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Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms

  • G. NawratilEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 3)

Abstract

In a recent publication [10] the author showed that self-motions of general planar Stewart Gough platforms can be classified into two so-called Darboux Mannheim (DM) types (I and II). Moreover, in [10] the author was able to compute the set of equations yielding a type II DM self-motion explicitly. Based on these equations we present a basic result for this class of self-motions.

Keywords

Self-motion Stewart Gough platform Borel Bricard problem 

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.

References

  1. 1.
    Borel, E.: Mémoire sur les déplacements à trajectoires sphériques, Mém. présenteés par divers savants, Paris(2), 33, 1–128 (1908).Google Scholar
  2. 2.
    Borras, J., Thomas, F., Torras, C.: Singularity-invariant leg rearrangements in doubly-planar Stewart-Gough platforms, In Proc. of Robotics Science and Systems, Zaragoza, Spain (2010).Google Scholar
  3. 3.
    Bricard, R.: Mémoire sur les déplacements à trajectoires sphériques, Journ. École Polyt.(2), 11, 1–96 (1906).Google Scholar
  4. 4.
    Husty, M.: E. Borel’s and R. Bricard’s Papers on Displacements with Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators, Int. Symp. on History of Machines and Mechanisms (M. Ceccarelli ed.), 163–172, Kluwer (2000).Google Scholar
  5. 5.
    Husty, M., Mielczarek, S., Hiller, M.: A redundant spatial Stewart-Gough platform with a maximal forward kinematics solution set, Advances in Robot Kinematics: Theory and Applications (J. Lenarcic, F. Thomas eds.), 147–154, Kluwer (2002).Google Scholar
  6. 6.
    Karger, A.: Architecture singular planar parallel manipulators, Mechanism and Machine Theory 38 (11) 1149–1164 (2003).zbMATHCrossRefGoogle Scholar
  7. 7.
    Karger, A.: New Self-Motions of Parallel Manipulators, Advances in Robot Kinematics: Analysis and Design (J. Lenarcic, P. Wenger eds.), 275–282, Springer (2008).Google Scholar
  8. 8.
    Karger, A.: Self-motions of Stewart-Gough platforms, Computer Aided Geometric Design, Special Issue: Classical Techniques for Applied Geometry (B. Jüttler, O. Röschel, E. Zagar eds.) 25 (9) 775–783 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Mielczarek, S., Husty, M.L., Hiller, M.: Designing a redundant Stewart-Gough platform with a maximal forward kinematics solution set, In Proc. of the International Symposion of Multibody Simulation and Mechatronics (MUSME), Mexico City, Mexico, September 2002.Google Scholar
  10. 10.
    Nawratil, G.: Types of self-motions of planar Stewart Gough platforms, under review.Google Scholar
  11. 11.
    Nawratil, G.: Basic result on type II DM self-motions of planar Stewart Gough platforms, Technical Report No. 215, Geometry Preprint Series, TU Vienna (2011).Google Scholar
  12. 12.
    Vogler, H.: Bemerkungen zu einem Satz von W. Blaschke und zur Methode von Borel-Bricard, Grazer Mathematische Berichte 352 1–16 (2008).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and Geometry, Vienna University of TechnologyViennaAustria

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