ADJOINT ANALYSIS OF GUIDED PROJECTILE TERMINAL PHASE

Transcription

1 9 ADJOIT AALYSIS OF GUIDED PROJECTILE TERMIAL PHASE Tio Silrnt nd Ari Siltvuori Fculty of Engineering nd Architecture Alto Univerity, School of Science nd Technology, Finlnd Sury Guided projectile terinl phe gint trget t ground level i invetigted uing n djoint iultion. A peudo-optil projectile nvigtion gin i looked for gint trget diturbing the projectile guidnce. The ue of counter-eure i odeled" uddenly detected trget brupt otion during the guidnce terinl phe. The i ditnce obtined re tudied nd the projectile optil nvigtion gin i choen bed on the xiu tolerted i ditnce. OTATIO C Accelertion Pitching oent coefficient C Pitching oent coefficient lope C Pitch dping oent q coefficient C Pitching oent control derivte C orl force coefficient C orl force coefficient lope C orl force control derivte d I y G l L M n q Dieter Moent of inerti Gin Length Lplce-trnfor denottion Pitching oent M Lod fctor Kinetic preure Angulr velocity M qsd C C Qd V C qs C C V Lplce - S t t go t F w V α ρ λ dj x Reference re Tie Tie-to-go Flight tie, finl tie Trget lterl velocity Velocity vigtion gin, norl force Angle of ttck Cnrd deflection ngle Dping rtio turl frequency Tie contnt Air denity Seeker-hed turning ngle Mi ditnce Adjoint vector Subcribt Airfre AP Autopilot c Cond, cloing Seeker-hed oie T Trget o Initil vlue d

2 9 ITRODUCTIO Beide the tie-forwrd direct iultion the djoint technique i often utilized in guided wepon end-ge nlyi. The ethod h prticulrly erit to quickly give perfornce projection of liner tie-vrint yte. So fr the ethod h not been ued widely one would expect bed on it flexibility nd ppliction potentil []. The objective of thi pper i to tudy the cpbility of the djoint ethod to predict the endge i ditnce. An optil yte gin i choen bed on the i ditnce obtined. However, t firt oe generic guided projectile erodynic propertie nd other chrcteritic etc re etited to find out oe repreenttive djoint iultion input dt. GEERIC GUIDED PROJECTILE A generic guided projectile tudied i depicted in Fig. nd it chrcteritic re lited in Tble. Tble. Fig.. A generic cnrd-controlled guided projectile. Phyicl nd geoetricl dt of generic cnrd-controlled guided projectile. Chrcteritic Vlue Chrcteritic Vlue dieter, d C length, l C center (CG) (fro the noe) C q -9 weight kg C -. oent of inerti, I y. kg C wing pn Actutor dynic ω = rd/; ζ =. cnrd control fin pn Ref re, S A iple ccelertion utopilot odel w ued to find out the projectile tep cond repone in order to chooe relitic utopilot (AP) pole nd projectile irfre () qudrtic pole loction. The utopilot block digr i depicted in Fig.. d

3 9 c G G G G Actutor Airfre q Fig.. Autopilot block digr. The gin vlue olved re G =.9, G =., G =. nd G =-. (M=.9 t flight ltitude ). Lod fctor [g] Repone Cond,,,,,, Tie [] Fig.. Projectile repone to g lterl ccelertion cond (M=.9 t flight ltitude ). The equivlent tie contnt ( % of cond reched) without eeker-hed contribution i bout.. The projectile cond repone including the utopilot nd ctutor lg t Mch nuber.9 nd t ltitude i depicted in Fig.. Correponding ngle of ttck hitory with conded nd true fin deflection re depicted in Figure nd. The cond updte frequency ued w Hz. The totl equivlent tie contnt i pproxitely. (ee Fig. ) bout which the irfre contribution i. if the ctutor portion i tken to be negligible. With the dping fctor. choen the reulting nturl frequency ω i bout rd/. Angle of ttck [deg],, Tie [] Fig.. Angle of ttck tie hitory.

4 9, Fin deflection [deg],, Repone Cond,,,,, Tie [] Fig.. Fin deflection hitorie obtined. ADJOIT SIMULATIO MODEL The beline projectile linerized guidnce loop t the bckground of the djoint odel i depicted in Fig.. In thi tudy th order guidnce loop with three rel pole nd qudrtic ditribution odel the guided projectile yte. The three rel pole tie contnt in the loop re for eeker-hed lg (τ for eeker-hed nd τ for noie filter) nd for utopilot (τ AP ). The tie contnt re. for ech of firt order coponent. The projectile irfre () inerti i odeled with econd order repone. The nturl frequency ω nd the dping rtio ζ re thoe bove-entioned rd/ nd. repectively. The projectile yte totl tie contnt obtined i bout τ tot =.. T V c t go - - V c AP Fig.. The tie-forwrd iile guidnce loop ued in thi tudy. The djoint odel i bed on thi originl yte. Projectile neuvering cpbility w not liited for the ke of the djoint yte linerity. o erodynic dt w explicitly preent in the iple loop of Fig.. The tndrd proportionl nvigtion lgorith w ued in thi tudy. The cloing peed V c in the nvigtion forul i prcticlly the e the projectile velocity t the decending prt of trjectory. Thi i ued typiclly to be bout the peed of ound. Contnt vlue / w ued for the projectile velocity in the coputtion.

5 The djoint yte cn be tie-vrying nd for exple the nvigtion gin y chnge during the iultion. However, the gin of the originl yte ut be generted bckwrd for the djoint yte []. In the trditionl preenttion with the inverted block-digr ignl flow the originl yte output of interet (the i ditnce) i een to becoe n ipulive input to the djoint yte. Correpondingly the originl yte input turn into n djoint output []. However, the trditionl djoint contruction i not perfored in thi tudy. Inted of tht the preenttion follow the text of Ref. [] with the djoint ethod derived in the generl etting of tte-pce odel. The block digr of Fig. yte cn be written in tte pce for ) ( ) ( t B t A u x x () y Cx(t) () nd we obtin by inpection T AP AP AP c go c go c AP n V t V t V dt d / / / / / / / () The input to yte i the trget neuvering n T (t) which i tken to be in thi tudy. Vrible t go = t F - t i the tie-to-go fro the ipule initition (= reolution of the trget oveent) to the interception nd t F i the finl tie or the tie of flight. The eeker-hed turning ngle nd irfre re denoted λ nd. The i ditnce i wihed the reult nd the output i choen to be y= [ ] AP () where the trix C i [ ]. The djoint of the tie-forwrd tte-pce odel i dj T dj go A t x x d d () T dj C () x () 9

6 The ignl flow in the djoint loop i phyiclly eningle. The reult wihed re obtined by proceing the outcoe uitbly outide the loop. For exple the i ditnce ξ due to the trget contnt lterl velocity w i obtined fro dj ( F F o 9 t ) wx ( t t ) () The yte i liner nd oe trget coplex neuvering effect on the i ditnce cn be obtined utilizing the uperpoition principle for iple neuver. ED-GAME GEOMETRY The trget i t ground nd i diturbing the terinl phe of the projectile flight. The projectile pproche the trget bout fro bove (ee Fig. ). Only the finl econd of flight re invetigted. The projectile reolve the true trget otion t oe point of the terinl phe. The trget oveent conit of contnt lterl velocity / to Et ocited with ocillting longitudinl (orth-south) velocity (pek vlue /). The trget poe either coineditributed or inuoidl velocity ocilltion with ngulr velocity rd/. The poitive direction re to Et nd orth (right nd forwrd repectively, ee Fig. ). The projectile pitch nd yw guidnce loop re identicl nd re tudied eprtely. In prctice thi i to cobine the djoint outcoe of the e loop uing the Pythgor forul in order to get D i ditnce reult. Once the djoint vector x dj (or error trck ) i obtined it will be redily vilble for tudie to find out vriou neuver cobintion effect on the i ditnce. Fig.. The end ge geoetry tudied. The projectile pproche bout fro bove nd the trget i locted t the Origin t ground level. The true trget brupt otion i detected t oe point of the terinl phe. In the djoint iultion the udden oveent will tke plce t ll t go -vlue (ll ditnce) in one run.

7 9 The inuoidl neuver cn be repreented n ipule through econd-order hping network [] ince w L ( w in( t)) () The correponding Lplce trnfor of coine-ditributed neuver i w L ( w co( t)) (9) where w nd ω re the trget velocity nd the ngulr velocity. The i ditnce wihed re dj coputed by integrting the error trck x through the network. The trget evive noeuvre preented i perhp not relitic but it i hoped to be illutrtive in the context. The trget pth on the ground een fro bove re depicted in Fig.. Soe lnt trjectory ngle effect could obviouly be introduced in the vlue of longitudinl velocity ocilltion plitude. Coine ocilltion Sinuoidl ocilltion orth [] Et [] Fig.. The trget end ge noeuvre reult of velocity ocilltion for... otion. The projectile i pproching the coordinte yte Origin fro bout bove. The trget otion fro the Origin i detected during the terinl phe. RESULTS At firt in Fig. 9 i depicted the linerized tie-forwrd iultion nd the revered djoint yte reult obtined in ce of pure trget lterl velocity /. In ce the nvigtion rtio w. The projectile-trget initil ditnce w vried in the tie-forwrd run to find out the i ditnce function of flight tie t f. The correponding djoint t go -grph w obtined in ingle coputtion. The i ditnce i iply the lterl velocity tie the redily vilble error trck dj x. The reult re een to tch perfectly.

8 9 Mi ditnce [] - Adjoint 9 - Flight tie or tie-to-go [] Fig. 9. The i ditnce in ce of pure trget lterl velocity function of the flight tie or tie-to-go. Secondly the djoint reult re copred with the one obtined uing nonliner tieforwrd bed ethod []. The totl i ditnce obtined re depicted in Figure nd. The greeent of reult i een to be t let fir. Mi Ditnce [] Adjoint onliner t go [] Fig.. The i ditnce in ce of trget contnt lterl velocity nd coine ditributed longitudinl velocity. The reult re preented function of the tie-to-go nd the djoint nd nonliner reult re copred (=).

9 9 Mi Ditnce [] Adjoint onliner t go [] Fig.. The i ditnce in ce of trget contnt lterl velocity nd inuoidl longitudinl velocity function of the tie-to-go. The reult re preented function of the tie-to-go nd the djoint nd nonliner reult re copred (=). It i worth of noting tht the projectile pek ccelertion obtined t ll t go -vlue re firly high (up to g) nd y exceed the true neuvering cpbility vilble. The projectile ccelertion h not o fr been liited in thi tudy ince it i not poible in the djoint nlyi. The third work phe w to crry out the nonliner coputtion once ore with g-liit for both chnnel eprtely. The liittion effect i depicted in Fig. for the ce with coine ditributed longitudinl velocity. The pttern i till recognizble even though the new reult re een to be coniderbly lrger when t go <. Mi Ditnce [] Adjoint onliner onliner+ g liit t go [] Fig.. The i ditnce in ce of trget contnt lterl velocity nd coine ditributed longitudinl velocity. The reult re preented function of the tie-to-go. The djoint nd nonliner reult lo with g-liit effect included re copred (=).

10 Mi Ditnce [] 9 9 t go [] Lt Velocity+Long Co Velocity Lt Velocity+Long Sin Velocity Fig.. The i ditnce obtined for the = ce. The gin i too ll to ke the projectile neuver enough to hit the trget. Mi Ditnce [] 9 t go [] Lt Velocity+Long Co Velocity Lt Velocity+Long Sin Velocity Fig.. The i ditnce obtined for the = ce. Mi Ditnce [] 9 t go [] Lt Velocity+Long Co Velocity Lt Velocity+Long Sin Velocity Fig.. The i ditnce obtined for the = ce. The gin give ot hit opportunitie.

11 9 Mi Ditnce [] 9 t go [] Lt Velocity+Long Co Velocity Lt Velocity+Long Sin Velocity Fig.. The i ditnce obtined for the = ce. The obtined i ditnce with the nvigtion gin vrible re depicted in Figure. The reult re preented function of tie-to-go t go. Only four run were needed for the four different nvigtion gin (,, nd ) conidered to obtin the reult preented. The totl i ditnce cued by the contnt lterl nd ocillting longitudinl velocity were obtined uing the Pythgor forul utilizing the e error trck for the both chnnel eprtely. The hit-criterion in thi pper i defined to be eter or le. The t go window to hit for different nvigtion gin (,, nd ) were copred nd bed on thi very liited tudy it ee tht the ce with = give ot hit opportunitie. COCLUDIG REMARKS The djoint ethod w ued to obtin the i ditnce gint trget locted t ground. The nvigtion gin w vried nd the peudo-optil vlue w found to be for the wepon yte nd end-ge ce tudied. With oe iplifiction nd uption done on ind the ethod prove to be cpble to produce the projectile perfornce projection quickly. The ce tudied, expnded fro the preenttion of Ref. [] including now ie the two chnnel guidnce loop nd the econd order projectile repone, i till very iplitic. The ethod flexibility llow invetigting fr ore coplex yte. REFERECES [] Pul Zrchn, 99, Tcticl nd Strtegic Miile Guidnce. AIAA Progre in Atronutic nd Aeronutic,. [] Mrtin Wei,, Adjoint Method for Miile Perfornce Anlyi on Stte-Spce Model. AIAA Journl of Guidnce, Control nd Dynic, (). [] Tio Silrnt nd Ari Siltvuori, 9, A Siplified Miile Model Agint Mneuvering Trget. Proceeding of SCM-: the nd ordic Seinr on Coputtionl Mechnic, October -, Alborg, Denrk. [] Tio Silrnt nd Ari Siltvuori,, Adjoint Siultion of Guided Projectile Terinl Phe. Proceeding of SCM-: rd ordic Seinr on Coputtionl Mechnic, October -, Stockhol, Sweden.