Regular and Chaotic Dynamics

, Volume 22, Issue 3, pp 239–247 | Cite as

The inertial motion of a roller racer

Article

Abstract

This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.

Keywords

roller racer invariant measure nonholonomic mechanics scattering map 

MSC2010 numbers

37J60 37C10 

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References

  1. 1.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S, On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 752–766.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borisov, A.V. and Mamaev, I. S, Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.CrossRefMATHGoogle Scholar
  3. 3.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A, The Rolling Motion of a Ball on a Sureface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 200–219.Google Scholar
  4. 4.
    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.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chaplygin, S.A, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376; see also: Mat. Sb., 1912 vol. 28, no. 2, pp. 303–314.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Eckhardt, B. and Jung, Ch., Regular and Irregular Potential Scattering, J. Phys. A, 1986, vol. 19, no. 14, L829–L833.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Krishnaprasad, P. S. and Tsakiris, D.P, Oscillations, SE(2)-Snakes and Motion Control: A Study of the Roller Racer, Dyn. Syst., 2001, vol. 16, no. 4, pp. 347–397.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tophøj, L. and Aref, H, Chaotic Scattering of Two Identical Point Vortex Pairs Revisited, Phys. Fluids, 2008, vol. 20, no. 9, 093605, 10 pp.CrossRefMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Kozlov, V.V, On the Existence of an Integral Invariant of a Smooth Dynamic System, J. Appl. Math. Mech., 1987, vol. 51, no. 4, pp. 420–426; see also: Prikl. Mat. Mekh., 1987 vol. 51, no. 4, pp. 538–545.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Martynenko, Yu.G, Motion Control of Mobile Wheeled Robots, J. Math. Sci. (N. Y.), 2007, vol. 147, no. 2, pp. 6569–6606; see also: Fundam. Prikl. Mat., 2005 vol. 11, no. 8, pp. 29–80.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rocard, Y., L’instabilité en mécanique: Automobiles, avions, ponts suspendus, Paris: Masson, 1954.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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