Nearest optimal repeatable control strategies for kinematically redundant manipulators

Kinematically redundant manipulators, by definition, possess an infinite number of generalized inverse control strategies for solving the Jacobian equation. These control strategies are not, in general, repeatable in the sense that closed trajectories for the end-reflector do not result in closed trajectories in the joint space. The Lie bracket condition (LBC) can be used to check for the possibility of integral surfaces, also called stable surfaces, which define regions of repeatable behavior. However, the LBC is only a necessary condition. A necessary and sufficient condition for the existence of stable surfaces is used to illustrate that such surfaces are much rarer than previously thought. A technique for designing a repeatable control that is nearest, in an integral norm sense, to a desired optimal control is presented. The desired optimal control is allowed to take the form of any generalized inverse. An example is presented that illustrates the capability of designing repeatable controls that approximate the behavior of desired optimal inverses in selected regions of the workspace