Semianalytical minimum-time solution for the optimal control of a vehicle subject to limited acceleration

A semianalytic solution of the minimum-time optimal control problem of a carlike 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 yields a nonlinear differential equation of the Riccati kind. The associated analytic solution exists in closed form, but it is numerically ill conditioned in some situations; hence, it is reformulated using asymptotic expansions that keep the numerical computations accurate and robust in all cases. Therefore, the proposed algorithm is designed to be fast and robust in sight to be the fundamental module of a more extended optimal control problem that considers a given clothoid curve as the trajectory and the presence of a constraint on the lateral acceleration of the vehicle. The numerical stability of the computation for limit values of the parameters is essential, as shown in the numerical tests.