Manipulators Singularities , Stable Surfaces , and the Repeatable Behavior of Kinematically Redundant

There has been significant interest in the periodic behavior, generally referred to as repeatability, exhibited by a kinematically redundant manipulator while performing a cyclic end-effector motion. Much of the early work in this area has been restricted to planar manipulators whose configuration is described in terms of absolute joint angles to simplify the problem. Unfortunately, this has resulted in the observation of certain phenomena that are unique to this special case and that do not describe the behavior of more complicated manipulators. The goal of this work is to clarify some possible misconceptions concerning the limiting behavior of a redundant manipulator under nonconservative control strategies, with particular emphasis on pseudoinverse control. In particular, stable surfaces are shown to be extremely rare, and a weaker property, referred to as repeatable trajectories, is responsible for the repeatable behavior observed in previous work. It is also shown that the Lie bracket condition need not be satisfied for this type of repeatable behavior to occur and that such trajectories need not have zero torsion, as has been previously suggested.