Variable Structure Model-following Control of a Robot Manipulator

Copyright © IFAC Computer Aided Design in Control Systems. Beijing. PRC. 1988

V ARIABLE STRUCTURE MODEL-FOLLOWING CONTROL OF A ROBOT MANIPULATOR A. S. I. Zinober Department of Applied and Computational Mathematics, University of Sheffield, Sheffield, UK

Abs t ract. The design of a vari able structure model - following controller is considered. By spec ify i ng t he structure of a model plan t and using the variable struc t ure con t r ol (VSC ) approac h, an adaptive controller whi ch forces the error between t he model and pl ant states to zero, can be designed. A continuous nonUnear control yi elds robust performance, whic h is insensitive to a c lass of variations in the plant pawameters and external disturban ces. The design technique incorporates an eigenvaluel eigenvec t or assignment algorithm available in an on-line CAD package, VASSYD. A cont ro ller is designed for a typi cal robot arm to demonstrate the method. Using t he nominal robo t parameters a deterministic controller is obtained whi c h yields effec ti ve decoupling bet ween the robot manipulator's subsystems. Simulation results for various values of plan t parameters are presented. The robot tracks the states of t he model plant, whic h in turn follows the desired trajectories, very closel y . This i ndic at es the i nherent robustness and effec t iveness of the VSC approach (and related t ec hniques) in the deterministic control of uncertain systems. Keywords . Variab l e s t ucture control; model - following; robust control; uncertain control sys t ems .

INTRODUCTION

nonlinear control; robots;

Thereafter a suitable control law is computed to ensure that sliding motion is achieved sufficiently rapidly. This deslgn technique ensures that the desired error transients are obtained.

The design of model-following control systems has received considerable attention in recent years. The general design objective for robotic manipulator systems is to develop a controller which forces the robot dynamics to track a desired trajectory. Here the tracking behaviour will be achieved by controlling the robot to follow the state trajectories of an ideal model. There has been much work on the deterministic control of uncertain systems, inc luding applications to robotic control using model-following and direct tracking techniques (Bales t rino, De Maria and Sciavicco, 1983; Guzzella and Geering, 1986; Hiroi and colleagues, 1986; Ryan, Le i tmann and Corless, 1985; Slotine and Sastry, 1983; Young, 1978a).

Here the model - following VSC design approach using a CAD package will be described for the control of a robot arm. Simulation results are presented to demonstrate the inherent robustness of the design technique. A continuous nonlinear controller yields trajectories arbirarily close to the sliding mode with the complete elimination of chattering motion, which may be undesirable in certain physical systems such as a robot manipulator. MODEL-FOLLOWING CONTROL SYSTEMS The technique of Model Reference Adaptive Control (MRAC) has been discussed in the literature by numerous authors, using two main approaches; Lyapunov and hyperstability theory (Landau, 1979). The actual plant is required to follow the dynami C behaviour of a specific model plant. The model is included as part of the system and it specifies the main design objectives. The adaptive controller should force the error between the model and the plant states to zero as time tends to infinity.

Various approac hes using stability criteria have been developed for model-following control systems (see, for instance, Grayson, 1965; Hang and Park, 1973; Landau, 1979 ) . In this paper we shall solve the model-following problem using variable structure control (Young, 1978; Zinober, 1984; Dorling and Zinober, 1985 ) . The controller will be designed using a user - friendly CAD package, VASSYD (Dorling, 1985; Dorling and Zinober, 1986, 1988). The basic philosophy of Variable Structure Control (VSC) design ( Itkis, 1976; Utkin, 1977; Zinober, 1979) is that the structure of' the feedback control is altered as the state crosses sliding surfaces in the state space. The control in the sliding mode remains on the intersection of these surfaces, yielding total invariance to a class of system parameter variations and disturbances . Precise closed- loop eigenvalue placement can be achieved in time - varying and uncertain systems.

Consider the plant and mode l described by x(t)

A x(t)

+

w( t)

F w(t)

+

B u(t)

+

f(x,u)

(1)

G r(t)

(2)

where x is the plant state n-vector, w is the model state n- vector, u is the plant control m-vector, f represents nonlinear additive terms and r is the model input. The tracking error vector is

The designer of a model-following VSC system first chooses the stable eigenvalues of the closed- loop error system; directly specifying the sliding hyperplanes. Eigenvectors may also be assigned.

e(t) = w(t) - x(t) We shall assume that the pairs

391

(3)

(A, B), (F, G) and

392

A. S. I. Zinober

(F,B) are stabilizable, i.e . any uncontrollable modes lie in the left- hand half of the s - plane (Balestrino , De Maria and Zinober, 1984 ; Young, 1978b; Zinober , 1984) . The model matrix F is assumed to be stable. The plant matrices A and B, and the vector f may be uncertain and time- varying . the The upper and lower magnitude bounds of elemen ts of these matrices and f are assumed to be known to the designer . It can be easily shown that

of the surface are not away from the switching surface . The state then slides and remains for some finite time on the surface s(e) = O. Before studying the nature of the cont rol enforcing sliding for the case of constant C, i.e. fixed sliding hyperplanes, the behaviour of the system dynamics during sliding needs to be considered. Suppose that the sliding mode exists on all the hyperplanes . Then

e (t) = Fe(t) + [(F - A)x(t) - f + Gr(t)J - Bu(t)(4) s = Ce =

°

and

s

Ce =

°

(11)

Matching conditions are necessary for perfect model-following to ensure that the equality w = x c an be attained. The error e and its derivative should be zero for any input r and plant state x, for e(O) = 0. Consider the linear control law

The equations governing the system dynamics may be obtained by substituting an equivalent control U for the original control u . Assuming CB is nonsingular, we have from (4) and (11)

u =

s = C[Fe + (F- A)x -f + Gr - B uJ =

Kx

+

Rr

+

Pf

(5)

and

Substituting (5) into (4) gives

e=

Fe + (F - A - BK)x - (l+BP)f + (G - BR)r

(6)

A - BK)x

This can be conditions satisfied

- (I+BP)f + (G - BR)r

=

°

A- FJ = rank B

(7)

Equivalently G, f and A-F should lie in the range space of B, R(B). Then R, K,P will exist and have the values

R

= B*G

K

B• (F-A )

P

=-B•

(8)

where B*= (B TB) - l B is the pesudo-inverse of B. VARIABLE STRUCTURE MODELFOLLOWING CONTROL SYSTEMS Into the framework of the above we shall now add a Variable Structure nonlinear control which ensures that all error states are attracted to a particular sliding subspace in the error state space, and thereafter approach the origin (e=O) on this subspace (Young , 1978b; Zinober, 1984). Variable structure model - following control systems are characterised by control discontinuities on m switching hyperplanes, si' in the tracking error space where s = (sl

s2'"

srn) T

Ce(t)

( 9)

The controller takes the form u = Le + uN(e) + Kx + Rr

( 12)

u=U into (4) yields an

nth

e = [I - B(CB) - l CJ [Fe + (F- A)x - f + GrJ

achieved i f the following matching (Chan, 1973 ; Erzberger, 1973) are

rank[B GJ = rank[B fJ = rank[B

U = (CB) - lC[Fe + (F- A)x - f + GrJ Substitution of system equation

We require that (F

o.

( 10)

where uN is a suitable discontinuous (or smoothed nonlinear) control to be described below. The linear feedback Le ensures rapid approach towards the sliding subspace, s=O, and also maintains the error state in this subs pace in the case of the nominal system . The nonlinear term is requ ired to attract the state into the subspace and maintain it on s =O in the presence of uncertain disturbances and parameter variation s (Ryan and Corless, 1984) . The variable structure model - following design philosophy is similar to that of the multi variable VSC re~ula tor (Dorling and Zinober, 1986 ; Young, 1978b ; Zinober, 1979). The analysis is , however, carried out in the error state space . Idealised sliding occurs if, at a point on the sliding subspace , s (e) =O, the directions of motion along the error state trajectories on either side

orde r ( 13)

For the perfect model - following case, since the conditions (7) correspond to the matching invariance conditions of VSC systems (Young, 1978) we obtain e =

[I - B(CB) - l CJ Fe

( 14)

= Ae e

where A is the equivalent error system matrix. The system Is formally equivalent to a system of lower order which has dimension (n- m), since m state variables can be expressed in terms of the remaining n- m variables. During sliding the state error vector e remains on the n- m dimensional manifold of the intersection of the m sliding surfaces . It can be read fly shown that m eigenvalues of the equivalent system are equal to zero . The error system response is then determined only by the other n-m eigenvalues (EI - Ghezawi, Zinober and Billings, 1982). Since the matching conditions hold , we our system using simply

e = F e - Bu

can

design ( 15)

because the terms in J in (4) lie in R(B) . These terms do, however, affect the gain requirements of the nonlinear control as will be mentioned below . Therefore, given F,B and assuming the pair ( F,B) is stabilizable, a matrix C can be found such that the error tends to zero with increasing time. Sufficiently fast error decay in the sliding subs pace can be ensured by placing n-m eigenvalues deep in the left-hand hal f of the complex plane . Equation (14) is independent of the actual values of the control u during the sliding mode and depends only on the choice of C. The function of the nonlinear control is to drive the error state into s =O, and thereafter to maintain it within the sliding subspace . The determination of th e matrix C may be completed without prior knowledge of the form of the cont ro l vector u (although the reverse is not true). The independence of the sliding motion from th e actual control , arises from the nonsingularity of CB. In geometrical terms, the condition that CB is nonsingular implies that the null space of C, N(C) , and the range space of B, R(B), are complementary subspaces ; i .e. N(C)n R(B) = {a }. If a system satisfi es the perfect model - following condi tions, the error state lies entirely within N(C) during the sliding mode , and the behaviour of the ideal system in N(C) is

Variable Structure Model-following Control of a Robot Manipulator unaffected by the form of the control. On the other hand, if ICBI= O, then N(C) and R(B) have points in common , and motion in N(C) will no longer be independent of u . Since we are concerned with the selection of the hyperplane matrix C, we may reasonably demand that CB is nonsingular in our design process. In the user-friendly CAD package, VASSYD, (Dorling , 1985; Dorling and Zinober, 1986, 1988) two methods; namely direct eigenvalue/eigenvector assignment, and robust eigenvalue assignment methods, are available for the design of the hyperplane matrix C. The designer can select some or all of the elements of the closed - loop eigenvectors corresponding to the non zero sliding mode eigenvalues, to achieve desired modal error responses during the sliding mode . In this study this method will be employed to ensure suitable decoupling of the robot subsystems. The robust eigenstructure assignment approach employed in VASSYD is based on a method proposed by Kautsky and Nichols (1983) in which the eigenvectors are rotated within the assignab]e subs paces to maximise the sum of the angles between each pair of vectors. For further details, see Dorling and Zinober (1988) . To ensure that the error state approaches the sliding surface in a suitably short time , the m range space eigenvalues should be specified to determine the linear part of the control , Le . The time in which the error state reaches N(C) depends on the values of these eigenvalues; i.e. the further left the eigenvalues are in the complex plane, the shorter the time in which the system will reach the sliding subspace s=O, from an error state not on the subspace. The equivalent control (12) is the effective control function which yields sliding motion. The value of U is in practice the average value of u which maintains the state on the surface s=O. The actual control consists of a low-frequency (average) component and a high-frequency (chatte r) component. The occurrence of chattering motion is due to the state continually passing backwards and forwards through the sliding manifold as the control switches repeatedly between two (or more) distinct values. This chattering is undesirable in many physical situations in which the actuation mechanism may be damaged by rapid switching. The simplest way of removing the high frequency chattering is to "soften" the action of the nonlinearity by substituting a continuous nonlinear approximation for the discontinuous part of the control. A control incorporating a softened non-linearity is termed a smoothing control and the "unit-vector" form (Balestrino, De Maria and Sciavicco, 1983; Balestrino, De Maria and Zinober, 1984 ; Burton and Zinober, 1986) has been found to be easily implemented. Its form is

CONTROL OF A ROBOT ARM The robot manipulator is a two link robot arm, which moves in a horizontal plane, with control torques u. applied at each joint (see Fig. 1) . J is the mom~nt of inertia of link i about axis i; and m. is the mass of link i. v. is the viscous fricli6n constant for axis i and I Is the moment of inertia of the axis 1 motor. The manipulator is described by the equations

fP 1 0

0] P2

N e I( IIM ell+ d)

( 16)

where th~ .gains P1' P2 are positive, and d is a small posItIve constant. The error state remains in the neighbourhood of N(C) during the sliding mode. that for d=O we obtain a control (Note discontinuous on s=O.) The details of the calculation of M, Nand L can be found in Ryan and Corless (1984) and Dorling and Zinober (1986) . C,M and N relate to the same sliding subspace. CONTROL OF A ROBOT ARM The robot manipulator considered is shown in Fig. 1 and is described by the equations

2 g [ - J vl ;,~ + av 0. + abQ'2 + Jb0' + 2JbQ0 + JU l 2 2 g [av l Q - hv 0 - bhQ2_ ab0 + 2abQ0 - aU l + 2

o

where the nominal parameters determined from experimental observation have the values J 2 = 0.000412, a = J 2 + 2m211l2 cos(0) , b = 2m 1 l sin(0) = 0 . 000559 sin(Q) ,h = J +J + 22 1 2 1 2 4m211 + I + 4m21112 cos(0) =0.008819 + 2

0 . 00118cos(0) , g = 1/(Jh - a ), v l = 0 . 0025 , v = 0 . 0000607 , J=J 2 · 2 From the above differential equations the matrix B in equation (1) is

13~.83

B

1

-308 .33

o

315~.39

[ - 308 . 33

T for the state vector x = ( Q o o ). The parameters, m and J , are assumed to have upper 2 2 and lower .limits four times and 0.1 times the nominal va.lues quoted above, depending upon the mass of the load to be carried . The state trajectories to be tracked by the actual robot arm are chosen to be the states of the model plant represented by (2) : 1

-20

F

o o

o o o -36

and G = B. This linear model has the same basic structure as the robot arm and its input r(t) represents the preCise trajectory desired. The model states Q and 0 follow the input vector r( t) very accurately, because the non zero eigenvalues of the model are [-10,-10,-6,-6] and the model response to r(t) is rapid. It has been found that controlling the robot to track the model output rather than r(t) directly, leads to better robot trajectories with smaller control values, paricularly when r(t) has large step changes. We shall consider r

l

393

l

(t)

r

= 94 - 100( t-l.5) 54 54 + 25sin[5(t-2.5)] 77 .45

o~

t t 2 , t 2.5 ~ t 1.5~

4

~

t

< 1.5 <2 < 2.5 <4 ~

5

r (t) = 0 2

in the simulations below with P=O in (5) . The CAD program VASSYD has been used to design first the sliding hyperplanes and then the cont rol function. The designer needs to choose 2 ( = n-m) eigenvalues for the closed-loop poles of the sliding system and the se have been taken to be -5, -7 to ensure rapid decay of the error states to zero. Specification of the two respective eigenvectors as

394

A. S. 1. Zinober nonlinear systems .

deterministic

control

of

uncertain

REFERENCES with X arbitrary, yields the desired decoupling in the sliding mode between the two angles Q and 0 . The control function needs now to be computed. The variation of the physical nonlinear robot from its approximate mathematical description and the effect of the tracking input r(t), can be considered as "d isturbance" terms in the error system. These variations all lie in the range space of B for this class of mechanical system . The non linear control can therefore be "tuned" to counteract the affect of these disturbances by suitably increasing the gain values of Pl and P2 in the control law (16) . be The necessary magnitude of the gains can determined using the VASSYD package by inputting the approximate maximum expected absolute values of the variation terms (see Dorling (1985)) . Clearly there is a practical upper limit to the gain values corresponding to the torque capabilities of the robot. Here both the gain values have been chosen to be 0 . 5 with satisfactory results. In addition to aid the rapidity of attraction of the error system towards the sliding subspace, one requires speedy range space dynamics . The range space eigenvalues have been chosen to be -12, - 14 for this problem and the matrices L, N, M of equation (16) are respectively - 3.680 [ - 0.364

- 0 . 236 - 0 . 471 - 0.0346 ] -0.0233 -0.160 - 0 . 0124

Balestrino , A. , G. De Maria and A.S.I. Zinober (1984) . Nonlinear adaptive model - following control . Automatica, 20, 559- 568. Burton, J.A. and A. S.I. Zinober (1986). Continuous approximation of variable structure control . Int. J Syst Sci, 22, 876-885. Chan, Y.T. (1973). Perfect model - following with real model. Proc JACC, 287-293. Dorling, C.M. (1985). The Design of Variable Structure Control Systems (Manual for the VASSYD CAD Package), University of Sheffield . Dorling, C.M. and A. S. I. Zinober (1985) . Hyperplane design in model - following variable structure control systems. Proc 7th IFAC/lFORS Symp on Identification and System Parameter Estimation (York), 1901 - 1905 . Dorling, C.M . and A. S.I. Zinober (1986) . Two approaches to hyperplane design in multivariable variable structure control systems. Int.J. Control,~, 65- 82. Dorling, C.M . and A. S.I. Zinober (1988) . Robust hyperplane design in multivariable variable structure control systems. To appear in Int J Control.

- 0.122 - 0.006420 - 0.00796 -0.000468 ] [ - 0 . 0118 - 0.000621 - 0.00448 - 0.000264

El - Ghezawi , O.M . E. , A. S.I.Zinober and S . A. Billings (1982) . Analysis and design of variable structure systems using a geometric approach. Int J Control , 38 , 657 - 671.

- 0 . 350 - 0 . 0184 - 0.133 - 0.00781] [- 0 . 161 - 0 . 00850 0.340 0.0200 Transient resposes are shown in Figs. 2, 3 and 4 for different values of the parameters, m. and J., to demonstrate the ability of th~ fix~d (deterministic) controller to handle a range of operating conditions. The desired trajectory for the angle Q(t), w~ich is the corresponding model state wl (t), is followed very closely by the actual angle Q. The state 0 should remain close to the zero value of the model state w3 for all time. The control values are within the physical constraints and the switching function values remain small, indicating that the error states remain satisfactorily in the neighbourhood of the sliding subspace. ACKNOWLEDGEMENT The CAD package VASSYD was developed support of the Science and Engineering Council grant GR/C/03119 .

Balestrino, A., G. De Maria and L. Sciavicco (1983) . An adaptive model - following control for robotic manipulators. Jnl of Dyn Syst Measurement and Control, 2Q2, 143.

with the Research

CONCLUSIONS The design of a variable structure model - following controller has been described . The continuous nonlinear control yields robust accurate tracking , is insensitive to a a wide range of which var iations in the plant parameters. The robot arm tr acks the s tates of the model plant, which in turn are specified by the desired trajectories. The design technique incorporates an eigenvalue/eigenvector assignment algorithm ava ilable in an on - line CAD package, VASSYD . Further research needs to be done to co~pare the above technique with other methods in the area of

Erzberger, H. (1968) . Analysis and design of model - following control systems by state space techniques. Proc JACC , Ann Arbor, 572- 581 . Grayson , L. (1965) . The status of synthesis using Lyapunov's Method . Automatica, ~, 91 - 125 . Guzzella, L. and H.P . Geering (1986) . Model -following variable structure control for a class of uncertain mechanical systems. Proc 25th IEEE Conf on Decision and Control , Athens, 312- 316. Hang , C.C. and P. C. Parks (1973) . Comparative studies of model reference adaptive control systems. Trans IEEE Autom Contr , ~, 419- 428. Hiroi , M. , M. Hasayuki, Y. Hashimoto, Y. Abse and Y. Dote (1986). Microprocessor- based decoupled control of manipulators using modified model - following method with sliding mode. IEEE Trans Industrial Electronics, 33, 110 . Itkis, U. (1976). Control Systems of Structure. Wiley, New York .

Variable

Landau, I . D. (1979) . Adaptive Control: The model reference approach . M. Dekker, New York. Kautsky, J. and N. K. Nichols (1983) . Robust eigenstructure assignment in state feedback controls . University of Reading , Dept. of Mathematics, Report NA/2/83. Ryan , E.P. and M. Cor less (1984) . Ultimate boundedness and asymptoti c stabi I i ty of

a

Variable Structure Model-fo llowing Control of a Robot Ma nipulator class of uncertain dynamical systems via continuous and discontinuous feeduacK control . IMA J Math Control Information , ~ , 223- 242 . Ry a n , E. P. , C. Practical dynamical tracking .

1.0

Leitmann and M. Corless (1985) . stabilizability of uncertain systems , Application to robotic J Opt Theory Applic , ~ , 235 .

0. 5

Slotine , J - J . E. and S . S. Sastry (1 983) . Tracking Control of Nonlinear Systems using sliding surfaces , with application to robot manipulators. Int J Control , 38 , 465 . Utkin , V. I . (1977) . Variable structure systems with sliding modes . IEEE Trans Autom Control , 22 , 212- 222 . Young, K- K. D. (1978a) . Controller design for a manipulator using theory of variable structure systems . IEEE Trans Systems , Man and Cyb, ~ , 101 .

•

Young , K. D. (1978b). Design of variable structure model - following control systems . IEEE Trans Autom Control , 23 , 1079- 1085 .

395

.j.L.----------------

0.0

0.0

2.5

5. 0

0.010 0.005 el

O. 000

~-+----=:"-"L-_+_.....,.._I____\_-__I_-::::::--

5.0

-0.005 -0. 010 0.00250

Zinober , A. S. I . (1979) . Controller design using the theory of variable structure systems . In S . A.Billing and C. Harris (Ed . ). Self- Tuning and Adaptive Control . Peter Peregrinus , Stevenage Chapter 9 , pp. 204 - 229. Zi nobe r, A. S.I . (1984) . Properties of adaptive discontinuous model - following control systems . Fou r th IMA International Conference on Control Theory , Cambridge , 337 - 346 .

0.00125

o 0.00000 -0.00125

0.2 0. 1 u

l

u

2

0.0

~--~~~~-P~~~~~~~~--

5.0 -0.1

0.004

sl

0.002 s2 O. 000

.f--'I5L,\-,...........-.......-d-~~_\\r_-I_+-".,...,:::::..,._ _

-0.002 Fig . i . Robot a r m

-0.004

Fig . 2 . Transient r espons es. nominal pa r ameters Q, u ' sl l

circled

w is mode l Q 1 mo del 0 ~I~

0

a ngl es in r ad i ans

A. S. 1. Zinober

396

1.0 g

1. 0

w,

w1

g

0.5

0.5

0.0

0. 0 0.0

2.5

5.0

0.0

o. ala

0.030

0.005 e1 0.000

0.015 e,

2.5

5. 0

0.000 -0.005 -0.015

-0. ala

0.050 0.075 0.025

0

0

0.000

0.000 -0.025

-0.075

-0.050

-0.150

0.2

O. 250

0.1 u 1 u2 0.0

0.125 u, u2 0.000 5.0

-0.1

5.0

-0.125 -0. 250

0.02 0.025 s, s2 0.000

0.01

s 1 s2 0.00

-0.025 -0.01

L ( sec) Flg . 3 . TransjenL responses .

parameLers

:

O. , x nomjnal va lues

ang l es jn radjans g, u, • s,

c j r e 1ed

w, js model g mode l 0 w 2

0

-0. 050 Flg.4. TransjenL responses .

parameLers

:

4 x nomjnal values

angles jn radj ans g, u, • s ,

ej r cl ed

w, js model g model 0 w 2

0