Line-symmetric motions with respect to reguli

Hamann, Marco

doi:10.1016/j.mechmachtheory.2011.02.004

Cited by 16 publications

(22 citation statements)

References 2 publications

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“…Krames [Kra37c, page 415]), it could also be called involutory motion. In this paper we use the name line-symmetry as it is probably the most commonly used term in today's kinematic community Further characterizations of line-symmetric motions (beside the cited one of Krames [Kra37b]) where given by Tölke [Töl75], Bottema and Roth [BR79, Chapter 9, § 7], Selig and Husty [SH11] and Hamann [Ham11]. If one uses the so-called Study parameters (e 0 : e 1 : e 2 : e 3 : f 0 : f 1 : f 2 : f 3 ) to describe isometries, then it is possible to characterize line-symmetric motions algebraically in the following way.…”

Section: Review On Line-symmetric Motionsmentioning

confidence: 99%

Mobile icosapods

Gallet

Nawratil

Schicho

et al. 2017

Advances in Applied Mathematics

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Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such 20-pods, but could not constrain the legs to have base and platform points with real coordinates. We show that Borel's construction yields all mobile 20-pods, and that it is possible to construct examples with all real coordinates.

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Section: Review On Line-symmetric Motionsmentioning

confidence: 99%

Mobile icosapods

Gallet

Nawratil

Schicho

et al. 2017

Advances in Applied Mathematics

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“…The case n = 2 (interpolation by conics) is even simpler and there is no need to address it here. It has been extensively treated in [1,6,12].…”

Section: Cubic Interpolation On the Study Quadricmentioning

confidence: 99%

“…, and (4, 5) represent valid combinations. Three of these pairs, (1,6), (2,4), and (3, 5), give rise to "general angle symmetric linkages" and six give rise to linkages of type "Waldron's double Bennett hybrid" [7,14].…”

Section: Synthesismentioning

confidence: 99%

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Four-Pose Synthesis of Angle-Symmetric 6R Linkages

Hegedüs

Schicho

Schröcker

2015

Journal of Mechanisms and Robotics

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We use the recently introduced factorization theory of motion polynomials over the dual quaternions for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. Our approach admits either no or a one-parametric family of solutions. We suggest strategies for picking good solutions from this family. IntroductionIn [1], Brunnthaler et al. presented a method for synthesizing closed kinematic chains of four revolute joints whose coupler can visit three prescribed poses (position and orientation). These linkages are also known under the name "Bennett linkage". By that time, it was already known that three poses in general position define a unique Bennett linkage. Other synthesis methods were available [2][3][4][5]. With hindsight, the most important novelty of [1] was the characterization of the coupler motions of Bennett linkages as conics in the Study quadric model of the direct Euclidean displacement group SE(3). Depending on the precise concept of "Bennett linkages", it might be necessary to exclude certain conics, see [6]. This interpretation allows to immediately construct the coupler motion from the three given poses-even before the linkage itself is determined.In this article, we generalize the method of [1] to the four-pose synthesis of special kinematic loops of six revolute joints (6R linkages). Our main tool is the recently developed technique of factoring rational motions [7], or more precisely, motion polynomials. This allows to decompose a rational parametrized equation in Study parameters into the product of linear factors. Each such factorization gives rise to an open kinematic chain. Suitable combinations of open chains produce closed loop linkages whose coupler follows the prescribed rational motion. If we combine this factorization of rational motions with rational interpolation techniques on a quadric, we obtain a framework for mechanism synthesis. It requires the completion of three steps:

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“…The kinematic image of this motion on the study-quadric is a conic [33,37]. Therefore, this motion also serves as a sort of 'primitive' in motion design [38].…”

Section: (D) the Bennett Isogrammentioning

confidence: 99%

On the flexibility and symmetry of overconstrained mechanisms

Stachel

2014

Phil. Trans. R. Soc. A.

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In kinematics, a framework is called overconstrained if its continuous flexibility is caused by particular dimensions; in the generic case, a framework of this type is rigid. Famous examples of overconstrained structures are the Bricard octahedra, the Bennett isogram, the Grünbaum framework, Bottema's 16-bar mechanism, Chasles’ body–bar framework, Burmester's focal mechanism or flexible quad meshes. The aim of this paper is to present some examples in detail and to focus on their symmetry properties. It turns out that only for a few is a global symmetry a necessary condition for flexibility. Sometimes, there is a hidden symmetry, and in some cases, for example, at the flexible type-3 octahedra or at discrete Voss surfaces, there is only a local symmetry. However, there remain overconstrained frameworks where the underlying algebraic conditions for flexibility have no relation to symmetry at all.

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Line-symmetric motions with respect to reguli

Cited by 16 publications

References 2 publications

Mobile icosapods

Mobile icosapods

Four-Pose Synthesis of Angle-Symmetric 6R Linkages

On the flexibility and symmetry of overconstrained mechanisms

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