Active target defense differential game

Abstract

A pursuit-evasion differential game involving three agents is discussed. This scenario considers an Attacker missile pursuing a Target aircraft. The Target is however aided by a Defender missile launched by, say, its wingman, to intercept the Attacker before it reaches the Target aircraft. Thus, a team is formed by the Target and the Defender which cooperate to maximize the distance between the Target aircraft and the point where the Attacker missile is intercepted by the Defender missile, and the Attacker which tries to minimize said distance. The solution to this differential game provides optimal heading angles for the Target and the Defender team to maximize the terminal separation between Target and Attacker and it also provides the optimal heading angle for the Attacker to minimize the said distance.

Cooperative target defense differential game with a constrained-maneuverable Defender

“…The active target defense differential game where all agents are described by simple motion models, that is, turning rate constraints are not incorporated into the Defender's dynamics, was addressed in [13]. The differential game in [13] was analyzed by attaching a rotating Cartesian frame in such a way that the extension to infinity of AD in both directions represents the X-axis and the orthogonal bisector line of AD represents the Y-axis. The solution of the differential game is obtained by solving the quartic equation in y(≥ 0)…”

“…In this reduced state space the coordinates of the Attacker are given by (x A (ϕ), 0) and the coordinates of the Target are given by (x T (ϕ),ȳ T (ϕ)). This equivalent scenario corresponds to the case in [13] and (14) can be solved to obtain y(ϕ) = y(x A (ϕ),x T (ϕ),ȳ T (ϕ)).…”

“…The complexity of the problem by including Defender turning rate constraints significantly increases compared to the simple motion case in [13]. It can be seen that a direct solution ϕ to the equation Y I (ϕ) = y(ϕ) is difficult to obtain because y(ϕ) is not explicitly given as a function of ϕ but the quartic equation (14) needs to be solved.…”

“…To facilitate the analysis, we will write the cost/payoff function (13) in terms of the variable ϕ, which is the angle subtended by the arc traveled while the defender is turning. Notice that we can transform the problem by adjusting the initial location as follows.…”

“…Our preliminary work involving the differential game formulation of the cooperative target defense problem assumed that all agents had simple motion ( [13], [14], [15]). This paper extends these previous results to consider the practically important case where the Defender's turning rate is bounded, and, therefore, it is not able to achieve its optimal heading instantaneously.…”