Computing All Solutions to a Discretization-Invariant Formulation for Optimal Mechanism Design

Kinematics is the first consideration in designing the mechanical structures that comprise robots. Of the many subcategories that exist under this umbrella, an often early design goal is to achieve some desired workspace. This goal applies to both single and multi-degree-of-freedom systems. Previous literature has applied the diversity of extant optimization techniques for achieving such design goals. A conceptually simple approach to single-objective optimization is to symbolically derive a gradient vector, then find all of its zeroes. This approach is easier said than done since the resulting system is nonlinear. For this reason, sophisticated optimization heuristics are more commonly employed. In this paper, we revitalize the former approach, offering a route to efficiently find all of the gradient zeroes, including the global minimum. Our approach is facilitated by homotopy continuation. We connect the theoretical results to practical problems by demonstrating the design of a mechanism for a humanoid walking gait and the finger of a robotic hand.