Time-Optimal Paths for Simple Cars with Moving Obstacles in the Hamilton-Jacobi Formulation

Abstract

We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2-and 3-spatialdimensions. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation. Many authors have developed similar models and work 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.