Factorization of rational curves in the study quadric

Hegedüs, Gábor; Schicho, Josef; Schröcker, Hans-Peter

doi:10.1016/j.mechmachtheory.2013.05.010

Cited by 71 publications

(196 citation statements)

References 12 publications

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“…Notations. We use the classical concepts and definitions of dual quaternions and the Study quadric for kinematics computation from [11,9,10,13]. Dual numbers are denoted by D := R + R, with multiplication defined by 2 = 0.…”

Section: Algebraic Modeling Of Kinematicsmentioning

confidence: 99%

Mechanism Singularities Revisited From an Algebraic Viewpoint

Mueller

2019

Volume 5B: 43rd Mechanisms and Robotics Conference

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It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal).Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert's Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.

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Section: Algebraic Modeling Of Kinematicsmentioning

confidence: 99%

Mechanism Singularities Revisited From an Algebraic Viewpoint

Mueller

2019

Volume 5B: 43rd Mechanisms and Robotics Conference

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“…• All linkages with k 14 = k 25 = 1 are known: there is one family [11] with k 36 = 1 and another family [14] with k 36 = 2. Both families are maximal, i.e.…”

Section: Bond Diagramsmentioning

confidence: 99%

A survey on the theory of bonds

Schicho

Schröcker

2016

IMA J Math Control Info

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Many researchers tried to understand/explain the geometric reasons for paradoxical mobility of a mechanical linkage, i.e. the situation when a linkage allows more motions than expected from counting parameters and constraints. Bond theory is a method that aims at understanding paradoxical mobility from an algebraic point of view. Here we give a self-contained introduction of this theory and discuss its results on closed linkages with revolute or prismatic joints.

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“…Rational motions are often represented by homogeneous transformation matrices of dimension four by four whose entries are rational functions. Our study is based on the dual quaternion model of SE (3) where rational motions appear as rational curves on the Study quadric S and are parameterized by certain polynomials with dual quaternion coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…However, exceptions to this relation of degrees do exist. The most famous example is probably the Darboux motion, [1,Chapter 9,§ 3] or [5]. It is represented by a polynomial C of degree three in dual quaternions while its trajectories are of degree two.…”

Section: Introductionmentioning

confidence: 99%

Rational motions with generic trajectories of low degree

Siegele

Scharler

Schröcker

2020

Computer Aided Geometric Design

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The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is lower. An algebraic criterion for this drop of degree is existence of certain right factors, a geometric criterion involves one of two families of rulings on an invariant quadric. Our characterizations allow the systematic construction of rational motions with exceptional degree reduction and explain why the trajectory degrees of a rational motion and its inverse motion can be different.

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Factorization of rational curves in the study quadric

Cited by 71 publications

References 12 publications

Mechanism Singularities Revisited From an Algebraic Viewpoint

Mechanism Singularities Revisited From an Algebraic Viewpoint

A survey on the theory of bonds

Rational motions with generic trajectories of low degree

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