An Adaptive Controller for Two Cooperating Flexible Manipulators

The adaptive control of robot manipulators has been presented as a solution for dealing with uncertainty and variation of the mass properties. Although these systems are nonlinear, globally stable tracking has been demonstrated both analytically and supported through experiments [1]. The key to these results has been the linear dependence of the model on the unknown parameters and, in many cases, the passivity of an appropriate input-output mapping [2]. For instance, with rigid robots it is well known that the map from joint torques to joint rates is passive and this property can be used to explain the stabilization properties of many motion control strategies including the ubiquitous proportional-derivative position feedback. Motivated by the utility of the passivity concept in rigid robot control and its occurrence in a single flexible link when the reflected tip rate is taken as the output [3], the present author introduced the -tip rate in [4] which generalized this output to open-chain flexible manipulators. Passivity using this output was demonstrated for when the manipulated payload was much more massive than the manipulator and a feedforward which permitted its use in tracking problems was constructed in [5]. The adaptive counterpart of that controller was detailed in [6] and variations of both controllers were implemented experimentally with great success in [7]. The extension of the -tip rate idea to cooperating robot arms forming a closed chain was treated in [8] where detailed stability proofs and experimental results were presented. An interesting feature of this development was the occurrence of a free-loading sharing parameter which was used to distribute the control torques between the two arms and form the generalization of the -tip rate. Beyond this, the dynamics and control of cooperating flexible arms has received little attention but there has been some [9,10]. This paper represents the dénouement for the line of research pursued in [4-8]. A fusion of the previous techniques is brought to bear on the problem of adaptive control for cooperating flexible robot arms in the case where the payload is significantly larger than the manipulators but otherwise unknown. Simulation results will be presented for a system of two planar arms each with three joints that manipulate a shared rigid payload. Each arm has two flexible links and a third rigid link which is cantilevered to the large rigid payload. Excellent tracking is demonstrated with simultaneous vibration suppression for the adaptive controller when the payload is sufficiently large for a variety of load-sharing parameters.