Optimal Control for Race Car Minimum Time Maneuvering

Abstract

Minimizing the time needed to travel a prescribed distance is the main development goal in motorsports. In racing car development, simulations are used to predict the effect of design parameter changes on vehicle performance. If approached as an optimal trajectory planning problem, a maneuver simulation can be used to determine not only the maneuver time, but also to identify the performance limitations on the system. This thesis describes the development and implementation of an optimal trajectory planning method using optimal control for short maneuvers. The requirements and modeling decisions are based on the application of the method to example problems related to TC design. The model for the method is based on a study of steady-state acceleration limits and stability. The rigid two-track model resulting from this study includes lateral and longitudinal load transfer, a nonlinear tire model, a limited-slip differential and aerodynamic downforce. An important contribution is the omission of wheel rotational velocities from the model, reducing the number of states by four and relaxing the requirements on the discretization interval. Possible misuse of this formulation is prevented by a constraint representing wheel rotational stability limitations. The formulation is validated by comparison to a reference model which includes wheel rotational velocities. The optimal trajectory planning method is formulated as an optimal control problem. The cost function is the maneuver time, and the constraints consist of the system dynamics and maneuver boundaries. The time-based dynamics are transformed into spatial dynamics, and a curvilinear coordinate system is used. The optimal control problem is discretized using a full collocation method, and the state and input trajectories are parametrized in terms of B-spline coefficients. The resulting problem is solved using a NLP solver. Interior-point solver IPOPT and SQP solver SNOPT are compared on various small problems. For this application IPOPT appears to be superior over SNOPT. The first order derivative information of the constraints required for IPOPT is approximated using sparse-finite differences, and the cost function gradient is calculated analytically. The precision of the method is assessed in a study of maneuver time dependency on mass. It appears that precision is mainly affected by convergence of the solver to various local minima. As such, the use of distance-dependent constraints and warm-start are employed for improving precision. The optimal trajectory for a hairpin with various radii is studied in detail. Special attention is paid to tire friction potential utilization and vehicle stability according the Lyapunov's First Method. For the given parameters it is shown that the optimal solution involves instances of overdriving either the front or rear axle. It is also shown that the vehicle is open-loop locally unstable on intervals along the optimal trajectory. In another simulation study, the reaction of the control inputs to temporary reductions in tire-road friction and perturbations to the yaw rate and body slip angle on turn-exit are evaluated. The most important result of this study is that the longitudinal control was found to be the primary means for rejecting such disturbances. The study also showed that steering angle changes are used as additional means for disturbance rejection if the perturbation is large enough to saturate the reduction of longitudinal control. The sensitivity of maneuver time and optimal trajectory to vehicle mass is studied by the use of so-called sensitivity differentials. This is done using a well-developed theoretical framework for parametric sensitivity for barrier methods, implemented in the software package sIPOPT. The sensitivity study can be seen as a proof of concept of the sensitivity differential approach for the race car MTM application.