Removing singularities of resolved motion rate control of mechanisms, including self-motion

Resolved motion rate control is an algorithm for solving the path-tracking problem in robotic control which can fail at singular points of the kinematic function. The questions of existence and smoothness of solutions to the path tracking problem at singular points have not heretofore been addressed. In this paper we find a new second-order condition which, when satisfied, ensures the existence of a solution path with continuous, bounded joint rates. The condition is related to the curvature of the path at the singular value. We prove that a modification of the usual resolved motion rate control algorithm can successfully compute this solution path. As an application, we give a sufficient condition for the existence of self-motion for redundant manipulators at singular points. We derive a simple formula for the rate of recovery of the manipulability measure near the singularity. Several realistic examples are presented, for which we compute exact solutions to typical path tracking problems passing through singular points