φ E V E E γ E V P γ P R θ Reference

Transcription

1 Dierential Games and Symbolic Programming to Calculate a Guaranteed Aircraft Evasion in Modern Aerial Duels Stephane Le Menec Institut National de echerche en Informatique et Automatique BP Sophia-Antipolis Cedex - FANCE Tel : (33) / Fax : (33) lemenec@sophia.inria.fr 33rd CDC Abstract The improvement of guidance possibilities of medium range missiles with new missiles like the Mica/Amraam 1 increases the number of phases in aerial duels and implies new and more complex shooting and escape strategies. We rst dene aerial medium range duels, before studying them as dierential sub-games. Then we explain how to design an evasion strategy to protect an aircraft against the missile of an opponent. This article describes this evasion strategy built with hypothetic reasoning and barriers of dierential sub-games. This evasion strategy is used in the several simulations of a decision support system developed with expert system techniques. Finally we give an example of this decision support system called Adam 2. This study results from a collaboration between MATA- DEFENSE, which is interested in new methodologies for pilot decision support system, and INIA (DET contract n 0 90=532 : \Decision Support System for Aerial Duels"). Keywords : Dierential game theory, hypothetic reasoning, simulations. 1 Introduction The object of game theory is the mathematical study of situations containing a conict of interests [5]. One of the applications of game theory is the study of pursuit games such as cat and mouse generic pursuit games. A problem exists however in computing the initial states of the game for which a player can be sure to win against any maneuver of its opponent (qualitative study of a game). For example, how far from its shelter can the mouse go and still be sure to get back safety as soon as 1 Missile d'interception de Combat et d'auto-defense/advanced Medium ange Air to Air Missile 2 Aide au Duel Aerien Moderne the cat arrives? Such a qualitative study with in the theory of dierential games consists of two parts. First we have to describe in continuous time the dynamics of the pursuit, then, we calculate the frontier, or barrier separating the state space related to the outcome of the optimal game played with the dierent initial conditions of the state space [2]. The solution of such a game gives controls that allow the players to keep the state of the game on the favorable side of the game barrier. In the case of the pursuit of an aircraft by a self guided short range missile, we consider the aircraft and the missile as two players in order to calculate a capture zone and an escape zone (or non capture zone) giving the congurations leading to the destruction of the aircraft or the loss of the missile. We calculate the initial conditions of the pursuit (characterized by variables of the game such as the missile speed, the range between the missile and its target, the angle of sight of the missile...) allowing the aircraft to evade any guidance law of the missile and allowing the missile to destroy any maneuvering target. The theory of games also solves air combat games between two aircrafts with re and forget missiles [3]. But these duels set new problems such as that of role determination of the aircraft, since an aircraft plays both as a pursuer and as an evader [7]. New medium range air to air missiles with more complex guidance modes have appeared, with undetectable shooting and early ight phases. These new missiles increase the number of phases of a modern aerial duel and introduce new more complex shooting and escape strategies. Therefore we are interested in developing algorithmic methods to study these new duels, which are dicult to study merely with the classical techniques of game theory. We have studied the dierent phases of a duel between two aircrafts with medium range missiles as sub-games to realize simulations of a modern aerial duel using information of sub-games barriers. These simulations developed with advanced programming languages allow us to design a decision support system called Adam advising strategies

2 to one of the two aircrafts to take advantage of the errors of the other player. This new method allows us to solve a complex game, studied up to now only with simulations and heuristic methods [8]. 2 The aerial duel We consider a medium range duel opposing two identical aircrafts. We will call the blue aircraft (BA) and the red aircraft (A). Each aircraft has a Mica/Amraam, called blue missile (BM) for BA and red missile (M) for A. At the beginning of the duel, each pilot is far enough from its opponent to decline the battle if he wants. There is engagement of the duel only if both aircrafts have the same chance to win, if they are in a head-to-head duel and at the same altitude. This explains why we restrict this study to a co-planar game. The aircraft begins the duel with a pre-launch phase at a range of about twice their shooting range. These medium range missiles use several guidance modes. Our model of the Mica/Amraam ies in informed mode { with its Ad 1 locked on the target { or in Lam 2 Moreover we say that an aircraft updates the information of the missile it has red by informing it about the target's position (for example, BM is in Lam mode if BM and A are in the radar's cone of BA). or in inertial guidance Then the missile ies towards the estimated target's position which is extrapolated from its previous ones. After the shooting a missile can y in Lam mode, then continue in inertial guidance before locking its Ad at the end of its ight. The interest of guiding a missile from the launching aircraft consists in taking advantage of the aircraft's powerful radar and therefore to shoot earlier. With a Mica/Amraam, an aircraft stays further away from its opponent. But just after shooting, it must play so that its radar cone still sees its missile and its target (post-launch phase). Hence the aircraft can no longer give up the duel just after shooting, but must keep on ying towards the opponent, until its missile locks on the target or is well positioned so that it can lock on by inertial guidance. We say that a Mica/Amraam ies in discreet mode when it uses a Lam or inertial guidance, and in non discreet mode when its Ad is locked. To shoot a missile in discreet mode makes the shooting undetectable. An aircraft sees the opponent's missile only when it locks on, by detecting its Ad. 1 \Auto-Directeur" or missile's active radar seeker 2 \Liaison Avion Missile" or uplink when the aircraft forwards information to the missile The Mica/Amraam is a coasting missile, which squanders more energy when maneuvering, and particularly when maneuvers are violent and numerous. 2.1 esult of the duel The outcome of the game is given for BA. A victory of BA corresponds to a defeat of A (win outcome) and a defeat of BA to a victory of A (lose outcome). A win outcome corresponds to the destruction of A with a successful evasion of BA, while in a lose outcome, BA is the destroyed aircraft. A duel can end by other outcomes too. We speak about a draw outcome when no missile reaches its goal and about a mutual kill outcome if both aircrafts are destroyed. In aerial duels with medium range missiles, the destruction of both aircrafts does not happen necessarily at the same time on account of the duration of the missiles' ight. 2.2 Necessity of simulations The complex roles of the aircrafts which introduce role determination problems, but also the multiplicity of game's phases (pre-launch phase, post-launch phase, evasion phase) and the number of possible outcomes (win, lose, evasion of both aircrafts, destruction of both aircrafts) make aerial duels with Mica/Amraam complex to study. We also have to decide whether a player prefers to end a duel by a draw outcome or by a mutual kill outcome, if it can not win. Moreover, both pilots can play cooperative strategies to obtain a draw outcome instead of ending by a mutual kill outcome, if both players prefer a draw outcome to a mutual kill outcome. The theory of dierential games is principally interesting in non cooperative zero sum games. These remarks make a duel like the Mica/Amraam duel practically untractable by techniques of game theory. It explains why we study this game by simulations using the barriers of the sub-games presented below [9]. 3 Sub-games First, we have calculated several ring domains of our Mica/Amraam model. These shooting domains describes below are presented in details in the next paragraphes. Since there exist dierent guidance modes of medium range missiles we dene several sub-games to study some parts of the complete duel. One of these sub-games corresponds to the nal short distance game when the missile has its Ad locked on its target. Other sub-games describe the initial phase with Lam guidance. Hypotheses on the duel (see section 3.1) allow one to dene from the nal phase with locked Ad the end conditions of previous sub-games. We study the pursuit sub-games between M and BA (These sub-games have to be seen also as pursuits between BM and A since the missiles and the aircrafts are identical) : 1. short range pursuit sub-game between M with its Ad locked on BA. 2

3 2. medium range sub-game with perfect information (each player knows the state of the game) giving the capture and evasion conditions of M in Lam guidance pursuing BA 3. sub-game identical to the previous one with a restricted evasion of BA. This sub-game describes the evasion of an aircraft staying in Lam guidance with its missile (realistic shooting domain) 4. sub-game identical to the sub-game 2 with no escape of BA. The barrier of this sub-game gives the maximum shooting range of a missile. All missiles with their targets in the escape zone of this sub-game are necessarily lost. These capture zones of pursuits between a missile and an aircraft are useful only for missiles in ight. Saying that BA is in the non capture zone of the sub-game 2 of a non red M makes no sense if A does not shoot immediately. 3.1 Hypotheses In the study of the Mica/Amraam duel, we have made the following assumptions : 1. An aircraft executes only one evasion which is denitive. 2. An aircraft can not shoot during its evasion maneuver. 3. An aircraft can not maintain the Lam during its evasion. 4. An aircraft evades systematically when it is locked by the enemy's Ad. 5. An aircraft does not shoot after the opponent's evasion. 6. In Lam mode, a missile has the same information than in autonomous guidance with its Ad locked. 7. An aircraft does not detect the opponent's Lam. The reasons of these hypotheses and the complete description of the sub-games are related in detail in [6]. 3.2 The short range sub-game The geometry of planar pursuit dening the state variables of the game is depicted in gure 1. The missile P possessing a velocity V P and a minimum admissible turning radius r c is pursuing in planar motion an aircraft E, assumed to be ying with a constant velocity V E and without constraint on its turning rate. The two constants a et b describe the missile drag. u (with?1 u 1) and E are respectively the control of the pursuer and the control of the evader. The game terminates with capture when the pursuer approaches the evader to the distance = f. P V P γ P φ P θ V E E γ E φ E eference Figure 1: Geometry of planar pursuit centered on P The kinematic equations are : _V P =?V 2 P (a + bu 2 ) (1) _ = V E cos ( E? )? V P cos ( P? ) (2) _ = [V E sin ( E? )? V P sin ( P? )] (3) _ P = V P r c u (4) The number of independent variables can be reduced to three variables, which is the minimum representation of the system, if use is made of the pursuer and evader relative angles with respect to the line of sight : P = P? (5) E = E? (6) The use of the reduced system complicates the analysis, but allows the representation of the capture zone and the non capture zone in a 3D state space. _V P =?V 2 P (a + bu 2 ) (7) _ = V E cos E? V P cos P (8) _ P = V P u [V E sin E? V P sin P ]? (9) r c In the reduced system, E uses the control E and the game target set is dened as a plane of equation = f, because no additional conditions are imposed on V P and P. The hamiltonian, which transcribes the energy of the system is : H =? VP V 2 (a + P ) bu2 + (V E cos E? V P cos P ) + P ( V P u rc? V E sin E?VP sin P ) (10) An investigation of an other version of such dynamic model is given in [4]. This other dynamic model looks like ours except that the authors consider a state variable more to constrain the minimum turning radius of the aircraft. Fortunately, as in [4], the adjoint equations of our game can be analytically integrated in terms of state variables and their nal values. The nal value of the adjoint vector on the game target is : At t = t f f = (0; 1; 0) (11) When exists, is the gradient of the barrier. Without loosing any generality, the nal line of sight is used as the 3

4 angular reference : f = 0 (12) The adjoint vector of optimal trajectories on the natural barrier is : VP = V E? (t f? t) V P (13) = cos (14) P = sin (15) The capture of the evader only occurs in the usable part of the game target. To capture an optimal evader, the pursuer must satisfy the compromise between its nal speed and its nal angle of attack given by the equation below : V Pf The limit of the usable part V Pf = V E cos Pf (16) V E cos P f denes the nal conditions of optimal trajectories of the natural barrier. Since E has no constraint on its turning rate, the optimal control strategy of the evader on the natural barrier is to take the nal line of sight direction. We note the optimal controls of the evader and the pursuer respectively E ( E in the 3D state space) and u. E = 0 (17) E =? (18) The analysis of the hamiltonian equation (10) with the analytic solution of the adjoint vector (equations (13) to (15)) give u on the natural barrier. u = max [?1; min(1; u 0 )] (19) 8 t < t f u 0 =? sin 2V F br c (t f? t) (20) This expression is not available on the game target, where the following expression of u 0 must be use. u 0 =? tan Pf 2br c (21) The natural barrier separates the capture zone and the non capture zone in the closeness of the game target, but the natural barrier is not sucient to close the capture zone for P small and superior to a value 1. To close the barrier of this pursuit game we have built a focal line starting in backward time at P = 0 and = 1. On the focal line, the pursuer plays the control u focal (equation (22)) to keep its velocity ~V P in the direction of the line of sight, i.e. to keep P = 0. u focal = V Er c V P sin E (22) The barrier of the short range sub-game is closed with optimal trajectories reaching tangentially the focal line in forward time. Figure 2: Barrier of the short range sub-game The gure 2 represents the barrier of this pursuit game in the 3D state space (V P ; ; P ). The focal line and the trajectories reaching it appears on the gure 2 at the front of the barrier. In the reduced state space (V P ; ; P ), the focal line is unique, but this focal line summarizes two dierent behaviors of E and P. If the evader turns left optimally, then the pursuer turns left with the control u focal (equation (22)). E can also turn right optimally and then P turns right as explained in equation (22). The gure 3 shows the focal line with E turning at right in the earthly referential (x; y). Other optimal pursuits of the short range sub-game barrier are also drawn on the picture 3 (P 1 pursuing E 1, P 2 pursuing E 2 and P 3 pursuing E 3). The optimal behavior of P at the beginning y (m) FOCAL LINE OF SHOT ANGE GAME E_1 P_1 ligne_focale_p ligne_focale_e E_2 P_2 E_3 P_3 x (m) Figure 3: Trajectories of P and E in the earthly referential reaching the focal line of the short range game of the pursuit is to turn with its maximum turning rate to reach the focal line. When P arrives near the focal line ju j decreases. During this time, E ies in straight line ( E = constant) towards the focal line. On the focal line, ju j increases again to keep ~V P in the direction of E. Therefore the trajectories of P and E are curved in the earthly referential when the state of the game is on the 4

5 focal line. At the end of the focal line, E continues in straight line in the direction of the nal line of sight and forces P to turn with its maximum turning rate. We use the part of the barrier of the short range subgame corresponding to = lock when the missile locks on the aircraft to design the game target of medium range sub-games. We call this new game target a \surcible". V P u= 1 φ P α θ V E φ E eference BAIE OF SHOT ANGE GAME AT = _LOCK surcible Figure 6: Geometry of medium range sub-game with constraint on evader direction 20 Phi_P (deg.) V_P (m/s) Figure 4: Game target for medium range sub-games 3.3 The medium range sub-games To simplify the analysis of the pursuit, we modify the kinematics of medium range sub-games. We change the equation (1) into the equation (23). j? E j (24) When the missile of the evader is in Lam guidance, E must y in the direction of the other aircraft, which corresponds approximately to the direction of the missile P. The sub-game 4 is identical to the sub-game 3 with = 0, i.e. : E = (25) The gure 7 presents the part of the barriers of the subgames 3 and 4 corresponding to V P = V Pmax MEDIUM ANGE GAMES maximum_range_of_a_missile_at_vp_max barrier_medium_range_constraint_game_at_vp_max _V P =?av 2 P (23) The gure 5 represents a part of the optimal medium range subgame barrier corresponding to the missile shooting time when V P = V Pmax (m) BAIE OF MEDIUM ANGE GAME AT VP = VP_max barrier_medium_range_game_at_vp_max PhiP (deg.) (m) Figure 7: Capture domain of the sub-games 3 and 4 at the missile shooting instant Figure 5: PhiP (deg.) Capture domain of the sub-game 2 at the missile shooting instant The sub-game 3 is an optimal dierential game with perfect information as the sub-game 2 except we put a constraint on the domain of the evader control E (gure 6) : 3.4 The evasion strategy of BA We want to help BA to take its decisions in the duel, in particular to choose its evasion time. The aircraft does not detect the opponent's shooting, it sees the enemy missile only when the enemy Ad locks on. When the Ad of the opponent missile locks, it can be too late to evade. To be sure of escaping from M, BA must not enter into the capture zone of M of the subgame 2, when BA has not yet detected the M shooting. A secure evasion of BA corresponds to an evasion started before entering in this capture zone. Since BA does not see M, BA protects itself against a M that A can shoot at the present time and against 5

6 all Ms that A may have shot in the past. That is why we introduce in all Adam's simulations some hypothetic Ms to perform the evasion strategy of BA : At each time step in the simulation, A res a hypothetic missile (hypothetic M) as soon as BA crosses the barrier of the sub-game 4 of a M supposed not yet to be shot. BA evades as soon as it reaches the capture zone of the sub-game 2 of a hypothetic M red or not. On one hand, because of the duel hypotheses and in particular of the hypothesis 5 (section 3.1), this evasion preserves BA from losing and assures BA of an outcome at least equal to a draw outcome and the BA evasion does not depend on the real M trajectory. On the other hand, with such an evasion strategy, BA takes no risk and can miss a possible win outcome. The BA evasion strategy looks like \the principle of min-max certainty equivalence" which states that one must look for what is actually the worst possible state with the available information and to play the strategy which would be optimal if we were certain that the state is eectively this state [1]. Of course, we prove no optimality of this strategy in the present context. A does not use hypothetic BM for its escape maneuver and does not know where BM is. Therefore A can not choose its evasion instant to make an evasion as ecient as that of BA, even if A uses the barriers of sub-games that it can manipulate because of the imperfect knowledge of the position of BM. If both A and BA were to evade considering hypothetic enemy missiles, Adam would be without interest, since the duel would always lead to a draw outcome. The gure 8 represents a duel in Adam. This decision support system simulates complex kinematics for the aircrafts and the missiles too. The aircrafts use an evasion strategy designed by us considering the barriers calculated previously, which give a good approximation of realistic shooting domains. The barriers determine the time to evade for BA and the side of its turn. On the gure 8, A res several hypothetic missiles. Some drawings on the trajectories explain the aircrafts positions in the state spaces of the sub-games. Other drawings explain the guidance mode of the missiles Conclusion This study illustrates the use of game theory to analyze some partial games (or sub-games) of a more complex situation we study by simulation. As an example of this approach, we have chosen the Mica/Amraam duel and we have used symbolic programming to build a decision support system. This example shows how game theory proposes in particular a secure evasion to an aircraft against a missile as a Mica/Amraam with only few hypotheses. eferences [1] P. Bernhard and Alain apaport. Min-max Certainty Equivalence & Dierential Games. INIA - esearch eport : -2019, August [2] Pierre Bernhard. A View on the Dierential Games Theory..A.I..O., Automatique/Systems Analysis and Control - Volume 15 - Bordas-Dunod, [3] W. Grimm and K. H. Well. Modelling air combat as dierential game recent approaches and future requirements. Fourth International Symposium on Dierential Games and Applications - Finland - Helsinki - University of Technology, August [4] M. Guelman, J. Shinar, and A. Green. Qualitative Study of a Planar Evasion Game in the Atmosphere. Guidance, Navigation and Control Conference, Minneapolis, Minnesota, August [5] ufus Isaacs. Dierential Games, a Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. The SIAM series in Applied Mathematics, New York, [6] Stephane Le Menec. Adam : Aide au duel Aerien Moderne. Theorie des jeux dierentiels et programmation symbolique appliquees au probleme de l'amraam. IN- IA - esearch eport : -2032, Septembre [7] G.J. Olsder and J.V. Breakwell. ole Determination in an Aerial Dogght. International Journal of Game Theory - Volume [8] Patrice Poyet, Philippe De La Cruz, Thierry Mileo et Jean-Noel Loiseau ecentes etudes en matiere de simulation tactiques intelligentes. 9ieme journees internationales d'avignon, Avignon, France, Mai-juin [9] J. Shinar and A.W. Siegel and Y.I. Gold On the Analysis of a Complex Dierential Game Using Articial Intelligence Techniques. 27th Conference on Decision and Control, Austin, Texas, December, Figure 8: right Duel in Adam with BA at left and A at 6