Smooth Sliding Mode Tracking Control of the Stewart Platform

Chin-I Huang1, student member, IEEE and Li-Chen Fu1,2, Fellow, IEEE Department of Electrical Engineering1

Department of Computer Science and Information Engineering2

National Taiwan University, Taipei, Taiwan, R.O.C. E-mail: lichen@csie.ntu.edu.tw

Abstract—This paper presents a smooth sliding modecontrol approach for the motion control of a Stewart platform.The control scheme is proposed given that the overall systemparameters are subject to uncertainties while only thepositions and velocities of links are measurable. To achieve high performance tracking control of a 6 DOF Stewartplatform normally requires the full knowledge of the systemdynamics. In this paper, some important properties of thedynamics of the Stewart platform have been derived andexploited to develop a sliding mode controller which can drivethe motion tracking error to zero asymptotically. Stabilityanalysis based on Lyapunov theory is performed to guaranteethat the controller design is stable. Finally, the experimentalresults confirm the effectiveness of our control design.

I. INTRODUCTION

In recent years, the parallel link manipulators haveattracted much attention and many studies have been done on the kinematics or static analysis of the parallel linkmanipulators [12]. Generally speaking, the parallel linkmanipulators provide better accuracy, higher rigidity, higher load-to-weight ratio, and more uniform load distributionthan the serial manipulators. Such advantages of fully parallel manipulators [1] originate from the fact that theactuators act in parallel sharing the common payload. The Stewart platform manipulator is a 6DOF mechanism with two bodies connected together by six extensible legs [1, 2].This closed-loop structure makes the manipulator system far more rigid in proportion to size and weight than any seriallink robot, and yields a force-output-to-manipulator-weightratio more than one order of magnitude greater than seriallink robot. Practical usage of the Stewart platformmanipulator has generally been in the area of low speed and large payload conditions such as motion base of the classicalautomobile or flight simulator, and motion bed of a machinetool [4]. For the design and the control of the Stewartplatform manipulators, dynamics analysis is a crucial step. In recent years, many research works have been conductedon the dynamics of the Gough–Stewart platform

manipulator [3–11]. Several methods such as the Lagrangeequation, Newton–Euler equation and principle of virtualwork are proposed to derive dynamic equations of theGough–Stewart platform. The Lagrange formulation is wellstructured and can be expressed in closed form, but a largeamount of symbolic computation is needed to find partialderivatives of the Lagrangian in this method. The Newton–Euler approach requires computation of allconstraint forces and moments between the links. However, these computations are not necessary for the simulation andcontrol of a manipulator.

The method of virtual work is an efficient approach toderive dynamic equations for the inverse dynamics of theStewart platform [8, 9]. However, for the forward dynamics,the method of virtual work is not straightforward because of the complicated velocity transform between the joint-space and task-space.

In this paper, an approach based on a sliding-mode controltechnique has been successfully developed for motioncontrol of the Stewart platform system with havingparametric uncertainty. These schemes are designed toguarantee practical robustness and stability. The remainderof this article is organized as follows: The kinematics anddynamics models for the Stewart platform are discussed inSection 2. In Section 3, the smooth sliding mode controllerfor a Stewart platform system is developed and the stabilityanalysis is conducted. Section 4 shows some experimentalresults on controlling a realistic Stewart platform. Finally,some conclusions are made in Section 5.

II. KINEMATICS AND DYNAMICS OF A STEWART PLATFORM

The Stewart Platform is a parallel manipulator [1, 2]. It has a lower base platform and an upper payload platformconnected by six extensible legs with ball joints at both ends. In the following subsections, we first make the inversekinematics analysis of the Stewart platform, and then deriveits dynamics.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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0-7803-9354-6/05/$20.00 ©2005 IEEE 43

2.1 Inverse and Forward Kinematics Analysis of a Stewart Platform

The Inverse and Forward Kinematics Analysis of Stewart platform please refer to reference [17].

2.2 Dynamics Analysis of Stewart PlatformTo design a system with high operational performance, a

sound control method is crucially needed. However, tocontrol the Stewart Platform system well is very challengingdue to the high nonlinearity in system dynamics, systemuncertainties, and complex kinematics. In general, thedynamic equations [4] of the Stewart Platform system can be written as:

( ) ( , ) ( ) ( )mM q q C q q q G q K q q (1a)( )m m m dJ q K q q (1b)

where , are generalized coordinates of therobot and the rotor subsystem, respectively;

6q R 6mq R

( )M q is aninertia matrix, which is symmetric and positive

definite for all ; is the Coriolis/centripetalforce vector; and

6 66q R ( , )C q q q

( )G q ( m )K q q are 6 1 vectors containing gravity force and elastic force transmittedthrough joints, respectively; K is an 6 diagonal matrixrepresenting the joint stiffness;

6mJ is actuator inertial

matrix; is the input force vector,6R d is the externaldisturbance of the server system.Some pertaining propertiesare given below.Property 1: are bounded functions if

are bounded. is a bounded function if are bounded.

( , ) and ( )C q q M q

and q q ( , )C q q

, andq q q

Property 2: M is a symmetric and positive definite matrix.Moreover, for an appropriate choice of C, 2M C can be a skew-symmetric matrix, which means that

( 2 ) 0,T nx M C x x R . This property is well known inthe robotics literature.Property 3: The hyperbolic tangent function defined as

tanh e e e e satisfies thefollowing properties:

(a) , for , this further implies for any

odd positive integer with

tanh 0a 0a

b tanh 0ba

(b) | tanh | tanh | | 0

(c) if tanh | |c d 1, for , ,then0c 0 d

1 1| | ln2 1

dc d

must be satisfied.

Let such that the complete system (4a) (4b) is expressed in the new coordinates as

1 mz Kq

1Mq Cq G z B1

1 1m dJ K z z B

where B Kq .

III. SMOOTH SLIDING MODE CONTROL FOR STEWARTPLATFORM

In this section, a two-stage approach, or namely the integrator backstepping approach is applied to design the adaptive controller. First stage: For the rigid manipulator subsystem, design avirtual control input z such that and as d1 q qd q qd

t where is the desired trajectory.q td ( )Second stage: Design the actual control input to drive

the variable and . Where and willdefined later. Moreover, The smooth robust control technique imposed for each stage to handle the uncertaintyof the system. The detailed procedure is addressed below.

1 1dz z 1 2dz z 1dz 2dz

3.1 Design a Virtual Control Input for the First Stagez1

Define de q q and r dq q e , where e, q denote themotion error and an auxiliary signal vector, respectively. In addition, define an exponentially stable manifold

r

s be themeasurement of motion tracking error, denoted by

rs e e q q . If the error measure s is driven to zero, then the position and velocity tracking error will bothconverge to zero in an exponential behavior. To this end, theerror dynamics of the error measure is rewritten as

1

r

r

Ms Mq Mq

Mq Cq G z B

If let as a virtual control input, and consider the feedback control

1z

1 1dz z with defined as 1dz

1d r rz Mq Cq G B sBecause the system has the parametric uncertainty, the equation can not be implemented. Hence the LP property is used for solve above problems. Therefore, if assume theuncertain parameters can be partition into 0

which are known priori, then the virtual control input zcan be rewritten as

d1

1 0 1dz Y Y B s (2) where the control parameter is positive defined and

is the robust function designed asfollows:

1 11 12 1, , , Tr

,

1 1, 1

( + ) tanh ( )( )n n

i i i j kij k

u vt s Y 1, 2, ,i r with

1 , , 0i u v and | | , then the error dynamics can be rewritten as follow:

i i

1Ms Cs s Y YFurther, using the Lyapunov function

112

TV s Ms

44

is positive

1

1

121 ( 2 )2

T T

T T T T

V s Ms s Ms

s s s M C s s Y s Y

,

1 , 1

, ,

11 , 1 , 1

,

1 , 1

, ,

11 , 1 , 1

( )

( )( + ) tanh ( )( )

| || |

| ( )( + ) tanh ( )( ) |

n nrT

j ki ii j k

n n n nr

j ki i i j kii j k j k

n nrT

j ki ii j k

n n n nr

j ki i i j kii j k j k

s s s Y

s Y u vt s Y

s s s Y

s Y u vt s Y

which leads to ,

11 , 1

,

11 , 1 , 1

| || |

| |( + ) tanh ( ) |

n nrT

j ki ii j k

n n n nr

j ki i i j kii j k j k

V s s s Y

s Y u vt s Y,

|

after applying Properties 3 (a)-(b). ,

, 1 1

| |tanh ( ) | |

+

n ni

j kij k i i

u vt s Y (3)

we have1

TV s s (4) 1

Since is positive definite and if (3) sustains, it can

be concluded that all error signals 1V 1 0V

, and , ,s e s e are alluniformly bounded. Applying Property 3.(c), the measure ofthe ultimated ball can be described by the radius about the origin as

1

1

| |1| | ( ln2( ) | |

ni i i

j ki ijk i i i

s Y bu vt

)

Or/ ss b Y b

Equation (4) can be rewritten in the following form2

1 min

min1

2

M

V s

VM

2 21 1P.S. ( )2 2m MM s V s M s

and integrating both sides of the above inequality with respect to time. We obtain min2 /

1 0MM tV V e and this

means using the property again, the reaching transient response is shaped by

min

0 Mt

MM

m

Ms t s e

M(5)

As a result, the motion tracking error exponentially decay if ss b . Obviously, the ball sb will converge to zero

linearly in time. Therefore, the error signals s willasymptotically converge to zero.

( )

We now show that s also converge to zero by usingcontradiction argument. Assume that s t is small enough

and 0s t at some . Since 0t 11M Y Y s is

continuous, without loss of generality we can say 1

1( )M Y Y for some 0 in the timeinterval ,t t T . By Integrating error equation, it follows

that 11(

t T

t)s t T M Y Y dt . Thus

s t T T , which contradicts with the fact that sconverges to zero. Hence as . This mean

as . Accordingly, the force error 0s t

0e t will asymptotically converge to zero as .tNow, since is not the input, consists an error term relatedto

1z

1 1z dz z1 and is shown below the practical error dynamics is

1 1Ms Cs s Y Y z (6) If can be driven to zero, the stability can be obtained. 1z

3.2 Regulate z Converge to for the Second Stage. 1dzStep 2: We now have the error dynamics of as 1z

1 1dz z z1 (7)where is with the uncertainty of the manipulator, so is not realizable. To let the control input be realized byposition, velocity and force measurements only, thefollowing relations are introduced:

1dz 1dz

11 1( )s M Y Y z Cs s

11( )T Tq M J z J B Cq G

Then the terms Y , where and q s replaced by signals, yields

,q q

1 0 2( , , , , , ) ( , , , , ) ( , , , , )r r r r r r rY q q q q q q Y q q q q Y q q q qAdditionally, the time derivative of the virtual control input can be partitioned into the known part and unknownpart

0NN , and is expressed as

1 0dz N Nwhere

12 0 1

11 1

0 0

0

( , , , , , )( )

( )( )

r r

dN Y q q q q Y d

dt

d M Y Y z Cs s

d d

d d d

Note that 0 denote the uncertain term can notbe obtained. The term is realizable on-line, and 0N N is an uncertain term with unknown parameters which is not satisfying the LP property. Hence the typical adaptive

45

backstepping can not be applied in this problem. Accordingto the stability analysis in Step 1, the state variables and the time derivative of the constrained force are all bounded once the can be steered to zero. This means that when a suitable control is applied, the uncertain term has a upper bound as

1zN

1( , , , , , ) 1, ,6ii r rN q q q q z N for i

Next, let us consider the error dynamics (10) again,yields

(8)1 1 1 0dz z z N N z1

Now, we assume as the virtual control input in the form:1z

1 2 0 1 1 2 1( ) tanh[( )d ]z z N z s N u vt z (9)

with 1 2 21 2 1 2{ , , }; { , , }and , >0n nN diag N N diag i .Then (11) can be further rewritten as

1 1 1 2 1( ) tanh[( ) ]z z s N N u vt z (10) Consider 1

2 1 12TV V z z1 as a Lyapunov function candidate

for the error system (10) with the virtual input (9). Let . We have the time derivative of as 1

TT Te s z 2V

2 11 , 1 11

, ,

1 1 21 , 1 , 1 1

( ) +

( )( + )tanh ( )( ) ( )tanh[( ) ]

r n nT

j ki i ii j k in

n n n nr n

j ki i i j ki i i i ii j k j k i

IV e e sY z N

I I

1sY u vt sY z N u vt z

In the same way, if both the inequality (3) and followinginequality

12

| |tanh ( ) | |

+i

i

i i

Nu vt z

N(11)

are satisfied, it follows that

(12)21

T

n

IV e

I Ie

Since 0 and 1 0nI , is assured. Using thefacts that is upper bounded and , the Lyapunovfunction is decreasing. this implies all error signals

2 0V

2V 2 0V

2V s ,are uniformly bounded. From the analysis of step 1, this means all error signals are uniformly bounded. Moreover,the error signal e will asymptotically converge to an ultimated bound defined by (6) and (14) as follows:

1z

1 1

2

2

| |1max , , with ( ln )2( ) | |

i i is z z

i i i

N Ne b b b

u vt N N

Obviously, the ball 1

max ,s zb b will converge to zerolinearly in time. Therefore, the error signals e will asymptotically converge to zero as long as t . The remaining of the proof is to show the control law issmooth as . From the facts that satisfies theinequality as

t 2V

2 21 2 2

22 3

c e V c e

V c e

1 min 1 max 3 min1

with ,1 , ,1 andm Mn

Ic M c M c

I I,we

can easily show that the error signal e is shapped by3

222

1

0c

tcc

e t e ec

Hence, using the same technique

of the proof in step 1, the robust function and its timederivative are bounded and smooth.Since is not the true input, we look again at the error 1z

2 2dz z z1 , where defined in (10). The error dynamics of is rewritten in

2dz

1z

1 1 1 2 1( ) tanh[( ) ] + 2z z s N N u vt z zIn Step 3, the actual designing a control input will be determined to let converge to zero. 2z

Step 3: Before applying the same design scheme as Step 2, let us reformulate into a nominal part and a uncertain part toobtain actual control input only form position, velocity, and force. Assume

2dz

mJ is nominally known as 0mJ , and the unknown part mJ . Thus the dynamics of is 2z

1 1 12 2 1

10 0 1( )

m m d m

m

J K z J K z J K z

J K w w E N z B(13)

where0 1 1 0 0 0 2 1 0 1 1 0( , , , , , , ) [ ] ( ) { ( )( )} ( )r rw qqq q z z N s s N Pvz u vt N z N z

1 11 1 0 0 0 0 2( , , , , , , , ) [ ] ( )r r d m d m m m dwqqq q z z N s KJ J J J z

1

2 211 1{sec [( ) ], , sec [( ) ]}nP diag h u vt z h u vt z

2 1( )( ) nE u vt N P INote that is a known part in the error system and the term0w

21u vt z is bounded has been shown in step2. Once the

error can be drived to zero, which is achieved in this step. Moreover, the uncertain term is assumed with a upper bound

2zw

| ( ) |i iw w .Let the control input as 1

1 2 2 1 0mz B z z J K ,where will be defined later. Thus we obtain the overall error system:

1 1Ms Cs s Y Y z (14)

1 1 1 2 1( ) tanh[( ) ] 2+z z s N N u vt z z

1) (15)

1 12 2 2 0 0(m mJ K z z J K w w E N z (16)

In (16), the new control will be used to cancel w , N .Therefore let us consider the Lyapunov function

13 2 22

TV V z z2 . From the analysis of , applying the inequality (5) and (13), the time derivative of

yields

2V

3V

V 13 1 2 0 0

2

0( )

0

T Tn n m

n n

Ie I I I e z J K w w E N

I I

46

where 1 2

TT T Te s z z . To obtain , we choose 3 0V

0 3 2( ) tanh[( ) ] (w w u vt z E ) ,

550 450 470x x mm

where1 3 31{ , , }; { , ,n nw diag w w diag 3 } and

1 4 2 41

[ , , ]; ( )tanh{( )[ ]}, 0 for 1, ,n

n i i j ji ij

N u vt z E i n

In this setting, the derivative of can be rewritten as follows:

3V

3 1

2

10 2 2 2 3 1

1

0

0

(| | | | tanh{( ) | |}( ) )

Tn n

n n

n

m i i i i i ii

IV e I I I e

I I

2J K z w z u vt z w P P

1 2 2 2 21 1 1

where

| || |; | | tanh{( ) | |}(n n n

j ji i j ji j ji i ij j j

P z E N P z E u vt z E N 4 )

Once the following inequality is satisfied

2

32

3

2

12

| |1| | max(( ln )2( ) | |

| |1,( ln ))/ | |2( ) | |

i i ii z

i i i

ni i i

jiii i i

w wz b

u vt w w

N NE

u vt N N

then we have

3 1

2

0

0

Tn n

n n

I

V e I I II I

e

this means that all error signals are uniformly bounded, and the signal e will exponentially converge to the ultimatedball as

1 2max , ,s z ze b b b . Since the ball shrinks to the

origin e will converge to zero along the ball as t .Applying this fact, we have

2 21 3 2

23 3

d e V d e

V d e

1 2

3 min 1

2

with d min ,1 , max ,1

0and

0

m M

n n

n n

M d M

I

d I I I

I I

Hence3

222

1

0d

tdd

e t e ed

, this meaning that the

control law is smooth and bounded as . Therefore theresults of step1 is achieved such that the , will convergeto zero as t .

te e

IV. EXPERIMENTAL RESULTS

In this section, we make a series experiment on the slidingmode controller design which is proposed in Section 4.

Fig 2 The Experimental System Configuration DiagramFigure 2 show the experimental system configuration of

the Stewart platform system in this study. The motioncontrol system computer runs a drive logic to control a hydraulic system that drive a 6 degree-of-freedom Stewartplatform for creating realistic motion cue. The experimentare done with 6 DOF hydraulic Stewart platform system, and is manufactured by SGD Co.. The hydraulic servo valve inthe Stewart platform uses MOOG’s J076-104. The D/A carduses Adventech’s 726 and A/D card uses Adventech’s 818H.The Detailed parameters and specification will be found inTable 1. Table 1: Specification and parameter of the Stewart platform

MotionDegree Of Freedom 6(Heave,Sway,Surge,Pitch,Roll,Yaw)Net Loading 500kg

Accelebration 1g

Angular Acceleration 260 / secControl Servo Class Hydraulic Actuating

SystemHeave 0 ~ 221.518mm mmSway 211.198mmSurge 244.758mmPitch 12.960Roll 10.821Yaw 18.474

Motion Platform Dimension(L x W x

H)1350 1200 760x x mm

Net Weight 600kg

Hydraulic System Dimension(L x W x H) 1250 1250 690x x mmNet Weight 800 kg Electricity Power 380 / 660 / 3 / 50 / 60V V Hz

Rated Motor Power 7.5HPRated (Max.) Operation Pressure 4.5(10)Mpa

System Flow Rate 65 / minliter

Oil Type ISO VG46 Oil Operation Temperature 10 ~ 50 C

CoolerDimension (L x W x H)Cooling Capacity 6520 / @35cal hr C

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4.1 Results of Smooth Sliding Mode controller for pitchmotion tracking

The desired trajectories for roll motion tracking case are shown in Table 2.

Table 2 Circular Motion TrajectoriesHeave 0-10-0 cmSway 0 cmSurge 0 cmPitch 0 degreeRoll 10sin(ft) degree/sec; f=rad/secYaw 0 degreeThe desired roll trajectory is shown in Fig 3. The desired

heave, sway, surge, pitch and yaw trajectories are shown in Fig.4. The every desired and real trajectory of the leg is shown in Fig. 5-10.

Fig. 3 Desired Pitch Trajectory Fig. 4 Desired , , , ,x y z Trajectories

Fig. 5 The Leg1 Trajectories Fig. 6 The Leg2 Trajectories

Fig. 7 The Leg3 Trajectories Fig. 8 The Leg4 Trajectories

Fig. 9 The Leg5 Trajectories Fig. 10 The Leg6 Trajectories

V. CONCLUSIONS

In this paper we present a smooth sliding mode controlapproach for the motion control of a Stewart platform. The

control scheme is proposed given that the overall systemparameters are subject to uncertainties while only thepositions and velocities of links are measurable. To achievehigh performance tracking control of a 6 DOF Stewartplatform normally requires the full knowledge of the systemdynamics. In this paper, some important properties of thedynamics of the Stewart platform have been derived andexploited to develop a smooth sliding mode backsteppingcontroller which can drive the motion tracking error to zeroasymptotically. Stability analysis based on Lyapunov theory is performed to guarantee that the controller design is stable.Finally, the experimental results confirm the effectiveness ofour control design.

REFERENCES

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[10] K. E. Zanganek, R. Sinatra, and J. Angeles, “Kinematics and dynamicsof a six-degree-of-freedom parallel manipulator with revolute legs,”Robot, vol. 15, pp. 385–394, 1997.

[11] L. W. Tsai, “Solving the inverse dynamics of parallel manipulators bythe principle of virtual work,” in 1998 ASME Design Eng. Tech. Conf (DETC/MECH-5865), Sept. 1998, pp. 451–457.

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[13] J.J.E. Slotine and W. Li. Applied Nonlinear Control. EnglewoodCliffs, NJ: Prentice-Hall, 1991.

[14] J.-B. Pomet and L. Praly, “Adaptive Nonlinear Regulation: Estimationfrom the Lyapunov Equation,” IEEE Trans. Automat. Cont., vol. 37, no. 6, pp. 729-740, June 1992.

[15] R. Lozano and B. Brogliato, “Adaptive Control of Robot Manipulators with Flexible Joints,” IEEE Trans. Automat. Cont., vol. 37, no. 2, pp. 174-181, February 1992.

[16] N. Sadegh and R. Horowitz, “Stability and Robustness Analysis of aClass of Adaptive Controller for Robotic Manipulators,” Int. J. Robotic Research, vol. 9, no. 3, pp. 74-92, 1990.

[17] C. I. Huang, C. F. Chang, M. Y. Yu and L. C. Fu, “Sliding-ModeTracking Control of the Stewart Platform,” 5th Asian Control Conference, pp. 561-568, 2004.

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