Pursuit-Evasion of Two Aircraft in a Horizontal Plane

Abstract

Pursuit-evasion between two aircraft in a horizontal plane is analyzed as a differential game using point-mass aircraft models. A suitable choice of real-space coordinates confers open-loop optimality on the game. The solution in the small is described in terms of the individual aircraft's extremal trajectory maps (ETM). Each is independent of role, adversary, and capture radius. An ETM depicts the actual trajectories flown by the aircraft in real space. A template method of generating constant control arcs is described. This is used to investigate bank saturation and throttle switching behavior exhaustively. Sections of the barrier are obtained by iteratively choosing pairs of extremals from the two ETM's. NALYTICAL studies of aircraft pursuit-evasion techniques seek to determine the influence of aircraft and weapon system performance upon combat outcome and to derive idealized tactics. Aircraft pursuit-evasion is considerably more difficult to analyze than optimal maneuvers because a closed-loop solution is a must in a differential game situation and exceptional and discontinuous surfaces are the rule rather than an exception. Analyses of aircraft pursuit-evasion have simplified the situation along three lines. First, pursuit-evasion has been studied as variants of the Game of Two Cars. 1~7 In these, the aircraft are assumed to fly at constant altitude and speeds; their optimal paths are composed of circular arcs and straight line segments. Second, in the energy approach,8'11 the aircraft are modeled realistically in terms of their energies and relative heading. However, the relative positions of the aircraft are ignored. The differential game studied is again trimensional. Third, in the dynamic modeling12 and numerical approaches, 13'16 combat is formulated as a game of prescribed duration thus avoiding discontinuous and exceptional surfaces. Even then, near-optimal closed-loop solutions have only been obtained for simple examples.16 Thus an analysis of constant-altitude pursuit-evasio n with varying aircraft speeds portrays the situation realistically. It also takes on, for the first time, a free-time combat game with five state variables. The game is formulated in a real space whose origin and axes change from party to party. The equations decompose into two sets, one for each aircraft. These are coupled in terms of the terminal quantities alone. This simplifies the solution in the small and effects significant computational economy. Moreover, each aircraft's optimal motion in real space can be described in terms of an extremal trajectory map (ETM) which is independent of the adversary and of role. The ETM thus serves as a measure of an individual aircraft's combat performance.