Extrction nd Filter Algorithm of Guidnce Informtion for Fullstrpdown Seeker on Rottion Missile YongShn Liu 1, Li Song 1, nd JingLong Li 1 1 School of Aerospce Engineering, Beijing Institute of echnolog, Chin Astrct. Strpdown seekers re superior to pltform seekers for their simple structure, high reliilit nd light weight ut cnnot mesure the line-of-sight ngle rte informtion for the guidnce of rottion missile directl. his pper ims t the engineering ppliction of full-strpdown seekers on rottion missile prolem. Firstl, line-of-sight ngle rte solution model is estlished. Bsed on the MALAB, the extended Klmn filter (EKF) lgorithm nd unscented Klmn filter (UKF) lgorithm re used to estimte the line-of-sight ngle rte informtion of the full-strpdown seekers. he results show tht using EKF filter nd UKF filter oth cn otin effective guidnce informtion nd the UKF s effect is etter. 1 INRODUCION Unlike trditionl pltform seeker, the full-strpdown seeker ttches ll hrdwre to the projectile, removes the stle pltform nd the control mechnism. Which hs the importnt significnce on reducing the cost of seeker nd improving the sstem reliilit. Wheres the fullstrpdown seeker cnnot directl otin the line-of-sight rte its phsicl trcking loop nd need digitl clcultion methods. Rotr missile sstem hs the dvntges of smller size, lower cost, less control equipment, nd cn reduce the interference cused the smmetr of the projectile. However, the spin of the missile od will cuse cross coupling etween the pitch nd w chnnels. Cregh MA et l. got the results through simulting which shows when the rottionl speed of the projectile is high, the coupling cused rolling ecomes more serious, nd due to the spin of the rotting projectile, the pitching chnnel nd wing chnnel of the rotting projectile will e cross-coupled [1]. As mentioned previousl, the other estimtion method of inertil line-of-sight rte is designed sed on reltive motion etween missile nd trget, where mn estimtion lgorithms re proposed ccording to the different filters dopted. Ehrich et l. put the dditionl rte compenstion + differentil network method to uild single chnnel line-of-sight ngle nd then enter the guidnce sstem through low-pss filter []. Kim et l. [3] put forwrd method to derive such correltion nd finll otined the inertil line-of-sight rte. During the deriving process, the od line-of-sight rte ws clculted differentil network. Bi Rui et l. [4] sed on the tpicl lunch conditions of smll ir-tosurfce missile estlished n estimtion lgorithm nd used the cuture Klmn filter for the rte estimtion. Sun et l. [5] estimted the inertil line-of-sight rte of strpdown opticl seeker using unscented Klmn filter (UKF) method, consider the estimtion ccurc minl depends on the od line-of-sight ngle ccurc nd gro ccurc nd simulte these two fctors. Wldmnn J [6] descries the modeling of n imging seeker nd the formultion of n extended Klmn filter for the estimtion of line-of-sight rte from mesurements of reltive ngulr displcement etween seeker gimls nd low-cost strpdwn inertil unit. Li, J. J. [7] used the pursuit guidnce lw nd proportion guidnce lw in order to get the line-of-sight rte nd through mending innovtion in the equtions of Klmn filter, outliereliminting Klmn filtering lgorithm is chieved. In this pper, first sed on the informtion coupling prolem of rottion missile nd the full-strpdown seeker prolem, decoupled model is estlished to extrct the inertil the line-of-sight ngle rte informtion from the oservtion dt of full-strpdown seekers. And then using the extended Klmn filter (EKF) lgorithm nd the unscented Klmn filter (UKF) lgorithm, the rel-time estimtion of projectile guidnce informtion is simulted nd lst verified the effectiveness of the lgorithms. LINE-OF-SIGH ANGLE RAE MODEL In this section, define four coordinte sstems nd the trnsformtion etween ech coordinte sstem t the outset. hrough introducing the missile trget distnce nd the pproching speed, the stndrd stte equtions nd oservtion equtions re presented. Getting the reltion etween the stte equtions nd oservtion eqution, the inertil line-of-sight elevtion ngle rte nd the inertil line-of-sight zimuth ngle rte re otined. he Authors, pulished EDP Sciences. his is n open ccess rticle distriuted under the terms of the Cretive Commons Attriution License 4.0 (http://cretivecommons.org/licenses//4.0/).
.1 Coordinte sstem definition he following coordinte sstems nd ngles will e used in this pper[8]:the inertil coordinte sstem O e x e e z e, the od coordinte sstem Ox z,the line-of-sight coordinte sstem Ox s s z s, nd the od line-of-sight coordinte sstem Ox l l z l.he inertil lineof-sight elevtion ngle q,the inertil line-of-sight zimuth ngle q,the od line-of-sight elevtion ngle q,the od line-of-sight zimuth ngle q,the pitch ngle,the w ngle,the roll ngle. he conversion of ech coordinte sstem is shown in Fig.1. inertil coordinte sstem Fig 1. Conversion of ech coordinte sstem.. Line-of-sight Informtion stte equtions According to the kinemtics nd geometric reltions of the rottion missile nd trget in spce[9-10], the rottionl ngulr velocit of the line-of-sight coordinte sstem Oxs s zs reltive to the ground coordinte sstem Oe xe ez e is: s= q sin q i q cos q j q k (1) s s s r in Eq.() represents the chnge rte of the missiletrget distnt,nd in the line-of-sight coordinte sstem Ox z hs the following reltionship: s s s r rq j rq cos q k x r rq sin q k rq i r rq cos q i rq sin q j z L(,, ) L(,, ) L(q,q ) L(q, q ) L(q, q ) L(q, q ) line-of-sight coordinte sstem L(q c) od coordinte sstem od line-of-sight coordinte sstem Differentite the missile-trget reltive speed V to otin the missile-trget reltive ccelertion is: r rq rq cos q V rq Vq rq sin q cos q rqq sinq Vq cosq rq cosq () (3) Assume tht the trget does not motorize, tht is to s the missile-trget reltive ccelertion 0, the Eq. (3) cn e otined: r r( q q cos q ) Vq q q sin q cos q (4) r Vq q qq tn q r he stte vector of the sstem cn e ugmented s 1 3 4 = x x x x x q q q q.hus, the line-of-sight informtion stte equtions re derived comining Eq. (4) s follows: x1 x V x x xx4 tn x3 r x3 x4 V x x sin x cos x x r 4 3 3 4.3 Oservtion Equtions Assume the position of the trget in the line-of-sight coordinte sstem nd the od line-of-sight coordinte sstem re oth R 0 0.And the coordintes of the trget nd the missile in the inertil coordinte sstem re x z ndx z. According to the e e e ed ed ed conversion reltionship etween the line-of-sight coordinte sstem Oxs sz s nd the inertil coordinte sstem Oexe ez e, the reltive equtions cn e written s: x xe xed R cos q cos q e ed = Rsin q z z e z ed R cos q sin q he geometric reltive equtions re Eq. (7). q rctn x z z q rctn x In the similr w, define the coordintes of the trget nd missile in the od coordinte sstem re x z nd D D D (5) (6) (7) x z. he conversion reltionship etween the od coordinte sstem Ox z nd the od line-of-sight coordinte sstem Ox l l z l :
x x xd R cos q cos q D Rsin q z z z D R cos q sin q he geometric reltions re: q rctn x z z q rctn x (8) (9) 3 Rottion Missile Guidnce Informtion Estimtion nd Simultion In this pper, the Extended Klmn Filter (EKF) nd Unscented Klmn Filter (UKF) lgorithm re simulted nd verified under oth the nvigtion sstem mesurement dt nd the stpling trget coordintes hve errors. In the inertil coordinte sstem, the ctul position of the trget is (0m,.5m, 0m), the coordinte of stpling trget is (100m, 5.5m, 60m). ht is, the stpling trget mesurement error is (100m, 50m, 60m). he initil position of the missile in the ground coordinte sstem is (-7900m, 050m, 60m). Rij is the element of the conversion mtrix C e in the i th row nd j th column which is the coordinte trnsformtion mtrix from the ground coordinte sstem to the od coordinte sstem,. he conversion mtrix: C e cos cos sin cos sin sin cos cos sin sin cos cos sin sin cos cos sin sin cos sin sin cos cos sin sin sin sin cos cos (10) C e Comine the Eq. (6), Eq. (8) nd Eq. (10) cn get: Rcos q cos q R11 R1 R13 Rcos q cos q Rsin q = R1 R R 3 Rsin q Rcos q sin q R R R Rcos q sin q 31 3 33 (11) Angle informtion provided full-strpdown seeker includes the od line-of-sight ngle q nd q, which re considered s the oservtion vriles. Under this circumstnces, defining the line-of-sight oservtion vector q q = 1. he line-of-sight oservtion equtions of the full-strpdown seeker cn e written s: Fig. Missile X, Y, Z loction. It cn e seen from the Fig. tht the mesurement errors etween the ctul position nd mesurement position of the rottion missile in the inertil nvigtion sstem grdull increses with time. herefore, directl using the dt mesured the inertil nvigtion sstem to guide would miss trgets. In the process of the rottion missile fling to the trget, the od line-of-sight elevtion ngle q nd the od line-of-sight zimuth ngle q re shown in Fig. 3. 1 rcsin( R1 cos x3 cos x1 R sin x3 R3 cos x3 sin x1 ) R cosx R tnx R sin x 31 1 3 3 33 1 -rctn( ) R11cos x1 R1 tnx3 R13 sin x1 (1) So fr, the line-of-sight ngle informtion stte equtions nd the oservtion equtions of the of the fullstrpdown seeker on the rotting missile re estlished. From Eq. (5) nd Eq. (1), it cn e seen tht there is strong nonlinerit existing oth in the line-of-sight stte equtions nd oservtion equtions. Fig 3. Bod line of sight nd prtil enlrgement. Since the rotting missile will generte the pitch nd w directions coupling during the movement, the od line-of-sight elevtion ngle q nd the od line-of-sight zimuth ngle q in Fig. 3 re sme size nd 90 degrees phse difference. Due to the low repetition rte of the lser pulse code, the mesured signls of the od lineof-sight elevtion ngle nd the od line-of-sight zimuth ngle re not perfect s sine functions. 3
MAEC We of Conferences 14, 03008 (018) 3.1 Estimtion of Guidnce Informtion Bsed on EKF nd UKF Algorithms he stochstic nonliner discrete formul for the extended Klmn (EKF) filtering lgorithm [11] is: X k k, k 1 X k 1 ( Xˆ k 1, k 1 ) Wk 1 (13) Z k H k X k Vk (14) In the Eq.(13) nd Eq.(14), W is the process noise mtrix, V is the oservtion noise mtrix. he line-of-sight informtion of rotting missile hs strong nonliner chrcteristics. When ppling extended Klmn filter, the Jcoin mtrix is needed to otin the devition sed pproximte liner equtions of the stte equtions. J k, k 1 f (X k 1, k 1) f X k 1 X k* 1 Fig4. EKF filtering line-of-sight nd line-of-sight rte estimtion. (15) X k 1 X kn 1 Discretize the Jcoin mtrix: k, k 1 I J k, k 1 t (16) Unscented klmn filter (UKF) lgorithm firstl needs to designs series of Sigm points. hen clculting the result of f with the set sigm points, nd (Z, PZ) sed on i. ˆ k(0) 1 X k 1 (i) (17) k 1 Xˆ k 1 ( (n ) Pk 1 )i, i 1,,..., n (i) k 1 Xˆ k 1 ( (n ) Pk 1 )i, i n 1, n,..., Propgte k, k 1 through nonliner Fig 5. UKF filtering line-of-sight nd line-of-sight rte estimtion. 3. Computtionl error nlsis Clculte the EKF nd UKF errors etween the filter vlue nd the true vlue on the line-of-sight elevtion ngle, the line-of-sight elevtion rce, the line-of sight zimuth ngle nd the line-of-sight zimuth rce. oservtion function H ( ) rk. From k, k 1 cn get the output prediction Zˆ k, k 1, s well s the self-covrince mtrix PZk nd the mutul covrince mtrix PXkZk : k, k 1 H k ( i, k 1, vk ) rk, i 0,1,..., n Zˆ k, k 1 i (m) i,(k,k 1) i (m) (18) H k ( i, k 1, v k ) rk (19) PZk i (c) ( i,(k,k 1) Zˆ k, k 1 )( i,(k,k 1) Zˆ k, k 1 ) Rk (0) PXkZk i (c) ( i, (k, k 1) Xˆ k, k 1 )( i, (k, k 1) Zˆ k, k 1 ) (1) So fr, we cn use the EKF filter lgorithm nd UKF filter lgorithm to extrct the line-of-sight ngle rce of the full-strpdown lser seeker on the rottion missile s the guidnce informtion [1]. It is estimted nd simulted s shown in Fig.4 nd Fig.5. Fig 6. Errors etween EKF filter nd true vlue. 4
guidnce informtion of full-strpdown seeker for rottion missile more ccurte. In this pper, ecuse of time constrints we just trgeted reserch the full-strpdown on rottion missile, other tpes of seeker lgorithms, such s infrred rdr, cn e further studied. And if more time permits, the control sstem of rotting projectile cn e further studied to relize projectile od integrtion. References Fig 7. Errors etween UKF filter nd true vlue. It cn e seen from the Fig.6 nd Fig.7 tht if the nonlinerit is not strong, the results of the EKF filter nd the UKF filter re similr. But in the cse of high nonlinerit such s the line-of-sight elevtion rce, the UKF filtering is etter thn the EKF filtering. Clculte the EKF nd UKF men error etween the filter vlue nd the true vlue of the on the line-of-sight ngle nd the line-of-sight rte gets the results shown in le 1: le 1. Men error of EKF nd UKF filtering informtion. Men error EKF filter UKF filter LOS elevtion ngle 0.3446 0.088877 LOS elevtion rce 0.34748 0.133783 LOS zimuth ngle 0.1946 0.0901 LOS zimuth rce 0.318609 0.186944 Clculte the root-men-squre error ((RMSE) of the EKF filter nd UKF filter cn otin le.. le. RMSE of EKF nd UKF filtering informtion. RMSE EKF filter UKF filter LOS elevtion 0.095683 0.11506 ngle LOS elevtion 0.194481 0.354065 rce LOS zimuth 0.085619 0.104441 ngle LOS zimuth rce 0.04107 0.14643 4 CONCLUSION his pper ims t the engineering ppliction prolem of the full-strpdown lser seeker on rottion missile. First estlishing line-of-sight rte informtion extrction model. Bsed on the model, using EKF filter lgorithm nd UKF filter lgorithm to filter nd estimte the lineof-sight ngle nd rte informtion of the full-strpdown lser seeker. And verifing the filtering lgorithm MALAB simultion. he results show tht UKF filter is etter thn EKF filter nd cn e used to extrct the 1. Cregh MA, Mee D J. Attitude guidnce for spinning vehicles with independent pitch nd w control[j]. Journl of guidnce, control, nd dnmics, 33(3),(010). Ehrich R D, Vergez P. Strpdown seeker technolog for the terminl guidnce of tcticl wepons[c],agard Guidnce nd Control Aspects of cticl Air lunched Missiles 15p,1980, 81(10): 7-15. 3. Kim, D., Roo, C. K., Kim, Y., nd Kim, J., Guidnce nd Control for Missile with Strpdown Seeker, Proceedings of the 11th Interntionl Conference on Control, Automtion nd Sstems, Vol. 86, IEEE Pul.,Pisctw, NJ, 011, pp. 969 97. 4. Bi Rui,Xi Qunli, Zhng Dochi, echnolog of line-of-sight rte estimtion using SCKF for strpdown seeker, Infrred nd Lser Engineering,Vol.46 No.11,(017) 5. Sun, Chu H, Zhng B, et l. Line-of-sight rte estimtion sed on UKF for strpdown seeker [J].Mthemticl Prolems in Engineering, 015, 015(1): 1-14. 6. Wldmnn J.Line-of-sight rte estimtion nd linerizing control of n imging seeker in tcticl missile guided proportionl nvigtion. IEEE rnsctions on control sstems technolog.10, (00) 7. Li, J. J., Reserch of LOS Rte Estimtion Method for Strpdown Imging Seeker, M.S. hesis, Hrin Inst. of echnolog, Hrin, Heilongjing Province, PRC, (008). 8. Qin Xingfng, Lin Ruixiong, Zho Ynn, Missile flight mechnics (in Chinese), Beijing Institute of echnolog Press, Chin, (013). 9. Jmes, M. M., Line of Sight Rte Estimtion for Guided Projectiles with Strpdown Seeker, AIAA GNCC, (015). 10. Joongsup Yun, Chng-Kung Roo, ek-lul Song, Strpdown sensors nd seeker sed guidnce filter design, ICCAS, 468-47, (008). 11. Julier, S. J., nd Uhlmnn, J. K., Unscented Filtering nd Nonliner Estimtion, Proceedings of the IEEE, No. 3, (004) 1. Luo Yufeng, Liu Yong, he simultion reserch on the strpdown inertil nvigtion lgorithm sed on the trjector genertor (in Chinese), Journl of Henn Poltechnic Universit (Nturl Science), 06,(015) 5