Game%20Optimal%20Support%20Time%20of%20a%20Medium%20Range%20Air-to-Air%20Missile

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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

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Problem setup

• One-on-one air combat with missiles
• Phases of a medium range air-to-air missile
• Target position downloaded from the launching a/c
• In blind mode target position is extrapolated
• Target position acquired with the missiles own
  radar
• In phase 1 (support phase), the launching a/c
  must keep the target within its radars 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
Phase 1 support
The problem optimal support times tB, tR?
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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 , iB,R reflect the
  players risk attitudes

Blues probability of survival
Blue missiles probability of hit
Blue
Red
Blue missiles probability of guidance
Blue missiles probability of reach
Blue missiles prob. of hit

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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
closing velocity at distance df
optimize minimize closing velocity
extrapolate
predetermined support maneuver
Probability of guidance
tracking error at
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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)
• u Blue a/cs controls, x states of Blue a/c
  and Red missile,
• f state equations, g constraints
• Initial state vehicles states at the end of
  Blues support phase
• Direct multiple shooting solution method gt
• time discretization and nonlinear programming

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Solution of the support time game

• Reaction curve
• Players optimal reactions to the adversarys
  support times
• Solution Nash equilibrium
• Best response iteration
• Red player
• Blue player

Support time of Red
wB0
Support time of Blue
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Example trajectories

Red (left), wR0.5, supports 12.4 seconds Blue
(right), wB1.0, supports 5.0 seconds
y range, km
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Real time solution

• Off-line
• Solve the closing velocities and tracking errors
  for a grid of initial states
• In real time
• Interpolate CVs and TEs for a given
  intermediate initial state
• Apply best response iteration
• Red
• Blue

optimized
interpolated
Support time of Red
Support time of Blue
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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