Repeatability of redundant manipulators: mathematical solution of the problem

Consider a redundant manipulator whose hand is to trace a path in its workspace. A local control strategy that governs the manipulator is a law that assigns an infinitesimal change in the joint angles so that the hand will move infinitesimally in the direction designated by the path. Because of the redundancy, there can be many such control strategies. For some strategies it turns out that when the hand returns to its initial position, the joint angles do not always return to their initial values. To determine a priori whether a given local control strategy guarantees repeatability or not, it is necessary to deduce its global properties as well as its local properties. This is achieved by considering integral surfaces for a distribution in the joint space. This yields a necessary and sufficient condition, in terms of Lie brackets, for a control to be repeatable.<>