We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2- and 3-spatial-dimensions. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation. Many authors have developed similar models and employed grid-based numerical methods to solve the partial differential equation required to generate optimal trajectories. However, these methods can be inefficient and do not scale well to high dimensions. We describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be developed to solve similar problems very efficiently, even in high dimensions, while maintaining the Hamilton-Jacobi formulation. We demonstrate our method with several examples.

翻译：我们提出了一种基于偏微分方程的最优路径规划框架，用于处理曲率约束运动，并应用于二维和三维空间中的车辆问题。该框架依赖于最优控制理论、动态规划以及汉密尔顿-雅可比-贝尔曼方程。许多研究者已开发出类似模型，并采用基于网格的数值方法来求解生成最优轨迹所需的偏微分方程。然而，这些方法可能效率低下，且在高维情况下可扩展性不佳。我们阐述了如何开发高效且可扩展的算法，用于求解高维汉密尔顿-雅可比方程，从而在保持汉密尔顿-雅可比形式的同时，高效解决类似问题，即使在高维场景下也是如此。我们通过多个示例展示了该方法的有效性。