Winding number analysis of invertible workspaces for redundant manipulators

Conventional controllers for nonredundant manipulators include an algorithm for computing a kinematic inverse function, which gives joint coordinates corresponding to a desired position and orientation of the end-effector. In a generalization of this approach to redundant manipulators, the nonuniqueness of the inverse function can be used to move or eliminate kinematical singularities. However, it appears that for almost all manipulators, there is no inverse function that is free of singularities on the whole reachable workspace. This paper illustrates how a simple idea from topology, called a winding number, can be used to discover restrictions on invertible workspaces. A short tutorial introducing the engineer to the relevant aspects of winding numbers is included, and results for four planar manipulators are worked out in detail.