Linear Differential Game with Two Pursuers and One Evader

Abstract

The purpose of this paper is to compute two on one differential game No Escape Zones (NEZ) [8]. The main objective (over the scope of this paper) consists in designing suboptimal strategies in many on many engagements. 1x1 NEZ and 2x1 NEZ are components involved in suboptimal approaches we propose (i.e. "Moving Horizon Hierarchical Decomposition Algorithm" [6]). Several specific two pursuers one evader differential games have been already studied ([4], [7]). Nevertheless, we propose to compute 2x1 NEZ from 1x1 DGL/1 NEZ because DGL models are games with well defined analytical solutions [12].We consider head on scenarios with two pursuers (P1 and P2) and one evader (E). First we summarize results about one on one pursuit evasion game using DGL/1 models. DGL/1 differential games are co planar interceptions with constant velocities, bounded controls assuming small motion variations around the collision course triangle. Under this assumption the kinematics are linear. Each player is represented as a first order system (time lag constant). The criterion is the terminal miss distance (terminal cost only, absolute value of the terminal miss perpendicular to the initial Line Of Sight, LOS). DGL/1 are fixed time duration differential games, with final time defined by the closing velocity (assumed constant) and the initial pursuer evader range. The terminal projection procedure [3] allows to reduce the initial four dimension state vector representation to a scalar representation and to represent the optimal trajectories in the ZEM (Zero Effort Miss), Tgo (Time to Go) coordinate frame. The (ZEM, Tgo) frame is divided into two regions, the regular area and the singular one. For some appropriated differential game parameters (pursuer to evader maximum acceleration ratio μ and evader to pursuer time lag ratio ε), the singular area plays the role of capture zone so called also NEZ (leading to zero terminal miss), whilst the regular area corresponds to the non capture zone. The NEZ can be bounded (closed) or unbounded (open). The natural optimal strategies are bang-bang controls corresponding to the sign of ZEM (some refinements exist when defining optimal controls inside the NEZ). We start the 2x1 DGL/1 analysis with unbounded 1x1 NEZ as pictured in Figure 1 (NEZ delimited by the two plain red lines, non capture zone corresponding to the state space filled with optimal trajectories in dot blue lines). Moreover, we first assume same Tgo in each DGL/1 game (same initial range, same velocity for each pursuer).

Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications

“…As before, the symbol τ denotes the time constant. Games 2 × 1 with dynamics of types (25) and (26) were studied in works [20,30,36]. Games 2 × 1 with dynamics (27) have not been studied earlier.…”

Numerical Study of Two-on-One Pursuit-Evasion Game

“…Passing to a new dynamics in the "equivalent" coordinates x 1 and x 2 (see Le Ménec (2008Ménec ( , 2010)), we obtaiṅ…”

“…The works Le Ménec (2008Ménec ( , 2010 consider only cases with exact captures and pursuers "stronger" than the evader. The latter means that the parameters A P i , A E and l P i , l E (i = 1, 2) are such that the maximal stable bridges in the individual games (P 1 vs. E and P 2 vs. E) grow monotonically in the backward time.…”

Optimal Evading Strategies for Two-Pursuer/One-Evader Problems

“…Partitions of the state space were identified and categorized, which are similar to the degenerate and the nondegenerate regions discussed later on in this paper. A linear differential game formulation restricting the motion of the players to a straight line with the two pursuers coordinating to reduce the miss distance was extensively studied by several researchers [30][31][32]. An algorithm to numerically construct level sets of the value function with fixed final time was also discussed in [33].…”

“…Most of the existing work on two-pursuer/one-evader problems either deals with linear dynamics and then tries to optimize miss distance (with quadratic cost and fixed final time, [30][31][32]) or considers formulations involving particular tasks and constraints [28,35,36]. Though time-optimal solutions for these problems were briefly investigated in the book by Isaacs [1], to the best of the authors' knowledge, there is no rigorous work existing beyond that discussion.…”