Analyse structurelle des mécanismes par groupe des déplacements

Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity The bound is sharp (ie, equal to one) for reduced components The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials

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Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

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.

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Numerical algebraic geometry and algebraic kinematics

When the leg rods of a fully in-parallel manipulator are fixed in their lengths, it is usual that the device can be assembled in several distinct ways. Sometimes it happens that motion between such assemblies can take place such that the linkage is never at a special configuration; that is, a configuration where the moving-platform body acquires uncontrollable freedom relative to the base. The possibility of such motion has implications for control. Focusing on 3-3 devices, the authors present a geometric explanation of how these motions arise, and give a sufficient condition for their existence. For the 3-3 planar-motion device, they show that never-special assembly changing motions can be excluded by making platform and base triangles similar, and they conjecture that appropriate, perhaps identical, specialization for the octahedral manipulator has the same effect.

S. Roberts

Papers

On Three‐bar Motion in Plane Space