Meccanica

, Volume 48, Issue 5, pp 1177–1190 | Cite as

Types of self-motions of planar Stewart Gough platforms

Article

Abstract

We show that the self-motions of general planar Stewart Gough platforms can be characterized in the complex extension of the Euclidean 3-space by the movement of three platform points in planes orthogonal to the planar base (3-point Darboux motion) and a simultaneous sliding of three planes orthogonal to the planar platform through points of the base (3-plane Mannheim motion). Based on this consideration, we prove that all one-parametric self-motions of a general planar Stewart Gough platform can be classified into two types (type I DM and type II DM, where DM abbreviates Darboux Mannheim). We also succeed in presenting a set of 24 equations yielding a type II DM self-motion that can be computed explicitly and that is of great simplicity seen in the context of self-motions. These 24 conditions are the key for the complete classification of general planar Stewart Gough platforms with type II DM self-motions, which is an important step in solving the famous Borel Bricard problem.

Keywords

Self-motion Stewart Gough platform Borel Bricard problem Darboux motion Mannheim motion 

Notes

Acknowledgements

This research is supported by Grant No. I 408-N13 of the Austrian Science Fund FWF within the project “Flexible polyhedra and frameworks in different spaces”, an international cooperation between FWF and RFBR, the Russian Foundation for Basic Research.

Moreover, the author would like to thank the reviewers for their useful comments and suggestions which have helped to improve the quality of the paper.

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

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