A Constrained Homotopy Technique for Excluding Unwanted Solutions From Polynomial Equations Arising in Kinematics Problems

This paper presents a constrained technique which enables classical homotopy continuation solvers to exclude unwanted solutions from a system of polynomials. It is well known that extraneous and degenerate solutions often appear in polynomial equations derived from kinematics problems. These solutions usually do not have physical meanings and prevent us from understanding the intrinsic feature of the studied problem. Traditionally to exclude these solutions, we have to rely on a tedious post-processing technique which often suffers from numerical instability especially when the number of unwanted solutions is large. In the proposed constrained homotopy technique, the unwanted solutions are mapped to the solutions at infinity of a higher dimensional system. We show that the extra polynomials do not increase the homotopy path count when they are used with a classical homotopy solver. This convenient and user friendly technique does not require any changes to the homotopy codes or extra steps to post-process out unwanted solutions. Compared with the regeneration procedure, our approach offers more freedom in eliminating positive dimensional solutions. In addition, our numerical experiments show that the solutions obtained with the constrained technique seem to have a better accuracy than the post processing approach. Kinematics analysis and synthesis problems are employed to demonstrate that the technique is capable of eliminating unwanted isolated and positive dimensional solutions. In the revisit of the classical nine-point path generation of four-bar, we verify that the maximum number of distinct mechanisms is 1442 triplets.