This paper presents a unified formulation for the generation of mechanical manipulator workspaces, and the representation of the volume of an entity swept along another. A constraint function is developed and its Jacobian is studied. The analysis is based upon a row-rank deficiency criteria of the non-square constraint Jacobian. Singular geometric entities inside the resulting volume are identified and parametrized. These entities are characterized by a set of singular generalized coordinates (parameters), when substituted into the constraint vector function yield parametric equations. Singular geometric entities are assembled and intersected to identify sub-entities that are boundary to the volume. A local perturbation method is applied to determine which sub-entities are boundary to the workspace. Several examples are presented to verify the formulation.

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