Skip to main content
SpringerLink
Log in

On the Hadamard–Hamel problem and the dynamics of wheeled vehicles

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we develop the results obtained by J. Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.

    Article  MathSciNet  MATH  Google Scholar 

  2. Zakalyukin, I.V., Dynamics of a Beam with Two Sleights via Systems of Implicit Differential Equations, Trudy MAI, 2011, no. 42, 25 pp.

  3. Hadamard, J., Sur les mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, 4e série, 1895, vol. 5, pp. 397–417.

    Google Scholar 

  4. Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1–57.

    MATH  Google Scholar 

  5. 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.

    Article  MathSciNet  MATH  Google Scholar 

  6. Borisov, A. V. and Mamaev, I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 553–604.

    Article  MathSciNet  Google Scholar 

  7. Jean, F., The Car with N Trailers: Characterization of the Singular Configurations, ESAIM Control Optim. Calc. Var., 1996, vol. 1, pp. 241–266.

    Article  MathSciNet  MATH  Google Scholar 

  8. Agrachev, A.A. and Sachkov, Yu.L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., vol. 87, Berlin: Springer, 2004.

  9. Kozlov V. V. On the realization of constraints in dynamics, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 594–600; see also: Prikl. Mat. Mekh., 1992, vol. 56, no. 4, pp. 692–698.

    Article  MathSciNet  Google Scholar 

  10. Vierkandt, A., Über gleitende und rollende Bewegung, Monatsh. Math. Phys., 1892, vol. 3, no. 1, pp. 31–38, 97–116.

    Article  MathSciNet  MATH  Google Scholar 

  11. Appell, P., Les mouvements de roulement en dynamique, Évreux: Hérissey, 1899.

    MATH  Google Scholar 

  12. Bloch, A., Nonholonomic Mechanics and Control, New York: Springer, 2003.

    Book  MATH  Google Scholar 

  13. Hamel, G., Theoretische Mechanik: Eine einheitliche Einführung in die gesamte Mechanik, 2nd ed., Berlin: Springer, 1978.

    MATH  Google Scholar 

  14. Ehlers, K.M. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25–30 August 2006), pp. 469–480.

  15. Stückler, B., Über die Differentialgleichungen für die Bewegung eines idealisierten Kraftwagens, Arch. Appl. Mech., 1952, vol. 20, no. 5, pp. 337–356.

    Google Scholar 

  16. Stückler, B., Über die Berechnung der an rollenden Fahrzeugen wirkenden Haftreibungen, Arch. Appl. Mech., 1955, vol. 23, no. 4, pp. 279–287.

    Google Scholar 

  17. Rocard, Y., L’instabilité en mécanique: Automobiles, avions, ponts suspendus, Paris: Masson, 1954.

    Google Scholar 

  18. Bottema, O., Die Bewegung eines einfachen Wagenmodells, Z. Angew. Math. Mech., 1964, vol. 44, no. 12, pp. 585–593.

    Article  MATH  Google Scholar 

  19. Staicu, S., Dynamics Equations of a Mobile Robot Provided with Caster Wheel, Nonlinear Dynam., 2009, vol. 58, no. 1, pp. 237–248.

    Article  MathSciNet  MATH  Google Scholar 

  20. Giergiel, J. and Żylski, W., Description of Motion of a Mobile Robot by Maggie’s Equations, J. Theor. Appl. Mech., 2005, vol. 43, no. 3, pp. 511–521.

    Google Scholar 

  21. 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.

    Article  MathSciNet  Google Scholar 

  22. 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.

    Article  MathSciNet  Google Scholar 

  23. 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.

    Article  MathSciNet  MATH  Google Scholar 

  24. Campion, G., Bastin, G., and d’Andréa-Novel, B., Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots, IEEE Trans. Robot. Autom., 1996, vol. 12, no. 1, pp. 47–62.

    Article  Google Scholar 

  25. 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.

    Article  MathSciNet  MATH  Google Scholar 

  26. Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.

    Article  MathSciNet  MATH  Google Scholar 

  27. 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.

    Article  MathSciNet  MATH  Google Scholar 

  28. Vershik, A. M. and Gershkovich, V.Ya., Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems, in Dynamical Systems: VII. Integrable Systems Nonholonomic Dynamical Systems, V. I. Arnol’d, S.P. Novikov (Eds.), Encyclopaedia Math. Sci., vol. 16, Berlin: Springer, 1994, pp. 1–81.

    Google Scholar 

  29. Arnol’d, V. I., Kozlov, V.V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.

  30. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Geometrisation of Chaplygin’s Reducing Multiplier Theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318.

    Article  MathSciNet  Google Scholar 

  31. Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.

    Article  MathSciNet  MATH  Google Scholar 

  32. Borisov, A. V. and Mamaev, I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 553–604.

    Article  MathSciNet  Google Scholar 

  33. 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.

    Article  Google Scholar 

  34. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.

    Article  MathSciNet  MATH  Google Scholar 

  35. Bizyaev, I. A., Bolsinov, A. V., Borisov, A.V., and Mamaev, I. S., Topology and Bifurcations in Nonholonomic Mechanics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2015, vol. 25, no. 10, 1530028, 21 pp.

    Article  MathSciNet  Google Scholar 

  36. Altafini, C., Some Properties of the General n-Trailer, Internat. J. Control, 2001, vol. 74, no. 4, pp. 409–424.

    Article  MathSciNet  MATH  Google Scholar 

  37. Wagner, A., Heffel, E., Arrieta, A. F., Spelsberg-Korspeter G., Hagedorn P., Analysis of an Oscillatory Painlevé — Klein Apparatus with a Nonholonomic Constraint, Differ. Equ. Dyn. Syst., 2013, vol. 21, nos. 1–2, pp. 149–157.

    Article  MathSciNet  MATH  Google Scholar 

  38. Borisov, A. V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.

    Article  MathSciNet  Google Scholar 

  39. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.

    Article  MathSciNet  Google Scholar 

  40. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 104–116.

    Article  MathSciNet  MATH  Google Scholar 

  41. 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.

    Article  MathSciNet  Google Scholar 

  42. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.

    Article  MathSciNet  MATH  Google Scholar 

  43. 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. 17, no. 2, pp. 170–190.

    Article  MathSciNet  MATH  Google Scholar 

  44. de León, M., A Historical Review on Nonholomic Mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 2012, vol. 106, no. 1, pp. 191–224.

    Article  MATH  Google Scholar 

  45. Ivanov A.P., On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 355–368.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Dolgoprudnyi, 141700, Russia

    Alexey V. Borisov

  2. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexander A. Kilin

  3. Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991, Russia

    Ivan S. Mamaev

Authors
  1. Alexey V. Borisov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander A. Kilin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivan S. Mamaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey V. Borisov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borisov, A.V., Kilin, A.A. & Mamaev, I.S. On the Hadamard–Hamel problem and the dynamics of wheeled vehicles. Regul. Chaot. Dyn. 20, 752–766 (2015). https://doi.org/10.1134/S1560354715060106

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354715060106

MSC2010 numbers

Keywords

Search

Navigation