S ystems Analysis Laboratory Helsinki University of Technology 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

S ystems Analysis Laboratory Helsinki University of TechnologyContents 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

S ystems Analysis Laboratory Helsinki University of Technology Problem setup The problem: optimal support times t B, t R ? Phase 1: support Phase 2: extrapolation Phase 3: locked

S ystems Analysis Laboratory Helsinki University of Technology 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

S ystems Analysis Laboratory Helsinki University of Technology Support time game Gives game optimal support times t B and t R 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 Blue’s probability of survivalBlue 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 =

S ystems Analysis Laboratory Helsinki University of Technology Modeling the probabilities p r and p g Probability of reach p r : −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 p g : −Depends, i.a., on the launch range, radar cross section of the target, closing velocity, and tracking error

S ystems Analysis Laboratory Helsinki University of Technology p r and p g in this study predetermined support maneuver optimize: minimize closing velocity extrapolate Probability of reach closing velocity at distance d f Probability of guidance tracking error at

S ystems Analysis Laboratory Helsinki University of Technology Minimum closing velocities For each (t B,t R ), the minimum closing velocity of the missile against the a/c at a given final distance d f (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

S ystems Analysis Laboratory Helsinki University of Technology 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: w B =0 Support time of Blue Support time of Red

S ystems Analysis Laboratory Helsinki University of Technology support phase extrapolation phase locked phase x range, km altitude, km y range, km Example trajectories Red (left), w R =0.5, supports 12.4 seconds Blue (right), w B =1.0, supports 5.0 seconds

S ystems Analysis Laboratory Helsinki University of Technology 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: Support time of Blue Support time of Red interpolated optimized Real time solution

S ystems Analysis Laboratory Helsinki University of TechnologyConclusions 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