A Simple Mono-Dimensional Approach for Lap Time Optimisation

Abstract

Lap time minimisation methods have great relevance in the analysis of race tracks, and in the design and optimisation of race vehicles. Several lap time minimisation approaches have been proposed in the literature, which are computationally demanding because they need to either solve differential equations or to implement a forward–backward integration based on an apex-finding method. This paper proposes an alternative method, based on a mono-dimensional quasi-steady-state numerical approach. The proposed approach uses a simplified vehicle model accounting for combined tyre–road interactions, aerodynamic effects, and power limitations. The method exploits the knowledge of the curvature of the trajectory, which is worked out through a rigorous approach that allows for the use trajectories defined with respect to ageneric curve parameter and not necessarily the arc length. An iterative routine is implemented that exploits the vehicle dynamics, without solving differential equations or performing forward–backward integrations from the trajectory apexes. Simulations are carried out on three different tracks and are shown to be computationally efficient. Despite being intentionally simple, the proposed method allows to grasp key aspects of the problem, such as the effect of the combined tyre–road interactions on the acceleration profiles, and the effect of aerodynamic drag and downforce on the position of the braking point on the track and on the speed profile.

Real-Time Optimal Trajectory Planning for Autonomous Vehicles and Lap Time Simulation Using Machine Learning

“…This approach enables complex, fully transient models of the vehicle to be used (Kelly, 2008), and allows the path to be adjusted in response to a change in vehicle parameters (Dal Bianco et al, 2018) -thus resulting in a highly-accurate, vehicle-specific racing line. However, due to the complexity of solving a trajectory of the vehicle model for every discretisation point, the solution time for the OCP approach typically takes considerably longer than the lap time (Lenzo and Rossi, 2020) -with simulators comprising relatively simplistic vehicle models taking around 15 minutes to solve a lap (Brayshaw and Harrison, 2005;Perantoni and Limebeer, 2014). Lot and Dal Bianco (2015) solved a lap using a more complex model with 14 Degrees of Freedom (DoF) in approximately 28 minutes -though it should be noted that the accuracy of the model was not greatly improved over a simplified 10-DoF model -taking 16 minutes to solve.…”

“…Traditionally, the more rapid approaches to generating a target path tend to be purely based upon the geometry of the circuit, generating an approximation that does not offer any guarantee that it will be the time-optimal solution (Kapania et al, 2016). Many of the existing algorithms for finding the optimal racing line are based around solving multiple simulations of a vehicle model for a wide range of possible trajectories, before converging upon the path which results in the minimum time -thus the iterative or optimisation process requires considerable computational resources (Lenzo and Rossi, 2020). This can be time-consuming for traditional lap time simulation applications and renders these approaches largely unsuitable for real-time applications.…”

Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles

“…Minimum lap-time simulations (MLTS) of road vehicles have been in use for many years [1,2]: they were historically formulated using quasi-steady-state (QSS) vehicle models on pre-defined two-dimensional trajectories [3][4][5][6][7][8][9][10], and evolved to employ transient vehicle models on three-dimensional roads [11][12][13][14][15], where the trajectory is a result of the optimization. Free-trajectory methods with QSS models have also been proposed both for two-dimensional [16] and three-dimensional roads [17,18].…”