A new approach to numerical analysis of workspaces of multibody mechanical systems is developed. Numerical techniques that are based on manifold theory and utilize continuation methods are presented and applied to a variety of mechanical systems, including closed-loop mechanisms. Generalized coordinates that define kinematics of a system are classified and interpreted from an input-output point of view. Boundaries of workspaces, which depend on the classification of generalized coordinates, are defined as sets of singular points of Jacobians of the kinematic equations. Numerical methods for tracing one dimensional trajectories on a workspace boundary are outlined and example are analyzed using one dimensional manifold mapping computer programs, such as PITCON and AUTO.