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Sliding Mode Control for Trajectory Tracking on Mobile Manipulators Weimin Ge, Duofang Ye, Wenping Jiang

Xiaojie Sun

School of Mechanical Engineering Tianjin University of Technology Tianjin 300191, China [email protected]

School of Computer Science Tianjin University of Technology Tianjin 300191, China [email protected]

Abstract—This paper proposes a sliding mode control approach for a wheeled mobile manipulator. The wheeled mobile manipulator composed of a four-wheeled mobile platform and a three-degree of freedom onboard manipulator. The dynamic models are established and the sliding mode control methods are employed. The entire control system is decomposed into two subsystems, including the sliding mode control of the mobile platform and the non-singular terminal sliding mode control of the manipulator. The simulation results are shown at last and demonstrate the effectiveness of the presented algorithm.

large class of nonholonomic mechanical systems in the presence of uncertainties and disturbances. Sliding mode control, taking the advantages of fast response, reduced amount of information, and robustness with respect to system uncertainties and external disturbances, is much more suitable for the control of mobile manipulators. In this paper, we focus on a redundant actuated mobile manipulator. Two control subsystems are proposed to solve the trajectory tracking problem, including the sliding mode control of the mobile platform and the nonsingular terminal sliding mode control of the manipulator.

Keywords— mobile manipulator; sliding mode control; trajectory tracking

I.

II.

2.1 System Description The geometric model of the mobile manipulator is shown in Figure 1, which consists of a wheeled mobile platform and a manipulator. Two motors drive the two front wheels respectively, while the other rear wheels have the function of guiding and stabilizing. The manipulator, with three degree of freedom, is installed on the mobile platform. A world coordinates {I} ={x, y, z} is set up in Figure 1; M is the middle point of the two driven wheels. Main parameters are as follows: l1 , l2 and l3 are the length of the link 1, link 2 and link 3

INTRODUCTION

A mobile manipulator composes of a mobile platform and a manipulator mounted on the mobile platform. The mobile platform extends the manipulator’s workspaces and higher redundant motion, which makes the mobile manipulator superior to the mobile robot and normal manipulator; on the other hand, the degrees of freedom of the platform also add the redundancy to the whole system, and strongly coupled dynamics between the mobile platform and the manipulator arises. Therefore, the control of mobile manipulators has been an attractive research field in the past decades. H Hui and P.Y. Woo [1] addressed a control scheme which consisted of an SMC and NNC, and the contribution proportion of these two parts was determined by a fuzzy supervisory controller. Sheng lin and A.A. Goldenberg [2] developed a robust damping control (RDC) controller for the motion control of mobile manipulators subject to kinematics constraints, and the proposed RDC controller is capable of disturbance rejection in the presence of unknown bound disturbance, without requiring the knowledge of its bound. Choon-Young Lee et al. [3] proposed a neural network based adaptive controller using radial basis function network (RBFN) for a mobile manipulator to track the given trajectories in the workspace, and introduced an additional control input to overcome the approximation error due to the inexact approximation by RBFN. Z. Wang et al [4] developed an adaptive robust force control systematically for holonomic mechanical systems and a

978-1-4244-2342-2/08/$25.00 ©2008 IEEE.

SLIDING MODE CONTROL OF THE MOBILE PLATFORM

respectively; q1 , q 2 and q3 are the rotation angles of the link 1, link 2 and link 3 respectively; r2 , r3 are centroid distance of each links respectively. 2.2 Design of sliding mode controller The mobile platform is driven by two front wheels respectively, and the kinematic modeling of two front wheels is described in Figure 2. The main parameters are as follows: rw is the radius of the wheel1 and the wheel2; v1 and v2 are the velocities of the centres of the wheel1 and wheel2 respectively. Suppose the mobile platform wheels move under pure rolling without sliding, and we get:

v1 = rw ω1 ; v1 = rwω2 (1) (2) where ω1 and ω2 are angular velocities of the wheel1 ⋅

⋅

and wheel2 respectively, and ω1 = θ 1 ,ω2 = θ 2 .

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r3

From reference [5], the position error differential function of the mobile platform is

L3

q3

L 2 q2 r2

⎛ xe ⎞ ⎛ yeω − υ + υ r cos θ e ⎞ ⎜ ⎟ ⎜ ⎟ p e = ⎜ y e ⎟ = ⎜ − xeω + υr sin θ e ⎟ ⎜ θ ⎟ ⎜ ⎟ ωr − ω ⎝ e⎠ ⎝ ⎠

L1 M

q1

z

The sliding mode switch function can be designed as [6] s = θe (10) To reduce chattering, let

y

x Figure 1 Geometric model of the mobile manipulator

v1

xw1

Vp

θ

p

ow1

yw1

v2 Ld

.

s = k 2 sgn s (11) Based on the equations (9), (10) and (11), the control law is obtained: y ⎛ ⎞ ⎛ υ ⎞ ⎜ υ cos θ e + k 1 x e + e υ r sin θ e ⎟ q = ⎜⎜ ⎟⎟ = ⎜ r xe ⎟ (12) ⎟ ⎝ω ⎠ ⎜ ω + k sgn s r 2 ⎝ ⎠

xw 2 ow2

o p yw 2

Where k 1 > 0 , k 2 > 0 2.2 Simulation We verify the performance of the control based on the kinematical model of the mobile platform through simulation in MATLAB. Tracking the circular trajectory of the point M, it moves with uniform motion. Suppose ω r = 1.0, υ r = 1.0 , k1 = k 2 = 8.0 , [10 1 0] is the position error initial value, and adopt the control law as equation (12). Suppose the position instruction is

Figure 2 Kinematic modelling of driven wheels

Based on the rigid body kinematics, we get: . 1 1 v p = (v1 + v 2 ) θp = (v 2 − v1 ) Ld 2 . . . 1 x p = rw (θ 1 + θ 2 ) cos θ p 2 . . . 1 y p = rw (θ 1 + θ 2 ) sin θ p 2

(3) (4) (5) (6)

.

. . 1 θp = rw (θ 2 − θ1 ) Ld From above equations, we can get ⎡ ⎤ ⎡ . ⎤ ⎢ 1 rw cosθ p 1 rw cosθ p ⎥ x p 2 ⎢ ⎥ ⎢2 ⎥⎡ . ⎤ 1 ⎢ . ⎥ ⎢1 ⎥ ⎢θ1 ⎥ ⎢ y p ⎥ = ⎢ 2 rw sin θ p 2 rw sin θ p ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎣θ 2 ⎦ r rw ⎢z p ⎥ ⎢ − w ⎥ ⎣ ⎦ Ld Ld ⎣ ⎦ .

p r = ( x r y r θ r )T = (t t cos t t )T The simulation results are shown in Figure 4-Figure 5.

(7)

(8)

In Figure3, the mobile platform moves from the position T T p = ( x y θ ) to the position p r = (x r y r θ r ) , whose coordinate in the new coordinate T system X e − Ye is pe = ( xe y e θ e ) , where θ e = θ r − θ .

Y

Figure 4 Trajectory tracking of xr

θe

θr

yr

Pr

M

xe ye

θ

P

M

x

(9)

xr

X Figure 5 Trajectory tracking of yr

Figure 3 Position error of the mobile platform

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III.

r23 = r32 = I x 3 + m3 r32 + m3 L2 r3 c3 = 0.2 + 0.2 cos q3

SLIDING MODE CONTROL OF THE MANIPULATOR

2 3 3

r33 = I x 3 + m r = 0.2

3.1 Dynamic model of the manipulator Based on the Lagrange dynamics, the dynamic function of the 3-DOF manipulator shown in Figure 1 is established as: R(q)q + M (q)ξ + g (q ) = τ (13)    where q , q and q are the angle, angular velocity and angular acceleration of each joints respectively; τ is the driving torque vector; R(q) is the inertia matrix; M (q)ξ and g(q) are centrifugal & Coriolis matrixes and gravitational matrixes depending on the structure of the manipulator. where [7] ⎡ r11 (q ) r12 (q) r13 (q) ⎤ R(q) = ⎢⎢r21 (q) r22 (q) r23 (q)⎥⎥ (14) ⎢⎣r31 (q) r23 (q) r33 (q) ⎥⎦

m12 = Γ112 + Γ121 =

q 2 q 2

q 2 q 3

q 3 q 3 ]

m13 = Γ113 + Γ131

(15) (16)

link 3 respectively; ℜ1 , ℜ 2 , ℜ 3 are inertia matrixes of link 1, link 2 and link 3 respectively. and 1 ⎧⎪ ∂Rij ∂Rik ∂Rkj ⎫⎪ Γijk = ⎨ i = 1,2,3 + − ⎬ (17) ∂q j 2 ⎪⎩ ∂q k ∂qi ⎪⎭ . . . . . . Main parameters of the manipulator are as shown in table 1. Table 1: The table of the manipulator parameters i

mi / kg

Li / m

ri / m

I x /kgm2

I y /kgm2

I z /kgm2

1 13 0.5 0.35 — — — 2 10 0.4 0.2 0.3 0.3 0.15 3 5 0.3 0.1 0.15 0.15 0.10 Note: m, r and L are the quality, centroid distance and length of each links respectively; I x , I y and I z are the moment of inertia. After calculating, we get the nonzero parameters in equations (14) and (15) 2 2 r11 = I y 2 s 22 + I y 3 s 23 + I z1 + I z 2 c 22 + I z 3 c 23 + m 2 r2 s 22 +

m3 ( L2 s 2 + r3 s 23 ) 2 = 0.6 + 1.35 sin 2 q 2 +

= 1.7 + 0.4 cos q3

(23)

m 21 = Γ211 = −

1 ∂r11 1 = − m12 2 ∂q 2 2

(24)

m31 = Γ311 = −

1 ∂r11 1 = − m13 2 ∂q3 2

(25)

m 25 = Γ223 + Γ232 = m 26 = Γ233 =

∂r22 = −0.2 sin q3 ∂q3

∂r23 = −0.2 sin q3 ∂q3

1 ∂r22 = 0.2 sin q3 2 ∂q3 And the parameters of gravitational matrixes are: ∂g (q) g1 = = 0; ∂q1 m34 = Γ322 = −

g2 =

∂g (q ) = − m2 gr2 s 2 − m3 g ( L2 s 2 + r3 s 23 ) = ∂q 2

(26) (27) (28)

(29)

− 39.2 sin q 2 − 4.9 sin(q 2 + q3 ) g3 =

∂g (q ) = −m3 gr3 s 23 = −4.9 sin( q 2 + q3 ) ∂q3

(30)

3.2 Design of sliding mode controller Assume that qd is the quested state vector, e = q − q d is the error vector. Assume that the control input is [8] τ = g(q) + R (q)(qd − Ce(t ) + ηξ ) (31) Combining the equation (13) and the equation (31), we get e(t ) + Ce(t ) = [η − M ′(q)]ξ (32) Where M ′(q ) = [mij′ (q )] = R −1 (q) M (q) C = diag (c1 , c2 , c3 )

η = {γ ij }, i = 1,2,3; j = 1," ,6

(18)

0.1sin q 2 sin( q 2 + q 3 ) r22 = I x 2 + I x 3 + m3 L22 + m2 r22 + m3 r32 + 2m3 L2 r3 c3

∂r = 11 = 0.1sin 2(q 2 + q3 ) + ∂q3

0.4 sin q 2 cos(q 2 + q3 )

J 1 , J 2 and J 3 are Jacobian matrixes of link 1, link 2 and

joint

∂r11 = 1.35 sin 2q 2 + 0.1sin 2(q 2 + q3 ) ∂q 2

(22)

⎡ m11 (q) m12 (q) m13 (q) m14 (q) m15 (q) m16 (q) ⎤ ⎢ ⎥ M (q) = ⎢m21 (q) m22 (q) m23 (q) m24 (q) m25 (q) m26 (q)⎥ ⎢⎣m31 (q) m32 (q) m33 (q) m34 (q) m35 (q) m36 (q) ⎥⎦ ⎡Γ111 Γ112 + Γ121 Γ113 + Γ131 Γ122 Γ123 + Γ132 Γ133 ⎤ = ⎢⎢Γ211 Γ212 + Γ221 Γ213 + Γ231 Γ222 Γ223 + Γ232 Γ233 ⎥⎥ ⎢⎣Γ311 Γ312 + Γ321 Γ313 + Γ331 Γ322 Γ323 + Γ332 Γ333 ⎥⎦

q1 q 3

(21)

+ 0.4 sin q 2 con(q 2 + q3 )

= J 1T ℜ1 J 1 + J 2T ℜ 2 J 2 + J 3T ℜ 3 J 3

ξ = [q1 q1 q1 q 2

(20)

(19)

The sliding mode switch function can be designed as s = e(t ) + Ce . (33) and then we adopt the sliding mode control law as follows: ⎡γ + > mij′ max ξ j s i < 0 γ ij = ⎢ ij− (34) ⎢⎣ γ ij < mij′ min ξ j si > 0

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2 position tracking of link3

According to the equation (13),(14) and (15), we get ′ m13 ′ 0 0 0 ⎤ ⎡ 0 m12 ⎢ ′ ⎥ −1 ′ ′ ′ M (q) = R (q)M (q) = ⎢m21 0 0 m24 m25 m′26 ⎥ ⎢⎣m31 ′ ′ m35 ′ m36 ′ ⎥⎦ 0 0 m34 (35) 0 γ γ 0 0 0 ⎡ ⎤ 12 13 (36) η = ⎢⎢γ 21 0 0 γ 24 γ 25 γ 26 ⎥⎥ ⎢⎣γ 31 0 0 γ 34 γ 35 γ 36 ⎥⎦

1 0 -1 -2 -3 -4 -5 0

desired trajectory achieved trajectory 1

2

Based upon the equation (34) we obtain η + = {γ ij+ } =

IV. (38)

3.3 Simulation Initial conditions are defined as: q1 ( 0 ) = 1.0 , q2 (0) = 0.9 , q3 (0) = 0.4 q1 (0) = q 2 (0) = q3 (0) = 0.0

The sliding C = diag (15,15,15) .

mode

control

parameters

The most sincere thanks go to the Tianjin Natural Science Foundation (06YFJMJC03600) and Tianjin High Education Technology Developing Foundation (2006BA11), which support this paper and research.

are:

REFERENCES [1] H. Hui, P. Y. Woo., “Fuzzy supervisory sliding-mode and neuralnetwork control for robotic manipulators,” IEEE Transactions on Industrial Electronics, vol.53, no.3, pp.929-940, 2006. [2] Sheng Lin and A.A.Goldenberg , “Robust damping control of mobile manipulators”, IEEE Trans. on systems, man, and cybernetics-part B: cybernetics, vol.32 (1), pp:126-132, 2002. [3] Choon-Young Lee, et al, “Motion control of mobile manipulator based on neural networks and error compensation”, Proc. of IEEE Int. Conf. on Robotics & Automation, New orieans, pp: 4627-4632, 2004. [4] Z. Wang, S. S. Ge, and T. H. Lee, “Robust motion/force control of uncertain holonomic/nonholonomic mechanical systems,” IEEE/ASME Trans. Mechatron., vol. 9, no. 1, pp. 118–123, Mar. 2004. [5] Kanayama Y,Kimura Y,Miyazaki F,et al., “A stable tracking control method for autonomous mobile robot”, IEEE International Conference on Robotics and Automation, 1990. [6] Liu Jinkun, “MATLAB Simulation for Sliding Mode Control”, Tsinghua University Press, 2005. [7] Richard Murray, Zexiang Li, S. Sastry, “M Mathematical Introduction to Robotic Manipulation”, CRC Press, 1994 [8] Wang Fengyao, “Sliding Mode Variable Structure Control”, China Machine Press, 1995.

1.5 desired trajectory achieved trajectory

1 0.5 0 -0.5 -1 -1.5 0

1

2

3

4

5

time(s)

Figure 6 Position tracking of link 1

1 desired trajectory achieved trajectory

0.5

CONCLUSIONS

In this paper, the sliding mode variable structure control has been presented to control the mobile manipulator to track the trajectory. Two sub-controllers, the sliding mode trajectory tracking control of the mobile platform and the non-singular terminal sliding mode control of the manipulator, have been employed and the simulation results have verified the effectiveness of the approach. ACKNOWLEDGEMENT

q1 = sin(t ) , q2 = cos(t ) , and q3 = t cos(t ) are the position instructions of the three joints respectively, and the simulation results are shown in Figure 6- Figure8.

position tracking of link1

5

(37)

η − = {γ ij } = −η +

position tracking of link2

4

Figure 8 Position tracking of link 3

0 0 0 ⎤ ⎡ 0 12 0.8 ⎢0.4 0 0 0.04 0.045 0.02⎥ ⎢ ⎥ ⎢⎣1.5 0 0 0.88 0.07 0.04⎥⎦

0 -0.5 -1 -1.5 0

3 time(s)

1

2

3

4

5

time(s)

Figure 7 Position tracking of link 2

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