Computing Tensor Eigenvalues via Homotopy Methods

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve.

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Complementarity Enhanced Nash’s Mappings and Differentiable Homotopy Methods to Select Perfect Equilibria

Journal of Optimization Theory and Applications ◽

10.1007/s10957-021-01977-x ◽

2021 ◽

Author(s):

Yiyin Cao ◽

Chuangyin Dang ◽

Yabin Sun

Keyword(s):

Homotopy Methods ◽

Perfect Equilibria

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Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization

22nd Biennial Mechanisms Conference: Mechanism Design and Synthesis ◽

10.1115/detc1992-0308 ◽

1992 ◽

Author(s):

Anoop K. Dhingra ◽

Jyun-Cheng Cheng ◽

Dilip Kohli

Keyword(s):

Group Theory ◽

Matrix Method ◽

Numerical Results ◽

Path Generation ◽

Link Function ◽

Homotopy Methods ◽

Motion Generation ◽

The Matrix ◽

Function Generators ◽

Complete Solutions

Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

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Multiply Separated Synthesis of Six-Link Mechanisms Using Parallel Homotopy With M-Homogenization

Volume 1B: 25th Biennial Mechanisms Conference ◽

10.1115/detc98/mech-5929 ◽

1998 ◽

Author(s):

A. K. Dhingra ◽

M. Zhang

Keyword(s):

Group Theory ◽

Matrix Method ◽

Parallel Implementation ◽

Homotopy Methods ◽

Numerical Work ◽

Function Generation ◽

Connection Machine ◽

The Matrix ◽

Generation Problem ◽

Complete Solutions

Abstract This paper presents complete solutions to the function generation problem of six-link Watt and Stephenson mechanisms, with multiply separated precision positions (PP), using homotopy methods with m-homogenization. It is seen that using the matrix method for synthesis, applying m-homogeneous group theory and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked in obtaining all possible solutions to the synthesis problem can be drastically reduced. Numerical work dealing with the synthesis of Watt and Stephenson mechanisms for 6 and 9 multiply separated precision points is presented. For both mechanisms, it is seen that complete solutions for 6 and 9 precision points can be obtained by tracking 640 and 286,720 paths, respectively. A parallel implementation of homotopy methods on the Connection Machine on which several thousand homotopy paths can be tracked concurrently is also discussed.

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