On the determination of boundaries to manipulator workspaces

The accumulation of two independent, broadly applicable formulations for determining the boundary to manipulator workspaces, presented elsewhere, are compared in this paper. Insights gained from one method are used to explain behavior exhibited in the other. Results are also compared and validated. A numerical formulation based on continuation methods is used to map curves that are on the boundary of a manipulator workspace. Analytical criteria based on row rank deficiency criteria of the manipulator's analytical Jacobian are used to map a family of one-dimensional solution curves on the boundary. The other formulation, based on a similar rank-deficiency criteria, yields analytic boundaries parametrized in terms of surface patches on the boundary. Results concerning the applicability of the numerical method to open- and closed-loop systems are compared with those limited to the open-loop for the analytical method. Conclusions regarding the behavior of the manipulator on geometric entities characterized by singular curves, higher-order bifurcation points, and surfaces inside the workspace are drawn. Applicability of both methods and their limitations are also addressed.