Interior and exterior boundaries to the workspace of mechanical manipulators

Analytical methods for identifying the boundary to the workspace of serial mechanical manipulators and the boundary to voids in the workspace are presented. The determination of parametric equations of surface patches that envelop the workspace of serial manipulators was presented elsewhere and is extended in this paper to an analytical method for void identification. Because of the ability to identify closed-form surface patches that exist internal and external to the workspace, a mathematical formulation based on the concept of a normal acceleration function is introduced. Admissible motion in the normal direction to a point on a singular surface is delineated and characterized by definiteness properties of a quadratic form. An enclosure bound by surface patches that do not admit normal motion is identified as a void. Several examples are treated using this formulation to illustrate the method.