# Construction of the Barrier for Reach-Avoid Differential Games in Three-Dimensional Space with Four Equal-Speed Players

## TL;DR

This paper derives a complete analytical barrier for a three-dimensional reach-avoid game with four equal-speed players, enabling real-time prediction of game outcomes and considering all pursuer cooperation strategies.

## Contribution

It provides a closed-form, analytical barrier for the game, considering all pursuer cooperation scenarios, and introduces an efficient algorithm for computing barriers in multi-pursuer settings.

## Key findings

- Complete analytical barrier for the game is derived.
- The barrier allows real-time prediction of game outcomes.
- An algorithm for computing barriers for any number of pursuers is proposed.

## Abstract

This paper considers a reach-avoid differential game in three-dimensional space with four equal-speed players. A plane divides the game space into a play subspace and a goal subspace. The evader aims at entering the goal subspace while three pursuers cooperate to prevent that by capturing the evader. A complete, closed-form barrier for this differential game is provided, by which the game winner can be perfectly predicted before the game starts. All possible cooperations among three pursuers are considered and thus the guaranteed winning for each team is a prior. Furthermore, an algorithm is designed to compute the barrier for multiple pursuers of any numbers and any initial configurations. More realistically, since the whole achieved developments are analytical, they require a little memory without computational burden and allow for real-time updates, beyond the capacity of traditional Hamilton-Jacobi-Isaacs method.