In this paper we derive the dynamic equations of a race-car model via Lie-group methods. Lie-group methods are nowadays quite familiar to computational dynamicists and roboticists, but their diffusion within the vehicle dynamics community is still limited. We try to bridge this gap by showing that this framework merges gracefully with the Articulated Body Algorithm (ABA) and enables a fresh and systematic formulation of the vehicle dynamics. A significant contribution is represented by a rigorous reconciliation of the ABA steps with the salient features of vehicle dynamics, such as road-tire interactions, aerodynamic forces and load transfers. The proposed approach lends itself both to the definition of direct simulation models and to the systematic assembly of vehicle dynamics equations required, in the form of equality constraints, in numerical optimal control problems. We put our approach on a test in the latter context which involves the solution of minimum lap-time problem (MLTP). More specifically, a MLTP for a race car on the N\"urburgring circuit is systematically set up with our approach. The equations are then discretized with the direct collocation method and solved within the CasADi optimization suite. Both the quality of the solution and the computational efficiency demonstrate the validity of the presented approach.

翻译：本文通过李群方法推导了赛车模型的动力学方程。当前计算动力学家和机器人学家对李群方法已较为熟悉，但该方法在车辆动力学领域的推广仍十分有限。我们通过展示该框架与铰接体算法(ABA)的自然融合，实现了车辆动力学的创新性系统化表达，致力于弥合这一研究鸿沟。重要贡献在于严格统一了ABA算法步骤与车辆动力学的核心特征，包括轮胎-路面相互作用、空气动力学力和载荷转移等。所提方法既适用于直接仿真模型构建，又能系统化组装数值最优控制问题中等式约束形式的车辆动力学方程。我们在最小圈时问题(MLTP)背景下验证了该方法：具体而言，以纽博格林赛道上的赛车MLTP为例，采用本文方法系统构建了数学模型，随后通过直接配点法离散化，并在CasADi优化套件中求解。求解质量与计算效率共同验证了该方法的有效性。