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

Finding out the relative equilibria and analyzing their stability is of great significance for revealing the intrinsic characteristics of mechanical system and developing effective controllers to improve system performance. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a surface of revolution. We notice that the symmetry group existing in the bicycle configuration description is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints on the bicycle system. Therefore, the bicycle dynamics can be classified as a general Voronets system in which the Lagrangian and constraint distribution remain unchanged under the action of the symmetry group. Applying the Voronets equations to bicycle dynamics modeling, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. Theoretical analysis for the reduced dynamic system shows that it has the properties of time-reversal and lateral symmetries. In addition, two types of relative equilibrium points, the static equilibria and the dynamic equilibria, exist. Further theoretical analysis shows that the two kinds of relative equilibria both form one-parameter solution families, and the Jacobian matrix at an equilibrium point has some specific properties that support the relevant stability analysis. The necessary condition responsible for a stable static equilibrium point is that all the eigenvalues of the Jacobian matrix at the equilibrium point must lie on the imaginary axis of the complex plane. Due to the existence of zero eigenvalues of the Jacobian matrix, the stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold based on the conservation of system energy. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, symbolic computation combined with Whipple bicycle benchmark parameters was used for numerical simulations. We then use our numerical simulation to study how the parameter of the surface of revolution affects the relative equilibrium solution and its stability.