface feet/substrate. δcrit is defined from the following relationship after prior identification of tangential As explained above, the feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force acting at the interface feet/substrate.

Figure 14 : Coulomb-Orowan model

As explained above, the feet are positioned exactly at the antinodes (Figure 5) of the

stiffness kt: δcrit is defined from the following relationship after prior identification of tangential stiffness kt :

$$\mathcal{S}\_{crit} = \mu\_0 \text{CR}\_n = \frac{\mu\_0 R\_n}{k\_t} \tag{14}$$

where C (m/N) is the compliance and μ0, the maximum friction coefficient at the interface (static friction).

As a consequence, we are able to obtain μ(t) during the foot/substrate contact time, limited by tc and ts which are respectively the contact and separation instants during one period of our vibrating device. We can then write:

$$\text{if } t \in \left] t\_{c'} t\_s \right[ \text{ and } \delta < \delta\_c \text{ then } \mu(t) = \frac{\mu\_0}{\delta\_{crit}} \delta(t) \tag{15}$$

$$\text{if } t \in \left[ t\_{c'}, t\_s \right] \text{ and } \mathcal{S} > \mathcal{S}\_c \text{ then } \mu = \mu\_0 \tag{16}$$

The displacement δ(t) is computed from the tangential speed integration.

$$\mathcal{S}(t) = \int\_0^t V\_t(t)dt\tag{17}$$

where Vt is the relative sliding speed between the two surfaces in contact.

The determination of <*Rt* > is a key point for this study since it is the reactive force sensed by the device user and, equations (11) and (12) show that it is a function of the pre-load, displace‐ ment speed and wave amplitude. To this end, we will have to control the wave amplitude, the two other variables being not suitable for control: the tangential speed will be imposed externally by the user, and the normal pre-load is set at a fixed value.

#### **5. Control of the vibration amplitude**

Calculating *Rt*

constant.

stiffness kt:

friction).

**Figure 14.** Coulomb-Orowan model

steady state <*Rn* > =

*Fn n* + *m*

shows that in steady state g

to generalize that = µRR nt .

294 MATLAB Applications for the Practical Engineer

of a time variable friction coefficient.

average value over a vibratory period, if a constant friction coefficient *μ* is

*<sup>n</sup> <sup>g</sup>*. It follows hence that regardless the flight ratio, <*Rt* > is constant

<sup>n</sup> +>=< . It follows hence that regardless the

. This corresponds to the

:

considered, <*Rt* > =*μ* <*Rn* > (<f> denotes average value of f). But, Equation 10 shows that in

Calculating Rt average value over a vibratory period, if a constant friction coefficient µ is considered, < <sup>t</sup> >= µ < RR <sup>n</sup> > (<f> denotes average value of f). But, Equation 10

4.3.1.3.Actuator behaviour along the tangential axis

To calculate the tangential force variation, a classical approach would have consisted of applying at the contact surface Coulomb friction modelling. It turns out that if the device is manipulated by a user, non-zero relative tangential speed always exists between the foot and the substrate. Coulomb law suggests therefore that = µRR nt during contact phase and = 0 Rt otherwise. Since in flight event = 0 R<sup>n</sup> , it is possible

for constant *μ*. That is not what was experimentally observed, justifying thus the consideration

n <sup>F</sup> <sup>R</sup> <sup>n</sup>

n m

flight ratio, < R<sup>t</sup> > is constant for constant µ . That is not what was experimentally observed, justifying thus the consideration of a time variable friction coefficient.

The approach consists of considering in a more refined way the sliding triggering phenomena occurring over every vibratory period. Indeed it has been seen that feet – substrate contact can be intermittent. In such an instance, at every resuming contact, while the actuator is tangentially moving due to the user action, the feet are first in adhesion on the substrate, then speedily in partial slip and finally in total slip before flying again: the consideration of partial slip phase is source of tangential force average variation. That phase may be characterized by an elastic behaviour of the foot, characterized by its tangential stiffness kt. This corresponds to the definition of a

The approach consists of considering in a more refined way the sliding triggering phenomena occurring over every vibratory period. Indeed it has been seen that feet – substrate contact can be intermittent. In such an instance, at every resuming contact, while the actuator is tangen‐ tially moving due to the user action, the feet are first in adhesion on the substrate, then speedily in partial slip and finally in total slip before flying again: the consideration of partial slip phase is source of tangential force average variation. That phase may be characterized by an elastic

definition of a time varying friction coefficient μ obeying Coulomb – Orowan law [38].

time varying friction coefficient µ obeying Coulomb – Orowan law [38].

µ0

Figure 14 depicts friction coefficient as a function of δ and the plot is divided approximately in two parts. The first part, relative to tangential stiffness of the contact, is linear and describes the partial slip. Then, from a critical displacement (δcrit) corresponding to total slip, μ is

Figure 14 depicts friction coefficient as a function of δ and the plot is divided approximately in two parts. The first part, relative to tangential stiffness of the contact, is linear and describes the partial slip. Then, from a critical displacement

<sup>µ</sup> <sup>C</sup>

δcritic

Figure 14 : Coulomb-Orowan model

As explained above, the feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical Coulomb friction force acting at the interface feet/substrate.

As explained above, the feet are positioned exactly at the antinodes (Figure 5) of the vibrating plate and they are in contact with a plane steel substrate for example. Therefore, when no voltage is applied to the ceramics, if users move the actuator, they can feel the classical

δcrit is defined from the following relationship after prior identification of tangential

0

*t*

*n*

= = (14)

δcrit is defined from the following relationship after prior identification of tangential stiffness kt

*<sup>R</sup> CR k* m

where C (m/N) is the compliance and μ0, the maximum friction coefficient at the interface (static

0

*crit n*

 m

d

µ (µm)

behaviour of the foot, characterized by its tangential stiffness kt

(δcrit) corresponding to total slip, µ is constant.

Coulomb friction force acting at the interface feet/substrate.

The control of the vibratory amplitude may be achieved following different approaches. In [39], the wave amplitude control is done thanks to the phase control of the standing wave according to the voltage signal supply. The advantage of this method is its high robustness against resonance frequency variations. One drawback is a lower dynamic behaviour due to the response time imposed by a phase locked loop (PLL).

Another way to control the wave amplitude is to tune the supply frequency around the resonance value. This approach comes from the characteristic frequency – vibratory amplitude which shows that beyond the resonant frequency, the wave amplitude W decreases quasi – linearly, making possible its control [40]. The method presents the advantage of being easy to implement and the loop dynamic is fast. Conversely, it has the disadvantage that changes in temperature displace the resonant frequency and lead to discrepancies in the control. Rigor‐ ously, to avoid that inconvenience, an algorithm to track the resonant frequency should be implemented to anticipate the preload influence on the resonant frequency. Nevertheless we have chosen this approach, also easier to implement.

6. Features of friction forces

#### **6. Features of friction forces** Maxon® controlled in speed to which the plate is attached by means of an inextensible cable and a 10 mm diameter pulley. The measured motor current is therefore an image of torque

load. The obtained results are as shown in Figure 15.

compared to the experimental are shown in Figure 15.

From equations introduced in section 5, it was possible to compute the behaviour of the actuator for a given wave amplitude, a given tangential speed and a given normal load. The obtained results are as shown in Figure 15. friction force. An optical encoder is used to measure the motor rotational speed and the so constituted setup is controlled by a dSPACE DS1104 application. Several simulations

based on the contact conditions described all along were performed and the results as

and thus force developed by the motor. That force is in absolute value equal to the explored

From equations introduced in section 5, it was possible to compute the behaviour of

To characterize the friction forces we use, for different vibratory amplitudes, a DC motor

To characterize the friction forces we use, for different vibratory amplitudes, a DC motor Maxon® controlled in speed to which the plate is attached by means of an inextensible cable and a 10 mm diameter pulley. The measured motor current is therefore an image of torque and thus force developed by the motor. That force is in absolute value equal to the explored

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297

An optical encoder is used to measure the motor rotational speed and the so constituted setup is controlled by a dSPACE DS1104 application. Several simulations based on the contact conditions described all along were performed and the results as compared to the experimental

The experimental results presented in Figure 15 were obtained in such a way that a load Mext was applied on the top of the device to assume pre-load. This load lied on an elastic element whose stiffness was low enough to consider constant the force Fn due to Mext (Fn=9.81Mext). Also, a steel substrate (*μ* =0.2) was used for these trials. Finally the time variable displacement

These results illustrate the overall behaviour of the structure and show the existence of a critical

Finally the time variable displacement w is imposed by the plate vibrations.

of a critical wave amplitude beyond which friction reduction is noticed.

to Mext (Fn=9.81Mext). Also, a steel substrate ( µ = 2.0 ) was used for these trials.

These results illustrate the overall behaviour of the structure and show the existence

The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (µ1, µ2) induced by vibratory amplitude enabled generation of "notches". In Figure 16, µ1 and µ2 correspond respectively to friction coefficient

Figure 16 : Profile of alteration of notches

x

x x

The other aspect of the evaluation was to determine if it was possible to discriminate

Figure 17 : Simulated notches (different spatial periods)

A preliminary psychophysical evaluation discussed in [42] showed how to assess the

The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric technology, piezoelectric materials and their effect in transducers technology mainly, together with several existing solutions using technologies other than piezoelectric actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting

validity of the structure to low force feedback application.

The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (μ1*,* μ2) induced by vibratory ampli‐ tude enabled generation of "notches". In Figure 16, μ1 and μ2 correspond respectively to friction

friction force.

 

are shown in Figure 15.

*w* is imposed by the plate vibrations.

the two profiles in Figure 17.

**Figure 16.** Profile of alteration of notches

µ1 µ2

8. Summary

wave amplitude beyond which friction reduction is noticed.

**7. Evaluation of the device for touch feedback application**

7. Evaluation of the device for touch feedback application

coefficient before and beyond the critical wave amplitude identified in Figure 12.

before and beyond the critical wave amplitude identified in Figure 12.

µapparent

tangential displacement speed, b) a given tangential speed as a function of the wave amplitude and the normal pre-load. **Figure 15.** Friction force for a) a fixed preload as a function of vibratory amplitude and tangential displacement speed, b) a given tangential speed as a function of the wave amplitude and the normal pre-load.

The experimental results presented in Figure 15 were obtained in such a way that a load Mext was applied on the top of the device to assume pre-load. This load lied on an elastic element whose stiffness was low enough to consider constant the force Fn due

Figure 15 : Friction force for a) a fixed preload as a function of vibratory amplitude and

To characterize the friction forces we use, for different vibratory amplitudes, a DC motor Maxon® controlled in speed to which the plate is attached by means of an inextensible cable and a 10 mm diameter pulley. The measured motor current is therefore an image of torque and thus force developed by the motor. That force is in absolute value equal to the explored friction force.

An optical encoder is used to measure the motor rotational speed and the so constituted setup is controlled by a dSPACE DS1104 application. Several simulations based on the contact conditions described all along were performed and the results as compared to the experimental are shown in Figure 15.

The experimental results presented in Figure 15 were obtained in such a way that a load Mext was applied on the top of the device to assume pre-load. This load lied on an elastic element whose stiffness was low enough to consider constant the force Fn due to Mext (Fn=9.81Mext). Also, a steel substrate (*μ* =0.2) was used for these trials. Finally the time variable displacement *w* is imposed by the plate vibrations.

These results illustrate the overall behaviour of the structure and show the existence of a critical wave amplitude beyond which friction reduction is noticed. to Mext (Fn=9.81Mext). Also, a steel substrate ( µ = 2.0 ) was used for these trials. Finally the time variable displacement w is imposed by the plate vibrations.

These results illustrate the overall behaviour of the structure and show the existence

#### **7. Evaluation of the device for touch feedback application** of a critical wave amplitude beyond which friction reduction is noticed.

7. Evaluation of the device for touch feedback application

The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (μ1*,* μ2) induced by vibratory ampli‐ tude enabled generation of "notches". In Figure 16, μ1 and μ2 correspond respectively to friction coefficient before and beyond the critical wave amplitude identified in Figure 12. The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (µ1, µ2) induced by vibratory amplitude enabled generation of

"notches". In Figure 16, µ1 and µ2 correspond respectively to friction coefficient

Figure 16 : Profile of alteration of notches

Figure 17 : Simulated notches (different spatial periods)

x x

A preliminary psychophysical evaluation discussed in [42] showed how to assess the

The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric technology, piezoelectric materials and their effect in transducers technology mainly, together with several existing solutions using technologies other than piezoelectric actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting

validity of the structure to low force feedback application.

before and beyond the critical wave amplitude identified in Figure 12.

The other aspect of the evaluation was to determine if it was possible to discriminate **Figure 16.** Profile of alteration of notches

8. Summary

the two profiles in Figure 17.

 

**6. Features of friction forces**

friction force.

6. Features of friction forces

296 MATLAB Applications for the Practical Engineer

obtained results are as shown in Figure 15.

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

b)

Friction force (N)

0.5

1

1.5

Friction force (N)

2

2.5

load. The obtained results are as shown in Figure 15.

compared to the experimental are shown in Figure 15.

0 0.2 0.4 0.6 0.8 1 W (10-6 m)

0 0.2 0.4 0.6 Vibratory amplitude (10-6 m)

speed, b) a given tangential speed as a function of the wave amplitude and the normal pre-load.

Figure 15 : Friction force for a) a fixed preload as a function of vibratory amplitude and tangential displacement speed, b) a given tangential speed as a function of the wave amplitude and the normal pre-load.

**Figure 15.** Friction force for a) a fixed preload as a function of vibratory amplitude and tangential displacement

The experimental results presented in Figure 15 were obtained in such a way that a load Mext was applied on the top of the device to assume pre-load. This load lied on an elastic element whose stiffness was low enough to consider constant the force Fn due

a)

From equations introduced in section 5, it was possible to compute the behaviour of the actuator for a given wave amplitude, a given tangential speed and a given normal load. The

An optical encoder is used to measure the motor rotational speed and the so constituted setup is controlled by a dSPACE DS1104 application. Several simulations based on the contact conditions described all along were performed and the results as

> Fn=10N; V=2cm/sexperience Fn=10N; V=2cm/s-

Fn=3N, V=2cm/s-

Fn=3N, v=2cm/sexperience Fn=8N; V=2cm/s-

Fn=8N; V=2cm/sexperience

Fn=3N; V=1cm/s experience Fn=3N; v=2cm/s experience Fn=3N; v=5cm/s experience Fn=3N; V=1cm/s

Fn=3N; V=2cm/s

Fn=3N; V=5cm/s

theory

theory

theory

theory

theory

theory

From equations introduced in section 5, it was possible to compute the behaviour of the actuator for a given wave amplitude, a given tangential speed and a given normal

To characterize the friction forces we use, for different vibratory amplitudes, a DC motor Maxon® controlled in speed to which the plate is attached by means of an inextensible cable and a 10 mm diameter pulley. The measured motor current is therefore an image of torque and thus force developed by the motor. That force is in absolute value equal to the explored the two profiles in Figure 17.

The other aspect of the evaluation was to determine if it was possible to discriminate the two profiles in Figure 17. µ1 µ2 x

**Appendix**

%

%

%

%

%

%

%

%

mg=g\*m;

%

dn=200; %[kg/m]

% is taken constant kn=9.7e6; % [N/m]

Fn=10; % [N/m]

% gravity is constant g=9.81; % [N/kg] % Calculation of the weight 'mg'

% Function activebrakepm

% could be called activebrake.mdl

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % declaration of the pre-load % %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % The pre-load is constant

% Input data: Li plate width [m] % l plate length [m]

function [m,kn,dn,Fn,h,n] = planeactuatpm(Li,l,h\_s,h\_p,rho\_s,rho\_p)

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% h\_s thickness of the metal layer of the plate [m] % h\_p thickness of the ceramic layer of the plate [m]

% with the parameters of the Simulink simulation window not set here

% Call: [m,kn,dn,Fn,h,n]=planeactuatpm(Li,l,h\_s,rho\_s,rho\_p)

% MATLAB function for parametrizing the Simulink system

% rho\_s metal layer density [kg/m^3] % rho\_p ceramic layer density[kg/m^3]

% Output data: the parameters of Equation (11) and (12)

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % declaration of the foot length and number of feet % %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % The length of the foot and the number of the feet are constant

h=0.0040; % length of the foot [m] n=3; % chosen number of the feet

% Calculation of the mass of the plate m=rho\_s\*(h\_s\*l\*Li)+rho\_p\*(h\_p\*l\*Li);

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % Damping and stiffness coefficient % %\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*% % normal damping coefficient is constant

% The stiffness coefficient function of the pre-load

Figure 16 : Profile of alteration of notches

The other aspect of the evaluation was to determine if it was possible to discriminate

to Mext (Fn=9.81Mext). Also, a steel substrate ( µ = 2.0 ) was used for these trials.

These results illustrate the overall behaviour of the structure and show the existence

The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (µ1, µ2) induced by vibratory amplitude enabled generation of "notches". In Figure 16, µ1 and µ2 correspond respectively to friction coefficient

Finally the time variable displacement w is imposed by the plate vibrations.

of a critical wave amplitude beyond which friction reduction is noticed.

before and beyond the critical wave amplitude identified in Figure 12.

7. Evaluation of the device for touch feedback application

Figure 17 : Simulated notches (different spatial periods)

A preliminary psychophysical evaluation discussed in [42] showed how to assess the **Figure 17.** Simulated notches (different spatial periods)

validity of the structure to low force feedback application. 8. Summary A preliminary psychophysical evaluation discussed in [42] showed how to assess the validity of the structure to low force feedback application.

The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric

together with several existing solutions using technologies other than piezoelectric

#### technology, piezoelectric materials and their effect in transducers technology mainly, **8. Summary**

actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric technology, piezo‐ electric materials and their effect in transducers technology mainly, together with several existing solutions using technologies other than piezoelectric actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting friction force even if this force highly depends on normal pre-load and tangential speed.

Finally, the proposed device may be a solution to cope with the lack of compactness and simplicity often encountered in haptics interfaces. Complementary experiments are needed to assess its response to touch feedback. Also required is a study of the device behaviour over the time to consider feet wear, or at least variation of contact conditions so that the initial vibratory amplitude control can anticipate such changes. Consequently a direct comparison between the solution proposed in this study and the demonstrated high-performance and practical electromagnetic mouse described in section 3.1.1 e.g. may not be easily sustainable. However, apart from simplicity and compactness characteristics common to both of them, it is an additional advantage in terms of behaviour that this design anticipated issues like the observed rotation of the electromagnetic mouse which resulted from the localized magnetic force.

The other aspect of the evaluation was to determine if it was possible to discriminate the two

x

x x

Figure 16 : Profile of alteration of notches

The other aspect of the evaluation was to determine if it was possible to discriminate

Figure 17 : Simulated notches (different spatial periods)

A preliminary psychophysical evaluation discussed in [42] showed how to assess the

A preliminary psychophysical evaluation discussed in [42] showed how to assess the validity

The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric technology, piezo‐ electric materials and their effect in transducers technology mainly, together with several existing solutions using technologies other than piezoelectric actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting friction

force even if this force highly depends on normal pre-load and tangential speed.

Finally, the proposed device may be a solution to cope with the lack of compactness and simplicity often encountered in haptics interfaces. Complementary experiments are needed to assess its response to touch feedback. Also required is a study of the device behaviour over the time to consider feet wear, or at least variation of contact conditions so that the initial vibratory amplitude control can anticipate such changes. Consequently a direct comparison between the solution proposed in this study and the demonstrated high-performance and practical electromagnetic mouse described in section 3.1.1 e.g. may not be easily sustainable. However, apart from simplicity and compactness characteristics common to both of them, it is an additional advantage in terms of behaviour that this design anticipated issues like the observed rotation of the electromagnetic mouse which resulted from the localized magnetic

The concept of friction coefficient reduction has been presented to design a 2Dof passive low force feedback device in this chapter. For a design utilizing piezoelectric technology, piezoelectric materials and their effect in transducers technology mainly, together with several existing solutions using technologies other than piezoelectric actuation in the field of touch feedback were briefly discussed. Modelling of the device with an emphasis on the normal reaction force, leading to the expression of the tangential force felt by a user moving the actuator the way he moves a common computer mouse was also presented. In particular, some details were given on a way to use Stateflow® to deal with modelling and simulation aspects of the normal reaction force in the system, regarded as finite state machine when restricted to its normal movement component. We also showed how results from experimental and theoretical investigations agree on the fact that it is possible to control the resulting

validity of the structure to low force feedback application.

to Mext (Fn=9.81Mext). Also, a steel substrate ( µ = 2.0 ) was used for these trials.

These results illustrate the overall behaviour of the structure and show the existence

The aim of the evaluation of the device for tactile feedback is to determine whether the device qualifies or not for the dedicated application. In his study, U. Spälter [41] indicates that there is no standard evaluation procedure for haptic devices. In this specific case, the profile in Figure 16 that shows an alternation of "notches" was considered. One aspect of the evaluation was to know if the alternation of apparent friction coefficient (µ1, µ2) induced by vibratory amplitude enabled generation of "notches". In Figure 16, µ1 and µ2 correspond respectively to friction coefficient

Finally the time variable displacement w is imposed by the plate vibrations.

of a critical wave amplitude beyond which friction reduction is noticed.

before and beyond the critical wave amplitude identified in Figure 12.

µapparent

7. Evaluation of the device for touch feedback application

profiles in Figure 17.

8. Summary

**8. Summary**

force.

the two profiles in Figure 17.

298 MATLAB Applications for the Practical Engineer

µ1 µ2

**Figure 17.** Simulated notches (different spatial periods)

of the structure to low force feedback application.

 

```
function [m,kn,dn,Fn,h,n] = planeactuatpm(Li,l,h_s,h_p,rho_s,rho_p)
%
% Function activebrakepm
%
% Call: [m,kn,dn,Fn,h,n]=planeactuatpm(Li,l,h_s,rho_s,rho_p)
%
% MATLAB function for parametrizing the Simulink system 
% could be called activebrake.mdl
%
% Input data: Li plate width [m]
% l plate length [m]
% h_s thickness of the metal layer of the plate [m]
% h_p thickness of the ceramic layer of the plate [m]
% rho_s metal layer density [kg/m^3]
% rho_p ceramic layer density[kg/m^3]
%
% with the parameters of the Simulink simulation window not set here
% 
% Output data: the parameters of Equation (11) and (12)
%
%*******************************%
% declaration of the pre-load %
%*******************************%
% The pre-load is constant
Fn=10; % [N/m]
%*****************************************************%
% declaration of the foot length and number of feet %
%*****************************************************%
% The length of the foot and the number of the feet are constant
h=0.0040; % length of the foot [m]
n=3; % chosen number of the feet
%
% Calculation of the mass of the plate
m=rho_s*(h_s*l*Li)+rho_p*(h_p*l*Li);
% gravity is constant
g=9.81; % [N/kg] 
% Calculation of the weight 'mg'
mg=g*m;
%**************************************%
% Damping and stiffness coefficient %
%**************************************%
% normal damping coefficient is constant
dn=200; %[kg/m]
%
% The stiffness coefficient function of the pre-load
% is taken constant
kn=9.7e6; % [N/m]
```
A call for the parameters of our actuator then yields:


[4] L. Garbuio, F. Pigache, J.F. Rouchon, B. Semail, Ultrasonic Friction Drive for Passive

Stateflow® Aided Modelling and Simulation of Friction in a Planar Piezoelectric Actuator

http://dx.doi.org/10.5772/57569

301

[5] L. Winfield, J. Glassmire, J. E. Colgate, M. Peshkin, T-PaD: Tactile Pattern Display through Variable Friction Reduction, Second Joint Eurohaptics Conference and Sym‐ posium on Haptic Interfaces for Virtual Environment and Teleoperator Systems

[6] M. Biet, F. Giraud, B. Semail. Squeeze film effect for the design of an ultrasonic tactile plate, in: IEEE Transactions on Ultrasonic, Ferroelectric and Frequency Control, De‐

[7] R I Leine, D. H. Van Campen, and A. De Kraker, Stick-slip Vibrations Induced by Al‐

[9] B. Nogarède, Moteurs piézoélectriques. Les techniques de l'ingénieur, DFB(D3765),

[10] K A Cook-Chennault, N Thambi and A M Sastry, Powering MEMS portable devices — a review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems, Smart Mater. Struct. 17 (2008)

[11] R. A. Wolf and S. Troiler-McKinstry, Temperature dependence of the Piezoelectric

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[13] S Singh, O P Thakur, C Prakash and K K Raina, 2005, Dilatometric and dielectric be‐

[15] S E Park and T R Shrout 1997 Ultrahigh strain and piezoelectric behavior in relaxor

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[17] M. B. Khoudja, M. Hafez, J. M. Alexandre, A. Kheddar, "Electromagnetically Driven High-Density Tactile Interface Based on a Multi-Layer Approach", Best Paper in In‐ ternational Symposium on Micromechatronics and Human Science, pp.147\_152, Na‐

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doi: 10.1002/j.1538-7305.1958.tb03884.x

ception, PhD Thesis (2006), USTL

goya, Japan, 2003

Force Feedback Devices, Electromotion, Vol.14, 2007, n°2, April-June 2007

(WHC'07), 2007.

cember 2007.

juin 1996.

04300.

#### **Author details**

3

G. M'boungui1\*, A.A. Jimoh1 , B. Semail2 and F. Giraud2

\*Address all correspondence to: mboungui@yahoo.fr

1 Department of Electrical Engineering, Tshwane University of Technology, Pretoria, RSA

2 Laboratoire d'Electrotechnique et d'Electronique de Puissance de Lille, université Lille IR‐ CICA, Villeneuve d'Ascq, France

#### **References**


[4] L. Garbuio, F. Pigache, J.F. Rouchon, B. Semail, Ultrasonic Friction Drive for Passive Force Feedback Devices, Electromotion, Vol.14, 2007, n°2, April-June 2007

A call for the parameters of our actuator then yields:

300 MATLAB Applications for the Practical Engineer

m=

kn=

dn= 200 Fn= 10 h=

0.0040

**Author details**

**References**

G. M'boungui1\*, A.A. Jimoh1

CICA, Villeneuve d'Ascq, France

n= 3

0.0723

9700000

[m,kn,dn,Fn,h,n]=planeactuatpm(0.038,0.064,0.003,0.00065,8250,7650)

, B. Semail2

\*Address all correspondence to: mboungui@yahoo.fr

vol. 55, no 6, pp. 535-536, Nov. 2002.

[3] http://www.fcs-cs.com HapticMaster

and F. Giraud2

1 Department of Electrical Engineering, Tshwane University of Technology, Pretoria, RSA

2 Laboratoire d'Electrotechnique et d'Electronique de Puissance de Lille, université Lille IR‐

[1] E. J. Berger, Friction Modelling for Dynamic System Simulation, Appl. Mech. Rev

[2] G. Casiez, P. Plenacoste, C. Chaillou, B. Semail, Elastic Force Feedback With a New

Multi-finger Haptic Device : The DigiHaptic, Eurohaptics 2003


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**Chapter 11**

**Modelling and Simulation Based Matlab/Simulink of a**

**Strap-Down Inertial Navigation System' Errors due to**

Inertial navigation is a dead reckoning positioning method based on the measurement and mathematical processing of the vehicle absolute acceleration and angular speed in order to estimate its attitude, speed and position related to different reference. Due to the specific operation principle, the positioning errors for this method result from the imperfection of the initial conditions knowledge, from the errors due to the numerical calculation in the inertial system, and from the accelerometers and gyros errors. Therefore, the inertial sensors perform‐ ances play a main role in the establishment of the navigation system precision, and should be considered in its design phase frames (Bekir, 2007; Farrell, 2008; Grewal et al., 2013; Grigorie,

Amazing evolution of physics and manufacturing technologies to improve the optical an electronic fields have made possible the development of opto-electronic rotation and transla‐ tion sensors in parallel with the mechanical sensors. The Ring Laser Gyros (RLG) have entered the market only in 1980's even if in 1963 was first demonstrated in a square configuration. Mechanical gyroscopes dominated the market and the RLG were required in military appli‐ cations, because these are ideal systems for high dynamics strap-down inertial navigation, used in extreme environments. The RLG has excellent scale-factor stability and linearity, negligible sensitivity to acceleration, digital output, fast turn-on, excellent stability and repeatability across the range, and no moving parts. Present day RLG's (Ring Laser Gyros) is considered a matured technology and its development efforts are to reduce costs more than

> © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**the Inertial Sensors**

Teodor Lucian Grigorie and Ruxandra Mihaela Botez

http://dx.doi.org/10.5772/57583

**1. Introduction**

Additional information is available at the end of the chapter

2007; Salychev, 1998; Titterton and Weston, 2004).
