Pursuit, Evasion and Defense in the Plane Selina Pan † , Haomiao Huang † , Jerry Ding † , Wei Zhang † , Duˇ san M. Stipanovi´ c, and Claire J. Tomlin Abstract— Multi-player games are important for analyzing complex real-world applications that involve both cooperative and adversarial agents, but computational complexity compli- cates solving such games. We study a modified pursuit-evasion game with multiple pursuers and a single evader, played in a convex domain with an exit through which the evader may escape. We present a strategy whereby one pursuer acts as a defender, utilizing a multi-mode switching strategy to prevent the evader from escaping while the other pursuers subsequently capture the evader. The strategy requires each pursuer to have knowledge only of its Voronoi neighbors and the evader, and runs in real time. The existence and uniqueness of the players’ trajectories are proved using non-smooth analysis, and it is also shown that the evader can never reach the exit regardless of its control inputs, resulting in eventual capture. Simulation results are presented demonstrating the algorithm. I. I NTRODUCTION Complex adversarial games involving multiple agents have many applications in robotics and automation. In addition to security and military applications, game solutions are useful in robust control systems with external disturbances; for example, pursuit-evasion strategies for both aircraft collision avoidance [1], [2] and autopilot safety analysis [3]. An important variant of pursuit-evasion games are situ- ations in which the evader may win by exiting the game region, analogous to such games in reachability analysis, where the objective is not merely to avoid some portion of the state-space, but also arrive safely in a goal set [4], [5]. Some complete solutions may be computed using Hamilton-Jacobi equations [1], [5], [6]; however, such methods are computa- tionally very expensive. Model predictive control (MPC) [7], [8] and optimization [9] have also been employed, assuming certain opponent behavior, minimizing a cost with respect to the player states over a time horizon, and obtaining solutions more quickly than Hamilton-Jacobi-based methods. However, there are no optimality or correctness guarantees due to prediction uncertainty. This work is supported in part by ONR under HUNT Award Number 550740 and NSF under grant CNS-0931843 S. Pan is with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA slpan@berkeley.edu H. Huang is with the Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA haomiao@stanford.edu W. Zhang is with the Department of Electrical and Computer En- gineering, The Ohio State University, Columbus, OH 43210, USA zhang@ece.osu.edu J. Ding and C. J. Tomlin are with the Department of Electrical Engineer- ing and Computer Sciences, University of California, Berkeley, CA 94720, USA {jding, tomlin}@EECS.Berkeley.edu D. M. Stipanovi´ c is with the Control and Decision Group, Coordi- nated Science Laboratory, University of Illinois, Urbana, IL 61801, USA dusan@illinois.edu † These authors contributed equally to this work. One recent approach to obtaining fast, feasible solutions in pursuit-evasion is to use the evader’s Voronoi cell area as a value function for the pursuers to minimize [10]. The Voronoi cell of a player corresponds to the set of all points that player can reach before any other player, assuming equal speeds [11]. Early Voronoi pursuit formulations required knowledge of the evader’s control law [12]; however, [10] guaranteed capture for multiple pursuers without this knowl- edge. There, the pursuers jointly minimized the area of the evader’s Voronoi cell, guaranteeing capture by reducing the area to zero. In this work, we present a Voronoi-based pursuit-evasion- defense game with many pursuers and a single evader. The game occurs in a planar convex polytope with an exit on the boundary through which the evader may escape to win the game. The pursuit strategy we propose assigns one pursuer the objective of defense, and the rest of the pursuers the objective of capture. Defense is performed by switching between the Voronoi strategy described in [10] and an exit guarding strategy, while capture is performed solely using the Voronoi strategy. The combined strategy generates pursuer control laws in real-time, and requires each pursuer to have knowledge of the states of only its Voronoi neighbors and the evader. We show that, if the game begins in a configuration such that the evader’s Voronoi region does not intersect the exit, an evader victory will always be prevented. A single pursuer guarantees that the evader cannot escape, and more than one pursuer guarantees capture. Empirical results also suggest that a single pursuer is sufficient for capture, although a formal proof cannot be currently provided. It is worth noting that there is strong interest in the control community to use biological systems to inspire automation design. In particular, one may draw inspiration from the animal kingdom in designing multi-player pursuit- evasion strategies. Our work has been inspired by research on predatory behavior of African lions [13]: this paper, with its theme of regions with exits, finds a parallel in the territorial defense employed by African lions [14]. This paper is organized as follows: First, we provide a con- cise problem formulation in Section II. We go on to describe the pursuit strategy in Section III. Sections IV and V show the existence of strategies and proofs of guaranteed escape prevention and capture, respectively. Simulation results are presented in Section VI, with conclusions and future work presented in Section VII. II. PROBLEM FORMULATION We consider a multiple-player pursuit-evasion game with N p pursuers and an evader that occurs in the interior of a convex polytope D in R 2 , with an exit region E . The goal