Lion and man game in the presence of a circular obstacle

Abstract

Abstract-In the lion and man game, a lion tries to capture a man who is as fast as the lion. We study a new version of this game which takes place in a Euclidean environment with a circular obstacle. We present a complete characterization of the game: for each player, we derive necessary and sufficient conditions for winning the game. Their (continuous time) strategies are constructed using techniques from differential games and arguments from geometry. Our main result is a decision algorithm which takes arbitrary initial positions as input, declares one of the players as the winner of the game and outputs a winning strategy for that player. We extend our approach to explicitly construct, in closed form, the decision boundary that partitions the arena into win and lose regions. I. OVERVIEW AND RELATED WORKIn a game of pursuit and evasion, one player (the pursuer) tries to get close to, and possibly capture the other (the evader). The evader, in turn, tries to avoid being captured. Pursuit-evasion games are of fundamental importance to researchers in the field of robotics. Consider the task of surveillance, where a guard (pursuer) has to chase and capture an intruder (evader). Another scenario is search-andrescue, where a rescue worker has to locate a lost hiker. Since the actions of the hiker are not known a priori, worstcase pursuit and evasion strategies guarantee that the hiker is found no matter what he does. Problems arising from diverse applications such as collision-avoidance [9], search-andrescue [6], [14], air-traffic control [3], and surveillance [9] have been modeled as pursuit-evasion games.A classical pursuit-evasion game is the Lion and Man game. It was originally posed in 1925 by Rado as follows A lion and a man in a closed arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal? The first solution was generally accepted by 1950: the lion moves to the center of the arena and then remains on the radius that passes through the man's position. Since they have the same speed, the lion can remain on the radius and simultaneously move toward the man. Although this strategy works in discrete-time, it was later shown by Besicovitch that exact capture in continuous time takes infinitely long in a bounded arena [13]. However, if the capture distance is set to some c > 0, Alonso et al. [2] The lion and man game in the presence of obstacles remains a challenge. In this paper, we take an important step for solving the lion and man game in an environment with obstacles. We present full characterization of the game in the presence of a single circular obstacle. That is, we present a decision algorithm which determines the winner of the game. We also construct the winner's strategy.As in the original version of the game, we assume that the players know exact locations of each other at all times and have equal maximum speeds.An important line of research is to study the effect of sensing limitations. Recent progress in this direction includes the study of range-based limitation...

Multi-robot target detection and tracking: taxonomy and survey

“…The authors distinguish different areas in the geometric decomposition of the scene, each area corresponding to a specific optimal strategy (see Figure 8). Karnad and Isler proposed similar results for a circular obstacle, and given any initial position can determine which protagonist will win the game (Karnad and Isler, 2009). This approach has a drawback: it barely scales with the number of obstacles.…”

On Optimal Pursuit Trajectories for Visibility-Based Target-Tracking Game

“…Therefore, research on pursuit evasion games currently are limited to environment with simple obstacles which can potentially shed light on solutions in more complicated environments. For example, in [18], the authors present optimal solutions to the lion and man game around a circular obstacle. Visibility-based pursuit-evasion game between two holonomic agents in an environment with obstacles are investigated in [19], [7], [20].…”

Capture bounds for visibility-based pursuit evasion

“…Surprisingly, the man can evade the lion indefinitely as shown by Besicovitch [18]-the lion fails to reach the man in any finite time although it can get arbitrarily close to him [2]. Extending this result to environments with obstacles, however, has proved difficult, and the only relevant result seems to be a recent work of Karnad-and-Isler [13] that deals with a single circular obstacle! Pursuit evasion is also studied as a form of differential games and solved using the Hamilton-Jacobi-Isaacs equation.…”