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

1. 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

2. Contents • Problem setup • Support time game • Modeling the probabilities related to the payoffs • Numerical example • Real time solution of the support time game • Conclusions

3. 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 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

4. Problem setup Phase 3: locked Phase 2: extrapolation Phase 1: support The problem: optimal support times tB, tR?

5. 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

6. Support time game • Gives game optimal support times tB and tRas 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 Blue’s probability of survival Blue missile’s probability of hit Blue: Red: Blue missile’s probability of guidance Blue missile’s probability of reach Blue missile’s prob. of hit = ´

7. 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

8. 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

9. 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/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

10. Solution 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: Support time of Red wB=0 Support time of Blue

11. support phase altitude, km extrapolation phase locked phase x range, km Example trajectories Red (left), wR=0.5, supports 12.4 seconds Blue (right), wB=1.0, supports 5.0 seconds y range, km

12. Real time solution • 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: optimized interpolated Support time of Red Support time of Blue

13. 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