Six-Bar Linkage Design System with a Parallelized Polynomial Homotopy Solver

Glabe, Jeffrey; McCarthy, J. Michael

doi:10.1007/978-3-319-93188-3_16

Cited by 3 publications

(2 citation statements)

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“…Since kinematic synthesis problems, such as path synthesis and motion generation [9,17], are classical but challenging problems that yield polynomial systems [28], the following provides some related work using numerical algebraic geometry [4,26]. A variety of homotopy methods have been used to solve a collection of synthesis problems, such as [10,15,20,27]. The estimation method in [19] uses a coupon collector model based on the success ratio of finding new solutions using parameter homotopies [18].…”

Section: Related Workmentioning

confidence: 99%

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Using monodromy to statistically estimate the number of solutions

Hauenstein¹,

Sherman²

2020

Preprint

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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.

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Section: Related Workmentioning

confidence: 99%

“…The N = 6 case was considered in [20], [10], and [2] reporting 5735, 5743, and 5754 solutions, respectively. The trace test confirms the number of solutions is indeed 5754 with results of our monodromy loops and estimations presented in Figure 5.…”

Section: Watt I Six-bar Mechanismmentioning

confidence: 99%