Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh

Original Paper
  • 17 Downloads

Abstract

In this paper, we investigate the behavior of an underactuated mixed-dynamic nonholonomic system, a Chaplygin sleigh, subjected to viscous dissipation and sinusoidal forcing. The viscous dissipation is in the allowable directions of motion and preserves the nonholonomic constraint. The inclusion of such dissipative effects produces limit cycle oscillations in a reduced velocity space. We find analytical approximations to such limit cycles and use these to determine sinusoidal inputs to control the speed of the sleigh. We further show small changes to the sinusoidal input can steer the sleigh to any desired direction. Invariant structures like limit cycles can be expected to be seen in the dynamics of other nonholonomic systems when the effects of viscous dissipation are included. The findings we report here are therefore applicable to a broad class of both terrestrial and aquatic locomotion systems with nonholonomic constraints.

Keywords

First keyword Second keyword More 

Notes

Acknowledgements

This paper is based upon work supported by the National Science Foundation under Grant Number CMMI 1563315.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest to report.

References

  1. 1.
    Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The theorem on the reducing multiplier. Math. Sb. 1, 303–314 (1911)Google Scholar
  2. 2.
    Caratheodory, C.: Der schlitten. J. Appl. Math. Mech. 13, 71–76 (1933)MATHGoogle Scholar
  3. 3.
    Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. AMS, Providence (1972)MATHGoogle Scholar
  4. 4.
    Blochm, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin (2003)CrossRefGoogle Scholar
  5. 5.
    Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn. 13(4), 369–376 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ostrowski, J.: Computing reduced equations for robotic systems with constraints and symmetries. IEEE Trans. Robot. Autom. 15(1), 111–123 (1999)CrossRefGoogle Scholar
  7. 7.
    Osborne, J.M., Zenkov, D.V.: Steering the Chaplygin sleigh by a moving mass. In: Proceedings of the American Control Conference (2005)Google Scholar
  8. 8.
    Kelly, S.D., Fairchild, M.J., Hassing, P.M., Tallapragada, P.: Proportional heading control for planar navigation: the Chaplygin beanie and fishlike robotic swimming. In: Proceedings of the American Control Conference (2012)Google Scholar
  9. 9.
    Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration. Regul. Chaotic Dyn. 22(8), 955–975 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bizyaev, I.A., Borisov, A.V., Kuznetsov, S.P.: Chaplygin sleigh with periodically oscillating internal mass. Europhys. Lett. 119(6), 60008 (2017)CrossRefGoogle Scholar
  11. 11.
    Tallapragada, P., Fedonyuk, V.: Steering a Chaplygin sleigh using periodic impulses. J. Comput. Nonlinear Dyn. 12(5), 054501 (2017)CrossRefGoogle Scholar
  12. 12.
    Borisov, A.V., Mamaev, I.S.: An inhomogeneous Chaplygin sleigh. Regul. Chaotic Dyn. 22(4), 435–447 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: Dynamics of the Chaplygin sleigh on a cylinder. Regul. Chaotic Dyn. 21(1), 136–146 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: The Jacobi integral in nonholonomic mechanics. Regul. Chaotic Dyn. 20(3), 383–400 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kuznetsov, S.P.: Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint. EPL (Europhys. Lett.) 118(1), 10007 (2017)CrossRefGoogle Scholar
  16. 16.
    Fedonyuk, V., Tallapragada, P.: The stick-slip motion of a Chaplygin sleigh witth a piecewise smooth nonholonomic constraint. In: Proceedings of the ASME DSCC (2015)Google Scholar
  17. 17.
    Fedonyuk, V., Tallapragada, P.: Stick-slip motion of the chaplygin sleigh with a piecewise smooth nonholonomic constraint. J. Comput. Nonlinear Dyn. 12, 031021 (2017)CrossRefGoogle Scholar
  18. 18.
    Ostrowski, J.: Reduced equations for nonholonomic mechanical systems with dissipative forces. Rep. Math. Phys. 42(1), 185–209 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dear, T., Kelly, S.D., Travers, M., Choset, H.: Snakeboard motion planning with viscous friction and skidding. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 670–675 (2015)Google Scholar
  20. 20.
    Borisov, A.V., Kuznetsov, S.P.: Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts. Regul. Chaotic Dyn. 21(7–8), 792–803 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Tallapragada, P.: A swimming robot with an internal rotor as a nonholonomic system. In: Proceedings of the American Control Conference (2015)Google Scholar
  22. 22.
    Tallapragada, P., Kelly, S.D.: Integrability of velocity constraints modeling vortex shedding in ideal fluids. J. Comput. Nonlinear Dyn. 12(2), 021008 (2017)CrossRefGoogle Scholar
  23. 23.
    Pollard, B., Tallapragada, P.: An aquatic robot propelled by an internal rotor. IEEE/ASME Trans. Mechatron. 22(2), 931–939 (2016)CrossRefGoogle Scholar
  24. 24.
    Fedonyuk, V., Tallapragada, P., Wang, Y.: Limit cycle analysis and control of the dissipative Chaplygin sleigh. In: ASME Dynamic Systems and Control Conference (2017)Google Scholar
  25. 25.
    Ijspeert, A.J.: Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21(4), 642–653 (2008)CrossRefGoogle Scholar
  26. 26.
    Murray, R., Sastry, S.S.: Steering nonholonomic systems using sinusoids. In: Proceedings of the 29th Conference of Decision and Control (1990)Google Scholar
  27. 27.
    Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992)CrossRefMATHGoogle Scholar
  28. 28.
    Basso, M., Genesio, R., Tesi, A.: A frequency method for predicting limit cycle bifurcations. Nonlinear Dyn. 13(4), 339360 (1997)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ostrowski, J.: Steering for a class of dynamic nonholonomic systems. IEEE Trans. Autom. Control 45, 14921497 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Bullo, F., Lewis, A.D.: Kinematic controllability and motion planning for the snakeboard. IEEE Trans. Robot. Autom. 9(3), 494–498 (2003)CrossRefGoogle Scholar
  31. 31.
    Chakon, O., Or, Y.: Analysis of underactuated dynamic locomotion systems using perturbation expansion: the twistcar toy example. J. Nonlinear Sci. 27, 1215–1234 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringClemson UniversityClemsonUSA

Personalised recommendations