Characteristics of optimal solutions in kinematic resolutions of redundancy

As a redundancy resolution method, the conventional extended Jacobian method (EJM) has two problems: One is the algorithmic singularity, and the second is nonexistence of a sufficient condition. Their impact on the inverse kinematic performances are exemplified. To remedy the related problems, we propose the notion of solution characteristics. It is based on an analytic sufficient condition for the EJM. This serves to confirm the right direction of optimization in the EJM and to characterize the algorithmic singularity problem. The above local characterization of an optimal solution by solution characteristics is globally extended by the invariance of the solution characteristics of the EJM. Specifically the characteristic of a solution with the EJM is invariant before crossing an algorithmic singularity. The ideas in this article also have practical implications since the EJM can now be fully analyzed with the theoretical results. To demonstrate its exactness and usefulness, planar three degrees of freedom (DOF) and spatial 4-DOF regional redundant manipulators are analyzed using the proposed solution characteristics.