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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 76 (36 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.
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 27 (9 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.
Solving the kinematics of planar mechanisms by Dixon determinant and a complexplane formulation
 ASME J. Mech. Design
, 2001
"... This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute and slider joints. The method combines the complex plane formulation of Wampler (1999) with the Dixon determinant procedure of Nielsen and Roth (1999). The result is simple t ..."
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Cited by 15 (4 self)
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This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute and slider joints. The method combines the complex plane formulation of Wampler (1999) with the Dixon determinant procedure of Nielsen and Roth (1999). The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equations are addessed. NOMENCLATURE ℓ Number of kinematic loops. θj eiΘj,whereΘjisan angle, in radians. z ∗ Complex conjugate of z. 1
On closedform solutions to the position analysis of Baranov trusses
 Mechanism and Machine Theory 50 (2012) 179
"... The exact position analysis of a planar mechanism reduces to compute the roots of its characteristic polynomial. Obtaining this polynomial usually involves, as a first step, obtaining a system of equations derived from the independent kinematic loops of the mechanism. Although conceptually simple, t ..."
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Cited by 2 (1 self)
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The exact position analysis of a planar mechanism reduces to compute the roots of its characteristic polynomial. Obtaining this polynomial usually involves, as a first step, obtaining a system of equations derived from the independent kinematic loops of the mechanism. Although conceptually simple, the use of kinematic loops for deriving characteristic polynomials leads to complex variable eliminations and, in most cases, trigonometric substitutions. As an alternative, a method based on bilateration has recently been shown to permit obtaining the characteristic polynomials of the threeloop Baranov trusses without relying on variable eliminations or trigonometric substitutions. This paper shows how this technique can be applied to solve the position analysis of all catalogued Baranov trusses. The characteristic polynomials of them all have been derived and, as a result, the maximum number of their assembly modes has been obtained. A comprehensive literature survey is also included.
Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety
, 2012
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Displacement Analysis of Spherical Mechanisms Having Three or Fewer Loops
"... Spherical linkages, having rotational joints whose axes coincide in a common center point, are sometimes used in multidegreeoffreedom robot manipulators and in onedegreeoffreedom mechanisms. The forward kinematics of parallellink robots, the inverse kinematics of seriallink robots and the inp ..."
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Spherical linkages, having rotational joints whose axes coincide in a common center point, are sometimes used in multidegreeoffreedom robot manipulators and in onedegreeoffreedom mechanisms. The forward kinematics of parallellink robots, the inverse kinematics of seriallink robots and the input/output motion of singledegreeoffreedom mechanisms are all problems in displacement analysis. In this article, loop equations are formulated and solved for the displacement analysis of all spherical mechanisms up to three loops. We show how to solve each mechanism type using either a formulation in terms of rotation matrices or quaternions. In either formulation, the solution method is a modification of Sylvester’s elimination method, leading directly to numerical calculation via standard eigenvalue routines. �DOI: 10.1115/1.1637653� 1
Direct Position Analysis of a Large Family of Spherical and Planar Parallel Manipulators with Four Loops
"... Abstract: The direct position analysis (DPA) of a manipulator is the computation of the endeffector poses (positions and orientations) compatible with assigned values of the actuatedjoint variables. Assigning the actuatedjoint variables corresponds to considering the actuated joints locked, wh ..."
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Abstract: The direct position analysis (DPA) of a manipulator is the computation of the endeffector poses (positions and orientations) compatible with assigned values of the actuatedjoint variables. Assigning the actuatedjoint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator onetoone correspond to the assembly modes of the structure that is generated by locking the actuatedjoint variables of that manipulator. Determining the assembly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topology which are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links and four binary links) connected through twelve revolute pairs to form four closed loops. 1
Contents lists available at ScienceDirect Mechanism and Machine Theory
"... journal homepage: www.elsevier.com/locate/mechmt Distancebased position analysis of the three sevenlink Assur ..."
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journal homepage: www.elsevier.com/locate/mechmt Distancebased position analysis of the three sevenlink Assur
Formulating Assur Kinematic Chains as Projective Extensions of Baranov Trusses
"... The real roots of the characteristic polynomial of a planar linkage determine its assembly modes. In this work it is shown how the characteristic polynomial of a Baranov truss derived using a distancebase formulation contains all the necessary and sufficient information for solving the position ana ..."
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The real roots of the characteristic polynomial of a planar linkage determine its assembly modes. In this work it is shown how the characteristic polynomial of a Baranov truss derived using a distancebase formulation contains all the necessary and sufficient information for solving the position analysis of the Assur kinematic chains resulting from replacing some of its revolute joints by slider joints. This is a relevant result because it avoids the casebycase treatment that requires new sets of variable eliminations to obtain the characteristic polynomial of each Assur kinematic chain.
Position analysis of Assur kinematic chains using coupler curves
, 2005
"... Abstract: The configurations of several Assur kinematic chains (AKCs) are analysed with the aid of coupler curves. An analysed linkage is dismantled into two constituent linkages. The intersection points of two coupler curves generated by the constituent linkages are the solutions of the dismantlin ..."
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Abstract: The configurations of several Assur kinematic chains (AKCs) are analysed with the aid of coupler curves. An analysed linkage is dismantled into two constituent linkages. The intersection points of two coupler curves generated by the constituent linkages are the solutions of the dismantling point. All possible configurations of the linkage can then be obtained with the dismantling point being found by solving two coupler curve equations. The coupler curve equations of WattI and StephensonI sixbar are derived with orders and circularities being emphasized. All three AKCs with sevenlink and several AKCs with ninelink and even with 11link are analysed. The maximum number of solutions can also be determined easily on the basis of the orders and circularities of the coupler curves. The chains with prismatic joints included are also considered.