Comparative Research on Multi-missile Cooperative Attack. Between the Differential Game and Proportional Navigation Method

Transcription

1 6 Sxh Inernaonal Conerence on Insrumenaon & easuremen, Compuer, Communcaon and Conrol Comparave Research on ul-mssle Cooperave Aac Beween he Derenal Game and Proporonal Navgaon ehod Guangyan Xu, Guangpu Sh Absrac hs paper nroduces an approach ha nvolved wh mul-mssle cooperave aac based on he derenal game. By comparng wh he radonal proporonal navgaon mehod, he derenal game shows beer perormance. Frsly, a mahemacal model o mul-mssle cooperave aac s esablshed; Secondly, hess nroduces he derenal game and proporonal navgaon mehod by solvng he gven problems; Fnally, a numercal smulaon has been made n alab programmng envronmen, whch shows ha he derenal game has more applcably and cooperave racng ably. Keywords Derenal Game; Proporonal Navgaon; ul-mssle; I. INRODUCION Because o beer perormance n deecng and aacng enemy arcra, he long-range ar-o-ar mssle becomes an eecve weapon n dealng wh elecronc reconnassance arcra, AWACS and oher hgh-value arges n modern war. In a long erm, he mssle pah plannng s an mporan ssue o sudy, whch can be seen as derenal games or boh conronaon sdes. In 95s, he derenal game s rsly proposed by Isaacs o solve he sngle-pursuer and sngle-evader problem[]. In he pas ew decades, a large number o research have been made n hs eld and acheve grea success. In he leraure[], he derenal-game verson o he coplanar al-chase aeral comba problem s analyzed by he numercal derenal dynamc programmng mehod. A coplanar pursu-evason game o nd n he amosphere beween a coasng pursuer and a maneuverng evader o consan speed s consdered n leraure[3]. he complex pursu-evason games wh sae varable nequaly consrans are nvesgaed n leraure[4][5]. I proposed a novel approach o he desgn o opmal rajecory nvolvng mulple objecve uncons and mulple players n leraure[6]. And n leraure[7][8][9][], derenal games had been appled o solve he pursu-evason problem under deren condons and acheve success. hs paper nroduces an approach ha nvolved wh mul-mssle cooperave aac based on he derenal game. By comparng wh he radonal proporonal navgaon mehod, he derenal game shows beer perormance II. DIFFERENIAL GAES DESCRIPION o smply he problem, a mahemacal model o wo mssles (, and sngle arge ( s esablshed n wo-dmensonal space. y y y y x LOS v x LOS v Fg. he model n wo-dmensons x V Fgure shows a geomercal model, where ( x, y ( x, y, ( x, y are he coordnaes o,, respecvely; v, (,, are, respecvely, he velocy and he aspec angles o velocy wh respec o he neral reerence lne. LOS (, s he lne-o-sgh beween he mssle (, and he arge. x *Resrach suppored by ABC Foundaon under gran # Guangyan Xu Auhor s wh he School o Auomaon, Shenyang Aerosapce Unversy, Shenyang, 36 Chna (phone: 39494;e-mal:guangyan_xu@63.com. Guangpu Sh Auhor s wh he School o Auomaon, Shenyang Aerosapce Unversy, Shenyang, 36 Chna (e-mal: shguangpu39@63.com. he dynamc equaons o moon are gven by: x v cos y vsn (,, he nal condons are assumed o be gven by ( /6 $3. 6 IEEE DOI.9/ICCC

2 x( x y( y (,, ( ( Where x [,, ] x y,,, are, respecvely, he sae varables and aspec angle rae o boh conronaon sdes. III. HE RANSFORAION OF DIFFERENIAL GAES he nonzero-sum mul-player derenal game s esablshed on he condon o conrol varables unconsraned. he dynamc equaons o moon are gven by equaon (. Accordng o he model n wo-dmensonal plane shown n Fgure, he relave dsance beween he mssle and he arge s dened as ollows equaon: ( x x ( y y ( (, 3 Now or opmzaon, he ollowng perormance ndex J s proposed: mn J ( ( K d 4 mn J ( ( K d 5 Where K, K, K are posve real number ha reaed as weghng acor relecng he relave mporance o he commanded acceleraon wh respec o he nercepon duraon. Based upon he perormance ndex, he Hamlonan o he sysem s consruced as: H λ λ L L v cos v sn 3 K v cos v sn 3 K where λ,, 3,,, are 3 he cosae varables o (, Accordng o he varaonal mehod, he cosae equaons are gven by: H H ; (, x y 7 H 3 sn cos v v Accordng o he varaonal mehod, he ransversal condons are gven by: (6 ( ( x x 3( ( y y 8 3( ; On he condon o conrol varables unconsraned, he conrol varables should be sased by: H H K 3 ; K 9 Equaons (9 can be solved mmedaely wh he help o he cosae varables: / K ; /K 3 3 Subsue ( no ( and solvng he resulng derenal, we have he equvalen opmal conrol model wh respec o he orgnal derenal equaons. he perormance ndex J s gven by: mn J ( K d P P P P he governng equaons are gven by: ( x v cos x v cos (, y v sn y vsn 3 v sn 3 K v cos he orgnal condons and he ransversal condons are gven by: x ( ( ( ( x x ( x x x y ( y y ( y ( y ( y ( ( ( 3( (, o aclae he numercal calculaon, he equaons above are ranslaed by applyng Legendre Pseudo-specral mehod no he nal orm. N N mn J ( x ( N, N KPP( s.. x ( x,, (,, x x λ D λ h( x, λ, x ( x, λ, (,, λ h x λ 3 x ( x x ( x ( N x ( N x ( N y ( y y ( y ( N y ( N y ( N ( ( 3( N 94

3 Where marx o LGL., D are he negral wegh and derenal, D are unque when he conguraon pons o LGL are dened. IV. SIULAION COPARISON A. Proporonal Navgaon ehod Prncple Proporonal navgaon s, by ar, he mos mporan o he classcal gudance laws. he proporonal navgaon gudance law s dened as law ha generaes a gudance command whch ensures ha he rae o roaon o he mssle velocy vecor s proporonal o he rae o roaon o he LOS. he relave poson o he mssle and arge s shown n Fgure. r q q Fg. he model o proporonal navgaon mehod Where r s he relave dsance o he mssle and he arge, q s a arge lne o sgh angle, whch s he angle beween arge lne and one daum lne n aac plane, counerclocwse roaon rom daum lne o he arge lne o sgh s posve. s a angle beween he mssle's velocy vecor and daum lne. s a angle beween he arge's velocy vecor and daum lne. s a mssle preva angle, whch s a angle beween he mssle's velocy vecor and arge lne o sgh. s a arge preva angle, whch s a angle beween he arge's velocy vecor and arge lme o sgh. he gudance relaon equaon s gven by: Nq Where, N s he navgaon gan. Accordng o he proporonal navgaon mehod, he relave moon equaons beween he mssle and he arge are gven by: dr v cos v cos d dq r v sn v sn d q( N ( Nq From ormula (4, we can ge equaon (6 N q v v Where, N whch usually ranged rom 3 o 6 s he proporonal navgaon consan o he mssles. q s he angular rae o LOS ha wh respec o he neral reerence lne. B. Smulaon A numercal smulaon made n ALAB programmng envronmen se he nal poson o,, s ( 5 m,, (5 m, and (, m. he nal angel o he mssle s degree, he nal angel o he mssle s 8 degree, and he nal angel o he arge s degree. he speed o mssle s v v =.35m/s, he speed o arge s v =.3m/s. he proporonal navgaon gan s N = N =6. he weghng acors are K = K = K =. he smulaon me s =45s, LGL conguraon pon number s se o 6. Smulaon ha use he SNOP uncon n he OLAB oolbox has been made o analyss he derenal game model based on he mul-mssle cooperave aac. In order o mprove he accuracy o he soluon, he sae varables are normalzed, he reerence lengh s 5Km, he reerence angle s, he normalzed nal sae o he mssle and he arge s gven by: [ x, y,, x, y,, x, y, ], [.,,,.,,,,.,] Smulaon calculaon s made n ALAB accordng o he derenal game and he proporonal navgaon mehod. For he purpose o comparson, he wo algorhms adop he same nal parameers, and he rajecory o reman he same oo. Fnally, we ge he smulaon resuls as shown n gure 3. cross range y(m Derenal Game Derenal Game Proporonal Navgaon Proporonal Navgaon arge down range x(m Fg3. Comparson o he move rajecory beween he proporonal navgaon mehod and he derenal game Accordng o he smulaon resul, shows ha he arge res o escape rom mssles, and a he same me, 95

4 he mssles ry o shoren he dsance rom he arge. Obvously, he derenal game s applcable and eecve n he case o mul-mssle cooperave aac. headng angle (deg Derenal Game Derenal Game Proporonal Navgaon Proporonal Navgaon arge cooperave aac whou conrol consrans. hs paper proposes he sae equaons and perormance ndex o arge and mssle, and by nroducng he assocaed Hamlon-Jacob equaon, he orgnal game s convered no a regular nonlnear opmzaon problem. he LP mehod s used o approxmae he problem by a NLP problem. he mehod can reduce he compuaonal nensy. Smulaons are perormed by placng he wo mssles n a peer o peer poson wh respec o he arge or he game usng he snop solver n he OLAB oolbox, and compared wh he radonal proporonal navgaon mehod vered he mehod has good applcably and good synergy ably o hur me (sec Fg4 Comparson o aspec angle beween he proporonal navgaon mehod and he derenal game I shows he comparson o aspec angle beween he proporonal navgaon mehod and he derenal game n gure 4, rom whch we can see boh he mssles and arge hope o move along he LOS. Compared wh he proporon navgaon mehod, he derenal game has grea advanage n he rend o angle. he average relave dsance m he average relave dsance o Derenal Game he average relave dsance o Proporonal Navgaon me (sec Fg 5. Comparson o he relave dsance beween he proporonal navgaon mehod and he derenal game I shows ha he relave dsance beween he mssles and he arge would be decrease connuously. Obvously, he derenal game has vsble advanage compared wh he proporonal navgaon mehod. In concluson, compared wh he proporonal navgaon mehod, he derenal game perorms beer n solvng he problem o mul-mssle cooperave aac. V. CONCLUSION hess presens a novel mehod based on he varaon mehod and he Legendre Pseudo-specral mehod(lp o solve he derenal game problems abou mul-mssle Reerences [] R. Isaacs, Games o Pursu, Rand Corp., Sana onca, 95, P-57. [] B. Jarmar, A.W. erz, J. Breawell, he Varable Speed al-chase Aeral Comba Problem, Journal o Gudance, Conrol, and Dynamcs. 4 ( [3]. Guelman, J. Shnar, A. Green, Qualave sudy o a planar pursu evason game n he amosphere, Journal o Gudance, Conrol, and Dynamcs. 3 ( [4].H. Brener, H.J. Pesch, W. Grmm, Complex Derenal Games o Pursu Evason ype wh Sae Consrans, Par : Necessary Condons or Open-Loop Sraeges, Journal o Opmzaon heory and Applcaons. 78 ( [5].H. Brener, H.J. Pesch, W. Grmm, Complex Derenal Games o Pursu-Evason ype wh Sae Consrans, Par : Numercal Compuaon o Open-Loop Sraeges, Journal o Opmzaon heory and Applcaons. 78 ( [6] Vasle,., Z. F. Bernell. Drec ulphase Opmsaon o ulobjecve rajecory Desgn Problems[C]. AIAA h Space Flgh echancs eeng, -5,. [7] Dave W. Oyler, Perre. Kabamba, Anouc R. Grard., Pursu evason games n he presence o obsacles, Auomaca, Volume 65, arch 6, Pages -, ISSN 5-98, [8] Ponan,., B. A. Conway. Numercal soluon o he hree-dmensonal orbal pursu-evason game. Journal o Gudance, Conrol, and Dynamcs 3(: , 9. 96

5 [9] Andrew K.Sun,Hugh H..Lu,ul-Pursuer Evason,AIAA Gudance, Navgaon and Conrol Conerence and Exhb,8- Augus 8. [] G.Y. Xu, C.. Sun, Vbraon suppressve conrol o lexble-jon space manpulaors based on Legendre pseudospecral mehod, Chnese Conrol and Decson Conerence, Changsha, Chna, 3 ay- June, 4. 97