Caishan Liu
Publications:
Xiong J., Jia Y., Liu C.
Symmetry and Relative Equilibria of a Bicycle System
2021, Vol. 17, no. 4, pp. 391411
Abstract
In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying
the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients
of these equations appeared to depend on the lean and steer angles only, and in one such
equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would
always vanish. These properties indeed arise from the symmetry of the bicycle system. From the
point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal
fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian
and constraint distribution invariant. We analyze the dimension relationship between the space
of admissible velocities and the tangent space to the group orbit, and then employ the reduced
nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties
of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local
representative of the reduced dynamic system on the reduced constraint space, whose relative
equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full
rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several
families of oneparameter solutions. Finally, we prove that these relative equilibria are Lyapunov
(but not asymptotically) stable under certain conditions. However, an isolated asymptotically
stable equilibrium may be achieved by restricting the system to an invariant manifold, which is
the level set of the reduced constrained energy.
