PROXIMITY OPERATIONS OF ON-ORBIT SERVICING SPACECRAFT USING AN ECCENTRICITY/INCLINATION VECTOR SEPARATION

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1 PROXMY OPERAONS O ON-ORB SERVCNG SPACECRA USNG AN ECCENRCYNCLNAON VECOR SEPARAON J. Sprmnn 1, S. D Amco 1 DLR, Grmn Spc Oprtons Cntr, 83 Wsslng, [email protected] DLR, Grmn Spc Oprtons Cntr, 83 Wsslng, [email protected] Abstrct: hs ppr nvstgts promt oprtons concpt bl to rlz th dmndng rqrmnts for on-orbt srvcng spccrft, whl provdng n ffcnt collson vodnc strtg nd rdcd propllnt consmpton. n contrst to trdtonl rndzvos pproch whr th vlton of th collson rsk s onl bsd on consdrtons don n th orbtl pln, hr rltv ccntrctnclnton vctor sprton concpt s proposd whch promss mprovd flblt, robstnss nd rdcd ntr-spccrft sprtons t mnml collson rsk. n ths ppr th novl promt oprtons concpt, orgnll dvlopd for gosttonr stllts nd lrd sd to oprt low Erth orbt formtons, s doptd to spport fr nd md rng oprtons of on-orbt srvcng non-cooprtv vhcls. Aftr ddrssng th ffcts of th rlvnt dffrntl prtrbtons, stbl nomnl rltv moton gomtrs r drvd bsd on rl-world nvgton ccrc nlss. Rdr trckng dt nd GPS nvgton soltons for th clnt nd srvcr stllts rspctvl r sd to trd mnmm sft sprtons nd chrctrstcs of th rndzvos ntr gt. nll fl-ffcnt formton kpng nd rconfgrton strtg s dsgnd bsd on nltcl soltons of th proposd lnr moton modl. Kwords: Eccntrctnclnton Vctor Sprton, Atonomos ormton Control, ormton lng, On-Orbt Srvcng, Rltv Nvgton. 1 ntrodcton h cpblt to tonomosl rndzvos nd dock two spccrft wold provd bnfts to mltpl msson scnros. h corrcton of lnch flrs s on rlvnt mpl. Usll, pon mlfncton of th ppr stg of th lnchr th spccrft dos not chv ts dsrd lttd or nclnton. Sch n njcton rror cn n th most css not b corrctd b th spccrft onbord proplson sstm nd th msson s lost or t lst nbl to chv ts prmr objctvs. n cs n on-orbt srvcng spccrft s lrd n orbt wthn flt mngmnt progrm for mpl th njcton rror cold b corrctd nd th ovrll sccss rt of spc mssons cold b ncrsd. Anothr pplcbl r for tonomos rndzvos nd dockng s th rmovl of spc dbrs. Evn f wthn th nt rs no frthr lnchs of spccrft wold b prformd cscd of collsons strts nhncng th nmbr of collson frgmnts nd thrb th ovrll nmbr of spc dbrs prtcls [1]. n ths contt n ctvt prvntng stllts from bcomng ncontrolld dbrs wold mprov th stton. Emplr t cold b thoght of rpr or rflng ctvts clos to th spccrft nd of lf. Lftm of stllts cold b tndd b tkng ovr th tttd nd orbt control fnctonlts n dockd confgrton. n sch n chvd combnd confgrton r-ntr of th copld spccrft cold b prformd s wll. Accordngl, lso th cpblt to rmov spc dbrs n gnrl wold b nbld. As rslt of th formntond msson pplctons th gol mst b to prform rndzvos nd dockng wth non-cooprtv spccrft nd propr d-orbtng for dbrs rmovl. or ths prpos ftr lnch nd phsng tlord formton flng confgrton hs to b stblshd bfor gong nto th two dmnsonl pproch for promt oprtons. h concpt of ccntrctnclnton vctor sprton [] s th most stbl to chv ths objctv, snc t

2 provds pssvl sf formton flng confgrton nd strghtforwrd cpblt for formton kpng nd rconfgrton. Prvos rsrch n ths fld hs lrd dmonstrtd th potntl of ths mthod for th GRACE [3] or th rrsar-xndem-x msson [4]. rthr, th tchnolog dmonstrton msson PRSMA hs lrd dmonstrtd tonomos formton control bsd on rltv ccntrctnclnton vctors n low Erth orbt [5]. Howvr, th formntond formton flng mssons rl on cooprtv spccrft. GPS nvgton soltons nd rw dt r vlbl on-bord for rl-tm nvgton nd control tsks s wll s on-grond for prcs post-fcto rconstrcton of th rltv moton. h rl chllng wth non-cooprtv spccrft n th frm of n on-orbt srvcng msson s rprsntd b th vlblt of bsolt nvgton dt onl. h rltv moton cn onl b rconstrctd ongrond bsd on msrmnt dt tps of bsolt ntr. As consqnc ffcts gvn b common rror cncllton cnnot b plotd lk tpcll don n formton flng stllts mbrkng hrdwr of sm bld nd tp. o ths nd dffrnt dt tps r nlzd n ths ppr obtnd from stllts msson flng n low Erth orbt. Among th dt tps r th GPS nvgton solton, rdr trckng dt provdd b rnhofr nsttt for hgh frqnc phscs nd rdr tchnqs HR, formr GAN [6] nd ngl trckng dt. rom th prformd nlss th most promsng cnddts wll b sd for rltv nvgton ccrc nlss rprsnttv for n on-orbt srvcng msson. h prsntd rltv nvgton ccrc nlss drvs th slcton of th dmnsons of th formton flng gomtr. or ths prpos th ccntrctnclnton vctor sprton mthod s ntrodcd. ts mn dvntg s tht rdl nd long-trck moton r no longr consdrd lon, bt nstd complt 3D moton wth sprton n th rdlcross-trck pln. h lrgr ncrtnts n th long-trck componnt rsltng from nvgton, tmosphrc dnst ncrtnt nd mnvr cton rrors cn ths b vodd. Wthn th prsnt work th prmtrzton of th rltv moton n trms of rltv ccntrctnclnton vctors s sd for th frst tm to trnsfr th srvcr to th rndzvos ntr gt from whch th two dmnsonl pproch s nttd. Wth rspct to ths trnsfr formton rconfgrton strtg wll b drvd wth rspct to mnmm collson rsk, mnmm fl consmpton nd th possblt to llow for frqnt tonomos rconfgrtons of th formton. An ddtonl constrnt rgrdd wthn th slcton of th formton gomtr s th vsblt constrnt for rltv nvgton snsors, whch wll b sd for th clos rng pproch drng n on-orbt srvcng msson. nll, n ffctv nd fl ffcnt w for formton kpng t frthr dstncs s drvd bsd on th -vctor sprton. Rltv Moton h proposd formton flng concpt s bsd on th gnrlzton of th ccntrctnclnton vctor sprton mthod. or th dscrpton of ths concpt frst th prmtrzton of th rltv moton wll b dscrbd. t wll bld p from th nprtrbd, prtrbd nd controlld moton dscrpton..1 Unprtrbd Moton or convnnc th clnt spccrft s tkn s rfrnc for th formton. n ths contt th srvcng spccrft wll orbt rond th clnt nd ts moton mppd n th rltv coordnt frm. h bsolt nd rltv orbt prmtrztons n th Erth-Cntrd-nrtl EC frm r gvn b th followng qtons [].

3 Ω Ω Ω Ω Ω Ω M S S S S S S S sn cos nd sn cos λ ω ω ω α α 1 h frst hlf of th qton gvs prmtrzton n Kplrn lmnts. hs s modfd b doptng th ccntrct vctor, nd th mn rgmnt of lttd. h scond hlf of qton 1 gvs th prmtrzton of th rltv moton n st of rltv orbtl lmnts ROE drvd from non-lnr combnton of th bsolt prmtrzton. h sbscrpt S s ntrodcd dnotng qntts of th srvcng spccrft. h ROE st conssts of normlzd sm-mjor s dffrnc nd th rltv mn longtd btwn th two spccrft λ. Addtonll, th rltv ccntrct nd nclnton vctors r ntrodcd for whch th followng Crtsn nd polr nottons r ppld []. ϑ ϑ sn cos nd sn cos h phss of th rltv -vctors r trmd rltv prg nd rltv scndng nod ϑ bcs th chrctrz th gomtr of th rltv orbt s sn b th Clnt spccrft. h rltv moton s dscrbd n th RN orbtl frm cntrd on th Clnt spccrft. h coordnt frm s dfnd b th nt vctor n rdl drcton postv otwrds, R, th nt vctor n long-trck drcton of th stllt moton postv n flght drcton, nd th nt vctor norml to th orbtl pln n drcton to th postv nglr momntm vctor cross-trck, N. h orbtl moton wthn ths frm cn b lnrzd for nr crclr orbts nd smll sprtons comprd to th orbt rds b th Hll-Clohss-Wltshr HCW qtons. hos cn b prssd n rltv orbtl lmnts wth th mn rgmnt of lttd s ndpndnt vrbl []. cos cos 3 sn sn sn 3 cos ϑ ϑ λ v v v v v v r r r N R N R 3 Accordng to Eq. 3 th gnrl bondd rltv moton wth zro dffrntl sm-mjor s s tltd llps gr 1. ts dmnson s dfnd b th rltv ccntrct vctor norm n th R- pln nd b th rltv nclnton vctor norm n cross-trck drcton. h mn long-trck sprton btwn th two spccrft s dfnd b λ

4 gr 1. Projctons of bondd rltv moton n th long-trckrdl lft nd crosstrckrdl rght drctons for rbtrr rltv orbtl lmnts. Hr Chf stnds for Clnt spccrft nd Dpt for Srvcr spccrft.. Prtrbd Moton h ntrl moton from Eq. 3 cn b tndd b prtrbtons rsng from Erth s obltnss ffcts J nd dffrntl drg. Accordngl, mor ffcnt formton kpng strtg cn b dfnd. h two prtrbton trms, vrng wth tm, r ncorportd b smmng thm wth constnt kplrn trm. D α J α α α 4 h nltcl trtmnt of th Erth s obltnss ffcts s bsd on th thor dvlopd b Browr nd Lddn [6][8]. f onl frst-ordr trms n J nd r consdrd n th srs pnson th ROE cn b prssd s fncton of th mn rgmnt of lttd [] frst trm n Eq. 5. or th tmosphrc drg th mn forc s prdomnntl nt-prlll to th vloct of th spccrft. As rslt th dffrntl drg hs lnr mpct on th rltv sm-mjor s nd qdrtc mpct on th rltv mn rgmnt of lttd scond trm n Eq sn 3 sn cos sn 1 B B ρ ρ γ γ λ α 5 or th rmnng coffcnts n Eq. 5 B s th dffrnc btwn th spccrft bllstc coffcnts B C D Am, C D th rodnmc drg coffcnt, A th cross-scton r, m th stllt mss, ρ th tmosphrc dnst nd

5 5cos 1 3 J R d γ E nd d γ. 6 Eq. 4 nds onl to b sbstttd n Eq. 3 to obtn frst ordr solton of th qtons of moton n th prsnc of J nd dffrntl drg ffcts..3 Controlld Moton or nclson of mnvrs nto th rltv moton modl nothr trm cn b ddd to Eq. 4. α α α α α 7 J D v Mor spcfc, ths trm rslts b nvrson of th lnr rltv moton modl. hs th consqnt chng of ROE from n mplsv mnvr s gvn b v λ v R 3 M v 1 v R snm v cosm α v n v R cosm v sn. 8 M v N cosm v N snm Eq. 8 shows th rslt of n ppld thrst t mn rgmnt of lttd m. n ddton nt chng of mn longtd wthn th tm ntrvl btwn th mnvr cton nd th poch of th rltv orbtl lmnts rslts. wo fndmntl spcts shold b notd from Eq. 8 for propr choc of th formton kpng nd rconfgrton strtg. rst of ll th n-pln nd ot-of-pln rltv motons r dcopld. Mnvrs n cross-trck drcton onl ffct th rltv nclnton vctor, whl th othr mnvr tps onl vr th n-pln componnt. Scondl, rdl mnvrs r two tms mor pnsv thn long-trck plss for chngs of th rltv ccntrct vctor nd do not ffct th sm-mjor s. 3 Pssv Sft nd Stblt h rltv -vctor sprton concpt s drvd from th formntond lnrzd rltv moton modl nd cn b sd to dsgn promt oprtons gomtrs chrctrzd b mnmm collson rsk pssv sft nd mnmm corrcton ffort pssv stblt. h concpt of -vctor sprton hs orgnll bn dvlopd for th sf collocton of gosttonr stllts [9], bt cn qll b ppld to promt oprtons for formton flght n LEO GRACE, rrsar-xndem-x, PRSMA []. t s bsd on th consdrton tht th ncrtnt n prdctng th long-trck sprton of two spccrft s gnrll mch hghr thn for th rdl nd cross-trck componnt. D to th coplng btwn sm-mjor s nd orbtl prod, smll ncrtnts n th ntl poston nd vloct rslt n corrspondng drft rror nd ths sclrl growng long-trck rror. Prdctons of th rltv moton ovr tndd prods of tm r thrfor prtclrl snstv to both orbt dtrmnton rrors nd mnvr cton rrors. o vod collson hzrd n th prsnc of long-trck poston ncrtnts, propr sprton of th two spccrft n rdl nd cross-trck drcton s dsrbl. Mor spcfc th collson rsk s dfnd b th mnmm sprton prpndclr to th flght drcton Eq. 9 for n rbtrr st of ROE [1].

6 r mn mn {, } or bondd rltv moton, th mnmm collson rsk s provdd b prlll or ntprlll rltv -vctors ϑ or ϑπ. or ths confgrton th formton s lws sprtd n th NR pln n contrst to prpndclr rltv -vctors, for whch rdl nd cross-trck sprton vnshs t th sm tm. Concrnng Eq. 9 nothr ffct on formton rconfgrton shll b notcd. Usng long-trck mnvrs, th rltv orbt llps s shftd n rdl drcton ql to th vrton of th smmjor s ndcd b th mnvr. n th cs tht th formton gomtr rmns sf, whrs for < th mnmm sprton dcrss. h lttr stton s tpcl of formton kpng scnros. hs phnomnon mks long-trck plss nhrntl lss sf thn rdl ons, bt cn b compnstd b stbl ncrsd rltv -vctor sprton. Concrnng J wthn th dsgn of pssvl sf formton th bsolt nclnton of th formton shold dll b dntcl to vod sclr moton of th rltv nclnton vctor Eq. 5. Nvrthlss, sprton of th two orbtl plns cn b chvd b smll offst n th rght scnsons of thr scndng nods. h rsltng rltv nclnton vctor hs phs ngl ϑ ±π, nd th sm or oppost phs mst b slctd for th rltv ccntrct vctor to obtn pssvl sf formton. Addng th ncssr condton for bondd rltv moton of th srvcr wth rspct to th clnt on obtns th followng convnnt nomnl confgrton λnom α nom ± 1 nom ± nom whch provds pssv stblt nd pssv sft to th formton. h ctl choc of th nomnl rltv orbtl lmnts hs to b don sch tht mn { } d nom, nom 9 11 whr d rprsnts sft thrshold for th mnmm sprton prpndclr to th flght drcton. or th dfnton of d th followng mn contrbtons hv to b consdrd: 1. Rltv ccntrct nd nclnton vctors cn onl b controlld down to crtn ccrc dpndng on th ppld control schm, th nvgton rrors nd th mnvr cton rrors.. ormton rconfgrtons n long-trck drcton, or rndzvos mnvrs rqr th cton of long-trck plss whch ffct th rltv sm-mjor s. 3. h mnmm llowd sprton btwn th cntr of mss shll tk nto ccont th phscl dmnsons of th spccrft. 4. A hrstc mrgn of 5% s tpcll ppld for sft rsons. h ovrll contrbton bdgt for th comptton of d cn b sl stmtd bsd on th followng consdrtons. h phscl dmnsons of th two spccrft shold not tnd 1m on

7 th rltv scl. rthr, th mmm sm-mjor s vrton s proportonl to th ppld mnvr nd ths strctl dpndnt on oprtonl constrnts lk rndzvos tm. h contrbton wll b drvd wthn th dsgn of th formton kpng nd rconfgrton mnvrs ltr on. nll, th pctd orbt control ccrc wll b drvd bsd on th nvgton ccrc nlss dscrbd n th nt scton. 4 Absolt Nvgton Accrc h gol of ths scton s to drv th ccrc of post-fct bsolt orbt dtrmnton bsd on msrmnt dt tps whch m b vlbl n n on-orbt srvcng scnro. Hr GPS nvgton soltons, rdr trckng nd ngl trckng dt r consdrd nd thr prformnc comprd for tpcl msson scnro. h nlss s prformd on flght dt of th crrntl oprtd mssons t th Grmn Spc Oprtons Cntr GSOC CHAMP, GRACE nd rrsar-x. h nlss ws rptd for mltpl mssons to ld frst ndcton of n lttd profl. At th tm of nlss CHAMP ws flng t 35 km lttd, GRACE t 485 km nd rrsar-x t 55 km. Snc dl-frqnc GPS rw dt from rcvrs of godtc qlt r vlbl for ll mssons prcs orbt dtrmnton POD wth ccrc t cm lvl cn b sd s rfrnc. h nvgton ccrc cn b drvd from th comprson of th phmrs rsltng drng nlss to th POD. or th dffrnt trckng dt tps n orbt dtrmnton s prformd ovr 4 hors, s n tpcl flght oprtons scnro. An orbt prdcton rc s ppndd to th nd of th orbt dtrmnton rc. h phmrs from propgton wll b comprd wth th POD ftr 9mnts, 1 hors nd 4 hors. Hnc, n ndcton of th rror dvlopmnt cn b rtrvd. h poston nd vloct rrors of th comprson r gvn blow bl 1. h rrors r gvn s th root mn sqr RMS n ch componnt of th RN frm for th mntond mssons nd th corrspondng trckng dt tp. Orbt dtrmnton rrors bsd on ngl trckng dt r roghl on ordr of mgntd wors comprd to GPS or rdr trckng lrd ftr orbt dtrmnton. Accordngl, ths msrmnt tp hs not bn consdrd n frthr drng th nlss. rthrmor, ths dt tp wold rqr n ctv loggng of th spccrft onbord trnspondr b th grond stton. Hnc, th pplcblt for on-orbt srvcng mssons s dfnd bov s rthr low snc t rqrs rmnng fnctonlt of th clnt spccrft. Orbt dtrmnton rrors bsd on GPS nd rdr trckng show rrors n th sm ordr of mgntd for poston nd vloct. Rdr trckng rslts r slghtl wors comprd to GPS spcll t hghr lttds. n gnrl th ovrll rror trnd dcrss wth hghr lttds. As pctd th lrgst contrbton to th rror bdgt s gvn b th long-trck componnt. n fct th nvgton rror n long-trck drcton ftr 4 hor propgton s on X1 or two CMP ordrs of mgntd hghr thn th orbt dtrmnton ccrc dpndng on th orbt lttd. or th rdl nd cross-trck componnt th nvgton rror sts wthn th sm ordr of mgntd for both consdrd trckng dt tps. h rslts from ths scton show tht orbt dtrmnton cn b prformd bsd on rdr trckng dt for th clnt nd on GPS nvgton solton for th srvcr, whl ldng to smlr nvgton rrors. Hnc, n dqt choc for n on-orbt srvcng rprsnttv scnro cn b md. rthrmor, th hghr ccrc n rdl nd cross-trck componnt ovr long-trck jstfs onc mor th dsgn choc bsd on th -vctor sprton concpt.

8 bl 1. Absolt Nvgton Errors Poston RMS [m] GPS RADAR Angl CMP GRA X1 CMP X1 X1 Rdl OD Acc mn h h Along-trck OD Acc mn h h Cross-trck OD Acc mn h h Vloct RMS [mms] GPS RADAR Angl CMP GRA X1 CMP X1 X1 Rdl OD Acc mn h h Along-trck OD Acc mn h h Cross-trck OD Acc mn h h Wht s stll nknown s th dgr of corrlton nd th lvl of common rror cncllton tht cn b pctd from th sbtrcton of orbt dtrmnton rslts n ordr to obtn th dsrd rltv stt. Sch problm wll b ddrssd n th nt scton. So fr th propgton rror trnds from th bsolt orbt dtrmnton nlss show hgh smlrt for GPS nd rdr trckng prformnc. Possbl vn smllr nvgton rrors cn b pctd from th rltv nvgton whch m ld to frthr rdcton of th sft thrshold d for th slcton of th nomnl formton flng confgrton. 5 Rltv Nvgton Accrc h rltv nvgton ccrc nlss mks s of th sm flght dt s sd n th prvos scton. Smlrl th orbt dtrmnton ccrc s vltd b comprng rslts bsd on dffrnt trckng dt wth th vlbl prcs orbt dtrmnton prodcts. Snc flght dt from rdr trckng nd GPS r not crrntl vlbl from co-orbtng stllts, th rltv nvgton ccrc nlss s hr prformd s sort of zro-bsln tst. On-grond orbt dtrmnton nd prdcton r prformd for th sm spccrft bsd on dffrnt dt tps.

9 orbt dtrmnton nd propgton s prformd for th sm spccrft bsd on dffrnt trckng dt tps. GPS nvgton soltons r sd to rprsnt th srvcr nd rdr trckng dt rprsnt th clnt. Snc th sm spccrft s consdrd th sm POD prodct s sd for both nd cncls ot drng sbtrcton of th obtnd phmrs. h rltv nvgton rrors obtnd from th grond-bsd orbt dtrmnton r lstd n bl. bl. Rltv Nvgton Errors Poston Vloct RMS [m] RMS [mms] CMP X1 CMP X1 Rdl OD Acc mn h h Along-trck OD Acc mn h h Cross-trck OD Acc mn h h As shown n bl, rltv rdl nd cross-trck componnts r chrctrzd b rrors blow 1 m RMS. t s notd tht th propgton rror s comprbl wth th orbt dtrmnton rror nd bscll dos not ncrs ovr tm. A frthr comprson wth th bsolt orbt dtrmnton ccrc fgrs cf. bl 1 shows tht th orbt prdcton s chrctrzd b hgh lvl of common rror cncllton, whl th orbt dtrmnton rrors r n-corrltd. As pctd th rltv nvgton rrors n long-trck drcton r grdll ncrsng ovr tm bt wth mgntd whch s 1 tms smllr thn th bsolt rrors. Howvr t hs to b notd tht for ths nlss orbt dtrmnton nd propgton hv bn prformd for th sm spccrft. h ctl rltv nvgton rror m b ffctd b msmodllng of dffrntl cclrtons csd.g. b dffrntl drg, spcll t low lttds s n th cs of CHAMP. bl 3. Dffrntl Eccntrct nd nclnton RMS [m] RMS [m] CMP X1 CMP X1 OD Acc mn h h OD Acc mn h h As dscrbd n scton 3, th dsgn of nomnl formton flng confgrtons s drvn b th rltv control trckng rrors nd n trn b th rltv nvgton rrors. n vw of th doptd

10 prmtrzton of th rltv moton, th rltv nvgton rrors r frthr prssd n trms of rltv ccntrct nd nclnton vctors. bl 3 shows tht th pctd rltv nvgton rrors from on-grond orbt dtrmnton bsd on GPS nd rdr trckng dt r of th ordr of 1 m RMS. As consqnc t cn b fnll concldd from th rslts tht th mmm nvgton rror n nd s lws smllr thn 1 m for prlll or nt-prlll rltv --vctors. 6 Gdnc nd Control h rltv nvgton rslts prsntd n th prvos scton gv s th possblt to slct trd-off formton flng confgrton whch s nhrntl sf nd stbl cf. scton 3. Onc th ntl nd fnl dsrd formton gomtrs r ssgnd, formton kpng nd rconfgrton strtg cn b dsgnd whch tks dvntg of th rltv orbtl lmnt prmtrzton. 6.1 Nomnl ormton Gomtr h frst objctv to drv th nomnl formton gomtr s to fnlz th dscsson from scton 3 on th dfnton of th sft thrshold d. h dffrnt contrbtons r gn shown n Eq. 1. h contrbtons rs s follows: [ { NvAcc, NvAcc} Ctrl v DPhs ] mrg d m 1 1. h mmm nvgton rror n nd s drvd to b 1m.. An mplsv orbt control schm s consdrd whr th nvgton ccrc cn b consdrd s 1% of th controlld rng Ctrl Gvn dsrd mmm long-trck drft for rndzvos of 1.7 kmrv, th mmm ncssr long-trck mpls monts to v 1 cms. h rsltng mmm smmjor s vrton monts to v 185 m. 4. h mnmm llowd cntr of mss sprton s tkn s D Phs 1 m to ccont for th spccrft phscl dmnsons. 5. A hrstc mrgn of 5% s tpcll ppld for sft rsons mrg 1.5. Accordng to Eq. 1, th mnmm sprton sft thrshold monts to 45 m. rom th drvd sft thrshold th strtg to rconfgr th formton from n ntl nomnl formton confgrton to so clld rndzvos ntr pont cn now b dvlopd mplong th rltv -vctor sprton concpt. h ntl formton confgrton s dntfd b Eq. 1. h rndzvos ntr pont s n rbtrr rltv poston E E R, E, E N dfnd n th RN orbtl frm. rom th ntl condtons t s lws possbl to dfn bondd rltv orbt throgh nt-prlll rltv -vctors whch contns th rndzvos ntr pont gr. B constrcton, th fnl formton gomtr wll provd th possblt to rvst th rndzvos ntr pont t ch orbtl rvolton gr.

11 gr. Rltv orbt gomtrs for th formton rconfgrton problm. h ntl rltv orbtl lmnts r dntfd b th sbscrpt, th fnl rltv orbtl lmnts r dntfd b th sbscrpt, th rndzvos ntr pont s dntfd b E, whl th rd crcl of rds d rprsnts th collson vodnc zon. h ntl formton flng confgrton cn b chosn ccordng to Eq. 1 s { } { } { } m 156,9,9,.577,,, d d d λ. 13 h slcton of th nomnl mn long-trck sprton s mnl drvn b vsblt constrnts gvn b th rltv nvgton snsors drng vros oprtonl scnros. n th sql th followng choc hs bn md { } 577. tn3, m nom nom nom o λ. 14 h fnl ROE st shll contn th rndzvos ntr pont wthn thr dmnsonl llps chrctrzd b nt-prlll rltv ccntrct nd nclnton vctors. or n rbtrr rndzvos ntr pont E 58,,154 otsd th collson vodnc zon th solton s smpl gvn b { } { } { } m 37,6,6,,,, R N R E E E E ± λ 15 Hvng dfnd thr of th s prmtrs ncssr to dfn rltv orbt b th qtons bov th mssng prmtrs r th rltv sm-mjor s, whch s st to zro to vod dvrgng drfts, nd th phss nd ϑ whch shold grnt th nt-prlllsm of th rltv -vctors.., -ϑ π. n ccordnc wth Eq. 1 th ntl nd fnl rltv orbtl lmnts for th rprsnttv scnro r gvn b m m λ λ α α 16

12 6. ormton Rconfgrton h bsc concpt for th drvd rprsnttv formton rconfgrton n cs of n on-orbt srvcng msson s llstrtd n gr 3. h ntndd trnsfr of th srvcr to th rndzvos ntr gt from whch th two dmnsonl pproch s nttd s shown n th lft sbplot b th corrctons ncssr to th rltv -vctors. gr 3. Ant-prlll rltv ccntrct nd nclnton vctors for th formton rconfgrton lft nd corrspondng rltv moton n th pln prpndclr to th flght drcton rght. h corrctons rqrd for rconfgrton r gvn b α α α 133 λ λ λ m n-pln nd ot-of-pln rltv orbt control problms r dcopld. On cross-trck mnvr s ncssr nd sffcnt to control th rltv nclnton vctor, whl two n-pln mnvrs r ncssr nd sffcnt to corrct rltv ccntrct vctor nd rltv sm-mjor s. h n-pln mnvrs m b mplmntd s rdl, long-trck or md rdllong-trck plss. n gnrl th dvntgs of rdl mnvrs rsd n thr lrgr sz snc rsltng n smllr cton rrors for smll mpls bts nd n th fct tht rdl mnvrs do not ffct th smmjor s. Sch dvntgs cn b plotd drng on-orbtng srvcng phss wth smll ntrspccrft sprtons nd hghr control ccrc rqrmnts. h dvntgs of long-trck mnvrs l n thr smllr sz hlf of th rdl mnvrs nd n th cpblt to chng th sm-mjor s for fl-ffcnt rndzvos oprton phss. A dsgn of th otlnd trnsfr sng on rdl nd on cross-trck mnvr pr, whl rdcng th mn long-trck sprton t th sm tm rslts n th followng V mnvr pln.

13 v v R1 R n n v v N1 N n n nλ ms nλ 4.171ms.16ms.16ms R1 tn N1 R tn N R1 N1 π π π π π π π n th qton bov rprsnts th locton of th mnvrs long th orbt. or th slctd vctors th rdl mnvrs r lws loctd t th scndng nd dscndng nods.., π, π, whl th cross-trck mnvrs r lws loctd t th trm northrn or sothrn lttds.., ±π. hs s lso shown wth th llstrton of th trnsfr blow gr gr 4. Complt formton rconfgrton sng dobl rdl nd cross-trck mplsv mnvrs from Eq. 16. h rsltng rltv poston s mppd n th RN orbtl frm blck rltv trjctors. h lbls 1 to 4 ndct th locton of th mnvrs. h lbls nd E ndct th rltv poston t strt nd t th rndzvos ntr pont. h dsrd fnl rltv orbt nd th collson vodnc rgons r llstrtd n sold nd dsh rd lns. 6.3 ormton Kpng h rltv moton of th two spccrft s mnl ffctd b Erth s obltnss prtrbtons. Mor spcfc th rltv ccntrct vctor follows crclr moton n th -vctor pln, whl th rltv nclnton vctor prncs lnr drft proportonl to th nclnton dffrnc. h lttr ffct cn b rmovd b slctng rltv scndng nod of zro.., ϑ, lk slctd for th nomnl rltv orbtl lmnts Eq. 16. B dfnng control wndows cntrd on th nomnl vls t s possbl to kp th formton b plotton of th formntond ntrl sclr moton gr 5.

14 gr 5. Grphcl rprsntton of th control wndows for th rltv ccntrct vctor lft nd th rltv nclnton vctor rght. h sbscrpt nom stnds for nomnl confgrton, m for mmm llowd dvton from nomnl, nd for qntts bfor nd ftr th cton of mnvrs. h dsrd rltv ccntrct nd nclnton vctors ftr prformng formton kpng mnvrs r gvn b nom cos nom sn nom sn cos sgn. 19 nom nom nom m h sbscrpt nom rprsnts th nomnl confgrton Eq. 1 whl nd m r mmm tolrtd dvtons. Eq. 19 cn b sd to dtrmn th rltv -vctors ftr ch orbt control mnvr. As dscrbd bov th control mnvrs onl hv to corrct th rltv - vctor for th dvlopd on-orbt srvcng rprsnttv scnro. hs cn b chvd b pr of long-trck mnvrs t th scndng nd dscndng nods Eq.. Evn thogh copl of rdl mnvrs cold b sd s wll th fl ffcnt vrson s prfrrd ths tm snc th mnvrs r smll nd th sprton drng formton flght s rthr lrg. v v 1 n n 4 4 n sn n sn 1 tn 1 π π or sch strtg th V bdgt s dpndnt on th lngth of th rltv ccntrct vctor nd th llowd dvton. or th scnro bov wth 9 m th dl V sms p to v v1 v n sn.68 ms 1

15 7 Conclson Wthn th prsntd work th bsolt nvgton ccrc nlss ld to th slcton of rdr trckng msrmnts for clnt nd GPS nvgton solton dt for grond bsd srvcr nvgton. Movng forwrd, th rltv nvgton nlss rsltd n th drvton of mnmm sprton sft thrshold n th NR-pln, whch montd to 45 m. hs n th contt of formr formton flng mssons [3][4][5] nomnl formton gomtr s dsgnd bsd on th concpt of ccntrctnclnton vctor sprton. Wth th mplmntton of th formton rconfgrton strtg sng mnvr copls n rdl nd cross-trck drcton n nhrntl sf cpblt to trnsfr to th rndzvos ntr pont s stblshd. n ths contt th -vctor sprton s ppld to promt oprtons of on-orbt srvcng spccrft for th frst tm. tr rsrch hs to consdr rltv nvgton snsors nd th dfnton of corrspondng pproch strtg. Espcll th hndovr btwn bsolt nd rltv nvgton wll b of ntrst. Anothr fld for ftr work s th drvton of mor lbort sft thrshold b nclson of dffrntl drg ffcts n th nvgton ccrc nlss. nll rl flght dt wll b sd to vlt th prformnc of dffrnt rltv moton modls nd thr mpct on th formton flng dsgn. 8. Rfrncs [1] Klnkrd, H., Spc Dbrs: Modls nd Rsk Anlss, Sprngr-Prs Books n Astrontcl Engnrng, Sprngr-Vrlg, 6. [] D'Amco, S., Atonomos ormton lng n Low Erth Orbt, PhD thss, chncl Unvrst of Dlft, SBN , 1. [3] Montnbrck, O. nd Krschnr, M. nd D'Amco, S. nd Bttdpr, S., E-vctor sprton for sf swtchng of th GRACE formton, Arospc Scnc nd chnolog, Vol. 1, No. 7, pp , 6. [4] D Amco, S. nd Montnbrck, O., Promt oprtons of ormton-lng Spccrft Usng n Eccntrctnclnton Vctor Sprton, Jornl of Gdnc, Control nd Dnmcs, Vol. 9, No. 3, pp. 554, 6. [5] D Amco, S. nd Ardns, J.S. nd D loro, S., Atonomos ormton lng Bsd on GPS PRSMA lght Rslts, 6th ntrntonl Workshop on Stllt Constllton nd ormton lng, p, wn, Novmbr, 1. [6] Ad, S. nd Ptzlt,. nd Lshck, L. nd Krschnr, M. nd Khlng, R.,, 1st ntrntonl Smposm on Spc lght Dnmcs, olos, rnc, 8. Sp.. Oct., 9. [7] Browr, D., Solton of th problm of rtfcl stllt thor wthot drg, h Astronomcl Jornl, Vol. 64, No. 174, pp , [8] Lddn, R.H., Smll ccntrcts or nclntons n th Browr thor of th rtfcl stllt, h Astronomcl Jornl, Vol. 68, No. 8, pp , [9] Eckstn, M.C. nd Rjsngh, C.K. nd Blmr, P., Colocton strtg nd collson vodnc for th gosttonr stllts t 19 dgrs wst, ntrntonl Smposm on Spc lght Dnmcs, olos, rnc, Novmbr, [1] D loro, S., D Amco, S., Collson Rsk Assssmnt for Co-Orbtng Spccrft, DEOS-DLR-N-, Vrson 1., 8. Aprl, 1.