Time-Optimal Collaborative Guidance Using the Generalized Hopf Formula

TL;DR

This paper introduces a novel method for calculating time-optimal guidance controls in multi-vehicle pursuit-evasion scenarios using the generalized Hopf formula, enabling efficient solutions in high-dimensional spaces.

Contribution

The paper develops a new approach to solve high-dimensional Hamilton-Jacobi-Isaacs equations without grid-based methods, improving computational efficiency for multi-vehicle guidance.

Findings

Efficient computation of value functions in high-dimensional spaces.

Successful derivation of optimal controls from value function gradients.

Application to multi-vehicle pursuit-evasion scenarios.

Abstract

Presented is a new method for calculating the time-optimal guidance control for a multiple vehicle pursuit-evasion system. A joint differential game of k pursuing vehicles relative to the evader is constructed, and a Hamilton-Jacobi-Isaacs (HJI) equation that describes the evolution of the value function is formulated. The value function is built such that the terminal cost is the squared distance from the boundary of the terminal surface. Additionally, all vehicles are assumed to have bounded controls. Typically, a joint state space constructed in this way would have too large a dimension to be solved with existing grid-based approaches. The value function is computed efficiently in high-dimensional space, without a discrete grid, using the generalized Hopf formula. The optimal time-to-reach is iteratively solved, and the optimal control is inferred from the gradient of the value function.