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27
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... 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 decomposi ..."
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Cited by 56 (26 self)
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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 byproducts, 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.
Matrices in Elimination Theory
, 1997
"... The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in ..."
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Cited by 45 (17 self)
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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...
A SubdivisionBased Algorithm for the Sparse Resultant
 J. ACM
, 1999
"... Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. ..."
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Cited by 33 (7 self)
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Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra.
Advances in Polynomial Continuation for Solving Problems in Kinematics
, 2004
"... 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 m ..."
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Cited by 16 (8 self)
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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 higherdimensional 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.
Resultant Over the Residual of a Complete Intersection
, 2001
"... In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula fo ..."
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Cited by 10 (4 self)
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In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F : G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples.
Systems of Bilinear Equations
"... How hard is it to solve a system of bilinear equations? No solutions are presented in this report, but the problem is posed and some preliminary remarks are made. In particular, solving a system of bilinear equations is reduced by a suitable transformation of its columns to solving a homogeneous sys ..."
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Cited by 6 (0 self)
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How hard is it to solve a system of bilinear equations? No solutions are presented in this report, but the problem is posed and some preliminary remarks are made. In particular, solving a system of bilinear equations is reduced by a suitable transformation of its columns to solving a homogeneous system of bilinear equations. In turn, the latter has a nontrivial solution if and only if there exist two invertible matrices that, when applied to the tensor of the coefficients of the system, zero its first column. Matlab code is given to manipulate threedimensional tensors, including a procedure that finds one solution to a bilinear system often, but not always. Contents 1 Introduction 3 2 Bilinear Systems with m = 1 6 2.1 Homogeneous Systems with m = n = 1 : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Homogeneous Systems with m = 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.3 NonHomogeneous Systems with m = 1 : : : : : : : : : : : : : : : : : : : : : : ...
MARS: A Maple/Matlab/C Resultantbased Solver
, 1998
"... The problem of computing zeros of a system of polynomial equations has been well studied in the computational literature. Anumber of algorithms have been proposed and many computer algebra and public domain packages provide the capability of computing the roots of polynomial equations. Most of these ..."
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Cited by 5 (0 self)
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The problem of computing zeros of a system of polynomial equations has been well studied in the computational literature. Anumber of algorithms have been proposed and many computer algebra and public domain packages provide the capability of computing the roots of polynomial equations. Most of these implementations are based on Grobner bases which can be slow for even small problems. In this paper, we present a new system, MARS, to compute the roots of a zero dimensional polynomial system. It is based on computing the resultant of a system of polynomial equations followed by eigendecomposition of a generalized companion matrix. MARS includes a robust library of Maple functions for constructing resultant matrices, an e cient library of Matlab routines for numerically solving the eigenproblem, and C code generation routines and a C library for incorporating the numerical solver into applications. We illustrate the usage of MARS on various examples and utilize di erent resultant formulations. 1
Five Precision Points Synthesis Of Spatial Rrr Manipulators Using Interval Analysis
 ASME J. Mech. Des
, 2002
"... In this paper, the geometric design problem of seriallink 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 ..."
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Cited by 5 (1 self)
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In this paper, the geometric design problem of seriallink 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 3R manipulator are computed so that the manipulator will be able to place its endeffector at these prespecified 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
A Three Degree Of Freedom Parallel Manipulator With Only Translational Degrees Of Freedom
, 1997
"... In this dissertation, a novel parallel manipulator is investigated. The manipulator has three degrees of freedom and the moving platform is constrained to only translational motion. The main advantages of this parallel manipulator are that all of the actuators can be attached directly to the base, c ..."
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Cited by 4 (0 self)
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In this dissertation, a novel parallel manipulator is investigated. The manipulator has three degrees of freedom and the moving platform is constrained to only translational motion. The main advantages of this parallel manipulator are that all of the actuators can be attached directly to the base, closedform solutions are available for the forward kinematics, the moving platform maintains the same orientation throughout the entire workspace, and it can be constructed with only revolute joints.