Near Time-Optimal Feedback Instantaneous Impact Point (IIP)

Transcription

1 1 Ne Time-Otiml Feedbck Instntneous Imct Point (IIP) Guidnce Lw fo Rocket Byeong-Un Jo 1 nd Jemyung An Koe Advnced Institute of Science nd Tecnology (KAIST) 91 Dek-Ro Dejeon Reublic of Koe Abstct Tis e ooses feedbck guidnce lw to move te instntneous imct oint (IIP) of ocket to desied loction. Anlytic exessions elting te time deivtives of n IIP wit te extenl cceletion of te ocket e intoduced. A ne time-otiml feedbck-fom guidnce lw to detemine te diection of te cceletion fo guiding te IIP is develoed using te deivtive exessions. Te effectiveness of te oosed guidnce lw in comison wit te esults of oen-loo tjectoy otimiztion ws demonstted toug IIP ointing cse studies. Keywods Instntneous Imct Point (IIP) Guidnce Ne Time-Otiml Rocket I. Intoduction Te instntneous imct oint (IIP) of ocket given its osition nd velocity is defined s its toucdown oint ssuming fee-fll fligt (witout oulsion) [1]. Te IIP is consideed s vey imotnt infomtion fo sfe lunc oetion of ocket nd it sould be clculted nd monitoed in el-time on te gound fcility o on-bod of te ocket. Te tesssing of IIP tjectoy coss destuction line (DL) is one imotnt citeion fo nge sfety decision ctivtion of te fligt temintion 1 Gdute Resec Assistnt Detment of Aeosce ngineeing 91 Dek-Ro. Associte Pofesso of Aeosce ngineeing 91 Dek-Ro; jemyung.n@kist.c.k. (Coesonding Auto).

2 system (FTS) fo fligt sfety oetion. A numbe of studies on ediction of te IIP nd tei lictions fo fligt sfety oetion could be found in te litetue. Tese studies include te tecniques fo comuting te IIP in vious coodinte systems [-5] metods on comenstion fo te effects of gvity etubtion nd tmoseic dg [6] exessions fo te time deivtives of IIP [7] nd intoduction of new fligt sfety citeion [8]. In ddition to fligt sfety oetions te IIP cn be used fo e-fligt nlysis nd oen-loo otimiztion of ocket ticully to obtin nd secify te imct oint of seted stges. Yoon nd An oosed tjectoy otimiztion ocedue consideing te IIPs of te fist-stge nd ylod fiing segments of lunc veicle s exlicit constints [9]. Using te disesion nlysis Mndic intoduced guidnce nd contol lgoitm tt cn stee ocket so tt its imct oint eces tget loction [10]. Te IIP cnge is imotnt fo ecent lnding guidnce of seted stge of eusble lunc veicle. Fo exmle it is known tt te lnding guidnce fo te seted fist stge of Flcon 9 involves te boostbck bun using tee out of nine engines wic cnge te IIP of stge towd te lnding site (bge si o lunc site) [11]. Tis e ooses new ne time-otiml feedbck guidnce lw tt moves te IIP of ocket to tget oint wose scemtic digm is sown in Fig. 1. Te nlytic fomultion tt descibes te time deivtives of n IIP fo given extenl cceletion vecto (imily oduced by te oulsion system) ws estblised. An otimiztion oblem tt detemines te comonents of te extenl cceletion vecto to lign te IIP deivtive vecto wit te desied diection nd mximize its mgnitude ws fomulted nd it cn be solved nlyticlly by intoducing te Lgnge multilies. Te oosed guidnce lw ws vlidted toug cse study nd comed wit te esults of n oen-loo tjectoy otimiztion to minimize te finl time. Tee key contibutions of tis study e summized s follows. Fist te oosed guidnce lw is feedbck fom tt cn exlicitly secify te IIP t te finl time. Since te guidnce lw is feedbck fom it is obust to te eo coming fom vious souces (e.g. te osition nd velocity eos t te beginning of te guidnce). Second its efomnce is ne time-otiml nd ne fuel-

3 3 otiml ssuming te cceletion ofile of te ocket is given. Lstly te oosed lw does not involve ny itetive ocedue wic is vey ttctive oety fo its otentil on-bod imlementtion. Tis eminde of tis e is ognized s follows. Section II intoduces te metodology to clculte te Kelein IIP nd its time deivtives. Section III ooses feedbck guidnce lw to move te IIP of ocket to desied oint in ne time-otiml mnne wic is obtinble by solving constined otimiztion oblem estblised bsed on te esults of Section II. Cse studies fo vlidting te guidnce lw e esented in Section IV. Finlly Section V discusses te comeensive conclusions of tis study nd otentil ootunities fo futue wok. Figue 1. Scemtic digm exlining IIP guidnce

4 4 II. Clcultion of IIP nd Its Time Deivtives Tis section intoduces te ocedues to clculte te IIP nd its time deivtives in n inetil (t centeed inetil CI) nd otting (t centeed t fixed CF) fmes wic ovides te fundmentls of te feedbck IIP guidnce lw discussed in tis e. Note tt Subsections II-A nd II- B e witten by summizing te esults of io studies conducted by An nd Ro [5 7]. Te metes nd geomety used to comute te IIP nd its time deivtives e sown in Fig.. Figue. Pmetes nd geomety fo comuting IIP nd its deivtives A. Clcultion of Kelein IIP [5] Conside te tnsltionl motion of ocket subject to gvity ( g ) nd n extenl cceletion ( ) s follows:

5 5 v (1) v g g i i i () () In te dynmic equtions nd v e osition nd velocity of te ocket esectively; nd e te comonents of cceletion vecto in osition tngentil nd line momentum diections esectively. In ddition te unit vectos i i nd i e defined s i (3) i v v v v i i i (4) (5) If te Kelein two-body motion is ssumed te gvittionl cceletion is exessed s g() (6) 3 Given cuent osition ( v 0 ) of te ocket its IIP in te CI coodinte fme is exessed s follows: 0 ) nd velocity ( cos( ) sin i i i (7) cos 0 cos v 0 In tis eqution 0 nd e esectively te fligt t ngle nd te ngle of fligt of te ocket exessed s 0 sin v v v v sin ( ) sin ( 0 0 ) c1c 3 c1 c3 ( c1 c )( c3 c ) c1 c (8) (9) wee c 1 c nd c 3 e exessed s c ( ) c 1 c 1 v (10)

6 6 Te time of fligt of te lunc veicle between te cuent time nd te imct time is exessed s follows [1] t F 1 tn (1 cos ) (1 ) sin cos tn v cos 1 cos cos( ) ( ) 1 cos cot sin cos cos 0 0 (11) wee ( ( v / v ) v / ) is defined s te sque of te tio between te cuent velocity nd 0 c 0 0 te cicul obit velocity wit given dius ( vc / 0 ). Te IIP ltitude nd longitude in te CI coodinte system cn be exessed using te comonents of te IIP unit vecto in q. (7) s 1 Lt sin ( iz ) (1) 1 Lon tn ( iy ix) (13) Te IIP longitude in te CF coodinte system is obtined by eflecting te t s ottion duing te time of fligt s Lon Lon ( t t t ) Lon t (14) e ef F e wee e indictes te ottionl te of te t t is te cuent time nd t ef is te time wen te CI nd CF coodintes coincide. Fo detils on te ocedue to comute te fligt time efe to efeences [5 13]. B. Time Deivtives of Kelein IIP [7] Cnge in te IIP of ocket occus wen te extenl cceletion is not zeo ( 0). Te time deivtive of te IIP unit vecto ( d( i ) / dt ) is exessed s line combintion of comonents of te extenl cceletion vecto s follows: T d dix diy diz i d d d (15) dt dt dt dt

7 7 In tis eqution d d nd d e te diections of te IIP deivtive ssocited wit tee comonents of extenl cceletion ( nd ) defined s follows: D d ( sin ( 0 v0)cos ) i0 ( 0 v0 cos ) i v0 (16) D d [ sin ( 0 v0)cos i0 ( 0 v0 cos ) i v0] (17) 1 d 0 sin i (18) In qs. (16)-(18) D nd D eesent te influence of nd on nd e exessed s D sin D ( c sin c cos ) 1 v 0 / e cos 0 0 sin ( c sin c cos ) 1 (19) (0) Note tt owing to te oety tt IIP moves long te sufce of te t te time deivtive comonents d d nd d e ll tngentil to i. In ddition qs. (16) nd (17) indicte tt d nd d wic e te in-lne cceletion comonents ssocited wit nd e llel. On te ote nd genetes te lne cnge motion of te ocket wose contibution in te IIP deivtive is diected to d. Te time deivtive of te time of fligt ( t F ) is exessed s follows e T e tf tf tf tf tf tf t 1 D D 1 D D D D D D D D 1 t (1) F e e F wee F D D F t t e D D e D nd e D in q. (1) e given s D 1 cos 1 cos t 3t 1 e e tf F F e 0 0 en sin sin 0 ()

8 8 t 1 cos 1 ecos cos 1 ecos tf F 0 0 De sin sin 0 e n esin esin 0 (3) D v0 sin0 e v0 sin0 D e (4) D v0cos 0 0 v0 cos e 0 D (5) e In qs. (4) nd (5) e nd n e te semimjo xis eccenticity semimete nd men motion of te obit esectively nd 0 nd e te eccentic nomly vlues of te ocket t te cuent nd imct oints exessed s follows 1 1/ v / 0 0 (6) (7) n 3 (8) / e 1 / (9) 0 cos e 1 0 (30) 1 cos e (31) Note tt t F is comosed of two ts: te t cused by gvity (-1) nd te deivtive ceted by extenl cceletion ( t F ). Te second t ( t otting (CF) fme wic is discussed in te next subsection. F ) is used to define te time deivtive of IIP in

9 9 C. Time Deivtives of Kelein IIP in CF Fme Te time deivtive of te Kelein IIP in te CF coodinte fme is exessed s follows d R d i TI ( i ) tf ( e i ) (3) dt dt wee i is te IIP unit vecto exessed in te CF fme T I is te tnsfom mtix fom te CI to CF coodinte system nd s e is te vecto eesenting te ottion of te t exessed i [ i i i ] T [cos(lt ) cos(lon ) cos(lt )sin(lon ) sin(lt )] T (33) x y z cos( et) sin( et) 0 T I sin( et) cos( et) 0 (34) [0 0 ] T (35) e Combining qs. (15) (1) nd (3) one cn obtin te following exession fo time deivtive of IIP in CF fme. d i d d d (36) dt wee i y t F d ed i x TI d (37) 0 i y t F d ed ix I T d (38) 0 I e d T d (39)

10 10 III. Feedbck Lw fo Ne Time-Otiml IIP Guidnce Tis section ooses ne time-otiml feedbck guidnce lw fo cnging te cuent IIP to te desied loction. Te guidnce lw ensues tt te IIP te vecto is ligned wit te diection of cicul c deting fom te cuent IIP nd iving t te tget osition wit its mximum mgnitude ossible. As eesented in Fig. 3 te sotest t between te cuent nd desied IIP long te sufce of te t is te c connecting te two oints. Let i denote te tget IIP vecto in CF coodi- t nte system. Ten q is defined s vecto noml to te lne secified by i nd t t ( i i) nd i q is defined s its unit vecto s follows: q i i i (40) t ( ) i q / q (41) q Figue 3: Definitions of unit vectos used fo IIP guidnce commnd genetion

11 11 Te unit vecto in te diection of te c (sotest t) between te cuent nd te tget IIPs ( defined s follows. i u ) is i i i (4) u q Te oblem detemines te cceletion vecto [ ] T wose mgnitude is given s m. Te IIP deivtive owing to (= d( i ) / dt d d d ) sould be llel to i u nd its mgnitude sould be mximized. Tt is we wnt to mximize [ d( i ) / dt] i subject to two u constints: 1) [ d( i ) / dt] i 0 nd ) q m 0. Tis oblem (P) cn be fomulted s constined otimiztion oblem descibed s follows. [P CG: IIP guidnce cceletion genetion] subject to T mx J( x) ( c x ) c x c x c x (43) x T xx m x1 x x3 m 0 (44) T f x f1x1 fx f3x3 0 (45) wee x is te decision vecto nd c T nd f T e te oblem metes defining te objective function nd constint esectively defined s x [ ] T (46) T c [ iu d iu d iu d ] [ c c c ] (47) 1 3 T f [ iq d iq d iq d ] [ f f f ] (48) 1 3 Te constined otimiztion oblem defined by qs. (43)-(48) cn be solved by intoducing L- T gnge multilies ssocited wit qs (44) nd (45) λ ( [ 1 ] ). Te ugmented cost function J ( ) x λ cn be defined s J ( x λ) c x ( x x ) ( f x) T T T 1 m ( c x c x c x ) ( x x x ) ( f x f x f x ) m (49) By lying J / x 0 (fo i = 1 3) te following equtions e obtined. i

12 1 J ( f c ) c x f 0 x x J ( f c ) c x f 0 x x 1 J ( f c ) c x f 0 x x (50) (51) (5) Putting qs. (50)-(5) into q. (45) nd solving fo te Lgnge multilie yields cf ( c f c f c f ) T T f f f1 f f3 (53) In ddition by combining qs. (50)-(5) wit q. (44) 1 is detemined s ( f1 c1 ) ( f c) ( f3 c3) T ( if c ( f c) 0) m 1 ( f1 c1 ) ( f c) ( f3 c3) T ( if ( ) 0) c f c m (54) Te cceletion comonents e given s f c 1 1 x1 (55) 1 f c x (56) 1 f c 3 3 x3 (57) 1 Te inuts to te oosed guidnce lw e te cuent osition ( 0) nd velocity (v 0) wic e used to clculte te cuent IIP nd diections fo IIP deivtives nd te oututs of te guidnce lw e te comonents of te guidnce cceletion vecto ( ). In ddition it sould be noted tt te ocedue does not involve ny itetion loos wic is n imotnt oety fo in-fligt imlementtion. Te ovell stuctue of te feedbck IIP guidnce lw esented in tis section is sown s block digm (wit elted eqution numbes) in Fig. 4. Te oosed guidnce is feedbck fom

13 13 tt uses te osition () nd velocity (v) s te inuts comutes te IIP ( i ) nd its diection vectos ( d d d ) s intemedite metes nd genetes te cceletion commnd () s te outut. In ddition ll te ocedues e sequentil witout ny involvement of itetions; even te otiml commnd genetion oblem (P CG) is solved nlyticlly. Figue 4. Block digm descibing te oosed feedbck IIP guidnce ocedue

14 14 IV. Cse Study Te vlidity of te oosed IIP guidnce lgoitm esented in Section III is demonstted toug cse study. Te IIP cnge mneuve of te seted fist stge of eusble lunc veicle (Flcon 9 of Sce X) ws selected s te scenio fo te cse study. Tble 1 summizes te configution of te ocket (seted stge) nd te initil condition used fo te cse study. Tble 1. Rocket configution nd initil condition fo cse study Rocket Configution Pmete Vlue Dy mss ton. Poellnt mss ton 57 Tust tonf 79.6 Secific imulse s 311 Fligt Condition Vlue Initil ltitude km 15.5 Initil seed inetil ( Initil fligtt ngle ( v 0 0 ) km/s ) deg 3.9 Initil osition ( 0) [km km km] [ ] Initil velocity (v 0) [m/s m/s m/s] [ ] Refeence time (t ef) s -40 Fig. 5 sows te cuent loction te oiginl IIP nd te desied imct loction fo te cse study. Te fist nd second cses decese te IIP distnce long te downnge diection by 100 km / 00 km. Te tid nd fout cses incese te nges by 100 km / 00 km. Te fift cse cnges te IIP to te cossnge diection by 150 km. To demonstte te qulity of te solution te esults of te feedbck IIP guidnce simultion wee comed wit te oen-loo tjectoy otimiztion. Te Genel Pseudosectl Otiml Contol Softwe (GPOPS) develoed by Ro et l. [14] ws used to cete te oen-loo otiml solutions fo cses. Te objective of te oen-loo tjectoy otimiztion ws cosen s minimiztion of te finl time (t f) te diection of te cceletion wee used nd te contol vibles nd te IIP t t f ws t imosed s constint ( i i ( t )). Bot te IIP guidnce simultion nd oen-loo otimiztion f

15 15 wee imlemented using MATLAB R015b nd un on mcine wit 3.5 GHz Intel Coe i7 CPU 16 GB RAM nd Windows 7 oeting system. Figue 5: Cuent nd imct loctions fo cse study Tble summizes te esults of te cse study obtined by using te oosed IIP guidnce lw nd te oen-loo tjectoy otimiztion. It cn be seen tt te diffeences in te objective function (finl time) between te esults obtined by two metods e vey smll less tn 1 % fo tee cses nd less tn 5 % even fo te wost cse (Cse ). Wen te esults e inteeted s te vlues of te velocity incements duing te mneuve te efomnce diffeence ws just 6.1 % t te wost cse. Bsed on tis comison esults we cn conclude tt te oosed IIP guidnce lw is ne time-otiml wit eltively smll mount of otimlity g. It is obseved tt te efomnce g (between te IIP guidnce nd te otiml esults) is eltively lge wen te cnge in IIP is not ligned wit te downnge diection. Cse equies tt te IIP cnge to te oosite of te downnge diection nd Cse 5 cnges te IIP in diection eendicul to te downnge diection.

16 16 Tble. Cse study esults (comison wit oen-loo otimiztion) Cse Finl Time (tf) s Poellnt Consumtion ton Otimiztion IIP Guidnce Diffeence % Otimiztion IIP Guidnce Otimiztion IIP Guidnce Diffeence % V m/s Figs. 6-8 sow te gound tjectoies of te ocket osition nd IIP obtined by te two metods fo Cses 4 nd 5 wee te IIPs e moving towd te desied loctions successfully. Tee is little noticeble diffeence between te two tjectoies. Figue 6. Tjectoies of ocket nd its IIP obtined by two metods (Cse )

17 17 Figue 7. Tjectoies of ocket nd its IIP obtined by two metods (Cse 4) Figue 8. Tjectoies of ocket nd its IIP obtined by two metods (Cse 5)

18 18 Te tjectoies using te two metods lmost efectly mtc. Howeve te cceletion ofiles ve meningful diffeences ticully in Cses nd 5. Figs come te istoies of cceletion comonents ( ) obtined by te IIP guidnce nd te oen-loo otimiztion fo Cses 4 nd 5. Te diffeences in cceletion comonents fo Cse 4 esented in Fig. 10 wic moves te IIP in te downnge diection wee eltively low. On te conty te ofiles of cceletion comonents fo te cses wee te diection of IIP cnge is significntly diffeent fom te downnge diection (Cse : te oosite diection Cse 5: 90 degee diffeence). It ws obseved tt te diffeences in comonents ssocited wit in-lne mneuve ( ) e eltively lge comed wit te diffeence in wic govens te lne cnge mneuve. Figue 9: Histoies of cceletion comonents obtined by two metods (Cse )

19 19 Figue 10. Histoies of cceletion comonents obtined by two metods (Cse 4) Figue 11: Histoies of cceletion comonents obtined by two metods (Cse 5)

20 0 Wile te esults of te cse study esented in tis section demonstte te ne time-otiml efomnce of te oosed feedbck IIP guidnce lw (oximtely 5 % diffeence in finl time fo IIP cnge of 00 km) some dditionl nlyses on te lgoitm e equied to justify its cticbility. Additionl test cses wit vious initil conditions nd IIP cnge tsks sould be conducted to undestnd te efomnce ccteistics nd obustness of te oosed lgoitm. In-det nlysis on te comuttionl lod of te ocedue is otentil subject fo futue wok. Studies on te detemintion of time to stt te oosed guidnce lw nd te imlementtion of te lgoitm witout cutoff cbility would be inteesting subjects fo otentil futue esec to imove its licbility. V. Conclusion A ne otiml feedbck guidnce lw to cnge te IIP of ocket to desied osition ws oosed in tis e. Te oosed lw is develoed by incooting te nlytic exessions fo te IIP nd its time deivtives nd solving constined otimiztion oblem wic ovides te cceletion commnd tt ligns te IIP deivtive vecto wit te desied diection nd mximizes its mgnitude. Cse studies on simultion of te IIP guidnce lw fo ocket (seted stge of lunc veicle) wit vious tget oints nd tei comisons wit numeiclly obtined oen-loo tjectoy otimiztion esults wee conducted. Te esults of te cse study demonstted te ne time-otiml efomnce of te oosed guidnce lw. Futemoe it cn be otentilly used s n on-bod lgoitm fo moving te IIP of ocket fte dditionl veifiction/vlidtion nd studies fo efomnce imovement. Acknowledgements Tis wok ws eed t te Koe Advnced Institute of Science nd Tecnology Detment of Aeosce ngineeing unde esec gnt fom te Ntionl Resec Foundtion of Koe (NRF- 013M1A3A3A004461). Autos tnk te Ntionl Resec Foundtion of Koe fo te suot of tis wok.

21 1 Refeences [1] Licensing nd Sfety Requiements fo Oetion of Lunc Site FAA Code of Fedel Regultions 14 Wsington D.C. Oct. 000 [] J. An W. Ro J. Pk G. Co Instntneous imct oint clcultion lgoitm of sounding ockets nd its liction to nge sfety system Jounl of te Koen Society fo Aeonuticl & Sce Sciences 8(6) (000) [3] O. Montenbuck M. Mkgf W. Jung B. Bull W. ngle GPS bsed ediction of te instntneous imct oint fo sounding ockets Aeosce Science nd Tecnology 6 (00) [4] O. Montenbuck M. Mkgf Globl ositioning system senso wit instntneous-imctoint ediction fo sounding ockets Jounl of Scecft nd Rockets 41 (004) [5] J. An W. Ro Nonitetive instntneous imct oint ediction lgoitm fo lunc oetions Jounl of Guidnce Contol nd Dynmics 35() (01) [6] J. An J. Seo Instntneous imct oint ediction using te esonse sufce metod Jounl of Guidnce Contol nd Dynmics 35() (01) [7] J. An W. Ro Anlytic time deivtives of instntneous imct oint Jounl of Guidnce Contol nd Dynmics 37() (014) [8] Y. Nm J. An T. Seong Adjusted instntneous imct oint nd new fligt sfety decision ule Jounl of Scecft nd Rockets 53(4) (016) [9] N. Yoon J. An Tjectoy otimiztion of lunc veicle wit exlicit instntneous imct oint constints fo vious nge sfety equiements Jounl of Aeosce ngineeing 9(3) (016) [10] S. Mndic Guidnce of gound to gound ockets using fligt t steeing metod Scientific Tecnicl Review 59(3-4) (009) [11] Te wy nd ow of lnding ockets SceX News June 015 URL: tt:// (Accessed on Nov ).

22 [1] A. D. Weelon. Fee fligt of bllistic missile ARS Jounl 9(1) (1959) [13] P. Zcn Tcticl nd Sttegic Missile Guidnce sixt edition Pogess in Aeonutics nd Astonutics AIAA Wsington D. C [14] A. V. Ro D. Benson C. Dby M. A. Ptteson C. Fncolin I. Sndes G. T. Huntington Algoitm 90: GPOPS MATLAB softwe fo solving multile-se otiml contol oblems using te Guss seudosectl metod ACM Tnsctions on Mtemticl Softwe 37() (010) 9-36.