Stability analysis of adaptively controlled flexible joint manipulators

The composite slow/fast control strategy, consisting of a slow adaptive controller designed for a rigid robot together with a fast control to damp the elastic oscillations of the joints, was previously derived by the authors. In this paper, mathematical details of the algorithm are presented. By using the composite Lyapunov theory for singularly perturbed systems, sufficient conditions are obtained for adaptive trajectory tracking. For point-to-point motion it is shown that there is always a range of joint stiffness for which convergence is achieved, and the region of convergence is quantified. For tracking (smooth and bounded) reference trajectories, sufficient conditions for closed-loop stability and uniform boundedness of the tracking error are given. A residual set to which the tracking error converges is quantified. It is also shown that for special classes of trajectories, which include step responses generated from reference models and certain joint interpolated trajectories, asymptotic tracking can be achieved