03-09-2012, 11:32 AM
A TWO-TARGET GAME ANALYSIS IN LINE-OF-SIGHT COORDINATES
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Abstract
The engagement between two aggressively operating similar aircraft armed with boresight
limited "all-aspect" missiles is modelled as a two target differential game between "two identical cars".
By using a line-of-sight coordinate system the symmetry of the problem can be exploited leading to a
reduced complexity in the game of kind analysis. The barrier trajectories are obtained in a closed form
allowing us to generate closed barrier surfaces with a reasonable computational effort. These barrier
surfaces enclose the "winning zone" of each player and the "region of mutual kill". The analysis reveals
several new features not encountered in previous studies.
INTRODUCTION
The motivation of the present paper is to use a relatively simple mathematical model for the
qualitative analysis of an air combat engagement between two aggressively operated similar fighter
aircraft equipped with modern guided missiles. The classical pursuit-evasion game formulation [1],
embedding the notion of an a priori role determination [2] is not suitable for such a scenario.
Fortunately, the "two-target game" concept, briefly mentioned in earlier works [3] and reintroduced
a few years ago [4], provides the appropriate mathematical frame for meaningful analysis
of such engagements. In the past several works [5 8] used this formulation.
In a two-target game each player wishes to drive the state of the game to his own target set,
while avoiding the target set of the opponent. This definition implies the qualitative (game of kind)
nature of the solution, namely, partitioning the game space into the respective "winning zone" of
each player, the region of "mutual kill" (if the intersection of the target sets is reached) and the
zone of inconclusive "draw" (if neither target set reached in finite time). The qualitative two-target
game solution is a necessary step towards the more complex quantitative (game of degree) combat
game analysis [9], having the objective to find "winning strategies" by optimizing different pay-off
functions in the different regions.
TWO-TARGET GAME SOLUTION
In solving the two-target game, formulated in Section 2 of this paper, the method of systematical
construction of semipermeable surfaces from the BUP of the target set (used in the solution of the
pursuit-evasion game of kind in Section 3) will be further applied.
In the two-target game the respective target set are defined by equation (8). Compared to the
previously solved p--e game the target set of player 1 is modified by excluding from it the line
segment ~be = 0. This exclusion generates in fact two new BUPs in T~ (~bt = 0; 4) 2 = 0 + and ~b~ = 0;
qS:=0 with I ~< R ~<L). From these BUPs, additional semipermeable surfaces have to be
constructed.
CONCLUSIONS
In the present paper a planar two-target game between two constant speed identical vehicles with
identical target sets is analyzed using a line-of-sight coordinate system. The target sets, intending
to serve as a simplified model of operational guided missiles, are characterized by having a non-zero
minimum range limit. In this respect the present model is different from the one analyzed in a recent
paper [12] in a conventional coordinate system attached to one of the players. For this reason the
comparison of the two studies has only a limited scope. The comparison confirms, however, a great
part of the results, such as the determination of the "maximum range barrier section" and the
"region of mutual kill".