SPACECRAFT FORMATION CONTROL AND ESTIMATION VIA IMPROVED RELATIVE-MOTION DYNAMICS

Transcription

1 AFRL-RV-PS- TR-7-5 AFRL-RV-PS- TR-7-5 SPACECRAFT FORMATION CONTROL AND ESTIMATION VIA IMPROVED RELATIVE-MOTION DYNAMICS Davd A. Ccc Abrn Unvers 7 Samford Hall Abrn, AL Mar 7 Fnal Repor APPROVED FOR PUBLIC RELEASE; DISTRIBUTION IS UNLIMITED. AIR FORCE RESEARCH LABORATORY Space Vehcles Drecorae 355 Aberdeen Ave SE AIR FORCE MATERIEL COMMAND KIRTLAND AIR FORCE BASE, NM

2 DTIC COPY NOTICE AND SIGNATURE PAGE Usng Governmen drawngs, specfcaons, or oher daa nclded n hs docmen for an prpose oher han Governmen procremen does no n an wa oblgae he U.S. Governmen. The fac ha he Governmen formlaed or sppled he drawngs, specfcaons, or oher daa does no lcense he holder or an oher person or corporaon; or conve an rghs or permsson o manfacre, se, or sell an paened nvenon ha ma relae o hem. Ths repor s he resl of conraced fndamenal research deemed eemp from pblc affars secr and polc revew n accordance wh SAF/AQR memorandm daed Dec 8 and AFRL/CA polc clarfcaon memorandm daed 6 Jan 9. Ths repor s avalable o he general pblc, ncldng foregn naonals. Copes ma be obaned from he Defense Techncal Informaon Cener DTIC hp:// AFRL-RV-PS-TR-7-5 HAS BEEN REVIEWED AND IS APPROVED FOR PUBLICATION IN ACCORDANCE WITH ASSIGNED DISTRIBUTION STATEMENT. //SIGNED// THOMAS A. LO VELL Program Manager //SIGNED// PAUL HAUSGEN, Ph.D. Techncal Advsor, Spacecraf Componen Technolog //SIGNED// JOHN BEAUCHEMIN Chef Engneer, Spacecraf Technolog Dvson Space Vehcles Drecorae Ths repor s pblshed n he neres of scenfc and echncal nformaon echange, and s pblcaon does no conse he Governmen s approval or dsapproval of s deas or fndngs.

3 REPORT DOCUMENTATION PAGE Form Approved OMB No Pblc reporng brden for hs collecon of nformaon s esmaed o average hor per response, ncldng he me for revewng nsrcons, searchng esng daa sorces, gaherng and mananng he daa needed, and compleng and revewng hs collecon of nformaon. Send commens regardng hs brden esmae or an oher aspec of hs collecon of nformaon, ncldng sggesons for redcng hs brden o Deparmen of Defense, Washngon Headqarers Servces, Drecorae for Informaon Operaons and Repors 74-88, 5 Jefferson Davs Hghwa, Se 4, Arlngon, VA Respondens shold be aware ha nowhsandng an oher provson of law, no person shall be sbjec o an penal for falng o compl wh a collecon of nformaon f does no dspla a crrenl vald OMB conrol nmber. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.. REPORT DATE DD-MM-YYYY 3. DATES COVERED From - To. REPORT TYPE Fnal Repor TITLE AND SUBTITLE Spacecraf Formaon Conrol and Esmaon Va Improved Relave-Moon Dnamcs 8 Jl 4 7 Mar 7 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 66F 6. AUTHORS 5d. PROJECT NUMBER Davd A. Ccc 889 5e. TASK NUMBER PPM943 5f. WORK UNIT NUMBER EF38 7. PERFORMING ORGANIZATION NAMES AND ADDRESSES 8. PERFORMING ORGANIZATION REPORT NUMBER Abrn Unvers 7 Samford Hall Abrn, AL SPONSORING / MONITORING AGENCY NAMES AND ADDRESSES. SPONSOR/MONITOR S ACRONYMS Ar Force Research Laboraor Space Vehcles Drecorae AFRL/RVSV 355 Aberdeen Ave, SE. SPONSOR/MONITOR S REPORT Krland AFB, NM NUMBERS AFRL-RV-PS-TR-7-5. DISTRIBUTION / AVAILABILITY STATEMENT 3. SUPPLEMENTARY NOTES FA ABSTRACT Ths research focsed on mproved spacecraf relave dnamcs modelng and analss, esmaon of spacecraf relave moon, and conrol of spacecraf relave moon, wh parclar emphass on esmaon.e. navgaon of spacecraf relave rajecores from angles-onl daa. 5. SUBJECT TERMS Orbal Mechancs; Saelle Formaon Flng; Saelle Rendevos; Saelle Prom Operaons 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT a. REPORT Unclassfed b. ABSTRACT Unclassfed 8. NUMBER OF PAGES c. THIS PAGE Unclassfed Unlmed 7 9a. NAME OF RESPONSIBLE PERSON Thomas A. Lovell 9b. TELEPHONE NUMBER nclde area code Sandard Form 98 Rev Prescrbed b ANSI Sd. 39.8

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5 Secon TABLE OF CONTENTS Page Ls of Fgres... Ls of Tables... v Preface... v Acknowledgemen and Dsclamer...v Smmar... Inrodcon... 3 Mehods, Assmpons, and Procedres Basc Descrpon of IROD Approach Garaneed Ambg Woffnden s Dlemma Ulaon of Nonlnear Dnamcs Ulaon of Nonhomogeneos Dnamcs Dealed Descrpon of IROD Algorhms Lnear Mar Mehod LMM Mar Reslan Mehod MRM Nonhomogeneos Observer Mehod NOM Praccal Consderaons Descrpon of Scenaros Grond-Based IROD GIROD Close-Prom IROD CIROD Delneaon of IROD es cases Daa Reqremens for Scenaros, wh Descrpon of Daa Flow GIROD CIROD Op Mercs and Verfcaon Relave Orb Elemens Resls and Dscsson GIROD GIROD Scenaro GIROD Scenaro GIROD Scenaro Assessmen of GIROD Resls CIROD CIROD Scenaro CIROD Scenaro CIROD Scenaro CIROD Scenaro CIROD Scenaro

6 4..6 CIROD Scenaro CIROD Scenaro CIROD Scenaro Conclsons Recommendaons References Ls of Smbols, Abbrevaons, and Acronms... 6

7 LIST OF FIGURES Fgre Page Fgre. LVLH Coordnae Frame, wh X and Y As Drecons Depced and Z Gong no he page.... Fgre. Varos Famles of Ambgos Trajecores all 5 Assmpons n Force... Fgre 3. Noonal Depcon of Two Trajecores Propagaed wh Nonlnear Dnamcs... Fgre 4. Noonal Depcon of LOS Measremens Taken b Maneverng Observer...5 Fgre 5. Flow Char of Grond-based IROD Algorhmc Process... 8 Fgre 6. Flow Char of Close-prom IROD Algorhmc Process... 3 Fgre 7. GIROD Scenaro, wh Observed Lnes of Sgh From Grond Sensor, Spermposed wh Tre Propagaed Trajecor Ble and IROD Solon Trajecor Green Fgre 8. LVLH Trajecor of IROD Solon for CIROD Scenaro vs., wh LOS Measremens and Tre Relave Orb Dsplaed...4 Fgre 9. LVLH Trajecor of IROD Solon for CIROD Scenaro vs., wh Tre Relave Orb Dsplaed...4 Fgre. LVLH Trajecor of IROD Solon for CIROD Scenaro vs., wh LOS Measremens and Tre Relave Orb Dsplaed Fgre. LVLH Trajecor of IROD Solon for CIROD Scenaro vs., wh Tre Relave Orb Dsplaed Fgre. LVLH Trajecor of IROD Solon for CIROD Scenaro 3 vs., wh LOS Measremens and Tre Relave Orb Dsplaed Fgre 3. LVLH Trajecor of IROD Solon for CIROD Scenaro 4 vs., wh LOS Measremens and Tre Relave Orb Dsplaed Fgre 4. LVLH Trajecor of IROD Solon for CIROD Scenaro 5 vs., wh LOS Measremens and Tre Relave Orb Dsplaed Fgre 5. LVLH Trajecor of IROD Solon for CIROD Scenaro 5 vs., wh Tre Relave Orb Dsplaed Fgre 6. LVLH Trajecor of IROD Solon for CIROD Scenaro 6 vs., wh LOS Measremens and Tre Relave Orb Dsplaed... 5 Fgre 7. LVLH Trajecor of IROD Solon for CIROD Scenaro 6 vs., wh Tre Relave Orb Dsplaed... 5 Fgre 8. LVLH Trajecor of IROD Solon for CIROD Scenaro 7 vs., wh LOS Measremens and Tre Relave Orb Dsplaed... 5 Fgre 9. LVLH Trajecor of IROD Solon for CIROD Scenaro 7 vs., wh Tre Relave Orb Dsplaed Fgre. LVLH Trajecor of IROD Solon for CIROD Scenaro 8 vs., wh LOS Measremens and Tre Relave Orb Dsplaed Fgre. LVLH Trajecor of IROD Solon for CIROD Scenaro 8 vs., wh Tre Relave Orb Dsplaed... 55

8 LIST OF TABLES Table Page Table. The Three Possble Delneaons of Tes Case Parameers n he Epermens o Follow... 6 Table. Descrpon of GIROD Scenaro, Incldng Table Parameer Vales Table 3. Orbal Elemens from Scenaro for a Consrced Reference Orb, b Frs IROD Solon, c Second IROD Solon, d Tre Orb Table 4. Descrpon of GIROD Scenaro, Incldng Table Parameer Vales Table 5. Orbal Elemens from Scenaro Table 6. Descrpon of GIROD Scenaro 3, Incldng Table Parameer Vales Table 7. Orbal Elemens from Scenaro Table 8. Descrpon of CIROD Scenaro, ncldng Table parameer vales... 4 Table 9. Tre Relave Orb of CIROD Scenaros -4, n Terms of ROE Vales... 4 Table. RMS Resdal Angle Error and ROE Raos for CIROD Scenaros Table. Descrpon of CIROD Scenaro, Incldng Table Parameer Vales... 4 Table. Descrpon of CIROD Scenaro 3, Incldng Table Parameer Vales Table 3. Descrpon of CIROD Scenaro 4, Incldng Table Parameer Vales Table 4. Descrpon of CIROD Scenaro 5, Incldng Table Parameer Vales Table 5. Tre Relave Orb of CIROD Scenaros 5-6, n Terms of ROE Vales Table 6. RMS Resdal Angle Error and ROE Raos/Dfferences for CIROD Scenaros Table 7. Descrpon of CIROD Scenaro 6, Incldng Table Parameer Vales Table 8. Descrpon of CIROD Scenaro 7, Incldng Table Parameer Vales... 5 Table 9. Tre Relave Orb of CIROD Scenaros 7-8 Pre-manever, n Terms of ROE Vales. 5 Table. RMS Resdal Angle Error and ROE Raos/dfferences for CIROD Scenaros Table. Descrpon of CIROD Scenaro 8, Incldng Table Parameer Vales v

9 PREFACE Ths fnal repor represens he work condced nder Conrac FA from 7/8/4 o /5/6. Ths projec was dreced b Prncpal Invesgaors PI Dr. Davd A. Ccc of Abrn Unvers. The PI was asssed b gradae sden Ms. Lara Heber. The parclar emphass s on esmaon of relave moon, dealng he developmen and es of algorhms collecvel ermed nal relave orb deermnaon. v

10 ACKNOWLEDGEMENT Ths maeral s based on research sponsored b Ar Force Research Laboraor nder agreemen nmber FA The U.S. Governmen s ahored o reprodce and dsrbe reprns for Governmenal prposes nowhsandng an coprgh noaon hereon. DISCLAIMER The vews and conclsons conaned heren are hose of he ahors and shold no be nerpreed as necessarl represenng he offcal polces or endorsemens, eher epressed or mpled, of Ar Force Research Laboraor or he U.S. Governmen. v

11 SUMMARY Spacecraf neral navgaon, or he means b whch an acve spacecraf deermnes s ephemers, s he fondaon for mos space mssons []. For nacve or non-cooperave space objecs, he same ask can be performed, b n hs case ms be done b passve means, and s ofen ermed orb deermnaon []. These asks, based on he phscs of neral orbal moon, each have an analog n he realm of relave orbal moon. In he case of navgaon, a spacecraf can esmae s ephemers or mprove s crren esmae of s ephemers based on measremens beween self and a reference objec whose orb s accrael known [3-5]. In he case of orb deermnaon, a spacecraf whose own orb s accrael known can esmae he ephemers of an nknown objec based on measremens beween self and ha objec [6-7]. Mahemacall, hese wo asks are essenall he same; n boh cases, measremens beween he spacecraf and a Resden Space Objec RSO are processed o esmae he relave rajecor beween he wo objecs. Ths research effor anales he processng of opcal angle or lne-ofsgh sensor daa from an acve spacecraf o esmae js sch a relave rajecor. Varos msson scenaros are nvesgaed, o nclde boh close-prom scenaros when he spacecraf mages a nearb RSO and long-range scenaros when he spacecraf mages oher RSOs. The mehods dealed heren are mean o generae an nal gess of he relave orb and can be consdered deermnsc n nare;.e. he mehods assme no modelng error he relave dnamcs on whch he mehods are based are assmed eac and no measremen error he measremens obaned are assmed eac. Therefore, he general erm o be sed for hese mehods s Inal Relave Orb Deermnaon IROD. The IROD solon can serve as an nal gess or sarer solon for a sascal e.g. bach leas-sqares or Eended Kalman Fler esmaor. In addon, he IROD approach can be appled o classcal grond-based rackng. Scenaros nvolvng grond sensor daa of eher he spacecraf or he RSO wll be eplored. Ths docmen conans he followng secons: a general descrpon of IROD heor s gven, followed b a dealed descrpon of he varos IROD echnqes developed wh an np-op descrpon of he daa flow, delneaon of each scenaro, resls, and conclsons.

12 INTRODUCTION I s well-known ha when rng o deermne an objec s orb sng srcl opcal measremens angles-onl, observabl can be an sse. Generall speakng, an orb deermnaon scenaro s observable f he objec s orb can be nqel deermned from he gven measremens. Unobservabl mples ambg,.e. here es more han one orb ha fs he gven measremens. Ths s mos easl demonsraed n a relave orb deermnaon scenaro [8], where boh he observer and RSO are orbng he Earh. Sppose we model he relave moon beween he observer and RSO n a Caresan coordnae frame. The logcal frame hen o emplo s he Local-vercal-local-horonal LVLH frame. Ths frame enals defnng a reference orb abo he Earh, defnng a chef objec on hs orb, and aachng a coordnae frame o he chef s cener of mass. Ths s depced n Fgre. Noe ha he LVLH frame s no predcaed on he esence of wo space objecs, or even one for ha maer, as he chef and s orb can be fabrcaed. The frame s also no predcaed on he chef orb havng an parclar eccenrc hogh for mos praccal applcaons, hs orb shold be closed,.e. e <. The LVLH coordnae drecons are hen defned as follows: he or radal drecon s algned wh he chef s neral poson vecor.e. he vecor from Earh s cener o he chef, he or crossrack drecon s algned wh he chef s anglar momenm vecor.e. perpendclar o he chef s orb plane, and he drecon s he cross prodc of and. If he chef orb s crclar, he drecon s hen algned wh he chef s neral veloc vecor and s ofen called along-rack. Noe ha he LVLH frame ranslaes and roaes arond he Earh wh he chef objec, herefore hese drecons are defned nsananeosl. Fgre. LVLH Coordnae Frame, wh X and Y As Drecons Depced and Z Gong no he page. Noe on Fgre : Orbal Moon s Clockwse and R Denoes Chef s Ineral Poson Vecor

13 If we choose he observer o be he chef, hen he sae vecor consss of he relave poson and veloc of he RSO epressed n LVLH coordnaes: T of he RSO relave o he reference orb s descrbed b lnear dnamcs,.e.. Sppose he moon, where represens he vales of he saes a an me, s a 6 vecor represens he vales of he saes a he nal me, and, s he 6 6 sae ranson mar represenng he lnear dnamcs of he moon. Two common choces for he sae ranson mar are ha derved from he Tschaner-Hempel solon [] or ha derved from he Clohess-Wlshre solon []. Boh solons assme he space objecs are sbjec onl o wo-bod Kepleran grav force, wh he Clohess-Wlshre solon conanng he added assmpon ha he chef s orb s crclar. Le s represen as r v The nsananeos Lne-of-sgh LOS from observer o RSO s he n vecor along he relave poson vecor: r ˆ r 3 r The nsananeos relave poson vecor s relaed o he relave poson vecor a b 3

14 r rr, rv, 4 where rr, and rv, are he pper lef and pper rgh sbmarces of,, respecvel. Inserng Eqaon 4 no Eqaon 3 elds ˆ r rr, rv, rr, rv, 5 Consder wo rajecores, one whose vales a are gven b and he oher whose nal vales are, where s a posve real nmber. A an gven me, he lne-of-sgh vecor o an RSO on he frs rajecor s ˆ r rr, rv, rr, rv, 6 whle he lne-of-sgh vecor o an RSO on he second rajecor s ˆ r rr rr rr, rv, rr, rv,, rv,,, rv ˆ r rr rr, rv,,, rv 7 Becase me s arbrar, clearl he wo rajecores possess he same lne-of-sgh hsor for all me. Ths, for a rajecor nall a, an rajecor wll possess he same lne-ofsgh hsor. Ths, a gven lne-of-sgh hsor represens an nfne faml of ambgos rajecores. 4

15 I s mporan o ndersand eacl wha assmpons lead o he above ambg. Reference 8 lss he followng assmpons o garanee sch ambg: Lnear dnamcs sed o model he relave moon beween he objecs Angle/LOS measremens onl 3 No manevers b eher objec However, hese assmpons do no fll conve he reqremens or consrans on he msson scenaro ha wll garanee ambg. The analss below reveals a more accrae se of assmpons ha wll resl n he ambg descrbed above. Consder he fac ha each LOS measremen above onl conans wo ndependen peces of nformaon, whch can be epressed as angles e.g. amh/elevaon A/El or rgh ascenson/declnaon RA/Dec or slopes. Epressng he measremens as slopes, we have a each measremen me :, 8 Nong ha,, and r r, we have r, 9 Sbsng for,, and sng Eqaon elds,,, 3,, where, s he frs row of he sae ranson mar, ec. We can rearrange each and measremen eqaon as,,,,, 3 or,,,,, 3 5

16 Sppose we oban LOS measremens a p dfferen mes, collecng p oal peces of nformaon n n. Or p measremen eqaons are hen 3 3,,,,,,,, n, n n,,, 3 n n n 3 These eqaons can be wren as A, where A s a p 6 mar whose elemens are all consans, fncons of he observer s orb, or fncons of he measremen mes herefore all elemens of A are known. I s well known from lnear algebra ha f A s fll rank, he onl solon s he ero vecor. Phscall, hs corresponds o a saon where here s no rajecor nder he assmed dnamcs ha eacl sasfes he consrans of he gven measremens.e. ha possesses ha parclar LOS hsor. If A s less han fll rank.e. snglar, here s one nonero solon for each degree of he nll space of A. However, hese nonero solons are nonnqe,.e. f s a solon, so s, where s an real consan. Ths verfes he * garaneed * ambg shown n Eqaon 7 above. Inspecon of he above dervaon shows ha ambg s garaneed f he measremen eqaons are lnear and homogeneos n he nal saes. Ths sffcenc condon s argabl he mos fndamenal wa o eplan he ambg dealed above. For reasons ha wll be seen laer, s sefl o decompose hs condon no he followng wo condons ha eqvalenl garanee ambg: The relave moon beween he objecs are descrbed b lnear homogeneos dnamcs.e. he relave saes a an me can be epressed enrel as lnear combnaons of he nal relave saes. The relaonshps beween he measremens and saes a each measremen me are lnear and homogeneos.e. hese relaonshps can be wren as homogeneos eqaons ha are lnear n he nsananeos relave saes. These wo assmpons qe evdenl lead o he resl n Eqaon 7. Agan, for reasons ha wll be seen below, hese assmpons can be frher decomposed no he followng: a The relave moon beween he objecs are descrbed b lnear dnamcs. 6

17 b No eernal non-homogenos forces ac on he objecs. Ths ncldes manevers, non-conservave forces, and rgd-bod conac forces sch as hose eered on a camera ha s separaed from he observer s cener of mass. c Measremens are aken b a sngle observer. a Angle/LOS measremens onl b The relave moon s modeled n a Caresan coordnae frame. Noe ha, whle hese assmpons are eqvalen o he sngle assmpon orgnall saed above, he ma no be nqe;.e. even f no all fve of hese assmpons are me, ma sll be possble o garanee he ambg of Eqaon 7 b formlang a dfferen se of assmpons ha are also eqvalen o he orgnal assmpon. However, for he scenaros dealed n hs repor, hs parclar se of assmpons wll serve as he chosen se o garanee ambg. 7

18 Heren les he cr of hese scenaros. Gven ha ambg s garaneed f all 5 oal assmpons are me, sands o reason ha relang an one or more of hese resrcons.e. volang one or more of he assmpons ma poenall eld observabl,.e. a nqe relave sae solon. Eamples of sch relaaons nclde he followng: Use of nonlnear dnamcs o descrbe he relave moon beween he objecs relaaon of Assmpon a Eecng and acconng for one or more known manevers b eher he observer or RSO relaaon of Assmpon b Acconng for non-conservave forces e.g. drag or Solar Radaon Pressre SRP n he relave moon dnamcs relaaon of Assmpon b Insallng he camera a ceran dsance apar from he observer s cener of mass relaaon of Assmpon b Processng oher measremen pes besdes Angles/LOS relaaon of Assmpon a Modelng he relave moon n a crvlnear clndrcal or sphercal coordnae frame relaaon of Assmpon b Processng measremens b mlple observers relaaon of Assmpon c Noe ha hese relaaons are mall eclsve, e.g. one ma accon for non-conservave forces whn a lnear relave moon model and sll have a hope of observabl. For hese scenaros, IROD algorhms are devsed based onl on he frs wo relaaons lsed above. These algorhms are esed on smlaed opcal measremen daa. A vare of dfferen accrac mercs are emploed. A major par of hs nvesgaon s o evalae how well each relaaon adds o he observabl and accrac of angles-onl IROD n varos msson scenaros. Before proceedng, he movaon for a relave dnamcs approach o nal orb deermnaon as opposed o an neral dnamcs approach, or classcal Inal Orb Deermnaon IOD shold be poned o. One advanage of a relave approach s ha several closed-form eplc relave moon solons es. These solons provde ecellen nsgh no he moon, eas vsalaon, ec, and allow he possbl of closed-form IROD algorhms,.e. algorhms ha are non-erave n nare. Sch algorhms have been developed and are dealed n he secons ha follow. These algorhms do no reqre an specfc knowledge of he RSO oher han he LOS measremens hemselves n order o deermne s relave orb. The algorhms are aracve for aonomos/on-board mplemenaon or grond mplemenaon drng he hecc pace of msson operaons becase he do no reqre hman-n-he-loop spervson. Compare hs o classcal IOD schemes [ 9-], whch end o be erave n nare. 8

19 9 3 METHODS, ASSUMPTIONS, AND PROCEDURES 3. Basc Descrpon of IROD Approach In hs secon, he condon of garaneed ambg all assmpons n force wll be frher descrbed, hen he effec of he frs wo relaaons n he blleed ls above wll be descrbed and how each of hese relaaons leads o a canddae IROD algorhm. 3.. Garaneed Ambg Woffnden s Dlemma For he scenaro of garaneed ambg n Secon, as governed b Eqaon s -7, all fve assmpons lsed above are n force. In order o llsrae he nare of he IROD algorhms ha follow, here we derve he measremen eqaons a slghl dfferen wa han n Secon. Consder a measred LOS vecor a me n he LVLH frame,.e. k j r ˆ ˆ ˆ ˆ. The measremen eqaons can be formed b reqrng ha he relave poson vecor s parallel o each measred LOS: ˆ r r r U r 4 or 5 Noe ha onl wo of he hree above eqaons are ndependen, b we ma nclde all hree eqaons who loss of general. Frher, he sae-ranson mar can be sed o relae he relave poson vecor o he nal sae vecor:

20 ,, rr rv 6 So f we oban LOS measremens a n dfferen mes, he measremen eqaons can be wren * * as A, as n Secon. Once agan, f here ess a nonero solon, hen an s a solon as well. Ths, a drecon sae vecor can be deermned ha sasfes he measremen ~ a non-nqe solon, b he magnde of he sae vecor denoe eqaons denoe hs as hs as canno be deermned. Ths s ofen referred o as range ambg and s depced n Fgre where he Clohess-Wlshre solon s sed o propagae he relave moon. Each ~ plo shows mlple rajecores sharng he same b each wh a dfferen of. Noe ha he rajecores shown nclde some ha mgh be consdered common or praccal for a prom operaons msson, as well as more obscre rajecores. The pon s ha he relave sae vecor ~ ma conss of an s real vales represenng an nfne space of 6 n, and for an gven ~, can ake on an posve real vale represenng an nfne faml of ambgos ~. Ths scenaro of garaneed ambg wll be referred o as Woffnden s rajecores for ha Dlemma becase was frs descrbed n Reference 8. Noe ha hs garaneed ambg s no a fncon of how man measremens are aken;.e. f all he assmpons garaneeng ambg are n force, one canno somehow creae observabl b akng more measremens. Fgre. Varos Famles of Ambgos Trajecores all 5 Assmpons n Force

21 3.. Ulaon of Nonlnear Dnamcs The frs relaaon o be descrbed s ha of Assmpon #a above. Here all aspecs of he scenaro descrbed n he prevos secon are n force, ecep ha he relave moon beween he observer and RSO are modeled wh nonlnear dnamcs. The nonlnear relave moon solon o be led s fond [3-5]. Ths solon s smlar o he Clohess-Wlshre solon n ha assmes wo-bod grav and a crclar chef orb. However, nsead of reanng onl erms lnear n he nal relave saes, reans second-order erms as well. The eac solon wll no be repeaed here, b s of he followng form: E E D D C C C C C C 7 Sbsng for,, and no Eqaon s 4 a measremen me elds D C D C E C E C E D E D 8 If we oban LOS measremens a n dfferen mes, or 3n measremen eqaons are copled second-order polnomals n s nknowns he s nal relave saes. Noe ha hese eqaons are no lnear n he nal relave saes,.e. he canno be wren as A. Ths, f * s a solon o hese eqaons, * s no,.e. we have escaped Woffnden s dlemma b emplong nonlnear relave dnamcs. Ths saon s depced n Fgre 3. Whle hese plos are noonal.e. no acal propagaed rajecores, he represen wo nal sae vecors and propagaed forward wh nonlnear dnamcs. The posons of he wo objecs relave o a chef a he orgn are shown a for dfferen mes. Whle he LOS vecors of he objecs are nall algned, he devae over me. Whereas f he objecs moon were propagaed wh lnear dnamcs, her LOS hsores wold be dencal for all me.

22 Fgre 3. Noonal Depcon of Two Trajecores Propagaed wh Nonlnear Dnamcs The sse ha remans hen s wheher an effcen closed-form approach can be fond o solve hese eqaons. Ths s lef for Secon Ulaon of Nonhomogeneos Dnamcs The ne relaaon o be descrbed s ha of Assmpon #b. Here all aspecs of he scenaro descrbed n he prevos secon are n force, ecep ha he observer s no resrced o le on he reference chef orb drng all he measremen mes. As allded o earler, he case of a maneverng observer s sbsmed nder hs scenaro. Consder an RSO ha performs a known manever v a me m. Frher consder a LOS measremen of he RSO colleced a me afer he manever. The solon for he relave poson a ms now ake no accon he manever, hs Eqaon 4 becomes r,, v rr, rv rv m 9

23 The scenaro more relevan o mos prom mssons s ha where he observer whose moon s assmed known performs he manever. Ths s easl accommodaed n he same framework, b reqres clarfcaon of some deals. Ths far, he sae has been assmed o descrbe he poson and veloc of he RSO relave o he observer a he orgn of he LVLH frame. Alernavel, he orgn of he LVLH frame can be consdered o be an arbrar objec ravelng along a wo-bod orb. In sch cases, he objec wll be referred o as he vral chef, and s orb as he reference orb. Whle an orb ma serve as he reference orb, s ofen convenen o defne he reference orb based on he nsananeos ephemers of he observer, hen propagang ha orb forward o each measremen me wh wo-bod dnamcs. Obvosl he observer wll gradall devae from hs reference orb.e. he orgn of he LVLH frame de o perrbaons, b for a reasonable span of me, wll reman close o he LVLH orgn nl/nless manevers. Consder an observer nall locaed on he reference orb. The observer hen performs a known manever v a me m. Consder hen a LOS measremen of he RSO colleced a me afer he manever. The observer s poson a s gven b rv, m v, so he relave poson beween observer and RSO a s shown below. r,, v rr, rv rv m From Eqaons 9 and, we see ha a known manever performed b eher he RSO or observer wll generall prodce some nonhomogeneos change r n he relave poson. For general, boh cases wll be smmared as follows: r r rr rv,, Sbson no he measremen eqaons 4 prodces he followng: rr,, rv r 3

24 Agan, onl wo of he hree eqaons are ndependen, b we ma nclde all hree eqaons who loss of general. Collecon of measremens a n nsans prodces a ssem of eqaons * * ha can be wren as A b. Noe ha f s a solon o hese eqaons, s no,.e. we have escaped Woffnden s dlemma b emplong nonhomogeneos relave dnamcs. Ths saon s depced n Fgre 4. Here LOS measremens are aken a,, 3, and 4, and he observer manevers a m beween 3 and 4. Becase he manever akes he observer off he reference orb, hs wll generall eld a dfferen LOS a 4 han wold be obaned f he observer had no manevered, as shown b he green rajecor. In sch a case, he observer s poson vecor n he LVLH frame.e. relave o he orgn s n fac he r represened n Eqaon. If, however, he observer s pos-manever locaon happens o eld he same LOS a 4 as f he observer had no manevered as shown b he red rajecor, hen he rgh-hand sde of Eqaon s ero becase ˆ r 4 r 4, hs he measremen eqaons reman homogeneos and no observabl s ganed. Ths pe of manever wll be referred o as a snglar manever. Of corse, he reslng r s a fncon of he relave rajecor, he manever me/magnde/drecon, and he pos-manever measremen me. Therefore, f a gven scenaro resls n a red rajecor a 4, an addonal measremen a 5 wold lkel eld observabl. For p LOS measremens, A s a 3p 6 mar. If he mar A s snglar, here sll wll no be a nqe solon here wll be eher no solon or an nfne nmber of solons, b f enogh measremens are obaned sch ha A s fll rank, a nqe solon can be obaned va psedonverse: T T A A A b 3 I s worh re-emphasng he pon made a he begnnng of hs sbsecon: ha a manever s no reqred o rela Assmpon #b. Raher, f he poson hsor of eher he RSO or observer relave o he reference orb over he span of measremen mes s sch ha, for whaever reason, canno be accrael descrbed b homogeneos lnear dnamcs, hen he measremen eqaons wll be as gven n Eqaon. Secon lsed mehods of relang Assmpon #b who maneverng. 4

25 Fgre 4. Noonal Depcon of LOS Measremens Taken b Maneverng Observer 3. Dealed Descrpon of IROD Algorhms Each IROD mehod esmaes he 6 saes correspondng o he rajecor of he observed objec relave o he defned reference orb a he epoch me, whch s chosen o be he me of he frs LOS measremen. In all cases, he observer s neral ephemers s assmed o be precsel known. The IROD algorhms ake advanage of he relaaons of he condons for garaneed ambg ha were descrbed n Secon. As descrbed, he se of nonlnear dnamcs prodces a ssem of second-order polnomal eqaons, and he se of nonhomogeneos dnamcs prodces a ssem of nonhomogeneos lnear eqaons. Solvng a ssem of second-order polnomals s nonrval, and wo dfferen mehods were mplemened and are descrbed below n Sbsecons 3.. and 3... Solvng a ssem of nonhomogeneos lnear eqaons s farl sraghforward, and he mplemenaon of he mehod s descrbed n Sbsecon Lnear Mar Mehod LMM The Lnear Mar Mehod s a farl ad-hoc approach o fnd an appromae solon o he ssem of second-order polnomals comprsng he measremen eqaons n Eqaon 8. The mehod s a wo-sep procedre. In he frs sep, each of he 7 possble lnear and nd -order combnaons of he s nknown varables.e., he nknown nal poson and veloc componens are reaed as ndependen nknowns. These 7 erms or monomals can be assembled no a vecor : 5

26 T where, conssen wh he orderng of erms n Eqaon 7 and 8, corresponds o, corresponds o, ec, and 7 corresponds o. The measremen eqaons of Eqaon 8 are lnear n he elemens of, and can hs be recas as A, where for n measremen mes, A s a 3n 7 mar. From Eqaon 8 we see ha A A A 3 ec... C D C E E, D,,...,...,..., A A,7 A,7 3,7 C D 7 C 7 7 E E 7 7 D 7 5 If A s less han fll rank.e. snglar, here s one nonero solon for each degree of he nll space of A. A frs glance, hs approach has no allevaed he range ambg becase for each * * canddae solon, he scalng s also a solon of A. However, hs gnores he * qadrac relaons beween he elemens of,.e. n order for o be a vable solon o he measremen eqaons, he 7 h * elemen of ms eqal he sqare of he s elemen, ec. The second sep of he mehod ncorporaes hese consrans. Afer solvng a nll space vecor of A call ~, s reformlaed as: ~ T ~ ~ 3 ~ ~ ~ ~ ~ ~ ~ where remans o be solved. Noe ha hs sep nvolves dscardng he 7 h hrogh 7 h elemens of ~.e. he elemens peranng o qadrac combnaons of he nknowns. Each 6

27 7 elemen of he reformlaed vecor ~ herefore conans eher lnearl or qadracall. Sbsng hs vecor back no he measremen eqaons,.e. pre-mlplng b A, elds ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A A 7 Here, 6 A s a 3n 7 mar composed of he frs 6 colmns of A, and 7 7 A s a 3n mar composed of he 7 h hrogh 7 h colmns of A. Ths represens a se of 3n qadrac eqaons for, he onl remanng nknown. Each eqaon acall conans a rval solon a. Facorng o leaves a se of lnear eqaons for : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A A 8 A hs pon shold be noed ha f here were no model error.e. f he reference orb were perfecl known and he solon n Eqaon 7 were an eac represenaon of spacecraf dnamcs and no measremen error.e. f Eqaon 4 represened he eac relaonshp beween he sae vales and measremen vales, we wold be garaneed o fnd a solon o he measremen eqaons Eqaon 8 b he wo-sep process above. Tha s, a leas one nll space vecor of A wold be garaneed o es call ~ *, and here wold be a vale of call * ha, combned wh ~ *, wold eacl solve each of he 3n eqaons n Eqaon 8. Ths solon wold correspond o he eac vales of he relave saes a he epoch me,. Tha s, he followng eqales wold hold:

28 * ~ * ~.... * * * * * * ~ 6... * * ~ 6 * ~ * ~ ~ * 9 where,, ec, are he re nal sae vales. However, de o model error and measremen ~ * * error, we are no garaneed ha sch a or ess. The ahors have smlaed several cases wh model error and/or measremen error, and a snglar vale decomposon of A shows ha he mar ends o have several snglar vales of appromael ero. Tha s, here are several rgh-snglar vecors of A of he form ~ ha appromael solve A. I s desred o eplore whch of hese vecors leads o he bes solon of Eqaon 8, where bes s here defned as he solon ha elds he mnmm RMS resdal angle error defned below. Therefore, he LMM algorhm proceeds as follows: Gven he LOS measremens and assmng he dnamcs of Eqaon 7, consrc he A mar accordng o Eqaon 5 Perform a snglar vale decomposon of A For each of he 7 rgh-snglar vecors ~ of A, consrc accordng o Eqaon 6 and calclae he leas-sqares solon for n Eqaon 8 descrbed below ~ For hs canddae solon 6, calclae he Roo Mean Sqare RMS resdal angle error b sbsng hese nal condons no he second-order dnamcs of Eqaon 7 specfcall he,, and epressons o prodce a predced poson vecor a each measremen me, T r, and evalae he followng epresson: n n cos r r r ˆ 3 8

29 In Eqaon 3, û r s he acal measred LOS a a, and n s he nmber of measremen mes. Ths, hs resdal merc s he RMS angle dfference beween he predced and acal LOS vales. The leas-sqares solon for s calclaed as follows: p p T T q p 3 where ~ ~ ~ p A and 77 ~ 6 q A 6 ~ ~ ~ 3 ~. Noe ha, snce p and q are boh 7 vecors, 4 ~ 5 ~ 6 Eqaon 3 represens a rao of wo scalars. Ths, of he 7 canddae solons evalaed n he above process, he one seleced as he IROD solon s ha prodcng he lowes vale of, whch hs comes closes o sasfng he measremen eqaons, Eqaon 8. Whereas, he lnear mar mehod does no have a rgoros heorecal bass, smlaon has shown s capable of generang reasonable solons n he presence of errors. Whle he ahors concede ha he A mar wll lkel have several snglar vales sgnfcanl dfferen han ero whch lkel do no serve as good canddaes for a solon, he reason for png all 7 snglar vales hrogh he above process s o emphase he aonomos nare of he algorhm: raher han relng on a hman-n-he-loop o evalae wheher each snglar vale of A s close enogh o ero o mer consderaon as a canddae solon, all snglar vales are gven eqal opporn as canddae solons. Fnall, some commen shold be made regardng he nmber of LOS measremens he LMM algorhm reqres. Techncall, an nmber of measremens wll allow he above seps o be followed n deermnng an IROD solon; becase A s a 3n 7 mar, wll alwas have 7 rgh-snglar vecors regardless he vale of n. However, smlaons have shown ha for an adeqae solon, measremens from a leas 3 dfferen mes shold be processed; gven ha each measremen me prodces ndependen measremens, hs elds a nmber of ndependen measremens eqal o he nmber of saes. Addonal measremens shold provde ncremenal mprovemen. 9

30 3.. Mar Reslan Mehod MRM Anoher approach mplemened o solve he ssem of qadrac eqaons was based on he heor of reslan marces, poplared b he work of Macala [6] and oher mahemacans. Two sgnfcan dfferences beween hs and he lnear mar mehod are ha he MRM algorhm solves a sqare ssem of eqaons nmber of measremens eqal o nmber of relave saes and he MRM algorhm s based on rgoros heorecal developmen n he lerare. To prodce a sqare ssem of polnomal eqaons, wo of he hree eqaons generaed a each measremen me Eqaon 4 are seleced. A mehod was mplemened o selec he mos ndependen par of eqaons. Each row of U s defned as a vecor. Then, he norm of he cross prodc of each par of vecors s calclaed. The par of componens assocaed wh he wo vecors wh he larges cross-prodc norm.e. he par ha s mos dfferenl dreced s chosen for nclson n he mar reslan mehod. The specfc mplemenaon of Macala reslans o solve a ssem of n polnomals n n varables was based on [7]. However, a general overvew of he mehod s provded here. The mehod begns b selecng a roo varable ha wll be solved for frs. The mehod s somewha smlar o he lnear mar mehod, n ha he hgher-order combnaons of he remanng nknown varables are reaed as ndependen nknowns and he eqaons are herefore reaed as lnear. Addonal eqaons o solve for hese remanng nknowns are generaed b mlplng he orgnal eqaons b varos polnomal combnaons of he orgnal nknowns, nl a sqare ssem s reached. Ths rases he overall order of he ssem, and can resl n a large ssem of eqaons. However, he generaon of addonal eqaons s carefll desgned so ha he solons for he addonal nknowns aomacall sasf he polnomal consrans among hem.e., no dscard & recombne sep s necessar, as s performed n he lnear mar mehod. The resl s a homogeneos ssem of lnear eqaons for he polnomal eqaons of he remanng nknowns, where he coeffcens are fncons of he roo varable. The esence of a solon reqres ha he mar of coeffcens s snglar, whch can be posed as a generaled egenvale problem. Solvng hs generaled egenvale problem prodces mlple solons for he roo varable, man of whch are nfne. Sbsng each fne vale of he roo varable one a a me no he mar of coeffcens and solvng for he nll vecor of he ssem provdes solons for he remanng nknown varables agan, for each vale of he roo varable. For he solon of s qadrac eqaons for he s nknown nal saes, he reslng generaled egenvale problem s Solon of hs problem sng he sandard precson n MATLAB has been seen o resl n large errors n he reslng solon. Therefore, an alernave mehod was mplemened based on conceps nrodced n [8]. If he nal epoch me s chosen as he me of he frs LOS measremen, hen he followng lnear ransformaon can be sbsed for he nal poson componens:

31 ˆ r r r or r 3 Ths, he hree nknowns,, and, can be lnearl replaced b he sngle nknown r. Then, he mos ndependen par of componens a wo addonal measremen mes are sed o solve for he remanng for nknowns: r and he hree nal veloc componens,, and. The hree eqaons generaed a each measremen are smlar o Eqaon 8, b reformlaed n erms of he new nknowns: D C r D C D C D C E C r E C E C E C E D r E D E D E D 33 Ths mplemenaon s referred o as he separaon-magnde formlaon. The solon of for qadrac eqaons for hese for nknowns resls n a generaled egenvale problem of. Solon of hs problem resls n sgnfcanl less loss of precson han he,584,584 scenaro descrbed above.

32 Some dscsson s n order regardng he nmber of solons elded b he mar reslan mehod. Accordng o Beo s Theorem [9], here are a b solons o a ssem of copled polnomals, where a represens he order of each polnomal and b represens he nmber of polnomals.e. nmber of varables. For for nd -order polnomals, he nmber of solons s hen 6. However, hese solons are n he comple doman,.e. s possble ha each solon ma conss of one or more vales wh magnar pars. There s no known heorem for how man real solons ma es, so he bes ha can be sad s ha he mamm nmber of real solons for a gven scenaro s 6. Unlke he nfne ambg assocaed wh Woffnden s dlemma, here we have a fne ambg o deal wh. Dsambgaon of he poenall mlple real solons s farl sraghforward. Frs, he measremen eqaons reqre ha he relave poson vecor a each measremen me be parallel o he LOS. Ths can resl n solons ha conan relave poson vecors ha are ponng he wrong drecon 8 o opposed o he LOS measremen a one or more measremen mes. For each solon, he propagaed relave poson vecor can be checked, and f pons n he wrong drecon a an measremen me, he solon can be dscarded. A hs pon, here ma sll be mlple real solons ha sasf he measremen eqaons. However, eperence shows ha all b one of he remanng solons are phscall nrealsc,.e. conanng separaon magndes.e. observer-o-rso range vales far beond he doman of applcabl of he second-order solon. Ths, he one realsc solon wold be consdered he wnner for hs algorhm Nonhomogeneos Observer Mehod NOM The fnal IROD algorhm akes advanage of he observabl provded b a nonhomogeneos observer, as descrbed n he prevos secon. The mehod ses daa for he LOS vecor from he observer o he RSO and he relave poson vecor of he observer relave o he reference orb boh epressed n he reference orb s LVLH componens. The mehod assmes ha he observer does no le on he reference orb drng all he measremen mes. If hs condon s me for eample b he observer performng manevers drng he span of he measremens, hen he nmber of manevers, eac seqencng of he manevers whn he measremens, and magnde and drecon of he manevers do no need o be eplcl np no he algorhm. Raher, one smpl needs he observer s poson relave o he reference orb a each measremen me reslng from he manevers; hs s r as represened n Eqaon. As descrbed prevosl, he mehod resls n a lnear ssem of eqaons, A b, for he nknown nal condons, whch can be solved b akng a psedonverse. So, whereas hree ndependen LOS measremens reslng n s measremen eqaons are reqred for A and herefore s psedonverse o be fll rank, here s no pper lm on how man LOS measremens can be ncorporaed. An nmber of measremens beond hree resls n a leassqares solon for. I shold be noed ha n a real scenaro, error wll es n he observer s poson. Becase he NOM mehod does no accon for hs error, ma be falsel arbed o nonhomogeneos moon and hs creae a false sense of observabl. Tha s, f he observer devaes from he defned reference orb de o poson knowledge error, he NOM mehod wll assme hs devaon s de o a manever. Therefore, n order for NOM o be effecve, he manever or oher nonhomogeneos observer acv ms ndce re nonhomogeneos

33 moon ha s greaer han he false nonhomogeneos moon ndced b he aforemenoned errors Praccal Consderaons Praccal sses arse when performng he varos IROD schemes dealed above. Some of hese sses are as follows: Eccenrc n he reference orb: Whle he NOM mehod allows for an ellpcal reference orb, he LMM and MRM mehods assme he reference orb s crclar. For mos scenaros of neres, hs s no epeced o be a problem for he LMM and MRM mehods. Tpcall, relave moon s drven more b he eccenrc dfference beween he chef and dep orbs han b he eccenrc n he chef orb self n fac, s he eccenrc dfference ha cases he relave rajecor o resemble a ellpse n he radal/along-rack plane. Smlaons show ha when he chef and dep orbs boh have small nonero eccenrc, he relave rajecor closel resembles ha for a crclar chef orb e.g. he ellpse. Ths, s epeced ha orb eccenrc wll no be a major sse when applng he LMM and MRM mehods heren. Spaal and emporal regons of applcabl for each mehod: I s epeced ha here are boh an pper and lower lm on he me span of measremens for whch he above mehods wll be effecve. Ths s becase he shorer he me beween frs and las measremen, he less of a look he mehod ges a he orb, whle he longer hs me nerval, he more probabl here s for an propagaon error o bld p beween he dnamc model assmed b hese mehods and real moon n space. Smlarl, here s epeced o be an pper and lower lm on he separaon beween he chef and dep orbs specfcall, beween he reference orb and he orb of he RSO beng observed, n erms of where he above mehods wll be effecve. Ths s becase he closer hese wo orbs are, he more he relave moon beween hem appears lnear. In sch qas-lnear condons, wll lkel be dffcl o deermne he proper magnde.e. scalng of he relave rajecor, even wh mehods desgned o preven Woffnden s dlemma sch as he above mehods. Conversel, he more dfferen he wo orbs are, he more nonlnear he relave moon beween hem wll be. For eample, here cold be a scenaros where a closedform epresson of he relave moon beween wo objecs ma reqre 3 rd -order or hgher erms o accrael represen he moon. In sch cases, he above mehods lkel wold no perform well becase he models assmed n hose mehods do no accrael represen he moon. Ths, n boh he spaal and emporal sense, here s an accepable regon, or swee spo, where he mehods shold perform well. 3.3 Descrpon of Scenaros Before delneang he varos scenaros, s sefl o reerae he followng properes of he IROD algorhms: Each algorhm assmes knowledge of he observer locaon a each measremen me, and n mos cases he reference orb n whose frame he relave solon s epressed s defned based on hs knowledge. 3

34 The algorhms reqre no knowledge of he observed objec orb, shape/geomer, ec., oher han he lne-of-sgh measremens o he objec; an ephemers nformaon of he observed objec e.g., on-board elemer or eernal solons s sed srcl for verfcaon. Two of he algorhms do no ncorporae manevers b he observer, whle one of he algorhms does; all hree algorhms assme he observed objec s no maneverng. There are hree general caegores of scenaros eplored heren: Relave approach o classcal grond-based IOD: Performance of IROD algorhms lng grond sensor daa Close-prom IROD: Performance of IROD algorhms lng space-based sensor daa 3.3. Grond-Based IROD GIROD In Sbsecon 3..3, was descrbed how observabl can be ndced f he observer does no le along he defned reference orb drng all he measremen mes. Ths wll be referred o here as he nonhomogeneos condon. The NOM mehod descrbed prevosl was consrced based on hs condon. Specfcall, he nonhomogeneos condon s ha he observer s moon relave o he reference objec vral chef, when epressed n he LVLH frame of he reference objec, does no obe lnear dnamcs. Ths condon was frs descrbed n he cone of a maneverng observer, snce hs s qe an obvos eample of he condon. I rns o ha an observer acall mees hs condon wheher s maneverng or no, snce s real moon relave o an defned wo-bod reference orb s no lnear. However, becase hs relave moon s of ver small magnde.e. an observer s orb s generall ver close o a neghborng wo-bod reference orb, hs s a qas-homogeneos scenaro. Tha s, he observer s moon relave o he vral chef wold add lle o no observabl.e. he proper scale facor of he moon cold lkel no be deermned wh an accrac. One pe of scenaro ha mees he nonhomogeneos condon qe well s ha of convenonal grond-based rackng, where he observer s a sensor on he Earh. 4

35 To appl IROD o grond-based rackng daa, he measremen eqaons are of he form n Eqaon. Tpcall, daa for an opcal grond sensor consss of rgh ascenson and declnaon of he lne-of-sgh from he sensor o he objec epressed n some famlar coordnae frame, e.g. Earh-Cenered Ineral ECI a each measremen me. In addon, he sensor s locaon on he Earh s known, pcall n erms of lade, longde, and alde. Gven hs nformaon, once a reference objec and orb are defned, he vecor from he sensor o he reference objec vral chef n he reference objec s LVLH frame can be knemacall derved a each measremen me. Ths vecor s hen r n Eqaons and. The NOM mehod can hen be sed o deermne he orb of he observed objec n he reference objec s LVLH frame,.e. he solon. Noe: Tha GIROD conses a relave approach o he classcal grond-based IOD problem Close-Prom IROD CIROD For hese scenaros, varos ses of LOS measremens from a space-based observer o an RSO are chosen, and each of he IROD algorhms wll be sed o process he measremens. Cases are smlaed ha nclde one or more manevers nerspersed beween he measremens, as well as measremen ses nvolvng no manevers. Cases are chosen nvolvng varos me spans from nal o fnal measremen, as well as varng degrees of separaon beween observer and RSO. The prpose here s no onl o eplore he general accrac of he IROD algorhms, b also he lms of he accepable regon boh spaall and emporall descrbed above Delneaon of IROD es cases The varos es cases o be demonsraed n he above wo scenaros are characered b varaon of he followng parameers, as dealed n Table : Wheher s -order NOM or nd -order LMM or MRM relave dnamcs are assmed n he IROD algorhm The choce of observer: eher an observer saelle or a grond sensor The bass for choosng he reference orb: eher he observer ephemers or a vral reference orb 5

36 Table. The Three Possble Delneaons of Tes Case Parameers n he Epermens o Follow Dnamcs order Observer Observed objec Reference objec/orb # of measremens melne s NOM grond sensor RSO vral varable varable s NOM observer RSO observer varable varable saelle saelle nd LMM or MRM observer saelle RSO observer saelle varable varable Regardng he hrd blle, he reference orb ma be consrced n wo was. The frs mehod s b obanng an ephemers for he observer js pror o he frs measremen me, and propagang ha ephemers forward wh wo-bod dnamcs. An ad-hoc mehod of reference orb consrcon s based onl on he LOS measremens and he assmpon ha he observed objec s n a near-geosnchronos objec. Usng he observer s locaon, he nersecon s calclaed of he frs LOS measremen wh a sphere cenered a he cener of he Earh and rads eqal o he geosnchronos sem-major as. The reference orb a he nal epoch s posoned a he locaon of hs nersecon: R R ˆ obs r 34 R T R a Geo 35 ˆ T r T R R R a obs obs obs Geo 36 The posve solon for corresponds o he desred nersecon. The reference orb a he nal epoch s posoned a he locaon of hs nersecon. The veloc of he reference orb a hs epoch s chosen wh magnde eqal o ha of a crclar geosnchronos orb and drecon perpendclar o he nal poson vecor and parallel o he eqaoral plane: 6

37 V T 37 a V Geo V a Geo Kˆ R Kˆ R 38 where Kˆ s he n vecor n he drecon of Earh s polar as. To selec from he wo possble parallel drecons, he veloc s chosen o be generall n he drecon of he second LOS measremen. Ths mehod s eqvalen o selecng he reference orb o be crclar and geosnchronos, wh nclnaon eqal o he lade of he observed objec a he nal epoch,.e. he reference orb s assmed o be a s mamm absole lade a he nal epoch. The resls shown n Secon 4 llsrae he se of IROD echnqes for grond-based rackng scenaros and close-prom scenaros, each as a fncon of he varos parameer choces dealed above and n Table. 3.4 Daa Reqremens for Scenaros, wh Descrpon of Daa Flow Ths secon descrbes he specfc nps reqred for each of he scenaro caegores, he op solon nformaon provded, and he flow of daa hrogh each IROD algorhm from nps o op. Ths srcre s dcaed largel b he delneaon of IROD es cases dealed above GIROD Inp Srcre Followng are he nps reqred for he grond-based IROD scenaros: LOS measremens from grond sensor o observed objec, epressed as rgh ascenson and declnaon n he ECI coordnae frame Tme of each aforemenoned LOS measremen Grond sensor lade, longde, and alde a me of each LOS measremen Reference orb nformaon, consrced from he LOS measremens va Eqaon s

38 3.4.. Daa Flow from Inps o Op Fgre 5 llsraes how he above nps are manplaed for processng o eld an IROD solon. The GIROD envronmen consss of wo man MATLAB scrps: Pre_Process_Daa.m, and Grond_Based_IROD.m, whch are eeced seqenall. These scrps are frher descrbed n he followng paragraphs. Fgre 5. Flow Char of Grond-based IROD Algorhmc Process Pre_Process_Daa.m: The man prposes of Pre_Process_Daa.m are o compe grond sensor LOS rgh ascenson and declnaon measremens of he observed objec n he LVLH frame of he reference orb and o compe he vecor from he sensor o he reference objec vral chef n he reference objec s LVLH frame a each measremen me. The reference orb s derved from he LOS measremens va Eqn s The reference objec s hen propagaed o each of he grond sensor measremen mes sng wo-bod Kepleran dnamcs so ha he grond sensor measremens can be convered from he ECI frame o he LVLH frame. Ths scrp also loads a hgh fdel ephemers for he locaon of he grond sensor n he ECI frame, whch also s convered a each measremen me o he LVLH frame. The op of Pre_Process Daa.m s daa fles n e forma wh he reference orb ECI saes, he LVLH relave poson vecor of he grond sensor a each measremen me, and he LOS vecors epressed n LVLH a each measremen me Grond_Based_IROD.m: Grond_Based_IROD.m loads he aforemenoned daa fles from Pre_Process Daa.m. The IROD solon s hen calclaed sng he Nonhomogeneos Observer Mehod. The specfc op daa s descrbed n a laer sbsecon. 8