self-tuning control of robot manipulators

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING. VOL. 7,405-416 (1993)

SELF-TUNING CONTROL OF ROBOT MANIPULATORS

SALVATORE NICOSIA AND PATRIZIO TOME1 Dijmrtimento di Ingegneria Elettronica, Universitb di Roma ‘Tor Vergata’, Via della Ricerca Scientifrca, 1-00133

Roma, Italy

SUMMARY We consider the tracking control problem for robot manipulators whose parameters are not exactly known. It is assumed that some unmodelled friction is acting at the actuated joints. The proposed solution guarantees that the disturbances on the outputs (the joint positions) can be attenuated by an arbitrary factor k. For the setpoint regulation problem a self-tuning controller is proposed. In this case the disturbances (uncertain parameters and unmodelled non-linearities) on the outputs are completely rejected asymptotically. Some simulation results are included.

KEY WORDS Robot manipulators Self-tuning control Disturbance attenuation

1. INTRODUCTION

Several model-based control laws have been proposed in the literature for trajectory tracking of robot manipulators. Basically they can be classified into two categories: computed torque (or inverse dynamics or feedback linearization) controllers and passivity-based controllers. Both these approaches are based on exact knowledge of the robot parameters. Even though it has been proved that such controllers have a certain degree of robustness with respect to uncertainties on the parameters, it is apparent that when the parameters are not precisely known, control laws specifically designed to deal with uncertainties are to be preferred.

While the adaptive control strategy is suitable and feasible when all non-linearities can be exactly parametrized,‘ in the case of unmodelled non-linearities the use of robust controllers is the only strategy which can be reasonably followed. Some approaches to robust control of robots have been proposed in the literature. In Reference 5 non-linear gains are used which depend on the tracking errors. In Reference 6 the stable factorization approach is applied along with an approximate feedback-linearizing control; robust tracking is achieved if the deviations of the true parameters from the nominal ones are not too large. A dynamic state feedback controller which includes an integral term has been proposed in Reference 7.

In this paper we propose a robust tracking controller which guarantees arbitrary attenuation of the effects of unmodelled non-linearities (such as friction) and of uncertainties on the parameters (whose bounds are known). For the setpoint regulation problem we propose a dynamic state feedback self-tuning controller which guarantees that asymptotically the desired position is exactly reached. In this case the knowledge of the parameter bounds is not required. The stability proofs are given using simple Lyapunov arguments. Some simulations have been carried out on a three-link robot to show the performance of the proposed controllers.

0890-6327/93/050405-12$11 .oO 0 1993 by John Wiley 8c Sons, Ltd.

406 S. NICOSIA AND P. TOME1

2. ROBUST TRACKING CONTROLLER WITH DISTURBANCE ATTENUATION

Consider the model of a rigid robot with n + 1 links interconnected by n rotational joints:

B(q, + c(q, 4, + h(q, e) + m e ) = u (1)

where qE I?" is the vector of joint displacements, 8 denotes a vector of constant unknown parameters belonging to a compact set 0 C I R p , B is the symmetric positive definite inertia matrix, C is the matrix which takes into account Coriolis and centripetal terms, h is the vector of gravity torques, f is the vector of friction torques and u is the vector of applied torques. We define as system outputs the joint positions

Y ' 4

The model (1) has the following properties.

(P4) llf(4,e) II Q f M II 4 II 9 ve E 0, v4 E R". (PS) The matrix B(q,8)-2C(q,q,O) is skew-symmetric provided that the matrix C is

suitably defined.'

We consider the problem given in the following definition.

DeJnition 1

system (l), (2) if there exists a parametrized state feedback control law (independent of 0) The robust tracking problem with disturbance attenuation is said to be globally solvable for

24 = u(k, q, 4,o with k denoting the attenuation factor, such that for any desired bounded output reference q d ( f ) with bounded time derivatives id(f) , dd(f) and for any initial condition q(O), q(0) the state q(t), q ( t ) is bounded for any t 2 0 and the tracking error q(t) - q d ( f ) converges to the region

s m = { ( q - q d ) c m " : I I q - q d I ) < e(k)) with

lim &(k)=O k-r m

0

We give in the next theorem a procedure to construct a control algorithm which solves the previous problem.

SELF-TUNING CONTROL OF ROBOT MANIPULATORS 407

Theorem 1

and a control law which solves the problem is given by The tracking problem with disturbance attenuation is globally solvable for system (l), (2)

(3)

where 80 and 81 are suitable positive constants depending on the attenuation factor k and q d ( f ) is the desired trajectory in joint co-ordinates.

u = -6O(k)&- 81(k)&(lI i l l 2 + I I d d 1 1 2 ) , g2 = (4 - i d ) + k(q - q d )

Proof. Define the errors

el = q - qd, & = d - d d (4)

( 5 )

From (1) and (4) we have

)i = 4, & = B-'(-Cd -1- h) + B-'U - & where the arguments of the matrices are suppressed for brevity. Let e: be defined as

(6) * e2 = -kel

and define the change in co-ordinates el,^) + (el, &) with

(7) *

Z2 = e2 - e2

In new co-ordinates we obtain

kl = kel + A, iz = B-'(-Cq - f - h ) - i jd + k 4 +B-'u (8)

By virtue of properties (Pl)-(P4) we can write

I ( E-'( - Cd - f - h) - dd) + ke2 11

Consider the function

The time derivative of "I: taking (9) into account, is such that

V= 3 (e:el+ dei)

Choose

u = - i?2bM[k+p( t ) ]


Defining p such that

p 2 a2 + f a: i.e.

C = OO(k) + Pl(k)( 11 4 11’ + 11 i d 11’) where 00 and PI are suitable positive constants depending on the attenuation factor k, we finally obtain

+Q -(k- 1)II [el,z21 I I ’ + I I z ~ I I ~ O Since a0 is bounded (the desired trajectory and its time derivatives are bounded), we can write

I aO(qd, id, dd) I Q 7 0 , YO > 0 which implies

For every 11 el 11 2 E we have

II [::I I1 I I1 [;:I II

+ - Q - 2 ( k - l ) + -

Q - 2(k - 1) + 2yo

Y


Therefore with k > 1, following Definition 1, the region of convergence is characterized by

'YO r ( k ) > - k- 1

Since yo is independent of the choice of k, the proof is concluded. 0

3. SELF-TUNING SETPOINT REGULATION

In this section we assume we are interested only in setpoint regulation. We will show that in this case it is possible to obtain asymptotic convergence to the desired constant reference. Moreover, the knowledge of lower bounds and upper bounds on the parameters is not required. The parameters are only required to belong to a bounded compact set. The problem we will consider is formalized in the following definition.

Dejnition 2

exists a (possibly dynamic) state feedback control law (independent of 8) The setpoint regulation problem is said to be globally solvable for system (l), (2) if there

= u(q, q , z , o , i = +GI, 492, t )

such that for any desired constant output reference q d , for any initial condition q(O), q(0) and for any 8 E fl the extended state q(t ) , &), z ( t ) is bounded for any I 2 0 and

lim 11 q( t ) 11 = 0

The next theorem gives a solution to the problem in Definition 2. t - m

lim 11 q(t ) - qd 11 = 0, t - m

0

Theorem 2

feedback controller which solves the problem is given by The setpoint regulation problem is globally solvable for system ( l ) , (2) and a dynamic state

u = -El [ 4 + K ( q - q d ) ] (1 + 11 4 1 1 2 ) + f i , b = - [q + K(q - q d ) ]

k1=<1 IIqIIz>IIq+K(q-4d)()Z

where K is a symmetric positive definite matrix.

Proof, Define the errors el = q - q d , ez = q

From (1) we obtain

15 = ez, t% = el + q d ) [ - C ( e l + q d , ede2 - f (e2) - h(el+ 4 d ) l

where the dependence on 8 is suppressed for brevity. Let e? be defined as

e?= -Kel

where K is a symmetric positive definite matrix, and define the change in co-ordinates (el , ez) + (el , h) with

ZZ = e2 - e? = e2 + Kel


where eii is the ith component of el and the functions $1 can be computed as

Since f(el,qd,O) is bounded for any el E IR" and for any B € 0, it follows that the functions $i are bounded for any el c IR" and for any B € 0. Therefore a positive constant *M can be found such that

Choose the control law u = -rG,(t)&(l + I I ez 11’) +m

where I?] and fi are still to be defined. Substituting (20) into (19)’ we obtain

in which

From (21) it is apparent that there exists a positive constant kl such that for

LI 2 ki

the quadratic form on the right-hand side in (21) is negative definite. Define fl = kl - I?]

and f i = * ( o , q d , 8 ) - 8

Consider the function z/A= Y+:f?+;fili2

The time derivative of Y6, taking (21)-(24) into account, is such that

so that if we choose

R I = - RI = -(I + 11 ez 11’) 11 A 11’, p” - f i = &

we obtain

which implies the boundedness of all states in the closed-loop system. Equations (l), (20) and (27) imply that 11 61 I( and 11 $2 I ( are also bounded and consequently -i/A is uniformly continuous. From

lim I r - % ( T ) d7= %(O) - lim K( t ) < QO I + Q 0 I - -

and using Barbalat’s lemma (see Reference 10, p. 210), we finally obtain

lim % ( t ) = ~ I - Q


i.e.

lim 11 &(t ) 11 = 0 t + W

lim II e1W I1 = 0, I + =

which in turn implies that q( t ) - q d and q(t ) asymptotically tend to zero. 0

4. SIMULATION RESULTS

Some simulations have been carried out with reference to a robot with three degrees of freedom and three rotational joints whose model can be found in Reference 11. The task was that of following a straight line in Cartesian space with a trapezoidal velocity law having a maximum velocity of 1 - 5 m s-'. According to Theorem 1, the adopted control law was

(29) In the first simulation run a linear controller was considered given by (29) with ko = 10, k = lo00 and kl = 0. The absolute error of the end-effector in Cartesian co-ordinates and the corresponding torques delivered by the motors are illustrated in Figure 1. In Figures 2 and 3 are reported the results corresponding to control laws having the same values of k and ko but with kl # 0; specifically, Figure 2 refers to kl= 100 and Figure 3 to kl = 1o00. As can be seen, even though the error in the first part of the trajectory is the same (since the actual and the desired velocity are small), the absolute error along the remaining part of the trajectory decreases as the gain kl increases.

u = - [ko + kl(qTq + i $ q d ) ] &, g2 = - i d + k(q - q d )

3 s s

rn I I

S s Figure I . Linear controller (ko = 10, k = IOOO)

SELF-TUNING CONTROL OF ROBOT MANIPULATORS

J

413

3 s s

3 s s

Figure 2. Robust non-linear controller (ko = 10, k = 1O00, kl = 100)

P s I 0

s s

S s Figure 3. Robust non-linear controller (ko = 10, k = 1o00, kl = 1o00)


P E -

P cf

Joint 1 torque 400, I

0.2

- Q cf -

-0.4

I -200; 1 2 3

-

400

200

0

-200

E z

S

Joint 3 torque 1

1 2 3

Joint 2 torque I r n m

S

S

Figure 5. Self-tuning regulator: joint torques


A second set of simulations has been carried out to evaluate the performance of the self- tuning controller. The task was that of regulation about the joint positions

q = [+O, ./lo, ./lOlT

starting from the initial condition q = O , q = O . According to Theorem 2, the adopted self- tuning control algorithm was

= - g1[ 4 + k(q - q d )I (1 + qT4) + B, i l = (1 + 4’4) [ 8 + k(q - qd)] [4 + k(q - q d 11 8 3 = - 4 3 - k(q3 - qd3) 81 =o, b2 = - 4 2 - k(q2 - 4dZh

with the following values for parameters and initial conditions:

k = 5 , ll(0) = 200, Bl(0) = 0, &(O) = -250, B,(O)= -50

Notice that since the first joint is not affected by gravity, the first estimator BI was set to be identically zero. The numerical results are illustrated in Figures 4 and 5, which report the joint errors and applied torques respectively.

5 . CONCLUSIONS

In this paper a robust tracking controller has been proposed which is capable of guaranteeing arbitrary attenuation on the joint positions of the disturbances caused by uncertainties on the parameters and unmodelled non-linearities. Only lower and upper bounds on the parameters are required to be known. Differentiating from the adaptive control approach, the proposed controller does not require linear parametrization and can therefore be used when some non- linearities (such as friction) cannot be exactly modelled.

For the setpoint regulation problem it is possible to obtain asymptotic convergence to the desired position using a modified version which also includes a self-tuning algorithm. The objective of the self-tuning part is twofold: firstly, it has to asymptotically reject constant disturbances; secondly, it allows us to ignore the bounds on the parameters (they are only required to belong to a bounded compact set).

Future work will be concerned with extension of the proposed techniques to flexible joint and/or flexible link robots.

ACKNOWLEDGEMENT

This work was supported by the CNR under contract 90.00374.PF67.

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self-tuning control of robot manipulators

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