Self-Stabilization Mechanism Analysis of Bicycle Nonholonomic System

A bicycle is a typical nonholonomic system, and the nonholonomic constraints attribute to different non-straightforward dynamic phenomena to the bicycle. Self-stabilization, i.e., the bicycle can move in balance without external assistance, is an interesting phenomenon, yet its mechanism is unclear due to complex interactions of the constrained bicycle multibody dynamics. We study the self-stabilization from two aspects: stability analysis and mechanism analysis. In the stability analysis, by the physical understanding of the bicycle system, we establish a dimension-reduction method to calculate the nontrivial equilibria of the highly nonlinear and high-dimensional differential algebraic equations (DAEs). Furthermore, we propose an implementable procedure to conduct stability analysis of the equilibria, where linearization of the DAEs is performed first and then the dimensionality reduction is followed. In the mechanism analysis, we obtain a reduced bicycle dynamic model based on the geometric symmetries and cyclic coordinates, which theoretically transforms the complex DAEs to a clear model structure without constraints. Based on the reduced model structure, we develop a bicycle surrogate model and establish comprehensive and quantitative understanding of the self-stabilization mechanism. The analysis shows that the nonholonomic constraints play important roles by providing anti-falling torques, equivalent stiffness and damping factors in the stabilization of the bicycle system.