Integral Sliding Mode Control for Trajectory Tracking Control of an Omnidirectional Mobile

Robot

Abstract—This paper considers the problem of trajectory tracking control of an omnidirectional mobile root. First, the dynamic model of an omnidirectional mobile robot is derived to facilitate the control design. In the controller design, state feedback is used to linearize the system. Based on the linearized system, integral sliding mode control is then designed for trajectory tracking control. This control strategy is robust to model uncertainties and exogenous disturbances. The designed control laws are implemented and tested on an experimental setup. The effectiveness of the control scheme is demonstrated through simulation and experimental studies, and is compared with the well-tuned PID control. It is shown that the control system with the integral sliding mode controller has better tracking performance.

Keywords-omnidirectional mobile root ; integral sliding mode control

I. INTRODUCTION (HEADING 1) Due to structural simplicity and highly static stability,

wheeled mobile robots have been very popular in industrial and service robotics. There are various types of wheeled mobile robots. Among them, the omnidirectional wheeled mobile robot [1] is able to provide translational movement along any desired path combined with any rotational movement. This wheeled mobile robot has superior maneuverability and has been widely studied in the dynamic environmental applications.

A number of studies [4]-[7] have been used sliding mode control strategies for controlling the wheeled mobile robots. The basic idea of sliding mode control is to use a high speed switching control law such that the state trajectories of the closed-loop motion are forced to reach the pre-defined sliding surface, and to maintain their motion on the sliding surface for all subsequent time. When the motion of the state trajectories is confined to the sliding surface, the dynamics of the system are forced to constrain its evolution on the pre-defined sliding surface such that the system is insensitive to parametric variation and external disturbance. However, a drawback of the classical sliding mode control is that the system is not robust in the reaching phase in which the system has not yet reached the sliding surface. In [2, 3], integral sliding mode control was proposed to solve the problem of the lack of robustness during the reaching phase. Thus, robustness can be guaranteed throughout the entire response of the system starting from the

initial time instance. This paper presents the design and implementation of integral sliding mode control for tracking control of an omnidirectional mobile root. The mathematical model of an omnidirectional mobile robot is derived. State feedback is first used to linearize the system. Based on the linearized system, integral sliding mode controller is then designed for the trajectory tracking control problem. The effectiveness of the designed control system is demonstrated through simulation and experimental studies.

The rest of this paper is organized as follows. In Section II, the mathematical model of an omnidirectional mobile robot is derived. Section III reviews the design methodology of integral sliding mode control. In Section IV, the design of the integral sliding mode controller for tracking control is given. A brief description of the implementation of the control system is given in Section V. Simulation and experimental results are presented in Section VI. Finally, Section VII contains some concluding remarks.

II. MATHEMATICAL MODEL This section presents the system modeling of an

omnidirectional mobile robot. This mathematical model is derived by using the Newton’s laws [8]. Fig. 1 illustrates the basic features of the robot. The mobile robot is equipped with three omnidirectional wheels, which are equally spaced 120 apart. It is assumed that there is no slip in all the wheels.

LL

L

1f

2f

3f

mx

myY

X

fyf

xfy

x

M

Figure 1. Schematic of the omnidirectional mobile robot.

Sho-Tsung Kao Wan-Jung Chiou Ming-Tzu Ho Department of Engineering Science Department of Engineering Science Department of Engineering Science

National Cheng Kung University National Cheng Kung University National Cheng Kung University Tainan, Taiwan Tainan, Taiwan Tainan, Taiwan

N9897103@mail.ncku.edu.tw bruceho@mail.ncku.edu.tw

Proceedings of 2011 8th Asian Control Conference (ASCC)

TuA1.1Kaohsiung, Taiwan, May 15-18, 2011

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The parameters of the omnidirectional mobile robot are defined as follows:

tM : mass of the robot

zI : moment of inertial of the robot

L : radius of the robot

,m mx y : moving frame

,X Y : world frame

: azimuth angle of the body

,x yF F : traction forces on the robot in the world frame

M : moment of around the center of gravity of the robot

,x yf f : traction forces on the robot in the moving frame

1 2 3, ,f f f : traction force of each wheel

wR : radius of the wheel

1 2 3, , : angular displacement of each wheel

1 2 3, , : angular velocity of each wheel

In the world frame by the Newton's second law, we have

,xw x

yw y

a Fa H F

M (1)

where

1 0 0

10 0 ,

10 0

t

t

z

M

HM

I

.xw w

yw w

a xa y

From Fig. 1, xf , yf and M are given by

1 2 3

1 2 3

1 2 3

cos(0) sin( ) sin( ) ,sin(0) cos( ) cos( ) ,

.

x

y

f f f ff f f fM Lf Lf Lf

(2)

For notational convenience, (2) is written in the following form

1

2

3

,x

y k

f ff B fM f

(3)

where

1 sin( ) sin( )0 cos( ) cos( ) .kBL L L

Since

cos sinsin cos

x x

y y

F fF f

,

and from (3), we obtain

1

2

3

,xw

wyw m k

a fa H R B f

f (4)

where

cos( ) sin( ) 0sin( ) cos( ) 0 .

0 0 1

wmR

Based on geometrical relationships, we have kinematic velocity equations:

1 1

T 12 2

3 3

(( ) ) ( ) ,w

ww m k

xy R R B T (5)

where

2cos( ) 3 sin( ) cos( ) 3 sin( ) cos( )3 3 3

2sin( ) 3 cos( ) sin( ) 3 cos( ) sin( )( ) .3 3 31 1 1

3 3 3

T R

L L L

The control force is generated by a DC motor. The relation

between the motor torque m and control voltage u is given by

2

,t tm m

a a

K Ku

R R (6)

where tK is the motor torque constant, m is the angular velocity of the motor, and aR is the armature resistance. The relationship between the motor torque and the force f is given by

1 ,mw

fR

(7)

From (6) and (7), we have

1 1 12

2 2 2

3 3 3

.t t

w a w a

f uK K

f uR R R R

f u (8)

From (4), (5), and (8), the dynamics of the robot in the world frame can be written as

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( ) ,w w w w CP A P B U (9) where

T ,w w wP x y

T

1 2 3 ,CU u u u

1

1

2

0 00 0 ,0 0

w

aA a

a

1 1 1 1 1

1 1 1 1 1

2 2 2

2 cos( ) cos( ) 3 sin( ) cos( ) 3 sin( )

( ) 2 sin( ) sin( ) 3 cos( ) sin( ) 3 cos( ) ,w

b b b b b

B b b b b bb b b

2

1 2

3,

2t

w t a

Ka

R M R

2 2

2 2

3,t

w z a

K La

R I R 1 .

2t

w t a

KbR M R

III. INTEGRAL SLIDING MODE CONTROL Consider the following non-linear system

( ) ( ) ( , ),x f x B x u d x t (10) where nx R are the states of the system, mu R are the control input, ( ) nf x R and ( ) n mB x R are smooth functions. where ( , ) nd x t R are the perturbation due to parameter variations, un-modeled dynamics, and external disturbances, and we assume that ( , )d x t satisfies the matching condition [3], which is

( , ) ( ) md x t B x d , .mmd R (11)

Moreover, we assume md are bounded

im id , 1, , i m , (12)

where i is a known and bounded positive number. For system (10), the control input is chosen as

0 1u u u (13) where 0u is the ideal control input. In general, 0u can be obtained by LQR control, PID control, pole placement or other control laws. 1u is a discontinuous control which is designed to force the system to constrain states trajectories on the sliding surface and to eliminate the external disturbance. The sliding function is chosen as

0 ( ) ss s x z , (18) where ms R , 0 ( ) ms x R and m

sz R . The function 0 ( )s x is designed as the linear combination of the system states. The function sz is an unknown integral function to be determined. From (18), we have

0s

ss x z

x. (19)

Form (16) and (19), and we obtain

0 [ ( ) ( )( )]m ss

s f x B x u d zx

. (20)

According to the idea of the integral sliding mode, the system states have to remain on the sliding surface. Thus, we have

00[ ( ) ( ) ].s

sz f x B x u

x (21)

To guarantee the state trajectories to be confined on the sliding surface starting from the initial instant, from (18), we can choose (0)sz as follow

0(0) ( (0)).sz s x (22) According to (18), (21) and (22), we obtain

0 0

000

( ) ( (0))( )

[ ( ( )) ( ( )) ( )] .t

s s x s xs x

f x B x u dx

(23)

In order to confine the system states on the sliding surface and eliminate the external disturbance and noise effectively, 1u is chosen as

1 sgn( )u s , (24) where m mR is a diagonal matrix. In other words,

diag{ }i , 1, , i m , and i is a positive real number.

IV. CONTROLLER DESIGN FOR THE ONMIDIRECTIONAL MOBILE ROBOT

A. State Feedback Linearization From (9), the system can be rewritten as follows:

( ) .w w w w CP A P B U (28) Let

T( ) ( ) ( ) ( )des des des desP t x t y t t (29) be the desired trajectory. Define the tracking error ( )e t as

( ) ( ) ( ),des we t P t P t (30) then

.des we P P (31) From (28) and (31), we have

[ ( ) ].des w w w Ce P A P B U (32) By choosing

[ ( ) ]des w w w CU P A P B U , (33) the tracking error dynamics become linear and can be written as follow

0 0

.0 0

e I ed Ue e Idt

(34)

From (33), the state feedback control law is

1( ) [ ].C w des w wU B P A P U (35)

B. PID Controller Design The PID control law is given as

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,e .p d iU K e K e K (36) From equation (35), the system control input is

1( ) [ ].C w d w w p d iU B P A P K e K e K (37)

C. Integral Sliding Mode Controller Design Because the actual system usually has external disturbance

or un-modeled uncertainty, the tracking error dynamic equations (34) can be rewritten as

e U d , (42) where d are the perturbations and

x

y

dd d

d, (43)

where xd , yd , and d are the perturbations in the X axis, Y axis, and , respectively. Moreover, the control input U is defined as

x

y

UU U

U (44)

where xU , yU , and U are the control input to the X axis, Y axis, and , respectively. According to (42), we can rewrite the tracking error dynamic equations as

0 0 00 0

e I ed U de e I Idt

(45)

With respect to the identical structure of controllers for the X axis, Y axis, and . In the following only the controller for X axis will be discussed. The tracking error dynamic equations

of X axis is given by

0 1 0 00 0 1 1

x xx x

x x

e ed U de edt

, (46)

where xe is the tracking error in the X axis. According to the design of integral sliding mode control, we first use the PD control law for the ideal system, i.e., without perturbations. The PD controller is given as

0x px x dx xU K e K e , (47) where pxK and dxK are positive real numbers. Then, the control input is chosen as

0 1x x xU U U (48) We choose the sliding function as

0( )

(0) (0)

t

x x dx x px x

dx x x

s e K e K e d

K e e (49)

And the switching control input 1xU is given by

1 sgn( )x x xU s . (50) Because xd is bounded, xd is chosen to satisfy the following condition:

x xd , (51)

where x is a real number. Moreover, we use sat xs to replace sgn( )xs to mitigate the chattering problem [3]. Therefore, we modify the control input as

sat( )xx px x dx x x

Ix

sU K e K e . (52)

By proceeding along similar lines, we can obtain the controllers for the Y axis and .

V. SYSTEM IMPLEMENTATION An omnidirectional mobile robot has been constructed as

shown in Fig. 2. All of controllers are implemented on a digital signal processor (DSP) board. This DSP board is based on the Texas Instruments TMS320F2812 (150 MHz/32-bit) processor. The sampling frequency of the system is chosen to be 1 kHz. The schematic overview of the experimental setup is shown in Fig. 3.

Figure 2. Omnidirectional mobile robot.

Figure 3. Schematic overview of the experimental setup.

In order to obtain the position of the omnidirectional mobile robot in the world coordinate, we have to obtain the angular

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displacement of the wheel. Thus, an optical encoder with a resolution 500 pulses/rev is attached to the shaft of each dc motor to measure the angular displacement of the omnidirectional wheel. The QEP (quadrature encoder pulse) circuits are implemented on an FPGA. The PWM (pulse width modulation) signals are generated by the designed control laws and supplied to the PWM driver to drive each dc motor.

VI. SIMULATION AND EXPERIMENTAL RESULTS This section is to present the simulation and experimental

results. The planned trajectories are linear and circular paths.

First, we set a linear path command 0.16 (m)desx t , 0.2 (m)desy t and 0 (deg)des in the world coordinate,

and the initial conditions are set to be TT(0) (0) (0) 0.0865 -0.1455 9.263x y . Fig. 4 and

Fig. 5 are the simulation results that use PID control and integral sliding mode control.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4. Simulation results of tracking of an straight line path.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.1

-0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5 3 3.5 4-0.05

0

0.05

0.1

0.15

0 0.5 1 1.5 2 2.5 3 3.5 4-20-10

0102030

error of X

error of Y

error of

Figure 5. Tracking error of PID(--) and ISMC ( ) (straight line).

Here, we set a circular path command of 0.2cos(0.5 ) (m),desx t 0.2sin(0.5 ) (m)desy t and

0 (deg)des in the world coordinate and the initial conditions are

TT(0) (0) (0) 0.0319 -0.0865 8.125x y . Fig. 6 and Fig. 7 are the simulation results of circular path that use the PID control and integral sliding mode control.

-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Cart displacement in X axis (meter)

Cart displacem

ent in Y axis(meter)

Command

PID ISMC

Figure 6. Simulation results of tracking of a circular path.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.02

00.020.040.060.08

0 0.5 1 1.5 2 2.5 3 3.5 4-0.02

00.020.040.060.08

0 0.5 1 1.5 2 2.5 3 3.5 4-15-10

-505

10

error of X

error of Y

error of

Figure 7. Tracking error of PID(--) and ISMC ( ) (circular path).

From Fig.4 to Fig.7, we know that the control system with the integral sliding mode controller has better transient tracking performance. The system designed by the PID controller has more oscillation and slower convergence to the trajectory command.

Then, we test the designed control laws on the experimental setup. The planned paths are the same as those of the simulation and the initial conditions of the robot are also the same as those of the simulation, too. Fig. 8 and Fig. 10 show the experimental results of tracking linear and circular paths, respectively. We find that integral sliding mode control has better transient performance. Moreover, Fig. 9 and Fig. 11 show the steady-state tracking error of linear and circular paths, respectively. From the experimental results, we can see that integral sliding mode control has less steady-state error.

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 8. Experimental results of tracking of a straight line.

1 1.5 2 2.5 3 3.5 4-101234

x 10-3

1 1.5 2 2.5 3 3.5 4-10123

x 10-3

1 1.5 2 2.5 3 3.5 4-1

-0.50

0.51

1.5

error of X

error of Y

error of

Figure 9. Tracking errors of PID(--) and ISMC ( ) (straight line).

-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Cart displacement in X axis (meter)

Cart displacem

ent in Y axis(meter)

Command

PID ISMC

Figure 10. Experimental results of tracking of a circular path.

1 1.5 2 2.5 3 3.5 4-3-2-10123

x 10-3

1 1.5 2 2.5 3 3.5 4-3-2-10123

x 10-3

1 1.5 2 2.5 3 3.5 4-0.5

0

0.5

1

error of X

error of Y

error of

Figure 11. Tracking errors of PID(--) and ISMC ( ) (circular path).

VII. CONCULSIONS In this paper, an integral sliding mode control scheme to

achieve trajectory tracking of an omnidirectional mobile robot was proposed. The mathematical model of the robot was derived to facilitate the controller design. In controller design, we first used state feedback to cancel out the nonlinear terms of the system, and PID control and integral sliding mode control are then used to achieve trajectory tracking. From simulation and experimental results, it is shown that integral sliding mode control has better performance.

REFERENCES [1] K. Watanabe, Y. Shiraishi, and S. G. Tzaffestas, “Feedback Control of

An Omnidirectional Autonomous Platform for Mobile Serve Robots,” Journal of Intelligent and Robotic Systems, vol. 20, pp. 315-330, 1998.

[2] V. Utkin and J. Shi, “Integral Sliding Mode in Systems Operating Under Uncertainty Conditions,” in Proc. Conf. Decision and Control, Kobe,Japan, Dec. 1996, pp. 4591–4596.

[3] V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical System, CRC Press, Boca Raton, 1999.

[4] J. A. Chacal B. and H. Sira-Ramirez, “On the Sliding Mode Control of Wheeled Mobile Robots,” in Proc. IEEE Int. Conf. Syst. Man. Cybern., Sept. 1994, pp. 1938–1943.

[5] Guldner and V. I. Utkin, “Sliding Mode Control for Gradient Tracking and Robot Navigation Using Artificial Potential Fields,” IEEE Trans. Robot. Automat., vol. 11, pp. 247–254, Apr. 1995.

[6] H.S. Shim, J.H. Kim and K. Koh, “Vdriable Structure Control of Nonholononiic Wheeled Mobile Robots,” Proc. IEEE Con.f Robot. Automat., pp. 1694-1699, May 1995.

[7] L. E. Aguilar, T. Hamel, and P. Soueres, “Robust Path Following Control for Wheeled Robots via Sliding Mode Techniques,” in Proc. 1997 IEEE/RSJ Int. Conf. Intell. Robot. Syst. (IROS’97), Sept. 1997, pp. 1389–1395.

[8] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, Addison Wesley, Upper Saddle River, NJ, 2002.

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