The Design of the Best Penetration Maneuvers Parameter of Variable Trajectory Maneuvers of Target Missiles

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1 Modeling Simulation Optimiation Technologies Applications (MSOTA 6) The Design of Best Penetration Maneuvers Parameter of Variable Trajector Maneuvers of Target Missiles Kaibo Bi Yifei Zhang Xianhui Sui Dept. of Missile Dalian aval Academ Liaoning Dalianchina68 Abstract To solve design problem of variable trajector maneuver of target missiles an integrated design model of variable trajector maneuver is presented. B closel cooperating between control signals of displacements control signals of overloads this integrated design model can make target missiles to realie maneuver of jump trajector maneuver of snake-wriggling trajector maneuver of pendulum trajector maneuver of spiral trajector so on. On this basis penetration effect models of variable trajector maneuvers of target missiles are researched. The design methods of best penetration maneuvers parameter is put forward. According to analsis target missiles should enlarge maneuver amplitude A A as much as possible within overload limit. At same time we select imal parameter k k according to various maneuver modes. According to se parameters variable trajector maneuver of target missiles can realie best penetration effect promote train effect of air defense missile. Kewords-target missile; penetration effect formatting I. variable trajector maneuvers; ITRODUCTIO Currentl man countries with developed militar force have alread equipped with terminal maneuver anti-ship missiles[-4]. For eample American Harpoon Missile is a kind of anti-ship missiles with terminal jump maneuver. Russian Moskit Missile is equipped with terminal snakewriggling maneuver[5]. Besides re are some ballistic missiles possess maneuvers of spiral trajector or maneuvers of pendulum trajector when enter into aerosphere. These maneuvers of trajector have greatl improved penetration abilit of se missiles. In a sense we can assume that as long as missiles undertake maneuver flights at certain intension penetration abilit can be enhanced. Therefore in militar training resisting target missiles with terminal maneuver trajector abilit plas an important role in testing real combat capabilit of aerial defense[6-9]. Compared with attitude control sstem overload control sstem has faster tracing speed higher accurac so overload control sstem can facilitate whole process of controlling terminal maneuver trajector. This stud delves into four kinds of integrated design models such as maneuver of jump trajector maneuver of snake-wriggling trajector maneuver of pendulum trajector maneuver of spiral trajector on base of which we establish penetration effect models of variable trajector maneuvers of target Missiles proposed various design methods of best penetration maneuvers parameter. II. GEERAL DESIG OF FORMS OF MAEUVER TRAJECTORY TARGET MISSILES According to target missiles as maneuver of jump trajector maneuver of snake-wriggling trajector maneuver of pendulum trajector maneuver of spiral trajector we can design ir displacement order signal overload order signal respectivel. Generall individual order signal cannot be used in common which is not beneficial for engineering application. In order to facilitate designing process of variable trajector maneuvers of target missiles this stud proposes an integrated design model of variable trajector. Trajector maneuvers of target missiles is trajector of missile centroid relative to ground coordinate O. Set longitudinal displacement of target projectile as independent variable height comm signal course signal are all functions. We can get general design form of variable trajector of maneuver through summing up height comm signals course signals of maneuver of jump trajector maneuver of snake-wriggling trajector maneuver of pendulum trajector maneuver of spiral trajector: sign(sin( ))l l sin k sign(cos( ))l l cos k () In this formula l l k k are design parameters of variable trajector. Copright 6 Authors. Published b Atlantis Press. This is an open access article under CC BY-C license ( 8

2 Through calculating coefficients of general form of variable trajector maneuver target missiles () for two times we can get projection quantum of maneuver acceleration on three aes in ground coordinate: a cos k a l k l k sin k a l k sin k l k cos k missile; nl is maneuver acceleration of interception missile. It is based on conjugated sstem method Figure conjugated sstem is shown in Figure II. (t ) AALYSIS OF PEETRATIO MODEL III. In order to enhance anti-ship missiles aerial defense abilit combat capabilit under complicated conditions it is necessar to anale penetration effect of variable trajector maneuver of target missiles deepl. This section takes effort miss of interception as an inde to evaluate penetration effect of target missiles. We emplo interception-penetration sstem which has been showed in figure as a model in this stud. am s s(ts ) t FIGURE II. THE COJUGATED SYSTEM OF SIMPLIFIED MODEL OF ITERCEPTIO-PEETRATIO MODEL In Figure t t F t is time variable of conjugated () In order to realie variable trajector maneuver of target missiles cooperation coordination of overload comm signal displacement comm signal is a necessit selection of design parameters should compl with real situation. miss(t ) s h(t ) f (tf t ) s sstem; (t ) is impulse input; h(t ) is impulse response of conjugated sstem.; f (t F t ) is function of maneuver acceleration of target missiles; miss (t ) is convolution of product is miss (t ) of t h(t ) f (t F t ) that f (t F ) h( )d miss (t ) is terminal maneuver of target missile at time t t F t lasting for t evocate miss effort at time. Accordingl re eists such kind of relationship in comple domain: M ( s) F (s) H (s) (3) In this formula M ( s ) F ( s ) H ( s ) are laplace transformation of miss (t ) f (t F t ) h(t ). It can be deduced from Figure II: al Ts ac c q s c ( tf t ) FIGURE I. SIMPLIFIED MODEL OF ITERCEPTIO-PEETRATIO SYSTEM In figure I am f (t ) (t ) is acceleration of target missile maneuver; is deviation of interception missile against height of target missile s position; s is vc tf t simplified transfer function of ; vc adjacent speed between target missile interception missile; vc is proportional guidance law for intercepting missile; q is line of sight rate of interception missile; is simplified Ts transfer function of interception missile in controlling process; nc is comm acceleration of interception Ts H (s) s Ts (4) In this formula is effective navigation ratio of interception missile. For first-order guidance sstem transfer function H ( s ) shows transitive relation from maneuvering acceleration of target missile to miss efforts of interception missile. Although analing method onl suits for maneuver of target missiles in a flat we can etend this method to analing maneuver of three dimensions ( maneuver of pendulum trajector maneuver of spiral trajector) mainl through dividing maneuver of three dimensions into longitudinal flat maneuver course plat maneuver. According to projection quantum of maneuver acceleration on three aes in ground coordinate as shown in () assuming that target missile undertakes uniform motion on ais X 9

3 that is for convenience of research we record quantum of target missiles maneuver acceleration as: a a A sin( t ) a A cos( t ) In view of influence of H ( s ) maneuver acceleration f (tf t ) Asin( t ) causes stable miss efforts: miss (t ) K H A sin( H t ) (5) (9) When target missile undertakes longitudinal flat snakewriggling stable miss effort is: In this A l k k V A l k formula k V miss (t ) K H A sin( H t ) () In this formula V t. K H Therefore when we take longitudinal flat maneuver of target missile into account f (t F t ) A sin( t ) ; when we take course miss (t ) miss (t ) miss (t ) (6) H ( ). tan( T ) In this formula: K H miss (t ) K H A sin( H t ) () interception missile miss (t ) is: ( T ) When target missile undertakes course flat snakewriggling stable miss effort is: of navigation flat maneuver of target missiles into account f (t F t ) A cos( t ). If f (t F t ) A sin( t ) miss efforts of T T H ( ). tan( T ) ( T ) When target missile undertakes course three dimensions stable miss effort is: In this formula miss (t ) is temporal miss efforts miss (t ) is stable miss efforts. Because temporal miss efforts decreases eponentiall it will not affect final result of miss efforts so we can just take stable miss efforts into account. According to frequenc response characteristic of we can get amplitude K H of H ( j ) phase angle H : KH T ( T ) 3 H ( ) tan ( T ) (7) (8) Due to effective navigation ratio of current air defense missile can meet condition that 3 we onl anale situation with 3. miss (t ) miss (t ) miss (t ) () Through comparing formula() formula() formula()we can know that miss effort of three dimensions is larger than that of a flat maneuver. IV. SIMULATIO AALYSIS In order to elaborate effect which variable trajector maneuver of target missiles poses on penetration we compare change of interception missiles miss efforts causes with without maneuver of implementation of maneuver. The target missile undertakes longitudinal snake-wriggling maneuver parameters are: k 6 l k l V 3m / s 5m 35m T.9s 3 compared with situation without implementation of maneuver under same conditions result of miss efforts of interception missiles are shown in Figure III.

4 FIGURE III. SIMULATIO CURVE OF ITERCEPTIO MISSILES MISS EFFORTS Comparing simulation results in figure III we can know that real line is miss effort of interception missile when it undertakes longitudinal snake-wriggling maneuver; dashed line is miss effort of interception missile when it does not undertake longitudinal snake-wriggling maneuver. From results above we can know when target missile does not undertake maneuver flight earlier interception is implemented smaller miss effort is it is eas to intercept missiles; however when target missile undertakes snake-wriggling maneuver miss effort of interception missiles will show a sinusoidal variation it increases uncertaint to intercept missiles at same time probabilit of penetration will be higher. Therefore if we design missile into variable trajector maneuver target missile we can furr enhance combat training difficult master defense technolog of resisting missile penetration. Furr more we can compare variable trajector maneuver of three dimensions with variable trajector maneuver in flat we can trace discrepanc. If target missile undertakes maneuver of pendulum trajector we can divide it into longitudinal snake-wriggling maneuver course snake-wriggling maneuver. Comparing miss efforts of target missiles of maneuver of pendulum trajector maneuver of longitudinal snake-wriggling maneuver of course snake-wriggling we can get simulation results shown in Figure IV. FIGURE IV. SIMULATIO CURVE OF ITERCEPTIO MISSILES MISS EFFORTS Comparing simulation results in figure IV we can know that dashed line is miss effort of interception missile when it undertakes longitudinal snake-wriggling maneuver; dotted lineation is miss effort of interception missile when it undertakes course snake-wriggling maneuver. The real line is miss effort of interception missile when it undertakes pendulum trajector. Through comparing results we can know that when target missile undertakes longitudinal course snake-wriggling maneuver miss effort of interception missiles shows a fluctuant variation; when target missile undertakes maneuver of pendulum trajector maimum of fluctuant variation of interception missiles miss effort is much larger than one of flat snake-wriggling maneuver. There fore it shows to some etent probabilit of undertaking pendulum maneuver is lower than that of flat snake-wriggling maneuver. It proves variable trajector of three dimensions has greater penetration effects than variable trajector of flat maneuver it can furr enhance training performance of interception missiles. V. DESIG OF THE OPTIMAL VARIABLE TRAJECTORY MAEUVER PARAMETER Analing () () () we can know that factors affecting penetration of target missiles are amplitude of target missile maneuver A A maneuver frequenc interception controlling sstem time constant T effective guidance ratio. Because we cannot get accurate value of interception controlling sstem time constant T effective guidance ratio we can anale imal maneuver amplitude frequenc assuming are under unchangeable condition. A. Assuming effective guidance ratio of interception missils 3 taking longitudinal snake-wriggling

5 maneuver as an eample we can get stable miss effort as formula (). When we let stable miss effort of target missiles reach miss (t ) with a ma value. maimum which is to refer Obviousl larger maneuver amplitude A is larger stable miss effort will be. But value of maneuver amplitude A is limited b limitation of etreme overload we can enhance maneuver amplitude under miss (t ) is maimum of target missiles overload. Besides function of maneuver frequenc miss (t ) has to make sure maimum K H. We get first derivative of against K H. ma value. We select imal ( ) 3T ( T ) T ( T ) ( T ) K H According to when satisfies maimum. when K H nature 4 T T of K H K H (3) (3) reaches maimum K K miss (t ) A sin( H t ) H H A sin( H t ) (4) Considering maneuver of pendulum of target missiles maneuver k k according to definition frequenc we can know appl it into (4)we can get miss (t) KH A sin( H t) KH A cos( H t) miss (t ) has maimum k V (5) speed V is difficult to change range of is settled in imal parameter ( ) 4 int function int TV represents take complete part of stable miss effort is k is maimum. Of course in order to get imal parameter k B. When target missiles undertakes maneuver of pendulum stable miss effort is: derivatives variable trajector of target missiles advance. Therefore designed As for course snake-wriggling maneuver of targeted missiles we can ad same analing method. The maneuver of jump trajector of target missile can be regarded as a special eample of longitudinal snake-wriggling maneuver ad same method to anale it. K H can get formula peak value. According to formula of target missiles first realie less imal or imal snake maneuver. Such kind of parameter design method can maimie stable miss effort of interception missiles enhance probabilit of target missiles penetration improve performance of practical training. B K H ( ) 4 at TV least we can get less imal or imal parameter k thus interception missile appl it into we must know effective guidance ratio control sstem time constant T. There eist some difficulties in real combat. However variable trajector maneuver of target missiles aims at enhance difficult of training we can estimate value of T of As for formula(5) when maneuver amplitude A A reach maimum stable miss effort miss (t ) is higher; so under ma available limit overload condition we can enhance maneuver amplitude A A as much as possible. Besides miss (t ) is function of maneuver frequenc miss (t ) has ma value. We need to select imal to realie maimum of miss (t ) in unit of maneuver period it is obj ( ) ma miss (t )dt. In order to facilitate stud process we can select imal make balance obj ( ) ma (miss (t )) dt valid from this we can get

6 K A sin( t ) H H dt obj( ) ma KH A cos( t H ) ma f obj ( ) 5 X:.4 Y: In this formula f obj ( ) K A K A H H fobj( )/ (6) In order to get maimum of (6) we calculate first derivative of f obj ( ) on (H) f obj ( ) A K H K H A K H K H A K H A K H FIGURE V. RELATIO BETWEE FUCTIO f obj ( ) (7) We In this formula can find AD imal frequenc of pendulum maneuver accuratel through graphic construction method. Thus we can get K H ( ) 3T ( T ) T ( T ) ( T ) (8) 3 ( T ) K H ( ) T ( T ) T ( T ) (9) According to nature of first derivative when satisfies f obj ( ) f obj ( ) f obj ( ) can get maimum. Although it is difficult to get analsis formula about from direct calculating formula (7) we can emplo graphical construction method using graphical diagram to represent relation between f obj ( ). Supposing 3 T.9s maimum amplitude of target missile A 7. g A 5.6 g relation between function f obj ( ) is shown in Figure V. imal k parameter ( ) int V ( ) k int. V C. When target missile takes spiral maneuver k k l l according to maneuver amplitude A A definition of maneuver frequenc we can know K H K H A A refore miss effort of target missile of spiral maneuver is: miss (t ) K H A sin( H t ) () According to () higher maneuver amplitude A is higher miss (t ) is; refore under maimum of target missile overl condition we can enhance maneuver amplitude as much as possible. Besides miss (t ) is miss (t ) has maimum value we select imal which can make miss (t ) get maimum value in unit of maneuver function of maneuver frequenc period that is obj ( ) ma miss (t )dt. In order to facilitate stud process we can get imal which 3

7 can make obj ( ) ma (miss (t )) dt reb we can get obj( ) ma KH A sin( H t ) dt ma f obj ( ) In this formula f obj ( ) K A REFERECES []. [] In order to get maimum of formula () we calculate first derivative of f obj ( ) on we can get [3] K H [4] f obj ( ) 4 A K H A K H [5] () [6] satisfies f obj ( ) f obj ( ) [7] f obj ( ) can get maimum value. B calculating () we [8] 5 f obj ( ) can get 5T [9] When can get when maimum value n we can make sure stable miss effort miss (t ) get maimum value in unit of maneuver period. According to imal maneuver frequenc parameter is k k according to various maneuver modes. Under influence of se imal parameters variable trajector maneuver of target missiles can realie best penetration effects furr enhance combat training process of aerial defense missiles. Through analsis of penetration effects we can master technologies to defend missile penetration eert a great value for target missile sstem combat usage.. () H select imal parameter k k Ghawe S Ghose D. Pure Proportional avigation against TimeVaring TargetsManeuvers. IEEE Transactions on Aerospace Electronic Sstems [J].996;3(5): Yang C D Yang C C. Optimal Pure Proportional avigation for ManeuveringTargets. IEEE Transactions on Aerospace Electronic Sstems [J]. 997; 33(3): WAG Xiao-huCHE Han-fuZHAG Min-lian The model adaptive estimation algorithm for maneuvering target in terminal guidance section of air missile Journal of Astronautics [J]. ; (): 6-7. ZHAG TaoPEG Shao-iongSOG Gui-bao. Design imiation of anti-ship cruise missile. Aerodnamic Missile Journal [J].3(7): 6-8. QU Dong-caiLI Tao. Russia Moskit anti-ship missile. Ship borne weapon[j].997; 5(3):7-. JIAG Yu-ianCUI Jing. The effectiveness of missile swing maneuver strateg. Journal of Beijing universit of aeronautics astronautics [J].; 8(): CUI JingJIAG Yu-ian. Effect of dnamic characteristics of missile interception of swing maneuver penetration effect. Journal of Astronautics [J]. (5): ZHAO Hong-chaoGU Wen-jinPEG Wen-hua. Design of control sstem of anti ship missile based on overload control.tactical Missile Control Technolog [J]. 45(3):-4 WAG Ting. X tpe low lateral transfer flight target missile. Journal of projectiles rockets missile guidance [J]. 7(5): make sure imal 5T ( ) 5 int 5 TV VI. COCLUSIO Centering on matter of designing variable trajector maneuvers of target missiles this stud proposes four kinds of variable trajectories maneuver of jump trajector maneuver of snake-wriggling trajector maneuver of pendulum trajector maneuver of spiral trajector. On basis of this we undertake various kind of stud on variable trajectories of penetration effects. And we undertake imal parameter design method of variable trajectories. According to analsis target missiles should enlarge maneuver amplitude A A as much as possible within overload limit when target missiles undertake variable trajector maneuver of target missiles. At same time we 4