Minimum-lap-time of race vehicles

This work deals with the numerical-optimisation methods employed for assessing the minimum-lap-time of race vehicles. A particular emphasis is given to the simulations based on the solution of an optimal-control problem. First, a detailed review is devoted to the optimal-control theory, including direct and indirect methods, together with explicit and implicit formulations. Three vehicle models are proposed for the implementation in steady-state and dynamic simulations. A steady-state double-track race-car model is presented first, implementing aerodynamic forces, Pacejka-based tyre model, fixed brake-ratio and steady-state lateral-load transfer. This model is then extended including the transient effects for the implementation in a dynamic minimum-time simulation. An essential but comprehensive steady-state motorcycle model is also presented, retaining the peculiar aspects of the motorcycle dynamics, such as the wheelie and stoppie conditions, while assuming the optimal braking-bias. Then, an optimisation program is developed for computing the race car g-g envelope. A different approach is shown for the steady-state motorcycle model, which is employed for computing the g-g diagrams analytically. Finally, the most widespread minimum-lap-time simulations are discussed and developed for the presented models, including a quasi-steady-state fixed-trajectory apex-finding approach and a quasi-steady-state fixed-trajectory optimal-control simulation. In addition, a novel approach that combines a steady-state vehicle model with the trajectory optimisation is presented and compared to the discussed simulations. This approach builds upon the formulation of an optimal-control problem and employs the g-g diagrams for limiting the vehicle performance. The results are employed for a sensitivity analysis, that underlines the effect of different parameters on the resulting race line. A dynamic race car model is also implemented in a free-trajectory optimal-control simulation. This program is employed for analysing the effect of different steering geometries on the vehicle performance in different test manoeuvres and during a track lap.