AI 中文总结
该研究针对车辆动力学模拟的平面地球误差，用微分几何建模曲面道路上的车辆运动，结合多体力学与最优控制，在纳斯卡椭圆形赛道模拟中验证了三维效应的动力学显著性。
AI 中文摘要
将道路表面广泛理想化地视为水平面会给车辆动力学模拟带来显著误差，这一现象被称为“平面地球误差”。本文提供了一份说明性指南，介绍如何利用经典微分几何对车辆在曲面上的运动进行建模。通过度量张量、第二基本形式、形状算子和克里斯托费尔符号来表征道路曲率的影响。使用以椭圆锥为基准示例的可复现MATLAB脚本，展示测地线曲线的构建以及曲面上粒子动力学的模拟。将所得几何结构集成到伪谱最优控制框架内的单轨车辆-轨道模型中。利用达灵顿赛道的高密度移动激光雷达轮廓上的轨迹优化结果，生成高保真道路表面模型。该模型在hp自适应配置框架内用于研究最小单圈时间最优控制车辆轨迹。这些计算捕捉了轮胎的非光滑牵引力饱和极限以及引力的位置依赖性变化。将微分几何、多体力学和最优控制相结合，对于高保真的有人驾驶和自动驾驶车辆动力学模拟至关重要。优化后的速度和轮胎侧滑曲线表明，真实世界的赛道几何会产生具有动力学显著性的三维效应。这些结果对于纳斯卡椭圆形等具有超高倾斜度的赛道表面上的性能受限模拟尤为重要。
英文摘要
The widespread idealization of road surfaces as horizontal planes can introduce significant inaccuracies into vehicle dynamics simulations, a phenomenon termed the ``Flat Earth Error.'' This article provides an expository guide to the use of classical differential geometry to model vehicular motion on curved surfaces. The influence of road curvature is characterized using the metric tensor, the second fundamental form, the shape operator, and the Christoffel symbols. Reproducible MATLAB scripts using an elliptic cone as a benchmark example illustrate the construction of geodesic curves and the simulation of particle dynamics on curved surfaces. The resulting geometric structures are integrated into a single-track vehicle-and-track model within a pseudospectral optimal control framework. Trajectory optimization results over a high-density mobile LiDAR profile of Darlington Raceway are used to generate a high-fidelity road-surface model. This model is used within an hp-adaptive collocation framework to investigate minimum lap time optimal control vehicular trajectories. These computations capture the non-smooth traction saturation limits of the tyres alongside position-dependent variations in gravitational forcing. Integrating differential geometry, multibody mechanics, and optimal control is essential for high-fidelity driven and autonomous vehicle-dynamics simulations. The optimized velocity and tyre slip profiles show that real-world racing track geometries induce dynamically significant three-dimensional effects. These results are particularly relevant to performance-limited simulations on highly banked track surfaces such as NASCAR ovals.
Comments38 pages, 12 figures, 3 tables
