Semi-Analytical Minimum Time Solution for the Optimal Control of a Vehicle subject to Limited Acceleration

The basic module for the solution of the minimum time optimal control of a car-like vehicle is herein presented. The vehicle is subject to the effect of laminar (linear) and aerodynamic (quadratic) drag, taking into account the asymmetric bounded longitudinal accelerations. This module is studied and designed to be fast and robust in sight to be the fundamental building block of a more extended optimal control problem that considers a given clothoid as the trajectory and the presence of a constraint on the lateral acceleration of the vehicle. The nonlinear dynamics and the different possible boundary conditions yield different analytical solutions of the differential equations, hence they by themselves a particular attention. The study of the numeric stability of the computation for limit values of the parameters is essential as showed in the numerical tests.