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Journal of Nonlinear Science

, Volume 28, Issue 3, pp 943–983 | Cite as

Reduced Dynamics of the Non-holonomic Whipple Bicycle

  • Frédéric Boyer
  • Mathieu Porez
  • Johan MaunyEmail author
Article

Abstract

Though the bicycle is a familiar object of everyday life, modeling its full nonlinear three-dimensional dynamics in a closed symbolic form is a difficult issue for classical mechanics. In this article, we address this issue without resorting to the usual simplifications on the bicycle kinematics nor its dynamics. To derive this model, we use a general reduction-based approach in the principal fiber bundle of configurations of the three-dimensional bicycle. This includes a geometrically exact model of the contacts between the wheels and the ground, the explicit calculation of the kernel of constraints, along with the dynamics of the system free of any external forces, and its projection onto the kernel of admissible velocities. The approach takes benefits of the intrinsic formulation of geometric mechanics. Along the path toward the final equations, we show that the exact model of the bicycle dynamics requires to cope with a set of non-symmetric constraints with respect to the structural group of its configuration fiber bundle. The final reduced dynamics are simulated on several examples representative of the bicycle. As expected the constraints imposed by the ground contacts, as well as the energy conservation, are satisfied, while the dynamics can be numerically integrated in real time.

Keywords

Non-holonomic systems Principal fiber bundle Bicycle dynamics Reduction 

Mathematics Subject Classification

37J60 70F25 70G45 

Supplementary material

Supplementary material 1 (mp4 26277 KB)

References

  1. Appell, P.: Traité de mécanique rationnelle. Gauthier-Villars et Cie, Paris (1931)zbMATHGoogle Scholar
  2. Astrom, K.J., Klein, R.E., Lennartsson, A.: Bicycle dynamics and control: adapted bicycles for education and research. IEEE Control Syst. 25(4), 26–47 (2005)MathSciNetCrossRefGoogle Scholar
  3. Aström, K.J., Murray, R.M.: Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, Cambridge (2010)Google Scholar
  4. Basu-Mandal, P., Chatterjee, A., Papadopoulos, J.: Hands-free circular motions of a benchmark bicycle. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 463(2084), 1983–2003 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bloch, A.M.: Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, vol. 24. Springer, New York (2015)CrossRefGoogle Scholar
  6. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136(1), 21–99 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bourlet, C.: Étude théorique sur la bicyclette. Bull. Soc. Math. Fr. 27, 76–96 (1899)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Boussinesq, J.: Aperçu sur la théorie de la bicyclette. J. Math. Pures Appl. 5, 117–136 (1899)zbMATHGoogle Scholar
  9. Boyer, F., Belkhiri, A.: Reduced locomotion dynamics with passive internal dofs: application to nonholonomic and soft robotics. IEEE Trans. Rob. 30(3), 578–592 (2014)CrossRefGoogle Scholar
  10. Boyer, F., Belkhiri, A.: Erratum to “Reduced locomotion dynamics with passive internal dofs: application to nonholonomic and soft robotics” [Jun 14 578–592]. IEEE Trans. Rob. 31(3), 805–805 (2015)CrossRefGoogle Scholar
  11. Boyer, F., Porez, M.: Multibody system dynamics for bio-inspired locomotion: from geometric structures to computational aspects. Bioinspir. Biomim. 10(2), 1–21 (2015)CrossRefGoogle Scholar
  12. Boyer, F., Primault, D.: The Poincaré–Chetayev equations and flexible multibody systems. J. Appl. Math. Mech. 69(6), 925–942 (2005). http://hal.archives-ouvertes.fr/hal-00672477
  13. Campion, G., Bastin, G., D’Andréa-Novel, B.: Structural properties and classification of kinematic and dynamic models of wheeled mobile robots. IEEE Trans. Robot. Autom. 12(1), 47–62 (1996)CrossRefGoogle Scholar
  14. Carvallo, E.: Théorie du movement du monocycle part 2: Théorie de la bicyclette. J. l’École Polytech. 6, 1–118 (1901)Google Scholar
  15. Cendra, H., Marsden, J.E., Ratiu, T.S.: Geometric mechanics, lagrangian reduction, and nonholonomic systems. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited—2001 and Beyond, pp. 221–273. Springer, Beriln (2001)Google Scholar
  16. Chaplygin, S.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn. 13(4), 369–376 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Chitta, S., Cheng, P., Frazzoli, E., Kumar, V.: Robotrikke: a novel undulatory locomotion system. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation (ICRA), pp. 1597–1602 (2005)Google Scholar
  18. Consolini, L., Maggiore, M.: Control of a bicycle using virtual holonomic constraints. Automatica 49(9), 2831–2839 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Featherstone, R.: Rigid Body Dynamics Algorithms. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  20. Franke, G., Suhr, W., Riess, F.: An advanced model of bicycle dynamics. Eur. J. Phys. 11(2), 116–121 (1990)MathSciNetCrossRefGoogle Scholar
  21. Getz, N.H., Marsden, J.E.: Control for an autonomous bicycle. In: Proceedings of 1995 IEEE International Conference on Robotics and Automation, vol. 2, pp. 1397–1402 (1995)Google Scholar
  22. Hertz, H.: Die Prinzipen der Mechanik in neuem Zusammenhange dargestellt. Gesamelte Werke, Band III. Leipzig (1894)Google Scholar
  23. Jones, A.T.: Physics and bicycles. Am. J. Phys. 10(6), 332–333 (1942)CrossRefGoogle Scholar
  24. Kelly, S.D., Murray, R.M.: Geometric phases and robotic locomotion. J. Robot. Syst. 12(6), 417–431 (1995)CrossRefzbMATHGoogle Scholar
  25. Klein, F., Sommerfeld, A.: Über die theorie des kreisels. Über die Theorie des Kreisels, by Klein, Felix; Sommerfeld, Arnold. New York: Johnson Reprint Corp., 1965. Bibliotheca mathematica Teubneriana; Bd. 1 4 (1965)Google Scholar
  26. Kooijman, J.D.G., Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L.: A bicycle can be self-stable without gyroscopic or caster effects. Science 332(6027), 339–342 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Le Hénaff, Y.: Dynamical stability of the bicycle. Eur. J. Phys. 8(3), 207–210 (1987)CrossRefGoogle Scholar
  28. Letov, A.: Stability of an automatically controlled bicycle moving on a horizontal plane. J. Appl. Math. Mech. 23(4), 934–942 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Lobry, C., Sari, T.: Singular perturbation methods in control theory. Contrle Non Linaire et Applications. Cours du CIMPA, Collection Travaux en Cours, pp. 155–182. Hermann, Paris (2005)Google Scholar
  30. Meijaard, J., Papadopoulos, J.M., Ruina, A., Schwab, A.: Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 463(2084), 1955–1982 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Morin, P., Samson, C.: Control of nonholonomic mobile robots based on the transverse function approach. IEEE Trans. Robot. 25(5), 1058–1073 (2009)CrossRefGoogle Scholar
  32. Ostrowski, J., Burdick, J.: The geometric mechanics of undulatory robotic locomotion. Int. J. Robot. Res. (IJRR) 17(7), 683–701 (1998)CrossRefGoogle Scholar
  33. Ostrowski, J., Burdick, J., Lewis, A.D., Murray, R.M.: The mechanics of undulatory locomotion: the mixed kinematic and dynamic case. In: Proceedings of 1995 IEEE International Conference on Robotics and Automation (ICRA), vol. 2, pp. 1945–1951 (1995)Google Scholar
  34. Ostrowski, J., Lewis, A., Murray, R., Burdick, J.: Nonholonomic mechanics and locomotion: the snakeboard example. In: Proceedings of the 1994 IEEE International Conference on Robotics and Automation (ICRA), vol. 3, pp. 2391–2397 (1994)Google Scholar
  35. Ostrowski, J.P.: Computing reduced equations for robotic systems with constraints and symmetries. IEEE Trans. Robot. Autom. 15(1), 111–123 (1999)CrossRefGoogle Scholar
  36. Psiaki, M.: Bicycle stability: a mathematical and numerical analysis. Undergradute thesis. Physics Dept., Princeton University, Princeton (1979)Google Scholar
  37. Rankine, W.J.M.: On the dynamical principles of the motion of velocipedes. The Engineer 28(79), 129 (1869)Google Scholar
  38. Timoshenko, S.P., Young, D.H.: Advanced Dynamics. McGraw-Hill, London (1948)zbMATHGoogle Scholar
  39. Whipple, F.J.: The stability of the motion of a bicycle. Quarterly Journal of Pure and Applied Mathematics 30(120), 312–348 (1899)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.IMT Atlantique Bretagne-Pays de la Loire, Campus de NantesNantes cedex 3France

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