Pursuit-evasion: A tool kit to make applications more accessible

A pursuit-evasion game takes place between two players. The pursuer is charged with capturing the evader while the evader tries to avoid getting caught. Many robotics applications such as search, tracking, and surveillance can be modeled as pursuit-evasion games. Equally important, these games can be modeled as fun mathematics problems to inspire newcomers to the field of robotics. We have witnessed this firsthand during summer Research Experiences for Undergraduates programs at the Institute for Mathematics and its Applications, located at the University of Minnesota. The subject is accessible, with many open problems that require creativity, insight, and strong algorithmic thinking. Our summer students digested the basics of the field and developed results that evolved into research publications [1], [2]. The purpose of this article is to provide a "tool kit" so as to make pursuit- evasion games accessible to a broader audience. We focus on a classical game known as the lion and man game, where the lion pursues the man, moving with equal speed. Rather than a traditional survey of literature on the lion and man game, the article is organized in a tutorial fashion. We start from simple motivating examples whose solutions are accessible to anyone with a high-school background in geometry and trigonometry. Our journey takes us to open variants such as pursuit-evasion on surfaces. Along the way, we introduce tactics that can be used as building blocks in different settings. The material in the earlier sections of the article provides a starting point for science, technology, engineering, and mathematics (STEM) educators looking for an engaging robotics problem accessible to high-school and undergraduate students. Simultaneously, we provide an introduction for researchers who would like to tackle one of the most challenging path-planning problems. We conclude with open problems for the researchers and exercises to engage students. Let's dive right in!.