Stability analysis for the Whipple bicycle dynamics

It has been known that bicycle stability is closely linked to a pair of ordinary differential equations (ODEs). The linearization technique used to derive these ODEs, nevertheless, has yet to be thoroughly examined. For this purpose, we conduct an analysis of the dynamics of the Whipple bicycle, starting with the contact kinematics, using the Gibbs-Appell method. The effort results in a complete nonlinear model with minimal dimensions, from which equilibrium points during the bicycle's straight and circular motions can be determined. The model can be linearized around these points via a perturbation analysis under no additional assumptions. Given the non-hyperbolic nature of the equilibria, we apply the center manifold theorem to analyze their stability, offering a rigorous derivation of the (well-know) exponential stability of the bicycle in its leaning and steering motions. Finally, a dimensionless index is defined to characterize the influence of physical parameters on the bicycle stability.