Stability and Bifurcation of Longitudinal Vehicle Braking

The longitudinal braking dynamics of a two-wheel vehicle model on an incline are considered using techniques from nonlinear dynamics. The model is planar and incorporates the coupled dynamics of two independently braked wheels and the vehicle body, and takes into account the slip dynamics of each wheel. By using the wheel slip values and the vehicle speed as dynamic states, it is shown that the qualitative behavior of the system can be completely captured by studying a relatively simple phase plane problem described in terms of the slip values. A systematic bifurcation analysis is carried out in which the brake torques of the two wheels are varied, and it is shown how the system transitions from stable braking to the possibility of lockup in one or both wheels, to guaranteed lockup in both wheels. In this manner a quite complete picture of the dynamic behavior is obtained as a function of the two brake torques, including regions with multiple possible steady braking outcomes, depending on the initial conditions. This analysis provides new insights into the dynamics of vehicle braking, and it provides a correction to the standard result for the critical values of the brake torques at which the wheels undergo lockup. This approach may also prove useful for evaluating brake proportioning schedules, or for investigating anti-lock braking systems and other methods of traction control.