Advertisement

Regular and Chaotic Dynamics

, Volume 24, Issue 1, pp 114–126 | Cite as

The Dynamics of a Chaplygin Sleigh with an Elastic Internal Rotor

  • Vitaliy FedonyukEmail author
  • Phanindra Tallapragada
Article
  • 10 Downloads

Abstract

In this paper the dynamics of a Chaplygin sleigh like system are investigated. The system consists a of a Chaplygin sleigh with an internal rotor connected by a torsional spring, which is possibly non-Hookean. The problem is motivated by applications in robotics, where the motion of a nonholonomic system is sought to be controlled by modifying or tuning the stiffness associated with some degrees of freedom of the system. The elastic potential modifies the dynamics of the system and produces two possible stable paths in the plane, a straight line and a circle, each of which corresponds to fixed points in a reduced velocity space. Two different elastic potentials are considered in this paper, a quadratic potential and a Duffing like quartic potential. The stiffness of the elastic element, the relative inertia of the main body and the internal rotor and the initial energy of the system are all bifurcation parameters. Through numerics, we investigate the codimension-one bifurcations of the fixed points while holding all the other bifurcation parameters fixed. The results show the possibility of controlling the dynamics of the sleigh and executing different maneuvers by tuning the stiffness of the spring.

Keywords

nonholonomic systems Chaplygin sleigh passive degrees of freedom 

MSC2010 numbers:

37J60 70E55 70K50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Joshi, V. A. and Banavar, R.N., Motion Analysis of a Spherical Mobile Robot, Robotica, 2009, vol. 27, no. 3, pp. 343–353.CrossRefGoogle Scholar
  2. 2.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hubbard, M., Lateral Dynamics and Stability of the Skateboard, J. Appl. Mech., 1979, vol. 46, no. 4, pp. 931–936.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ostrowski, J., Lewis, A., Murray, R., and Burdick, J., Nonholonomic Mechanics and Locomotion: The Snakeboard Example, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (San Diego, Calif., May 1994), pp. 2391–2397.Google Scholar
  6. 6.
    Bloch, A. M., Krishnaprasad, P. S., Marsden, J.E., and Murray, R. M., Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal., 1996, vol. 136, no. 1, pp. 21–99.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Martynenko, Yu.G., The Theory of the Generalized Magnus Effect for Non-Holonomic Mechanical Systems, J. Appl. Math. Mech., 2004, vol. 68, no. 6, pp. 847–855; see also: Prikl. Mat. Mekh., 2004, vol. 68, no. 6, pp. 948–957.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kremnev, A. V. and Kuleshov, A. S., Nonlinear Dynamics and Stability of the Skateboard, Discrete Contin. Dyn. Syst. Ser. S, 2010, vol. 3, no. 1, pp. 85–103.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V., Mamaev, I. S., Kilin, A.A., and Bizyaev, I.A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 739–751.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bizyaev, I.A., The Inertial Motion of a Roller Racer, Regul. Chaotic Dyn., 2017, vol. 22, no. 3, pp. 239–247.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Murray, R. M. and Sastry, S. Sh., Nonholonomic Motion Planning: Steering Using Sinusoids, IEEE Trans. Automat. Control, 1993, vol. 38, no. 5, pp. 700–716.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tilbury, D., Murray, R. M., and Sastry, S. Sh., Trajectory Generation for the n-Trailer Problem Using Goursat Normal Form, IEEE Trans. Automat. Control, 1995, vol. 40, no. 5, pp. 802–819.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bravo-Doddoli, A. and García-Naranjo, L.C., The Dynamics of an Articulated n-Trailer Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 497–517.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence, R.I.: AMS, 1972.Google Scholar
  15. 15.
    Arnol’d, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Dynamical Systems 3, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1988.Google Scholar
  16. 16.
    Zenkov, D.V., Bloch, A. M., and Marsden, J.E., The Lyapunov–Malkin Theorem and Stabilization of the Unicycle with Rider, Systems Control Lett., 2002, vol. 45, no. 4, pp. 293–302.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Borisov, A.V. and Mamaev, I. S., The Dynamics of a Chaplygin Sleigh, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 156–161; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 2, pp. 219–225.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dickinson, M.H., Lehmann, F.-O., and Sane, S.P., Wing Rotation and the Aerodynamic Basis of Insect Flight, Science, 1999, vol. 284, no. 5422, pp. 1954–1960.CrossRefGoogle Scholar
  19. 19.
    Whitney, J.P. and Wood, R. J., Aeromechanics of Passive Rotation in Flapping Flight, J. Fluid Mech., 2010, vol. 660, pp. 197–220.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Beal, D.N., Hover, F. S., Triantafyllou, M. S., Liao, J. C., and Lauder, G. V., Passive Propulsion in Vortex Wakes, J. Fluid Mech., 2006, vol. 549, pp. 385–402.CrossRefGoogle Scholar
  21. 21.
    Fish, F.E. and Lauder, G.V., Passive and Active Flow Control by Swimming Fishes and Mammals, Annu. Rev. Fluid Mech., 2006, vol. 38, pp. 193–224.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pollard, B. and Tallapragada, P., Passive Appendages Improve the Maneuverability of Fish-Like Robots, IEEE/ASME Trans. Mechatronics, 2018, accepted.Google Scholar
  23. 23.
    Tallapragada, P., A Swimming Robot with an Internal Rotor As a Nonholonomic System, in Proc. of the American Control Conf. (ACC, Chicago, Ill., July 2015), pp. 657–662.Google Scholar
  24. 24.
    Tallapragada, P. and Kelly, S.D., Integrability of Velocity Constraints Modeling Vortex Shedding in Ideal Fluids, J. Comput. Nonlinear Dynam., 2016, vol. 12, no. 2, 021008, 7 pp.CrossRefGoogle Scholar
  25. 25.
    Pollard, B., Fedonyuk, V., and Tallapragada, P., Limit Cycle Behavior and Model Reduction of an Oscillating Fish-Like Robot, in Proc. of the ASME Dynamic Systems and Control Conference (Atlanta,Ga., Sept 30–Oct 3, 2018): Vol. 1, Paper No. DSCC2018-9016, pp. V001T04A006, 7 pp.Google Scholar
  26. 26.
    Fedonyuk, V. and Tallapragada, P., Chaotic Dynamics of the Chaplygin Sleigh with a Passive Internal Rotor, Nonlinear Dynam., 2019, vol. 95, no. 1, pp. 309–320.CrossRefGoogle Scholar
  27. 27.
    Fedonyuk, V. and Tallapragada, P., The Stick-Slip Motion of a Chaplygin Sleigh with a Piecewise Smooth Nonholonomic Constraint, J. Comput. Nonlinear Dynam., 2017, vol. 12, no. 3, 031021, 8 pp.CrossRefGoogle Scholar
  28. 28.
    Tallapragada, P. and Fedonyuk, V., Steering a Chaplygin Sleigh Using Periodic Impulses, J. Comput. Nonlinear Dynam., 2017, vol. 12, no. 5, 054501, 5 pp.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringClemson UniversityClemsonUSA

Personalised recommendations