On the Suicidal Pedestrian Differential Game

TLDR: It is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer, which can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories.

Abstract: We consider the following differential game of pursuit and evasion involving two participating players: an evader, which has limited maneuverability, and an agile pursuer. The agents move on the Euclidean plane with different but constant speeds. Whereas the pursuer can change the orientation of its velocity vector arbitrarily fast, that is, he is a “pedestrian” a la Isaacs, the evader cannot make turns having a radius smaller than a specified minimum turning radius. This problem can be seen as a reversed Homicidal Chauffeur game, hence the name “Suicidal Pedestrian Differential Game.” The aim of this paper is to derive the optimal strategies of the two players and characterize the initial conditions that lead to capture if the pursuer acts optimally, and areas that guarantee evasion regardless of the pursuer’s strategy. Both proximity-capture and point-capture are considered. After applying the optimal strategy for the evader, it is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer. Therefore, the well-known ZNP solution can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories. The results are directly applicable to collision avoidance in maritime and Air Traffic Control applications.