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Using Monodromy to Statistically Estimate the Number of Solutions

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2nd IMA Conference on Mathematics of Robotics (IMA 2020)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 21))

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Abstract

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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Authors and Affiliations

  1. University of Notre Dame, Notre Dame, IN, 46617, USA

    Jonathan D. Hauenstein & Samantha N. Sherman

Authors
  1. Jonathan D. Hauenstein
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  2. Samantha N. Sherman
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Correspondence to Jonathan D. Hauenstein .

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Editors and Affiliations

  1. Department of Engineering, Manchester Metropolitan University, Manchester, UK

    William Holderbaum

  2. School of Engineering, London South Bank University, London, UK

    J. M. Selig

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Hauenstein, J.D., Sherman, S.N. (2022). Using Monodromy to Statistically Estimate the Number of Solutions. In: Holderbaum, W., Selig, J.M. (eds) 2nd IMA Conference on Mathematics of Robotics. IMA 2020. Springer Proceedings in Advanced Robotics, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-91352-6_4

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