Affiliation:
1. University of Notre Dame Department of Aerospace and Mechanical Engineering, , Notre Dame, IN 46556
Abstract
Abstract
The method of kinematic synthesis requires finding the solution set of a system of polynomials. Parameter homotopy continuation is used to solve these systems and requires repeatedly solving systems of linear equations. For kinematic synthesis, the associated linear systems become ill-conditioned, resulting in a marked decrease in the number of solutions found due to path tracking failures. This unavoidable ill-conditioning places a premium on accurate function and matrix evaluations. Traditionally, variables are eliminated to reduce the dimension of the problem. However, this greatly increases the computational cost of evaluating the resulting functions and matrices and introduces numerical instability. We propose avoiding the elimination of variables to reduce required computations, increasing the dimension of the linear systems, but resulting in matrices that are quite sparse. We then solve these systems with sparse solvers to save memory and increase speed. We found that this combination resulted in a speedup of up to 250 × over traditional methods while maintaining the same accuracy.
Funder
Directorate for Engineering
Subject
Industrial and Manufacturing Engineering,Computer Graphics and Computer-Aided Design,Computer Science Applications,Software
Reference45 articles.
1. Newtons Method;Lipson,1976
2. An Interval Method for Global Nonlinear Analysis;Kolev;IEEE Trans. Circ. Syst. I Fundam. Theory Appl.,2000
3. Some Examples for Solving Systems of Algebraic Equations by Calculating Groebner Bases;Boege;J. Symbol. Comput.,1986
4. Numerical Solution of Systems of Nonlinear Equations;Freudenstein;J. ACM,1963
5. The Kinematic Design of Six-Bar Linkages Using Polynomial Homotopy Continuation;Plecnik,2015