  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 Hamilton-Jacobi-Bellman equations. We develop
efficient and scalable algorithms for solutions of high dimensional
Hamilton-Jacobi equations which can solve these types of path-planning problems
efficiently, even in high dimensions, while maintaining the Hamilton-Jacobi
formulation. Because our method is rooted in optimal control theory and has no
black box components, it has solid interpretability, and thus averts the
tradeoff between interpretability and efficiency for high-dimensional
path-planning problems. We demonstrate our method with several examples.
