Critical manifolds in the dynamics of a two-wheel model of an automobile

Abstract

Insight into steering and stability properties of automobiles in critical driving conditions is essential to advance driver assist systems and autonomous driving functions. To study the dynamic properties of an automobile with rear-wheel drive, methods of geometric singular perturbation theory are applied to a planar two-wheel vehicle model. Following a branch of periodic solutions bifurcating from the steady state of the vehicle at the limits of handling, a behaviour similar to a cycle near a homoclinic orbit is observed. The periodic orbit spends most of its period on a segment with an almost constant sideslip angle and slowly varying velocity. This behaviour can be explained by the nearby existence of a critical manifold consisting of a family of stationary solutions and a heteroclinic orbit connecting two points of this critical manifold. For slightly perturbed parameter values the critical manifold is replaced by a slow manifold with very slow dynamics, which governs the dynamics along the observed slow segment. The critical manifold and the heteroclinic orbit are calculated numerically, and good agreement with the derived approximations is obtained.