A SOLUTION OF MISSILE-AIRCRAFT PURSUIT-EVASION DIFFERENTIAL GAMES IN MEDIUM RANGE

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1 Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, A SOLUION OF MISSILE-AIRCRAF PURSUI-EVASION DIFFERENIAL GAMES IN MEDIUM RANGE FUMIAKI IMADO Department of Mechanical Systems Engineering Shinshu University Wakasato Nagano, Nagano JAPAN AKESHI KURODA Advanced echnology Laboratory Mitsubishi Electric Corporation Amagasaki sukaguchi-honmachi, Hyogo 661 JAPAN Abstract: his paper shows a method to obtain the exact solution for a missile and an aircraft differential games which minimizes and maximizes the essential pay-off of the problem: the miss distance, without employing any linearized approximation. For this purpose, we introduced the concept of energy maneuverability. Once the families of optimal terminal surfaces are obtained, the proposed method can solve mini-max solutions under several vehicle s acceleration constraints. Although it is impossible to obtain the solution in real time, however, if the results are incorporated into the missile guidance system as a knowledge base, the performance will be fairly improved. Keywords: - Optimal Controls, Differential Games, Interception Problems, Pursuit-Evasion, s, 1 Introduction Many studies have appeared about pursuit-evasion dynamic games between a missile and an aircraft, however, none of them has obtained the exact solution which minimizes and maximizes the essential pay-off of the problem: the miss distance (MD). his study shows a method to obtain solutions for the games with realistic models, and without employing any linearized approximation. Our former studies have shown the method to obtain the solutions for short range cases by employing the sub-optimality of the proportional navigation guidance. In this paper, a method to obtain more general solutions for medium range cases is shown by introducing the concept of energy maneuverability. As a perfect information game is assumed, the aircraft faint movement which is the important factor of short range cases has no use as the missile knows the aircraft strategy. As the result, both vehicles tries to maintain most advantageous states until just before the interception the aircraft takes the upward or downward maximum acceleration. he missile takes instantly maximum or minimum acceleration command corresponding to the aircraft movement. Once the families of optimal terminal surfaces are obtained, the proposed method can solve mini-max solutions under several vehicle s acceleration constraints. he detail of the problem, the algorithm for the solution and the feature of the result is shown in this paper. Problem Formulation Fig. 1 shows the relative geometry of the pursuer and the evader and symbols. Our studies have shown that a vertical maneuver is more effective than a horizontal maneuver for the aircraft, therefore, the motions are constrained in a vertical plane. In the paper, the pursuer is a missile, and the evader is an aircraft. Both vehicles are modeled as point masses, and the equations of motion in a vertical plane are as follows. v& t = ( t cosα D) / mt g sin γ t (1) & γ t = ( L + t sin α) /( mt vt ) ( g / vt ) cosγ t () x& t = vt cosγ t (3) h& t = vt sin γ t (4) L = ( 1/ ) ρtvt StCL D = ( 1/ ) ρtvt StCD (5) C L = CLα ( α α ) C D = CD + kcl (6) ρ = ρ h ), t = t (v t,h t ) (7) t t ( t

2 f Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, A constraint is imposed on the value of the aircraft lateral acceleration a t, which is treated as the aircraft control variable. at = L / mt at max (8) he missile lateral acceleration is approximated by a first-order lag to a lateral acceleration command a mc, which is treated as the missile control variable. v& m = 1/ mm ( m Dm ) (9) a& m = ( amc am ) / τ (1) γ& m = a m / vm (11) x& m = v m cosγ m (1) h& m = v m sin γ m (13) D m = k1 vm + k ( am / vm ) (14) k 1 1 = ( ) ρ m S mc Dm k = k m /( ρ S ) (15) m m m m m m for < t te = (16) for te < t Our purpose is to obtain the optimal control histories of the aircraft and missile, a t (t) and a mc (t) that maximize and minimize the final MD respectively. 3 Some Features of hese Game Solutions Some features of the aircraft optimal evasive maneuvers in a plane obtained in our previous study are as follows. (a) If the initial relative range is large, the aircraft takes the maximum normal acceleration to turn to the opposite direction, then flies away straight (ype 1). (b) If the initial relative range is small, the aircraft takes the maximum upward acceleration at first, then at an appropriate time, switches to maximum downward acceleration (ype ). (c) If the initial relative range is medium, the aircraft takes an optimal trajectory (which is able 1 Vehicle Parameters (afterburner) t max mt = 75kg h t = 457m S t = 6m vt = 9.m / s (.9M ) C L α = / rad C D =.4 k =.6 a t max = 9g Engine : PWF-1 max = max ( v, h) (Sustainer phase) m m = kg I SP = 5 s m = 6N t e = 8s S m =.34m h m = 457m v m = 644.6m / s (.M ) C L α = 35. / rad C D =.9 k =.3 ac max = 3g τ =.5s r = 3m (head on) able. Fig.1 Geometry and symbols thrust in relation to altitude and velocity (in N) H (m) V(m/s) explained in a later section) at first, then at an appropriate time, switches to its maximum upward or downward acceleration (ype 3). Maneuvers type and type 3 are called vertical-s and split-s, respectively. Mini-max solutions for type 1 and are shown in our former papers [1],[]. Obtaining the mini-max solution for type 3 is most difficult, and in this paper, we show the algorithm to solve for type 3, and show a result. 4 he Algorithm to Solve ype 3 able 1 shows initial conditions and some parameters of vehicles. In order to obtain the mini-max solution, the concept of energy maneuverability is introduced, that is, with a larger energy maneuverability, a vehicle can produce a larger g, therefore, with a larger velocity an aircraft can increase the MD, while a missile with a larger velocity can reduce the MD. Our former studies showed that, against the PNG missile, the aircraft gradually increases upward acceleration until it reaches its maximum value, then at an appropriate time, inverse the aircraft attitude and take the maximum g downward. On the other hand, in a differential game, as the missile does not know whether the aircraft will continue the maximum

3 Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, Y (m) 5 45 M 3.6s ψ a,i.s A.s ψ m,i 3.6s X(m) Fig. Optimal trajectories and terminal surfaces of the aircraft and missile upward g, or reverse the direction at any time, therefore the optimal missile control should be the one that minimizes the MD in both cases. As the result, both the missile and aircraft tries to keep their energy maneuverability maximum until very close point before the interception. he mini-max DG solution branched into two trajectories depending on whether the aircraft continues maximum upward acceleration, or switches to maximum downward acceleration. he steps to obtain this differential game solution for a medium range case (ype 3) are as follows. (1) Let the aircraft take the maximum upward acceleration and minimum downward acceleration throughout the time, while the missile is guided by PNG, and conduct simulations. In these cases the missile hits the aircraft at the time t f =3.4s at the altitude h f =49m and at the time t f =3.3s at the altitude h f =435m both without MD respectively. () In reference to the t f values above, define the time set { t i }= { t 1, t,..., t n 1, t n }. For each ti, ( i = 1 ~ n) calculate the set of optimal controls of the missile and aircraft which maximize the missile and aircraft velocity at t i,respectively. he above process produces the sets of terminal surfaces { ψ }( i 1 ~ ) for the missile, and m, i = n a, i ( i = 1 ~ n { } ) ψ for the aircraft in the x h plane, respectively. Some of these terminal surfaces and optimal trajectories of vehicles are shown in Fig.. (3) Assume we have selected a t i and from the arbitrary point A i on ψ a, i, the aircraft takes maximum upward or downward acceleration. he trajectories of the aircraft are shown as S a1 and S a in Fig.3, respectively. Calculate the missile optimal controls which start from ψ m, i, and minimize MD against S a1 and S a. Starting from i =1 and A i from the lowest altitude to the highest altitude of ψ a, i, also from the lowest altitude to the highest altitude of M on i ψ m,i corresponding to each A i (4) For a small value of i, for arbitrary A i on ψ a, i, we will find some M i s on ψ m, i both MD against S a1 and Sa are. In the case, the aircraft has not a chance to escape from the missile. However, by increasing t i values, we will find the i, and A i optimal missile maneuvers from any M i can not reduce MDs against both Sa1and S a to zero. hen we select the M i which produces the same value of MDs against both Sa1and S a. Work out the process on the arbitrary point of ψ a, i and ψ m, i for all { t 1, t,..., t n 1, t n }. hen we can write the contours of the values of the mini-max MD. In these parameter searches, the obtained aircraft control and corresponding missile control which produce the largest mini-max value of MD may be called the solution of this differential game. Fig.4 shows the final part of vehicle trajectories. Figure 5 shows the concept of a mini-max solution. Figure 6 shows the positions of vehicles corresponding to the MD of 1.m through.4m. he mini-max value of this case is ψ M o ψ m,i ψ m, i +1 M i S m1 S m S a ψ a, i+1 a,i : optimal (velocity maximum) surface at i ψ m,i : optimal (velocity maximum) surface at t i S a1 : takes maximum upward acceleration at A i S a : takes maximum downward acceleration at A i S m1: takes optimal control to minimize terminal miss against S a1 at Mi S m : takes optimal control to minimize terminal miss against S a at Mi Fig.3 he calculation of the mini-max solution for general medium range cases S a1 t A i ψ a,i A o

4 Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, m. Figs 7 and 8 shows the aircraft control history and parts of trajectories corresponding to this value. 5 Some Comments hroughout this study, nonlinear optimal control problems are solved by the steepest ascent code, which has been developed by authors. he principle of the algorithm is explained in the appendix.[3] Although the algorithm is an old one, however several techniques such as searching for the solution automatically, avoiding local optima, and accelerating convergence are introduced in this code. As a perfect information game is assumed, the aircraft faint movement (which is important factor of ype ) has no use as the missile knows the aircraft strategy. However, the aircraft has the initiative to take an action at any time while the missile can not take an action until the aircraft takes an active strategy. As the result, the aircraft select the most advantageous position (including all states of both vehicles) and takes the upward or downward maximum acceleration. he missile takes instantly maximum or minimum acceleration command corresponding to the aircraft movement, therefore the final situation always become like as shown in Fig.4. In the cases of this paper, the aircraft took Relative altitude (m) Relative distance (m) Fig.4 he enlarged figure near interception points M o V ( m t i ) max S a1 MD min MD max ( Max upward g) Ai s M is S a t is M i t is A i t i t i V ( a t i ) max MD min MD max ( Max downward g) Fig.5 he concept of a minimax solution A o Altitude (m) Horizontal range (m) Fig.6 he positions of vehicles corresponding to the MD of 1.m.4m

5 Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, 6 38 rather straight trajectories. he reason is considered as the aircraft maximum acceleration is limited. On the other hand, if the aircraft maximum angle of attack is limited, the aircraft tried to take a lower altitude because it can produce a larger acceleration in a lower altitude. One important result of this study is that, a PNG missile tends to take higher altitude than an aircraft, while an intelligent missile should maintain a little lower altitude than an aircraft with maximum energy maneuverability. A very short time before the interception, the missile can switch to PNG for real time guidance. 6 Necessary Conditions of he Solution Although the possibility of the local solution of this result can not be denied, necessary conditions of the solution are numerically verified as follows. H = or amc = ac max amc for t s t t f H = or at = at max at for t t (1) f t s is the time when the minimax trajectory of the vehicles branches into two curves, and the system Hamiltonian function H is defined by ( λ λ ν) = f φω ψω ( a mc, at ) H () u = (3) As for symbols employed in above equations, see the appendix of this. he above characteristics are automatically satisfied by the steepest ascent algorithm. he saddle point condition is satisfied in the result. < < J amc, at J amc( t), at J amc( t), at ( t ) for t t f (4) t amc and at ( ) are optimal solutions of a mc (t) and a t (t), respectively. he above characteristics are assured by the proposed algorithm. It is also verified by selecting arbitrary differential changes from the optimal control of one vehicle and calculating corresponding one-sided optimal control of the opponent vehicle and conducting simulations. normal acceleration (m/s ) Altitude (m) ime (sec) Fig.7 A typical optimal aircraft control against a DG missile (medium range) Horizontal distance (m) Fig.8 A final part of optimal aircraft trajectories against a DG missile in medium range 7 Conclusion A method to obtain exact solutions for missile-aircraft pursuit-evasion differential games in medium range is derived and explained in detail. Some results obtained by the method also are shown. he results are useful for designing guidance systems of a practical missile. References: [1] F. Imado and. Kuroda, Exact Solutions for - Pursuit-Evasion Differential Games, Proc. of 11th International Symposium on Dynamic Games and Applications, ucson, 4,CD-ROM. [] F. Imado and. Kuroda, A Method to Solve - Pursuit-Evasion Differential Games, Proc. of 16th IFAC World Congress, Prague, 5, CD-ROM [3] A. E. Bryson Jr. and W. E. Denham, A Steepest Ascent Method for Solving Optimum Programming Problems, Journal of Applied Mechanics, Vol.9, 196, pp

6 Proceedings of the 1th WSEAS International Confenrence on APPLIED MAHEMAICS, Dallas, exas, USA, November 1-3, Appendix A steepest ascent method is a well-known algorithm to solve nonlinear optimal control problems. A summary of the algorithm is shown here for the readers convenience. Find u (t) to maximize (for minimizing, change the sign of the following φ ) [ x( t f )] J = φ (A-1) x = f ( x, u, t) (A-) x ( t ) = x : specified (A-3) with terminal constraints ψ x( t ), t = (A-4) [ f f ] x (t) is an n -dimensional state vector, u (t) is an m -dimensional control vector, and ψ is a q -dimensional constraint vector. he terminal time t f is determined from the following stopping condition: [ ( t f ), t f ] = Ω x (A-5) he optimal control u (t) is obtained by the following algorithm. 1)Estimate a set of control variable histories u (t) (which is called a nominal control) )Integrate the system equations (A-) with the initial condition (A-3) and control variable histories from step 1 until (A-5) is satisfied. Record x (t), u (t), and ψ [ x( t f )]. Calculate the time histories of the ( n n) and ( n m ) matrices of functions: F = G( t) = (A-6) u 3)Determine n -vector influence functions λ φ (t), λ Ω (t) and ( n q ) matrix of influence functions λ ψ (t), by backward integration of the following influence equations, using x ( t f ) obtained in Step to determine the boundary conditions: λ φ = φ λ & λ & Ω = λω ψ = λψ λ & φ (A-7) λ φ ( t f ) = (A-8) (A-9) (A-11) Ω λ Ω ( t f ) = (A-1) ψ λ ψ ( t f ) = (A-1) Calculate the following influence functions & φ( t f ) λφω = λφ λω Ω& ( t f ) (A-13) ψ& ( t f ) λψω = λψ λψ Ω& ( t f ) (A-14) 4)Simultaneously with Step 3,compute the following integrals: 1 ψψ Ω t 1 ψφ = λ ψωgw G λφω t 1 φφ = λ φωgw G λφω t I I I = λ ψωgw G λψ (A-15) (A-16) (A-17) 5)Choose values of δ ψ to cause the next solution to be closer to the desired values ψ [ x( t f )] =. For example, one might choose δψ [ ( t f )], < 1 = εψ x ε (A-18) he proper choice of δ u (t), which increase the J is given as follows: 1 δ u( t) = (1/ µ ) W G ( λ φ Ω λψω v) (A-19) Iφφ I µ = ( dp) 1 ψψ 1 ψφ Iψψ Iψφ 1 dψ Iψψ d ψ 1 ψψ ψφ 1 (A-) v = µ I dψ + I I (A-1) dp and ( m m ) matrix of weighting functions W (t) are chosen to satisfy = ( dp) δ u W δu (A-) t 6)Repeat Steps 1-5, using an improved estimate of u (t) u = u old + δu (A-3) he key technique of the algorithm is the proper choice of the value dp, which must be changed every step, and how to avoid dropping into local optima, and to reach to the global optimum.

A METHOD TO SOLVE MISSILE-AIRCRAFT PURSUIT-EVASION DIFFERENTIAL GAMES. Fumiaki Imado* and Takeshi Kuroda**

A METHOD TO SOLVE MISSILE-AIRCRAFT PURSUIT-EVASION DIFFERENTIAL GAMES. Fumiaki Imado* and Takeshi Kuroda** A MEHOD O SOLVE MSSLE-ARCRAF PURSU-EVASON DFFERENAL GAMES Fumiaki mado and akeshi Kuroda Deparmen o Mechanical Sysems Engineering, Shinshu Universiy 4-17-1 Wakasao Nagano, Nagano 38-8553 JAPAN Advanced