A method to solve minimum-lap-time problems using quasi-steady-state models and free (i.e. not predetermined) trajectory on two-dimensional tracks has been recently proposed. This work describes the extension of the method to three-dimensional tracks and builds upon g-g-g diagrams (instead of the standard g-g), to account for the effects of three-dimensionality. The main features of car and motorcycle g-g-g diagrams are discussed, to get insight into the main effects of three-dimensionality and to suggest a convenient parametrisation for the subsequent optimal control problem (OCP), whose size is not affected by the complexity of the vehicle model employed to generate the g-g-g diagrams. The application of the method to the Mugello and Barcelona-Catalunya circuits is given, with vehicle datasets resembling those of a race motorcycle (MotoGP) and a race car (Formula One). The results obtained are in line with those reported in the literature using full-dynamic models, yet such dynamic models are generally associated to the solution of much larger OCP.

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 ($$\approx 10^\circ$$ ≈ 10 ∘ ) 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.

Minimum-lap-time simulations with quasi-steady-state models and predefined trajectory have been in use for many years. However, most of the published works deal with twodimensional roads and employ the so-called 'apex-finding' method. This paper focuses on the application to three-dimensional roads, both with an extension of the 'apex-finding' approach to three-dimensional scenarios and with an optimal control method. Both approaches are based on g-g-g diagrams, i.e. the three-dimensional extension of the well-known g-g maps. In addition, under the assumption that the predefined trajectory is determined from noisy data (e.g. logged from the real vehicle), the three-dimensional trajectory reconstruction problem is addressed, to find a smooth and drift-free racing line to be used in the minimum-time simulation -again an optimal control approach is employed for the trajectory reconstruction. Examples of application are given both for a car and a motorcycle.