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Research Papers

Finding Only Finite Roots to Large Kinematic Synthesis Systems

[+] Author and Article Information
Mark M. Plecnik

Biomimetic Millisystems Lab,
Department of Electrical Engineering
and Computer Sciences,
University of California,
Berkeley, CA 94720
e-mail: mplecnik@berkeley.edu

Ronald S. Fearing

Professor
Biomimetic Millisystems Lab,
Department of Electrical Engineering
and Computer Sciences,
University of California,
Berkeley, CA 94720
e-mail: ronf@eecs.berkeley.edu

1Corresponding author.

Manuscript received October 17, 2016; final manuscript received January 25, 2017; published online March 9, 2017. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 9(2), 021005 (Mar 09, 2017) (8 pages) Paper No: JMR-16-1319; doi: 10.1115/1.4035967 History: Received October 17, 2016; Revised January 25, 2017

In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess quantities of finite roots that are orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be precluded in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct startpoints and start systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.

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Figures

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Fig. 1

Flowchart of the Finite Root Generation method

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Fig. 2

Multihomogeneous homotopy applied to the four-bar benchmark problem tracks 97% of the paths to roots at infinity

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Fig. 3

The expected results of FRG found 95% of the roots in 91% less paths than multihomogeneous homotopy applied to the four-bar benchmark problem

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Fig. 4

The expected results of FRG found 100% of the roots in 71% less paths than multihomogeneous homotopy applied to the four-bar benchmark problem

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Fig. 5

A four-bar path generator displaced to position j

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Fig. 6

The number of roots obtained as FRG trials progressed for the four-bar benchmark problem. Vertical lines mark trials that resulted in a new root. Their frequency forms a solid colored section at the beginning of the algorithm. The computational effort to find 95% of roots and the final 5% of roots are dimensioned as percentages of the total number trials needed to find 99.988% of roots.

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Fig. 7

Difference between the estimated and actual percentage of roots obtained as trials progressed

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Fig. 8

The success ratio α plotted against the percentage of roots obtained n̂ for both the benchmark experiment and theoretical curve

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Fig. 9

(a) A defect-free mechanism and (b) a mechanism with a branch defect

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Fig. 10

Synthesis results that resemble Chebyshev linkages: (a) a defect-free mechanism and (b) a mechanism with a branch defect

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