Abstract
It is natural to employ an optimization algorithm for the approximate kinematic synthesis of linkages. The hope is to find some superior points in the design space that indicate dimensions that are practically useful. One way to achieve this is to find all minima of an objective, then to filter them so the best remain. However, the prospect of finding all minima is bleak unless the optimization problem at hand is particularly small. In this work, we show how to find nearly all minima for a large optimization problem using polynomial homotopy continuation in the approximate synthesis of a four-bar path generator. The system at hand has a Bézout bound of 543,848,665 and a Schnabel estimate to the maximum number of stationary points of 6 · (303,249 ± 713), within a 95% confidence interval. At least with regards to mechanism synthesis, this work represents the largest scale deployment to date of homotopy continuation to solve an unconstrained optimization problem. The challenges of scaling and suggestions for design are given. Example usage for the design of a leg mechanism is given. On the mechanism design front, this is the first presentation of a nearly complete (within the limitations of numerical discernment) solution of the general four-bar optimal path synthesis problem.
