On minimum time vehicle manoeuvring: the theoretical optimal lap

Abstract

This work is a research on the minimum time vehicle manoeuvring problem, with a particular application to finding the minimum lap time for a Formula One racing car. The proposed method allows to solve the general problem of evaluating the vehicle lateral and longitudinal controls which yield the minimum time required to traverse a lap of a circuit. The minimum time vehicle manoeuvring problem is formulated as one of Optimal Control and is solved using mathematical programming methods. Novel techniques are employed to solve the resulting non-linear programming problem which allow to achieve effective optimisation with satisfactory accuracy, robustness and computational efficiency. Particularly, the proposed solution strategy is generally applicable to any arbitrarily complex vehicle mathematical model. Car and circuit models are set up, and the optimisation program is applied to investigate the sensitivity of the vehicle performance with respect to vehicle design parameters, such as the yaw moment of inertia, the total mass and the weight distribution. Furthermore, the minimum time manoeuvring problem is solved for very different vehicle configurations. The optimisation program accurately quantifies the vehicle performance in terms of manoeuvre time, and the nature of the optimal solution is shown to be always in excellent agreement with the dynamic properties of the vehicle model. A part of the work is devoted to the development of a strategy to obtain an initial estimate of the racing line and of the vehicle lateral and longitudinal controls to be used at the start of the optimisation. Two algorithms to compute the racing line using on board measured data from the real car are presented. A new mathematical model for the vehicle steering control is derived. The model uses multiple preview information of the intended path. Its structure derives from linear optimal preview control theory, but it is adapted to deal with non-linear vehicle operations arising from the inevitable tyre force saturation in vigorous manoeuvring. The excellent path following capability of the model is demonstrated by solving various path following tasks involving moderate manoeuvring and racing speeds