Using monodromy to statistically estimate the number of solutions

Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for estimating the generic number of solutions of such parameterized polynomial systems. The new approach extends previous work on success ratios of parameter homotopies to using monodromy loops as well as the addition of a trace test that provides a stopping criterion for validating that all solutions have been found. Several examples are presented demonstrating the method including Watt I six-bar motion generation problems.运动学 (kinematics) 中的连杆机构综合问题往往会构造出大型结构化参数化多项式系统，此类系统在一般情形下的解数远低于传统上界的预估结果。本文针对这类参数化多项式系统，提出了用于估计其一般解数的统计模型。该新方法将此前关于参数同伦 (parameter homotopies) 成功率的研究拓展至单值回路 (monodromy loops) 的应用场景，同时新增了迹检验 (trace test) 模块，可提供用于验证所有解均已被找到的停止准则。文中给出了多个验证该方法有效性的示例，其中包括瓦特Ⅰ型六杆机构 (Watt I six-bar) 运动生成问题。