Pergiunon ooo5-1098(95)ooo39-9

A Simple Iterative Learning Controller with Flexible Joints*

DANWEI WANG?

Key Words-Iterative learning control; manipulators; flexible joints.

Aum~licu, Vol. 31, No. 9, pp. 1341-1344,199s Copyright @ 1995 Ekvier Science Ltd

Printed in Great Britain. All rights reserved Gals-1098/95 $9.50 + 0.00

Brief Paper

for Manipulators

Abstract-A simple iterative learning control scheme is proposed for flexible-joint manipulators to improve tracking accuracy. Insufficient knowledge of robot dynamics and joint flexibility for precise tracking control can be overcome by using this iterative learning law as the manipulation task is repeated. This law guarantees the convergence of the link trajectory to the desired one within bounds, and these bounds can be reduced by properly choosing the learning control parameters. Furthermore, this iterative learning control is computed off-line using link position, velocity and acceleration tracking errors.

I. Introduction The advantage of a robot manipulator over a human worker is that it can repeat the same operation tirelessly and with the designed accuracy. In many industrial applications, man- ipulators are deployed to carry out accurate operations that require precise tracking of a specified trajectory. However, uncertainties of robot manipulators in dynamics and parameters can make control designs difficult to implement. To achieve the objective of accurate tracking operation without having perfect knowledge of its dynamics and parameters, one solution is to modify its control action based on its previous tracking performance. A learning law in addition to a stabilization controller is considered feasible to meet this objective for industrial robot applications. Recently much effort has been directed to the learning control design for robot repetitive operations (Arimoto et al., 1984, 1990; Arimoto, 1986, 199Oa, b; Atkeson and McIntyre, 1986; Miyazaki et al., 1986; Togai and Yamamo, 1986; Hauser, 1987; Bondi et OZ., 1988; Heinzinger et al., 1989; Messner et al., 1991; Tae-Yong Kuc et al., 1991; De Luca and Ulivi, 1992).

Arimoto et al. (1984, 1990) and Arimoto (1986, 1990a, b) proposed learning control laws of the PID type. They showed that learning control laws work well in the vicinity of the desired trajectories by using a linearized dynamic model. Hauser (1987) and Heinzinger et al. (1989) designed learning control laws with nonlinear robot dynamic models. Tae-Yong Kuc et al. proposed a simple learning law using position and velocity tracking errors, and justified the convergence using Lyapunov’s direct method. Miyazaki et al. (1986) considered the iterative learning control problem for manipulators with joint elasticity. They proposed a two-stage control scheme consisting of two parts. First, the motor reference trajectory is learned iteratively, and then the required torque input is

learned iteratively. The limitation is that the convergence is

* Received 1 March 1992; revised 25 August 1993; revised 5 May 1994; revised 21 November 1994; received in final form 21 February 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Annaswamy under the direction of Editor C. C. Hang. Corresponding author Dr Danwei Wang. Tel. +65 7911744; Fax +65 7912687.

t School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263, Republic of Singapore.

1341

slow to achieve the objective. De Luca and Ulivi (1992) also studied the learning control design for robot manipulators with joint flexibility. They suggested that the design be done in the frequency domain using a linearized robot dynamics.

The stabilization problem of robots with flexible joints have been extensively studied by Khorasani and Spong (1985), Spong (1987), De Luca (1988), Fhorbel et al. (1989) and Tomei (1991): To achieve accurate tracking, .exact knowledge of the dvnamics is reauired (Swne. 1987). When

I

simple controllers are used, -global ‘sta&ity &n be guaranteed, but inaccuracy in regulation and hence tracking is inevitable. It is natural to use learning control to overcome such difficulties. So far, there has been little research, and there are no rigorous results for iterative learning control design for robot with flexible joints.

In this paper a simple iterative learning control law is proposed and shown to be able to improve the tracking performance for robot manipulators with flexible joints. It is shown that the link angles of a robot will track the specified trajectory with bounded errors. This iterative learning control is computed off-line using link position, velocity and acceleration tracking errors.

2. Learning control design Consider a robot manipulator with flexible joints. The

flexibility is assumed only on the rotary direction of the rotor and link angles, which are referenced to the same axis. The control input is applied to the motor, and the motor torque is passed through the flexible transmission to turn the link on ihe other side.

The dynamics of robot with flexible joints can be modeled by the following equations (Tomei, 1991):

+ ( P(4J 4) + w7~ 44 + N2(% 4P + g(q) + K(q - 0) + 44 wq, 414 + K(q - 0) + &Id, 1

= 0 ,” (1)

In these 2n differential equations, q E W is the vector comprising the n link angles, 4 is the link angular velocity vector, B E R” is the vector comprising the rotor angles, tj is the rotor angular velocity vector and u E R” comprises the control inputs applied at the n joint motors. M(q) and J are the inertia matrices of the manipulator and the joint motors respectively, while

P(S? 4) = [ $w?)]4 - ; [ gww ] combines the centrifugal and Coriolis terms, and g(q) is the gravity term. The stiffness of the elastic joints are characterized by the n x n matrix K, whose diagonal elements are the stiffnesses of the flexible joint transmissions. The gyroscopic coupling matrix R(q) is strictly upper- triangular, and the matrices Ni, i = 1,2,3, have linear dependences on velocity and are zero when R(q) is a

1342 Brief Papers

constant matrix. The matrices 4 and F, are diagonal positive-semidefinite, representing friction at the link and at the motor side of the transmissions.

To use a learning controller to improve the tracking performance of such robots with joint flexibility, a feedback controller is assumed available to guarantee the closed-loop stability. A simple PD feedback type with a gravity compensation has been shown to guarantee global stability of flexible joint robots (Tomei, 1991).

In this paper the objective is to design a learning controller for flexible-joint robots. In particular, qd(t), cjd(t) and ijd(t) are given as the link angle trajectories to be followed repeatedly in the time interval [0, T]. We assume that the rotor angles are availabIe for on-line feedback control purposes. We also assume that the link angles are available for off-line computation after an operation task is completed. We assume that the manipulators considered here satisfy all the basic assumptions made by De Luca and Ulivi (1992) to ensure that a learning control scheme is applicable.

We consider a controller consisting of two parts as follows, in the kth operation:

u=u1+u, (2)

where u, and v, are designed to guarantee stability and to improve tracking performance respectively.

The first part is a stabilization feedback controller that is the same at every operation cycle:

u1 = C(6 - qd, ti - &). (3)

The controller uses qd and & as reference trajectory for 0 and 0, because computation of 6d and 6d requires exact knowledge of robot dynamics and parameters, which are not available. But we know that qd and 6, are close if the joint transmissions are stiff. Design of such stabilization controllers has been successfully carried out in many papers (Khorasani and Spong, 1985; Spong, 1987; De Luca, 1988; Fhorbel et al., 1989; Tomei, 1991). Since such a feedback control can guarantee closed-loop stability (Tomei, 1991), it can be assumed the velocity tracking error is uniformly bounded, i.e. I/Q(t) - gdll I p, where p is a constant. A similar property has been verified by Arimoto (199Oa) for the case of a rigid-joint robot manipulator with a PD feedback control.

The second part of (2) is a biased function. It is updated between two consecutive cycles according to the following learning law:

uk+l = (I - y)% + ‘YVO + L&id - qk) + ‘%(4d - &)

+ La(ijd -ii& (4)

where the subscript k denotes the kth operation of the task, 0 < y < 1 is a forgetting factor, v. is the function of initial guess of the biased term, and the three n X n matrices L,, L, and L, are the learning gains and are chosen symmetric and positive-definite.

In proving the convergence of such a learning control, we use the following norm definitions:

(9 the norm of an n-vector w is

and the induced norm of a matrix A is

IlAll = v max ATA eigenvalue

(ii) the k norm for a function h: [0, T] -3 W” is

Remark. Ilh IIA I Ilh Ilm 5 eAT I/h II* for A > 0 implies that the A norm is equivalent to Ilhllm (~up~~[~,~~ llhll).

Define bLa = IlLa and m, >O for all q so that

m,I< WI) ( R(q) RT(s) ) 1 .

The convergence of the learning law can be stated as follows.

Theorem. Consider the feedback control robot system with joint flexibility as given by (l)-(4). Let the learning gain L, be an constant n x n matrix chosen to be symmetric and positive-definite and to satisfy the inequality

Then the tracking errors Ilqk(r) - qd(t)llA and Ilv,(r) - ~~(1) 11 A converge to certain bounds in the time interval [0, T] as the operation number k increases.

Proof. The closed-loop robot dynamics can be written as

( M(q) R(q) ii

>( 1 RT(q) J 8

P(4J 4) + N(qt a + %(% 418, + g(q) + w? - 0) + 64 %(% 4)d - K(q - 0) + Frnd - cw 6 qdt dd)

Define

y= ;, 0 8

Z= 0 e’

X= Y 0 z .

The system (5) can be rewritten in state-space form:

where

R = D_‘(q)F(x) + D-‘(q)Ev, (6)

F(x) =

/ 4 P(q, 4) + WI, 814 + N267,4) + g(q) + K(q - 0) + _ I e 9

0

E= ;, 0 x

At the kth operation, the dynamics is given by

& = D-‘(q,)F(xk) + Lr’(q#u,. (7)

De Luca and Ulivi (1992) showed that when the desired link angle trajectories qd are given, the desired motor angle variables e,(t), e,(t), e&) and the desired biased term vd(t) exist. The dynamics at the desired state trajectory is given by

id = D-‘(&-,)F(Xd) + D-‘(q,)&. (8)

Taking the difference of (7) and (8),

& - Xd = D-‘(q&F(Xk) - F(&) + E(Vk - lJd)l + [D-‘(q,) - D-‘(qd)][F(Xd) + Evdl* (9)

With the assurance of tracking stability, local Lipschitz conditions are assumed as follows:

Ilo-’ - o-‘(qd)!! <al h& - qdii, WI

IiF - Fbd) /I 5 a2 /hi - xdii 7 (11)

where (I, and a2 are the corresponding local Lipschitz constants.

Brief Papers 1343

Define 8u,(t) = t&(r) - &i(r), 8x,(r) =x&) - x,(r), SY,&) = ~~0) -Y&), Sqdt) = qk(t) - q&J and respective derivatives. By integrating both sides of (9) we have

- ~-‘(qdl[F(xd + -%+)I dz II

5 ll~~km + s

‘~ll~-‘(4*)II[IIF(x~) - FMII o

+ llwl1+ IID-’

- D-Yqdll IO%) + Eu,llld7 ,

5 ll~.dO)II + I( o

b, II&II +$ II&II ck 1

where b, = u2/ml + ala3, with ~3 = sup,.le,rl IIF + /&II. Using the Gronwall-Bellman inequality (Flett, 1980,

p. 96), we have

IIhkWll 5 ll~Q4II e 1 f

*I’+ - I ml o

e*l(‘-‘) l18ukIl dr. (12)

Taking the norm of (9) and using (12)

Il%(r)ll s l/D-‘(%)I1 [lb%,) - F(Xd)Ii + I16uklll

+ a3 IID-’ - D-‘(qd)/l

~(m,a2+44 IIWI +m~ lk%II

smI IIWI + mla2+ ala3

I ‘eblcrmr) II8ukII dr

ml 0

+ (mla2 + WZ~) IIWO)II e*l’. (13)

On the other hand, the learning update law (4) can be rewritten as

U k+l = (1 - r)Uk + -,‘UO + L&d - (j’k) + L(Yd -Ykh (I41

where

L = [L,, L”].

Define bL = I/L 11. From the inequality (14) using (12) and (13) we have

lb%+,ll~(l-Y) iIsuk/i +Y II6’Joll +b~,Il&fkll +bLIlhkll

5(1 -Y) II~“kII + ‘Y il~voli +ha 116-fkit +b, IIsxkIi

5 (1 - Y + bL,md li6vkiI + Y II6Uoll + b2 i16xk(o)Ii ebl’

+62

I

,

ml o ebl(‘-r) II&J, II dr,

where b2 = bL + bLBJmla2 + a,u3). Using the techmques of Heinzinger et al. (1989) and

Arimoto (199Oa), we multiply both sides by em*‘, with A > b,:

118uk+lll em*‘5 (1 - y + bL,ml) Ij6ukj( em”‘+ y I16uoll e-*’

+ b2 l16xk(0)II e@-*)’

b2 +- I

‘e@-*Wr) II&,~I( e-“rd7. ml 0

In the A norm, this inequality yields

I16uk+11iAs(1 - Y +b&ml) li8ukiI* + Y IIfkollA

+b, I16xk(0)I/ +~llfiukil*~e(*l-*)(‘-‘)dr

5 (l-y+b,~~m,)+m(~~b,)(l-e~b~~*~r)] I 1 x IIsuk~~* + Y II6Uoll~ + b2 II~xk(o)fI

Or, in shorthand,

~~6uk+,II,5~ li6vkil*+E, (15)

AUTO 3,-9-J

where

P=(l-v+bL,mi)+ ml(:s_ b,) (1 - e@-*)3,

e = Y II6voll* + Mxo,

with bxo = maxk IISxk(0)ll. Since 1 - y + bL,ml < 1, it is possible to find a A that is

large enough to make p = 1 - y + bq,m, < 1. This implies that (15) is a contraction-map inequahty. As k increases, we have

lim sup l16ukll*‘&e. k-m

(16)

This implies that uk converges to the neighborhood of ud with respect to the A norm. The radius of this neighborhood is proportional to the initial state errors and initial biased term contribution.

Similarly (12) yields

Ilsxk(l)h 5 I16xk(o)iI e Cb,-*P+L

I ’ e(bl-*)(r-r) 11 8uk )I * dr.

ml o

s II6xk(o)II +m (A1_ b,) t1 -e(blm*)r) I18ukII*~ 1

Using the inequality (16) we have

lim sup Il~Xkll,rb,,+~-~~~~,~~~. k-m

(17) 1

Remarks. In the case of perfect repeatability of initialization, i.e. 6xk(0) = 0 for all k, and exact knowledge of robot dynamics, which makes 8uo = 0, the tracking error bounds will be zeros, and this implies the convergence of the algorithm to the desired trajectories.

For most industrial robot manipulators, repeatability of initialization is quite accurate, i.e. bxO is small. When the knowledge of the robot parameters is limited, IISuoll* could be dominant in the tracking error bounds in (15) and (17). To reduce the effect due to this factor, we can (i) choose a small forgetting factor y and/or (ii) reduce II 6uo(lA from cycle to cycle of operation. The first way is to reduce the weighting of the inaccuracy of the initial guess u. in the bounds. The second way can be implemented by refreshing the memory content of u. by uk after the kth operation, where I16xkII* is observed to be less than II 6xoll,,, and resetting k = 0. In other words, when IISxk II* is less than I18xoll*, the biased term uk is considered a better guess than the initial guess uo.

3. The tracking error bounds remain the same if additional disturbances exist, as long as they are Lipschitz. In the presence of the disturbances that are discontinuous, additional terms will appear in the tracking error bounds.

3. Conclusions It has been shown that an iterative learning control can be

used to improve the tracking performance of robot manipulators with flexible joints. The simple learning law uses feedback of link angle tracking errors, and is computed off-line. Such a learning law guarantees the convergence of link angles and motor angles to their desired trajectories within a bound that tends to zero when better repeatability and knowledge of robot parameters are available. The bound can be reduced by choosing learning control parameters properly.

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Arimoto, S. (1990a). Learning control theory for robotic motion. ht. J. Adaptive Control and Signal Process., 4, 543-564.

Arimoto, S. (199Ob) Robustness of learning control for robot manipulators. In Proc. IEEE Znt. Conf: on Robotics and Automafion, Cincinnati, OH.

Arimoto, S., S. Kawamura and F. Miyazaki (1984). Bettering

Brief Papers

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