Game Optimal Support Time of a
Medium Range Air-to-Air Missile
Janne Karelahti, Kai Virtanen, and Tuomas Raivio
Systems Analysis Laboratory
Helsinki University of Technology
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Contents
• Problem setup
• Support time game
• Modeling the probabilities related to the payoffs
• Numerical example
• Real time solution of the support time game
• Conclusions
S ystems
Analysis Laboratory
Helsinki University of Technology
Problem setup
•
One-on-one air combat with missiles
•
Phases of a medium range air-to-air missile:
1. Target position downloaded from the launching a/c
2. In blind mode target position is extrapolated
3. Target position acquired with the missile’s own radar
•
In phase 1 (support phase), the launching a/c must
keep the target within its radar’s gimbal limit
•
Prolonging the support phase
−
Shortens phase 2, which increases the probability of hit
−
Degrades the possibilities to evade the missile possibly fired
by the target
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Problem setup
Phase 3: locked
Phase 2: extrapolation
t
Phase 1: support
R
B
tlock
 on
tB
tB
tR
The problem: optimal support times tB, tR?
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R
tlock
 on
Modeling aspects
• Aircraft & Missiles
− 3DOF point-mass models
− Parameters describe identical generic fighter aircraft and missiles
− Missile guided by Proportional Navigation
− Assumptions
−
−
−
−
−
−
Simultaneous launch of the missiles
Constant lock-on range
Target extrapolation is linear
Missile detected only when it locks on to the target
State measurements are accurate
Predefined support maneuver of the launcher keeps the target
within the gimbal limit
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Support time game
•
•
•
•
Gives game optimal support times tB and tR as its solution
The payoff of the game  probabilities of survival and hit
The probabilities are combined as a single payoff with weights
The weights wi  0,1 , i=B,R reflect the players’ risk attitudes
Blue’s probability of survival
Blue missile’s probability of hit
B
R B R
B
B B R
w
(
1

p
(
t
,
t
))

(
1

w
)
p
(t , t )
Blue: max
h
h
B
t
Red:
R
B B R
R
R B R
max
w
(
1

p
(
t
,
t
))

(
1

w
)
p
h
h (t , t ),
R
t
phB (t B , t R ) p gB (t B , t R ) prB (t B , t R )
Blue missile’s prob. of hit =
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Blue missile’s
Blue missile’s
probability of guidance ´ probability of reach
Modeling the probabilities pr and pg
• Probability of reach pr:
− Depends strongly on the closing velocity of the missile
− The worst closing velocity corresponding to different support
times  a set of optimal control problems for both players
• Probability of guidance pg:
− Depends, i.a., on the launch range, radar cross section of the
target, closing velocity, and tracking error
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pr and pg in this study
Probability of reach pr (t , t )  closing velocity at distance df
B
t
B
f
B
R
optimize: minimize closing velocity vc (t Bf )
df
t
B
extrapolate
t0
R
tlock
 on
predetermined
support maneuver
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tR

R
x BA (tlock
 on )
t0
Probability of guidance
R
xˆ BA (tlock
 on )
p (t , t )  tracking error  at
R
g
B
R
R
x RM (tlock
 on )
R
tlock
 on
Minimum closing velocities
• For each (tB,tR), the minimum closing velocity of the missile against
the a/c at a given final distance df (here for Blue aircraft):
B
min
v
(
t
c
f )
B B
u ,t f
s.t. x  f ( x, u B , t ), t  [t B , t Bf ]
g ( x, u B )  0
r (t Bf )  d f  0
• u = Blue a/c’s controls, x = states of Blue a/c and Red missile,
f = state equations, g = constraints
• Initial state = vehicles’ states at the end of Blue’s support phase
• Direct multiple shooting solution method =>
time discretization and nonlinear programming
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Solution of the support time game
w  0,0.1,0.2,...,1.0
15.3
Support time of Red
• Reaction curve:
− Player’s optimal reactions
to the adversary’s support
times
• Solution = Nash equilibrium
− Best response iteration
R
• Red player: w  0.5
• Blue player:
13.3
Reaction curve of Red
11.3
9.3
Reaction curves of Blue
7.3
wB=0
B
5.3
5.0
7.0
9.0
11.0
13.0
Support time of Blue
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15.0
Example trajectories
altitude, km
12.0
support phase
10.0
8.0
extrapolation phase
6.0
4.0
0
5.0
10.0
x range, km
locked phase
15.0
4.0
Red (left), wR=0.5, supports 12.4 seconds
Blue (right), wB=1.0, supports 5.0 seconds
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y range, km
20.0
0.0
Real time solution
• Off-line:
• In real time:
− Interpolate CV’s and TE’s
for a given intermediate
initial state
− Apply best response
iteration
optimized
Support time of Red
− Solve the closing velocities
and tracking errors for a
grid of initial states
15.0
• Red: w  0.5
x0R  18650, y0R  0, h0R  10000
B
• Blue: w  0.5
x0B  0, y0B  0, h0B  [9600,10000]
R
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13.0
h0B  10000
interpolated
11.0
9.0
7.0
h0B  9600
5.0
7.0
9.0
11.0
13.0
Support time of Blue
Conclusions
• The support time game formulation
− Seemingly among the first attempts to determine optimal support times
• AI and differential game solutions: the best support times based on
predefined decision heuristics
• Discrete-time air combat simulation models: predefined support times
• Pure differential game formulations are practically intractable
• Utilization aspects
− Real time solution scheme could be utilized in, e.g.,
• Guidance model of an air combat simulator
• Pilot advisory system
• Unmanned aerial vehicles
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Analysis Laboratory
Helsinki University of Technology