Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Finding the relative equilibria and analyzing their stabilities are of great significance to revealing the intrinsic properties of mechanical systems and developing effective controller. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a revolution surface. We note that the symmetry group of the bicycle is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints to the bicycle system. Therefore, we can classify the bicycle dynamics as a general Voronets system whose Lagrangian and constraint distribution are kept invariant under the action of the symmetry group. Applying the Voronets equations to the bicycle dynamics, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. This system takes time-reversal and lateral symmetries, and has two kinds of relative equilibria: the static equilibria and the dynamic equilibria. Further theoretical analysis shows that both kinds of relative equilibria form one-parameter solution families, and their Jacobian matrices take some specific properties. We then show that a static equilibrium cannot be stable unless all the eigenvalues of the Jacobian matrix are located at the imaginary axis of the complex plane. The stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold, which is established based on the conservation of energy of the system. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, we employ symbolic computation to carry out numerical simulations in conjunction with the benchmark parameters of a Whipple bicycle. How the revolution surface affects the relative equilibria and their stabilities is then investigated through our numerical simulations.