Exponential Convergence of a Learning Controller for Robot Manipulators

This note presents the proof for the exponential convergence of a class of learning and repetitive control algorithms for robot manipulators. The learning process involves the identification of the robot inverse dynamics function by having the robot execute a set of tasks repeatedly. Using the concepts of functional persistence of excitation (PE) and functional uniform complete observability (UCO), it is shown that, when a training task is selected for the robot which is persistently exciting, the learning controllers are globally exponentially stable. Repetitive controllers are always exponentially stable. I. INTR~DUCIION In this note we provide exponential convergence proofs for some of the learning and repetitive robot motion control algorithms introduced in [4]. In [4], the learning control problem was formulated as a functional identification problem. Unknown functions were described in terms of integral equations of the first kind consisting of known kernels and unknown influence functions. The learning process involves the indirect estimation of the unknown functions by estimating the influence functions. In the robot motion control context, robot tasks were associated with tracking a set of desired trajectories and velocities. The objective of the learning algorithm is to identify the robot inverse dynamic function which the control action must generate to achieve perfect tracking. Sufficient conditions for the existence of integral equations for Manuscript received December 8, 1989; revised June 8, 1990. This work was supported by the National Science Foundation under the PYI Grant MSM-8657520, the IBM Corporation under the SUR Contract SL 88086, D.S.T.O. of Australia, and by Boeing (BCAC). R. Horowitz and W. Messner are with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720. repetitive tasks were presented, and the asymptotic stability of the learning process was demonstrated. The robustness of the learning scheme to noise and unmodeled dynamics was not addressed. However, these properties were apparent from both simulation and experimental results. In adaptive systems theory, the role of persistence of excitation to achieve exponential stability and robustness in the presence of unmodeled disturbances and dynamics is well understood (cf. [l], [9]). In [5] a linear systems framework was proposed in which concepts such as functional persistence of excitation and functional uniform complete observability were developed to prove the exponential stability of a repetitive learning algorithm presented in [4], under the assumption that the unknown functions and kernels have a finite eigenfunction expansion. In this note, we will further extend this framework to a class of nonlinear mechanical systems which includes rigid link anthropomorphic robot arms. II. LEARNING CONTROL Using the results of Sadegh and Horowitz in [7], it was shown in [4] that the tracking error dynamics of an n degree-of-freedom robot manipulator, under the action of the desired compensation control law (DCCL), satisfy the following differential equation. e = f ( t , e ) + ~ ( t , e ) [ w ( u ) w ( t , u ) ] (2.1) where the nonlinear function f ( t , e ) is defined in [4] M( a , ) is the manipulator generalized inertia matrix. The tracking error state vector is given by e ( t ) = [e ; , e:]‘ e,( t ) = x d ( t ) x p ( t ) and e,( t ) = ep( t ) + Apep( t ) , A,> 0 (2.2) where x d ( t ) E R “ and x, ( t )ER” are the desired and actual joint positions as functions of time. The function w ( ) : A -+ R“ represents the unknown desired robot inverse dynamics. The function U ( * ) : R+-+ A is a known function of time, which is associated with the learning task. The compact subspace A depends on the specified learning tasks. In the general learning control problem, the robot is required to track an arbitrary finite set of desired trajectories S = { x , E C * I x d ( t ) €E p , i d ( t ) E E , , x d ( t ) €E , } where E,, E,,, and E, are compact subsets of R”. For this case, A = Ep x E,, x E, and U := [ x;, x;, x;] w ( u ) : = M ( x ~ ) ~ ~ + c ( x ~ , X ~ , X ~ ) + g ( x d ) . (2 .3 ) M ( . ) : R” -+ R”’” is the generalized inertia matrix, c ( * ; ; ): R3” -+ R” is the vector due to Coriolis, and centripetal accelerations and g ( . ) : R” -+ R” is the vector due to gravitational torque and forces. In the repetitive control problem considered in [6], [lo], [8] and their references, the robot is required to track a single periodic trajectory Xd(t) = xd( t + T ) E Cz, where T is the period. A j, B. WEE Log Number 9144617. is with the Department of Systems Engineering, Australian related problem is considered in [31, V I . In this case, w ( ) can be considered as an unknown periodic function of time w,(t + T) = w,(t) = w ( t mod T ) and A = [0, TI. National University, Canberra, ACT 2601, Australia. 0018-9286/91/0700-0890%01.00 01991 IEEE