Efficient and Scalable Path-Planning Algorithms for Curvature Constrained Motion in the Hamilton-Jacobi Formulation

TL;DR

This work develops a Hamilton-Jacobi-Bellman PDE framework for curvature-constrained path planning in two and three spatial dimensions, focusing on Dubins-type vehicles (car, airplane, submarine). It preserves interpretability by staying within the PDE/optimal-control structure while achieving scalability through a grid-free, Hopf-Lax-inspired saddle-point solver (PDHG) and level-set obstacle handling. The method formulates a value function $u(x,t)$ representing minimum distance to the goal, derives model-specific HJB equations, and provides detailed, analytically tractable proximal updates for the car and airplane and more involved updates for the submarine, including obstacle incorporation via a smooth indicator $O(x,t)$. Simulations demonstrate real-time-like performance across 2D/3D scenarios with stationary and moving obstacles, highlighting the approach’s practicality, interpretability, and potential for extension to more complex environments and multi-agent problems.

Abstract

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.

Figures (7)

• Figure 1: A simple self-driving car
• Figure 2: Three obstacles approximated by disjoint circles.
• Figure 3: A car navigating around obstacles to end point in the top right of each frame. The obstacles are the blue regions, and the collection of green circles is their computational representation.
• Figure 4: A car navigating around the same obstacles as in figure \ref{['fig:carM']}, but in this case the obstacles begin rotating clockwise about the center of the domain.
• Figure 5: A Dubins airplane circling down for a landing. The 3D view is displayed on the left of each panel, and the projection down to the $xy$-axis is displayed on the right.