**Part 6**

**Robot Applications** 

456 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Thomas Stockmeier. (2000). Power semiconductor packaging-a problem or a resource? From

M. Ayadi, M.A. Fakhfakh, M. Ghariani, & R. Neji. (2010). Electrothermal modeling of hybrid

M.A. Fakhfakh, M. Ayadi, and R. Neji. Thermal behavior of a three phase inverter for EV

U. Drofenik & J. Kolar. (2005). A Thermal Model of a Forced-Cooled Heat Sink for Transient

M. Usui, M. Ishiko, "Simple Approach of Heat Dissipation Design for Inverter Module,"

C. M. Ong. (1998). Dynamic simulation of electric machinery using MATLAB/Simulink. H. Roisse, M. Hecquet, P. Brochet. (1998). Simulation of synchronous machines using a

90431 Nürnberg (Germany)

Conference (PESC 2006), pp.2048-2052, 2006

*IPECNiigata* 2005, pp. 1169-1177, 2005

2010, pp. 170-177

2005.

pp.3656-3659, 1998.

the state of the art to future trends. *Semikron Elektronik GmbH*, Sigmundstr. 2000,

power modules. *Emerald, Microelectronics International (MI),* volume 27, issue 3,

(Electric Vehicle). *in Proc of 15th IEEE Mediterranean Electromechanical Conference (MELECON'10)*, Valletta, Malta, April 25-28, 2010, C4P-E24-3465, pp.1494-1498. Kojima, et al. Novel Electro-thermal Coupling Simulation Technique for Dynamic Analysis of HV (Hybrid Vehicle) Inverter. *Proceedings of PESCO6*, pp. 2048-2052, 2006. Hamada. Novel Electro-Thermal Coupling Simulation Technique for Dynamic Analysis of

HV (Hybrid Vehicle) Inverter," Proc. of 7thIEEE Power Electronics Specialists

Temperature Calculations Employing a Circuit Simulator. *Proceedings of* 

Proc. of International Power Electronics Conference (IPEC 2005), pp. 1598-1603,

electric-magnetic coupled network model. IEEE Trans. on Magneticss, vol.34,

**20** 

*Italy* 

**Design and Simulation of Legged Walking** 

It is well known that legged locomotion is more efficient,speedy, and versatile than the one by track and wheeled vehicles when it operates in a rough terrain or in unconstructed environment. The potential advantages of legged locomotion can be indicated such as better mobility, obstacles overcoming ability, active suspension, energy efficiency, and achievable speed (Song & Waldron, 1989). Legged walking robots have found wide application areas such as in military tasks, inspection of nuclear power plants, surveillance, planetary explorations, and in forestry and agricultural tasks (Carbone & Ceccarelli, 2005; González et

In the past decades, an extensive research has been focused on legged walking robots. A lot of prototypes such as biped robots, quadrupeds, hexapods, and multi-legged walking robots have been built in academic laboratories and companies (Kajita & Espiau, 2008). Significant examples can be indicated as ASIMO (Sakagami et al., 2002), Bigdog (Raibert, 2008), Rhex (Buehler, 2002), and ATHLETE (Wilcox et al., 2007). However, it is still far away to anticipate that legged walking robots can work in a complex environment and accomplish different tasks successfully. Mechanical design, dynamical walking control, walking pattern generation, and motion planning are still challenge problems for developing a reliable legged walking robot, which can operate in different terrains and environments with

Mechanism design, analysis, and optimization, as well as kinematic and dynamic simulation of legged walking robots are important issues for building an efficient, robust, and reliable legged walking robot. In particular, leg mechanism is a crucial part of a legged walking robot. A leg mechanism will not only determine the DOF (degree of freedom) of a robot, but also actuation system efficiency and its control strategy. Additionally, it is well understood that a torso plays an important role during animal and human movements. Thus, the aforementioned two aspects must be taken into account at the same time for developing

Computer aided design and simulation can be considered useful for developing legged walking robots. Several commercial simulation software packages are available for performing modeling, kinematic, and dynamic simulation of legged walking robots. In particular, Matlab® is a widely used software package. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. By using a flexible programming environment, embedded functions, and several useful simulink® toolboxes, it is relative

**1. Introduction** 

al., 2006; Kajita & Espiau, 2008).

speedy, efficient, and versatility features.

legged walking robots.

**Robots in MATLAB® Environment** 

Conghui Liang, Marco Ceccarelli and Giuseppe Carbone *LARM: Laboratory of Robotics and Mechatronics, University of Cassino* 

## **Design and Simulation of Legged Walking Robots in MATLAB® Environment**

Conghui Liang, Marco Ceccarelli and Giuseppe Carbone *LARM: Laboratory of Robotics and Mechatronics, University of Cassino Italy* 

#### **1. Introduction**

It is well known that legged locomotion is more efficient,speedy, and versatile than the one by track and wheeled vehicles when it operates in a rough terrain or in unconstructed environment. The potential advantages of legged locomotion can be indicated such as better mobility, obstacles overcoming ability, active suspension, energy efficiency, and achievable speed (Song & Waldron, 1989). Legged walking robots have found wide application areas such as in military tasks, inspection of nuclear power plants, surveillance, planetary explorations, and in forestry and agricultural tasks (Carbone & Ceccarelli, 2005; González et al., 2006; Kajita & Espiau, 2008).

In the past decades, an extensive research has been focused on legged walking robots. A lot of prototypes such as biped robots, quadrupeds, hexapods, and multi-legged walking robots have been built in academic laboratories and companies (Kajita & Espiau, 2008). Significant examples can be indicated as ASIMO (Sakagami et al., 2002), Bigdog (Raibert, 2008), Rhex (Buehler, 2002), and ATHLETE (Wilcox et al., 2007). However, it is still far away to anticipate that legged walking robots can work in a complex environment and accomplish different tasks successfully. Mechanical design, dynamical walking control, walking pattern generation, and motion planning are still challenge problems for developing a reliable legged walking robot, which can operate in different terrains and environments with speedy, efficient, and versatility features.

Mechanism design, analysis, and optimization, as well as kinematic and dynamic simulation of legged walking robots are important issues for building an efficient, robust, and reliable legged walking robot. In particular, leg mechanism is a crucial part of a legged walking robot. A leg mechanism will not only determine the DOF (degree of freedom) of a robot, but also actuation system efficiency and its control strategy. Additionally, it is well understood that a torso plays an important role during animal and human movements. Thus, the aforementioned two aspects must be taken into account at the same time for developing legged walking robots.

Computer aided design and simulation can be considered useful for developing legged walking robots. Several commercial simulation software packages are available for performing modeling, kinematic, and dynamic simulation of legged walking robots. In particular, Matlab® is a widely used software package. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. By using a flexible programming environment, embedded functions, and several useful simulink® toolboxes, it is relative

Design and Simulation of Legged Walking Robots in MATLAB®

the biped robot is improved.

mechanisms at LARM

**2.1 Mechanism description** 

A different methodology can be considered such as constructing a biped robot with reduced number of DOFs and compact mechanical design. At LARM, Laboratory of Robotics and Mechatronics in the University of Cassino, a research line is dedicated to low-cost easyoperation leg mechanism design. Fig. 1 shows a prototype of a single DOF biped robot fixed on a supporting test bed. It consists of two leg mechanisms with a Chebyshev-Pantograph linkage architecture.The leg mechanisms are connected to the body with simple revolute joints and they are actuated by only one DC motor through a gear box. The actuated crank angles of the two leg mechanisms are 180 degrees synchronized. Therefore, when one leg mechanism is in non-propelling phase another leg mechanism is in propelling phase and vice versa. A big U shaped foot is connected at the end of each leg mechanism with a revolute joint equipped with a torsion spring. The torsion spring makes the foot contact with the ground properly so that it has adaptability to rough terrain and the walking stability of

Fig. 1. A prototype of a single DOF biped robot with two Chebyshev-Pantograph leg

mechanism depends on the length of link HI and link IA or the ratio of PA and PB.

The built prototype in Fig. 1 consists of two single DOF leg mechanisms, which is composed of a Chebyshev four-bar linkage LEDCB and a pantograph mechanism PGBHIA, as shown in Fig. 2. The Chebyshev mechanism LEDCB can generate an ovoid curve for the point B, so that the leg mechanism can perform a rear-forth and up-down motion in sagittal plane with only one actuation motor. In Fig. 2, the crank is LE, the rocker is link CD, and the coupler triangle is EDB. Joints at L, C, and P are fixed on the body of the biped robot. The offsets a, p, and h between them will greatly influence the trajectory shape of point A. The pantograph mechanism PGBHIA is used to amplify the input trajectory of point B into output trajectory with the same shape at point A. In particularly, unlike the traditional design solution, the point P is fixed on the body of the robot instead of the point B in order to have a more compact robust design. However, drawbacks will exist and in this work the aim is to maintain them within certain limits. The amplify ratio of the pantograph

Environment 461

easy and fast to perform kinematic and dynamic analysis of a robotic mechanical system (Matlab manual, 2007). Additionally, motion control and task planning algorithms can be tested for a proposed mechanism design before implementing them on a prototype.

In this chapter, the applications of Matlab® tool for design and simulation of legged walking robots are illustrated through three cases, namely a single DOF biped walking robot with Chebyshev-Pantograph leg mechanisms (Liang et al., 2008); a novel biologically inspired tripod walking robot (Liang et al., 2009 & 2011); a new waist-trunk system for biped humanoid robots (Carbone et al., 2009; Liang et al., 2010; Liang & Ceccarelli, 2010). In details, the content of each section are organized as follows.

In the first section, operation analysis of a Chebyshev-Pantograph leg mechanism is presented for a single DOF biped robot. The proposed leg mechanism is composed of a Chebyshev four-bar linkage and a pantograph mechanism. Kinematic equations of the proposed leg mechanism are formulated and programmed in Matlab® environment for a computer oriented simulation. Simulation results show the operation performance of the proposed leg mechanism with suitable characteristics. A parametric study has been carried out to with the aims to evaluate the operation performance as function of design parameters and to achieve an optimal design solution.

In the second section, a novel tripod walking robot is presented as inspired by tripod gaits existing in nature. The mechanical design problem is investigated by considering the peculiar requirements of leg mechanism to have a proper tripod walking gait. The proposed tripod walking robot is composed of three leg mechanisms with linkage architecture. The proposed leg mechanism is modeled for kinematic analysis and equations are formulated for simulation. A program has been developed in Matlab® environment to study the operation performance of the leg mechanism and to evaluate the feasibility of the tripod walking gaits. Simulation results show operation characteristics of the leg mechanism and feasible walking ability of the proposed tripod walking robot.

In the third section, a new torso design solution named waist-trunk system has been proposed for biped humanoid robots. The proposed waist-trunk system is composed of a six DOFs parallel manipulator and a three DOFs orientation parallel manipulator, which are connected in a serial chain architecture. In contrast to the traditional torso design solutions, the proposed new waist-trunk system has a high number of DOFs, great motion versatility, high payload capability, good stiffness, and easy-operation design features. A 3D model has been built in Matlab® environment by using its Virtual Reality (VR) toolbox. Kinematic simulations have been carried out for two operation modes, namely walking mode and manipulation mode. Operation performances have been evaluated in terms of displacements, velocities, and accelerations. Simulation results show that the simulated waist-trunk system can be very convenient designed as the torso part for humanoid robots.

#### **2. A single DOF biped robot**

A suvery of existing biped robots shows that most of their leg mechanisms are built with an anthropomorphic architecture with three actuating motors at least at the hip, knee, and ankle joints. These kinds of leg mechanisms have an anthropomorphic design, and therefore they show anthropomorphic flexible motion However, mechanical design of these kinds of leg systems is very complex and difficult. Additionally, sophisticated control algorithms and electronics hardware are needed for the motion control. Therefore, it is very difficult and costy to build properly a biped robot with such kinds of leg mechanisms.

easy and fast to perform kinematic and dynamic analysis of a robotic mechanical system (Matlab manual, 2007). Additionally, motion control and task planning algorithms can be

In this chapter, the applications of Matlab® tool for design and simulation of legged walking robots are illustrated through three cases, namely a single DOF biped walking robot with Chebyshev-Pantograph leg mechanisms (Liang et al., 2008); a novel biologically inspired tripod walking robot (Liang et al., 2009 & 2011); a new waist-trunk system for biped humanoid robots (Carbone et al., 2009; Liang et al., 2010; Liang & Ceccarelli, 2010). In

In the first section, operation analysis of a Chebyshev-Pantograph leg mechanism is presented for a single DOF biped robot. The proposed leg mechanism is composed of a Chebyshev four-bar linkage and a pantograph mechanism. Kinematic equations of the proposed leg mechanism are formulated and programmed in Matlab® environment for a computer oriented simulation. Simulation results show the operation performance of the proposed leg mechanism with suitable characteristics. A parametric study has been carried out to with the aims to evaluate the operation performance as function of design parameters

In the second section, a novel tripod walking robot is presented as inspired by tripod gaits existing in nature. The mechanical design problem is investigated by considering the peculiar requirements of leg mechanism to have a proper tripod walking gait. The proposed tripod walking robot is composed of three leg mechanisms with linkage architecture. The proposed leg mechanism is modeled for kinematic analysis and equations are formulated for simulation. A program has been developed in Matlab® environment to study the operation performance of the leg mechanism and to evaluate the feasibility of the tripod walking gaits. Simulation results show operation characteristics of the leg mechanism and

In the third section, a new torso design solution named waist-trunk system has been proposed for biped humanoid robots. The proposed waist-trunk system is composed of a six DOFs parallel manipulator and a three DOFs orientation parallel manipulator, which are connected in a serial chain architecture. In contrast to the traditional torso design solutions, the proposed new waist-trunk system has a high number of DOFs, great motion versatility, high payload capability, good stiffness, and easy-operation design features. A 3D model has been built in Matlab® environment by using its Virtual Reality (VR) toolbox. Kinematic simulations have been carried out for two operation modes, namely walking mode and manipulation mode. Operation performances have been evaluated in terms of displacements, velocities, and accelerations. Simulation results show that the simulated waist-trunk system can be very convenient designed as the torso part for humanoid robots.

A suvery of existing biped robots shows that most of their leg mechanisms are built with an anthropomorphic architecture with three actuating motors at least at the hip, knee, and ankle joints. These kinds of leg mechanisms have an anthropomorphic design, and therefore they show anthropomorphic flexible motion However, mechanical design of these kinds of leg systems is very complex and difficult. Additionally, sophisticated control algorithms and electronics hardware are needed for the motion control. Therefore, it is very difficult and

costy to build properly a biped robot with such kinds of leg mechanisms.

tested for a proposed mechanism design before implementing them on a prototype.

details, the content of each section are organized as follows.

feasible walking ability of the proposed tripod walking robot.

and to achieve an optimal design solution.

**2. A single DOF biped robot** 

A different methodology can be considered such as constructing a biped robot with reduced number of DOFs and compact mechanical design. At LARM, Laboratory of Robotics and Mechatronics in the University of Cassino, a research line is dedicated to low-cost easyoperation leg mechanism design. Fig. 1 shows a prototype of a single DOF biped robot fixed on a supporting test bed. It consists of two leg mechanisms with a Chebyshev-Pantograph linkage architecture.The leg mechanisms are connected to the body with simple revolute joints and they are actuated by only one DC motor through a gear box. The actuated crank angles of the two leg mechanisms are 180 degrees synchronized. Therefore, when one leg mechanism is in non-propelling phase another leg mechanism is in propelling phase and vice versa. A big U shaped foot is connected at the end of each leg mechanism with a revolute joint equipped with a torsion spring. The torsion spring makes the foot contact with the ground properly so that it has adaptability to rough terrain and the walking stability of the biped robot is improved.

Fig. 1. A prototype of a single DOF biped robot with two Chebyshev-Pantograph leg mechanisms at LARM

#### **2.1 Mechanism description**

The built prototype in Fig. 1 consists of two single DOF leg mechanisms, which is composed of a Chebyshev four-bar linkage LEDCB and a pantograph mechanism PGBHIA, as shown in Fig. 2. The Chebyshev mechanism LEDCB can generate an ovoid curve for the point B, so that the leg mechanism can perform a rear-forth and up-down motion in sagittal plane with only one actuation motor. In Fig. 2, the crank is LE, the rocker is link CD, and the coupler triangle is EDB. Joints at L, C, and P are fixed on the body of the biped robot. The offsets a, p, and h between them will greatly influence the trajectory shape of point A. The pantograph mechanism PGBHIA is used to amplify the input trajectory of point B into output trajectory with the same shape at point A. In particularly, unlike the traditional design solution, the point P is fixed on the body of the robot instead of the point B in order to have a more compact robust design. However, drawbacks will exist and in this work the aim is to maintain them within certain limits. The amplify ratio of the pantograph mechanism depends on the length of link HI and link IA or the ratio of PA and PB.

Design and Simulation of Legged Walking Robots in MATLAB®

<sup>a</sup> B = cos<sup>α</sup> -

2 22 2 a +m -c +d a C = - cos<sup>α</sup>

2 22 2 a a +m -c +d D = cos<sup>α</sup> - c 2 m c

The five design parameters a, m, c, d, and f characterize the Chebyshev four-bar linkage, which have a fixed ration with each other as reported in (Artobolevsky, 1979). A numerical simulation can be carried out by using Eqs. (1), (2), and (3) with proper value of the design

The pantograph mechanism PGBHIA with design parameters is shown in Fig. 2. The point P is fixed and point B is connected to the output motion that is obtained by the Chebyshev four-bar linkage. The transmission angles γ1 and γ2 are important parameters for mechanism efficiency. A good performance can be ensured when |γi-90º|< 40º (i=1, 2) according to practice rules for linkages as reported in (Hartenberg and Denavit,

Referring to the scheme in Fig. 2, kinematic equations of the pantograph mechanism can be

1- 1+k -k = 2tan k -k <sup>ϕ</sup>

1- 1+k -k = 2tan

2 2

2 2

k -k <sup>ϕ</sup> (4)

2(l - b )(y - h) (5)



B

B x -p k = y -h

1 B B 22 B B

B

1 B <sup>b</sup> + y + x - (l - b ) + p + h - 2px - 2hy k = 2b (y - h)

> B <sup>p</sup> - x k = y -h

1 B B 22 B B

22B -b + y + x + (l - b ) + p + h - 2px - 2hy k =

Consequently, from Fig. 2 transmission angles γ1 and γ2 can be evaluated as γ1=φ1+φ2 and

1

2 2 2 222

3

222 222

3 4

1 2

formulated after some algebraic manipulation in the form, (Ottaviano et al., 2004),

1

2

γ2=π-θ-φ1, respectively. The coordinates of point A can be given as

2

4

2 m d d

and

parameters.

1964).

with

Environment 463

<sup>m</sup> (3)

Fig. 2. A kinematic scheme of the single DOF Chebyshev-Pantograph leg mechanism

#### **2.2 A kinematic analysis**

A kinematic analysis has been carried out in order to evaluate the operation performance of the single DOF leg mechanism by using Matlab® programming. Actually, the pantograph mechanism amplifies the input motion that is produced by the Chebyshev linkage, as well as parameters p and h affect location and shape of the generated ovoid curve. A kinematic study can be carried out separately for the Chebyshev linkage and pantograph mechanism. A scheme of the Chebyshev four-bar linkage LEDCB with design parameters is shown in Fig. 2. When the crank LE rotates around the point L an output ovoid curve can be traced by point B. Assuming a reference frame XY fixed at point L with X axis laying along in the direction of straight line LC, it is possible to formulate the coordinates of point B as a function of input crank angle α in the form, (Ottaviano et al., 2004),

$$\mathbf{x}\_{\text{B}} = \mathbf{m}\cos\alpha + (\mathbf{c} + \mathbf{f})\cos\theta$$

$$\mathbf{y}\_{\text{n}} = \text{-m}\sin\alpha \cdot (\mathbf{c} + \mathbf{f})\sin\theta \tag{1}$$

where

$$\mathbf{0} = \mathbf{2} \tan^{\circ} (\frac{\sin a \cdot (\sin^{\circ} a + \mathbf{B}^{\circ} \cdot \mathbf{D}^{\circ})^{\times}}{\mathbf{B} + \mathbf{D}}) \tag{2}$$

and

462 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Fig. 2. A kinematic scheme of the single DOF Chebyshev-Pantograph leg mechanism

function of input crank angle α in the form, (Ottaviano et al., 2004),

A kinematic analysis has been carried out in order to evaluate the operation performance of the single DOF leg mechanism by using Matlab® programming. Actually, the pantograph mechanism amplifies the input motion that is produced by the Chebyshev linkage, as well as parameters p and h affect location and shape of the generated ovoid curve. A kinematic study can be carried out separately for the Chebyshev linkage and pantograph mechanism. A scheme of the Chebyshev four-bar linkage LEDCB with design parameters is shown in Fig. 2. When the crank LE rotates around the point L an output ovoid curve can be traced by point B. Assuming a reference frame XY fixed at point L with X axis laying along in the direction of straight line LC, it is possible to formulate the coordinates of point B as a

<sup>B</sup> x = m cosα + (c + f) cosθ

y = -m sin <sup>B</sup> α - (c + f) sinθ (1)


2 2 2 1/2

**2.2 A kinematic analysis** 

where

$$\mathbf{B} = \cos a \cdot \frac{\mathbf{a}}{\mathbf{m}} \tag{3}$$

$$\mathbf{C} = \frac{\mathbf{a}^\circ + \mathbf{m}^\circ \cdot \mathbf{c}^\circ + \mathbf{d}^\circ}{2 \text{ m} \, \text{d}} \cdot \frac{\mathbf{a}}{\text{d}} \cos \mathbf{a}$$

$$\mathbf{D} = \mathbf{a} \begin{array}{c} \mathbf{a} \\ \cos \alpha \mathbf{a} \end{array} \begin{array}{c} \mathbf{a}^\circ + \mathbf{m}^\circ \cdot \mathbf{c}^\circ + \mathbf{d}^\circ \end{array}$$

$$\mathbf{D} = \frac{\mathbf{a}}{\mathbf{c}} \cos \alpha \mathbf{-} \frac{\mathbf{a}^z + \mathbf{m}^z \cdot \mathbf{c}^z + \mathbf{d}^z}{2 \text{ m} \, \text{c}}$$

The five design parameters a, m, c, d, and f characterize the Chebyshev four-bar linkage, which have a fixed ration with each other as reported in (Artobolevsky, 1979). A numerical simulation can be carried out by using Eqs. (1), (2), and (3) with proper value of the design parameters.

The pantograph mechanism PGBHIA with design parameters is shown in Fig. 2. The point P is fixed and point B is connected to the output motion that is obtained by the Chebyshev four-bar linkage. The transmission angles γ1 and γ2 are important parameters for mechanism efficiency. A good performance can be ensured when |γi-90º|< 40º (i=1, 2) according to practice rules for linkages as reported in (Hartenberg and Denavit, 1964).

Referring to the scheme in Fig. 2, kinematic equations of the pantograph mechanism can be formulated after some algebraic manipulation in the form, (Ottaviano et al., 2004),

$$\begin{aligned} \mathfrak{sp}\_{\circ} &= 2 \tan^{\circ} \frac{1 - \sqrt{1 + \mathbf{k}\_{\circ}^{\circ} - \mathbf{k}\_{\circ}^{\circ}}}{\mathbf{k}\_{\circ} - \mathbf{k}\_{\circ}} \\\\ \mathfrak{sp}\_{\circ} &= 2 \tan^{\circ} \frac{1 - \sqrt{1 + \mathbf{k}\_{\circ}^{\circ} - \mathbf{k}\_{\circ}^{\circ}}}{\mathbf{k}\_{\circ} - \mathbf{k}\_{\circ}} \end{aligned} \tag{4}$$

with

$$
\mathbf{k}\_{\circ} = \frac{\mathbf{x}\_{\text{B}} \cdot \mathbf{p}}{\mathbf{y}\_{\text{B}} \cdot \mathbf{h}}
$$

$$
\mathbf{k}\_{\circ} = \frac{\mathbf{b}\_{\circ}^{\circ} + \mathbf{y}\_{\circ}^{\circ} + \mathbf{x}\_{\circ}^{\circ} + \mathbf{x}\_{\circ}^{\circ} \cdot (\mathbf{l}\_{\circ} \cdot \mathbf{b}\_{\circ})^{\circ} + \mathbf{p}^{\circ} + \mathbf{h}^{\circ} \cdot 2\mathbf{p} \mathbf{x}\_{\circ} \cdot 2\mathbf{h} \mathbf{y}\_{\circ}}{2\mathbf{b}\_{\circ}(\mathbf{y}\_{\circ} \cdot \mathbf{h})}
$$

$$
\mathbf{k}\_{\circ} = \frac{\mathbf{p} \cdot \mathbf{x}\_{\circ}}{\mathbf{y}\_{\text{A}} \cdot \mathbf{h}}
$$

$$
\mathbf{k}\_{\circ} = \frac{-\mathbf{b}\_{\circ}^{\circ} + \mathbf{y}\_{\circ}^{\circ} + \mathbf{x}\_{\circ}^{\circ} + (\mathbf{l}\_{\circ} \cdot \mathbf{b}\_{\circ})^{\circ} + \mathbf{p}^{\circ} + \mathbf{h}^{\circ} \cdot 2\mathbf{p} \mathbf{x}\_{\circ} \cdot 2\mathbf{h} \mathbf{y}\_{\circ}}{2(\mathbf{l}\_{\circ} \cdot \mathbf{b}\_{\circ})(\mathbf{y}\_{\circ} \cdot \mathbf{h})}\tag{5}
$$

Consequently, from Fig. 2 transmission angles γ1 and γ2 can be evaluated as γ1=φ1+φ2 and γ2=π-θ-φ1, respectively. The coordinates of point A can be given as

$$\mathbf{x}\_{\text{A}} = \mathbf{x}\_{\text{a}} + \mathbf{b}\_{\text{z}} \mathbf{c} \text{cos}\mathbf{p}\_{\text{z}} - (\mathbf{l}\_{\text{i}} \cdot \mathbf{b}\_{\text{i}}) \mathbf{c} \text{cos}\mathbf{p}\_{\text{i}}$$

$$\mathbf{y}\_{\text{A}} = \mathbf{y}\_{\text{a}} - \mathbf{b}\_{\text{z}} \text{sin}\mathbf{p}\_{\text{z}} - (\mathbf{l}\_{\text{i}} \cdot \mathbf{b}\_{\text{i}}) \text{sin}\mathbf{p}\_{\text{i}} \tag{6}$$

Design and Simulation of Legged Walking Robots in MATLAB®

(a) (b)

(b) generated ovoid curve at point A

Fig. 3. Simulation results of one leg mechanism: (a) computed trajectories of points A and B;

A scheme of the biped motion and trajectories of critical points are shown in Fig. 5(a). Referring to Fig. 5(a), when the leg mechanism is in a non-propelling phase, it swings from rear to forth and the supporting leg propels the body forward. The swinging leg mechanism has a relative swing motion with respect to the supporting leg mechanism. Therefore, the velocity of point B1 in a non-propelling phase with respect to the global inertial frame is larger than that during a supporting phase. This is the reason why the size of curve a-b-c is larger than that the size of curve c-d-e in Fig. 5(b) even if the Chebyshev mechanisms

produce the motion with only 180 degs phase differences at the points B1 and B2.

Fig. 4. Simulation results for motion trajectories of the leg mechanisms during biped walking Fig. 5(b) shows the trajectories of points A1, A2, B1, and B2 in a biped walking gait. The trajectories are plotted with solid lines for the right leg mechanism and with dashed lines for the left leg mechanism, respectively. The motion sequences of points B1 and A1 are indicated with alphabet letters from a to e and a' to e', respectively. In Fig. 5(b), the trajectory segments a-b-c of point B1 and a'-b'-c' of point A1 are produced by the right leg mechanism while it swings from rear to forth. The trajectory segments c-d-e are produced while the right leg is in contact with the ground. Correspondingly, c', d', and e' are at the same point. The trajectories of points A2 and B2 for left leg mechanism are similar but have 180 degs

Environment 465

By using Eqs. (4), the transmission angles γ1 and γ2 can be computed to check the practical feasibility of the proposed mechanism. By derivating Eqs. (6), the motion velocities of point A can be easilty computed. Accelerations can be also computed through a futher derivative of the obbtained equations of velocities. Similarly, the velocities and accelerations at point B can be computed through the first and second derivatives of Eqs. (1), respectively.

By using velocity and acceleration analysis for the generated ovoid curve, kinematic performance of the proposed leg mechanism can be evaluated.

#### **2.3 Simulation results**

A simulation program has been developed in Matlab® environment to study kinematic performance of the proposed leg mechanism, as well as the feasible walking ability of the single DOF biped robot. The elaborated code in m files are included in the CD of this book. Design parameters of the simulated leg mechanism are listed in Table 1.


Table 1. Design parameters of a prototype leg mechanism at LARM with structure of Fig. 2 (sizes are in mm)

Examples of simulation results of one leg mechanism are shown in Fig. 3. When the input crank LE rotates around point L with a constant speed, the motion trajectories of point A and point B can be obtained in the form of ovoid curves. A scheme of the zoomed view of the ovoid curve in Fig. 3(a) is shown in Fig. 3(b) in which four characteristic angles of the input crank actuation are indicated. The dimension of the ovoid curve is characterized by the length L and height H. The generated ovoid curve is composed of an approximate straight-line and a curved segment with a symmetrical shape. The straight-line segment starts at the actuation angle α=90 degs and ends at α=270 degs. Actually, during this 180 degs interval the leg mechanism is in the non-propelling phase and it swings from rear to forth. During the next 180 degs interval the actuation angle goes from α=270 degs to α=90 degs corresponding to the coupler curve segment. In this period, the foot grasps the ground and the leg mechanism is in the propelling phase. The leg mechanism is in a almost stretched configuration when α=0 deg just as the leg scheme shows in Fig. 3(a).

Fig. 4 shows simulation results of the biped robot when it walks on the ground. In Fig. 4, the right leg is indicated with solid line when in contact with the ground and the crank actuation angle is at α=0 deg, the left leg is indicated with dashed line when the crank is at angle α=180 degs and it swings from rear to forth. The trajectories of points A and point B are also plotted as related to the non-propelling phase. It is noted that at the beginning and the end of the trajectory, the density of the points is higher than in the middle segment. Since the time periods are same between each plotted points, the velocity of the swinging leg mechanism in the middle is higher than that at the start and end of one step.

x = x + b cos - (l - b )cos A B 2 2 11 1 ϕ ϕ

By using Eqs. (4), the transmission angles γ1 and γ2 can be computed to check the practical feasibility of the proposed mechanism. By derivating Eqs. (6), the motion velocities of point A can be easilty computed. Accelerations can be also computed through a futher derivative of the obbtained equations of velocities. Similarly, the velocities and accelerations at point B

By using velocity and acceleration analysis for the generated ovoid curve, kinematic

A simulation program has been developed in Matlab® environment to study kinematic performance of the proposed leg mechanism, as well as the feasible walking ability of the single DOF biped robot. The elaborated code in m files are included in the CD of this book.

a b c d h m 50 20 62.5 62.5 30 25 f p l1 l2 b1 b2 62.5 30 300 200 75 150 Table 1. Design parameters of a prototype leg mechanism at LARM with structure of Fig. 2

Examples of simulation results of one leg mechanism are shown in Fig. 3. When the input crank LE rotates around point L with a constant speed, the motion trajectories of point A and point B can be obtained in the form of ovoid curves. A scheme of the zoomed view of the ovoid curve in Fig. 3(a) is shown in Fig. 3(b) in which four characteristic angles of the input crank actuation are indicated. The dimension of the ovoid curve is characterized by the length L and height H. The generated ovoid curve is composed of an approximate straight-line and a curved segment with a symmetrical shape. The straight-line segment starts at the actuation angle α=90 degs and ends at α=270 degs. Actually, during this 180 degs interval the leg mechanism is in the non-propelling phase and it swings from rear to forth. During the next 180 degs interval the actuation angle goes from α=270 degs to α=90 degs corresponding to the coupler curve segment. In this period, the foot grasps the ground and the leg mechanism is in the propelling phase. The leg mechanism is in a almost

Fig. 4 shows simulation results of the biped robot when it walks on the ground. In Fig. 4, the right leg is indicated with solid line when in contact with the ground and the crank actuation angle is at α=0 deg, the left leg is indicated with dashed line when the crank is at angle α=180 degs and it swings from rear to forth. The trajectories of points A and point B are also plotted as related to the non-propelling phase. It is noted that at the beginning and the end of the trajectory, the density of the points is higher than in the middle segment. Since the time periods are same between each plotted points, the velocity of the swinging

stretched configuration when α=0 deg just as the leg scheme shows in Fig. 3(a).

leg mechanism in the middle is higher than that at the start and end of one step.

can be computed through the first and second derivatives of Eqs. (1), respectively.

performance of the proposed leg mechanism can be evaluated.

Design parameters of the simulated leg mechanism are listed in Table 1.

**2.3 Simulation results** 

(sizes are in mm)

y = y - b sin - (l - b )sin A B 2 2 11 1 ϕ ϕ (6)

A scheme of the biped motion and trajectories of critical points are shown in Fig. 5(a). Referring to Fig. 5(a), when the leg mechanism is in a non-propelling phase, it swings from rear to forth and the supporting leg propels the body forward. The swinging leg mechanism has a relative swing motion with respect to the supporting leg mechanism. Therefore, the velocity of point B1 in a non-propelling phase with respect to the global inertial frame is larger than that during a supporting phase. This is the reason why the size of curve a-b-c is larger than that the size of curve c-d-e in Fig. 5(b) even if the Chebyshev mechanisms produce the motion with only 180 degs phase differences at the points B1 and B2.

Fig. 4. Simulation results for motion trajectories of the leg mechanisms during biped walking

Fig. 5(b) shows the trajectories of points A1, A2, B1, and B2 in a biped walking gait. The trajectories are plotted with solid lines for the right leg mechanism and with dashed lines for the left leg mechanism, respectively. The motion sequences of points B1 and A1 are indicated with alphabet letters from a to e and a' to e', respectively. In Fig. 5(b), the trajectory segments a-b-c of point B1 and a'-b'-c' of point A1 are produced by the right leg mechanism while it swings from rear to forth. The trajectory segments c-d-e are produced while the right leg is in contact with the ground. Correspondingly, c', d', and e' are at the same point. The trajectories of points A2 and B2 for left leg mechanism are similar but have 180 degs

Design and Simulation of Legged Walking Robots in MATLAB®

(a) (b)

X and Y axes; (b) accelerations of point P in X and Y axes

are plotted as function of parameter a as output of Matlab® m files.

of the biped robot.

back sliding motion.

and Y axis, respectively. Similarly, Fig. 7(b) shows the accelerations of point P on the body

In Fig. 7(a), the acceleration of point A at the end of leg mechanism is computed between -1 ms2 to 10 m/s2 along X axis and between -10.5 m/s2 to -3.5 m/s2 along Y axis. The acceleration along X axis reaches the maximum value when the input crank angle is at t=0.5

In Fig. 7(b), the acceleration at point P is computed between -2.3 m/s2 to 9 m/s2 along X axis and between -10.2 m/s2 to -0.2 m/s2 along Y axis. The acceleration in X axis reaches the maximum value when one leg mechanism is in the middle of supporting phase and acceleration in Y axis reaches the minimum value, correspondingly. The acceleration in X axis reaches the minimum value during the transition phase of leg mechanisms and the negative value shows that the biped robot in a double supporting phase and produces a

Fig. 7. Computed accelerations during one biped walking gait: (a) accelerations of point A in

An optimal design of the leg mechanism can perform an efficient and practical feasible walking gait. By using the flexibility of Matlab® environment with the elaborated simulation codes. A parametric study has been proposed to characterize the operation performance of the proposed single DOF biped robot as function of its design parameters. Actually, the lengths of the linkages determine a proper shape and size of the generated ovoid curve that is produced by the Chebyshev linkage through an amplification ration of the pantograph mechanism as shown in Fig. 2. Therefore, only three parameters a, p, and h can be considered as significant design variables. In Fig. 8, results of the parametric study

By increasing the value of parameter a, size of the ovoid curve is decreased in X axis and is increased in Y axis as shown in Fig. 8(a). Particularly, the ovoid curve with an approximately straight line segment is obtained when a=0.05 m. Fig. 8(b) shows the corresponding trajectories of COG (center of gravity) and the feet of swinging leg when the other leg mechanism is in contact with the ground. The step length L decreases and step

In Fig. 9, results of parametric study are plotted as function of parameter p. In Fig. 9(a), by varying parameter p the ovoid curve generated at point A has only displacements along X

height H increases as function of the value of parameter a, as shown in Fig. 8(b).

s (α=20 degs) and the minimum value when it is at t=7.3 s (α=325 degs).

Environment 467

differences with respect to the right leg mechanism. There are small circles in the trajectories of point B1 and point B2 during the transition of the two walking phases. This happens because there is a short period of time during which both legs are in contact with the ground and a sliding back motion occus for the body motion of the biped robot.

Fig. 5. Simulation results of biped walking: (a) motion trajectories of the leg mechanisms; (b) a characterization of the computed trajectories of points A1, A2, B1, and B2

Fig. 6(a) shows plots of the computed transmission angles γ1 and γ2 of the right leg mechanisms as function of the input crank angles α1. The value of the transmission angles are computed between 50 degs and 120 degs. The transmission angles for left leg mechanism have 180 degs time differences. Therefore, the proposed leg mechanism has an efficient motion transmission capability. Fig. 6(b) shows the computed plots for angles φ<sup>1</sup> and φ2 of the right leg mechanism. The value of φ1 is between 18 degs and 100 degs as a good contact with the ground. The value of φ2 is between -5 degs and 100 degs and there is no confliction between the legs and body.

Fig. 6. Characterization angles of the right leg mechanism as function of angle α1: (a) angles γ1 and γ2; (b) angles φ1 and φ<sup>2</sup>

The acceleration of point A is computed by using kinematics equations, which are computed in Matlab® m files. Fig. 7(a) shows the computed acceleration values of point A along X axis

differences with respect to the right leg mechanism. There are small circles in the trajectories of point B1 and point B2 during the transition of the two walking phases. This happens because there is a short period of time during which both legs are in contact with the ground

and a sliding back motion occus for the body motion of the biped robot.

(a) (b)

a characterization of the computed trajectories of points A1, A2, B1, and B2

(a) (b)

Fig. 6. Characterization angles of the right leg mechanism as function of angle α1: (a) angles

The acceleration of point A is computed by using kinematics equations, which are computed in Matlab® m files. Fig. 7(a) shows the computed acceleration values of point A along X axis

no confliction between the legs and body.

γ1 and γ2; (b) angles φ1 and φ<sup>2</sup>

Fig. 5. Simulation results of biped walking: (a) motion trajectories of the leg mechanisms; (b)

Fig. 6(a) shows plots of the computed transmission angles γ1 and γ2 of the right leg mechanisms as function of the input crank angles α1. The value of the transmission angles are computed between 50 degs and 120 degs. The transmission angles for left leg mechanism have 180 degs time differences. Therefore, the proposed leg mechanism has an efficient motion transmission capability. Fig. 6(b) shows the computed plots for angles φ<sup>1</sup> and φ2 of the right leg mechanism. The value of φ1 is between 18 degs and 100 degs as a good contact with the ground. The value of φ2 is between -5 degs and 100 degs and there is and Y axis, respectively. Similarly, Fig. 7(b) shows the accelerations of point P on the body of the biped robot.

In Fig. 7(a), the acceleration of point A at the end of leg mechanism is computed between -1 ms2 to 10 m/s2 along X axis and between -10.5 m/s2 to -3.5 m/s2 along Y axis. The acceleration along X axis reaches the maximum value when the input crank angle is at t=0.5 s (α=20 degs) and the minimum value when it is at t=7.3 s (α=325 degs).

In Fig. 7(b), the acceleration at point P is computed between -2.3 m/s2 to 9 m/s2 along X axis and between -10.2 m/s2 to -0.2 m/s2 along Y axis. The acceleration in X axis reaches the maximum value when one leg mechanism is in the middle of supporting phase and acceleration in Y axis reaches the minimum value, correspondingly. The acceleration in X axis reaches the minimum value during the transition phase of leg mechanisms and the negative value shows that the biped robot in a double supporting phase and produces a back sliding motion.

Fig. 7. Computed accelerations during one biped walking gait: (a) accelerations of point A in X and Y axes; (b) accelerations of point P in X and Y axes

An optimal design of the leg mechanism can perform an efficient and practical feasible walking gait. By using the flexibility of Matlab® environment with the elaborated simulation codes. A parametric study has been proposed to characterize the operation performance of the proposed single DOF biped robot as function of its design parameters. Actually, the lengths of the linkages determine a proper shape and size of the generated ovoid curve that is produced by the Chebyshev linkage through an amplification ration of the pantograph mechanism as shown in Fig. 2. Therefore, only three parameters a, p, and h can be considered as significant design variables. In Fig. 8, results of the parametric study are plotted as function of parameter a as output of Matlab® m files.

By increasing the value of parameter a, size of the ovoid curve is decreased in X axis and is increased in Y axis as shown in Fig. 8(a). Particularly, the ovoid curve with an approximately straight line segment is obtained when a=0.05 m. Fig. 8(b) shows the corresponding trajectories of COG (center of gravity) and the feet of swinging leg when the other leg mechanism is in contact with the ground. The step length L decreases and step height H increases as function of the value of parameter a, as shown in Fig. 8(b).

In Fig. 9, results of parametric study are plotted as function of parameter p. In Fig. 9(a), by varying parameter p the ovoid curve generated at point A has only displacements along X

Design and Simulation of Legged Walking Robots in MATLAB®

**3. A biologically inspired tripod walking robot** 

the following aspects:

walking tasks.

only.

parameters.

the walking operation.

the biped robot has displacements along Y axis as shown in Fig. 10.

without change of the step length L and step height H. The COG of the biped robot has corresponding displacements along X axis. Similarly, by varying parameter h, the COG of

Therefore, the position of point P determines the location of the ovoid curve without any shape change. Correspondingly, the location of COG of the biped robot can be as function of the position of point P since the mass center of the leg mechanism varies correspondingly. The parametric study have analyzed the shape of the generated ovoid curve as function of three parameters a, p, and h. The parametric study whose main results are shown in Fig. 8, 9, and 10 has been aimed to check the motion possibility and design sensitivity of the proposed leg mechanism. Interesting outputs of the parametric study can be considered in




Therefore, an optimized mechanical design for leg mechanism and an efficient walking gait for minimizing input crank torque can be determined by selecting proper design

Legged locomotion in walking robots is mainly inspired by nature. For example, biped robots mimic the human walking; quadruped robots perform leg motion like dogs or horses and eight legged robots are inspired to spider-like motion (Song & Waldron, 1989; González et al., 2006). Most of animals have an even number of legs with symmetry character. With this important character animals can move easily, quickly and stably. However, among legged walking robots, biped walking robots are the human-like solutions but sophisticated control algorithms are needed to keep balance during operation (Vukobratovic, 1989). Multi legged robots have a good stable walking performance and can operate with several walking gaits. However, the number of motors increases together with legs. How to

Actually, there are some tripod walking experiences in nature, even around our daily life. A significant example of tripod walking can be recognized in old men walking with a cane. Two human legs and a walking cane as a third leg can produce a special tripod walking gait. With this kind of tripod walking gait, old people with aged or illness nervous system can walk more stably since they always keep two legs in contact with the ground at the same time. Additionally, a standing phase is more stable since there are three legs on the ground and forms a rigid triangle configuration. By talking into account of the advantages of a tripod walking gait, a novel tripod walking robot has been proposed as shown in Fig. 11. In Fig. 11, the tripod walking robot consists of three single DOF Chebyshev-Pantograph leg mechanisms, a body frame, and a balancing mechanism, which is mounted on the top of body frame. Three leg mechanisms are installed on the body frame in a triangle arrangement with one leg mechanism ahead and two leg mechanisms rear in the same line.

coordinate control the motors and gaits synthesis are still difficult problems.

The main specifications of the designed model are listed in Table. 2.


Environment 469

Fig. 8. A parametric study of the leg mechanism as function of parameter a in Fig. 2: (a) generated ovoid curves at point A; (b) trajectories of COG and foot trajectories of the swinging leg

Fig. 9. A parametric study of the biped robot as function of parameter p in Fig. 2: (a) generated ovoid curves at point A; (b) trajectories of COG and foot point of swinging leg

Fig. 10. A parametric study of the biped robot as function of parameter h in Fig. 2: (a) generated ovoid curves at point A; (b) trajectories of COG and foot point of swinging leg

Fig. 8. A parametric study of the leg mechanism as function of parameter a in Fig. 2: (a) generated

ovoid curves at point A; (b) trajectories of COG and foot trajectories of the swinging leg

(a) (b)

(a) (b)

(a) (b)

Fig. 10. A parametric study of the biped robot as function of parameter h in Fig. 2: (a) generated ovoid curves at point A; (b) trajectories of COG and foot point of swinging leg

Fig. 9. A parametric study of the biped robot as function of parameter p in Fig. 2: (a) generated ovoid curves at point A; (b) trajectories of COG and foot point of swinging leg without change of the step length L and step height H. The COG of the biped robot has corresponding displacements along X axis. Similarly, by varying parameter h, the COG of the biped robot has displacements along Y axis as shown in Fig. 10.

Therefore, the position of point P determines the location of the ovoid curve without any shape change. Correspondingly, the location of COG of the biped robot can be as function of the position of point P since the mass center of the leg mechanism varies correspondingly.

The parametric study have analyzed the shape of the generated ovoid curve as function of three parameters a, p, and h. The parametric study whose main results are shown in Fig. 8, 9, and 10 has been aimed to check the motion possibility and design sensitivity of the proposed leg mechanism. Interesting outputs of the parametric study can be considered in the following aspects:


Therefore, an optimized mechanical design for leg mechanism and an efficient walking gait for minimizing input crank torque can be determined by selecting proper design parameters.

#### **3. A biologically inspired tripod walking robot**

Legged locomotion in walking robots is mainly inspired by nature. For example, biped robots mimic the human walking; quadruped robots perform leg motion like dogs or horses and eight legged robots are inspired to spider-like motion (Song & Waldron, 1989; González et al., 2006). Most of animals have an even number of legs with symmetry character. With this important character animals can move easily, quickly and stably. However, among legged walking robots, biped walking robots are the human-like solutions but sophisticated control algorithms are needed to keep balance during operation (Vukobratovic, 1989). Multi legged robots have a good stable walking performance and can operate with several walking gaits. However, the number of motors increases together with legs. How to coordinate control the motors and gaits synthesis are still difficult problems.

Actually, there are some tripod walking experiences in nature, even around our daily life. A significant example of tripod walking can be recognized in old men walking with a cane. Two human legs and a walking cane as a third leg can produce a special tripod walking gait. With this kind of tripod walking gait, old people with aged or illness nervous system can walk more stably since they always keep two legs in contact with the ground at the same time. Additionally, a standing phase is more stable since there are three legs on the ground and forms a rigid triangle configuration. By talking into account of the advantages of a tripod walking gait, a novel tripod walking robot has been proposed as shown in Fig. 11.

In Fig. 11, the tripod walking robot consists of three single DOF Chebyshev-Pantograph leg mechanisms, a body frame, and a balancing mechanism, which is mounted on the top of body frame. Three leg mechanisms are installed on the body frame in a triangle arrangement with one leg mechanism ahead and two leg mechanisms rear in the same line. The main specifications of the designed model are listed in Table. 2.

Design and Simulation of Legged Walking Robots in MATLAB®

mechanism.

Fig. 12(b).

between two legs contacting the ground.

another leg swings in the air.

**3.2 Simulation results** 

forth. The two legs in contact with the ground together with the robot body form a parallel

A scheme of the proposed leg mechanism for tripod walking robot is shown in Fig. 12(b). The tripod walking robot is mainly composed of three one-DOF leg mechanisms. The three leg mechanisms are the same design which is installed on the robot body to have a triangle configuration in horizontal plane. All the three legs are fixed on the body and actuated by

The basic kinematics and operation characters of the proposed leg mechanism are investigated in the work (Liang et al., 2009). This one-DOF leg mechanism is composed of a Chebyshev four-bar linkage CLEDB and a pantograph mechanism BGMHIA. Points L, C and M are fixed on the body. The Chebyshev mechanism and pantograph mechanism are jointed together at point B through which the actuation force is transmitted from the Chebyshev linkage to the pantograph leg. Linkage LE is the crank and α is the input crank angle. The transmission angles γ1 and γ2 of the leg mechanism are shown in the

When the crank LE rotates around point L, an ovoid curve with an approximate straight line segment and symmetry path as traced by foot point. Each straight line segment has a 180º phase in the crank rotation input. The straight line segment represents the supporting phase and the curve segment represents the swinging phase. When the leg mechanism operates in a supporting phase it generates a horizontal motion to points L, C and M which are fixed at the body. Therefore, the body of the robot is propelled forward without force conflict

In a tripod walking gait each leg must has 2/3 period of time in supporting phase and another 1/3 time in swinging phase. In order to avoid the problem of force conflict between legs, a solution is that the two legs on the ground can produce a straight line motion in horizontal plane with the same speed and without waving in vertical direction. A careful

A feasible solution requires that the actuation speed of the input crank is twice during swinging phase as compared with supporting phase. Ai (i=1, 2, 3) are the end points of three leg mechanisms. They trace the same ovoid curve but with 90ºactuation phase differences in supporting phase. Therefore, there will be always two legs in contact with the ground and

Simulations have been carried out in the Matlab® environment with suitable codes of the proposed formulation. The design parameters of the mechanisms for simulation are listed in Table.3. The rotation velocity of the input crank actuation angle is set at 270 degs/s. Each step lasts in 1/3 second for each leg, and numerical simulation has been computed for 2

Chebyshev mechanism (mm) Pantograph mechanism (mm) Leg location (mm)

Table 3. Simulation parameters of the single DOF leg mechanism for the tripod walking robot

d=62.5 m=25 l1=330 l2=150 H1=100 c=62.5 a=50 b1=110 b2=100 H2=100 f=62.5 p=230 p=230 \_ H3=240

analysis will help to define a propel operation of the leg mechanism.

seconds to evaluate a walking behavior in a stationary mode.

DC motors. The leg mechanism is sketched with design parameters in Fig. 12(b).

Environment 471

Fig. 11. A 3D model of the proposed tripod walking robot in SolidWorks® environment


Table 2. Main specifications of the 3D model for the tripod walking robot in Fig. 11

The tripod walking robot is developed for payload transportation and manipulation purposes. The proposed design of the tripod walking robot will be capable of moving quickly with flexibility, and versatility within different environments. Therefore, in the mechanical design, particular attentions have been focused to make the tripod walking robot low-cost easy-operation, light weight, and compact. Particularly, commercial products have been extensively used in the designed model to make it easy to build. Aluminum alloy is selected as the material of the tripod walking robot since it has proper stiffness, mass density, and cheap price.

#### **3.1 The proposed mechanical design**

The mechanism design problem can be started by considering a concept of a tripod waking robot model as shown in Fig. 12(a). The scheme of the mechanism in Fig. 12(a) is a simplified structure with two DOFs that can perform a required back and forth, up and down movement in saggital plane. Actuation motors are fixed at the point C1, C2 and C3. Two feet grasp the ground at point A1 and point A2 while the third leg swings from back to forth. The two legs in contact with the ground together with the robot body form a parallel mechanism.

A scheme of the proposed leg mechanism for tripod walking robot is shown in Fig. 12(b). The tripod walking robot is mainly composed of three one-DOF leg mechanisms. The three leg mechanisms are the same design which is installed on the robot body to have a triangle configuration in horizontal plane. All the three legs are fixed on the body and actuated by DC motors. The leg mechanism is sketched with design parameters in Fig. 12(b).

The basic kinematics and operation characters of the proposed leg mechanism are investigated in the work (Liang et al., 2009). This one-DOF leg mechanism is composed of a Chebyshev four-bar linkage CLEDB and a pantograph mechanism BGMHIA. Points L, C and M are fixed on the body. The Chebyshev mechanism and pantograph mechanism are jointed together at point B through which the actuation force is transmitted from the Chebyshev linkage to the pantograph leg. Linkage LE is the crank and α is the input crank angle. The transmission angles γ1 and γ2 of the leg mechanism are shown in the Fig. 12(b).

When the crank LE rotates around point L, an ovoid curve with an approximate straight line segment and symmetry path as traced by foot point. Each straight line segment has a 180º phase in the crank rotation input. The straight line segment represents the supporting phase and the curve segment represents the swinging phase. When the leg mechanism operates in a supporting phase it generates a horizontal motion to points L, C and M which are fixed at the body. Therefore, the body of the robot is propelled forward without force conflict between two legs contacting the ground.

In a tripod walking gait each leg must has 2/3 period of time in supporting phase and another 1/3 time in swinging phase. In order to avoid the problem of force conflict between legs, a solution is that the two legs on the ground can produce a straight line motion in horizontal plane with the same speed and without waving in vertical direction. A careful analysis will help to define a propel operation of the leg mechanism.

A feasible solution requires that the actuation speed of the input crank is twice during swinging phase as compared with supporting phase. Ai (i=1, 2, 3) are the end points of three leg mechanisms. They trace the same ovoid curve but with 90ºactuation phase differences in supporting phase. Therefore, there will be always two legs in contact with the ground and another leg swings in the air.

#### **3.2 Simulation results**

470 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Fig. 11. A 3D model of the proposed tripod walking robot in SolidWorks® environment

Table 2. Main specifications of the 3D model for the tripod walking robot in Fig. 11

The tripod walking robot is developed for payload transportation and manipulation purposes. The proposed design of the tripod walking robot will be capable of moving quickly with flexibility, and versatility within different environments. Therefore, in the mechanical design, particular attentions have been focused to make the tripod walking robot low-cost easy-operation, light weight, and compact. Particularly, commercial products have been extensively used in the designed model to make it easy to build. Aluminum alloy is selected as the material of the tripod walking robot since it has proper stiffness, mass

The mechanism design problem can be started by considering a concept of a tripod waking robot model as shown in Fig. 12(a). The scheme of the mechanism in Fig. 12(a) is a simplified structure with two DOFs that can perform a required back and forth, up and down movement in saggital plane. Actuation motors are fixed at the point C1, C2 and C3. Two feet grasp the ground at point A1 and point A2 while the third leg swings from back to

Weight 10 kg

Dimension 300×300×600 mm

Walking speed 0.36 km/h

Walking cycle 1 sec/step

density, and cheap price.

**3.1 The proposed mechanical design** 

Step size 300 mm/step

Degrees of freedom 7 (3 for legs, 1 for balancing mechanism, 3 for passive ankle joints)

> Simulations have been carried out in the Matlab® environment with suitable codes of the proposed formulation. The design parameters of the mechanisms for simulation are listed in Table.3. The rotation velocity of the input crank actuation angle is set at 270 degs/s. Each step lasts in 1/3 second for each leg, and numerical simulation has been computed for 2 seconds to evaluate a walking behavior in a stationary mode.


Table 3. Simulation parameters of the single DOF leg mechanism for the tripod walking robot

Design and Simulation of Legged Walking Robots in MATLAB®

completes one cycle of walking.

α3=270 degs

α1=180 degs

proposed leg mechanism.

A typical walking cycle for the proposed tripod walking robot can be described as following by referring to Fig. 13 and Fig. 15. The leg No.3 leaves the ground and swings from back to forth in the so-called swinging phase; at the same time the leg No.1 and the leg No.2 are in the supporting phase, since they are in contact with the ground and they propel the body forward. The speed of the input crank in leg No.3 is twice than in leg No.2 and Leg No.1. When the swinging leg No.3 touches the ground, it starts the propelling phase and the leg No.1 is ready to leave the ground. When Leg No.2 touches the ground, the tripod robot

Fig. 13. The tripod walking robot at initial configuration with α1=180 degs, α2=90 degs, and

(a) (b) (c) (d) (e) (f)

shown as function of time in Fig. 16(a) and Fig. 16(b), respectively.

Fig. 14. Walking snapshots of the tripod walking robot as function of the input for leg motion: (a) α1=270 degs; (b) α1=90 degs; (c) α1=180 degs; (d) α1=270 degs; (e) α1=90 degs; (f)

In order to investigate the operation characteristics and feasibility of the proposed mechanism, the plots of transmission angles γ1 , γ2 and leg angles φ1, φ2 for three legs are

The plots are depicted for each leg. It can be found out that the transmission angle γ1 varies between 60 degs and 170 degs and γ2 varies between 70 degs and 120 degs. According to the kinematics rule of linkages, a feasible and effective transmission can be obtained for the

Environment 473

Fig. 12. The proposed tripod walking robot: (a) configurations of three leg mechanisms; (b) a scheme of one leg mechanism with design parameters

In Fig. 13, the tripod walking robot is given at initial configuration with the input crank angles α1=180 degs, α2=90 degs, and α3=270 degs. At this initial time, the three legs are on the ground with two legs in supporting phase and the third leg is about to get into swinging phase.

In Fig. 14, a sequence of snapshots are shown for the tripod walking robot walks in three dimension space as computed in the numerical simulation. The trajectories of points Ai (i=1,2, 3) of the feet are depicted with small curves. In Fig. 15, the movements of the legs for tripod walking robot are shown in saggital plane. The positions of three feet are also shown in horizontal plane as referring to the computed snapshots.

As shown in Fig. 15, at each step, there are always two legs contacting the ground. Actually, a balancing mechanism can be installed on the body of the robot to adjust the gravity center between the two legs, which grasp the ground at each step. A simple rotation mechanism with a proper mass at end is likely to be installed on the body of robot as a balancing mechanism. Therefore, with a very simple control algorithm and specially sized balancing mechanism the tripod walking robot can walk with a static equilibrium even while it is walking.

Y

Leg No.1

B1

Z

X

C1

Motor

 (a) (b) Fig. 12. The proposed tripod walking robot: (a) configurations of three leg mechanisms; (b) a

A1

In Fig. 13, the tripod walking robot is given at initial configuration with the input crank angles α1=180 degs, α2=90 degs, and α3=270 degs. At this initial time, the three legs are on the ground with two legs in supporting phase and the third leg is about to get into swinging

In Fig. 14, a sequence of snapshots are shown for the tripod walking robot walks in three dimension space as computed in the numerical simulation. The trajectories of points Ai (i=1,2, 3) of the feet are depicted with small curves. In Fig. 15, the movements of the legs for tripod walking robot are shown in saggital plane. The positions of three feet are also shown

As shown in Fig. 15, at each step, there are always two legs contacting the ground. Actually, a balancing mechanism can be installed on the body of the robot to adjust the gravity center between the two legs, which grasp the ground at each step. A simple rotation mechanism with a proper mass at end is likely to be installed on the body of robot as a balancing mechanism. Therefore, with a very simple control algorithm and specially sized balancing mechanism the

tripod walking robot can walk with a static equilibrium even while it is walking.

scheme of one leg mechanism with design parameters

Leg No.2

B3

B2

C3

Motor

C2

A2

Leg No.3

A3

Body Joint

Motor

Torsional Spring

in horizontal plane as referring to the computed snapshots.

phase.

A typical walking cycle for the proposed tripod walking robot can be described as following by referring to Fig. 13 and Fig. 15. The leg No.3 leaves the ground and swings from back to forth in the so-called swinging phase; at the same time the leg No.1 and the leg No.2 are in the supporting phase, since they are in contact with the ground and they propel the body forward. The speed of the input crank in leg No.3 is twice than in leg No.2 and Leg No.1. When the swinging leg No.3 touches the ground, it starts the propelling phase and the leg No.1 is ready to leave the ground. When Leg No.2 touches the ground, the tripod robot completes one cycle of walking.

Fig. 13. The tripod walking robot at initial configuration with α1=180 degs, α2=90 degs, and α3=270 degs

Fig. 14. Walking snapshots of the tripod walking robot as function of the input for leg motion: (a) α1=270 degs; (b) α1=90 degs; (c) α1=180 degs; (d) α1=270 degs; (e) α1=90 degs; (f) α1=180 degs

In order to investigate the operation characteristics and feasibility of the proposed mechanism, the plots of transmission angles γ1 , γ2 and leg angles φ1, φ2 for three legs are shown as function of time in Fig. 16(a) and Fig. 16(b), respectively.

The plots are depicted for each leg. It can be found out that the transmission angle γ1 varies between 60 degs and 170 degs and γ2 varies between 70 degs and 120 degs. According to the kinematics rule of linkages, a feasible and effective transmission can be obtained for the proposed leg mechanism.

Design and Simulation of Legged Walking Robots in MATLAB®

(a) (b) Fig. 16. The transmission angles of the three leg mechanisms during a simulated walking as

(a) (b)

(a) (b) Fig. 18. The position of point A and point C in Saggital XY plane for three legs; (a) positions

Fig. 17. The transmission angles of three leg mechanisms as function of time; (a)

transmission angle φ1; (b) transmission angle φ<sup>2</sup>

of points A i (i=1, 2, 3); (b) positions of points Ci (i=1, 2, 3)

function of time; (a) transmission angle γ1; (b) transmission angle γ<sup>2</sup>

Environment 475

Fig. 15. Walking sequences and trajectories of the feet in saggital plane and position of the three feet in horizontal plane: (a) α1=270 degs; (b) α1=90 degs; (c) α1=180 degs; (d) α1=270 degs; (e) α1=90 degs; (f) α1=180 degs

The plots of leg angles φ1, φ2 are shown in Fig. 17. Angle φ1 varies in a feasible region between 45 degs and 95 degs. It reaches the maximum value at the transition point from swinging phase to supporting phase and the minimum value vice versa. Angle φ2 varies between 5 degs and 72 degs. Therefore, no conflict exists between pantograph mechanism and Chebyshev linkage in the proposed leg mechanism.

Fig. 18(a) shows plots the motion trajectories in saggital plane for points Ai (i=1, 2, 3). Dimension of the length and height for each step are depicted as L and H, respectively. These two dimension parameters are useful to evaluate walking capability and obstacles avoidance ability for the tripod walking robot. They have been computed as L=300 mm and H=48 mm for each step.

A tripod walking gait is composed of three small steps. Fig. 18(b) shows the positions of points of Ci (i=1, 2, 3) in saggital plane. It can be noted that the trajectories are approximate straight lines with very small waving. Therefore, the body of the tripod walking robot has a very small movement of less 5 mm in vertical direction and can be seem as an energy efficiency walking gait. It is computed that the body of robot is propelled forward 100 mm for each leg step. Therefore, the body is propelled forward 300 mm in a cycle of tripod walking gait. The walking speed can be computed as 0.3 m/s. However, there is a period of time that points C2 and C3 do not maintain the rigid body condition, but they move very slightly with respect to each other. Actually, this happens because the propelling speeds of two supporting legs are different. Therefore, a small difference of the motions between points C2 and C3 have been computed in the simulation of the walking gait.

Fig. 19 shows those differences between the positions of points Ci (i=1, 2, 3) as corresponding to Fig. 12(a), during the tripod walking. Fig. 19(a) shows the differences in X axis and Fig. 19(b) in Y axis, respectively. The difference in X axis is less than ∆X2=5 mm and difference in Y axis is less than ∆Y2=1.6 mm. The difference in Y axis can be used as compliance capability during the walking also to smooth the ground contacts. The difference in X axis can be compensated by installing a passive prismatic translation joint on the leg joints at the robot body.

(a) (b) (c) (d) (e) (f)

degs; (e) α1=90 degs; (f) α1=180 degs

H=48 mm for each step.

of the walking gait.

the leg joints at the robot body.

and Chebyshev linkage in the proposed leg mechanism.

Fig. 15. Walking sequences and trajectories of the feet in saggital plane and position of the three feet in horizontal plane: (a) α1=270 degs; (b) α1=90 degs; (c) α1=180 degs; (d) α1=270

The plots of leg angles φ1, φ2 are shown in Fig. 17. Angle φ1 varies in a feasible region between 45 degs and 95 degs. It reaches the maximum value at the transition point from swinging phase to supporting phase and the minimum value vice versa. Angle φ2 varies between 5 degs and 72 degs. Therefore, no conflict exists between pantograph mechanism

Fig. 18(a) shows plots the motion trajectories in saggital plane for points Ai (i=1, 2, 3). Dimension of the length and height for each step are depicted as L and H, respectively. These two dimension parameters are useful to evaluate walking capability and obstacles avoidance ability for the tripod walking robot. They have been computed as L=300 mm and

A tripod walking gait is composed of three small steps. Fig. 18(b) shows the positions of points of Ci (i=1, 2, 3) in saggital plane. It can be noted that the trajectories are approximate straight lines with very small waving. Therefore, the body of the tripod walking robot has a very small movement of less 5 mm in vertical direction and can be seem as an energy efficiency walking gait. It is computed that the body of robot is propelled forward 100 mm for each leg step. Therefore, the body is propelled forward 300 mm in a cycle of tripod walking gait. The walking speed can be computed as 0.3 m/s. However, there is a period of time that points C2 and C3 do not maintain the rigid body condition, but they move very slightly with respect to each other. Actually, this happens because the propelling speeds of two supporting legs are different. Therefore, a small difference of the motions between points C2 and C3 have been computed in the simulation

Fig. 19 shows those differences between the positions of points Ci (i=1, 2, 3) as corresponding to Fig. 12(a), during the tripod walking. Fig. 19(a) shows the differences in X axis and Fig. 19(b) in Y axis, respectively. The difference in X axis is less than ∆X2=5 mm and difference in Y axis is less than ∆Y2=1.6 mm. The difference in Y axis can be used as compliance capability during the walking also to smooth the ground contacts. The difference in X axis can be compensated by installing a passive prismatic translation joint on

Fig. 16. The transmission angles of the three leg mechanisms during a simulated walking as function of time; (a) transmission angle γ1; (b) transmission angle γ<sup>2</sup>

Fig. 17. The transmission angles of three leg mechanisms as function of time; (a) transmission angle φ1; (b) transmission angle φ<sup>2</sup>

Fig. 18. The position of point A and point C in Saggital XY plane for three legs; (a) positions of points A i (i=1, 2, 3); (b) positions of points Ci (i=1, 2, 3)

Design and Simulation of Legged Walking Robots in MATLAB®

Technology, China (Qiang et al., 2005).

mechanisms with a relative high number of DOFs.

more anthropomorphic characteristics.

**4.1 A new waist-trunk system** 

Therefore, they can be better accommodated in our daily life environment (home, office, and other public places) by providing services for human beings (Kemp et al., 2008). This research field has attracted large interests since two decades and a lot of prototypes have been built in the laboratories or companies. Significant examples of biped humanoid robots can be indicated for example in ASIMO developed by HONDA Corporation (Sakagami et al., 2002), HRP series developed at AIST (Kaneko et al., 2004), WABIAN series at the Takanishi laboratory in the Waseda University, Japan (Ogura et al., 2006), HUBO series built at KAIST in Korea (Ill-Woo et al., 2007), and BHR series built in the Beijing Institute of

A survey on the current humanoid robots shows that their limbs (arms and legs) are anthropomorphically designed as articulated link mechanisms with 6 or 7 DOFs. However, torsos of humanoid robots are generally treated as rigid bodies, which are passively carried by the biped legs. The torsos of the existing humanoid robots like ASIMO, HRP, and HUBO have almost a box shaped body with a small number of DOFs. A motivation of this kind of designs is that the torso is used to store the computer, battery, sensors, and other necessary devices, so that the whole system can be designed as compact, robust, and stiff. In addition, due to mechanical design difficulties and complexity of controlling multi-body systems, torsos have been designed by using serial mechanism architectures. However, this kind of designs introduces several drawbacks, which give limitations on the motion capability and operation performances for humanoid robots (Carbone et al., 2009). Therefore, it is promising to design an advanced torso for humanoid robots by adopting parallel

Actually, the human torso is a complex system with many DOFs, and plays an important role during human locomotion such as in walking, turning, and running. Humans unconsciously use their waists and trunks to perform successfully tasks like bending, pushing, carrying and transporting heavy objects. Therefore, an advanced torso system is needed for humanoid robots so that they can be better accommodated in our daily life environment with suitable motion capability, flexibility, better operation performances, and

In the literature, there are few works on design and control issues of the torso system for humanoid robots. A humanoid robot named WABIAN-2R has been developed at Takanishi laboratory in the Waseda University with a 2 DOFs waist and 2 DOFs trunk, (Ogura et al., 2006). The waist and trunk of WABIAN-2R is a serial architecture and it is used for compensating the moment that is generated by the swinging legs when it walks, and to avoid the kinematics singularity in a stretched-knee, heel-contact and toe-off motion. A musculoskeletal flexible-spine humanoid robot named as Kotaro has been built at the JSK laboratory in the University of Tokyo. Kotaro has an anthropomorphic designed trunk system with several DOFs and a complicated sensor system, and it is actuated by using artificial muscle actuators. However, it is not able to walk, (Mizuuchi, 2005). A 3 DOFs parallel manipulator named as CaPaMan2 bis at LARM has been proposed as the trunk module for a low-cost easy-operation humanoid robot CALUMA with the aim to keep balance during walking and for manipulation movements, (Nava Rodriguez et al., 2005). However, these torso

Human torso is an important part of human body. It can be recognized as the portion of the human body to which the neck, upper and lower limbs are attached. Fig. 21(a) shows a

systems are fundamentally different from the proposed waist-trunk system.

Environment 477

Fig. 19. The errors between points Ci (i=1, 2, 3) as function of time: (a) errors in X axis; (b) errors in Y axis

Fig. 20. The velocity of points Ai (i=1, 2, 3). as function of time: (a) velocity in X axis ; (b) velocity in Y axis

The plots of velocity at points Ai (i=1, 2, 3) in X and Y axis are shown in Fig. 20(a) and .Fig. 20(b), respectively. It can be noted that the velocity reaches the maximum value when the legs move to the highest point in a swinging phase in X axis. At the same point the velocity in Y axis is zero and the sign of velocity is changed. In the supporting phase because points Ai (i=1, 2, 3) are on the ground, the velocity is zero. Since the input crank speed is twice time in swinging phase than that in supporting phase, the plots are discontinuous at the transition point. Actually, this can be modeled as an impact between feet and ground that can be smoothed by the above mentioned differences in the paths of Ci points.

Matlab® programming has been suitable and indeed efficient both for performance computation and operation simulation by using the formulated model for the design and operation of the proposed tripod walking robot.

#### **4. A New waist-trunk system for humanoid robots**

Humanoid robots are designed as directly inspired by human capabilities. These robots usually show kinematics similar to humans, as well as similar sensing and behaviour.

(a) (b)

(a) (b)

can be smoothed by the above mentioned differences in the paths of Ci points.

operation of the proposed tripod walking robot.

**4. A New waist-trunk system for humanoid robots** 

Fig. 20. The velocity of points Ai (i=1, 2, 3). as function of time: (a) velocity in X axis ; (b)

The plots of velocity at points Ai (i=1, 2, 3) in X and Y axis are shown in Fig. 20(a) and .Fig. 20(b), respectively. It can be noted that the velocity reaches the maximum value when the legs move to the highest point in a swinging phase in X axis. At the same point the velocity in Y axis is zero and the sign of velocity is changed. In the supporting phase because points Ai (i=1, 2, 3) are on the ground, the velocity is zero. Since the input crank speed is twice time in swinging phase than that in supporting phase, the plots are discontinuous at the transition point. Actually, this can be modeled as an impact between feet and ground that

Matlab® programming has been suitable and indeed efficient both for performance computation and operation simulation by using the formulated model for the design and

Humanoid robots are designed as directly inspired by human capabilities. These robots usually show kinematics similar to humans, as well as similar sensing and behaviour.

errors in Y axis

velocity in Y axis

Fig. 19. The errors between points Ci (i=1, 2, 3) as function of time: (a) errors in X axis; (b)

Therefore, they can be better accommodated in our daily life environment (home, office, and other public places) by providing services for human beings (Kemp et al., 2008). This research field has attracted large interests since two decades and a lot of prototypes have been built in the laboratories or companies. Significant examples of biped humanoid robots can be indicated for example in ASIMO developed by HONDA Corporation (Sakagami et al., 2002), HRP series developed at AIST (Kaneko et al., 2004), WABIAN series at the Takanishi laboratory in the Waseda University, Japan (Ogura et al., 2006), HUBO series built at KAIST in Korea (Ill-Woo et al., 2007), and BHR series built in the Beijing Institute of Technology, China (Qiang et al., 2005).

A survey on the current humanoid robots shows that their limbs (arms and legs) are anthropomorphically designed as articulated link mechanisms with 6 or 7 DOFs. However, torsos of humanoid robots are generally treated as rigid bodies, which are passively carried by the biped legs. The torsos of the existing humanoid robots like ASIMO, HRP, and HUBO have almost a box shaped body with a small number of DOFs. A motivation of this kind of designs is that the torso is used to store the computer, battery, sensors, and other necessary devices, so that the whole system can be designed as compact, robust, and stiff. In addition, due to mechanical design difficulties and complexity of controlling multi-body systems, torsos have been designed by using serial mechanism architectures. However, this kind of designs introduces several drawbacks, which give limitations on the motion capability and operation performances for humanoid robots (Carbone et al., 2009). Therefore, it is promising to design an advanced torso for humanoid robots by adopting parallel mechanisms with a relative high number of DOFs.

Actually, the human torso is a complex system with many DOFs, and plays an important role during human locomotion such as in walking, turning, and running. Humans unconsciously use their waists and trunks to perform successfully tasks like bending, pushing, carrying and transporting heavy objects. Therefore, an advanced torso system is needed for humanoid robots so that they can be better accommodated in our daily life environment with suitable motion capability, flexibility, better operation performances, and more anthropomorphic characteristics.

In the literature, there are few works on design and control issues of the torso system for humanoid robots. A humanoid robot named WABIAN-2R has been developed at Takanishi laboratory in the Waseda University with a 2 DOFs waist and 2 DOFs trunk, (Ogura et al., 2006). The waist and trunk of WABIAN-2R is a serial architecture and it is used for compensating the moment that is generated by the swinging legs when it walks, and to avoid the kinematics singularity in a stretched-knee, heel-contact and toe-off motion. A musculoskeletal flexible-spine humanoid robot named as Kotaro has been built at the JSK laboratory in the University of Tokyo. Kotaro has an anthropomorphic designed trunk system with several DOFs and a complicated sensor system, and it is actuated by using artificial muscle actuators. However, it is not able to walk, (Mizuuchi, 2005). A 3 DOFs parallel manipulator named as CaPaMan2 bis at LARM has been proposed as the trunk module for a low-cost easy-operation humanoid robot CALUMA with the aim to keep balance during walking and for manipulation movements, (Nava Rodriguez et al., 2005). However, these torso systems are fundamentally different from the proposed waist-trunk system.

#### **4.1 A new waist-trunk system**

Human torso is an important part of human body. It can be recognized as the portion of the human body to which the neck, upper and lower limbs are attached. Fig. 21(a) shows a

Design and Simulation of Legged Walking Robots in MATLAB®

(a) (b)

the same structure of a Stewart platform (Tsai, 1999; Ceccarelli, 2004).

a biped leg system can be connected to the moving pelvis platform.

Matlab® environment

humanoid robot design

side in a downward architecture.

Fig. 22. A new waist-trunk system for humanoid robots: (a) a model for imitating the movements of human torso; (b) the proposed new waist-trunk system as modeled in

close to the human torso dimensions as reported in (Kawuchi and Mochimaru, 2005).

The proposed waist-trunk system is illustrated in Fig. 22(b) as a kinematic model that has been elaborated in Matlab® environment. The design sizes of the proposed waist-trunk system are

In Fig. 22(b), upper part of the proposed waist-trunk system is named as trunk module, which consists of a thorax platform, a waist platform, and six identical leg mechanisms to obtain a 6 DOFs parallel manipulator structure. Actually, the proposed trunk module has

In the trunk module, each leg mechanism is composed of a universal joint, a spherical joint, and an actuated prismatic joint. The trunk module has six DOFs with the aim to imitate the function of human lumbar spine and thorax to perform three rotations (flexion-extension, lateral-bending, and transverse-rotation movements) and three translation movements. In particular, head, neck, and dual-arm systems can be installed on the thorax platform in a

The lower part in Fig. 22(b) is named as waist module, which consists of a pelvis platform, a waist platform, and three identical leg mechanisms to obtain a 3 DOFs orientation parallel manipulator structure. This 3 DOFs orientation parallel platform is a classical parallel mechanism, which has been designed as the hip, wrist, and shoulder joints for humanoid robots as reported in a rich literature (Sadjadian & Taghirad, 2006). The waist module shares waist platform with the trunk module but the leg mechanisms are installed on the counter

The pelvis platform is connected to the waist platform with three leg mechanisms and a passive spherical joint. There are six bars connected with the passive spherical joint with the waist platform and pelvis platform with the aim to make it very stiff. The waist module is an orientation platform and has three rotation DOFs for yaw, pitch, and roll movements. The rotation center is a passive spherical joint, which plays a role like the symphysis pubis in the human pelvis to carrry the weight of the human body. The waist module is aimed to imitate the function of human pelvis during walking, running, and other movements. In particular,

Environment 479

scheme of the skeleton of human torso. It can be noted that the human torso consists of three main parts: thorax, waist, and pelvis (Virginia, 1999). The rib cages and spine column of the upper part contribute to thorax. In the thorax, the heart and lungs are protected by the rib cage. The human spine is composed of 33 individual vertebrae, which are separated by fibrocartilaginous intervertebral discs and are secured to each other by interlocking processes and binding ligaments. In particular, the lumbar spine, which is the waist segment, is the most important and largest part of human spine. The main function of the lumbar spine is to bear the weight of the human body. The spine is connected with the pelvis by sacrum and the pelvis is connected with two femurs in the lower part. Additionally, there are hundred pairs of muscles, flexible tendons, and ligaments, complex blood and nervous system with different functions to make a human torso an important part of the human body. Since the human torso is composed of three portions, namely the thorax, waist, and pelvis. In Fig. 21(a), three black rectangles have been indicated on the skeleton of the human torso as reference platforms for the thorax, waist, and pelvis, respectively. Fig. 21(b) shows the corresponding positions of these three parts on the human spine. Therefore, a simply model with three rigid bodies has been proposed as shown in Fig. 22(a), which is expected to imitate the function of human torso during difference human movements.

Fig. 21. Schemes of human torso with reference platforms: (a) skeleton structure; (b) S curve of a human spine

The proposed model in Fig. 22(a) is composed of three rigid bodies namely thorax platform, waist platform, and pelvis platform, from the top to the bottom, respectively. The thorax platform can be connected together to the waist platform by using parallel mechanism with suitable DOFs, which has been named as the trunk module. The thorax platform is expected to imitate the movements of human thorax. Arms, neck, and head of humanoid robots are assumed to be installed a connected to the thorax platform. The pelvis platform is connected to the waist platform with suitable mechanism, which has been named as the waist module. Two legs are expected to be connected to pelvis platform.

scheme of the skeleton of human torso. It can be noted that the human torso consists of three main parts: thorax, waist, and pelvis (Virginia, 1999). The rib cages and spine column of the upper part contribute to thorax. In the thorax, the heart and lungs are protected by the rib cage. The human spine is composed of 33 individual vertebrae, which are separated by fibrocartilaginous intervertebral discs and are secured to each other by interlocking processes and binding ligaments. In particular, the lumbar spine, which is the waist segment, is the most important and largest part of human spine. The main function of the lumbar spine is to bear the weight of the human body. The spine is connected with the pelvis by sacrum and the pelvis is connected with two femurs in the lower part. Additionally, there are hundred pairs of muscles, flexible tendons, and ligaments, complex blood and nervous system with different functions to make a human torso an important part of the human body. Since the human torso is composed of three portions, namely the thorax, waist, and pelvis. In Fig. 21(a), three black rectangles have been indicated on the skeleton of the human torso as reference platforms for the thorax, waist, and pelvis, respectively. Fig. 21(b) shows the corresponding positions of these three parts on the human spine. Therefore, a simply model with three rigid bodies has been proposed as shown in Fig. 22(a), which is expected to imitate the function of human torso during difference human movements.

(a) (b)

Two legs are expected to be connected to pelvis platform.

of a human spine

Fig. 21. Schemes of human torso with reference platforms: (a) skeleton structure; (b) S curve

The proposed model in Fig. 22(a) is composed of three rigid bodies namely thorax platform, waist platform, and pelvis platform, from the top to the bottom, respectively. The thorax platform can be connected together to the waist platform by using parallel mechanism with suitable DOFs, which has been named as the trunk module. The thorax platform is expected to imitate the movements of human thorax. Arms, neck, and head of humanoid robots are assumed to be installed a connected to the thorax platform. The pelvis platform is connected to the waist platform with suitable mechanism, which has been named as the waist module.

Fig. 22. A new waist-trunk system for humanoid robots: (a) a model for imitating the movements of human torso; (b) the proposed new waist-trunk system as modeled in Matlab® environment

The proposed waist-trunk system is illustrated in Fig. 22(b) as a kinematic model that has been elaborated in Matlab® environment. The design sizes of the proposed waist-trunk system are close to the human torso dimensions as reported in (Kawuchi and Mochimaru, 2005).

In Fig. 22(b), upper part of the proposed waist-trunk system is named as trunk module, which consists of a thorax platform, a waist platform, and six identical leg mechanisms to obtain a 6 DOFs parallel manipulator structure. Actually, the proposed trunk module has the same structure of a Stewart platform (Tsai, 1999; Ceccarelli, 2004).

In the trunk module, each leg mechanism is composed of a universal joint, a spherical joint, and an actuated prismatic joint. The trunk module has six DOFs with the aim to imitate the function of human lumbar spine and thorax to perform three rotations (flexion-extension, lateral-bending, and transverse-rotation movements) and three translation movements. In particular, head, neck, and dual-arm systems can be installed on the thorax platform in a humanoid robot design

The lower part in Fig. 22(b) is named as waist module, which consists of a pelvis platform, a waist platform, and three identical leg mechanisms to obtain a 3 DOFs orientation parallel manipulator structure. This 3 DOFs orientation parallel platform is a classical parallel mechanism, which has been designed as the hip, wrist, and shoulder joints for humanoid robots as reported in a rich literature (Sadjadian & Taghirad, 2006). The waist module shares waist platform with the trunk module but the leg mechanisms are installed on the counter side in a downward architecture.

The pelvis platform is connected to the waist platform with three leg mechanisms and a passive spherical joint. There are six bars connected with the passive spherical joint with the waist platform and pelvis platform with the aim to make it very stiff. The waist module is an orientation platform and has three rotation DOFs for yaw, pitch, and roll movements. The rotation center is a passive spherical joint, which plays a role like the symphysis pubis in the human pelvis to carrry the weight of the human body. The waist module is aimed to imitate the function of human pelvis during walking, running, and other movements. In particular, a biped leg system can be connected to the moving pelvis platform.

Design and Simulation of Legged Walking Robots in MATLAB®

humanoid robot.

elaborated codes included in the CD of this book.

virtual reality toolbox simulation environment

**4.3.1 Simulation of the walking mode** 

capability of well imitating different movements of human torso.

**4.3 Simulation results** 

depend on the locations of the manipulated objects.

Step 1. Movements of the moving platforms are computed according to assigned tasks. For

Step 2. The prescribed movements are the inputs of a motion pattern generator, where

Step 3. The computed reference trajectories of the actuated joints are the inputs of a direct

Step 4. The position and orientation for each component of the VRML model are updated for

robot are shown in animations, which are stored as videos in AVI format. Therefore, two different operation modes of the proposed waist-trunk system can be simultated and its operation performances can be conveniently characterized by using

Fig. 24. A scheme for a simulation procedure of the biped humanoid robot in Matlab®

In this section, simulation results of the simulated VRML biped humanoid robot are illustrated for a walking task and a bending-manipulation task. The movements of the waistrunk system are prescribed with suitable equations according to the assigned tasks. Operation performances of the simulated waist-trunk system have been characterized in terms of displacement, velocity, and acceleration. Simulation results show that the proposed waist-trunk system has satisfied operation characteristics as a mechanical system and has a

For a walking mode of the wais-trunk system, the waist platform is assumed to be the fixed base. Thus, positions and orientation angles of the thorax platform and the pelvis platform

positions and orientations of each component can be computed.

a walking mode, motion trajectory of the waist platform is determined as based on the prescribed ZMP (zero momentum point) and COM (center of mass) trajectories. The movements of the pelvis platform are functions of the walking gait parameters. For a manipulation mode, the movements of waist platform and thorax platform

walking pattern or manipulation pattern is generated for the simulated biped

kinematics solver. By solving the direct kinematics of the biped humanoid robot,

each step of simulation. The computed movements of the simulated biped humanoid

Environment 481

#### **4.2 A kinematic simulation**

Simulations have been carried out with the aim to evaluate the operation feasibility of the proposed waist-trunk system for a biped humanoid design solution. In Fig. 23(a) a biped humanoid robot with the proposed waist-trunk system has been modeled in Matlab® virtual reality toolbox environment by using VRML language (Vitual Reality Toolbox Users' Manual, 2007). VRML is a standard file format for representing 3D interactive vector graphics, which has been extensively used in robotic system simulation applications (Siciliano and Khatib, 2008). OpenHRP® is a simulation software package developed for performing dynamic simulation of the famous HRP series humanoid robots by using VRML language (Kanehiro et al., 2004). In a VRML file, the geometric sizes and dynamics parameters of the humanoid robot can be defined as a text-based format.

In Fig. 23(a), the modeled biped humanoid robot is composed of several balls, cuboids, and cylinders with the aim to avoid the complex mechanical design of a humanoid robot. Fig. 23(b) shows the modeling details of the waist-trunk system for a biped humanoid robot. In particular, universal joint and spherical joint are modeled by using balls. Motion constraints have been applied for each joint so that they have the proposed motion capability for a mechanical design solution. The geometry sizes and dynamics parameters of the modeled VRML model are close to the design specifications of most current humanoid robots.

Fig. 23. 3D models in VRML: (a) a biped humanoid robot; (a) modeling details of the waisttrunk system

In Fig. 24, a simulation procedure of the biped humanoid robot in Matlab® virtual reality toolbox is shown in several steps as described in the following:


Therefore, two different operation modes of the proposed waist-trunk system can be simultated and its operation performances can be conveniently characterized by using elaborated codes included in the CD of this book.

Fig. 24. A scheme for a simulation procedure of the biped humanoid robot in Matlab® virtual reality toolbox simulation environment

#### **4.3 Simulation results**

480 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Simulations have been carried out with the aim to evaluate the operation feasibility of the proposed waist-trunk system for a biped humanoid design solution. In Fig. 23(a) a biped humanoid robot with the proposed waist-trunk system has been modeled in Matlab® virtual reality toolbox environment by using VRML language (Vitual Reality Toolbox Users' Manual, 2007). VRML is a standard file format for representing 3D interactive vector graphics, which has been extensively used in robotic system simulation applications (Siciliano and Khatib, 2008). OpenHRP® is a simulation software package developed for performing dynamic simulation of the famous HRP series humanoid robots by using VRML language (Kanehiro et al., 2004). In a VRML file, the geometric sizes and dynamics

In Fig. 23(a), the modeled biped humanoid robot is composed of several balls, cuboids, and cylinders with the aim to avoid the complex mechanical design of a humanoid robot. Fig. 23(b) shows the modeling details of the waist-trunk system for a biped humanoid robot. In particular, universal joint and spherical joint are modeled by using balls. Motion constraints have been applied for each joint so that they have the proposed motion capability for a mechanical design solution. The geometry sizes and dynamics parameters of the modeled

(a) (b) Fig. 23. 3D models in VRML: (a) a biped humanoid robot; (a) modeling details of the waist-

In Fig. 24, a simulation procedure of the biped humanoid robot in Matlab® virtual reality

toolbox is shown in several steps as described in the following:

VRML model are close to the design specifications of most current humanoid robots.

parameters of the humanoid robot can be defined as a text-based format.

**4.2 A kinematic simulation** 

trunk system

In this section, simulation results of the simulated VRML biped humanoid robot are illustrated for a walking task and a bending-manipulation task. The movements of the waistrunk system are prescribed with suitable equations according to the assigned tasks. Operation performances of the simulated waist-trunk system have been characterized in terms of displacement, velocity, and acceleration. Simulation results show that the proposed waist-trunk system has satisfied operation characteristics as a mechanical system and has a capability of well imitating different movements of human torso.

#### **4.3.1 Simulation of the walking mode**

For a walking mode of the wais-trunk system, the waist platform is assumed to be the fixed base. Thus, positions and orientation angles of the thorax platform and the pelvis platform

Design and Simulation of Legged Walking Robots in MATLAB®

t=0s t=0.3s t=0.6s

t=0.9s t=1.2s t=1.5s

prismatic joints Sk (k=1,2,3) of the waist module.

procedure

modules, respectively.

Fig. 25. Simulation snapshots of the movements of a biped humanoid robot in a walking

The prescribed orientation angles of the trunk module and waist module are shown in Fig. 26(a) and Fig. 26(b). The solid and dashed lines represent rotation angle around the roll axis and yaw axes, respectively. The rotation magnitudes have been set as 20 degs and 10 degs, respectively. The dot-dashed line represents the rotation angle around the pitch axis, which has been set as a small value to avoid the computation singularity problem in the ZYZ orientation representation. The computed displacements of the prismatic joints Li (i=1,…,6) of the trunk module are shown in Fig. 27. Fig. 28 shows the computed displacements for the

Fig. 29 and Fig. 30 show the computed velocities and accelerations for the waist and trunk

It can be noted that the characterization plots are quite smooth. The proposed waist-trunk system shows a human-like behaviour for an assigned walking task because of the smooth time evaluation of the motion characteristics. The maximum velocity has been computed as 58 mm/s along Y axis for the trunk module and 120 mm/s along Y axis for the waist module. The maximum acceleration has been computed as 240 mm/s2 along Y axis for the trunk module and 460 mm/s2 along Y axis for the waist module. These values are feasible in proper regions for the operation of both the parallel manipulators and they properly simulate the operation of the human torso. Particularly, it can be noted that the velocity and acceleration curves of the trunk module and waist module have different signs, as an

indication of the counter rotation of thorax platform and pelvis platform.

Environment 483

can be conveniently prescribed. During a normal walking, the movements of the pelvis platform and thorax platform can be prescribed by using the equations listed in Table 4.

In Table 4, Aφ(v, h) is the magnitude of the rotation angle around roll axis as function of the walking parameters that can be determined by the walking speed v and step height l. A<sup>θ</sup> is the magnitude of the rotation angle around pitch axis as function of the slope angle α of the ground. In particular, for a flat ground it is Aθ(α) =0. Aψ(v, l) is the magnitude of the rotation angle around yaw axis as function of the walking parameters of the walking speed v and step length l. φW,0 and ψW,0 are the initial phase angles. ω=π/Ts is the walking frequency. Ts is the time period for one step of walking. The expressions in Table 4 can describe the periodical motion of the walking. A similar motion generation method is also presented in (Harada et al., 2009). The motion trajectories of the thorax platform can be prescribed similarly but in opposite motion direction in order to have a counter rotation with respect to the pelvis movements. This is aimed to preserve the angular monument generated by the lower limbs for walking stability. Particularly, only the orientation angles have been prescribed in the trunk module in the reported simulation. However, the position can be prescribed independently since the thorax platform has 6 DOFs.


Table 4. Prescribed movements for the moving platforms in a walking mode

Simulation time has been prescribed in 1.5s to simulate the function of waist-trunk system in a full cycle of humanoid robot normal walking (Ts=0.75 s/step). An operation has been simulated with 150 steps. In general, the range of motion of human pelvis is between 5 degrees and 20 degrees, and therefore, the orientation capability of the waist module has been designed within a range of 25 degrees. Thus, the waist module can imitate different movements of human pelvis through proper operations.

Fig. 25 shows the movements of the simulated biped humanoid robot, which have been simulated for two steps of walking in Matlab® environment by using the computed data in the previous analysis. The inverse kinematics analysis results have been imported to actuate the VRML model in Fig. 23. It is convenient to output the characterization values and annimations by using the flexible programming environment in Matlab®. The simulated humanoid robot shows a smooth motion which well imitates the movements of human thorax and pelvis during a walking task. In addition, it can be noted that the proposed waist-trunk system shows suitable motions to imitate the movements of the human torso during a normal walking.

can be conveniently prescribed. During a normal walking, the movements of the pelvis platform and thorax platform can be prescribed by using the equations listed in Table 4. In Table 4, Aφ(v, h) is the magnitude of the rotation angle around roll axis as function of the walking parameters that can be determined by the walking speed v and step height l. A<sup>θ</sup> is the magnitude of the rotation angle around pitch axis as function of the slope angle α of the ground. In particular, for a flat ground it is Aθ(α) =0. Aψ(v, l) is the magnitude of the rotation angle around yaw axis as function of the walking parameters of the walking speed v and step length l. φW,0 and ψW,0 are the initial phase angles. ω=π/Ts is the walking frequency. Ts is the time period for one step of walking. The expressions in Table 4 can describe the periodical motion of the walking. A similar motion generation method is also presented in (Harada et al., 2009). The motion trajectories of the thorax platform can be prescribed similarly but in opposite motion direction in order to have a counter rotation with respect to the pelvis movements. This is aimed to preserve the angular monument generated by the lower limbs for walking stability. Particularly, only the orientation angles have been prescribed in the trunk module in the reported simulation. However, the position can be

Positions (mm) Orientation angles (degs)

XW = 0 φW = Aφ(v, h)sin(ωt+φW,0)

ZW = 0 ψW = Aψ(v, l)sin(ωt+ψW,0)

XT = 0 φT = - Aφ(v, h)sin(ωt+φT,0)

ZT = 0 ψT = - Aψ(v, l)sin(ωt+ψT,0)

YW = 0 θW = Aθ(α)

YT = 0 θT = - Aθ(α)

Simulation time has been prescribed in 1.5s to simulate the function of waist-trunk system in a full cycle of humanoid robot normal walking (Ts=0.75 s/step). An operation has been simulated with 150 steps. In general, the range of motion of human pelvis is between 5 degrees and 20 degrees, and therefore, the orientation capability of the waist module has been designed within a range of 25 degrees. Thus, the waist module can imitate different

Fig. 25 shows the movements of the simulated biped humanoid robot, which have been simulated for two steps of walking in Matlab® environment by using the computed data in the previous analysis. The inverse kinematics analysis results have been imported to actuate the VRML model in Fig. 23. It is convenient to output the characterization values and annimations by using the flexible programming environment in Matlab®. The simulated humanoid robot shows a smooth motion which well imitates the movements of human thorax and pelvis during a walking task. In addition, it can be noted that the proposed waist-trunk system shows suitable motions to imitate the movements of the human torso

Table 4. Prescribed movements for the moving platforms in a walking mode

movements of human pelvis through proper operations.

prescribed independently since the thorax platform has 6 DOFs.

Waist platform

Thorax platform

during a normal walking.

Fig. 25. Simulation snapshots of the movements of a biped humanoid robot in a walking procedure

The prescribed orientation angles of the trunk module and waist module are shown in Fig. 26(a) and Fig. 26(b). The solid and dashed lines represent rotation angle around the roll axis and yaw axes, respectively. The rotation magnitudes have been set as 20 degs and 10 degs, respectively. The dot-dashed line represents the rotation angle around the pitch axis, which has been set as a small value to avoid the computation singularity problem in the ZYZ orientation representation. The computed displacements of the prismatic joints Li (i=1,…,6) of the trunk module are shown in Fig. 27. Fig. 28 shows the computed displacements for the prismatic joints Sk (k=1,2,3) of the waist module.

Fig. 29 and Fig. 30 show the computed velocities and accelerations for the waist and trunk modules, respectively.

It can be noted that the characterization plots are quite smooth. The proposed waist-trunk system shows a human-like behaviour for an assigned walking task because of the smooth time evaluation of the motion characteristics. The maximum velocity has been computed as 58 mm/s along Y axis for the trunk module and 120 mm/s along Y axis for the waist module. The maximum acceleration has been computed as 240 mm/s2 along Y axis for the trunk module and 460 mm/s2 along Y axis for the waist module. These values are feasible in proper regions for the operation of both the parallel manipulators and they properly simulate the operation of the human torso. Particularly, it can be noted that the velocity and acceleration curves of the trunk module and waist module have different signs, as an indication of the counter rotation of thorax platform and pelvis platform.

Design and Simulation of Legged Walking Robots in MATLAB®

(a) (b)

(a) (b)

**4.3.2 Simulation of the manipulation mode** 

have been prescribed by using the equations in Table 5.

Fig. 30. Computed accelerations in Cartesian space: (a) thorax platform; (b) pelvis platform

The movements of the waist module and trunk module are combined together in a manipulation mode. The waist-trunk system is a redundant serial-parallel structure with totally 9 DOFs. The inverse kinematics and motion planning are challenge issues for this peculiar serial-parallel structure. The pelvis platform has been assumed to be the fixed base, and the motion trajectories of the center point of the thorax platform and waist platform have been prescribed independently. A simulation has been carried out for a bendingmanipulation procedure in order to evaluate the operation performance for a simultaneous action of the two parallel manipulator structures. The movements of the moving platforms

Fig. 31 shows a sequence of snapshots of the simulated biped humanoid robot performing a bending-manipulation movement. The biped humanoid robot bends his torso and tries to manipulate the object that is placed on the top of a column on the ground. The doubleparallel architecture gives a great manipulation capability for the biped humanoid robot, which is a hard task for current humanoid robots to accomplish. From the motion sequences in Fig. 31, it can be noted that the proposed waist-trunk system shows a suitable motion

Fig. 29. Computed velocities in Cartesian space: (a) thorax platform; (b) pelvis platform

Environment 485

Fig. 26. Prescribed orientation angles for an operation of walking mode: (a) thorax platform; (b) pelvis platform

Fig. 27. Computed leg displacement of the trunk module: (a) for legs 1, 2, and 3; (b) for legs 4, 5, and 6

Fig. 28. Computed leg displacements of the waist module for the three legs

Fig. 26. Prescribed orientation angles for an operation of walking mode: (a) thorax platform;

(a) (b) Fig. 27. Computed leg displacement of the trunk module: (a) for legs 1, 2, and 3; (b) for legs

Fig. 28. Computed leg displacements of the waist module for the three legs

(a) (b)

(b) pelvis platform

4, 5, and 6

Fig. 29. Computed velocities in Cartesian space: (a) thorax platform; (b) pelvis platform

Fig. 30. Computed accelerations in Cartesian space: (a) thorax platform; (b) pelvis platform

#### **4.3.2 Simulation of the manipulation mode**

The movements of the waist module and trunk module are combined together in a manipulation mode. The waist-trunk system is a redundant serial-parallel structure with totally 9 DOFs. The inverse kinematics and motion planning are challenge issues for this peculiar serial-parallel structure. The pelvis platform has been assumed to be the fixed base, and the motion trajectories of the center point of the thorax platform and waist platform have been prescribed independently. A simulation has been carried out for a bendingmanipulation procedure in order to evaluate the operation performance for a simultaneous action of the two parallel manipulator structures. The movements of the moving platforms have been prescribed by using the equations in Table 5.

Fig. 31 shows a sequence of snapshots of the simulated biped humanoid robot performing a bending-manipulation movement. The biped humanoid robot bends his torso and tries to manipulate the object that is placed on the top of a column on the ground. The doubleparallel architecture gives a great manipulation capability for the biped humanoid robot, which is a hard task for current humanoid robots to accomplish. From the motion sequences in Fig. 31, it can be noted that the proposed waist-trunk system shows a suitable motion

Design and Simulation of Legged Walking Robots in MATLAB®

movement: (a) thorax platform; (b) waist platform

(a) (b)

prismatic joints Li (i=1,…,6) of the trunk module are shown in Fig. 35.

(a) (b)

manipulation movement: (a) thorax platform; (b) waist platform

Fig. 33. Prescribed positions in Cartesian space for a simulated operation of bending-

Fig. 36 shows the computed velocities for the waist and trunk modules. The maximum velocity has been computed as 600 mm/s along X axis of the thorax platform and 80 mm/s along X axis of the waist platform. Fig. 37 shows the computed accelerations in the Cartesian space with the maximum acceleration as 1000 mm/s2 along X axis of the thorax platform

Fig. 32. Prescribed orientation angles for a simulated operation of bending-manipulation

The prescribed orientation angles of the thorax platform and waist platform are shown in Fig. 32(a) and Fig. 32(b), respectively. The thorax platform rotates 60 degs around its pitch axis and the waist platform rotates 30 degs around its pitch axis. The prescribed positions are plotted in Fig. 33(a) and Fig. 33(b). The center point of the thorax platform moves 120 mm along X axis and 80 mm along Z axis. The center point of the waist platform moves 66 mm along X axis and 17 mm along Y axis. Particularly, since the waist module is an orientation parallel manipulator, the positions of the waist platform are coupled with its orientation angles and they can be computed when the orientation angles are known. Fig. 34 shows the computed displacements of the prismatic joints Sk (k=1,2,3) of the waist module. The computed displacements of the

Environment 487

which well imitates the movements of a human torso during a bending-manipulation procedure. It is remarkable the smooth behaviour of the overall operation that makes the waist-trunk system to show a human-like motion characteristic and it can be very convenient designed as the torso part for humanoid robots.


Table 5. Prescribed movements of the moving platforms for a bending-manipulation motion

Fig. 31. Simulation snapshots of the movements of a biped humanoid robot in a bendingmanipulation procedure

which well imitates the movements of a human torso during a bending-manipulation procedure. It is remarkable the smooth behaviour of the overall operation that makes the waist-trunk system to show a human-like motion characteristic and it can be very

XT = Xt,0 +120 sin(ωt) φT = 0

ZT = ZT,0 - 80 sin(ωt) ψT = 0

XW = XW(φW, θW, ψW) φW = 0

ZW = ZW(φW, θW, ψW) ψW = 0

Table 5. Prescribed movements of the moving platforms for a bending-manipulation motion

t=0s t=0.3s t=0.6s

t=0.9s t=1.2s t=1.5s

Fig. 31. Simulation snapshots of the movements of a biped humanoid robot in a bending-

Positions (mm) Orientation angles (degs)

YT = 0 θT = 30 sin(ωt+θ0)

YW = 0 θW = 30 sin(ωt+θ0)

convenient designed as the torso part for humanoid robots.

Thorax platform

Waist platform

manipulation procedure

Fig. 32. Prescribed orientation angles for a simulated operation of bending-manipulation movement: (a) thorax platform; (b) waist platform

The prescribed orientation angles of the thorax platform and waist platform are shown in Fig. 32(a) and Fig. 32(b), respectively. The thorax platform rotates 60 degs around its pitch axis and the waist platform rotates 30 degs around its pitch axis. The prescribed positions are plotted in Fig. 33(a) and Fig. 33(b). The center point of the thorax platform moves 120 mm along X axis and 80 mm along Z axis. The center point of the waist platform moves 66 mm along X axis and 17 mm along Y axis. Particularly, since the waist module is an orientation parallel manipulator, the positions of the waist platform are coupled with its orientation angles and they can be computed when the orientation angles are known. Fig. 34 shows the computed displacements of the prismatic joints Sk (k=1,2,3) of the waist module. The computed displacements of the prismatic joints Li (i=1,…,6) of the trunk module are shown in Fig. 35.

Fig. 33. Prescribed positions in Cartesian space for a simulated operation of bendingmanipulation movement: (a) thorax platform; (b) waist platform

Fig. 36 shows the computed velocities for the waist and trunk modules. The maximum velocity has been computed as 600 mm/s along X axis of the thorax platform and 80 mm/s along X axis of the waist platform. Fig. 37 shows the computed accelerations in the Cartesian space with the maximum acceleration as 1000 mm/s2 along X axis of the thorax platform

Design and Simulation of Legged Walking Robots in MATLAB®

characterization plots from Fig. 32 to Fig. 37.

**5. Conclusion** 

simulations.

this chapter can be indicated as follows.

and 170 mm/s2 along X axis of the waist platform. It can be noted that the characterization plots are quite smooth. The characterization values are feasible in proper regions for the operation of both the parallel manipulators and the proposed waist-trunk system has suitable and feasible operation performances for a robotic system as reported in the

(a) (b) Fig. 37. Computed accelerations in Cartesian space: (a) thorax platform; (b) waist platform From the reported simulation results, it is worth to note that a complex mechanical system such as a humanoid robot can be conveniently modeled and evaluated in Matlab®

In this chapter, design and simulation issues of legged walking robots have been addressed by using modeling and simulation in Matlab® environment. In particular, Matlab® is a powerful computation and simulation software package, which is quite useful for the design and operation performances evaluation of legged robotic systems. Three examples are illustrated and they have been studied for motion feasibility analysis and operation performances characterizations by taking advantages of Matlab® features. Contributions of

A kinematic study of a Chebyshev-Pantograph leg mechanism has been carried out, and equations are formulated in the Matlab® environment. From the reported simulation results, it shows that the practical feasible operation performance of the Chebyshev-Pantograph leg mechanism in a single DOF biped robot. Additionally, a parametric study has been developed by using the elaborated Matlab® analysis code to look for an optimized

A novel biologically inspired tripod walking robot is proposed by defining suitable design and operation solution for leg mechanism. Simulation results show the proposed design performs a tripod walking gait successfully. Operation performance of the leg mechanisms and the tripod walking robot are reported and discussed by using results from Matlab®

A new waist-trunk system for humanoid robots has been proposed by using suitable parallel architectures. The proposed system shows an anthropomorphic design and operation with several DOFs, flexibility, and high payload capacity. Simulation results show

mechanical design and to determine an energy efficient walking gait.

environment due to its flexible programming environment and its powerful toolbox.

Environment 489

Fig. 34. Computed leg displacements of the waist module

Fig. 35. Computed leg displacements of the trunk module: (a) for legs 1, 2, and 3; (b) for legs 4, 5, and 6

Fig. 36. Computed velocities in Cartesian space: (a) thorax platform; (b) waist platform

and 170 mm/s2 along X axis of the waist platform. It can be noted that the characterization plots are quite smooth. The characterization values are feasible in proper regions for the operation of both the parallel manipulators and the proposed waist-trunk system has suitable and feasible operation performances for a robotic system as reported in the characterization plots from Fig. 32 to Fig. 37.

Fig. 37. Computed accelerations in Cartesian space: (a) thorax platform; (b) waist platform

From the reported simulation results, it is worth to note that a complex mechanical system such as a humanoid robot can be conveniently modeled and evaluated in Matlab® environment due to its flexible programming environment and its powerful toolbox.

#### **5. Conclusion**

488 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Fig. 34. Computed leg displacements of the waist module

(a) (b)

4, 5, and 6

Fig. 35. Computed leg displacements of the trunk module: (a) for legs 1, 2, and 3; (b) for legs

(a) (b)

Fig. 36. Computed velocities in Cartesian space: (a) thorax platform; (b) waist platform

In this chapter, design and simulation issues of legged walking robots have been addressed by using modeling and simulation in Matlab® environment. In particular, Matlab® is a powerful computation and simulation software package, which is quite useful for the design and operation performances evaluation of legged robotic systems. Three examples are illustrated and they have been studied for motion feasibility analysis and operation performances characterizations by taking advantages of Matlab® features. Contributions of this chapter can be indicated as follows.

A kinematic study of a Chebyshev-Pantograph leg mechanism has been carried out, and equations are formulated in the Matlab® environment. From the reported simulation results, it shows that the practical feasible operation performance of the Chebyshev-Pantograph leg mechanism in a single DOF biped robot. Additionally, a parametric study has been developed by using the elaborated Matlab® analysis code to look for an optimized mechanical design and to determine an energy efficient walking gait.

A novel biologically inspired tripod walking robot is proposed by defining suitable design and operation solution for leg mechanism. Simulation results show the proposed design performs a tripod walking gait successfully. Operation performance of the leg mechanisms and the tripod walking robot are reported and discussed by using results from Matlab® simulations.

A new waist-trunk system for humanoid robots has been proposed by using suitable parallel architectures. The proposed system shows an anthropomorphic design and operation with several DOFs, flexibility, and high payload capacity. Simulation results show

Design and Simulation of Legged Walking Robots in MATLAB®

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that the proposed waist-trunk system can well imitate movements of human torso for walking and manipulation tasks. Additionally, the proposed design has practical feasible operation performances from the reported simulation results.

#### **6. Acknowledgment**

The first author likes to acknowledge Chinese Scholarship Council (CSC) for supporting his Ph.D. study and research at LARM in the University of Cassino, Italy for the years 2008- 2010.

#### **7. References**


that the proposed waist-trunk system can well imitate movements of human torso for walking and manipulation tasks. Additionally, the proposed design has practical feasible

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**6. Acknowledgment** 

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**21** 

**Modeling, Simulation and Control of a Power** 

Power assist robot is a human-robot cooperation that extends human's abilities and skills in performing works (Kazerooni, 1993). Breakthrough in power assist robots was conceived in early 1960s with "Man-amplifier" and "Hardiman" (Kazerooni, 1993), but the progress of research on this significant field is not satisfactory yet. It is found through literature that power assist systems are currently being developed mainly for sick, physically disabled and old people as healthcare and rehabilitation supports (Kong *et al*., 2009; Seki, Ishihara and Tadakuma, 2009). Few power assist systems have also been developed for other applications such as for lifting baby carriage (Kawashima, 2009), physical support for workers performing agricultural jobs (Tanaka *et al.*, 2008), hydraulic assist for automobiles (Liu *et al.*, 2009), skill-assist for manufacturing (Lee, Hara and Yamada, 2008), assisted slide doors for automobiles (Osamura *et al*., 2008), assist-control for bicycle (Kosuge, Yabushita and

We think that handling heavy objects, which is common and necessary in many industries, is another potential field of application of power assist robots. It is always necessary to move heavy objects in industries such as manufacturing and assembly, mining, construction, logistics and transport, disaster and rescue operations, forestry, agriculture etc. Manual manipulation of heavy objects is very cumbersome and it causes work-related disabilities and disorders such as back pain to humans. On the contrary, handling objects by autonomous systems may not provide required flexibility in many cases. Hence, it is thought that the uses of suitable human-robot cooperation systems such as power assist systems may be appropriate for handling heavy objects in industries. However, suitable power assist systems are not found in industries for this purpose because their design has

A power assist robot reduces the perceived heaviness of an object manipulated with it (Kazerooni, 1993), as illustrated in Fig.1. Hence, load force (manipulative force tangential to

Hirata,2004), assist for sports training (Ding, Ueda and Ogasawara, 2008), etc.

**1.2 Manipulating heavy objects in industries with power assist robots** 

**1. Introduction** 

not got much attention yet.

**1.3 Weight illusion for power assist robots** 

**1.1 Power assist robot and its current applications** 

**Assist Robot for Manipulating Objects** 

**Based on Operator's Weight Perception** 

S. M. Mizanoor Rahman1, Ryojun Ikeura2 and Haoyong Yu1

*1National University of Singapore* 

*2Mie University 1Singapore 2Japan* 


## **Modeling, Simulation and Control of a Power Assist Robot for Manipulating Objects Based on Operator's Weight Perception**

S. M. Mizanoor Rahman1, Ryojun Ikeura2 and Haoyong Yu1 *1National University of Singapore 2Mie University 1Singapore 2Japan* 

#### **1. Introduction**

492 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Kawauchi, M. & Mochimaru, M. (2010). AIST Human Body Properties Database, Digital

Tsai, L.-W. (1999). *Robot Analysis – The Mechanics of Serial and Parallel Manipulator*, John Wiley

Ceccarelli, M. (2004). *Fundamental of Mechanics of Robotic Manipulator*, Kluwer Academic

Sadjadian, H.; & Taghirad, H.D. (2006). Kinematic, Singularity and Stiffness Analysis of the

Visual Reality Toolbox User's Guide. (2007). The MathWorks, Inc. Available from http://www.mathworks.com/access/helpdesk/help/pdf\_doc/vr/vr.pdf Siciliano, B. & Khatib, O. (2008). *Springer Handbook of robotics*, Springer-Verlag, Berlin

Harada, K.; Miura, K.; Morisawa, M.; Kaneko, K.; Nakaoka, S.; Kanehiro, F.; Tsuji, T. &

2010.

& Sons, New York, USA.

Heidelberg, Germany.

MO, USA, pp. 1071-1077.

Publishers, Dordrecht, Germany.

*Robotics*, Vol. 20, n. 7, pp. 763–781.

Human Laboratory (AIST, Japan), Available on line: http://www.dh.aist.go.jp,

Hydraulic Shoulder: A 3-d.o.f. Redundant Parallel Manipulator, *Journal of Advanced* 

Kajita, S. (2009). Toward Human-Like Walking Pattern Generator. *In Proceedings of the 2009 IEEE/RSJ international Conference on intelligent Robots and Systems*, St. Louis,

#### **1.1 Power assist robot and its current applications**

Power assist robot is a human-robot cooperation that extends human's abilities and skills in performing works (Kazerooni, 1993). Breakthrough in power assist robots was conceived in early 1960s with "Man-amplifier" and "Hardiman" (Kazerooni, 1993), but the progress of research on this significant field is not satisfactory yet. It is found through literature that power assist systems are currently being developed mainly for sick, physically disabled and old people as healthcare and rehabilitation supports (Kong *et al*., 2009; Seki, Ishihara and Tadakuma, 2009). Few power assist systems have also been developed for other applications such as for lifting baby carriage (Kawashima, 2009), physical support for workers performing agricultural jobs (Tanaka *et al.*, 2008), hydraulic assist for automobiles (Liu *et al.*, 2009), skill-assist for manufacturing (Lee, Hara and Yamada, 2008), assisted slide doors for automobiles (Osamura *et al*., 2008), assist-control for bicycle (Kosuge, Yabushita and Hirata,2004), assist for sports training (Ding, Ueda and Ogasawara, 2008), etc.

#### **1.2 Manipulating heavy objects in industries with power assist robots**

We think that handling heavy objects, which is common and necessary in many industries, is another potential field of application of power assist robots. It is always necessary to move heavy objects in industries such as manufacturing and assembly, mining, construction, logistics and transport, disaster and rescue operations, forestry, agriculture etc. Manual manipulation of heavy objects is very cumbersome and it causes work-related disabilities and disorders such as back pain to humans. On the contrary, handling objects by autonomous systems may not provide required flexibility in many cases. Hence, it is thought that the uses of suitable human-robot cooperation systems such as power assist systems may be appropriate for handling heavy objects in industries. However, suitable power assist systems are not found in industries for this purpose because their design has not got much attention yet.

#### **1.3 Weight illusion for power assist robots**

A power assist robot reduces the perceived heaviness of an object manipulated with it (Kazerooni, 1993), as illustrated in Fig.1. Hence, load force (manipulative force tangential to

Modeling, Simulation and Control of a Power Assist

**1.5 Lifting, lowering and horizontal manipulation** 

characteristics in their control modeling (Hara, 2007).

**1.6 The chapter summary** 

of this chapter.

Robot for Manipulating Objects Based on Operator's Weight Perception 495

In industries, workers need to transfer objects in different directions such as vertical lifting (lift objects from lower to higher position), vertical lowering (lower objects from higher to lower position), horizontal manipulation etc. in order to satisfy task requirements. We assume that maneuverability, heaviness perception, load force and motions for manipulating objects among these directions may be different from each other and these differences may affect the control and the system performances. Hence, it seems to be necessary to study object manipulation in all of these directions, compare them to each other, and to reflect the differences in the control (Rahman et al., 2010a, 2011a). However, such study has also not been carried out yet in detailed.We studied lifting objects in vertical direction in our previous works (Rahman *et al*., 2009a, 2010c, 2011a), but manipulating objects in horizontal direction is still unaddressed though horizontal manipulation of objects is very common in practical fields. A few power-assist robotic systems consider manipulating objects in horizontal direction. But, they are not targetted to industrial applications and they have limitations in performances as they do not consider human

This chapter presents a power assist robot system developed for manipulating objects in horizontal direction in cooperation with human. Weight perception was included in robot dynamics and control. The robot was simulated for manipulating objects in horizontal direction. Optimum maneuverability conditions for horizontal manipulation of objects were determined and were compared to that for vertical lifting of objects. Psychophysical relationships between actual and perceived weights were determined, and load forces and motion features were analyzed for horizontal manipulation of objects. Then, a novel control scheme was implemented to reduce the excessive load forces and accelerations, and thus to improve the system performances. The novel control reduced the excessive load forces and accelerations for horizontal manipulation of objects, and thus improved the system performances in terms of maneuverability,safety, operability etc. We compared our results to that of related works. Finally, we proposed to use the findings to develop human-friendly

This chapter provides information to the readers about the power assist robot system- its innovative mechanical design, dynamics, modeling, control, simulation, application etc. Thus this chapter introduces a new area of applications of power assist robot systems and also introduces innovations in its dynamics, modeling, control etc. On the other hand, the readers will get a detailed explanation and practical example of how to use Matlab/Simulink to develop and simulate a dynamic system (e.g., a power assist robot system). The readers will also receive a practical example of how to measure and evaluate human factors subjectively for a technical domain (e.g., a power assist robot system). As a whole, this chapter will enrich the readers with novel concepts in robotics and control technology, Matlab/Simulink application, human factors/ergonomics, psychology and psychophysics, biomimetics, weight perception, human-robot/machine interaction, user interface design, haptics, cognitive science, biomechanics etc. The contents of this chapter were also compared to related works available in published literatures. Thus, the readers will get a collection of all possible works related and similar to the contents

power assist robots for manipulating heavy objects in various industries.

grip surfaces) required to manipulate an object with a power assist robot should be lower than that required to manipulate the object manually. But, the limitations with the conventional power assist systems are that the operator cannot perceive the heaviness of the object correctly before manipulating it with the system and eventually applies excessive load force. The excessive load force results in sudden increase in acceleration, fearfulness of the operator, lack of maneuverability and stability, fatal accident etc. Fig.2 further explains the interaction processes and phenomena between a power assist robot and its operator for object manipulation. A few power assist systems are available for carrying objects (Doi *et al*., 2007; Hara, 2007; Lee *et al*., 2000; Miyoshi and Terashima, 2004). But, their safety, maneuverability, operability, naturalness, stability and other interactions with users are not so satisfactory because their controls do not consider human characteristics especially weight illusion and load force features.

Fig. 1. A human manipulates (lifts) an object with a power assist robot and feels a scaleddown portion of the weight.

#### **1.4 Distinctions between unimanual and bimanual manipulation**

It is noticed in practices in industries that workers need to employ one or two hands to manipulate objects and they decide this on the basis of object's physical features such as shape, size, mass etc. as well as of task requirements (Bracewell *et al*., 2003; Giachritsis and Wing, 2008; Lum, Reinkensmeyer and Lehman, 1993;Rahman *et al*., 2009a). We assume that weight perception, load force and object motions for unimanual manipulation may be different from that for bimanual manipulation, and these differences may affect modeling the control. Hence, it seems to be necessary to study unimanual weight perception, load force and motion features and to compare these to that for bimanual manipulation, and to reflect the differences in modeling the power-assist control. We studied distinctions between unimanual and bimanual manipulation in our previous works though it is still necessary to deeply look into their differences to make the control more appropriate (Rahman *et al*., 2009a , 2011a).

Fig. 2. Interaction processes and phenomena between robot and human when manipulating an object with a power assist robot.

#### **1.5 Lifting, lowering and horizontal manipulation**

In industries, workers need to transfer objects in different directions such as vertical lifting (lift objects from lower to higher position), vertical lowering (lower objects from higher to lower position), horizontal manipulation etc. in order to satisfy task requirements. We assume that maneuverability, heaviness perception, load force and motions for manipulating objects among these directions may be different from each other and these differences may affect the control and the system performances. Hence, it seems to be necessary to study object manipulation in all of these directions, compare them to each other, and to reflect the differences in the control (Rahman et al., 2010a, 2011a). However, such study has also not been carried out yet in detailed.We studied lifting objects in vertical direction in our previous works (Rahman *et al*., 2009a, 2010c, 2011a), but manipulating objects in horizontal direction is still unaddressed though horizontal manipulation of objects is very common in practical fields. A few power-assist robotic systems consider manipulating objects in horizontal direction. But, they are not targetted to industrial applications and they have limitations in performances as they do not consider human characteristics in their control modeling (Hara, 2007).

#### **1.6 The chapter summary**

494 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

grip surfaces) required to manipulate an object with a power assist robot should be lower than that required to manipulate the object manually. But, the limitations with the conventional power assist systems are that the operator cannot perceive the heaviness of the object correctly before manipulating it with the system and eventually applies excessive load force. The excessive load force results in sudden increase in acceleration, fearfulness of the operator, lack of maneuverability and stability, fatal accident etc. Fig.2 further explains the interaction processes and phenomena between a power assist robot and its operator for object manipulation. A few power assist systems are available for carrying objects (Doi *et al*., 2007; Hara, 2007; Lee *et al*., 2000; Miyoshi and Terashima, 2004). But, their safety, maneuverability, operability, naturalness, stability and other interactions with users are not so satisfactory because their controls do not consider human characteristics especially

Fig. 1. A human manipulates (lifts) an object with a power assist robot and feels a scaled-

It is noticed in practices in industries that workers need to employ one or two hands to manipulate objects and they decide this on the basis of object's physical features such as shape, size, mass etc. as well as of task requirements (Bracewell *et al*., 2003; Giachritsis and Wing, 2008; Lum, Reinkensmeyer and Lehman, 1993;Rahman *et al*., 2009a). We assume that weight perception, load force and object motions for unimanual manipulation may be different from that for bimanual manipulation, and these differences may affect modeling the control. Hence, it seems to be necessary to study unimanual weight perception, load force and motion features and to compare these to that for bimanual manipulation, and to reflect the differences in modeling the power-assist control. We studied distinctions between unimanual and bimanual manipulation in our previous works though it is still necessary to deeply look into their

Fig. 2. Interaction processes and phenomena between robot and human when manipulating

Manipulative forces are incorrect, system behaviours are unexpected

Human programs feed-forward manipulative forces based on visually perceived weight

weight

Human manipulates the object with the robot

Human experiences that haptically perceived weight is different from visually perceived

**1.4 Distinctions between unimanual and bimanual manipulation** 

differences to make the control more appropriate (Rahman *et al*., 2009a , 2011a).

Human perceives the weight visually

weight illusion and load force features.

down portion of the weight.

an object with a power assist robot.

A novel control strategy is needed to overcome these problems

Object to be manipulated with a power assist robot This chapter presents a power assist robot system developed for manipulating objects in horizontal direction in cooperation with human. Weight perception was included in robot dynamics and control. The robot was simulated for manipulating objects in horizontal direction. Optimum maneuverability conditions for horizontal manipulation of objects were determined and were compared to that for vertical lifting of objects. Psychophysical relationships between actual and perceived weights were determined, and load forces and motion features were analyzed for horizontal manipulation of objects. Then, a novel control scheme was implemented to reduce the excessive load forces and accelerations, and thus to improve the system performances. The novel control reduced the excessive load forces and accelerations for horizontal manipulation of objects, and thus improved the system performances in terms of maneuverability,safety, operability etc. We compared our results to that of related works. Finally, we proposed to use the findings to develop human-friendly power assist robots for manipulating heavy objects in various industries.

This chapter provides information to the readers about the power assist robot system- its innovative mechanical design, dynamics, modeling, control, simulation, application etc. Thus this chapter introduces a new area of applications of power assist robot systems and also introduces innovations in its dynamics, modeling, control etc. On the other hand, the readers will get a detailed explanation and practical example of how to use Matlab/Simulink to develop and simulate a dynamic system (e.g., a power assist robot system). The readers will also receive a practical example of how to measure and evaluate human factors subjectively for a technical domain (e.g., a power assist robot system). As a whole, this chapter will enrich the readers with novel concepts in robotics and control technology, Matlab/Simulink application, human factors/ergonomics, psychology and psychophysics, biomimetics, weight perception, human-robot/machine interaction, user interface design, haptics, cognitive science, biomechanics etc. The contents of this chapter were also compared to related works available in published literatures. Thus, the readers will get a collection of all possible works related and similar to the contents of this chapter.

$$
\hbar m \dddot{x}\_d = f\_h + F\_0 \tag{1}
$$

$$
\dot{m}\_1 \dddot{x}\_d = f\_h + m\_2 g. \tag{2}
$$

$$
\vec{x}\_d = \frac{1}{m\_1} (f\_h + m\_2 g) \tag{3}
$$

$$
\dot{\boldsymbol{x}}\_d = \int \vec{\boldsymbol{x}}\_d \, d\boldsymbol{t} \tag{4}
$$

$$\mathbf{x}\_d = \int \dot{\mathbf{x}}\_d \, d\mathbf{t} \tag{5}$$

$$
\dot{\mathfrak{x}}\_c = \dot{\mathfrak{x}}\_d + G(\mathfrak{x}\_d - \mathfrak{x}) \tag{6}
$$


Modeling, Simulation and Control of a Power Assist

used with the servomotor in velocity control mode.

maneuverability plays the pivotal role.

power assist robot system.

robot system in horizontal direction.

**3.3 Objectives** 

Robot for Manipulating Objects Based on Operator's Weight Perception 499

Fig. 5. Block diagram of the power-assist control, where G denotes feedback gain, D/A indicates D/A converter and *x* denotes actual displacement. Feedback position control is

There are two requirements for power assist systems. The first requirement is amplification of human force, assistance of human motion etc. This is the realization of power assist itself, and there may have problems such as its realization method and stability of human-robot system. The second is safety, sense of security, operability, ease of use etc. This requirement does not appear as specific issue comparing to the first requirement, and it is difficult to be taken into account. However, in order to make power assist systems useful, the second requirement is more important than the first requirement. In case of lifting objects, we think that the main requirements for the power assist systems are maneuverability, safety and stability. Again, these requirements are interrelated where

Some basic requirements of a power assist system regarding its maneuverability have been mentioned by Seki, Iso and Hori (2002).However, we thought that only the light (less force required), natural and safe system can provide consistent feelings of ease of use and comfort though too light system may be unsafe, uneasy and uncomfortable. Hence, we considered operator's ease of use and comfort as the evaluation criteria for maneuverability of the

Objectives of experiment 1 were to (i) determine conditions for optimum maneuverability, (ii) determine psychophysical relationships between actual and perceived weights, (iii) analyze load force and determine excess in load force, (iv) analyze object's motionsdisplacement, velocity and acceleration etc. for manipulating objects with the power assist

Position based impedance control and torque/force based impedance control produce good results. Results may be different for force control for reducing excessive force. Our control as introduced above is limited to position based impedance control. We used position based impedance control for the following reasons (advantages):


However, there are some disadvantages of position control as the following:


If the difference between *m* and *m*1 is very large i.e., if (*m*-*m*1) is very big, the position control imposes very high load to the servomotor that results in instability, which is not so intensive for force control. Force control is better in some areas, but position control is better in some other areas. However, position control is to be effective for this chapter. Force control may be considered in near future.

The control shown in Fig.5 is not so complicated. However, there is novelty in this control that human's perception is included in this control. Again, another novel control strategy is also derived from this control that includes human features. This control can be recognized as an exemplary and novel control for human interactive robot control.

#### **3. Experiment 1: Analyzing maneuverability,heaviness perception, force and motion features**

#### **3.1 Subjects**

Ten mechanical engineering male students aged between 23 and 30 years were nominated to voluntarily participate in the experiment. The subjects were believed to be physically and mentally healthy. The subjects did not have any prior knowledge of the hypothesis being tested. Instructions regarding the experiment were given to them, but no formal training was arranged.

#### **3.2 Evaluation criteria for power assist robot systems**

Power assist can be defined as augmenting the ability or adjusting to the situation when human operates and works. In particular, in case of supporting for elderly and disabled people, the purpose of power assist is improvement of QOL (Quality of Life), that is, support for daily life. It has two meanings. One is support for self-help and the other is support for caring. The former is to support self-sustained daily life, and the latter is to decrease burden of caregiver.

Position based impedance control and torque/force based impedance control produce good results. Results may be different for force control for reducing excessive force. Our control as introduced above is limited to position based impedance control. We used position based

1. Position based impedance control automatically compensates the effects of friction, inertia, viscosity etc. In contrast, these effects are needed to consider for force control, however, it is very difficult to model and calculate the friction force for the force control. Dynamic effects, nonlinear forces etc. affect system performances for force control for

2. Ball-screw gear ratio is high and actuator force is less for position control. However, the

3. It is easy to realize the real system for the position control for high gear ratio. However,

1. Instability is high in position control. If we see Fig.5 we find that a feedback loop is created between *x* and *fh* when human touches/grasps the object for manipulation. This feedback effect causes instability. In contrast, force control has less or no stability

3. Value of G and velocity control also can compensate the effects of friction, inertia, viscosity etc., but the effects are not compensated completely, which may affect

If the difference between *m* and *m*1 is very large i.e., if (*m*-*m*1) is very big, the position control imposes very high load to the servomotor that results in instability, which is not so intensive for force control. Force control is better in some areas, but position control is better in some other areas. However, position control is to be effective for this chapter. Force control may

The control shown in Fig.5 is not so complicated. However, there is novelty in this control that human's perception is included in this control. Again, another novel control strategy is also derived from this control that includes human features. This control can be recognized

**3. Experiment 1: Analyzing maneuverability,heaviness perception, force and** 

Ten mechanical engineering male students aged between 23 and 30 years were nominated to voluntarily participate in the experiment. The subjects were believed to be physically and mentally healthy. The subjects did not have any prior knowledge of the hypothesis being tested. Instructions regarding the experiment were given to them, but no formal training

Power assist can be defined as augmenting the ability or adjusting to the situation when human operates and works. In particular, in case of supporting for elderly and disabled people, the purpose of power assist is improvement of QOL (Quality of Life), that is, support for daily life. It has two meanings. One is support for self-help and the other is support for caring. The former is to support self-sustained daily life, and the latter is to

However, there are some disadvantages of position control as the following:

2. Motor system delay affects the stability more intensively for position control.

as an exemplary and novel control for human interactive robot control.

**3.2 Evaluation criteria for power assist robot systems** 

impedance control for the following reasons (advantages):

multi-degree of freedom system.

human's weight perception.

be considered in near future.

**motion features** 

**3.1 Subjects** 

was arranged.

decrease burden of caregiver.

problem.

opposite is true for the force control.

the opposite is true for the force control.

Fig. 5. Block diagram of the power-assist control, where G denotes feedback gain, D/A indicates D/A converter and *x* denotes actual displacement. Feedback position control is used with the servomotor in velocity control mode.

There are two requirements for power assist systems. The first requirement is amplification of human force, assistance of human motion etc. This is the realization of power assist itself, and there may have problems such as its realization method and stability of human-robot system. The second is safety, sense of security, operability, ease of use etc. This requirement does not appear as specific issue comparing to the first requirement, and it is difficult to be taken into account. However, in order to make power assist systems useful, the second requirement is more important than the first requirement. In case of lifting objects, we think that the main requirements for the power assist systems are maneuverability, safety and stability. Again, these requirements are interrelated where maneuverability plays the pivotal role.

Some basic requirements of a power assist system regarding its maneuverability have been mentioned by Seki, Iso and Hori (2002).However, we thought that only the light (less force required), natural and safe system can provide consistent feelings of ease of use and comfort though too light system may be unsafe, uneasy and uncomfortable. Hence, we considered operator's ease of use and comfort as the evaluation criteria for maneuverability of the power assist robot system.

#### **3.3 Objectives**

Objectives of experiment 1 were to (i) determine conditions for optimum maneuverability, (ii) determine psychophysical relationships between actual and perceived weights, (iii) analyze load force and determine excess in load force, (iv) analyze object's motionsdisplacement, velocity and acceleration etc. for manipulating objects with the power assist robot system in horizontal direction.

Modeling, Simulation and Control of a Power Assist

1. Very Easy & Comfortable (score: +2) 2. Easy & Comfortable (score: +1)

4. Uneasy & Uncomfortable (score: -1) 5. Very Uneasy & Uncomfortable (score: -2)

3. Borderline (score: 0)

separately for each trial.

Robot for Manipulating Objects Based on Operator's Weight Perception 501

All subjects evaluated the system for maneuverability as above for small, medium, large object independently for each *m*1 and *m*2 set. Load force and motions data were recorded

Fig. 7. Setting appropriate parameters for the custom-derived blocks.

Each subject after each trial also manually manipulated a reference-weight object horizontally on a table using right hand alone for reference weights. Weight of the referenceweight object was sequentially changed in a descending order starting from 0.1 kg and ending at 0.01 kg maintaining an equal difference of 0.01 kg i.e., 0.1, 0.09,…0.02, 0.01kg.The subject thus compared the perceived weight of the PAO to that of the reference-weight object and estimated the magnitude of the perceived weight following the psychophysical method 'constant stimuli'. Appearance of PAO and reference-weight object were the same.

*m***1 (kg)** 2.0 1.5 1.0 0.5

*m***2 (kg)** 0.09 0.06 0.03

Table 1. Values of variables for the simulation

#### **3.4 Design of the experiment**

Independent variables were *m*1 and *m*2 values, and visual object sizes. Dependent variables were maneuverability, perceived weight, load force, and object's motions (displacement, velocity and acceleration).

#### **3.5 Experiment procedures**

The system shown in Fig.5 was simulated using Matlab/Simulink (solver: ode4, Runge-Kutta; type: fixed-step; fundamental sample time: 0.001s) for twelve *m*1 and *m*2 sets (Table 1) separately. The ranges of values of *m*1 and *m*2 were nominated based on our experience. The program for simulation is shown in Fig.6. We set the parameters of three custom-derived blocks such as counter, D/A converter and A/D converter before the simulation started. Fig. 7 shows what the parameters were and how they were set.

Fig. 6. The program for the simulation

The subject manipulated (from 'A' to 'B' as in Fig.4, distance between 'A' and 'B' was about 0.12 m) each size object with the robot system once for each *m*1 and *m*2 set separately. The task required the subject to manipulate the object approximately 0.1m, maintain the object for 1-2 seconds and then release the object. For each trial (for each *m*1 and *m*2 set for each size object), the subject subjectively evaluated the system for maneuverability as any one of the following alternatives:-


Independent variables were *m*1 and *m*2 values, and visual object sizes. Dependent variables were maneuverability, perceived weight, load force, and object's motions (displacement,

The system shown in Fig.5 was simulated using Matlab/Simulink (solver: ode4, Runge-Kutta; type: fixed-step; fundamental sample time: 0.001s) for twelve *m*1 and *m*2 sets (Table 1) separately. The ranges of values of *m*1 and *m*2 were nominated based on our experience. The program for simulation is shown in Fig.6. We set the parameters of three custom-derived blocks such as counter, D/A converter and A/D converter before the simulation started.

The subject manipulated (from 'A' to 'B' as in Fig.4, distance between 'A' and 'B' was about 0.12 m) each size object with the robot system once for each *m*1 and *m*2 set separately. The task required the subject to manipulate the object approximately 0.1m, maintain the object for 1-2 seconds and then release the object. For each trial (for each *m*1 and *m*2 set for each size object), the subject subjectively evaluated the system for maneuverability as any one of the

Fig. 7 shows what the parameters were and how they were set.

**3.4 Design of the experiment** 

velocity and acceleration).

**3.5 Experiment procedures** 

Fig. 6. The program for the simulation

following alternatives:-


All subjects evaluated the system for maneuverability as above for small, medium, large object independently for each *m*1 and *m*2 set. Load force and motions data were recorded separately for each trial.

Fig. 7. Setting appropriate parameters for the custom-derived blocks.

Each subject after each trial also manually manipulated a reference-weight object horizontally on a table using right hand alone for reference weights. Weight of the referenceweight object was sequentially changed in a descending order starting from 0.1 kg and ending at 0.01 kg maintaining an equal difference of 0.01 kg i.e., 0.1, 0.09,…0.02, 0.01kg.The subject thus compared the perceived weight of the PAO to that of the reference-weight object and estimated the magnitude of the perceived weight following the psychophysical method 'constant stimuli'. Appearance of PAO and reference-weight object were the same.


Table 1. Values of variables for the simulation

Modeling, Simulation and Control of a Power Assist

insignificant (*F*9,18<1 for each *m*1 and *m*<sup>2</sup> set).

gravity force is compensated.

**3.6.2 Relationship between actual and perceived weight** 

Robot for Manipulating Objects Based on Operator's Weight Perception 503

We determined the mean perceived weight for each size object separately for *m*1=0.5kg, *m*2=0.03kg (condition 1) and *m*1=1kg, *m*2=0.03 kg (condition 2) as shown in Fig.8. We assumed *m*2 as the actual weight of the power-assisted object. It means that the actual weight was 0.03kg or 0.2943 N for each size object for the two *m*1 and *m*2 sets. We compared the perceived weights of Fig.8 to the actual weight (0.2943 N) for each size object for *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03 kg. The figure shows and we also found in our previous research that *m*1 does not affect weight perception, but *m*2 does affect (Rahman et al., 2009a, 2011a). We also see that visual object sizes do not affect weight perception (Gordon *et al*., 1991). Results for two-way (visual object size, subject) analyses of variances separately analyzed on perceived weights for the two *m*1 and *m*2 sets showed that variations due to object sizes were insignificant (*F*2, 18 <1 for each *m*1 and *m*2 set).The reason may be that subjects estimated perceived weights using haptic cues where visual cues had no influences. Variations among subjects were also found statistically

The actual weight of the object was 0.2943 N, but the subjects felt about 0.052 N when the object was manipulated with the power assist robot system in horizontal direction. Hence, the results reveal that the perceived weight is about 18% of the actual weight if an object is manipulated horizontally with a power assist robot system. Its physical meaning is that the perceived weight of an object manipulated with power-assist in horizontal direction is 18% of the perceived weight of the same object manipulated in horizontal direction manually. This happens because the power assist robot system reduces the perceived weight through its assistance to the user. It is a well-known concept that a power assist robot system reduces the feeling of weight. However, it was not quantified. This research quantified the weight attenuation for horizontal manipulation of objects with the power assist robot system. As we found in our previous research, the perceived weight reduces to 40% and 20% of the actual weight if the object is vertically lifted (Rahman *et al*., 2011a) or vertically lowered (Rahman *et al*., 2011b) respectively. The weight perception is less for horizontal manipulation as the

Fig. 8. Mean (*n*=10) perceived weights for different object sizes for condition 1 (*m*1=0.5kg,

*m*2=0.03kg) and condition 2 (*m*1=1kg, *m*2=0.03 kg).

#### **3.6 Experiment results**

#### **3.6.1 Optimum maneuverability**

Mean evaluation scores of the system regarding its maneuverability for 12 *m*1 and *m*2 sets for each size object were determined separately. Table 2 shows the mean evaluation scores for the medium size object. Similar scores were also determined for large and small size objects. The results reveal that maneuverability is not affected by visual size of object. The reason may be that human evaluates maneuverability using haptic senses where visual size cue has no influence. However, haptic cues might influence the maneuverability.

The table shows that ten *m*1 and *m*2 sets got positive scores whereas the remaining two sets got negative scores. Results show that *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03kg got the highest scores. Hence, optimum maneuverability may be achieved at any one of these two conditions. We think that a unique and single condition for optimum maneuverability could be determined if more values of *m*1 and *m*2 were used for the simulation. The subjects felt very easy and comfortable to manipulate objects with the power assist system only when *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03kg. This is why these two sets were declared as the optimum conditions for maneuverability. Here, optimality was decided based on human's feelings following heuristics.

These findings indicate the significance of our hypothesis that we would not be able to sort out the positive sets (satisfactory level of maneuverability) of values of *m*1 and *m*2 from the negative sets (unsatisfactory level of maneuverability) of values of *m*1 and *m*2 for different sizes of objects unless we thought ݉ଵ ് ݉ଶ ് ݉,݉ଵ ا ݉ǡ ݉ଶ ا ݉,݉ଵݔሷ ௗ ് ݉ଶ݃.

We see that the optimum/best sets are also the sets of the smallest values of *m*1 and *m*2 in this experiment. If much smaller values of *m*1 and *m*2 are chosen randomly, the perceived heaviness may further reduce, but it needs to clarify whether or not this is suitable for human psychology. Again, in zero-gravity or weightless condition when *m*2=0, the object is supposed to be too light as it was studied by Marc and Martin (2002) in actual environment and by Dominjon *et al.* (2005) in virtual environment. It was found that the zero-gravity is not feasible because the human loses some haptic information at zero-gravity that hampers human's weight perception ability (Rahman *et al*., 2009b).It is still not known whether the optimum sets are optimum only for the particular conditions of this experiment or they will persist as the optimum for all conditions in practical uses in industries.


Table 2. Mean maneuverability scores with standard deviations (in parentheses) for the medium size object

#### **3.6.2 Relationship between actual and perceived weight**

502 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Mean evaluation scores of the system regarding its maneuverability for 12 *m*1 and *m*2 sets for each size object were determined separately. Table 2 shows the mean evaluation scores for the medium size object. Similar scores were also determined for large and small size objects. The results reveal that maneuverability is not affected by visual size of object. The reason may be that human evaluates maneuverability using haptic senses where visual size cue has

The table shows that ten *m*1 and *m*2 sets got positive scores whereas the remaining two sets got negative scores. Results show that *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03kg got the highest scores. Hence, optimum maneuverability may be achieved at any one of these two conditions. We think that a unique and single condition for optimum maneuverability could be determined if more values of *m*1 and *m*2 were used for the simulation. The subjects felt very easy and comfortable to manipulate objects with the power assist system only when *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03kg. This is why these two sets were declared as the optimum conditions for maneuverability. Here, optimality was decided based on

These findings indicate the significance of our hypothesis that we would not be able to sort out the positive sets (satisfactory level of maneuverability) of values of *m*1 and *m*2 from the negative sets (unsatisfactory level of maneuverability) of values of *m*1 and *m*2 for different

We see that the optimum/best sets are also the sets of the smallest values of *m*1 and *m*2 in this experiment. If much smaller values of *m*1 and *m*2 are chosen randomly, the perceived heaviness may further reduce, but it needs to clarify whether or not this is suitable for human psychology. Again, in zero-gravity or weightless condition when *m*2=0, the object is supposed to be too light as it was studied by Marc and Martin (2002) in actual environment and by Dominjon *et al.* (2005) in virtual environment. It was found that the zero-gravity is not feasible because the human loses some haptic information at zero-gravity that hampers human's weight perception ability (Rahman *et al*., 2009b).It is still not known whether the optimum sets are optimum only for the particular conditions of this experiment or they will

**m1 m2 Mean maneuverability score** 

Table 2. Mean maneuverability scores with standard deviations (in parentheses) for the

**1** 0.06 +0.83(0.41) **2** 0.06 +0.33(1.21) **0.5** 0.03 +2.0 (0) **1** 0.03 +2.0 (0) **1.5** 0.03 +1.5 (0.55) **2** 0.09 -0.17(0.98) **0.5** 0.06 +1.0 (0) **1.5** 0.09 -0.17(0.98) **0.5** 0.09 +0.17(0.75) **1** 0.09 +1.0 (0.63) **1.5** 0.06 +0.67(0.52) **2** 0.03 +1.17(0.41) ௗ ് ݉ଶ݃.

no influence. However, haptic cues might influence the maneuverability.

sizes of objects unless we thought ݉ଵ ് ݉ଶ ് ݉,݉ଵ ا ݉ǡ ݉ଶ ا ݉,݉ଵݔሷ

persist as the optimum for all conditions in practical uses in industries.

**3.6 Experiment results** 

**3.6.1 Optimum maneuverability** 

human's feelings following heuristics.

medium size object

We determined the mean perceived weight for each size object separately for *m*1=0.5kg, *m*2=0.03kg (condition 1) and *m*1=1kg, *m*2=0.03 kg (condition 2) as shown in Fig.8. We assumed *m*2 as the actual weight of the power-assisted object. It means that the actual weight was 0.03kg or 0.2943 N for each size object for the two *m*1 and *m*2 sets. We compared the perceived weights of Fig.8 to the actual weight (0.2943 N) for each size object for *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03 kg. The figure shows and we also found in our previous research that *m*1 does not affect weight perception, but *m*2 does affect (Rahman et al., 2009a, 2011a). We also see that visual object sizes do not affect weight perception (Gordon *et al*., 1991). Results for two-way (visual object size, subject) analyses of variances separately analyzed on perceived weights for the two *m*1 and *m*2 sets showed that variations due to object sizes were insignificant (*F*2, 18 <1 for each *m*1 and *m*2 set).The reason may be that subjects estimated perceived weights using haptic cues where visual cues had no influences. Variations among subjects were also found statistically insignificant (*F*9,18<1 for each *m*1 and *m*<sup>2</sup> set).

The actual weight of the object was 0.2943 N, but the subjects felt about 0.052 N when the object was manipulated with the power assist robot system in horizontal direction. Hence, the results reveal that the perceived weight is about 18% of the actual weight if an object is manipulated horizontally with a power assist robot system. Its physical meaning is that the perceived weight of an object manipulated with power-assist in horizontal direction is 18% of the perceived weight of the same object manipulated in horizontal direction manually. This happens because the power assist robot system reduces the perceived weight through its assistance to the user. It is a well-known concept that a power assist robot system reduces the feeling of weight. However, it was not quantified. This research quantified the weight attenuation for horizontal manipulation of objects with the power assist robot system. As we found in our previous research, the perceived weight reduces to 40% and 20% of the actual weight if the object is vertically lifted (Rahman *et al*., 2011a) or vertically lowered (Rahman *et al*., 2011b) respectively. The weight perception is less for horizontal manipulation as the gravity force is compensated.

Fig. 8. Mean (*n*=10) perceived weights for different object sizes for condition 1 (*m*1=0.5kg, *m*2=0.03kg) and condition 2 (*m*1=1kg, *m*2=0.03 kg).

Modeling, Simulation and Control of a Power Assist

0.12

0.1

0.08

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0.04

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0

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0.08

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0.02

0


Robot for Manipulating Objects Based on Operator's Weight Perception 505

End

10 10.5 11 11.5 12 12.5 13 13.5 14

Start

Peak

Time (s)

10 10.5 11 11.5 12 12.5 13 13.5 14

Time (s)

#### **3.6.3 Force analysis**

The time trajectory of load force for a typical trial is shown in Fig.9. We derived the magnitude of peak load force (PLF) for each object size for condition 1 (*m*1=0.5kg, *m*2=0.03kg) and condition 2 (*m*1=1kg, *m*2=0.03 kg) separately and determined the mean PLFs. The results are shown in Table 3. Results show that mean PLFs for condition 2 are slightly larger than that for condition 1. We found previously that both *m*1 and *m*2 are linearly proportional to peak load force. However, *m*1 affects load force, but it does not affect weight perception. On the other hand, *m*2 affects both load force and weight perception (Rahman *et al*., 2011a). Here, we assume that larger *m*1 in condition 2 has produced larger load force.

We have already found that subjects feel the best maneuverability at *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03 kg. On the other hand, actually required PLF to manipulate the powerassisted object should be slightly larger than the perceived weight (Gordon *et al*., 1991), which is 0.052 N. We compared the perceived weights from Fig.8 to the PLFs (Table 3) for the large, medium and small objects and determined the excess in PLFs. The results show that subjects apply load forces that are extremely larger than the actually required load forces for condition 1 and 2. We also see that the magnitudes of PLFs are proportional to object sizes (Gordon *et al*., 1991). We assume that the excessive load forces create problems in terms of maneuverability, safety, motions etc. that we discussed in the introduction.

#### **3.6.4 Motion analysis**

Fig.9 shows trajectories of displacement,velocity and acceleration for a typical trial. The figure shows that the time trajectories of load force and object's acceleration are synchronized i.e., when load force reaches the peak; acceleration also reaches the peak and so on. However, the trajectory of displacement is different from that of load force and acceleration i.e., the displacement is not entirely synchronized with load force and acceleration. Hence, we see that there is a time delay between PLF (peak acceleration as well) and peak displacement. Previously we assumed that the time delay is caused due to a delay in position sensing (Rahman et al., 2010b), but this research reveals that the time delay may be caused by the combined effects of the time constant of the position sensor and the delay in adjusting the situation and motions by the subject. We also assume that the time delay may cause the feeling of reduced heaviness of the object manipulated with the power assist robot system.

We derived peak velocity and peak acceleration for each trial and determined their means for each object size in each condition separately as shown in Table 4 and Table 5 respectively. The results show that the velocity and accelerations are large. We assume that the large peak load forces have resulted in large accelerations that are harmful to the system in terms of maneuverability, safety, motions etc.


Table 3. Mean peak load forces for different conditions for different object sizes

The time trajectory of load force for a typical trial is shown in Fig.9. We derived the magnitude of peak load force (PLF) for each object size for condition 1 (*m*1=0.5kg, *m*2=0.03kg) and condition 2 (*m*1=1kg, *m*2=0.03 kg) separately and determined the mean PLFs. The results are shown in Table 3. Results show that mean PLFs for condition 2 are slightly larger than that for condition 1. We found previously that both *m*1 and *m*2 are linearly proportional to peak load force. However, *m*1 affects load force, but it does not affect weight perception. On the other hand, *m*2 affects both load force and weight perception (Rahman *et al*., 2011a). Here, we assume that larger *m*1 in condition 2 has

We have already found that subjects feel the best maneuverability at *m*1=0.5kg, *m*2=0.03kg and *m*1=1kg, *m*2=0.03 kg. On the other hand, actually required PLF to manipulate the powerassisted object should be slightly larger than the perceived weight (Gordon *et al*., 1991), which is 0.052 N. We compared the perceived weights from Fig.8 to the PLFs (Table 3) for the large, medium and small objects and determined the excess in PLFs. The results show that subjects apply load forces that are extremely larger than the actually required load forces for condition 1 and 2. We also see that the magnitudes of PLFs are proportional to object sizes (Gordon *et al*., 1991). We assume that the excessive load forces create problems in terms of maneuverability, safety, motions etc. that we discussed in the introduction.

Fig.9 shows trajectories of displacement,velocity and acceleration for a typical trial. The figure shows that the time trajectories of load force and object's acceleration are synchronized i.e., when load force reaches the peak; acceleration also reaches the peak and so on. However, the trajectory of displacement is different from that of load force and acceleration i.e., the displacement is not entirely synchronized with load force and acceleration. Hence, we see that there is a time delay between PLF (peak acceleration as well) and peak displacement. Previously we assumed that the time delay is caused due to a delay in position sensing (Rahman et al., 2010b), but this research reveals that the time delay may be caused by the combined effects of the time constant of the position sensor and the delay in adjusting the situation and motions by the subject. We also assume that the time delay may cause the feeling of reduced heaviness of the object manipulated with the power

We derived peak velocity and peak acceleration for each trial and determined their means for each object size in each condition separately as shown in Table 4 and Table 5 respectively. The results show that the velocity and accelerations are large. We assume that the large peak load forces have resulted in large accelerations that are harmful to the system

*m*1=0.5kg, *m*2=0.03kg 2.9131(0.1307) 2.6020(0.1151) 2.4113(0.1091) *m*1=1.0kg, *m*2=0.03kg 2.9764(0.2009) 2.6554(0.1552) 2.4602(0.1367)

Table 3. Mean peak load forces for different conditions for different object sizes

Mean PLFs (N) with standard deviations (in parentheses) for different object sizes Large Medium Small

**3.6.3 Force analysis** 

produced larger load force.

**3.6.4 Motion analysis** 

assist robot system.

in terms of maneuverability, safety, motions etc.

*m*1, *m*2 sets

Modeling, Simulation and Control of a Power Assist

sizes for different conditions

object sizes for different conditions.

*m*1=6\**e*-6t + 1.0, *m*2=0.03 are the same.

equal-interval scale as follows:

1. Best (score: +3) 2. Better (score: +2)

**4.1 Experiment** 

Robot for Manipulating Objects Based on Operator's Weight Perception 507

**Large** 0.1497(0.0149) 0.1557(0.0209) **Medium** 0.1345(0.0157) 0.1399(0.0122) **Small** 0.1098(0.0121) 0.1176(0.0119) Table 4. Mean peak velocity with standard deviations (in parentheses) for different object

*m*1=0.5kg, *m*2=0.03kg *m*1=1.0kg, *m*2=0.03kg

*m*1=0.5kg, *m*2=0.03kg *m*1=1.0kg, *m*2=0.03kg

**Object size Mean peak velocity (m/s)** 

**Object size Mean peak acceleration (m/s2)** 

**4. Experiment 2: Improving system performances by a novel control** 

load forces and accelerations by applying a novel control method.

**Large** 0.2309 (0.0901) 0.2701 (0.0498) **Medium** 0.2282 (0.0721) 0.2542(0.0153) **Small** 0.1887(0.0298) 0.2134(0.0525) Table 5. Mean peak accelerations with standard deviations (in parentheses) for different

Table 3 and Table 5 show that subjects apply too excessive load forces and accelerations that cause problems as we discussed in section 1. Experiment 2 attempted to reduce excessive

The novel control was such that the value of *m*<sup>1</sup> exponentially declined from a large value to 0.5kg when the subject manipulated the PAO with the system and the command velocity of Eq.(6) exceeded a threshold. We found previously that load force is linearly proportional to *m*<sup>1</sup>and we also found that subjects do not feel the change of *m*1 (Rahman et al., 2011a). Hence, reduction in *m*<sup>1</sup> would also reduce the load force proportionally. Reduction in load force would not adversely affect the relationships of Eq. (2) because the subjects would not feel the change of *m*1. It means that Eq. (7) and Eq. (8) were used for *m*1 and *m*2 respectively to modify the control of Fig.5. The digit 6 in Eq. (7) was determined by trial and error. The novel control is illustrated in Fig.10 as a flowchart. The procedures for experiment 2 were the same as that for the experiment 1, but *m*1 and *m*2 were set as *m*1=6\**e*-6t + 0.5, *m*2=0.03 (condition 1.a) and *m*1=6\**e*-6t + 1.0, *m*2=0.03 (condition 2.a) for the simulation. Program for the simulation is shown in Fig.11. We here ignore presenting the simulation details for *m*1=6\**e*-6t + 1.0, *m*2=0.03 because the concept and procedures for *m*1=6\**e*-6t + 0.5, *m*2=0.03 and

 *m*2=0.03 (8) The system performances were broadly expressed through several criteria such as motion, object mobility, naturalness, stability, safety, ease of use etc., and in each trial in each scheme, the subjects subjectively evaluated (scored) the system using a 7-point bipolar and

 *m*1=6 \* *e*-6t + 0.5 (7)

Fig. 9. Time trajectories of displacement , velocity , acceleration and load force for a trial when a subject manipulated the small size PAO with the system at condition 1 (*m*1=0.5kg, *m*2=0.03kg).


Table 4. Mean peak velocity with standard deviations (in parentheses) for different object sizes for different conditions


Table 5. Mean peak accelerations with standard deviations (in parentheses) for different object sizes for different conditions.

#### **4. Experiment 2: Improving system performances by a novel control**

#### **4.1 Experiment**

506 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

Peak

10 10.5 11 11.5 12 12.5 13 13.5 14

Time (s)

Peak

Fig. 9. Time trajectories of displacement , velocity , acceleration and load force for a trial when a subject manipulated the small size PAO with the system at condition 1 (*m*1=0.5kg,

10 10.5 11 11.5 12 12.5 13 13.5 14

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*m*2=0.03kg).

0.2

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2.5

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0





Table 3 and Table 5 show that subjects apply too excessive load forces and accelerations that cause problems as we discussed in section 1. Experiment 2 attempted to reduce excessive load forces and accelerations by applying a novel control method.

The novel control was such that the value of *m*<sup>1</sup> exponentially declined from a large value to 0.5kg when the subject manipulated the PAO with the system and the command velocity of Eq.(6) exceeded a threshold. We found previously that load force is linearly proportional to *m*<sup>1</sup>and we also found that subjects do not feel the change of *m*1 (Rahman et al., 2011a). Hence, reduction in *m*<sup>1</sup> would also reduce the load force proportionally. Reduction in load force would not adversely affect the relationships of Eq. (2) because the subjects would not feel the change of *m*1. It means that Eq. (7) and Eq. (8) were used for *m*1 and *m*2 respectively to modify the control of Fig.5. The digit 6 in Eq. (7) was determined by trial and error. The novel control is illustrated in Fig.10 as a flowchart. The procedures for experiment 2 were the same as that for the experiment 1, but *m*1 and *m*2 were set as *m*1=6\**e*-6t + 0.5, *m*2=0.03 (condition 1.a) and *m*1=6\**e*-6t + 1.0, *m*2=0.03 (condition 2.a) for the simulation. Program for the simulation is shown in Fig.11. We here ignore presenting the simulation details for *m*1=6\**e*-6t + 1.0, *m*2=0.03 because the concept and procedures for *m*1=6\**e*-6t + 0.5, *m*2=0.03 and *m*1=6\**e*-6t + 1.0, *m*2=0.03 are the same.

$$m\_1 = 6 \,\, ^\ast e^{\ast 6t} + 0.5 \,\, \tag{7}$$

$$m\_2 = 0.03\tag{8}$$

The system performances were broadly expressed through several criteria such as motion, object mobility, naturalness, stability, safety, ease of use etc., and in each trial in each scheme, the subjects subjectively evaluated (scored) the system using a 7-point bipolar and equal-interval scale as follows:


Modeling, Simulation and Control of a Power Assist

Fig. 11. The program for the simulation for the novel control method

change significantly due to the control modification.

**4.2.2 Improvement in system performances** 

*m*1, *m*2 sets

*m*1=6 \* e-6t + 0.5,

*m*1*=*6 *\* e-6t +* 1.0,

control modification

Mean peak accelerations for different object sizes after the control modification are shown in Table 7. The results show, if we compare these to that of Table 5, that the peak accelerations reduced due to control modification. The reason may be that the reduced peak load forces after the control modification reduced the accelerations accordingly. The velocity did not

We determined the mean evaluation scores for the three objects separately. Fig.12 shows the mean scores for the small size object for both conditions. The scores for the large and medium size objects in each condition were almost the same as that shown in the figure. It

*<sup>m</sup>*2=0.03 1.3569 (0.1154) 1.1123(0.0821) 0.9901(0.0910)

*<sup>m</sup>*2*=*0.03 1.8646 (0.1707) 1.5761(0.1071) 1.0990 (0.0885)

Table 6. Mean peak load forces for different conditions for different object sizes after the

Mean PLFs (N) with standard deviations (in parentheses) for different object sizes Large Medium Small

means that the novel control was effective in improving the system performances.

Robot for Manipulating Objects Based on Operator's Weight Perception 509


#### **4.2 Experiment results**

#### **4.2.1 Reduction in peak load forces and peak accelerations**

We compared the mean PLFs of experiment 2 conducted at *m*1=6 \* *e*-6t + 0.5, *m*2=0.03 and *m*1=6 \* *e*-6t + 1.0, *m*2=0.03 to that of experiment 1 conducted at *m*1=0.5, *m*2=0.03 and *m*1=1.0, *m*2=0.03. The findings are shown in Table 6. Findings show that PLFs reduced significantly due to the control modification.

Fig. 10. Flowchart and hypothetical trajectory of inertial mass for the novel control technique.

We compared the mean PLFs of experiment 2 conducted at *m*1=6 \* *e*-6t + 0.5, *m*2=0.03 and *m*1=6 \* *e*-6t + 1.0, *m*2=0.03 to that of experiment 1 conducted at *m*1=0.5, *m*2=0.03 and *m*1=1.0, *m*2=0.03. The findings are shown in Table 6. Findings show that PLFs reduced significantly

*m*1=6.5

Start

ሶ => 0.005

*m*1=6\**e*-6t + 0.5

End

ݔ

Yes

No

Fig. 10. Flowchart and hypothetical trajectory of inertial mass for the novel control technique.

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>0</sup>

Time (s)

**4.2.1 Reduction in peak load forces and peak accelerations** 

3. Good (score: +1) 4. Alike (score: 0) 5. Bad (score:-1) 6. Worse (score:-2) 7. Worst (score:-3)

**4.2 Experiment results** 

due to the control modification.

1

2

3

**Value of** 

*m***1 (kg)**

4

5

6

7

Fig. 11. The program for the simulation for the novel control method

Mean peak accelerations for different object sizes after the control modification are shown in Table 7. The results show, if we compare these to that of Table 5, that the peak accelerations reduced due to control modification. The reason may be that the reduced peak load forces after the control modification reduced the accelerations accordingly. The velocity did not change significantly due to the control modification.

#### **4.2.2 Improvement in system performances**

We determined the mean evaluation scores for the three objects separately. Fig.12 shows the mean scores for the small size object for both conditions. The scores for the large and medium size objects in each condition were almost the same as that shown in the figure. It means that the novel control was effective in improving the system performances.


Table 6. Mean peak load forces for different conditions for different object sizes after the control modification

Modeling, Simulation and Control of a Power Assist

Technology of Japan for financial supports.

Vol. 83, No.3, pp. 477–482.

54, No. 1, pp.638-650.

*Communication*, pp.10-20.

*Man and Cybernetics*, pp.780–785.

Issue: 3, pp.512 – 521.

*Neuroscience*, Vol.18, No.8, pp.2396-2402.

*Biomedical Robotics and Biomechatronics*, pp.181 – 186.

environments', *In Proc. of IEEE Virtual Reality*, pp.19-25.

direction will be investigated.

**6. Acknowledgment** 

**7. References** 

Robot for Manipulating Objects Based on Operator's Weight Perception 511

simulation, Matlab/Simulink, psychology, human factors etc. We will verify the results using heavy objects and real robotic systems in near future. The system will be upgraded to multidegree of freedom system. Distinctions in weight perception, load forces and motion characteristics between unimanual and bimanual manipulation of objects in horizontal

The authors are thankful to the Ministry of Education, Culture, Sports, Science and

Bracewell, R.M., Wing, A.M., Scoper, H.M., Clark, K.G. (2003) 'Predictive and reactive co-

Ding, M., Ueda, J., Ogasawara, T. (2008) 'Pinpointed muscle force control using a power-

Doi, T., Yamada, H., Ikemoto, T. and Naratani, H. (2007) 'Simulation of pneumatic hand crane type power assist system', *In Proc. of SICE Annual Conf.*, pp. 2321 – 2326. Dominjon, L., Lécuyer, A., Burkhardt, J.M., Richard, P. and Richir, S. (2005) 'Influence of

Giachritsis, C. and Wing, A. (2008) 'Unimanual and bimanual weight discrimination in a desktop setup', M. Ferre (Ed.): *EuroHaptics 2008*, LNCS 5024, pp. 378–382. Gordon, A.M., Forssberg, H., Johansson, R.S., Westling, G. (1991) 'Visual size cues in the

Hara, S. (2007) 'A smooth switching from power-assist control to automatic transfer control

Kawashima, T. (2009) 'Study on intelligent baby carriage with power assist system and comfortable basket', *J. of Mechanical Science and Technology*, Vol.23, pp.974-979. Kazerooni, H. (1993) 'Extender: a case study for human-robot interaction via transfer of

Kong, K., Moon, H., Hwang, B., Jeon, D., Tomizuka, M. (2009) 'Impedance compensation of

Kosuge,K., Yabushita, H., Hirata, Y. (2004) 'Load-free control of power-assisted cycle', *In Proc. of IEEE Technical Exhibition Based Conference on Robotics and Automation*, pp.111-112. Lee, H., Takubo, T., Arai, H. and Tanie, K. (2000) 'Control of mobile manipulators for power

Lee, S., Hara, S., Yamada, Y. (2008) 'Safety-preservation oriented reaching monitoring for

assist systems', *Journal of Robotic Systems*, Vol.17, No.9, pp.469-477.

ordination of grip and load forces in bimanual lifting in man', *European Journal of* 

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control/display ratio on the perception of mass of manipulated objects in virtual

programming of manipulative forces during precision grip', *Exp. Brain Research*,

and its application to a transfer machine', *IEEE Trans. on Industrial Electronics*, Vol.

power and information signals', *In Proc. of IEEE Int. Workshop on Robot and Human* 

SUBAR for back-drivable force-mode actuation', *IEEE Trans. on Robotics*, Vol. 25,

smooth control mode switching of skill-assist', *In Proc. of IEEE Int. Conf. on Systems,* 


Table 7. Mean peak accelerations with standard deviations (in parentheses) for different object sizes for different conditions after the control modification

Fig. 12. Mean performance evaluation scores for small size object for condition 1.a (*m*1=6 \* e-6t + 0.5, *m*2=0.03) and condition 2.a (*m*1=6 \* e-6t + 1.0, *m*2=0.03) after the control modification.

#### **5. Conclusions**

In this chapter, we presented a 1-DOF power assist robot system for manipulating objects by human subjects in horizontal direction. We included human features in the robot dynamics and control. We determined optimum maneuverability conditions for manipulating objects with the robot system. We also determined psychophysical relationships between actual and perceived weights for manipulating objects with the robot system. We analyzed weight perception, load forces and motion characteristics. We implemented a novel control method based on weight perception, load forces and motion characteristics that improved the system performances through reducing the peak load forces and peak accelerations. The findings may help develop human-friendly power assist robot devices for manipulating heavy objects in industries such as manufacturing and assembly, mining, logistics and transport, construction etc. This chapter also provides a vivid example to the readers of how Matlab/Simulink is used to model and develop control system and interfaces between hardware and software for simulation and control of a robotic system. The findings of this chapter are novel and they enhance the state-of-the-art knowledge and applications of robotics, control system, simulation, Matlab/Simulink, psychology, human factors etc. We will verify the results using heavy objects and real robotic systems in near future. The system will be upgraded to multidegree of freedom system. Distinctions in weight perception, load forces and motion characteristics between unimanual and bimanual manipulation of objects in horizontal direction will be investigated.

#### **6. Acknowledgment**

The authors are thankful to the Ministry of Education, Culture, Sports, Science and Technology of Japan for financial supports.

#### **7. References**

510 MATLAB for Engineers – Applications in Control, Electrical Engineering, IT and Robotics

**Large** 0.1234 (0.0403) 0.1404 (0.0302) **Medium** 0.1038 (0.0233) 0.1220 (0.0107) **Small** 0.0884 (0.0311) 0.1008 (0.0164) Table 7. Mean peak accelerations with standard deviations (in parentheses) for different

Fig. 12. Mean performance evaluation scores for small size object for condition 1.a (*m*1=6 \* e-6t + 0.5, *m*2=0.03) and condition 2.a (*m*1=6 \* e-6t + 1.0, *m*2=0.03) after the control modification.

In this chapter, we presented a 1-DOF power assist robot system for manipulating objects by human subjects in horizontal direction. We included human features in the robot dynamics and control. We determined optimum maneuverability conditions for manipulating objects with the robot system. We also determined psychophysical relationships between actual and perceived weights for manipulating objects with the robot system. We analyzed weight perception, load forces and motion characteristics. We implemented a novel control method based on weight perception, load forces and motion characteristics that improved the system performances through reducing the peak load forces and peak accelerations. The findings may help develop human-friendly power assist robot devices for manipulating heavy objects in industries such as manufacturing and assembly, mining, logistics and transport, construction etc. This chapter also provides a vivid example to the readers of how Matlab/Simulink is used to model and develop control system and interfaces between hardware and software for simulation and control of a robotic system. The findings of this chapter are novel and they enhance the state-of-the-art knowledge and applications of robotics, control system,

**5. Conclusions** 

*m*1*=*6 *\* e-6t +* 0.5*, m*2*=*0.03 *m*1*=*6 *\* e-6t +* 1.0, *m*2*=*0.03

**Object size Mean peak acceleration (m/s2)** 

object sizes for different conditions after the control modification


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### *Edited by Karel Perutka*

The book presents several approaches in the key areas of practice for which the MATLAB software package was used. Topics covered include applications for: -Motors; -Power systems; -Robots; -Vehicles. The rapid development of technology impacts all areas. Authors of the book chapters, who are experts in their field, present interesting solutions of their work. The book will familiarize the readers with the solutions and enable the readers to enlarge them by their own research. It will be of great interest to control and electrical engineers and students in the fields of research the book covers.

MATLAB for Engineers - Applications in Control, Electrical Engineering, IT and Robotics

MATLAB for Engineers

Applications in Control, Electrical Engineering,

IT and Robotics

*Edited by Karel Perutka*

Photo by sakkmesterke / iStock