The theory of bonds: A new method for the analysis of linkages

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

doi:10.1016/j.mechmachtheory.2013.08.004

Cited by 34 publications

(60 citation statements)

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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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“…We note that there is another approach for computing exceptional sets for linkages via the theory of bonds, e.g., see [1,15,16]. The main difference is that this combinatorial technique is applicable to linkages while the fiber product [25] and our new approach are general purpose methods.…”

Section: Fn(x; P)mentioning

confidence: 99%

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Hauenstein

Liddell

2015

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation

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Exceptional sets are the sets where the dimension of the fiber of a map is larger than the generic fiber dimension, which we assume is zero. Such situations naturally arise in kinematics, for example, when designing a mechanism that moves when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products to promote such sets to become irreducible components. We propose an alternative approach using rank constraints on Macaulay matrices. Symbolic computations are used to construct the proper Macaulay matrices, while numerical computations are used to solve the rank-constraint problem. Various exceptional sets are computed, including exceptional RR dyads, lines on surfaces in C 3 , and exceptional planar pentads.

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“…Using computer algebra, a Gröbner basis for this set of equations can be computed. 2 The degree of X R can also be computed using computer algebra by the leading monomials of the Gröbner basis: it is 40.…”

Section: A New Compactificationmentioning

confidence: 99%

Bond theory for pentapods and hexapods

2014

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This paper deals with the old and classical problem of determining necessary conditions for the overconstrained mobility of some mechanical device. In particular, we show that the mobility of pentapods/hexapods implies either a collinearity condition on the anchor points, or a geometric condition on the normal projections of base and platform points. The method is based on a specific compactification of the group of direct isometries of $\mathbb{R}^3$.Comment: 17 pages, 1 figur

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The theory of bonds: A new method for the analysis of linkages

Cited by 34 publications

References 10 publications

Mechanism Singularities Revisited From an Algebraic Viewpoint

Mechanism Singularities Revisited From an Algebraic Viewpoint

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Bond theory for pentapods and hexapods

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