Computing saddle graphs via homotopy continuation for the approximate synthesis of mechanisms. (October 2022) - Baskar, Aravind; Plecnik, Mark; Hauenstein, Jonathan D.
Abstract: An approach for approximate kinematic synthesis of mechanisms is proposed in this paper 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. Relevant design specifications tend to be particular per application and are either unclear as how to incorporate into an objective, or clear but with great consequence to the complexity of function evaluation. 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. Highlights: A new method for the approximate kinematic design of linkages is introduced. An objective invariant to the number of motion specifications is formulated. Polynomial homotopy continuation finds all minima and saddles of the objective. Gradient descent paths from saddles connect minima, organized into a saddle graph. Saddle graphs are evaluated according to auxiliary considerations.