Dynamics of a Chaplygin sleigh with an unbalanced rotor: regular and chaotic motions

  • Ivan A. Bizyaev
  • Alexey V. BorisovEmail author
  • Ivan S. Mamaev
Original paper


This paper addresses the problem of the motion of a sleigh with a free rotor. It is shown that this system exhibits chaotic and regular motions. The case in which the system is balanced relative to the knife edge is of particular interest because it has an additional integral. In this case, the problem reduces to investigating a vector field on a torus and to classifying singular points on it.


Chaplygin sleigh Unbalanced rotor Nonholonomic mechanics Strange attractor Regular and chaotic trajectories Invariant measure Integrable systems system of two bodies 



The work of I. A. Bizyaev (Section 2) was supported by the RNF under Grant No. 18-71-00110. The work of I. S. Mamaev (Section 3) was supported by the RFBR Grant No. 18-29-10051_mk and was carried out at MIPT under project 5-100 for state support for leading universities of the Russian Federation. The work of A. V. Borisov (Section 1) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2404.2017/4.6) and was supported by the RFBR Grant No. 18-08- 00999_a.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn. 13(4), 369–376 (2008);see also:Mat. Sb. 28(2), 303–314 (1912)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borisov, A.V., Kilin, A.A., Mamaev, I.S.: On the Hadamard–Hamel problem and the dynamics of wheeled vehicles. Regul. Chaotic Dyn. 20(6), 752–766 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bloch, A.: Nonholonomic Mechanics and Control. Springer, New York (2003)CrossRefGoogle Scholar
  4. 4.
    Borisov, A.V., Mamaev, I.S.: The dynamics of a Chaplygin sleigh. J. Appl. Math. Mech. 73(2), 156–161 (2009);see also:Prikl. Mat. Mekh. 73(2), 219–225 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kozlov, V.V.: The phenomenon of reversal in the Euler–Poincare–Suslov nonholonomic systems. J. Dyn. Control Syst. 22(4), 713–724 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carathéodory, C.: Der Schlitten. Z. Angew. Math. Mech. 13(2), 71–76 (1933)CrossRefGoogle Scholar
  7. 7.
    Laumond, J.P., Jacobs, P.E., Taix, M., Murray, R.M.: A motion planner for nonholonomic mobile robots. IEEE Trans. Robot. Autom. 10(5), 577–593 (1994)CrossRefGoogle Scholar
  8. 8.
    Krishnaprasad, P.S., Tsakiris, D.P.: Oscillations, SE(2)-snakes and motion control: a study of the Roller Racer. Dyn. Syst. 16(4), 347–397 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hirose, S.: Biologically Inspired Robots: Snake-Like Locomotors and Manipulators, vol. 1093. Oxford University Press, Oxford (1993)Google Scholar
  10. 10.
    Borisov, A.V., Kilin, A.A., Mamaev, I.S.: Invariant submanifolds of genus 5 and a Cantor staircase in the nonholonomic model of a snakeboard. Int. J. Bifurc. Chaos. 29(3), 1930008 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bizyaev, I.A.: The inertial motion of a roller racer. Regul. Chaotic Dyn. 22(3), 239–247 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Martynenko, Y.G.: Motion control of mobile wheeled robots. J. Math. Sci. 147(2), 6569–6606 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: Exotic dynamics of nonholonomic roller racer with periodic control. Regul. Chaotic Dyn. 23(7–8), 983–994 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Bizyaev, I.A., Borisov, A.V., Kozlov, V.V., Mamaev, I.S.: Fermi-like acceleration and power-law energy growth in nonholonomic systems. Nonlinearity 32, 3209–3233 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bravo-Doddoli, A., Garcia-Naranjo, L.C.: The dynamics of an articulated \(n\)-trailer vehicle. Regul. Chaotic Dyn. 20(5), 497–517 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bizyaev, I.A., Borisov, A.V., Kuznetsov, S.P.: The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass. Nonlinear Dyn. 95(1), 699–714 (2019)CrossRefGoogle Scholar
  18. 18.
    Bizyaev, I.A., Borisov, A.V., Kuznetsov, S.P.: Chaplygin sleigh with periodically oscillating internal mass. EPL. 119(6), 60008 (2017)CrossRefGoogle Scholar
  19. 19.
    Fedonyuk, V., Tallapragada, P.: Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh. Nonlinear Dyn. 93(2), 835–846 (2018)CrossRefGoogle Scholar
  20. 20.
    Fedonyuk, V., Tallapragada, P.: Chaotic dynamics of the Chaplygin sleigh with a passive internal rotor. Nonlinear Dyn. 95(1), 309–320 (2019)CrossRefGoogle Scholar
  21. 21.
    Maciejewski, A.J., Przybylska, M.: Dynamics of constrained many body problems in constant curvature two-dimensional manifolds. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 376(2131), 20170425 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mokhamed, E.A., Smolnikov, B.A.: Free motion of a hinged two-body system. Izv. Akad. Nauk. Mekh. Tverd. Tela. 5, 28–33 (1987). (Russian)Google Scholar
  23. 23.
    Fedonyuk, V., Tallapragada, P.: The Dynamics of a Chaplygin Sleigh with an Elastic Internal Rotor. Regul. Chaotic Dyn. 24(1), 114–126 (2019) see also:Prikl. Mat. Mekh. 73(2), 219–225 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Bloch, A.M., Marsden, J.E., Zenkov, D.V.: Quasivelocities and symmetries in non-holonomic systems. Dyn. Syst. 24(2), 187–222 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Borisov, A.V., Mamaev, I.S.: Symmetries and reduction in nonholonomic mechanics. Regul. Chaotic Dyn. 20(5), 553–604 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bolsinov, A.V., Borisov, A.V., Mamaev, I.S.: Topology and stability of integrable systems. Russ. Math. Surv. 65(2), 259–318 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kozlov, V.V.: On the existence of an integral invariant of a smooth dynamic system. J. Appl. Math. Mech. 51(4), 420–426 (1987);see also:Prikl. Mat. Mekh. 51(4), 538–545 (1987)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: An invariant measure and the probability of a fall in the problem of an inhomogeneous disk rolling on a plane. Regul. Chaotic Dyn. 23(6), 665–684 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, New York (2013)Google Scholar
  30. 30.
    Arnol’d, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., 3rd edn., vol. 3. Springer, Berlin (2006)Google Scholar
  31. 31.
    Moshchuk, N.K.: Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form. J. Appl. Math. Mech. 51(2), 172–177 (1987);see also:Prikl. Mat. Mekh. 51(2), 223–229 (1987)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kaplan, H., Yorke, J.A.: Lecture Note in Mathematics. Springer, Berlin (1979)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Center for Technologies in Robotics and Mechatronics ComponentsInnopolis UniversityInnopolisRussia
  3. 3.Udmurt State UniversityIzhevskRussia
  4. 4.Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN)MoscowRussia

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