Workspace boundaries of serial manipulators using manifold stratification

Workspace boundaries of serial manipulator arms have traditionally been presented as numerical curves traced on a boundary or as analytical formulations for a specific class of manipulators. This paper presents a broadly applicable formulation to identify and visualize the workspace, based on methods adapted from differential geometry and topology. Because the position Jacobian for manipulators with more than three degrees of freedom is not square, manifold stratification of the vector function (a topological vector space) yields nonlinear analytic functions. These functions give rise to analytic and semi-analytic varieties, some of which characterize lower-dimensional manifolds (submanifolds). The process is repeated until all strata are identified. The problem of determining which stratum (or part thereof) is on the boundary is addressed by introducing an acceleration function, which yields a quadratic form whose definiteness properties delineate the exact boundary. The method yields the exact boundary of the workspace in the form of closed-form equations and is capable of identifying voids therein. A complete example is first illustrated, followed by numerous tabulated examples (some of which are compared with reported works in the literature).