A reduced-order lie group-based race car model for systematic trajectory optimization on 3D tracks

This paper derives the dynamic equations of a reduced-order race-car model using Lie-group methods. While these methods are familiar to computational dynamicists and roboticists, their adoption in the vehicle dynamics community is limited. We address this gap by demonstrating how this framework integrates smoothly with the Articulated-Body Algorithm (ABA) and provides a fresh and systematic formulation of vehicle dynamics. For the first time, we model the car body as the end effector of a serial robot with a floating base connected to the track via virtual revolute and prismatic joints. Our formulation also accounts for the effects of 3D track geometry, providing a natural embedding of the car into the 3D track. We rigorously reconcile the ABA steps with key aspects of vehicle dynamics, including road-tire interactions, aerodynamic forces, and load transfers. The resulting model, simple yet accurate, is a powerful tool to efficiently solve Minimum-Lap-Time Planning problems. To demonstrate the effectiveness of our approach, we show numerical results obtained on the Nürburgring circuit. Our optimization problem is formulated via a direct collocation method and solved using the CasADi optimization suite. To validate the results, we test our reduced-order model against a full-fledged multi-body model recently developed by the same authors. The comparison confirms the validity of our reduced-order model, proving both the accuracy of the solution and the computational efficiency achieved.