Curved-ribbon-based track modelling for minimum lap-time optimisation

Three-dimensional road models for vehicular minimum-lap-time manoeuvring are typically based on curvilinear coordinates and generalizations of the Frenet–Serret formulae. These models describe the road as a parametrized ‘ribbon’, which can be described in terms of three curvature variables. In this abstraction the road is assumed laterally flat. While this class of road models is appropriate in many situations, this is not always the case. In this research we extend the laterally-flat ribbon-type road model to include lateral curvature. This accommodates the case in which the road camber can change laterally across the track. Lateral-position-dependent camber is introduced as a generalisation that is required for some race tracks. A race track model with lateral curvature is constructed using high-resolution LiDAR measurement data. These ideas are demonstrated on a NASCAR raceway, which is characterized by large changes in lateral camber angle (≈10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 10^\circ$$\end{document}) on some parts of the track. A free-trajectory optimization is employed to solve a minimum-lap-time optimal control problem. The calculations highlight the practically observed importance of lateral camber variations.