A Game Between Two Identical Dubins Cars: Evading a Conic Sensor in Minimum Time

TL;DR

This paper analyzes a differential game involving two identical Dubins cars where one tries to evade a conic sensor held by the other, deriving optimal strategies and revealing key geometric surfaces influencing the game.

Contribution

It introduces a theoretical framework for the pursuit-evasion problem with conic sensors using differential game theory and identifies critical singular surfaces affecting optimal strategies.

Findings

Existence of Transition and Evader's Universal Surfaces.

Optimal strategies derived near the game's end.

Identification of barriers outside the playing space.

Abstract

A fundamental task in mobile robotics is keeping an intelligent agent under surveillance with an autonomous robot as it travels in the environment. This work studies a theoretical version of that problem involving one of the most popular vehicle platforms in robotics. In particular, we consider two identical Dubins cars moving on a plane without obstacles. One of them plays as the pursuer, and it is equipped with a limited field-of-view detection region modeled as a semi-infinite cone with its apex at the pursuer's position. The pursuer aims to maintain the other Dubins car, which plays as the evader, as much time as possible inside its detection region. On the contrary, the evader wants to escape as soon as possible. In this work, employing differential game theory, we find the time-optimal motion strategies near the game's end. The analysis of those trajectories reveals the existence of at least two singular surfaces: a Transition Surface (also known as a Switch Surface) and an Evader's Universal Surface. We also found that the barrier's standard construction produces a surface that partially lies outside the playing space.