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


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.


Non-holonomic systems Principal fiber bundle Bicycle dynamics Reduction 

Mathematics Subject Classification

37J60 70F25 70G45 

Supplementary material

Supplementary material 1 (mp4 26277 KB)


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Authors and Affiliations

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

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