Variable structure control of constrained dynamic systems

(1)

V A R I A B L E STRUCTURE C O N T R O L O F C O N S T R A I N E D DYNAMIC SYSTEMS

Han-Pang Huang Marlon Lin

Robotics L a b o r a t o r y

Department of Mechanical Engineering National Taiwan University Taipei, Taiwan 10764, R.O.C.

A B S T R A C T

A typical example of constrained dynamic systems is the constrained robot system in which the motion of the robot end- effector is restrained by environment. Such a system is usually composed of a set of differential equations and a set of algebraic equations. A modified computed torque controller has been developed by McClamroch and Wang. If the mathematical model of the robot is exact, the modified computed torque can simultaneously control the robot motion and contact force in an accurate way. However, there may exist uncertainties in the model, such as flexibility of joints and links, joint friction, and inexact surface model. It will be shown that the modified computed torque controller may result in an unstable closed- loop system for the system with uncertainties. This difficulty can be overcome by using variable structure controller. The controller is robust in that i t is insensitive t o variations in the plant parameters and to external disturbances of contact force.

In many caaes, the robot end-effector interacts with environment. Hence, the robot motion is constrained. Such system is a typical example of constrained dynamic systems. For a constrained system, simultaneous control of position and in- teracting contact force is required. In order t o investigate this problem, the contact force can be incorporated into the robot model t o form a swxlled singular system [1,2]. Several con- trol strategies have been developed on the basis of this singular model [3,10,11]. In those approaches, the models for the robot system and environment are assumed exact; however, it is not generally true. For instance, the system may be subjected to uncertainties due to joint friction, joint and link flexibilities, surface friction, and compliant surface. Under these circum- stances, the approaches described in [3,10,11] may be inade- quate or even result in unstable system. In order to overcome the above difficulty, a robust controller is proposed for the constrained dynamic system. The controller is variable structure type per se. It is insensitive to the variation of system param- eters and external disturbance; moreover, the chattering effect can be eliminated in the controller design.

This paper is organized BS follows: The constrained dynamic system and some preliminary results are first presented. Next, the variable structure controller is designed on the ba- sis of nonsingular system model. Then the variable structure controller is designed for simultaneous control of position and contact force with the consideration of system uncertainties. Finally, an example for equality constrained system is given. Comparisons between the variable structure controller and the

modified computed torque controller are also made. It will be shown that the variable structure controller is robust t o un- modelled dynamics, while the computed torque controller may result in unstable systems.

II.

Problem Formulation and Preliminaries The robot system under consideration is described by equa- tions of the type [1,2,10]

+((I) = 0 (2)

where q E R" is the generalized displacement; M ( q ) is an n x n inertial matrix function; F ( q , q ) is an n dimensional vector function, containing the Coriolis, the centrifugal and the grav- itational t e r m ; U E

R"

is the generalized control input; 4(q)

is the m dimensional constraint vector function; J ( q ) =

-

a4(4

aq

is an m x n Jacobian matrix; X 6 R"is the generalized contact force vector associated with the constraints.

For a linear timeinvariant differential system, the variable structure controller I481 is directly applicable. However, the system under consideration is nonlinear as well as differential- and-algebraic type. These characteristics make the concept of variable structure controller fail t o apply. In order to facil- itate the following developments, the system (1)-(2) is first converted into a nonsingular model by eliminating the contact force; then the design of variable structure controller is based on this nonsingular model. The other approach is t o convert the system into two subsystems; then the variable structure controller is designed for both position and force control.

The nonsingular model will be first derived. Since the constraint

4(q)

is twice continuously differentiable, the first and second time derivatives of the constraint are ala0 equal t o zero; i.e.,

and

J ( d Q

= 0 (3)

(4)

J ( q ) i

+

j ( q ) i = 0

where j ( q ) = $ J ( q ) . From Eqns. (1) and (4), the contact force can be obtained as

x

= [JM-'J=]-'

.

[ J M - ' F

-

JM-'u - jq] ₍₅₎ Here we assume that [JM-'JT] is nonsingular. Note that the argument in equations are omitted hereafter for simplicity. Define

(2)

R = K u

'Then the singular system (1)(2) is converted into the following nonsingular system

The first controller will be developed in terms of this nonsingular model.

Next, a transformed model for the system can be derived by using a nonlinear transformation [3] as

Our problem turns out t o design the variable structure con- troller for the transformed system (8)-(10).

III.

Deeien of Variable Structure Controller

The theory of variable structure system was proposed by Itkis [4]. It utilizes a -called sliding surface t o integrate two unstable systems into a stable system. When the representative point slides t o origin along the sliding surface, i.e., the system enters the sliding mode, the trajectory is kept on the sliding surface on account of continuous switch of the gain between positive and negative values. In the mean time, the system response follows the motion of sliding mode instead of original system equations. Therefore, the variable structure system is robust to the variation of system parameters and external disturbances. The change of system dynamics in the variable structure system is quite similar to the constrained dynamic system. In order t o design an adequate variable structure controller, the design parameters should be suitably chosen ( must satisfy Routh-Hurwitz criterion ) to form the sliding surface; furthermore, the controller should satisfy the sliding condition

8 . i

5

- q l S l (13)

where s=O is the sliding surface, and 11 is a positive scalar. If the sliding condition is met, then the sliding mode exists and the representative point can reach the sliding surface in finite time.

Since the constrained dynamic system is a singular system in nature, the typical design of variable structure controller can not be directly applied. In order to resolve this problem, two approaches are adopted: One is to convert the constrained dynamic system into a nonsingular system, the other is to utilize the transformed subsystems.

Controller Desian bv Usina Nonsinaular Model

The nonsingular model of the system has been obtained in Eq.(7). According to Eq.(7), the design of variable structure controller can be proceeded in the following way.

Let q d ( t ) denote the desired trajectory. Assume the sliding surface is chosen as

s =

[8181..'Sn]' = c ( q - q d ) + ( q - q d ) = o (14)

where C = diag[cl cz * . c,], ci > 0 , i = 1 , 2 , - .

,

n. By differentiating,

Suppose that M S has the form

where

A = diag[Pi(q,q)

-

sgn(si)/si] Pi(q,q)

>

0, i = 1 , 2 , . - - , n

1, 8 > 0

0, 8 = 0

Define the Lyapunov function

V

as

V = S T M S (18)

Since the matrix

M

is symmetric and positive definite, from Eq.(17) we obtain

.

V = ST MS

+

S T M S

+

ST M S

The sliding surface S=O will be asymptotically stable as long

as (A

-

$) matrix is positive definite. The condition can be derived as

n

p,

-

sgn(si)/.i

>

I

Mij/2

I

(20) j=1

Let A M and A H denote the system uncertainties due to pa- rameter variations. Then M = MO

+

A M , H = Ho

+

A H , where M and H are estimated yalues;

MO

and H o are nominal values. Suppose that A M i j , Mij and AHi have the following bounds

1

A M i j

I <

i f i j ,

I

AH^

I <

ri,,

I

ilkij

I <

liTij

i , j = i , 2 , . . - , n (21) Combine Eqns.(l6) (17), we have

M[C(q - i d ) - i d ]

-

H R = (22)

Choose the controller

R

as

where b = [c(q - q d )

-

i d ] . Substitute (23) into (22), we get

(3)

or equivalently in scalar form as n

x A M ; , b ,

-

AH;

+

g

= -E8gn(ai) (25) j= 1

By observing condition (20), we conclude that when Si

<

0 ,

Eq.(25) becomes

n n

Z A M j j b ,

-

AH;

+

<

-8;

j= 1 j= 1

when S;

>

0, Eq.(25) becomes

n n c A M j b j

-

AH;

+

g

>

-8; j= 1 j= 1 Therefore, n 8gn(8;)[XAMijbj

-

AH;

+e

<

-

18;

I

c

I

M;,/2

I

j= 1 j= 1

The auxiliary controller Ri can be further chosen as

Use Eq.(26) in Eq.(23), the final controller can be obtained as

or equivalently in the vector form as

The controller (28) includes the sgn(s) function. Since sgn(s) is discontinuous a t s=O, chattering phenomenon may occur in the neinhborhood of the sliding surface. In order t o eliminate the chattering effect, the boundary layer concept [6,7] can be applied; namely, sgn(s) is replaced by saturation function sat(8/r), where

1, r > l

-1, r < - 1

E > 0 denotes the configuration of boundary layers. Larger E results in better elimination of chattering effect. However, too large E will reduce the system accuracy. A suitable E may be selected according to the following criterion.

(29)

E =

{

€ 1 2 i f ' % I q - q d ( < € l

if €2

1

q - qd

I>

€1 €2

I

q

-

qd

1,

where el

>

0, C

>

€2 2 0 are constants. If the controller (28) takes into account the elimination of chattering effect, the matrix S2 should be modified as

S2 = diag[eat(81/el), t U t ( S 2 / € 2 ) , * * * , s a t ( 8 , J ~ ~ ) ] (30) Since R is only a pseudo controller, the original controller U can be recovered by using Eqns. (7)(28). If the matrix

K

is nonsingular, the controller U is

U = K-'(q)R (31)

U = K+(q)R (32)

otherwise

where K + ( q ) is the generalized inverse of K(q). The above results can be summarized as

Theorem 1:

Consider the robot system (1)(2). kssume matrix K(q) is nonsingular. Given any initial conditions q(0)

,

q(0) satisfying

4(q) = O,&q) = 0, then the controller (31) will result in a stable closed-loop control; i.e., q ( t ) -+ qd(t) as t -+ CO, and the

contact force X is indirectly controlled.

Controller Desinn bv Usinn Nonlinear Transformation In the last section, the controller design is based on a nonsingular model of the system. The contact force X haa been eliminated from the model; hence, the force can only be con- trolled through the control of q and q. In many cases, simul- taneous control of both position and contact force is required. This objective can be achieved by using transformed subsys- t e m , Eqns. (8)-( 10).

Eqns.(8)(9) are rewritten below

EIM(zP)E;Jz

+

E i P ( ~ 2 , 2 2 ) = EiT"(Zz)U

+

E 1 T T ( ~ 2 ) J T ( ~ 2 ) X (33) EaM(zz)G22

+

E ~ P ( z ~ , Z ~ ) = &T'(Z~)U (34) The controller is selected as

TT(z2)u=ETGIElTT(~2)J'(~2)(X - Ad)

-

TT(z2)JT(z2)Xd

+

P o ( z 2 , 2 9 )

+

p(zZ)E;fuN

(35)

where GI is an m x m, symmetric and non-negative definite matrix; Ad k the desired contact force;

Po

and

@

are nominal values of

P

and M, M-&P = AM,

P - P o

=

AP; UN E Rn-m is the variable structure controller which will be determined.

Substitute the controller (35) into Eqns.(33)(34), we obtain

(36) ElliiOg(39

-

U N )

+

ElAMiiE;P32

+

E i A P

= (1,

+

Gf)EITTJT(X - Ad)

EaWE;(& - UN)

+

EaAME;&

+

G A P = 0 (37) Let

where

W

E

P-".

Since @ is nonsingular, i t has n indepen- dent column vectors. ( & f " ~ ) , x l n - m ~ is also a full column AfiE;Zz

+

AP

= (MoEF)W (38)

(4)

rank matrix. Thus, W can be found from Eq.(38) by least square solution [Q].

is ignored; the friction between the robot end-effector and the constraint surface is not taken into account. It is not true for practical applications. In the following, those unmodelled dynamics will be regarded as an extra input; its influence t o the controller design and the position and force control will be

W

= [(U"E,')'(n;l"E,')]-'

(lii"g).

(39)

[hag22

+

AP] d i s c U d .

Suppose the joint and link compliance, friction, exter- F.,(q,i). F.,(q,q) E

R"

is an unknown vector function with into account the unmodelled dynamics becomes

From the above equation, W is function of 2 2 , 3 2 and 22. If

I

qi

I, I

4;

I

and

I

&

are bounded, then

I

22i

I, I

.&

I

and

1

2,; are

(n-m)'

use

(38) in

disturbance, and system uncertainty be as

a known bound. Then the constrained dynamic system taking Hence,

1

W i

I<

w;

for i=1,2,

(36) and (37), we have

Eilii"EZ[33

-

U N

+

w ]

= (1,

+

G , ) E I T T J T ( X - A d ) (40)

M ( q ) q

+

F ( q , 9 ) = U

+

JT ( q ) A

+

Fe,(%

i )

(48)

& n ; i " g ( Z p - U N

+

W ] = 0 ₍₄₁₎

4 ( d

= 0 (49)

Suppose that (&n;i"g) is a nonsingular matrix, then Eq.(41) gives

2 2 - U N

+

W

= 0 (42f

If the controller design is in terms of the transformed method, then the transformed subsystems becomes

In order to design the variable structure controller, the slidinf surface is chosen as

si

=

(z

+

C;)'

{l'

[z2; - %ad;] d t }

E i w g [ 2 2

-

U N ]

+

E l [ A M G &

+

A P - Fe,]

(50)

d = (I,

+

G ~ ) E ~ T T J T ( X - A,)

E2a0E,'[i2 - U N ]

+

& [ A a g Z 2

+

AP - Fe,] = 0 (51)

(43: Let

Then follow the same procedures as in the previous section, A M g 2 2 + A P - F e q = ( n ; l " g ) . W

where ci

>

0, i=1,2,

. .

.,(n-m). The integral control in the and dlsturbance. Differentiate (43) and use (42), we obtain

the Controller U N is Similar t o Eq.(47) except t h a t

@;

beCOme6 of W; will depend on the bound of Feq.

Corollary 3:

Consider the constrained robot system (1)(2). Suppose the bound of the unmodelled dynamics is given. By adjust- ing the magnitude of

@;,

the controller defined in Eq.(35) will guarantee q(t) -+ qd(t) and X(t) -+ &(t) as t -+

M.

Eq.(43) is used t o reduce the steady state error due to friction larger in Order to compensate the uncertainty Feq- The range

si

= -Wi -k U N ;

-

2 2 d ;

+

2c;[h2; - z a d i ]

(44)

+

(2% - % d i ]

By observing this equation, the controller can be selected as

Since

I

W;

I C

@;,

Eq.(46) is always lees than zero. The condi- tion for sliding mode existence is satisfied. Namely, the trajectory will reach the sliding surface for any initial conditions. In addition, the chattering effect can be eliminated by replacing sgn(e;) with sat(s;/c;). Then the controller U N can be denoted

as

U N = Z 2 d - 2 c l ( z 2 - z 2 d ) - c 2 ( 2 2 - z 2 d ) - s 3 ' w (47)

where C1 = diag[cl, c2 ,

.

,

c n - , ] , C2 = diag[cf, c i ,

-

,

cz-,,,I, and S3 = diag[eat(el/c1), - ~ - , s a t ( s , - , , , / ~ , - ~ ] . The original controller U can be obtained by substituting U N into Eq.(35). This variable structure type controller U will guarantee z2 -+

2 2 d and q(t) -+ qd(t) as t --+ CO. From Eqns.(33)(34), x ( t ) -+ Xd(t) as t -+ W. The result is summarized as follows: Theorem 2:

Consider system (1)(2). For any given initial conditions q ( O ) , g ( O ) satisfying 4(q) = 0 and $ ( q ) = 0, the controller defined by (35) results in a stable closed-loop system. Namely, q(t) + qd(t) and X(t) -+ X ( t ) as t -+ CO.

Effects of-unmodelled Dynamics

In the previous development, the system model is an ideal one. The robotic arm is assumed to be rigid; the joint friction

In this section, a second order nonlinear constrained dy- namic system will be used to illustrate the design and ro- bustness of the variable structure controller (V.S.C.). The results are alm compared with the computed torque controller (C.T.C.) proposed by McClamroch and Wang. It will be shown that the computed torque controller may cause unstable closed- loop system under system parameter variations. While the variable structure controller performs well for all cases.

Consider the following second order nonlinear constrained system

aq1

+

10q1q2 = U 1

+

x

The constraint equation is defined by

4 ( q l * q 2 ) = q I -qaa=o (53) Note that constants a,/3,7 are system parameters; their nom- inal values are a = 1,

B

= 1, and 7 = 1.

Let q

7

[q! q2IT and z = [zl 2 2 I T . Then the nonlinear transformation IS given by

(54)

(55) Using this nonlinear transformation, the transformed subsys- tems are obtained as

(5)

2a2222

+

[ZOz;

+

2a(i2)'] = u1

+

x

₍₅₆₎ (57)

2 1

= o

(58)

(4az;

+

a)&

+

(202:

-

8072;Za

+

4 a ~ ~ ( & ) ~ ] = 222Ul

+

%

T w o typea of controllers will be constructed in terms of the transformed subsystems: one is the computed torque controller, the other is the variable structure controller.

First, the computed torque controller (C.T.C.) developed by McClamroch and Wang can be found as

TTU =h$Er&d - TTJTXd

+

P

+

q G , E I T T (A - Ad)

+

U g [ G , ( & d - ; a ) +Gd(zad - za)]

(59) Since the C.T.C. does not consider the variation of system pa- rameters, the controller is obtained by setting a = /3 = 7 = l

(60)

(61) U1 =22a[i&d

+

Gm(&d - Za)

+

G.i(Zpd

-

Za)]

+

[G,(X - Ad)

-

Ad]

+

[loz;

+

2 ( 4 ) ] Ua =[bad

+

Gt.(&d

-

2 2 ) -k Gd(%d - za)]

- 22,[Gj(X

-

Ad)

-

Ad]

-

8oZiZa Thus, the closed-loop system is given by

(62) 2ffZaia

+

2(a - 1)Zz - 221[82d

+

G,(Zid - 2 0 )

+

Gd(Zad

-

2 ~ ) ] = (1

+

G,)(X - Ad) (4aZ;

+

a)&

-

(42;

+

1)[Zzd

+

G. (+ad - 2 2 )

+

Gd(2a.j

-

Za)] -k 4(U

-

i ) Z p Z i

-

80(7 - 1)Z;Za = 0

(63) Next, the variable structure controller (V.S.C.) will be constructed. The sliding surface is chosen aa

s

= (& -

&)

+

- zad) (64) The V.S.C. controller can be found as

(65)

(66) U1 =22a[&d

-

c(& - &ad) -

k

' 8d(s/€)]

+

[G,(A

-

Ad)

-

Ad]

+

[102;

+

22:] =[%ad - .(&a - &ad) -

@

* 8d(S/e)]

-

222[G,(X - Ad) - Ad] - 802i22

Using this controller in the original system, the closed-loop system is obtained as

2a22&

+

2 ( a

-

I)*: - 222[&d - c(ip - *ad)

(67)

- k

* S d ( S / € ) ] = (1

+

Gj)(X

-

Ad)

(4QLZi

+a)&,

-

(42;

+

l)[kad - c(& - &ad)

-

k

* 8d(S/e)]

+

4 ( a - l)z,Z;

-

80(7

-

l)z;Za = 0

(68) In this example, the desired trajectory is planned as qld =

(t/2- I)', qad = t/2- 1, and the desired contact force is Ad = 3.

The initial conditions are a ( 0 ) = -1, qa(0) = 0.5. The design parameters are selected as G.

=

8, Gd = 32, G, = 100, and

C

= 5. The parameters of boundary layer are set aa el = 0.01, €2 = 1. Furthermore, the robot motion is bounded by Iq21 5 1,

1

6

1

I 1,

I&]

5

1; and the range of parameter variation are 0.1

5

a

5

2, 0.1

5 5

2, 0.1

5

7

5

2. Then the bound of

k

can be computed as

k

= 16. Consider the real parameters are larger than the nominal values; i.e. U = = 7 = 2. In this case, C.T.C. controller becomes unstable in position and force control. While V.S.C. k e e p excellent performance in position control; the maximum force error is only 0.2. The reaults are shown in Fig.1 to Fig.5.

V.

Conclusion

This paper presents two typea of variable structure con- troller design for the constrained dynamic system. One is based on the nonsingular system model, the other is based on the sin- gular model. The former can not directly control the contact force; the later allows simultaneous control of position and contact force. The variable structure controller is compared with the computed torque controller. According to the simulation results, C.T.C. controller is sensitive to the parameter variation; while V.S.C. can handle various parameter variations. Namely, V.S.C. is robust to the system unmodelled dynamics. The implication is that the constrained dynamic system and the variable structure system may poseess some common features in nature.

REFERENCE

Huang, H.P., "The unified Formulation of Constrained Robot Systems," Proc. of IEEE Conf. on Robotics and Automation, Philadelphia, 1988.

McClamroch, N.H., H.P. Huang, "Dynamics of a Closed Chain Manipulator," Proc. of American Control Confer- ence, Boston, 1985.

McClamroch, N.H., D. Wang, "Feedback Stabilization and Tracking of Constrained Robots," Proc. of American Con- trol Conference, 1987.

Itkis,

U.,

Control System of Variable Structure. New York: Wiley, 1976.

Utkin, V.I., "Variable Structure System with Sliding Modes," IEEE Trans. on Automatic Control, Vol. AC22, April, 1977.

Slotine, J.E., S.S. Sastry, "Tracking Control of Nonlinear Systems using Sliding Surface with Application to Robot Manipulators," Int. J. of Control, Vol. 38, No.2, 1986. Slotine, J.E., "The Robust Control of Robot Manipula- tors," Int. J. of Robotics Research, Vol. 4, 1985. Yeung, K.S., Y.P. Chen, "A New Controller Design for Manipulators using the Theory of Variable Structure S y s tems," IEEE Trans. on Automatic Control, Vol. AC33, No. 2, Feb. 1988.

Noble, B., J.W. Daniel, Applied Linear Algebra. New York: PrenticeHall, 1977.

[lo] Huang, H.P., N.H. McClamroch, "Time-Optimal Control for a Contour Following Problem." IEEE J. of Robotics and Automation, Vo1.4, No.2, 140-149, April 1988. [ll] Yoshikawa, T., "Dynamic Hybrid Position Force Control

of Robot Manipulators-Description of Hand Constrained and Calculation of Joint Driving Force," IEEE J. of Robotics, and Automation, Vol. RA-3, No.5, 1987.

[12] Young, K.D., "Controller Design for a Manipulator wing Theory of Variable Structure Systems," IEEE Trans. on System, Man, and Cybernetics, Vol. SMC-8, Feb. 1978. (131 Huang, H.P., "Simulation of Constrained Robot Systems,"

National Science Council Report, No. NSC76-0401-EO02- 14, Taiwan, R.O.C., July, 1988.

(6)

d c 4 -

-

9 1 4

_.-..

C.T.C. 3 -

_ _ _ _ -

V.S.C. ! - I I @a F 5

-

1 .

---

V.S.C.

...

C.T.C. M

i-i

-

k

---

,,<L---;

'..J

\

U !x

,.i

\

0 3 ---+:---:-- 2 2 ,- i

-

q s 4 I

____-

--

V.S.C. 1 -

_.._....

-

C.T-C. 0 1' 2 3 4 TIME Fig.2. Displacement qz 5 0 , I -1 0 0 ' 1 2 3 4 0 TIME Fig.4. Input 1000

-

V.S.C.

1

-1000' J 0 1 2 3 4 TIME Fig.5. Input ua