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 itsmultiplicity. The bound issharp (i.e., 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 zeroesof a finite number of polynomials.

CONTENTS 1 Introduction 1 2 The Basics of Predictor-Corrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 Piecewise-Linear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (1881--1886), Klein (1882-- 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS-9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the

Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most favorable equations to represent the geometric problem. Since a well-conditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been verified. The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

. Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed. AMS Subject Classification. 14N10, 14M15, 52A39, 52B20, 52B55, 65H10, 68Q40. Keywords. polynomial system, numerical algebraic geometry, homotopy, continuation, deformation, path following, dense, sparse, determinantal, B'ezout bound, Newton polytope, mixed volume, root count, enumerative geometry, numerical Schubert calculus. Contents 1. Introduction 1 2. Three Classes of Polynomial Systems 3 3. The Principles of Polynomial Homotopy Continuation Methods 5 4. The Geometry of the Deformations 8 5. Root Counts and Start Systems 10 5.1. Dense Polynomials modeled by Highest Degrees 10 5.2. Mixed Subdivisions of Newton Polytopes to compute Mixed Vo...

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed, it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or re-run paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

Abstract—Given some geometric elements such as points and lines in Q, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branch-and-prune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one. Index Terms—Branch-and-prune, Cayley–Menger determinant, direct and inverse kinematics, distance constraint, interval method, kinematic and geometric constraint solving, octahedral manipulator. I.

Abstract — This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology, and complete, in the sense that all possible solutions get accurately bounded, irrespectively of whether the analyzed linkage is rigid or mobile. The problem is tackled by formulating a system of linear, parabolic, and hyperbolic equations, which is here solved by a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions at the desired accuracy in short computation times. I.

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In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these pre-specified locations. Denavit and Hartenberg parameters and 4x4 homogeneous matrices are used to formulate the problem and obtain the design equations and an interval method is used to search for design solutions within a predetermined domain. At the time of writing this paper, six design solutions within the search domain and an additional twenty solutions outside the domain have been found. KEYWORDS Geometric Design, Robot Manipulators, Interval Analysis