A Coarse-Grained Parallel Homotopy for Mechanism Design

Abstract

Polynomial systems of equations arise frequently in many scientific and engineering applications such as solid modeling, kinematics, and robotics and chemical process de-sign. Locally convergent iterative solution methods may diverge or fail to find all possible solutions to a system of polynomial equations. Recently, homotopy algorithms have been proposed for solving polynomial equations. These methods are globally convergent from any arbitrary starting point, can reliably compute all possible solutions, and are inherently parallel in nature. For problems arising in mechanism design, it is shown in this paper that through a use of m-homogenization and by defining auxiliary equations in addition to the design equations, the number of homotopy paths to be tracked (and the associated computational effort) to obtain all possible solutions can be reduced significantly. Further computation time gains are realized by using the data-parallel nature of these methods, which involved implementing them on Connection Machines CM-2/5. Numerical examples dealing with the synthesis of a slider-crank mechanism for six and eight finitely and multiply separated precision positions are presented.