Nonlinear Stability of Hybrid Control

Abstract

A theoretical and experimental investigation on the stability proper ties of the hybrid control scheme was performed using Lyapunov's theory for both the original scheme, which uses the Jacobian in verse for mapping Cartesian errors to joint errors, and a scheme using the Jacobian pseudoinverse. Both schemes result in position and force controllers that are statically coupled in the task space. Stability analysis shows that the pseudoinverse scheme is asymp totically stable, whereas the inverse scheme may become unstable depending on the manipulator attitude and the environmental stiff ness. In the manipulator workspace, where kinematic instabilities have been reported to exist even away from kinematic singularities, the Jacobian inverse affects negatively the Lyapunov function's posi tive definiteness and the negative sign of its derivative; this effect may become dominant when the environmental stiffness is zero or very low. Experimental results for a 2- and 3-degrees-of-freedom planar manipulator using a PUMA 560 were performed both in free space where stiffness is zero and in contact with a stiff surface. Experi mental results in free space have confirmed the stability properties of the two schemes as predicted by the theoretical analysis and are in agreement with previously reported simulation and experimental results. Experimental results in contact with a stiff wall gave stable results for both schemes.