A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems

Tari, Hafez; Su, Hai‐Jun; Li, Tien Yien

doi:10.1016/j.mechmachtheory.2010.01.002

Cited by 19 publications

(11 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, they showed that there are 1442 distinct four-bar coupler curves which pass through nine general points which, together with Roberts cognates [35], yields 4326 distinct four-bar linkages. Although this computation has been repeatedly confirmed using various homotopy continuation methods, e.g., [5,12,21,22,34,38], these numerical computations do not preclude the existence of additional solutions. In fact, one of the distinct four-bar linkages was missed by the homotopy continuation solver in [40] but was reconstructed using the cognate formula.…”

Section: Introductionmentioning

confidence: 77%

Probabilistic Saturations and Alt’s Problem

Hauenstein

Helmer

2020

Experimental Mathematics

View full text Add to dashboard Cite

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

show abstract

Section: Introductionmentioning

confidence: 77%

Probabilistic Saturations and Alt’s Problem

Hauenstein

Helmer

2020

Experimental Mathematics

View full text Add to dashboard Cite

show abstract

“…Wampler, Morgan, and Sommese [16] presented the complete solutions of nine-position four-bar path synthesis problem solved with polynomial continuation method. The revised constrained homotopy method was presented in [23]. Raghavan [17] presented the forward kinematics of general Stewart Platform with continuation method.…”

Section: Homotopy Methodsmentioning

confidence: 99%

“…Raghavan [17] presented the forward kinematics of general Stewart Platform with continuation method. Other applications have also been found in [18][19][20][21][22][23][24][25][26]. …”

Section: Homotopy Methodsmentioning

confidence: 99%

The Application of General Homotopy Method on Planar Four-Bar Path Synthesis

Chin¹,

Hsu²,

Fu³

2016

Proceedings of the 2016 International Conference on Mechatronics, Control and Automation Engineering

View full text Add to dashboard Cite

Abstract-Homotopy method is a numerical continuation method that can locate all the zeros of any given function without the specifying of initial estimates. With the assistance of proper programming technique, homotopy method can be efficient and reliable. The application of homotopy method on kinematic synthesis can resolve the inherent shortcomings that most numerical methods possess. This work presents a demonstration of homotopy method on the path synthesis of planar four-bar mechanisms. The example of nine point-path synthesis is provided. The results of all real solutions are tabulated and the corresponding mechanisms are graphically displayed so comparisons can be made with past publications.

show abstract

“…(When run tracking homotopy paths in parallel using a system with 65 processors, the regeneration method was proportionately faster.) More tests comparing regeneration to polyhedral methods are reported in [107] along with a complementary method that helps sort the solution sets more automatically.…”

Section: Equation-by-equation Methodsmentioning

confidence: 99%

Numerical algebraic geometry and algebraic kinematics

Wampler¹,

Sommese²

2011

Acta Numerica

View full text Add to dashboard Cite

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

show abstract

A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems

Cited by 19 publications

References 38 publications

Probabilistic Saturations and Alt’s Problem

Probabilistic Saturations and Alt’s Problem

The Application of General Homotopy Method on Planar Four-Bar Path Synthesis

Numerical algebraic geometry and algebraic kinematics

Contact Info

Product

Resources

About