Fuel- or Time-Optimal Transfers Between Coplanar, Coaxial Ellipses. Using Lambert s Theorem. Chang-Hee Won

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1 Fuel- o Te-Optl Tsfes Betwee Copl, Col Ellpses Usg Lbet s Theoe Chg-Hee Wo Electocs Telecouctos Resech Isttute, Tejo , Republc of Koe Abstct Uoubtely, u-fuel -te obt tsfe e the two jo gols of the op tl obt eue. Ths ppe coses two copl ellptc obts whe the psl l es coce. We lytclly f the cotos fo the two-pulse u-te t sfe obt usg Lbet s theoe. I the u-te tsfe, the tsfe te s ecesg fucto of ble elte to the tsfe obt s sejo s. Coseque tly thee ests o uque u-te soluto. Thus fo the u-te cse, th ee s ltg soluto oly, howee, thee ests uque soluto the cse of u-fuel tsfe, fo whch we f the ecessy suffcet cotos. Futheo e, s specl cse, we cose whe the tsfe gle s oe hue eghty ege es. I ths cse we show tht we obt the fuel-optl Hoh tsfe obt. We ls o ee the Hoh tsfe te elt-elocty equtos fo oe geel equ tos, whch e lso obte usg Lbet s theoe. Thee s teoff betwee u-te -fuel tsfe. Flly, we popose optl copl obt eue lg oth fo tg off the u-te gol gst the u-fuel gol. Itoucto Seo Resech Egee, TT&C Syste Te, Stellte Co. Syste Dept., ETRI, Yusog P.O. Bo 06, Tejo, Koe, , AIAA Mebe

2 Tjectoy optzto wth espect to fuel ws pefoe s ely s 96 by Lw e usg the pe ecto, but the optl-te tsfe ws ot esse. The te e sus fuel teoff fo the eezous poble ws esse by Pussg Chu, who obte the u-fuel, ultple-pulse, te fe solutos fo copl ccul eezous poble. They lso showe tht the Hoh tsfe s the te-ope solut o fo the optl eezous. Pussg Chu eelope tete zto p oceue to etee the optl ube of pulses the postos te usg Lwe s pe ecto. It s well-kow tht the Hoh tsfe s the u-fu el two-pulse tsfe betwee two copl ccul obts. Fo tjectoy optzto poble, see Byso Ho the efeeces cte thee. Betts 5 ge ecellet geel suey of uecl ethos fo tjectoy optzto. A typcl pplcto of Lbet s theoe s to etee the tsfe obt fo the coectg two posto ectos the tsfe te. See 6,7 the efeeces cte wt h. It s possble to pefo, howee, the optl-fuel o -te tsfe usg Lbet s theoe. I ths ppe we foulte the tsfe te elt-elocty equtos usg Lbet s theoe. The we ee the u-te tsfe coto fo the two- pulse ellptc-to-ellptc tsfe. The ecessy suffcet cotos e lso ee fo u-fuel tsfe whe the tsfe gle s ywhee betwee zeo thee h ue sty egees. Futheoe, we show tht the Hoh tsfe s the u-f uel two-pulse soluto whe the tsfe gle s oe hue eghty egees. Ths optl tsfe s beg eelope wth stellte obt eues, especlly st ellte lttue eues,. Moe th two pulse eues e ot beg co see becuse whe the tl obt us s.9 tes slle th the tget obt us, the bsolute u-fuel ccle-to-ccle copl tsfe s the Hoh tsf

3 e the obts we e ly teeste e ll slle th tht 8,9. Thus, we cose just the two-pulse solutos. O the othe h, the optl tsfe te s othe jo fcto ou eelopet becuse t s pott to kow the te spcecft tk es to ech ptcul pot the obt. I ths cse, we f the u-te soluto ssug tht two-pulses e use. Ths s ecessy the cse of spcecft tec epto eezous. We cose the theoes lgoths ole the fuel es us te teoff. I ths ppe, we f the two-pulse u-te tsfe obt lytclly. Mo eoe, we ee the ecessy suffcet cotos fo the u-fuel obt ts fe poble usg Lbet s theoe. We lso stte lgoth to f the teoff bet wee u-fuel -te obt tsfe. Poble Foultos We ssue two ellptc copl obts whe the psl les coce. Fg. show s the tl obt us, tget obt us, the tl obt sejo s, the t get obt sejo s, the ffeece the tue oly (tsfe gle f, the co legth, c. The se-peete of the spce tgle s ge by Followg Btt s foulto 0, efe costt by c s. ( s c. ( s Note tht [, ] t s closely elte to the tsfe gle, f. Thus, eltes to the co legth. Fo eple, 0 ples f π ± ples f 0.

4 Also, the ble s efe by, ( whee s the se-jo s of the u-eegy ellptc tsfe obt ge s s, ( s the se-jo s of the tsfe obt. Tsfe Te Equto Usg Lbet s theoe, we he the tsfe te equto fo ellptc (hypebolc tsfe obts s 0 α sα β s β t (5 α β s s whee k /s s the gttol costt of the Eth. Fo hypebolc o bts, we eplce s wth sh ultply Eq. (5 by. Followg Btt 0 we efe α ccos(, (hypebolc; coshα / (6 β ccos( y, (hypebolc; y cosh β / (7 y (. (8 Thus we he epesse the tsfe te equto tes of the ew ble. Eq. (5 c be utlze to f the u-te tsfe obt. I the et secto, we ffee tte t wth espect to to f the u tsfe te obt. Befoe we cose ths ete, howee, t s helpful to epess α β te

5 s of geoetc petes, s, c,. Whe Eqs. ( s substtute to Eq. (, we obt s ±. (9 Equtg the boe equto wth Eq. (6 usg sple tgooetc etty we o bt s α s. (0 Slly, substtute Eq. ( (9 to Eq. (8 to obt y ( s c /(. Also f o the efto (7 sple tgooetc etty we obt s β s c. ( Flly, we he α β tes of s, c, Eqs. (0 ( Delt Velocty Equto The elto betwee the tsfe obt s sejo s the fuel usge c be fou fo the totl elt-elocty ecto, utlzg Lbet s theoe 0. I ths secto, the totl elt-elocty ecto shll be wtte s fucto of the ble, to pefo fu el optzto the sequel. Let be the tsfe obt elocty t P be the tl obt elocty t P (Fg.. The the elt-elocty s efe s. ( f ( s, (

6 whee s ut ecto efg the ecto of fo the foce cete s u t ecto ol to the obtl ple. Defe P by ( s ψ ( fo ellptc obt. Moeoe, ψ s ge s β α ψ (5 whee α β e ge (0 (. Also, becuse we ssue two ellptc copl obts, the tl obt elocty s ge by. (6 Slly, let be the tsfe obt elocty t be the tget obt elocty t : P f P f, (7 s ( f, (8 f. (9 The the totl s ge s the su of Eqs. ( (7 s totl totl (0 whee s ( ( f (

7 ( ( f s ( To ze the fuel usge of spcecft, totl ust be ze. Cosequetly, we ze Eq. (0 wth espect to f the u-fuel tsfe. Note tht o-copl obts y be cosee ths e to f the optl-fu el -te ellptc tsfe obt. I the o-copl cse, Eq. (5 woul e the s e, howee, Eqs. ( (8 woul chge to, Eq. (6 woul chge to, Eq. (9 woul chge to. Hee s ut ecto ol to the t sfe obtl ple, s ut ecto ol to the tl obtl ple, s ut ecto ol to the tget obtl ple. t t Mu-Te Tsfe Cotos Deto of the Tsfe Te Equto The cotos to chee two-pulse optl-te tsfe e ee. We show th t the tsfe te s ecesg fucto of. Thus u-te tsfe oes ot he efte soluto, but t hs ltg soluto the sese tht t s possble to f the ltg such tht the u tsfe te ppoches ths lt. The esults e stte ths secto the poofs e ge the ppe. Tkg ete of Eq. (5 wth espect to, we obt 0 t F[,;5 /, ( / ] F[,;5 /, ( y / ] ( whee F[,;5/ ; z] s the hypegeoetc fucto whose cotue fcto epso s

8 ( ( 5 o ( ( F[,;5/ ; z] γ γ z. ( ( ee γ z ( ( Futheoe F[,;5/ ; z] c be ewtte s F[,;5/ ; z] γ zg( whee z the ete wth espect to z s ge s 0 G( z γ z γ z, (5 F[,;5/ ; z] 9G( z / 5. (6 z ( z( 6zG( z / 5 To elute the fst te o the ght h se of (, we let z ( /. The z (/ F [,;5/ ; z] F[,;5/ ;( / ]. (7 z Slly, fo the seco te o the ght h se of (, we let z ( y / to obt z ( F [,;5/ ; z] y F[,;5/ ;( y / ]. (8 z Thus, we c elute Eq. ( usg Eqs. (7 (8. I the ppe, usg Eqs. (, (7, (8, we poe tht the tsfe te equt o s ootoclly ecesg fucto wth espect to the ble,,.e., t / < 0. Cosequetly the lgest lue of ges the fstest tsfe te. Fo e ple, to f the fstest ellptc tsfe obt, we let ppoch uty wthout ctully

9 echg t ( coespos to pbolc obt. Futheoe, ths ples tht the p bolc tsfe obts e fste th ellptc tsfe obts, hypebolc tsfe obts e fste th pbolc tsfe obts. Itutely ths sttes tht f fuel s ulte, the oe elt-elocty oe epes the shote the tsfe te. Necessy Suffcet Cotos fo Mu-Fuel Tsfe The ecessy suffcet cotos fo the u-fuel tsfe the cse of two-pulse ellptc tsfe obt e ee ths secto. The tl tget obt s e ssue to be copl col ellptc obts. We f the coto o the ble (equlet to the sejo s such tht the tsfe obt zes the fuel usge. The gtues of the elt eloctes, e ge by the followg E ( qs. ( (. By efto, s poste 0 t c be ewtte s y ( (9 Ge f, the ecessy suffcet cotos fo u-fuel tsfe obt e ge by totl 0 totl > 0, whee totl. Necessy Coto Thus, to f the ecessy suffcet cotos, we tke the fst ete of th e elt-eloctes ge Eqs. ( ( wth espect to. Also, we f the fst e te of Eq. (9 wth espect to. We obt

10 f f s s (, (0 f f s s (, ( y. ( The fst ete of the totl elt elocty, totl, ( c be tlly obte fo Eqs. (0 (. Thus the ecessy coto fo u eques 0 totl. Suffcet Coto Fo the suffcet coto, we f the seco ete of Eq. ( wth espect to. y y y ( The we tke the seco ete of the totl elt elocty wth espect to. Afte othe log pulto, we obt

11 totl s s f f s s f f (5 Ths seco ete ust be gete th zeo fo totl to be u. Thus we h e the suffcet coto fo locl u; 0 > totl. (6 Equtg ( to zeo ges the ecessy coto (6 ges the suffcet co to to chee u-fuel tsfe fo y two pots spce (.e., c be ywh f

12 ee betwee zeo 60 egees. Copl Hoh Tsfe Fo Lbet s Theoe Necessy Suffcet Cotos fo Mu To efy the esults of the peous secto, we ssue ellptc copl col obt s ee the ecessy suffcet cotos fo the Hoh tsfe cse. By Hoh tsfe, we e tht tgetl pulses e pple t opposg pses. Cos equetly, the tl tget pots ust be o the le of pses fo Hoh tsfe. We show tht fo the Hoh tsfe s equl to zeo. I the Hoh tsfe, w e he f π c. Thus fo Eqs. ( (, we get 0. Moeoe, bec use 0, fo Eq. ( we obt ( s poste by efto. Substtutg the se lues to Eq. ( ges the followg esult: totl. (7 Thus / totl s equl to zeo whe s equl to zeo. Cosequetly, fo the cse of Hoh tsfe ( f π, we obt the optl fuel tsfe whe s equl to ze o. Ths shows tht coto (7 s specl cse of the geel coto (. To f the suffcet coto, we elute Eq. (5 fo the Hoh tsfe. The we obt, totl > 0. (8 Thus, ths ges the suffcet coto fo u. Moeoe, ths shows tht fo th e ellptc-to-ellptc, copl, col obts, f tsfe whe 0. f π the we he u-fuel

13 Note tht 0 coespos to the u eegy tsfe 0, fo Hoh t sfe the u eegy tsfe coespos to the u-fuel tsfe, but s wll b e see the sulto secto, whe f π ths s ot the cse. Deto of Tsfe Te Delt Velocty Equtos We ow show tht the well-kow Hoh Tsfe esults e obte fo the L gge fo of the tsfe te the elocty equtos. Fo the Hoh tsfe c s. Thus fo Eqs. (0 (, we f tht α π β 0. Thus usg the fct tht 0 Eq. (5, we obt ( π t. 8 I suy, fo the Lgge fo of the tsfe te Eq. (5, we obt the Hoh tsfe te equto, ( π t (9 8 whe f s equl to 80 egees. We ote tht ths tsfe te equto s l fo ell ptc-to-ellptc col obts s well s the ccle-to-ccle obts. Now, we ee totl fo the Hoh tsfe usg Eq. (0 to efy tht we eco e the Hoh tsfe totl. Oce g, fo the Hoh tsfe, we he c s, α π, β 0, 0,. The lst equto ples ψ α β 0. Becuse ( s ψ fo ellptc obt, we obt

14 π ψ. Substtutg these lues to Eq. (, we obt s π. (0 Usg Eq. ( we get. ( By sl lyss, we obt. ( Usg the eltos ( e ( e, we obt / ( / ( e. ( / ( e. ( The gtue of the su of the boe two equtos ges the ese esult. As fo the specl cse of ccle-to-ccle tsfe, we he, we get. (5 By sl lyss, we lso obt. (6 Fo the Eq. (0, we ecoe the well-kow Hoh totl elt-elocty, totl

15 totl, whe f totl (7 s equl to 80 egees. Note tht the Hoh tsfe s specl cse fo f equl to 80 egees. Thus fo f 80 egees, the Hoh tsfe cot be use, howee, we c stll obt th e u-fuel o u-te tsfe obt usg the etho escbe the pe ous secto. Optl Copl Obt Meue Algoth Hee, etho to pefo the optl-fuel -te eue usg Lbet s theo e s pesete. The poble s to f the tsfe obt s sejo s such tht the tsfe te the fuel use s u. Fo the esults of the peous secto, we ote tht the Hoh tsfe s the optl soluto whe f π. Howee, f we ws h to put the spcecft t ptcul pot the tget obt whee f π, the the Ho h tsfe cot be pefoe the u-fuel tsfe s obte fo Eq. (. It s pott to ote tht Eq. ( s l ee f the tl tget pots e ot o the le of pses. Ths type of tsfe ght be ecessy the cse of tecept o eezous. Now, we use the esults of the peous sectos to obt the optl-fuel o -te t sfe obt. The optl lgoth s ge s follows:. Ge tl obt us,, tget obt us,, the ffeece the tue o ly (tsfe gle, f, we y to f the tsfe te, t, totl elt elo

16 cty, totl.. F the co legth, c, fo the equto, c cos f. (8. F the sepeete of the spce tgle, s, fo Eq. (.. F fo Eq. (. 5. F costt fo Eq. (. Note tht the co legth, c, c lso be epesse tes of ths ble s Thus s elte to the co legth. 6. F α fo Eq. (6. 7. F β fo Eq. (7. c (. (9 8. F the tsfe te fo the Lgge fo of the tsfe te Eq. (5. 9. F the totl elt elocty fo Eqs. (0, (, (. Eq. (5 c be use to f the tsfe obt s sejo s such tht the tsfe t e s ze. We c plot t (whch s elte to the tsfe obt se-jo s esus, f such tht t s u. Actully t s kow fo the peous s ecto tht t s ecesg fucto of. Thus fo eple, fo ellptc tsfe o bt the sllest t s chee by lettg ppoch uty wthout ctully echg t ( coespos to pbolc obt. Thus, just to pefo the ellptc u-t e obt tsfe we let be ey close to uty (fo ellptc tsfe obt <. Thee s othe fcto to tke to coseto whe we pefo optl eu e. We woul lke to ze fuel usge whle tsfeg to the tget obt. We c

17 plot (whch s elte to the tsfe obt se-jo s esus usg Eq. ( totl 0, f such tht s u to f the optl-fuel tsfe obt. Thus t totl hee s teoff betwee the te the spcecft tkes to get to the tget obt the f uel ecessy to get to the tget obt. Nuecl Sulto Results The uecl sulto s pefoe ths secto usg the optl eue lg oth eelope the peous secto. Hee we let 6700 k, 670 k, 6800 k, 6900 k. The fst eple s whe f π (Hoh tsfe the seco eple s whe f π / 0. Cse ( f π : Hoh Tsfe We plot totl esus fo Hoh tsfe Fg.. Note tht u totl, whch hs the lue 0.69 k/s, coespos to 0, whch epesets the Hoh t sfe. Usg Eqs. ( (5, we he efe tht totl / 0 totl / > 0 whe 0. Thus 0 coespos to the u-fuel soluto. Fg. shows tsfe te esus. Note tht t s ootoclly ecesg fuct o. Whe s equl to zeo, the tsfe te s 7.0 secos. It s possble to obt tsfe obt tht wll ge slle tsfe te, but the fuel use wll cese. Thus thee s teoff betwee the fuel use u-te. Although t s ot show o t he plot, the pbolc tsfe obt hs slle tsfe te th ellptc tsfe obt, the hypebolc tsfe obt hs slle tsfe te th pbolc tsfe obt. To see the eltoshp betwee the tsfe obt s sejo s, see Fg..

18 Note tht ee though thee s oly oe fo the ge, thee e two s fo th e ge. The sejo s of the tsfe obt coespog to equl to zeo s 67.8 k. Cse ( f π / 0 I the cse of eezous, we y eque tht f π. Thus the questo s wht s t he optl-fuel o -te tsfe obt whe f π. Hee we ssue tht the two pots e septe by f π / 0. Fg. 5 shows totl esus. Note tht the u totl s chee whe s ppotely Note lso tht the u eegy o bt ( 0 oes ot coespo to the u-fuel obt. Usg Eqs. ( (5, we he efe tht totl / 0 totl / > 0 whe Thus 0.69 coespos to the u-fuel soluto. See the totl / esus gp h Fg. 6. We ote tht totl / s egte whe < t becoes poste whe > 0.69, whch ples tht s u. Fg. 7 shows the tsfe te esus. Note tht t s ootoclly ecesg fu cto s befoe. Thus fo ths poble the optl fuel te tsfe obt y be w he 0.65, whch coespos to k. Fo 0. 65, totl 0.70 k/s t 7.0 sec. To cope these lues wth the Hoh tsfe cse, see Tble. Note tht totl s slle fo the Hoh tsfe (Cse s epecte but the tsfe te s uch lge. Tble Copso of Cse Cse

19 totl (k/s t (seco (k Cse f π Cse f π / Dffeece Coclusos The tsfe te elt-elocty equtos e stte usg Lbet s theoe. Us g these equtos, u-te u-fuel cotos tes of the se jo s of the ellptc tsfe obt e ee. Moeoe, t s show tht the u -te tsfe obt ppoches ltg lue. It s lso show tht whe the tsfe gle s 80 egees the optl fuel tsfe s ge by the well-kow Hoh ts fe. A lgoth wth the fuel te teoff s pesete. Ths poceue c be us e to f the optl-te -fuel tsfe obt the cse of eezous. Appe: Poof tht Tsfe Te s Decesg Fucto of I ths ppe, we poe tht the tsfe te Eq. ( s ootoclly eces g fucto wth espect to. We show tht the fst te o the ght h se of Eq. ( s egte. The we show tht the seco te s gtue s slle th the fst te, thus showg tht t / < 0. Now, we e ey to show tht F [,;5/,( / ] s less th zeo. Fst we f F[,;5/ ; z] usg the lgoth ge by Btt 0, whch s epete hee fo the ske o f copleteess:

20 Itlze δ u Σ clculte δ, (A, u u ( δ, Σ Σ u γ zδ whee γ s ge (. The fo z <, F[,;5 / ; z] l Σ. Seco we f G(z f o Eq. (5. Th we f F / z fo Eq. (6. Flly we f F [,;5/ ;( / ] usg Eq. (7. The esult s show Fg. 8. Although t s ot cl ely show Fg. 8, ll the lues e less th zeo fo betwee, see Tbl e. Thus ths shows tht fo < <, F[,;5/ ;( / ] < 0. (A Tble Hypegeoetc fucto lues F [,;5/ ;( / ] Uefe We cope the gtue of the fst seco tes o the ght h se of Eq. (. Fo the ellptc tsfe obt, (, [, ]. Usg ths foto, we

21 5 plot / y esus Fg. 9 fo seel lues of gg betwee. We ote tht 5 y. (A 5 Substtute Eq. (8 to / y, obt 5 y 5 (. (A 5 5 The we ote tht / y whe / y whe fo ll. Us g Iequlty (A we obt 5 F z y The we use Eqs. (7 (8 the boe equlty to obt F z. (A5 F [,;5/ ;( / ] F[,;5/ ;( y / ]. (A6 Becuse of Iequltes (A (A6 we he the followg, F[,;5/ ;( / ] F[,;5/ ;( y / ] 0. (A7 Flly, becuse of ( (A7 we he the ese esult, t 0. (A8 Thus, t s ecesg fucto wth espect to. Ackowleget The utho woul lke to thk Sguk Lee fo toucg Lbet s theoe to the utho. The utho woul lso lke to thk the eewes fo the costucte coet

22 s tht le to poe eso of the ppe. Refeeces Lwe, D.F., Optl Tjectoes fo Spce Ngto, Buttewoths & Co., Lo o, 96, pp Pussg, Joh E., Chu, Jeg-Hu, Optl Multple-Ipulse Te-Fe Re ezous Betwee Ccul Obts, Joul of Guce, Cotol, Dycs, Vol. 9, No., Juy-Febuy 986, pp. 7-. Pussg, Joh E., Sple Poof of the Globl Optlty of the Hoh Tsfe, Joul of Guce, Cotol, Dycs, Vol. 5, No., July-August 99, pp Byso, Athu E., Ho, Yu-Ch, Apple Optl Cotol: Optzto, Estt o, Cotol, Hesphee Publshg Co., New Yok, 975, pp Betts, Joh T., Suey of Nuecl Methos fo Tjectoy Optzto, Joul of Guce, Cotol, Dycs, Vol., No., Mch-Apl Btt, Rch H., Vugh, Rob M., A Elegt Lbet Algoth, Jou l of Guce, Cotol, Dycs, Vol. 7, No. 6, Noebe-Decebe 98, pp Ploe, Jul, A Eleety Poof of the Optlty of Hoh Tsfes, Jo ul of Guce, Cotol, Dycs, Vol. 7, No. 5, Septebe-Octobe Pussg, Joh E., Cowy, Buce A., Obtl Mechcs, Ofo Uesty Pe ss, New Yok, 99, pp Wetz, Jes R., Lso, Wley J. (etos, Spce Msso Alyss Desg, Spce Techology Lby, Kluwe Acec Publshes, Doecht, Nethels, 99,

23 pp. 9-0 Btt, Rch H., A Itoucto to the Mthetcs Methos of Astoy cs, ete by J. S. Pzeeeck, AIAA Eucto Sees, New Yok, 987, pp

24 Lst of Fgue Cptos Fg. Copl obt tsfe g Fg. esus ( f π totl Fg. Tsfe te esus ( f π Fg. Tsfe obt se-jo s esus Fg. 5 Fg. 6 esus ( f π / 0 totl totl / esus ( f π / 0 Fg. 7 Tsfe te esus ( f π / 0 Fg. 8 F [,;5/ ;( / ] esus Fg. 9 5 / y esus fo ous lues of