A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems

TL;DR

This paper introduces a primal-dual method leveraging the generalized Hopf formula to efficiently solve high-dimensional Hamilton-Jacobi equations for time-optimal control, enabling rapid trajectory generation without grid-based discretization.

Contribution

It develops a novel primal-dual algorithm that solves the convex optimization formulation of the Hopf formula, allowing scalable and parallelizable computation for high-dimensional control problems.

Findings

Execution times are on the order of milliseconds.

Computation scales approximately polynomially with dimension.

Method enables direct trajectory generation from the solution.

Abstract

Presented is a method for efficient computation of the Hamilton-Jacobi (HJ) equation for time-optimal control problems using the generalized Hopf formula. Typically, numerical methods to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension. The generalized Hopf formula avoids the use of grids and numerical gradients by formulating an unconstrained convex optimization problem. The solution at each point is completely independent, and allows a massively parallel implementation if solutions at multiple points are desired. This work presents a primal-dual method for efficient numeric solution and presents how the resulting optimal trajectory can be generated directly from the solution of the Hopf formula, without further optimization. Examples presented have execution times on the order of milliseconds and experiments show computation scales approximately polynomial in dimension with very small high-order coefficients.