Computing minimum lap-time trajectories for a single-track car with load transfer

In this paper we address the minimum lap-time problem for a single-track rigid car which includes tire models and load transfer. Given a planar track including lane boundaries, our goal is to find a trajectory of the car minimizing the lap time subject to tire and steering limits. By using a new set of coordinates, the time-dependent system is transformed into a \squote{space-dependent} (and space-variant) system. The choice of a suitable set of coordinates and the partition of the dynamics into a ``longitudinal'' one and a ``transverse'' one, allows us to convert the minimum time problem into a fixed horizon constrained optimal control problem. Based on a projection operator nonlinear optimal control technique, we propose a minimum lap-time strategy to push the rigid car to the limit of its handling capabilities. Finally, we provide numerical computations that: (i) show the effectiveness of the proposed strategy, and (ii) allow us to highlight important features of minimum lap-time trajectories.