Topological Analysis of Global Inverse Kinematic Solutions for Redundant Manipulators

The sub-optimality of redundant manipulators inverse kinematics solutions deriving from calculus of variations is discussed through some considerations of topology. This paper clarifies the relations of homotopy linking together, on one hand, self-motion manifolds and, on the other, joint space paths. With an example, it proves that the sub-optimality does not depend on the homotopy classes of self-motions but is linked to both the presence of distinct self-motions and the existence of different homotopy classes of joint space paths. This paper also clarifies the notion of pre-image of a workspace path, showing that more complex surfaces than deformed tori can be generated in the Cartesian configuration space depending on some topological properties of the path.