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Symmetry and invariants of kinematic chains and parallel manipulators

Published online by Cambridge University Press:  21 March 2012

Roberto Simoni ,
Celso Melchiades Doria  and
Daniel Martins
Roberto Simoni*
Affiliation:
Centro de Engenharia da Mobilidade – Campus Joinville, Universidade Federal de Santa Catarina – 88040-900 – Florianópolis, Santa Catarina, Brasil
Celso Melchiades Doria
Affiliation:
Departamento de Matemática – Campus Trindade, Universidade Federal de Santa Catarina – 88040-900 – Florianópolis, Santa Catarina, Brasil
Daniel Martins
Affiliation:
Departamento de Engenharia Mecânica – Campus Trindade, Universidade Federal de Santa Catarina – 88040-900 – Florianópolis, Santa Catarina, Brasil
*
*Corresponding author. E-mail: [email protected]
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Summary

This paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis of mechanisms and parallel manipulators in early stages of mechanisms design.

Keywords

Kinematic chainParallel manipulatorsGraph symmetryAutomorphism groupActionsOrbits
Type
Articles
Information
Robotica , Volume 31 , Issue 1 , January 2013 , pp. 61 - 70
Copyright
Copyright © Cambridge University Press 2012

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