The first phase of an investigation of a two-target game, representing an air combat with boresight limited all-aspect guided missiles is summarized. The results, obtained by using a line of sight coordinate system, are compared to a similar recently published work. The comparison indicates that improved insight, gained by using line of sight coordinates, allows to discover important new features of the game solution

Shinar, J.

English

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TECIANION RESEARCH AND DEVELOPMENT FOUNDATION,L-iD.
TECFINION CITY, FIAIFA 32000
ISRAEL
An Extended Two-Target Differential Game Model -for
Medium-Range Air Combat Game Analysis
Semiannual Scientific Status Report, covering the period of
1 Nov. 1984 - 30 April 1985
.IUL 1D85
RECEIVED
*'M'S71'FACILITy,,
DEPT;
Principal Investigator: Prof. Josef Shinar,	 1
Department of Aeronautical L:ngineering
BY
NASA STI
DATE:
DCAF ND. P,
BY
AIM,
1
(NASA-CR 175968)AN EXTENDED 1R0-TARGET
	 N65-25667
DIFFERENTIAL GAME MODEL FOR MEDIUM-RANGE AIR
COMBAT GALE ANAIYSIS Semiannual Status
Report, 1 Nov. 1984 - 30 Apr. 1985 (Technion G3	 UnclasResearch and Development Foundation) 14 p W166 21628
NASA STI
ESA • SDS
NASA Cooperative Agreement No. NCCW-5
Technion Research No. 160-973.
An Extended Two—Target Differential Game Model for
Medium
—Range Ai r Combat Gane Analysis*
ABSTRACT
This interim report summarizes the first phase of an
investigation of a two —target game, representing an air combat
with boresighL• limited all—aspect guided missiles. The results,
's
obtained by using a line o •f sight coordinate system, are compared
	 q
to a similar recently published work. The comparison indicates
that improved insight, gained by using line of sight coordinates,
allows to discover important new -Features of the game solution.
Detailed results will be presented in a verbal briefing at
j	 NASA Ames Research Center, Moffett rield, Cal. in August 1905,
NASA Cooperative Agreement No. NCCW-4
Principal InvestigaL• or- Prof. Josef Shinar, Department of
Aeronautical Engineering. Technion, Israel Institute of
Technology, Haifa, Israel.	 f(
1	 ^i
ORIGINAL VAQE 15'
OF POOR QUALITY
1. 1r 1:roduction
'rho research activity in the first six months of the
investigation was oriented to evaluaL• e the usefullness of a .line
of sight coordinate system and to develop a computer program
which generates barrier trajectories in these coordinates. As a	 j
first example, a simple pwr si',t-evasion game (the game of two 	 g
t
identical cars) with a circular target set solved. This sa1uL•ion	
ppi
was presented at the 26 Israel Annual Conference on Aviation and	 k
}
Astronautics (Feb. 19M, and demonstrated the advantages of
using , a line of sight coordinate system. The copy of the paper
(Ref. 1) is enclosed with this report. 'rho convenient coordinate
(
system allowed to discover a new type o•f singular surface, which
was overlooked in at previous investigation of the same game. 	 1
Guided by the SUCCeSsfUl results of the above step the next
phase was oriented to use the line of sight coordinate system in
a two-target game analysis. The example selected -For this phase
represents an air combat between -two aircraft with boresi.ghl•
`i
limited all aspect guided missiles. One o •f the reasons for -this
selection was the information obtained from Prof. J.V. Lrealwell 	 1
at the beginning of the investigation, that a comparable work
(Ref. 2) had been submitted for publication. It se-_med that
testing the validity of the new approach, used in the present
investigation, b ang	 y	 appropriate comparison is of a major
importance.
;in this interim report 'the main findings of  "first-firing"
two-target game analysis, including the results o•f the comparison
2
with Ref. 2 1 are sumoarized. The detailed solution will be
presented in a verbal briefing at NASA Ames Research Center in
Y
Auiust 1985, as well as in a scientific paper, which is now in
preparation. (The paper has been invited for a special issue of
Computer and Mathematics with Applications on Pursuit- 	
i
Evasion Differential Games by the Guest Editor Dr. Y. Yavin.)
2. Problem Formulation
The dynamic model of the "first •-firing" two-target game to
be investigated is of the game of two identical cars, used in
Ref.1 and expressed by Egs.(12)-^14) there. The target setsset  of
the game are line segments aligned with the respective velocity
vectors and limited by the minimum and maximum effective ranges
of the guided weapon system. The variables 
of 
the game and the
target sets are depicted in a fixed (inertial) coordinate system
in Fig. 1. In Fig. 2 the usable parts o•f the target sets are
shown in a line of sight coordinate system. As it is shown in
the figure, a part of the min mum range boundary is determined by
the rate of turn constraint,
Ri = sin 0_, :• R,nt,	 i * i	 i"i=1,2.
The objective of the two-target game analysis is to
determine the winning zones of each player (1,2) as well as the 	 E1
regions of draw and eventual mutual kill. The -First phase of
L	
's
{r. .
such an analysis is the solution of the respective qualitative
pursuit-evasion games (1 vs. 2 and 2 vs. 1).
3. Pursuit-Evasion Same Solution (i pursuing 2)
The starting point is to determine the optimal strategies at
the boundary of the usable parts of the target set.
It turns out that on the "maximum-range boundary" (R = Sm.,)
only . a single point A, (0, = 0 1 0= = ± n) can serve as a terminal
point for optimal strategies. It indicates that all barrier
trajectories terminating at maximum range do it in a "tail-
chase".
On the "minimum range boundary" every point can serve as an
end point of an optimal barrier trajectory. Moreover, optimal
strategies can terminate an both sides of the zero-thickness
target set.
Based on the optimal strategies, determined along the
boundary of the usable parts of the target sets, optimal
trajectories can be integrated backwards in time. In the line of
sight coordinate system the backward integration can be performed
analytically yielding a closed form solution, as outlined in
Section 3.4 of Ref. 1 (see Egs.(45)-(71) there).
The only end point (A,) on the "maximum-range" boundary is a
junction of 4 universal lines 1 2 for each player). These
universal lines are also optimal trajectories And can be obtained
4
ti
"0 . 
 a,
by backward integration. Moreover, at any point of a universal
line two optimal trajectories coming from opposite directions
meet. These regular barrier trajectories are also obtained in a
closed form by similar backward integration. A part o•f these
trajectory Families, ending on the 4 universal lines, intersect
each other along dispersal lines and generate the "maximum-
range barrier section".
The barrier trajectories, emanating backwards from both
sides of the "minimum-range boundary" generate two symmetrical
surface sections (one for
	 0 and the other for
	 Ox < 0) and
the majority of them itersect wit!i trajectories coming from the
"maximum-range barrier". Thes2 two surface sections can be
called the "minimum-range barrier section."
The maximum-range and minimum-range barrier section do not
form yet a closed barrier surface. Near to the "tail-chase" zone
(I0=I '4 n) the "minimum-range barrier section" ends before
reaching the "maximum-range barrier section". This gap is closed
by a third type of barrier section, baptized as the
"interconnecting barrier section". The origin of this third
barrier section is a strategy change (switching line) of the
pursuer. The optimal trajectories generating this
"interconnecting barrier section" emanate from an evader
dispersal line and continue towards a "tail-chase" due to
the above mentioned "strategy switch" of the purs!,ier.
5
,6
u ►^PGP;ZAL f^F,t; is	 por POOR QUACI Y.
It can be thus concluded that the barrier of the 1 against 2
pursuit-evasion game is a closed surface generated by three
different types of sections. It encloses the "winning zone" of
the pursuer and separates it from the other part Of the state
space.
The barrier of -the second pursuit-evasion game (2 pursuing
1) is identical if the roles of (As.rh1 and V)- are interchanged.
4. Two-Target Same Solution
The respective barriers of the two pursuit-evasion games
intersect along lines in the planes of symmetry 4M1 = ch- and
(G1 = -d)-. These planes are in -Fact semipermeable surfaces of
the "first-firing" two-target game and intersect along the R
axis (y• 1 = $, = 9). The consequence of barrier° intersection is
that a part of the respective "winning zones", being e,tclosed by
the opponents' barrier, disappears. The remaining regions have
to be considered in the i•,wo-target games analysis as only
candidai:es for an eventual winning. A part of each "candidate
winning zone" may turn out to belong to the "region of mutual
kill".
Simultaneous mutual kill occurs at any point along the
target set intersection Wit = 4)- _ 01 Am1n w R $ RmW„). Optimal
trajectories leading to this "mutual kill target set” generate
a barrier separating the "region of mutual kill" from the
`0%
remaining "winning zones". At the maximum range point "F"
(R = A," .,,, 4), = cP= = 0) four singular trajectories, characterized
by "pure-pursuit" strategies the = (h.+ = 0 U = 1 1 2) meet.	 1
The singular trajectories serve also as universal lines
for one of the players. At any point along the universal lines
two regular trajectories meet. These trajectories -farm a surface
which intersects the "maximum-range barrier sections" of the
original pursuit-evasiun games along new dispersal lines.
This pyramid type surface, generated by all the trajectories
ending at the point " P" separate the "region of mutual kill" from
the respective " winning zones" and can be called as the "internal
mutual kill barrier sections."
H second surface, section that separates the "region of
mutual kill" from the "zone of draw", consists of parts of
"maximum-range barrier sections" of the pursuit -evasion games.
It also has a pyramid shape and can be called as the "cl;ter
mutual kill barrier section". At any point on this barrier
section one of the players can select either to be an evader - and
in this case the game terminates with a marginally successful
•t-ail chase escape - or to act agressively and then the trajectory
enters to the "region o•f mutual kill". T!ie aggressive strategy
in 'the "region of mutual kill" is leading towards a "head-on"
(fig = r^ 2 = 0) encounter at R = R, 	 Inside this region the
players are committed to a mutual kill and any of them who tries
to evade will be killed by the opponent. A mutually agreed
disengagement leading to a draw does not seem to be likely,
7
abecause in this case any of the players can deceive his opponent
at any time and drive the trajectory closer to his own "winning
zone"
It can be summarized that the "region of mutual kV.1" is
enclosed by two pyramid shape barrier sections. The remaining
"winning zones" of the two target game are substantially redurod
compared to the pursuit—evasion game solution. They are rather
limited to initial conditions of clear directional advantage with
J
respect to the opponent.
l	 I
1
5. Comparison to Ref. 2
The results of the above outlined two — target game analysis
were compared to a recently published paper (Ref. 2). In that
parrer a similar problem is investigated, but without any
restriction on the minimum firing range, allowing there-Fore
collision type mutual kills. In all other respects 'the game
models are identical.	 j
I,
Most of the results of the two independent investigations
confirm each other. There are, however, two elements discovered
in the present analysis which could not be found in Ref. 2. It
has been verified that these elements are not connected to the
difference in the minimum range definition.
The first- element• is the existence of a strategy switch of
the pursuer in the pursuit—evasion game. It leads to generate
r
lkw: vt
the "interconnecting barrier section" and enlarges the "winning
zone" of the pursuer compared to the results of Ref. 2.
The second element relates to the "mutual kill barrier"
The "pure pursuit" strategy Fo.v,d to be optimal along the
universal lines of the internal "mutual kill barrier
section" is not mentioned in Ref. 2. This strategy leads to
reduce the "region of mutual kill" and consequently enlarges the
remaining "winning zones".
'	 b
6. Conclusions.
In the first six months of the investigation two ma.ior steps
were accomplished. The methodology of game analysis in a line of
sight coordinate system was developed and tested by comparing
results with similar game models analyzed in previous studies.
f
The new methodology allows to generate barrier trajectories in a
closed -form and to identify dispersal lines and other types of
singularities. It provides an improved insight for game analysis
as it was demonstrated by discovering features which had been
tC
overlooked in previous works.
The validated methodology can be thus considered as a ready
tool for the forthcoming phases of the three years research
program outlined in the original research proporsal.
9
(K" "
References
1. Davidowitz, A., and 8hinar, J.: "Pursuit-evasion dame
analysis in a line of sight coordinate sysLem". Proceedings
of the 27 uh Israel Annual Conference on Aviation and
Astronautics, Feb. 1985. pp. 108-116.
2. Merz, A.W.: "To pursue or to evade? - That iG the Question".
J. of Guidance and Control, Vol. 8, Ila. 2 7 1905 1 pp.16i-166.
10
it
LIST OF FIGURES
Fl.g. 1. Two-target game geome # -y in a fixed coordinate system.
Fig. 2. Target sets in the line of sight coordinate system.
11
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An extended two-target differential game model for medium-range air combat game analysis

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