GTOC9: Results from the National University of Defense Technology

Transcription

1 GTOC9: Rsults om th Natoal Uvsty o Ds Tchology Yazhog Luo *, Yuh Zhu, Ha Zhu, Zh Yag, Shua Mou, J Zhag, Zhjag Su, ad Ju Lag Collg o Aospac Scc ad Egg, Natoal Uvsty o Ds Tchology, Chagsha 4173, Cha Abstact: Th th dto o th Global Tajctoy Optmzato Comptto (GTOC) ss was succssully ogazd Apl 17, wh th compttos w calld to dsg a ss o mssos abl to mov a st o 13 obtg dbs pcs whl mmzg th ovall cumulatv cost. A th-lvl optmzato amwok s pstd to solv th complx poblm, whch combs th dyamc Tavllg Salsma Poblm (TSP), mxd-tg squc optmzato ad ptubd tajctoy dzvous optmzato. Th amwok was mployd by th NUDT tam dug GTOC9, ad th cospodg sult akd scod th vtual ladboad. I. Itoducto Th dsg o spac tajctos ca b potably appoachd as a global optmzato poblm. Th optmal tajctoy, whch s sgcat o pactcal spac msso dsg, s usually vy dcult to b obtad. Th Global Tajctoy Optmzato Comptto (GTOC) ss [1], was bo wth th objctv o ostg sach ths aa by lttg th bst aospac gs ad mathmatcas woldwd challg thmslvs to solv o, dcult, wll-dd, poblm o tplatay tajctoy dsg. Th GTOC9 [] was succssully ogazd Apl 17 by th Advacd Cocpts Tam (ACT) o Euopa Spac Agcy (ESA), who wo th GTOC8. Fo callg to potct th vomt o ath obts, th backgoud o th GTOC poblm s put th a-ath spac o th st tm. Th compttos a calld to dsg a ss o mssos abl to mov a st o 13 obtg dbs pcs whl mmzg th ovall cumulatv cost. To d th optmal soluto o such a complx poblm, th sub-poblms d to b xtactd ad solvd. 1) Fst, th st o 13 dbs pcs ds to b dvdd to sval goups. Each goup o dbs s movd by o msso. Optmzato s pomd to mmz th ovall cumulatv cost. Ths ca b appoachd as a dyamc TSP ad solvd by som voluto algothm [3]. ) Scod, gv a goup o th dbs, o msso s dsgd by mxd-tg optmzato to mov thm whl costg mmal vlocty cmt [4]. 3) Fally, gv th cut ad xt dbs as wll as th dzvous duato, th mpulsv mauv statgy s dsgd to poduc th optmal lght tajctoy [5]. Ths pap psts a th-lvl optmzato amwok to solv th abov poblm. Th amwok was mployd by th tam o Natoal Uvsty o Ds Tchology(NUDT) GTOC9. Th sult akd scod th vtual ladboad, oly bhd th amazg sult om Jt Populso Laboatoy (JPL). II. Poblm Statmt A. Poblm Dscpto Sc th lauch o th st satllt, Sputk, 1957, makd has placd coutlss spaccat obt aoud th Eath. Today, lss tha 1% o th tackabl objcts obtg th Eath a opatoal satllts. Th mad s smply juk, ad ths spac dbs s bcomg a casgly sous poblm. Followg th upcdtd xploso o a Su-sychoous satllt, th Kssl ct tggd uth mpacts ad th Su-sychoous obts vomt was svly compomsd []. Sctsts om all ma spac agcs ad pvat spac compas solatd a st o 13 obtg dbs pcs that, movd, would sto th possblty to opat that pcous obtal vomt ad pvt th Kssl ct to pmatly compoms t. Tho, th poblm o GTOC9 s to dsg mssos abl to cumulatvly mov all th 13 obtg dbs whl mmzg th ovall cumulatv cost o such a davo. O msso s a multpl-dzvous spaccat tajctoy wh a subst o sz N o th 13 obtg dbs s movd by th dlvy ad actvato o N d-obt packags. Th ollowg cost ucto has to b mmzd: J C c m m dy 1 1 tsubmsso tstat c cm ( cm cm ) t t d stat (1) * Cospodg autho: E-mal: luoyz@udt.du.c. 1 / 9

2 wh C s th cost chagd by th cotactd lauch suppl o th th msso. At th bgg o th th msso, s th spaccat mass ad m dy m ts dy mass. Each spaccat tal mass s th sum o ts dy mass, th wghts o th d-obt packags to b usd ad th popllat mass: m m Nm m. All spaccat hav a dy mass o dy d p mdy [kg] ad a maxmum tal popllat mass o m N 1 m 5 p [kg] (lss popllat may b usd, whch cas th lauch costs wll dcas). Each d-obt packag has a wght o [kg]. m d 3 6 Th paamt s st to b. 1 [ MEUR / Kg ]. s th poch at whch th th t submsso msso s valdatd, ad ad a th d ad th bgg pochs o th GTOC9 comptto. Th mmal basc cost s 45 MEUR, whl th maxmum cost s 55 MEUR. Th oth dtald costats dd by th ogaz a gv []. Dug ach tas btw two succssv dbs, th spaccat dyamcs s dscbd, by a Kpla moto ptubd by ma cts o a oblat Eath,.. wh J. c M t d c m t stat x vx, y vy, z vz x 3JR E 5z vx x x y 3JR E 5z () vy 1 y 3 5 y z 3JR E 5z vz 3 z 3 5 z [ x, y, z] T ad v [ v, v, v ] T a th x y z spaccat s posto ad vlocty vcto dscbd th ma quato tal coodat systm o th ct body,, dots th Euclda om o a vcto,, ad J a th gavtatoal costat, ma quato adus ad J -ptubato coct o th ctal body spctvly. Γ s th thust acclato. Th oly mauvs allowd to cotol th spaccat tajctoy a stataous chags o th spaccat vlocty (ts magtud bg dotd by V. At ach such mauv, th spaccat mass s to b updatd usg Tsolkovsky quato: V m m xp, (3) v wh v Isp g. A maxmum o 5 mpulsv vlocty chags s allowd dug ach tas btw two succssv dbs. Ths do ot clud th dpatu ad aval mpuls. Accodg to Eq. (1), th ollowg th stps R E a qud to comput th cost ucto o GTOC9 poblm. Stp 1): Dtm th umb o mssos (.. ) ad th umb o dbs ach msso (.. ), so that N Stp ): Dtm th dzvous squc om dbs to dbs o msso. Stp 3): Dtm th tas tajctoy om dbs to dbs o msso,.., to comput th mauv tm ad mpulss so that th spaccat ca sucssully tas om th st dbs to th last o ud th gv costats. As a tmdat pocdu, Stp ) ad b hadld togth wth Stp 1) o wth Stp 3). I od to ctly obta a a-optmal soluto, w solv th GTOC9 poblm usg all o ths th stps, thy a toducd Scs. IV, V, ad VI, spctvly. B. Poblm Aalyss Th GTOC9 poblm s to dsg a ss o mssos wth mmal popllat costs, o th objctv o movg th gv 13 obtg dbs. I od to d th optmal dbs-moval pla, th ollowg th sub-poblms must b addssd: 1)How to pla th succssv moval mssos? )How to mmz th cost o a sgl msso? 3)How to optmz th tajctoy btw ach two dbs? Th st sub-poblm s a lag-scal mult-squc combatoal optmzato poblm, whch s smla to th combato o th classc tavllg salsma poblm (TSP) ad b packg poblm (BPP). Th TSP s to d a closd mma-dstac path, wth th costats that all ods should b vstd oc ad oly oc. Th BPP s to d th mmum umb o packags, so that all th tms ca b packd. Howv, compad wth TSP ad BPP, th ollowg dcs mak ths sub-poblm mo dcult. I th BPP, oly wght costats d to b satsd, ad th squc o ach tm dos ot act th wght o a packag. Howv, o th GTOC9 poblm, both th tms (dbs) thmslvs ad th packag squc wll sgcatly act th cost ucto, whch maks th soluto spac much lag tha BPP, ad thus lads to dcults o optmzato. Addtoally, compaso to th TSP, th GTOC9 poblm ca b dvdd to a ulmtd umb o mssos, ad th out o dbs moval ca b dscotuous btw vy two mssos. Oc aga t maks th soluto spac o ths poblm much lag tha th TSP, ad cass th dcults o optmzg as wll. I th TSP, all th ods to b vstd a xd th pla ad th cost o gog om o od to a- N / 9

3 t V V oth ca b asly calculatd accodg to th Catsa dstac th pla. Whl o th GTOC9 poblm, th cost o gog om o dbs to aoth dpds o th spctv statg dat ad aval dat. Ths uth maks th poblm tm-dpdt ad dcult. Th scod sub-poblm s a mxd-tg ola-pogammg (MINLP) poblm, whch both th squc (tg vaabls) o th dbs ad th tas tms (al vaabls) btw vy two dbs d to b gadd as dsg vaabls. Tho, t bcoms mo dcult to solv tha th mxd-tg la-pogammg (MILP) cass ad ola-pogammg (NLP) cass. Th thd sub-poblm s a obtal tas poblm. Wth th authos bst kowldg, t s almost mpossbl to accuatly comput th ad th dtald mauv pla ( t, v ), 1,,..., K sub-poblms 1) ad ) a psoal comput, du to th hug computato tm. Th bst choc s to appoxmatly stmat th cost sub-poblms 1) ad ) to ctly dtm th umb o mssos ad th dzvous squc ach msso, th to optmz th accuat mauv pla ( t, v ), 1,,..., K. Th a two challgs solvg th thd sub-poblm. Fst, t s dcult to stmat th V cost ad th lght tm wth a latvly hgh pcso o a sgl dzvous msso btw two dbs. Ths s bcaus that a bg th ght ascso o ascdg od (RAAN) dc may xst btw two dbs, ad th coomc way to coct ths bg out-o-pla dc s to mak us th atual obtal pcsso at du to th ctal body s J -ptubato [7]. As a sult, th -pla mauvs coupls wth th out-o-pla mauvs, ad a compomsg xsts btw th dzvous tm t ad th V cost. I o optmzato s t V pmttd o th cosdato o computato tm, a accuat stmato o ad s a challgg wok. W wll toduc ou stmato mthod Sc. IV.C. Scodly, t s dcult to d th optmal soluto o th log-duato ( s up to 3 days) J -ptubd dzvous poblm. Wh th J -ptubato s tak to accout, th wll-kow obtal tagtg algothms such as th Lambt algothm wll b ald obtag th asbl solutos, ad th costad optmzato mthods whch ca dctly copoat al stat costats, such as th squtal quadatc pogammg (SQP), wll also cout covgc poblms o log-duato dzvous. Fom th scop o obtal dyamcs, at last two mpulss a dd to tagt th al posto ad vlocty vctos. Howv, th total vlocty cmt o th -mpuls mauvs wll b vy lag o a dzvous msso, spcally o log-duato, lag o-coplaa dzvous poblms. Tho, a dzvous msso usually uss mo tha two mpulss. Du to th log-duato, mult-mpuls chaactstcs, th dsg vaabls (.g. th mauv tm) hav lag sach spac, ad may sub-optmal solutos may xst, thus t s dcult to d th global optmal soluto o ths poblm v though th stat-o-at optmzato algothm s usd. I addto, umcal tgato o th J -ptubd tajctoy s qud th optmzato pocss, whch maks th optmzato tm-cosumg. Th asbl optmzato appoach w usd wll b pstd Sc. VI. t Dvd all o th dbs to sval mssos Sval dbs chas wth a ogal squc Roptmz th dbs cha o ach msso O dbs cha wth a optmzd squc ad tas tm Optmz th accuat tas tm ad chaactstc vlocty Top Lvl Mddl Lvl Bottom Lvl Fgu 1 Itacto pocss amog th th lvls III. Optmzato Famwok Basd o th aalyss Sc. II, th whol optmzato amwok s maly dvdd to th lvls, whch s llustatd Fg. 1: Top Lvl: Dvd all th dbs to sval mssos, wth ach msso havg a goup o squtally umbd dbs. Mddl Lvl: Fo a sgl msso, optmz th dzvous squc o th dbs, ths dzvous squc ca b dt om th o obtad th up-lv. Bottom Lvl: Optmz th accuat tas tm ad chaactstc vlocty o ach dzvous om o dbs to aoth. Th spaatd optmzato modls ad algothms a dvlopd o ach lvl, whch a all codd C++ ad som cods a paallld usg 3 / 9

4 MPI. Fo th top-lvl poblm, w us th Dbs_Cha_Buchg At Coloy Optmzato (DCB_ACO) algothm, t s mpovd basd o th classc ACO usd o th succssv dbs-moval msso. A hybd-codg Gtc Algothm (HEGA) s usd th mddl lvl to smultaously dtm th squc ad th cd tas tm a sgl dzvous msso. Fo th bottom-lvl poblm, a dtal voluto (DE) algothm togth wth a J -ptubd obtal tagtg tchqu s usd to optmz th obtal tas tajctoy. IV. Multpl Mssos Optmzato Th ma task o ths lvl s to dvd th dbs to sval dbs chas. A ACO vaat o buchg dbs chas s mpovd to solv ths optmzato sub-poblm. A. Classc ACO Th ACO algothm was ogally spd by th ablty o bologcal ats to d th shotst path btw th st ad a ood souc [7]. Th udamtal wokg pocdu o th Classcal ACO, kow as At Systm (AS), s show Algothm 1. Algothm 1 At Systm Phomo tal talzato; whl Tmato cta ot mt do Soluto costucto; Phomo updat; d whl B. DCB_ACO Th udamtal pocdu o DCB_ACO s smla to th classc ACO. Th most mpotat atu o a ACO s th dsg o th hustc, whch s vtually combd wth th phomo omato to buld solutos. W maly pst th hustc ad soluto costucto mthod o DCB_ACO ths pat. Evy at stats wth a sam st o dbs (th dbs that hav ot b movd a stod a aay ). Fo ach at, buldg a asbl soluto should tak th ollowg stps. Stp 1: St T= T (MJD) as th stat tm. Stp : Italz a w mpty dbs cha. Radomly slct a dbs om V ad st t as th had o th cha. Dlt th cospodg dbs V. I V bcoms mpty, go to stp 8; othws, go to stp 3. Stp 3: Statg om T, Estmat th obtal tas tm Tl ad th chaactstc vlocty V l btw th last dbs V l th cut cha ad all mag dbs V. Go to stp 4. V Stp 4: Copy all o th caddats that qualy o bg buchd to th tal o th cut cha om ad dpost thm to a aay. I s mpty, go to stp 7; othws, go to stp 5. Stp 5: Radomly slct a dbs om, dlt th cospodg dbs at bg buchd to th tal o th cut cha. Go to stp 6. Stp 6: Cla up U, st ad tu to stp 3. Stp 7: Clos th cut cha. I V bcoms mpty, go to stp 8; othws, st ( TM ad[35 day,65 day] ) ad tu to stp. Stp 8: Fsh buchg, collct all o th accomplshd dbs chas. I stp 3, th optmal tas tm ad chaactstc vlocty btw ach two dbs a stmatd basd o th mthod pstd Sc. IV.C. I stp 4, th caddat s to all o th dbs that satsy th total ul costats o o msso at bg buchd to th tal o th cha. I stp 5, th pobablty that a at k wll choos a dbs j as th xt dbs o th cut cha b th patal soluto s s gv by bj j k, j U ( s, b) k p ( )= bg g bj s (4) k gu ( s, b) othws V V V T = T + T ld U V d U U T = T + T M k wh U ( s, b ), comg om stp 4, s th st o dbs that qualy o buchg bhd th cut cha b ad = V s th hustc valu. Th paamt j lj Eq. (4) s xd to 1 h bcaus usg th paamt s suct to lct th wght btw th phomo omato ad hustc omato. s th phomo om dbs l to dbs j, wh bj dbs l s th last dbs th cut cha b. I stp 6, s th stmatd optmal tas tm btw th last dbs th cut cha ad th slctd dbs. I DCB_ACO, th vapoato paamt ρ s st as.5ad th cas o th phomo s T ld V d V l k j lmtd to th maxmum valu o.1* j to avod pmatu covgc. C. Aalytcal Estmato o Tas Tm ad ΔV Cost To ctly valuat th objctv ucto o ach pack ad th ovall cumulatv cost, a aalytcal stmato modl s pstd basd o th Gauss s om o vaatoal quatos [8]. 4 / 9

5 a s v 1 cos vt 1 1 s v cos cos E vt a cosu v h a 1 su v h a 1 s 1 cos v 1 s vt cos a p 1 M cos v 1 s vt a p p (5) wh th ma moto, th smlatus c- tum p a1. 3 a 1) Adjustmt o th RAAN dc Th dc o th RAAN dt vlocty btw th spaccat ad th dbs should b ully usd. I th RAAN dc caot b coctd atually dug th maxmum dzvous duato, a mpuls ppdcula to th obtal pla ca b mplmtd at th oth o south vtx o th obt. a 1 s vh (6) ) Adjustmt o th clato dc As th J -ptubato dos ot chag th obtal clato, th clato dc must b coctd by mauvs. A mpuls ppdcula to th obtal pla ca b mplmtd at th ascdg od o dscdg od. 5 / 9 h v s (7) wh h, s th agumt o lattud. 3) Adjustmt o th smmajo axs ad cctcty At th spaccat tass to th sam obtal pla wth th dbs, th smmajo axs ad cctcty a adjustd by two tagtal mpulss, spctvly at th pg ad apog. Fo a a ccula obt, omttg th hgh od tms o, th vlocty cmt o th two mpulss s omulatd as ollows. a a v (8) 4) Estmato o dzvous duato Th dzvous duato should maly com om th adjustmt o th RAAN dc so as to mak ull us o th atual RAAN dt du to J -ptubato. Th ollowg adjustmts do ot d too much tm. To b cosvatv, th dzvous duato s oughly stmatd as th duato o RAAN adjustmt plus o day. What s mo, t may b otcd that th adjustmt o phas dc s ot cludd. Cosdg that th dzvous duato usually lasts o sval days, th vlocty cmt costd by phas dc adjustmt ps to b much lss tha th vlocty cmts o th oth adjustmts. To b cosvato, a upp boud o phas dc adjustmt ca b dctly addd o th stmato sult. V. Sgl Msso Optmzato A. Optmzato Modl Th a two typs o dsg vaabls. A soluto Y s mad up o a goup o sal tgs ad a st o al umbs that cossts o dzvous obtal tas tms ad svc tms. (9) wh Y 1 ( p1, p,... p N ) ad Y ( du1, du,... du N ; s1, s,... s N ). Though th squc o ts lmts th sal tg vcto psts a svc od. Th sach spac o s tho dsct ad ts lmts must b mapulatd combato. Th objctv s to mmz th popllat cosumd by obtal mauvs: m ( m m Qm ) (1) wh m dy Y 1 Y 1 Y Y ( Y, Y ) 1 s th spaccat s mass at th last movg msso ad also dots th spaccat s dy mass. B. Solvg Statgy Dot th stat o a spaccat as E ( a,, q1, q,, ) T (11) wh a s th sm-majo axs, s th obtal clato, s th RAAN, s th ma agumt o lattud, s th cctcty, s th agumt o pg, ad q1 cos ad q s a th modd obtal lmts sutabl o dscbg a-ccula obts. Th stat vaabl usd to xpss obtal dcs btw th spaccat ad o dbs s X ( a / a,, 1,,, ) T q q (1) wh th subscpt dots th c obt, a s th dc sm-majo axs, s th dc agumt o lattud, s th dc obtal clato, s th dc RAAN, ad q1 ad q gv th dcs cctcty vcto. Usg st od appoxmatos th qth dzvous opato, th stat tastos o obtal lmt dcs ud th J ptubato a gv by [9] dy d Y 1

6 q q v qj qj qj j1 X( t ) Φ( t ) X Φ ( t, u ) v (13) tq C (3 4s ) tq 1 4 C s( ) tq cos( t ) s( t ) Φ ( ) (14) J q J q tq s( J tq) cos( J t q) 1 7 C costq C s tq C(3 4s ) tqj 4C s( ) cosuqjtqj s( uqj J t ) cos( ) qj uqj J t qj Φv ( tqj, uqj ) cos( uqj J t ) s( ) qj uqj J t (15) qj cosuqj s uqj 7C costqj C s cosuqjt qj s 7 3JR valu o, a goup o valus o ad E wh C a, s th ma ca b obtad, ad s d to as ad 5 v q( tq 1). Th goups o v q1( tq 1) ad agula moto at, ad J C s s v q( tq 1) total a calculatd ad th a compad wth ach oth to d th goup wth th local th dt at o pg. tq tq tq duq s obtal tas tm, tqj tq tqj, s th agumt o lattud o th mauv, ad valus o th ad ths goup a mmum valu o vq 1( tq 1) v q( tq 1), ad th usd T jth a v v, v, v s th mpuls vcto. Th qj qjx qjy qjz obtal coodat systm usd to dscb th mpuls s gv as ollows: s alog th obtal adal dcto, s alog th -tack dcto, ad s alog th obtal omal dcto ad complts th ad-hadd systm. Th last mauv s xcutd at,.. tq tq. Eq. (13) s a la latv dyamc quato ud th ptubato. I ths study, oly two mauvs a cosdd o ach obtal tas that sx ukow mpuls compots cospod to sx quatos, ad th th soluto to Eq. (13) ca b asly obtad usg Gaussa lmato. Th dtals o ths la dyamcs modl ca b oud th cs [1] ad [9]. Log-duato dzvous poblms ud th J ptubato hav multpl local mma both th duato o o obtal pod ad th duato o multpl obtal pods [9]. I od to ovcom th popty o multpl local mma o obtal pod, th bu tm o th st mauv t q1 s umatd om t q to tq T wth a stp o T / N um, wh T s th c obtal pod ad N s th umb o umatos. Fo ach z um J t q y x 3 u qj t q1 v q1 Num 1 v q v q1 v ( t ) q1 q1 v q as th mpulss o th obtal tas o th qth dzvous. Basd o th modl povdd abov, th mauv mpulss o ach dzvous obtal tas a oly uctos o th tal stat, th qud dg stat ad th obtal tas tm, ad th th popllat cost ca b valuatd wth small computato cost. C. Optmzato Algothm A hybd-codg gtc algothm (HEGA) s adoptd to solv th omulatd appoxmatd sub-poblm. Ths algothm s a combato o a tg-codd GA o classcal TSPs [11] ad th al-codd gtc algothm [1]. Th dsg vaabl vcto Y ( Y1, Y ) s usd dctly as th chomosom o a dvdual. Th tss assgmt pocss o a sgl-objctv poblm s staghtowad. Th athmtcal cossov ad o-uom mutato opatos a appld to [1], whl th odd cossov (OX) ad th-pot dsplacmt mutato opatos a both appld to Y 1 [11]. A ltst statgy s mployd alogsd touamt slcto dug th algothm s slcto phas. Ths ca hlp pvt th loss o good solutos oc thy hav b oud [1]. Y 6 / 9

7 VI. Sgl Rdzvous Tajctoy Optmzato Oc th dzvous squc o ach msso s dtmd usg th mthods gv Scs. IV ad V, a optmzato modl s qud o mmzg th cost o vy sgl dzvous btw two dbs. Th obtal tas om dbs to dbs s a mult-mpuls, ptubd dzvous poblm ud th dyamcs o Eq. (). Obtal tagtg algothms basd o two-body dyamcs, such as th latv-obt-lmt basd latv moto quatos ad th Lambt algothm, caot dctly obta a asbl soluto ulss som dtal coctos o smpl tatos a mployd. I od to ctly d a a-optmal soluto o th gv log-duato (up to 3 days) dzvous poblm, a asbl tato optmzato modl s usd, whch th homotopc ptubd Lambt algothm [5, 14] s mployd as obtal tagtg algothm. Th optmzato modl s gv ths scto. V A. Dsg vaabls Th a 4 dsg vaabls o a -mpuls mauv pla: D [ t, v, v, v ], 1,,..., K (16) x y z wh K s th mauv umb o tms, t s th th mauv tm, ad v [ v, v, v ] T s th x y z th mauv mpuls vcto. I ou sults, 4-mpuls mauv pla s adoptd,.. K = 4. B. Objctv ucto Th objctv s to mmz th total vlocty cmt: C. Costats K m J = v v (17) 1 Th duato btw two adjact mauvs should b lag tha a gv valu,.., t t 1 T, (18) t [ t, t ], 1,,..., K wh t, t 3 days, T1 5 days, T ca b st as zos o,..., K. I addto, at th al tm, th dvato btw th spaccat s stat vcto x [, ] T v ad th stat vcto x xt o th xt dbs should b small tha th gv tolat o,.., 1 m, v v 1 m/s (19) xt D. Fasbl Itato Optmzato Appoach xt To dal wth th la qualty costats pstd Eq. (18), a goup o popotoalty cocts,, [,1] 1 s usd to substtut th mauv tms t,, 1 t as optmzato vaabls. Th, th mauv tms ca b calculatd as t t ( t t )+ T, wh 1 1 T 5 days, 1 () T,,..., K To dal wth th ola qualty costats pstd Eq. (4), th last two mpulss v 1 ad a chos to satsy th ola qualty costats, ad that thy a obtad by solvg a ptubd two-pot bouday valu poblm. H th homotopc ptubd Lambt algothm poposd by Yag t al. [5,14] s mployd to comput ths two mpulss as dscbd by: ( v, v ) Lambt_ p x( t ), x ( t ), t t v K 1 K K 1 K K K 1 (1) wh, ad th posto ad vlocty o tolacs o th ptubd Lambt algothm a spctvly st as 1 m ad 1 m/s. Ths ptubd Lambt algothm allowd th ptubd solutos that cludd th succssul computato o th J -ptubato though a homotopc tagtg tchqu whch th two-body Lambt soluto was usd as a tal valu ad th Rug-Kutta tgato o Eq. () was usd as a ptubd tajctoy popagato. Ovall, th optmzato modl o a multpl-mpuls dzvous poblm ca b wtt as ollows basd o th asbl tato appoach: x( t ) x K m J X xt 1 xt, subjct to 1 m, v v 1 m/s, wh v 1 1 ( vk 1, vk ) Lambt_ p x( tk 1), x( tk ), tk tk 1 xt X 1,, K, v1,, vk, [,1], t t ( t t )+ T, 1,..., K () Although th optmzato poblm pstd Eq. (8) s omulatd as a ucostad poblm, t s vy dcult to solv dctly wth a gadt-basd mthod such as SQP algothm bcaus ths algothms a vy sstv to th tal guss. Moov, th log duato ad pobably lag o-coplaa dc o RAAN mak t vy dcult to obta th optmal soluto o th dzvous poblm. Tho, t s hghly dsabl ad cssay to mploy a global optmzato algothms that do ot qu ay gadt omato o tal guss to solv th poblm. W us a dtal voluto (DE) algothm [16] to solv th optmzato modl o Eq. (). VII. Smulato Rsults 7 / 9

8 Th stat ad d poch as wll as th dbs moval squc o ach msso a lstd Tabl 1. It ca b oud that all th 13 dbs a movd 1 mssos. Th umbs o dbs movd Msso Od Stat Epoch (MJD) Ed Epoch (MJD) ach msso a maly btw sv ad twlv xcpt o 17 o th st msso. Tabl 1 Evts pochs ad dbs moval squcs Dbs Dbs Rmoval Squc Numb Stat Mass (kg) , 115, 1, 67, 19, 48, 1, 7, 63, 61, 8, 17, 41, 11, 45, 85, , 8, 9, 51, 7, 69, 1, 66, 73, 64, , 86, 13, 16, 11, 9, 49, 3,, 54, 7, , 43, 9, 55, 95, 14, 1, 39, 113, , 75,, 35, 119, 4, 18, 37, 11, 14, 3, , 65, 74, 5, 94, 1, 97, 79, , 1, 4, 76, 89, 99, 15, 59, 98, , 91, 93, 7, 18, 15, 88, , 53, 33, 68, 71, 8, 57, 6, , 81, 96, 6, 1, 3, 34, , 9, 11, 31, 38, 5, 4, 77, 13, , 111, 56, 78, 17, 19, V (m/s) V (m/s) msso ob Fgu 1 Total V o ach msso msso ob Fgu Avag V o ach msso V (m/s) RAAN () msso ob Fgu 3 Mmum ad maxmum V o ach msso MJD (day) x 1 4 Fgu 4 Hstoy o th RAAN o th actv spaccat ad cospodg dbs o msso #1 Th total vlocty cmts o dzvous o ach msso a pstd Fg. 1. As s show, th total vlocty cmts o most mssos a btw 15 m/s ad 5 m/s whl oly that o th th msso s byod 3 m/s. Howv, t should b otcd that th st msso has also movd th most dbs. Th avag vlocty cmts o ach 8 / 9 msso a btt dcs to valuat th pomac o ach msso. As s show Fg., th avag vlocty cmts o th st ou mssos a blow 5 m/s, whl o most o oth mssos th avag vlocty cmts a a 3 m/s, whch dcats that th pomacs o th st ou mssos a btt tha oths. It s cla that th avag

9 vlocty cmt o th ghth msso s th lagst wth a umb o a 4 m/s, whch dcats that th msso s ot optmal. Th mmum ad maxmum vlocty cmts o ach msso a llustatd Fg. 3. Accodg to Fg. 3, th smallst vlocty cmt o all 1 mssos s 38.6 m/s whl th lagst o s m/s. It ca b dcatd that th ag o vlocty cmts o a sgl dzvous pocss o ach msso s vy wd. Th hstos o th RAAN o th actv spaccat ad cospodg dbs movd th last msso a show Fg. 4, wh th d l wth ccls dcatd th hstoy o RAAN o th spaccat. It ca b oud that th RAAN o th spaccat cass gadually as t dzvouss th dbs o by o. Th RAAN o dbs 4, as show th Fg. 4, s ot clos to oth dbs ths squc, but th s a tscto btw t ad that o dbs 19 at th MJD aoud Cosqutly, t s cla that th spaccat wats to that poch to dzvous wth dbs 4 om dbs 19 so as to duc th vlocty cmts o mauvs causd by a lag tal RAAN o. V. Coclusos A th-lvl optmzato amwok s pstd to solv th poblm o GTOC9, wh th compttos a calld to dsg a ss o mssos abl to mov a st o 13 obtg dbs pcs whl mmzg th ovall cumulatv cost. Th top lvl s smla to a dyamc TSP, wh th dbs pcs a dvdd to sval goups ad ach goup o dbs s movd by o msso. Th mddl lvl s a mxd-tg optmzato poblm, wh th mpulss ad duatos o ach dzvous o msso a dsgd. Ad th bottom lvl s th pcs ad dtald optmzato o th lght tajctoy o dzvous. Th sult o GTOC9 obtad by ths amwok s th llustatd. Th sult dcats that th th-lvl optmzato amwok s ct ad ca obta good solutos cosdabl tm. Rcs [1] GTOC Potal, [] GTOC9 Hompag, [3] Izzo D, Gtz I, Hs D, t al. Evolvg solutos to TSP vaats o actv spac dbs moval, Pocdgs o th 15 Aual Coc o Gtc ad Evolutoay Computato, ACM, 15: [4] Zhag, J., Paks, G. T., Luo, Y., Tag, G. Multspaccat Rulg Optmzato Cosdg th J Ptubato ad Wdow Costats, Joual o Gudac, Cotol, ad Dyamcs, Vol. 37, 14, pp [5] Yag, Z., Luo, Y.-Z., Zhag, J., Tag, G.J. Homotopc Ptubd Lambt Algothm o Log-Duato Rdzvous Optmzato, Joual o Gudac, Cotol, ad Dyamcs, Vol. 38, No. 11, 15, pp [6] Yag, Z., Luo, Y. Z., Zhag, J., Two-lvl Optmzato Appoach o Mas Obtal Log-Duato, Lag No-Coplaa Rdzvous Phasg Mauvs, Advacs Spac Rsach, Vol. 5, 13, pp [7] Dogo M., Gambadlla L.M., At Coloy Systm: A coopatv lag appoach to th tavlg salsma poblm, IEEE Tasactos o Evolutoay Computato, Vol. 1, No. 1, 1997, pp [8] Luo Y.Z., L H.Y., Tag G.J. "Hybd Appoach to Optmz a Rdzvous Phasg Statgy", Joual o Gudac, Cotol, ad Dyamcs, Vol. 3, No. 1, 7, pp [9] Zhag, J., Luo, Y. Z., ad Tag, G. J., Hybd Plag o LEO Log-Duato Mult-Spaccat Rdzvous Msso, Scc Cha Tchologcal Sccs, Vol. 55, No. 1, 1, pp [1] Laboudtt, P., ad Baaov, A. A., Statgs o o-obt dzvous cclg Mas, Advacs th Astoautcal Sccs, Vol. 19,, pp [11] Chattj S., Caaa, C., ad Lych L. A., Gtc Algothms ad Tavlg Salsma Poblms, Euopa Joual o Opatoal Rsach, Vol. 93, No. 3, 1996, pp [1] Db, K., Patap, A., Agawal, S., ad Myava, T., A Fast ad Eltst Mult-Objctv Gtc Algothm: NSGA-II, IEEE Tasactos o Evolutoay Computato, Vol. 6, No.,, pp [13] Yag, Z., Luo, Y. Z., ad Zhag, J., Robust Plag o Nola Rdzvous wth Uctaty, Joual o Gudac, Cotol, ad Dyamcs, pss. [14] Luo, Y. Z., Zhag, J., ad Tag, G. J., Suvy o Obtal Dyamcs ad Cotol o Spac Rdzvous, Chs Joual o Aoautcs, Vol. 7, No. 1, 14, pp [15] Luo, Y. Z., Tag, G. J., L, Y. J., t al., Optmzato o Multpl-Impuls, Multpl -Rvoluto, Rdzvous-Phasg Mauvs, Joual o Gudac, Cotol ad Dyamcs, Vol. 3, No. 4, 7, pp [16] Zhu Y. H., Wag H., ad Zhag J., Spaccat Multpl-Impuls Tajctoy Optmzato Usg Dtal Evoluto Algothm wth Combd Mutato Statgs ad Bouday-Hadlg Schms, Math. Pobl. Eg., / 9