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Computational Algorithm for Determining the Generic Mobility of Floating Planar and Spherical Linkages

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Computational Kinematics

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 15))

Abstract

It is well-known that structural mobility criteria, such as the Chebychev-Kutzbach–Grbler (CKG) formula, fail to correctly determine the mobility of mechanisms with special geometries. Even more, any known structural mobility criteria also fail to determine the generic (i.e. topological) mobility since they are prone to topological redundancies. A computational algorithm is proposed in this paper, which always finds the correct generic mobility of any planar and spherical mechanism. Its foundation is a novel representation of constraints by means of a constraint graph. The algorithm builds on the ‘pebble game’, originally developed within combinatorial rigidity theory for checking the rigidity of graphs. An extension of Laman’s theorem is introduced that enables application of the algorithm to any planar or spherical mechanism with higher and lower holonomic kinematic pairs and multiple joints. The novel algorithm further yields the redundantly constrained sub-linkages of a mechanism. In addition this algorithm naturally leads to a decomposition of a mechanism into Assur graphs, however this is beyond the scope of this paper.

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Authors and Affiliations

  1. Mechanical Engineering School, Tel-Aviv University, Tel-Aviv, Israel

    Offer Shai

  2. Institute of Mechatronics, Chemnitz, Germany

    Andreas Müller

Authors
  1. Offer Shai
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  2. Andreas Müller
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Corresponding author

Correspondence to Andreas Müller .

Editor information

Editors and Affiliations

  1. Llorens Artigas, Inst Robótica e Informática Industrial, Barcelona, Spain

    Federico Thomas

  2. Mechanical Engineering, Idaho State University, Pocatello, Idaho, USA

    Alba Perez Gracia

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Shai, O., Müller, A. (2014). Computational Algorithm for Determining the Generic Mobility of Floating Planar and Spherical Linkages. In: Thomas, F., Perez Gracia, A. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7214-4_22

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  • DOI: https://doi.org/10.1007/978-94-007-7214-4_22

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-7213-7

  • Online ISBN: 978-94-007-7214-4

  • eBook Packages: EngineeringEngineering (R0)

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