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

  • Offer Shai
  • Andreas MüllerEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 15)


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.


Mobility  Topological redundancy  Pebble game  Assur graphs  


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Mechanical Engineering SchoolTel-Aviv UniversityTel-AvivIsrael
  2. 2.Institute of MechatronicsChemnitzGermany

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