S ystemsAnalysis LaboratoryHelsinki University of Technology

Game Optimal Support Time of a Game Optimal Support Time of a Medium Range Air-to-Air MissileMedium Range Air-to-Air Missile

Janne Karelahti, Kai Virtanen, and Tuomas RaivioSystems Analysis Laboratory

Helsinki University of Technology

S ystemsAnalysis LaboratoryHelsinki University of Technology

ContentsContents

• Problem setup

• Support time game

• Modeling the probabilities related to the payoffs

• Numerical example

• Real time solution of the support time game

• Conclusions

S ystemsAnalysis LaboratoryHelsinki University of Technology

Problem setupProblem 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/c2. In blind mode target position is extrapolated3. 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

S ystemsAnalysis LaboratoryHelsinki University of Technology

Problem setupProblem setup

RtBt

Bt

The problem: optimal support times tB, tR? R

onlockt

Bonlockt

RtPhase 1: support

Phase 2: extrapolation

Phase 3: locked

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Modeling aspectsModeling 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 gameSupport 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 , i=B,R reflect the players’ risk attitudes

),(),(),(

),,()1()),(1(max

),()1()),(1(max

RBBr

RBBg

RBBh

RBRh

RRBBh

R

t

RBBh

BRBRh

B

t

ttpttpttp

ttpwttpw

ttpwttpw

R

B

Blue’s probability of survival Blue missile’s probability of hit

Blue:

Red:

1,0iw

Blue missile’s probability of guidance

Blue missile’s probability of reach

Blue missile’s prob. of hit =

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Modeling the probabilities Modeling the probabilities pprr and and ppgg

• 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

S ystemsAnalysis LaboratoryHelsinki University of Technology

pprr and and ppg g in this studyin this study

)( Ronlock

RM tx

),( RBBr ttp

fd

Bt

predeterminedsupport maneuver

optimize: minimize closing velocity )( Bfc tv

),( RBRg ttp

Rt

extrapolate

Probability of reach closing velocity at distance df

0t

0tProbability of guidance tracking error at

Ronlockt

)(ˆ Ronlock

BA tx

)( Ronlock

BA tx

Ronlockt

Bft

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Minimum closing velocitiesMinimum 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):

• 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

..

)(min,

ts

tv Bfc

tu Bf

B

0)(

0),(

],[),,,(

fBf

B

Bf

BB

dtr

uxg

ttttuxfx

S ystemsAnalysis LaboratoryHelsinki University of Technology

5.0 7.0 9.0 11.0 13.0 15.05.3

7.3

9.3

11.3

13.3

15.3

Reaction curves of Blue

Reaction curve of Red

Solution of the support time gameSolution of the support time game

• Reaction curve:− Player’s optimal reactions

to the adversary’s support times

• Solution = Nash equilibrium− Best response iteration

• Red player:• Blue player:

5.0Rw

0.1,...,2.0,1.0,0Bw wB=0

Support time of Blue

Sup

port

time

of R

ed

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0

5.0

10.0

15.0

20.0 0.0

4.0

4.0

6.0

8.0

10.0

12.0

support phase

extrapolation phase

locked phasex range, km

altit

ude,

km

y range, km

Example trajectoriesExample trajectories

Red (left), wR=0.5, supports 12.4 secondsBlue (right), wB=1.0, supports 5.0 seconds

S ystemsAnalysis LaboratoryHelsinki University of Technology

5.0 7.0 9.0 11.0 13.0

7.0

9.0

11.0

13.0

15.0• Off-line:− Solve the closing velocities

and tracking errors for a grid of initial states

• In real time:− Interpolate CV’s and TE’s

for a given intermediate initial state

− Apply best response iteration

• Red:

• Blue:

5.0Rw

5.0Bw10000,0,18650 000 RRR hyx

]10000,9600[,0,0 000 BBB hyx Support time of Blue

Sup

port

time

of R

ed96000 Bh

100000 Bh

interpolated

optimized

Real time solutionReal time solution

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ConclusionsConclusions• 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