Supplemental Software for Computing Saddle Graphs via Homotopy Continuation for the Approximate Synthesis of Mechanisms

An approach for approximate kinematic synthesis of mechanisms which computes a graph that identifies minima of an objective function as vertices and connections between them as edges. Such a graph is interactively presented to a designer, whereby edges are continuously traversed to navigate families of design candidates in between minima. Candidates are evaluated continuously according to auxiliary considerations for the exploration of design trade-offs. Computing the aforementioned graphs begins with finding all minima and saddles of an objective function through polynomial homotopy continuation. Connections between minima that minimize their maximum objective value must pass through a saddle to do so. Therefore, after gathering saddles, each is perturbed both ways in its least eigendirection to seed gradient descent paths which connect two minima when pieced together. Discovered connections between minima are organized into a graph, where edges correspond to gradient descent paths.