Adaptive backstepping tracking control of the Stewart platform

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Adaptive Backstepping Tracking Control of the

Stewart Platform

Chin-I Huang1, student member, IEEEandLi-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: [email protected]

Abstract—This paper presents an adaptive backstepping

control approach for the motion control of a Stewart platform. The control scheme is proposed given that the overall system parameters are subject to uncertainties while only the positions and velocities of links are measurable. To achieve high performance tracking control of a 6 DOF Stewart platform normally requires the full knowledge of the system dynamics. In this paper, some important properties of the dynamics of the Stewart platform have been derived and exploited to develop an adaptive backstepping controller which can drive the motion tracking error to zero asymptotically. Stability analysis based on Lyapunov theory is performed to guarantee that the controller design is stable. Finally, the experimental results confirm the effectiveness of our control design.

I. INTRODUCTION

In recent years, the parallel link manipulators have attracted much attention and many studies have been done on the kinematics or static analysis of the parallel link manipulators [12]. Generally speaking, the parallel link manipulators provide better accuracy, higher rigidity, higher load-to-weight ratio, and more uniform load distribution than the serial manipulators. Such advantages of fully parallel manipulators [1] originate from the fact that the actuators 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 serial link robot, and yields a force-output-to-manipulator-weight ratio more than one order of magnitude greater than serial link robot. Practical usage of the Stewart platform manipulator has generally been in the area of low speed and large payload conditions such as motion base of the classical automobile or flight simulator, and motion bed of a machine tool [4]. For the design and the control of the Stewart platform manipulators, dynamics analysis is a crucial step. In recent years, many research works have been conducted on the dynamics of the Gough–Stewart platform

manipulator [3–11]. Several methods such as the Lagrange equation, Newton–Euler equation and principle of virtual work are proposed to derive dynamic equations of the Gough–Stewart platform. The Lagrange formulation is well structured and can be expressed in closed form, but a large amount of symbolic computation is needed to find partial derivatives of the Lagrangian in this method. The Newton–Euler approach requires computation of all constraint forces and moments between the links. However, these computations are not necessary for the simulation and control of a manipulator.

The method of virtual work is an efficient approach to derive dynamic equations for the inverse dynamics of the Stewart 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 control technique has been successfully developed for motion control of the Stewart platform system with having parametric uncertainty. These schemes are designed to guarantee practical robustness and stability. The remainder of this article is organized as follows: The kinematics and dynamics models for the Stewart platform are discussed in Section 2. In Section 3, 4 the Smooth Projection Algorithm and the adaptive backstepping controller for a Stewart platform system is developed and the stability analysis is conducted, respectively. Section 5 shows some experimental results on controlling a realistic Stewart platform. Finally, some conclusions are made in Section 6.

II. KINEMATICS AND DYNAMICS OF A STEWARTPLATFORM

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

43rd IEEE Conference on Decision and Control December 14-17, 2004

Atlantis, Paradise Island, Bahamas

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A. Inverse and Forward Kinematics Analysis of a Stewart Platform

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

B. Dynamics Analysis of Stewart Platform

To design a system with high operational performance, a sound control method is crucially needed. However, to control the Stewart Platform system well is very challenging due to the high nonlinearity in system dynamics, system uncertainties, and complex kinematics. In general, the dynamic equations [4] of the Stewart Platform system can be written as: 1 ( ) ( , ) ( ) M q q C q q q G q Kz (1a) J qm m B qm m Kz1 W (1b) and z₁ q_mq,

where M q( ) is an inertia matrix, which is a symmetric and positive definite for all ; is the Coriolis/Centripetal vector;

6 6u

6

q R C q q q( , ) ( ) and

G q W are 6 1u vectors containing gravity torques and input torques, respectively; is generalized coordinates of the actuator subsystem; and Kz are elastic force. We can see this system like a flexible joint manipulators system. Some pertaining properties are given below.

6

m

q R

1

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 , and q q q

Property 2:

M

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

( 2 ) 0,

T n

x M C x x R . This property is well known in the robotics literature.

III. SMOOTHPROJECTIONALGORITHM

The smooth projection algorithms are studied or used in [14, 15]. For our application, a similar convex domain defined in Lozano and Brogliato [15] is considered and is approximated by a new convex set with a proper choice of a convex function. Then the smooth projection algorithm is appropriately defined such that the singularity of estimation will be avoided in the adaptive control and thus the backstepping design can proceed smoothly.

First, using the facts Tkitki (ki is the lower bound of stiffness coefficient), the convex domain of parameters Tki,

Td 1, and Td 2 will be defined in a general form. Let us define a subspace D spanned by the known function as ranges over and expressed as

( ) d q q n R

^

h| ( ), n

`

D dR d d q q R

in which the corresponding function d( ) of Tki,Td 1, and Td 2 are 1, , and , respectively. The region of true parameters is defined in

1( )

d q d q₂( )

^

_{R d}h| T , _d _D

`

T T T

3 t

where T is the concerned parameters, such as Tki,Td 1, and Td 2, with a known constant T is the lower bound of Tki. To avoid the singularity due to parametric estimations, the region of possible values of estimated parameters is defined by

^

h| T ,

`

a T R d T T G d D

3 t

where G! 0 is a thickness about the neighborhood of T and satisfies T G ! 0 . For the smooth projection of parameter estimation, the convex domain 3 concerning T will be arbitrarily close to a new compact convex set constructed by

^

_T _Rh| ( ) 1_T

`

3 A d

A

where the smooth function from to satisfies the following conditions:

A( ) 3a R

(1) The true parameter vector T satisfies that A( )T d 0, which means T lies in 3 and also be restricted in the interior of 3A.

(2) The set

^

_T_Rh| ( )_A_T _{d p}

`

_{is convex and contains 3} for each real number p in [0,1].

(3) The vector p( )T w _T( )T w { A A

is nonzero for all T satisfying A( )T

_>

0, 1

_@

.

From the definition of a convex 3A, an example of A( ) is given as ( ) ( ) ( ) T d P P P P T T T T T G A , for Pt 2 .

It also noted that ( ) 1AT when _dT

T T G . In this case, the defined convex set 3A is concerning as a union of the set

3 and a boundary layer O( )G around it.

The smooth projection algorithm is then constructed by

2 if ( ) 0 if ( ) 0 and 0 Proj( , )= ( ) otherwise T T h p p p I p \ T \ T \ T \ T \ d ° t d ° ® § · ° _¨ _¸ ° ¨ ¸ © ¹ ¯ A A A A A A A

with the initial condition (0)T chosen in 3 . This results the following useful properties:

A

(M1) Proj( , )T \ is Lipschitz continuous;

(M2) If \( )t is continuously differentiable and Proj( , )

T T \ , T(0)3A

(3)

remains in 3A;

(M3) Proj( , ) Proj( , )T T

T \ T \ d\ \;

(M4) (_{T T}) Proj( , T _{T \}) (_{T T \})T _, _.

d , T 3 T 3A

These properties will be used later in the stability analysis. IV. ADAPTIVEBACKSTEPPINGCONTROL FOR STEWART

PLATFORM

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 a virtual control input z such that1d qoqd and qoqd as

t o f where q td( ) is the desired trajectory.

Second stage: Design the actual control input W to drive the variable z1 converges to z .1d

All the system parameters are unknown in the design phase, whereas only positions and velocities of the manipulator are measurable.

A. Design a Virtual Control Input z1d for the First Stage

Define and , where e, denote

the motion error and an auxiliary signal vector, respectively. In addition, define an exponentially stable manifold

d

e q q q_r q_d /e qr

s be

the measurement of motion tracking error, denoted by

r

s e / e q q . If the error measure s is driven to zero, then the position and velocity tracking error will both converge to zero in an exponential behavior. To this end, the error dynamics in terms of the error measure is written as:

1 1 r 1d (2)

where Y( ) Mq_rCq_r ; and

K K K are the estimated flexibility constants and its error, respectively; and ~z1 z1z1d where is the virtual control input. The virtual control law is set as

z1d z1d = () r Ms Mq Mq Kz Cs Kz Y T Kz G K diag

_{^ `}

T_ki

2

1 1d d r ( s sp) p

z K YT K e K s K e , (3)

where the control gains K K are positive-definite matrix. Then substituting (3) into (2) results in the error dynamics in the form:

K s, sp, p 2 1 ( _s _sp) _p _{D rk} _r Ms Cs Ke K s K e YT 'YTKz. (4)

In above equation, we have some notions as

~ ~ ~ TrkT Tr T T k T { with Tr TrTr _and k k k, ,' {Y Y( ) Yd( ) . T T T YD{ Yd( ) Dz( )

Based on a similar argument in [Sadegh and Horowitz, 16], the compensation matrix is proven to be bounded above through the following inequality:

'Y 

bg

2 1 2 3 4 r YT U s U s e U e U ' d e (5)

where Ui,i 1, ,"4 are positive constants which depend on the desired trajectories, i.e., , , , the upper bounds of system parameters, and control gain

d

q q_d q_d

/ .

To select the adaptation laws for ˆT_r and ˆT_k , let us consider a Lyapunov function candidate:

1 1 1 1 1 1 1 2 2 2 2 T T T T p r r r k k k V s Ms e K e T _*T T _*T _,

where *r and *k are symmetric positive-definite

matrices. The time derivative of along the error dynamics (4) is 1 V

2 ₁ 1 1 1. T T T T T s sp p p r r r T T T k k k z r V s K s e s K s s K e e K e Y s D s s Y s Kz T T T T T * * ' T d

Then the update law of T_r is chosen as

T r r dY s

T * . (6a)

According that K is a diagonal matrix with strictly positive entries and K is required to be invertable, i.e.,T_ki z0, for everyt t0, the adaptive law of Tˆk is set by

using the smooth projection:

Proj , , with (0)

ki ki kD sz i ki ki

T T * T 3 (6b)

where the smooth projection algorithm, Proj

Tki, Zi

withZ *kD sz , is given by the corresponding proper

convex function A_ki

T_ki in Section 3, and the initial conditionTki 0

are set in the interior of . The property (M4) of a smooth projection for (6b) can be presented as

ki 3

1 1 1 1 1 0, Proj( , ) n T k _k _k _z ki ki _ki _i i n ki ki ki i i i D s T _T T _T _Z T T Z Z * * * d

¦

(7) Consequently,V1 is reduced to 2 1 1. T T T T T s sp p r V s K s e s K s e /K e s 'YT s Kz (8)

Since the compensation matrix satisfies (5), the time derivative of V in (8) leads to

Y ' 1

2 2 2 2 2 2 2 1 1 2 3 2 2 2 2 2 1 2 3 1 2 2 ₁ ₁ 2 2 2 4 2 4 4 4 1 T s sp p T s sp p T T V k s k e s k e s s e e s s e s Kz k s k e s k e s e e s s Kz e s e e s x Qx s Kz U U U U U U U U U U d d d 4 1 (12)

where k_s , k_sp , and k_p are the minimum norm ofK Ks, , sp and/Kp, respectively, and

s x e e s ª º « » « » « » ¬ ¼ 1 1 1 2 3 2 2 11 12 1 1 3 4 2 4 21 22 1 2 4 2 0 0 s p T sp k Q Q Q k Q Q k U U U U U U U ª º ª º « » _{{ «} _» « » ¬ ¼ « » ¬ ¼ . If the control gain can be suitably chosen such

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that and , i.e., matrix Q is positive-definite, then and

11 0

Q ! 1

22 12 11 12T 0

Q Q Q Q _!

e s are uniformly

asymptotically stable once is steered to zero. Hence the first stage is completed. The control problem has been transferred into an equivalent control problem for stabilization of .

1

z

1

z

B. Regulate z₁ Converge toz_1d for the Second Stage

To stabilize , we differentiate to yield z₁ z₁

, (10)

1 1 1d

z z z

where is defined in (3). Since the virtual control input is composed of the integral positions and velocities of links, its time derivative depends not only on q

and , but also . To implement using only measurable quantities, such as positions and velocities, the right-hand

side of equation is

substituted to appearing in . Hence it become possible to avoid using the link acceleration. It is also noted that the desired trajectory compensation is introduced in .

1d z 1d z 1d z q q z_1d 1 1 ( ( , ) ( )) q M Kz C q q q G q q z_1d ( ) d Y  z1d

The first and the second time derivative of Y _d( ) are all exactly-known and is beneficial to the replacement of q . Since det( )M appears in the denominator, care must be taken to express as an LP form. We multiply both sides of (10) by det

1d

z

( )M and obtain

1 1

det( )M z det( )M z det( )M z1_d. (11)

This ensures that det( )M z₁_d is an LP form and depends on reasonable measurements q, , and .q z₁

Since the actual control input does not appear in (11), the integrator backstepping procedure continues to introduce a new virtual control input z2d and define

. (12)

1 2 2d

z z z

In light of stability analysis using Lyapunov method, adding and subtracting 1

1

2( det( ))dtd M z to the right-hand side

of (11), the error dynamics of in (11) is represented as follows 1 z 1 1 1 1 2 2 det( ) ( T ) ( det( ))d d d dt 1 M z d T z M z (13) where 1 1 1 det( ) 1 2( det( ))d 1 z z d dt Y T M z M z and Y_z₁( ) is a known function of q , q, and ;z₁ T_z₁ is an unknown parametric vector with appropriate dimension. Therefore, the virtual control law z_2d is designed in the form:

1 2d ( 1T d1) ( z1 z1 z1 1 ) z d T Y T K z K s (14)

with the estimated parameters Td1

andTz1

; the positive definite control matrix gainsK_z₁; and an additional term

Ks for the purpose to eliminate the cross-term that

appeared in (9). Let 1 1 d1, denote

the estimated parameter errors. The error dynamics (13) becomes

1

1 1 1 2 2 1 1 1 1

det( ) (T ) ( ( det( ))d ) det( )

d d dt n z z z 2 M z dT z M I K z Y T Ks M z. (15) d d T T T Tz1 Tz1Tz1

To select update laws of Td1

andTz1 , let us consider a Lyapunov-like function 1 1 1 1 1 2 1 2det( ) 1 1 2 1 1 1 2 1 1 T T T d d d z z z V V M z z T *T T *T1 (16)

where*d1,*z1 are positive-definite symmetric matrices.

From the error dynamics of in (15), the time derivative of is 1 z 2 V 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 det( ) ( det( )) ( ) ( det( ) T d T T T d d d z z z dt T T T T T T z d d d d z z z z T V V M z z M z z ) x Qx z K z z z d Y z M z z T T T T T T T T * * d * *

The update laws is then given by

1 1 1 T z zY zz T * ₁ (17a) 1 Proj( d1, ),1 1 1T 2 1 1, 1(0) d with z zd d d d F1 T T F F * T 3 (17b)

where 3F1 is a convex set for a chosen convex function of 1

d

T defined in Section 3.

According to the smooth projection (17b), Td1

satisfies -1 -1 1 1( d1 1) 1 1(Proj( d1, )1 1) T T d d d d T T F T T F F 0. * * d (18)

This yields that V₂ hold the condition

(19) 1 2 2 2 1 0 det( ) 0 T T z Q V x x M z z K ª º d _« _» ¬ ¼ 2 with 2 1 T T T x _¬ªx z º_¼.

Thus, the error signal is required to be stabilized such that

2

z

2 0

V d _.

C. Design the Control W (Continuing the Second Stage)

Differentiatingz₂ leads to (20) 1 2 1 2d m ( 1 m m) ( 2 z z z J WKz B q q q ₎z d

To stabilize , the time derivative of is investigated as a function of ,

2

z z2d

1d

z s,z1, and Td1 . Since the smooth

projection is applied in the update law of K, the second derivative is well defined. Similar to the argument of stabilization of , pre-multiplying by

1d

z

1

z z₂ det( )M , which

results that the parametric uncertainties with LP property can be compensated by using measurable signals. In light of this, the new strategy for linear parameterized is to multiply both sides of (20) by _{(det( ))}2

m

M J and yield

2 1 2

2 2 2 2 2 1 2

(det( )) ( T ) ( (det( )) )d det( )

m d dt m z

M J z dT W M J z M z Y T_z₂

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2 1 2 2 1 2 2 2 1 1 ( ( )) ( ) ( ( (det( )) ) det( )( ( ) ) d z z m m m m d dt m m a Y det M Kz B q J q J z M J M J M Kz Cq G z T 2 _z

Meanwhile, we have added and subtracted

2 1

2

2( (det( )) )dtd M J zm and det( )M z1 for eliminating

cross-term based on the same approach for stabilization of . Accordingly, the actual control law is set as follows

1 z 1 2 2 2 2 2 2 ( T ) ( d z z z d Y K W T T z ) (21)

and the estimated parameters are turned by

2 2 2 T z z Y zz T * 2 _(22a) 2 Proj( d2, 2), 2 2 2T 2, 2(0) d with d z d d F2 T T F F * W T 3 (22b)

which result in the error dynamics

2 1 2

2 2 2 2 2 2 1 2 2. (23)

(det( )) (T ) ( ( (det( )) )d ) det( )

m d dt m z z z

M J z dT W M J K z M z Y T

Consider the finial Lyapunov candidate function as

2 1 1 1 1 2 2(det( )) 2 2 2 2 2 2 2 2 T T T m d d d z z V V M z J z 1 2 z2 T T T T * * .(24)

After differentiating above equation with respect to time along the error dynamics (23) and accompanying with the results in (19), it results in

T

Vd e Pe (25)

with_eT _ª_xT _z₁T _z₂T_º

¬ ¼ and P diag Q K K{ , z1, z2}.

In the derivation, the property (M4) of smooth projection (22b) is used again like the presentation in (18). Since (25) is without containing perturbation terms, it means that the integator backstepping procedure is completely accomplished. Thus the stability results of overall tracking systems are in turn to be addressed.

Theorem : Consider the Stewart Platform system (1a) with actuator dynamics (1b). Using the control law (21) with virtual control inputs designed in (7) and (14) and the update laws defined in (6), (20), and (22), the following statement will be truly satisfied provided that the control gains are proper chosen.

(A1) All signals in the overall dynamic systems are uniformly bounded.

(A2) will asymptotically

track as t . ( ), ( ) and ( ) q t q t q t ( ), ( ) and ( ) d d d q t q t q t o f V. EXPERIMENTALRESULTS

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

Fig. 1 The Experimental System Configuration Diagram Figure 1 show the experimental system configuration of the Stewart platform system in this study. The motion control system computer runs a drive logic to control a hydraulic system that drive a 6 degree-of-freedom Stewart platform for creating realistic motion cue. The experiment are done with 6 DOF hydraulic Stewart platform system, and is manufactured by SGD Co.. The hydraulic servo valve in the Stewart platform uses MOOG’s J076-104. The D/A card uses Adventech’s 726 and A/D card uses Adventech’s 818H. The Detailed parameters and specification will be found in Table 1.

Table 1: Specification and parameter of the Stewart platform Motion

Degree Of Freedom 6(Heave,Sway,Surge,Pitch,Roll,Ya w)

Net Loading 500kg Accelebration r1g Angular Acceleration _r_{60 / sec}_D 2

Control Servo Class Hydraulic Actuating System Heave 0mm~ 221.518mm Sway r211.198mm Surge r244.758mm Pitch _r_12.960D Roll _r_10.821D Yaw _r_18.474D 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 mm Net Weight 800 kg Electricity Power 380 / 660 / 3 / 50 / 60V V I Hz

Rated Motor Power 7.5HP Rated (Max.) Operation

Pressure

4.5(10)Mpa System Flow Rate 65liter/ min

Oil Type ISO VG46

Oil Operation Temperature _{10 ~ 50 C}D D

Cooler

Dimension (L x W x H) 550 450 470x x mm

Cooling Capacity ₆₅₂₀_{cal hr}_/ _@35D_C

A. Results of Adaptive Backstepping Controller Design for circular motion tracking

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

Table 2 Circular Motion Trajectories

Heave 0-10-0 cm

Sway 10sin(ft); f=rad/sec

Surge 10cos(ft); f=rad/sec

Pitch 0 degree

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Yaw 0 degree

The desired circular trajectories on x-y plant are shown in Fig 2. The desired heave, pitch, roll and yaw trajectories are shown in Fig.3. The every desired and real trajectory of the leg is shown in Fig. 4-9.

Fig. 2 Desired Circular Trajectory Fig. 3 Desired z, , ,_{D E J} Trajectory

Fig. 4 The Leg1 Trajectories Fig. 5 The Leg2 Trajectories

Fig. 6 The Leg3 Trajectories Fig. 7 The Leg4 Trajectories

Fig. 8 The Leg5 Trajectories Fig. 9 The Leg6 Trajectories VI. CONCLUSIONS

In this paper we present an adaptive backstepping control approach for the motion control of a Stewart platform. The control scheme is proposed given that the overall system parameters are subject to uncertainties while only the positions and velocities of links are measurable. To achieve high performance tracking control of a 6 DOF Stewart platform normally requires the full knowledge of the system dynamics. In this paper, some important properties of the dynamics of the Stewart platform have been derived and exploited to develop an adaptive backstepping controller which can drive the motion tracking error to zero

asymptotically. Stability analysis based on Lyapunov theory is performed to guarantee that the controller design is stable. Finally, the experimental results confirm the effectiveness of our control design.

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