Variable Structure Control of Spherical Robots with Exponential Reaching Law

Studies in System Science (SSS) Volume 2, 2014 www.as‐se.org/sss

Variable Structure Control of Spherical Robots with Exponential Reaching Law Tao Yu*1, Hanxu Sun2 1

Faculty of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou, Liaoning, China

2

School of Automation, Beijing University of Posts and Telecommunications, Haidian District, Beijing, China

*1

yutaolanjie@163.com; 2hxsun@bupt.edu.cn

Abstract In this study, the dynamics and control aspects of the longitudinal motion of pendulum‐driven spherical mobile robots are investigated. A simplified dynamic model is established for the longitudinal motion of a pendulum‐driven spherical robot by using Lagrangian dynamics. By appropriate definitions the equations of motion for the robotic system are transformed into the state space form. A hierarchical sliding mode controller based on a new exponential reaching law is proposed to achieve set‐point regulation of the longitudinal motion. The asymptotic stability of the whole system is verified through Lyapunov analysis, and the validity of the proposed approach is illustrated through numerical simulations.

Keywords Spherical Robot; Longitudinal Motion; Hierarchical Sliding Mode Control; Exponential Reaching Law

Introduction The extensive research on spherical mobile robots has the foundation in the belief that there are certain applications where spherical mobile robots are more advantageous than traditional wheeled mobile robots. Spherical mobile robots are more versatile, less exposed to physical conditions, and they have greater resistance levels towards object collisions. Moreover, the spherical shell of this class of mobile robots provides an efficient cover for the internal driving mechanisms and sensory equipments.

Spherical mobile robots can be categorized into different types according to their internal driving mechanisms [1‐9].

Compared with other types of spherical mobile robots [1‐6], a pendulum‐driven spherical mobile robot [7‐9] has a simpler structure, making it easier to be manoeuvred. The schematic diagram of a pendulum‐driven spherical mobile robot with dual inputs is illustrated in Fig. 1. Longitudinal motion is a basic form of locomotion of pendulum‐driven spherical mobile robots, and it is realized by moving a motor‐controlled pendulum forwards or backwards. In this paper, a hierarchical sliding mode control approach based on a novel exponential reaching law is presented for stable control of the longitudinal motion. In the proposed controller, a double layer structure is used to guarantee the stability of the whole system, and the sub‐sliding surfaces are utilized to drive the tracking errors to zero.

Y

O'

m1



 B(D), m2





R

l

E, m3

O

X

FIG. 1 STRUCTURE OF A PENDULUM‐DRIVEN SPHERICAL ROBOT FIG. 2 SIMPLIFIED MODEL OF THE LONGITUDINAL MOTION

Dynamic Analysis We start with a simplified planar model, only considering no slip longitudinal motion on flat surfaces. Fig. 2 illustrates the simplified model with a side view of a pendulum‐driven spherical robot. It represents the spherical shell with its center of mass B, the internal mechanism with its center of mass D, which coincides with that of the

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spherical shell, and the pendulum (composed of a massless link and a counterweight at its end) with its center of mass E and the axis attached at the center of the sphere. The definition of the model parameters is listed in Table 1.

TABLE 1 PARAMETER DEFINITION OF THE PLANAR MODEL

m1, m2, m3 R, l θ,  I1, I2, I3 τ

mass of the spherical shell, the internal mechanism and the pendulum, respectively radius of the spherical shell and length of the pendulum, respectively roll angle of the spherical shell and sway angle of the pendulum, respectively moment of inertial of the spherical shell, the internal mechanism and the pendulum, respectively torque applied to the pendulum

We first choose the roll angle of the sphere  and the sway angle of the pendulum  as the generalized coordinates of the robotic system, and then we develop the equations of motion by calculating the Lagrangian L  T  P of the system, where T and P are the kinetic energy and potential energy of the system respectively.

The kinetic energy and potential energy of the whole robotic system are given by T

1  2 1 2   cos  J1 + J 2  m3 Rl 2 2

P  m3 gl cos  (1)

where J1  Mt R 2  I1 , Mt  m1  m2  m3 , J 2  m3l 2  I 2  I3 ; g denotes the gravitational acceleration.

Using the extended Lagrange equation [10], the dynamics of the longitudinal motion can be expressed as M  q  q  N  q , q   E  q  (2)

where M  q  is the inertia matrix, N  q , q  is the nonlinear terms, and E  q  is the input transformation matrix.

J1 m3 Rl cos    M q    J2 m3 Rl cos  





     m3 Rl sin 2   N  q , q          m3 gl sin    





1 E q     1

Here  represents the viscous damping coefficient associated with the pendulum‐sphere bearing. Using the control



input u   and the state vector X   , ,  , 



T

, we can rewrite Eq. (2) as follows

 x1  x 2   x 2  f1  X   b1  X  u (3)   x 3  x 4  x  f  X   b  X  u 2 2  4

where f1  X  

m22n1  m12n2 m12 m21  m11m22

f2  X  

m11n2  m21n1 m12 m21  m11m22

b1  X  

m22  m12 m11m22  m12 m21

b2  X  

m11  m21

m11m22  m12m21

Here mij denotes the element in the i‐th row and j‐th column of the matrix M  q  , and nk represents the k‐th component of the vector N  q , q  .

Controller Design In this section, we investigate the set‐point regulation scheme of the longitudinal motion of a spherical robot, and a hierarchical sliding mode controller [11] based on a new exponential reaching law is derived to asymptotically stabilize the robot around its desired equilibrium.

Considering the system represented by Eq. (3), we construct the following first layer sliding surfaces s1  1e1  e1

s2  2 e2  e2 (4)

where 1 and 2 are positive constants; e1  x1   d , e2  x3   d ,  d and  d are desired values of  and  respectively.

Differentiating Eq. (4) and equalizing the result to zero, we can obtain the equivalent control as

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f1  1e1  d b1

ueq1  

ueq 2  

f 2  2 e2  d (5) b2

We design the second layer sliding surface as a linear combination of the two sub‐sliding surfaces S   s1   s2 (6)

where  and  are positive constants.

The control input of the system is assumed to take the following form u  ueq1  ueq 2  usw (7)

where usw is the switching control.

To construct the switching component usw , we propose the following exponential reaching law

 S   sgn  S    S (8) N S  where  and  are positive constants.

N  S    0  1   0  e

 S

p

Here  0 is a positive constant that is less than one, p is a positive integer, and  is also a positive constant.

The proposed ERL given by Eq. (8) is composed of a variable rate reaching term [12] and an exponential term.

Comparing with the conventional exponential reaching law [13], we can see from Eq. (8) that if S increases, N  S  approaches  0 , and therefore  N  S  converges to   0 , which is larger than  . This means that  N  S  increases in the reaching phase, and consequently the attraction to the sliding surface S is faster. On the other hand, if S decreases, then N  S  approaches one, and  N  S  converges to  . This means that, when the system state approaches the sliding surface S ,  N  S  gradually decreases to reduce the chattering. Therefore, the proposed ERL allows the controller to dynamically adapt to the variations of the switching function S by letting  N  S  vary between  and   0 .

Differentiating Eq. (6) and using Eq. (8), we can obtain the switching control as usw  

where N  S  



N S 

 b2 ueq1   b1ueq 2  N  S  sgn  S    S  b1   b2

(9)

.

Substituting Eq. (9) into Eq. (7), we can obtain the following sliding mode control law u

 b1ueq1   b2 ueq 2  N  S  sgn  S    S  b1   b2

(10)

Theorem 1: Supposing that the robotic system represented by Eq. (3) is controlled by the sliding mode controller given

by Eq. (10), then the system defined by Eq. (3) is asymptotically stable.

Proof: Considering the Lyapunov function candidate V 

1 2 S , then V can be given by 2

V  SS   N  S  S   S 2   S   S 2  0 (11)

Integrating both sides of Eq. (11), we have

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V t  



1 2 S  V 0   2



lim 0t  S   S 2 d  V  0   

t 

(12)

According to Eq. (12), we have S  L , S  L2 . And according to Eq. (11), we have S  L . Consequently, by applying Babalat’s lemma we can conclude that the sliding surface S is asymptotically stable, i.e.

lim S  0

t 

Then the system can be guaranteed to be asymptotically stable.

Simulation Study In this simulation, the following physical parameters of the spherical mobile robot [14] and design parameters of the sliding mode controller are used. m1  1.2 kg m2  1.85 kg m3  2.05 kg R  0.15 m l  0.12 m I1  0.018 kg  m 2

  0.03 N  m   rad s 

1

I 2  0.0017 kg  m 2

I3  0.0006 kg  m 2

g  9.81 m s 2   7.9   0.5 1  4 2  0.6   3  0  0.1   10

p  1   7.8

In addition, the initial and desired values of the system states are chosen as x0   0, 0, 0, 0  , x d   , 0, 0, 0  .

T

T

The simulation results are depicted in Fig. 3 and Fig. 4. As it is theoretically expected, we can find that both the roll angle of the sphere and the sway angle of the pendulum are asymptotically stabilized to their desired values, and the anti‐sway control is achieved in a rapid manner after only one oscillation.

200

30 20

100

φ(deg)

θ(deg)

150

50

−50

0

Response Desired

0 0

2

4 t(s)

6

10

−10

8

0

2

4 t(s)

6

8

FIG. 3 TRACKING RESULTS OF THE ROLL AND SWAY ANGLES OF THE SPHERICAL ROBOT 100

150

˙ φ(deg/s)

˙ θ(deg/s)

100 50

50

0

0 −50

−50 0

2

4 t(s)

6

8

0

2

4 t(s)

6

8

FIG. 4 TIME EVOLUTION OF THE ROLL AND SWAY ANGULAR VELOCITIES OF THE SPHERICAL ROBOT

Conclusions In this paper, we present a variable structure control strategy for set‐point regulation of the longitudinal motion of pendulum‐driven spherical robots. The control development is based on the construction of a cascade sliding mode controller and a new exponential reaching law, and the proposed control approach consists of designing a nonlinear reaching law by using a switching component that dynamically adapts to the variations of the controlled system. The asymptotic stability of the sliding surface of the whole system is theoretically proved, and the simulation results further verify the validity of the proposed controller.

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ACKNOWLEDGMENT

The authors wish to acknowledge the financial support provided by the National Natural Science Foundation of China (No. 51175048) and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708011).

REFERENCES

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[2]

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Otani, Toshiaki et al., “Position and Attitude Control of a Spherical Rolling Robot Equipped with a Gyro.” Proceedings of the 9th IEEE International Workshop on Advanced Motion Control, Istanbul, Turkey, 416‐421, 2006.

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Michaud, Francois et al., “Autonomous Spherical Mobile Robot for Child‐development Studies.” IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 35(4), 471‐480, 2005.

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[9]

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[10] Abbott, M. S. “Kinematics, Dynamics and Control of Single‐axle, Two‐wheel Vehicles (Biplanar Bicycles).” MS diss., Virginia Polytechnic Institute and State University, 2001.

[11] Wang, W., Yi, J., and Zhao D. “Design of Sliding‐mode Controller Based on Stable Analysis for a Class of Underactuated Systems.” Information and Control, 34(2), 232‐235, 2005.

[12] Fallaha, C. J., Saad, M., and Kanaan, H. Y. “Sliding‐mode Robot Control with Exponential Reaching Law.” IEEE Transactions on Industrial Electronics, 58(2), 600‐610, 2011.

[13] Gao, W. and Hung, J. C. “Variable Structure Control of Nonlinear Systems: A New Approach,” IEEE Transactions on Industrial Electronics, 40(1), 45‐55, 1993.

[14] Yu, T. “Study on Control Methodology for the Slope Motion of a Spherical Robot.” PhD diss., Beijing University of Posts and Telecommunications, 2014.

Tao Yu was born in Jinzhou, Liaoning, China, in 1980. He received the B.S. and M.S. degrees in mechanical engineering from Jilin University, Changchun, Jilin, China, in 2002 and 2005, respectively, and the Ph.D. degree in mechanical engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 2014.

He is currently an Associate Professor with Faculty of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou, Liaoning, China. His main research interests include sliding mode control, intelligent control, and robotics.

Hanxu Sun was born in Hanzhong, Shanxi, China, in 1960. He received the M.S. degree in mechanical engineering from Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin, China, in 1986, and the Ph.D. degree in mechanical engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1989.

He is currently a Professor with School of Automation, Beijing University of Posts and Telecommunications, Beijing, China. His main research interests concern the dynamics and control of industrial robots, space robots and special robots.

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