An Efficient Semi-Real-Time Algorithm for Path Planning in the Hamilton-Jacobi Formulation

TL;DR

This paper introduces a grid-free, semi-real-time algorithm for optimal path planning based on Hamilton-Jacobi equations, offering a scalable and interpretable alternative to traditional PDE methods for high-dimensional and real-time applications.

Contribution

The paper proposes a novel grid-free numerical method using Hopf-Lax formulas for Hamilton-Jacobi path planning, enabling efficient, interpretable, and semi-real-time solutions in higher dimensions.

Findings

Algorithm successfully applied to 2D synthetic examples

Retains interpretability of PDE-based methods

Avoids curse of dimensionality in path planning

Abstract

We present a semi-real-time algorithm for minimal-time optimal path planning based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ) equations. Partial differential equation (PDE) based optimal path planning methods are well-established in the literature, and provide an interpretable alternative to black-box machine learning algorithms. However, due to the computational burden of grid-based PDE solvers, many previous methods do not scale well to high dimensional problems and are not applicable in real-time scenarios even for low dimensional problems. We present a semi-real-time algorithm for optimal path planning in the HJ formulation, using grid-free numerical methods based on Hopf-Lax formulas. In doing so, we retain the intepretablity of PDE based path planning, but because the numerical method is grid-free, it is efficient and does not suffer from the curse of dimensionality, and thus can be applied in semi-real-time and account for realistic concerns like obstacle discovery. This represents a significant step in averting the tradeoff between interpretability and efficiency. We present the algorithm with application to synthetic examples of isotropic motion planning in two-dimensions, though with slight adjustments, it could be applied to many other problems.