Regular and Chaotic Dynamics

, Volume 17, Issue 2, pp 191–198 | Cite as

Two non-holonomic integrable problems tracing back to Chaplygin

Article

Abstract

The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.

Keywords

non-holonomic constraint integrability invariant measure gyroscope quadrature coupled rigid bodies 

MSC2010 numbers

76M23 34A05 

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References

  1. 1.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 8, no. 2, pp. 170–190.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chaplygin, S.A., On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls, Regul. Chaotic Dyn., 2012, vol. 8, no. 2, pp. 199–217 [Russian original: Mat. Sb., 1897, Vol. 20; reprinted in: Collected Works: Vol. 1, Moscow-Leningrad: Gostekhizdat, 1948, pp. 26–56].CrossRefGoogle Scholar
  3. 3.
    Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Systems and Control Engineering, 2003, vol. 217, pp. 457–467.Google Scholar
  4. 4.
    Bhattacharya, S. and Agrawal, S.K., Design, Experiments and Motion Planning of a Spherical Rolling Robot, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (San Francisco, CA, April 2000), IEEE, 2000, pp. 1207–1212.Google Scholar
  5. 5.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465–483.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Camicia, C., Conticelli, F., and Bicchi, A., Nonholonimic Kinematics and Dynamics of the Sphericle, in Proc. of the 2000 IEEE/RSJ Internat. Conf. on Intelligent Robots and Systems (Takamatsu, Japan, Oct. 31–Nov. 5 2000), IEEE, 2000, pp. 805–810.Google Scholar
  8. 8.
    Chung, W., Nonholonomic Manipulators, Springer Tracts in Advanced Robotics, vol. 13, Berlin: Springer, 2004.Google Scholar
  9. 9.
    Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh, PA, 2006.Google Scholar
  10. 10.
    Goncharenko, I., Svinin, M., and Hosoe, S., Dynamic Model, Haptic Solution, and Human-inspired Motion Planning for Rolling-based Manipulation, J. of Computing and Information Science in Engineering, 2009, vol. 9, no. 1, 011004, 10 pp.CrossRefGoogle Scholar
  11. 11.
    Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Michaud, F. and Caron, S., Roball, the Rolling Robot, Autonomous Robots, 2002, vol. 12, pp. 211–222.MATHCrossRefGoogle Scholar
  13. 13.
    Mukherjee, R., Minor, M.A., and Pukrushpan, J. T., Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-plate Problem, J. Dyn. Syst. Meas. Control, 2002, vol. 124, pp. 502–511.CrossRefGoogle Scholar
  14. 14.
    Wilson, J. L., Mazzoleni, A.P., DeJarnette, F.R., Antol, J., Hajos, G.A., and Strickland, C. V., Design, Analysis, and Testing of Mars Tumbleweed Rover Concepts, J. of Spacecraft and Rockets, 2008, vol. 45, no. 2, pp. 370–382.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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