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Determination of Motion Extrema in Multi-Satellite Systems

Permanent Link: http://ufdc.ufl.edu/UFE0043286/00001

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Title: Determination of Motion Extrema in Multi-Satellite Systems
Physical Description: 1 online resource (151 p.)
Language: english
Creator: Allgeier, Shawn E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: astrodynamics -- extrema -- formation -- groebner -- multi-body -- satellite
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Spacecraft, or satellite formation flight has been a topic of interest dating back to the Gemini program of the 1960s. Traditionally space missions have been designed around large monolithic assets. Recent interest in low cost, rapid call up mission architectures structured around fractionated systems, small satellites, and constellations has spurred renewed efforts in spacecraft relative motion problems. While such fractionated, or multi-body systems may provide benefits in terms of risk mitigation and cost savings, they introduce new technical challenges in terms of satellite coordination. Characterization of satellite formations is a vital requirement for them to have utility to industry and government entities. Satellite formations introduce challenges in the form of constellation maintenance, inter-satellite communications, and the demand for more sophisticated guidance, navigation, and control systems. At the core of these challenges is the orbital mechanics which govern the resulting motion. New applications of algebraic techniques are applied to the formation flight problem, specifically Groebner basis tools, as a means of determining extrema of certain quantities pertaining to formation flight. Specifically, bounds are calculated for the relative position components, relative speed, relative velocity components, and range rate. The position based metrics are relevant for planning formation geometry, particularly in constellation or Earth observation applications. The velocity metrics are relevant in the design of end game interactions for rendezvous and proximity operations. The range rate of one satellite to another is essential in the design of radio frequency hardware for inter-satellite communications so that the doppler shift can be calculated a priori. Range rate may also have utility in space based surveillance and space situational awareness concerns, such as cross tagging. The results presented constitute a geometric perspective and have utility to mission designers, particularly for missions involving rendezvous and proximity operations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shawn E Allgeier.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fitz-Coy, Norman G.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043286:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043286/00001

Material Information

Title: Determination of Motion Extrema in Multi-Satellite Systems
Physical Description: 1 online resource (151 p.)
Language: english
Creator: Allgeier, Shawn E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: astrodynamics -- extrema -- formation -- groebner -- multi-body -- satellite
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Spacecraft, or satellite formation flight has been a topic of interest dating back to the Gemini program of the 1960s. Traditionally space missions have been designed around large monolithic assets. Recent interest in low cost, rapid call up mission architectures structured around fractionated systems, small satellites, and constellations has spurred renewed efforts in spacecraft relative motion problems. While such fractionated, or multi-body systems may provide benefits in terms of risk mitigation and cost savings, they introduce new technical challenges in terms of satellite coordination. Characterization of satellite formations is a vital requirement for them to have utility to industry and government entities. Satellite formations introduce challenges in the form of constellation maintenance, inter-satellite communications, and the demand for more sophisticated guidance, navigation, and control systems. At the core of these challenges is the orbital mechanics which govern the resulting motion. New applications of algebraic techniques are applied to the formation flight problem, specifically Groebner basis tools, as a means of determining extrema of certain quantities pertaining to formation flight. Specifically, bounds are calculated for the relative position components, relative speed, relative velocity components, and range rate. The position based metrics are relevant for planning formation geometry, particularly in constellation or Earth observation applications. The velocity metrics are relevant in the design of end game interactions for rendezvous and proximity operations. The range rate of one satellite to another is essential in the design of radio frequency hardware for inter-satellite communications so that the doppler shift can be calculated a priori. Range rate may also have utility in space based surveillance and space situational awareness concerns, such as cross tagging. The results presented constitute a geometric perspective and have utility to mission designers, particularly for missions involving rendezvous and proximity operations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shawn E Allgeier.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fitz-Coy, Norman G.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043286:00001


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DETERMINATIONOFMOTIONEXTREMAINMULTI-SATELLITESYSTEMS By SHAWNE.ALLGEIER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011ShawnE.Allgeier 2

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InlovingmemoryofFrederickCharlesAllgeier,forquietlyleadingbyexample,and GeorgeMaxwellIsenhardt,whomIneverhadtheprivilegeofmeeting,andyetknowso intimately 3

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ACKNOWLEDGMENTS Fewendeavorsaretrulysolitary,proceedingwithoutsignicantassistanceand interactionswithothers.Iwouldliketoexpressmygratitudetoasubsetoftheindividuals whomhavebeeninstrumentalinfacilitatingmyeortstobringthisresearchtofruition.I amfortunatetohavehadsuchnepeopletoprevailuponforassistance. IwouldliketothankDr.NormanFitz-Coyforhisguidance,navigation,and occasionalcontroloverthepastsixyears.Perhapsunintentionally,heconvincedme tostayingraduateschoolfarlongerthanIhadplanned.UnderhistutelageIhavebeen exposedtoaplethoraofexcitingtopics,andhadtheopportunitytoexploremyinterests. Hisdemandforrigorandexactitudehastaughtmewhatisrequiredtoproducequality work.Iamaghastinrecallingmyignoranceduringourinitialmeetings.Icanonly hopethatshouldIndmyselfinapositiontomentorsomeoneinalikemanner,thatI demonstratecomparablepatience. Ifoundthetaskofselectingasupervisorycommitteetobestraightforward.Isimply chosethefacultymemberswhomhaveenhancedmygraduateexperiencethemost,beit throughcourseinstruction,researchcollaboration,orpersonalinteractions.Drs.Scott Banks,PrabirBarooah,andJaniseMcNairrepresentneexamplesofthetypeofmentors whichUniversityofFloridafacultyshouldallaspiretoemulate.Dr.ScottErwinatthe AirForceResearchLaboratoryhasalsogenerouslydonatedagreatdealoftimetohelp fosterthisresearch,andIamparticularlygratefulforbothhiscontributionsandthe timingthereof.Althoughnotamemberofmycommittee,Dr.DavidHahnwastherst facultymemberthatIobserveddemonstratingagenuineconcernfortheplightofgraduate students.Duringmyrstsemesterinthedepartmentheassistedinsmoothingoutsomeof thetransientsignals.Hislackofhesitationinhelpingmethroughthemachinerestoredmy faithinthedepartmentandpreventedaprematuredeparture.Ihaveneverforgottenthat actofbenevolence. 4

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Myfather,mother,andsisterhavealsodisplayedcopiouslevelsofpatiencewhile waitingforthecompletionofthiswork.Iwouldliketopubliclythankthemfortheir endlesssupport.Thisjourneywouldnothavebeenpossiblewithoutthem,orBarbara JacksonAllgeierandMaryLouiseIsenhardt.Thisdocumentisasmuchatestamentto yourpatienceasitistomine.IwillbeevergratefulfortheopportunitiesthatIhavebeen abletotakeadvantageof. Ihavehadtheprivilegeofknowingmanyhardworkinganddedicatedengineersinthe spacesystemsgroup.AndrewTatschhasbeenarolemodelandanexampletoaspireto.I amgratefulforthelatenightsspenttogetherpunchingkeyboards.Andrew,Ai-Ai,Bungo, Dante,Fred,Jimmy,Josue,Katie,Kunal,Sharan,Sharath,Shawn,Takashi,andVivek haveallcontributedtoauniqueandraucousride.Icannotimagine,norwishitwereany otherway.Insolidarityalways;Ilookforwardtoourpathsintersectingthroughoutour careers. Wemakethebestdecisionswecanwiththeinformationthatisavailabletous.I wouldliketothankJonLawrence,LesterPinera,andGuneetaSinghforhelpingmeto makesomeofmine.WehaveallcomealongwayandIwouldnotbewritingthisifnot forthesespecialindividuals.IamgratefulforthesupportprovidedbymyfriendsMorgan Baldwin,KyleDeMars,andJennySanchezduringthenalpush;sometimestoughlove istherightmedicine.Itmaybefairtosaythatitisthejourneyandnotthedestination thatmatters.Evenso,Iamgladthatthisjourneyisover. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................8 LISTOFFIGURES....................................9 LISTOFSYMBOLS....................................11 ABSTRACT........................................13 CHAPTER 1INTRODUCTION..................................15 1.1Background...................................15 1.2Motivation....................................16 2KEPLERIANDYNAMICS.............................20 2.1Gravitation...................................20 2.2CentralBodywithSphericalSymmetry....................20 2.3CentralBodywithAzimuthalSymmetry...................21 2.4TwoBodyMotion...............................23 3RELATIVEMOTION................................27 3.1PhysicalDescription..............................28 3.2LiteratureReview................................29 4FORMATIONFLIGHT...............................34 4.1EquationsofMotion..............................34 4.2FormationFlightMetrics............................36 4.2.1AlgebraicTheory............................37 4.2.1.1Polynomialalgebra......................40 4.2.1.2Grobnerbasis.........................43 5POSITIONMETRICS................................47 5.1LocalVerticalPositionMetric.........................51 5.1.1Grobnerbasisfordeputyeccentricanomaly..............52 5.1.2Backsolvingforchieftrueanomaly...................55 5.2LocalHorizontalPositionMetric.......................57 5.2.1Grobnerbasisfordeputyeccentricanomaly..............58 5.2.2Backsolvingforchieftrueanomaly...................59 5.3CrossTrackPositionMetric..........................60 5.3.1Crosstrackpositionmeanvalue....................62 6

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5.3.2ComparisonwithClohessy-Wiltshireformulation...........63 5.4PositionMetricSimulations..........................65 5.5PositionManifoldDescription.........................81 6VELOCITYMETRICS...............................82 6.1RelativeSpeedMetric.............................82 6.1.1Extremalequations...........................83 6.1.2Grobnerbasisforrelativespeed....................84 6.1.3Backsolvingforchieftrueanomaly...................85 6.2LocalVerticalVelocityMetric.........................87 6.2.1Grobnerbasisfordeputytrueanomaly................88 6.2.2Backsolvingforchieftrueanomaly...................89 6.3LocalHorizontalVelocityMetric........................89 6.3.1Grobnerbasisfordeputytrueanomaly................90 6.3.2Backsolvingforchieftrueanomaly...................91 6.4CrossTrackVelocityMetric..........................92 6.5RangeRateMetric...............................92 6.5.1PolynomialFormulation........................95 6.5.2Case1..................................98 6.5.3Case2..................................99 6.5.4Grobnerbasis..............................102 6.6VelocityMetricSimulations..........................104 6.7EvaluationofMotionExtremaMethodology.................121 7STATESPACEGEOMETRY............................123 7.1StateSpaceEvolution.............................123 7.2LineofSightObstruction...........................127 8CONCLUSION....................................129 APPENDIX AREFERENCEFRAMES...............................132 BASTRODYNAMICRELATIONS..........................134 CTRIGONOMETRICPOLYNOMIALFORCIRCULARRANGERATE....136 REFERENCES.......................................141 BIOGRAPHICALSKETCH................................150 7

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LISTOFTABLES Table page 2-1ConicSections....................................23 3-1RelativeMotionScenarios..............................28 5-1OrbitalParametersforPositionSimulations:1Commensurate........66 5-2OrbitalParametersforPositionSimulationsIncommensurate..........74 5-3GlobalExtremaforPositionSimulationsIncommensurate...........74 5-4OrbitalParametersforPositionSimulationsDisparateSizes..........79 6-1GlobalExtremaforVelocitySimulationsIncommensurate...........111 6-2OrbitalParametersforRangeRateSimulations:3CommensurateandCircular113 6-3IridiumExample...................................119 6-4GlobalExtremaforIridiumSimulation.......................120 7-1StateSpaceVariables.................................123 8

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LISTOFFIGURES Figure page 1-1DART.........................................18 1-2OrbitalExpress....................................18 1-3CanX45........................................19 1-4XSS-11.........................................19 2-1GravitationalForces.................................20 2-2TwoBodyGeometry.................................21 2-3PerifocalReferenceFrame..............................24 2-4EarthCenteredInertialReferenceFrame......................24 3-1RelativeMotionGeometry..............................28 4-1LocalVerticalLocalHorizontalReferenceFrame..................35 5-1RelativePositionManifolds.............................48 5-2RangeUpperandLowerBoundsIncommensurateSystem...........49 5-3RangeUpperandLowerBounds:3CommensurateSystem..........50 5-4CrossTrackSeparation................................61 5-5EarthCenteredInertialTrajectories.........................67 5-6PositionUnivariatePolynomials:1Commensurate...............68 5-7LocalVerticalPositionManifold:1Commensurate...............69 5-8LocalHorizontalPositionManifold:1Commensurate.............70 5-9CrossTrackPositionManifold:1Commensurate................71 5-10RelativePositionTrajectory:1Commensurate.................72 5-11LocalVerticalPositionManifoldIncommensurateSystem............75 5-12LocalHorizontalPositionManifoldIncommensurateSystem..........76 5-13CrossTrackPositionManifoldIncommensurateSystem.............77 5-14RelativePositionTrajectoryIncommensurateSystem..............78 5-15DisparateOrbitGeometry..............................79 9

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5-16RelativePositionTrajectoryDisparateOrbitSizes................80 6-1CircularCoplanarGeometry.............................99 6-2RelativeSpeedUnivariatePolynomial........................104 6-3RelativeSpeedManifold:1Commensurate...................105 6-4VelocityUnivariatePolynomials:1Commensurate...............106 6-5LocalVerticalVelocityExtrema:1Commensurate...............107 6-6LocalHorizontalVelocityExtrema:1Commensurate..............108 6-7CrossTrackVelocityExtrema:1Commensurate................109 6-8RelativeVelocityTrajectory:1Commensurate.................110 6-9TimeHistoryofRelativeSpeedIncommensurateSystem............112 6-10RangeRateCircularCoplanarExtrema:3Commensurate...........114 6-11RangeRateCircularCoplanarGeometry:3Commensurate..........115 6-12EarthCenteredInertialTrajectoriesforRangeRateSimulation.........116 6-13RangeRateUnivariatePolynomial.........................117 6-14RangeRateCircularManifold............................118 6-15IridiumSimulationGeometry............................120 7-1IntegralCurveofEccentricAnomalyStateSpace:2Commensurate.....125 7-2GradientofRangeFunction.............................126 7-3PointsofMinimumDistanceBetweenEarthandLineofSightVector......128 A-1OrbitalElements...................................132 A-2OrbitalPlaneGeometryProgradeExample...................133 B-1EccentricAnomalyGeometry............................135 10

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LISTOFABBREVIATIONS,SYMBOLS,ORNOMENCLATURE AOSAcquisitionofSignal DCMDirectionCosineMatrix ECIEarthCenteredInertial GEOGeosynchronousOrbit.Orbitswitharadiusofapproximately42,467 km. HEOHighEccentricityOrbit LEOLowEarthOrbit.Orbitswitharadiusofapproximately6,600to 9,000km. LOSLineofSight LVLHLocalvertical,localhorizontal MEOMidEarthOrbit.Orbitswitharadiusofapproximateley24,000km. NASANationalAeronauticsandSpaceAdministration PQWPerifocalreferenceframe RSOResidentSpaceObject.Anobjectinaknownreferenceorbitwhich couldbeaspacecraft,debris,ornaturallyoccurringbody. S/CSpacecraft 11

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v anelementofalinearvectorspace.Specicallyanalgebraic structure p L ; F ; q where L R 3 isanadditivegroup, F R isa commutativeeld,theexternalcompositionlaw isanactionof F on L ,andtheinternallaw )]TJ/F15 11.9552 Tf 13.005 0 Td [(in L isrelatedto bymixedleftand rightdistributivelaws. ^ v aunitvector scalarinnerproductoftwovectors crossproductoperatoroftwovectorsin R 3 a v acolumnmatrixdirectioncosinematrixrepresentingthe componentsofvector v resolvedintoframe F a a pq b o aquantitydescribingbody b ,coordinatizedinframe F a ,abouta pointofinterest O r thelocaltimederivativeofthevector r .Thisderivativeaccountsfor thechangeofthecomponentsofthevectorintheparticularframe thatitisresolvedin. 9 r theinertialderivativeofthevector r ,includingthemotionofthe referenceframebasisvectors. ~ theskewsymmetricmatrixof .If 1 2 3 T ,then ~ 0 3 2 3 0 1 2 1 0 C anelementaryDCM.ThethreeelementaryDCMsare: C p 1 ; q 100 0cos sin 0 sin cos C p 2 ; q cos 0 sin 010 sin 0cos C p 3 ; q cos sin 0 sin cos 0 001 R ba thedirectioncosinematrixDCM R P R 3 3 ,transformingavector coordinatizedin F a into F b ,sothat R ij ^ b i ^ a j 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DETERMINATIONOFMOTIONEXTREMAINMULTI-SATELLITESYSTEMS By ShawnE.Allgeier December2011 Chair:NormanG.Fitz-Coy Major:AerospaceEngineering Spacecraft,orsatelliteformationighthasbeenatopicofinterestdatingbacktothe Geminiprogramofthe1960s.Traditionallyspacemissionshavebeendesignedaround largemonolithicassets.Recentinterestinlowcost,rapidcallupmissionarchitectures structuredaroundfractionatedsystems,smallsatellites,andconstellationshasspurred renewedeortsinspacecraftrelativemotionproblems.Whilesuchfractionated,or multi-bodysystemsmayprovidebenetsintermsofriskmitigationandcostsavings,they introducenewtechnicalchallengesintermsofsatellitecoordination.Characterization ofsatelliteformationsisavitalrequirementforthemtohaveutilitytoindustryand governmententities. Satelliteformationsintroducechallengesintheformofconstellationmaintenance, inter-satellitecommunications,andthedemandformoresophisticatedguidance, navigation,andcontrolsystems.Atthecoreofthesechallengesistheorbitalmechanics whichgoverntheresultingmotion.Newapplicationsofalgebraictechniquesareappliedto theformationightproblem,specicallyGrobnerbasistools,asameansofdetermining extremaofcertainquantitiespertainingtoformationight. Specically,boundsarecalculatedfortherelativepositioncomponents,relativespeed, relativevelocitycomponents,andrangerate.Thepositionbasedmetricsarerelevant forplanningformationgeometry,particularlyinconstellationorEarthobservation applications.Thevelocitymetricsarerelevantinthedesignofendgameinteractions 13

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forrendezvousandproximityoperations.Therangerateofonesatellitetoanotheris essentialinthedesignofradiofrequencyhardwareforinter-satellitecommunicationsso thatthedopplershiftcanbecalculatedapriori.Rangeratemayalsohaveutilityinspace basedsurveillanceandspacesituationalawarenessconcerns,suchascrosstagging.The resultspresentedconstituteageometricperspectiveandhaveutilitytomissiondesigners, particularlyformissionsinvolvingrendezvousandproximityoperations. 14

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CHAPTER1 INTRODUCTION Astrodynamicsisanoldandwelltravelledeldofresearch.Originallymotivated byadesiretounderstandthecosmosandcelestialobjects,ithasinthelasthalfcentury becomeanactivearenaofappliedresearch.Manymathematicaldevelopmentsintheareas ofvectormechanics,dynamics,estimation,numericalanalysis,anddierentialequations havebeenmotivatedbyproblemsofinterestintheeldofastrodynamics.Abriefsurvey ofpresentdayinterestsandareasofactivityisdescribed,followedbyadiscussionof currentmotivatingfactors.Thetermssatelliteandspacecraftareusedsynonymously throughout. 1.1Background Rigidbodyspacecraftdynamicsareoftencategorizedintotwogroups,translational androtational.Thetranslationaldescriptionisoftenreferredtoasastrodynamicsandthe rotational,orattitudedynamicsareoftentreatedseparately,althoughinrealitythetwo arecoupled.Theresearchpresentedherewillfocusentirelyonastrodynamics,treating satellitesaspointmasses.Activeareasinastrodynamicsincludeorbitdetermination, prediction,andcorrection.Optimalcontrolistypicallyusedasabasisfordesigning trajectoriestotransferaspacecraftorsatellitefromoneorbittoanother,andinsome missionsfromanorbitaroundonegravitationalbodytoanother.Whilethemajority ofspacemissionsaredesignedaroundasinglespacecraftarchitecture,therepresently existsaninterestinthecapabilitiesofmissionsutilizingmultiplespacecraft.Such missionsgeneratenewresearchquestionsintheareasofsatelliteservicese.g.,inspection, rendezvous,anddocking,whicharereferredtoasrendezvousandproximityoperations RPOs.Inaddition,constellationmaintenanceandspacebasedobservationofassetsare notionswhichariseoutofthedynamicsofmultiplerigidbodies. 15

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1.2Motivation Aninterestinspacemissionsbasedontherelativeposition,orientation,and/or dynamicsofmultiplespacecrafthasexistedsincetheGeminiprogram[1].Afterthe completionoftheApolloprogram,interestwanedinrelativemotionresearchasthe UnitedStatesspaceprogramfocusedheavilyonmissionarchitectureswhichhavebeen centeredonlargemonolithicspacecraftwithamultitudeofcapabilities,e.g.,theSpace TransportationSystemshuttleplatform[2].Recentinterestinlow-cost,rapidcall-up missionarchitecturesstructuredaroundfractionatedsystems[3],smallsatellitesi.e., pico-,nano-,micro-satellites,andconstellationshasspurredrenewedeortsintherelative motionproblem. Spaceassetsaretraditionallyengineeredaroundacomplexmonolithicbuswith amyriadofcapabilities.Thelongdevelopmenttimes-15yearsandhighcosts [4]associatedwiththeacquisitionofthesetraditionalsystemshasproducedarisk aversephilosophy[5]thatbuildsredundancyintoeveryaspectofthespacecraft,further increasingthedevelopmenttimeandcost.Consequently,newertechnologicaladvancesare typicallyavoidedinfavoroftechnologiespossessingightheritage,furthersupportinga deciencyinideasandinnovationswithinthespaceindustry.Onemechanismforaltering thisaspectofthespaceindustryistherecentpushtoutilizesmall"pico-,nano-,and micro-classsatellitestocomplementthefunctionalityofthetraditional,largersatellites. Smallsatellites,orsmallsats,areanenablingtechnologyforaddressingthecurrent U.S.NationalSpacePolicyneedsoflower-cost,more-responsivespacesystems,asis evidencedbygovernmentprogramssuchasOperationalResponsiveSpaceORS[6], theDefenseAdvancedResearchProjectsAgency'sDARPASystemF6Program[3], andnumerousrequestsforsmallsatellitesystemsand/ortechnologiese.g.,NRO's ColonyII[7].TherecentlaunchofTACSAT-3furtherhighlightstheDepartmentof Defense'sDoDinterestinsmallsatellitesandthepotentialutilitythesesatellitesbring toDoDmissions[8].Thenotionofdecomposinglargemonolithicspaceassetsintosmaller 16

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fractionatedassets,whereanumberofsmallsatellitemodulesclustertogethertocreatea larger,virtualsatelliteisgainingacceptancewithinthespacecommunity. SmallsatellitescanbeusedtoconductlowEarthorbitLEObasedmissions involvingcommunicationsnetworking,remotesensing,andcanserveastestbedsfor on-orbitvalidationofemergingtechnologies.Furthermoresmallsatellitesmayhaveutility asremotesensorsforspacesituationalawarenessSSAinamannersimilartotheMSX satellite[9,10].ThepotentialutilityofsmallsatstotheDoDe.g.,ORSandcommercial interestsisyettobedetermined.Recentdiscussions[6,7]havefocusedonexploringwhat functionalitycanberealizedfromsmallsats. Smallsatsarenottheonlyplatformforfractionatedormulti-agentspaceassets. MostnotableareconstellationssuchastheGlobalPositionSystemGPS,Defense MeteorologicalSatelliteProgramDMSP,andtheNationalOceanicandAtmospheric Agency'sNOAAweathersatellites.Anumberoftraditionallysizedsatellitemissions haverecentlybeendesignedaroundsatelliteinteractions.NASA'sDemonstrationof AutonomousRendezvousTechnologyDART,showninFigure1-1,in2005wasdesigned toincludeanautonomousrendezvousbetweentwosatellites.OrbitalExpress,aDARPA missionshowninFigure1-2,wasintendedtodemonstrateautonomousservicingof spacecraft.TheUniversityofToronto'sCanX45mission,depictedinFigure1-3,isa multi-bodysatellitemissionusingtwosmallsatsinformationight.In2005theAirForce launchedXSS-11,showninFigure1-4.ThesatelliteperformedRPOswithanexpended rocketbodyandcertaindecommissionedspaceassets. Spacedebrisconstitutesyetanothertypeofmulti-bodysystem.Currentlythereare tensofthousandsofobjectson-orbitwhichareclassiedasdebris[11].TherecentChinese A-satoperation[12,13],Iridium/Cosmoscollision[14],andU.S.[15]operationproduced tensofthousandsofpiecesofdebrisinlowEarthorbit,whichposesathreattomany spaceassets,bothmannedandunmanned.Currentlythereisnoactivemechanismfor debrisremovalandthenumberofobjectsinspacewillincreaseuntilsuchacapabilityis 17

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Figure1-1.DART:ImageCourtesyofNASA http://mediaarchive.ksc.nasa.gov/detail.cfm?mediaid=23642 Figure1-2.OrbitalExpress:ImageCourtesyofDARPA implemented[16].Consequentlyitisincreasinglyimportanttounderstandhowobjects inorbitmoverelativetooneanother.Tothatend,thisresearchfocusesonthenatural, oruncontrolledmotionofasatelliterelativetoanotherobjectbeitdebrisoranother spacecraft,takeninthiscontexttobeasecondsatellite. 18

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Figure1-3.CanX45: http://www.utias-sfl.net/nanosatellites/CanX45/ Figure1-4.XSS-11:ImageCourtesyofUnitedStatesAirForce http: //www.kirtland.af.mil/shared/media/document/AFD-070404-108.pdf 19

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CHAPTER2 KEPLERIANDYNAMICS Forcompleteness,anecessarilybriefdiscussionofKepleriandynamicsisnow presented.Onlyessentialmaterialrelevanttothedissertationresearchiscovered. 2.1Gravitation Newton'slawofgravitationstatesthatthegravitationalforceactingonabodyof mass m duetothepresenceofabodyofmass M locatedadistance r awayasshownin Figure2-1A,isgivenby F g GMm r 2 ^ r ; {1 where r pointsfrom M to m and G 6 : 669 10 11 m 3 kgs 2 istheuniversalgravitational constant.When M representsabodyofdistributedpointmassestheforceon m isthe sumoftheforcesduetoeachinnitesimalpointmass dM insidethevolume V of M ,as illustratedinFigure2-1B.Theresultantforceis F g dF V Gm r 2 dM ^ r dM dM: {2 APointMasses BDistributedMassBody Figure2-1.GravitationalForces 2.2CentralBodywithSphericalSymmetry Whenthemassdistributionofbody M issphericallysymmetricitcanbeshown [17]thatEquation2{2reducestoFigure2{1andthevector r pointsfromthecenter ofmassof M to m .Toanalyzethemotionofmass m relativetomass M asaresultof 20

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thegravitationalforcedescribedbyEquation2{1,thedierentialequationsofmotion foreachmassareconstructedusingNewton'ssecondlaw.Letmass M belocatedata position R 1 relativetoaninertialpoint O andmass m islocatedataposition R 2 as showninFigure2-2.Thenthemotionofthetwomassesisgovernedby M : R 1 GMm || R 1 R 2 || 3 p R 1 R 2 q {3 m : R 2 GMm || R 2 R 1 || 3 p R 2 R 1 q : {4 Figure2-2.TwoBodyGeometry DividingEquations2{3and2{4by M and m respectivelyandsubtractingthemyields thefamiliartwobodymotiondierentialequation: : r r 3 r {5 where G p M )]TJ/F21 11.9552 Tf 11.216 0 Td [(m q GM and r R 2 R 1 isafreevector.ForEarth C 3 : 986 10 5 km 3 s 2 [18]. 2.3CentralBodywithAzimuthalSymmetry Sincethegravitationalforcevectoreldhaszerocurl,itcanbewrittenasthe gradientofascalarpotentialfunction[19] V as g p r q rV B V B r T : {6 21

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Whenthemassdistributionof M isnolongersphericallysymmetricthegravitational forceisperturbed.ForlowEarthorbitstheperturbationsareduetooblatenesseectsi.e., abulgeatitsequator.Inthiscasethegravitationalforceperunitmass g thatabody m experiencesduetobody M isgivenbyamoregeneralexpressionandthescalarpotential functionisgivenby[20]: V r 1 8 k 2 J k r eq r k P k p cos q ; where r eq isthemeanequatorialradiusoftheEarthand P k istheLegendrepolynomial oforder k ,therstsixofwhichareseeninEquation2{7,withhigherorderscalculated usingRodriques'formula[21,22].Thecoecients J k aretheunnormalizedzonalharmonic coecients,ofwhichtherstthreevaluesoftheEGM-96model[20,23]aregivenin Equation2{8. P 0 p x q 1 P 1 p x q x P 2 p x q 1 2 p 3 x 2 1 q P 3 p x q 1 2 p 5 x 3 3 x q P 4 p x q 1 8 p 35 x 4 30 x 2 )]TJ/F15 11.9552 Tf 11.76 0 Td [(3 q P 5 p x q 1 5 p 63 x 5 70 x 3 )]TJ/F15 11.9552 Tf 11.76 0 Td [(15 x q {7 J 2 0 : 00108263 J 3 0 : 00000254 J 4 0 : 00000161 {8 WhentheoblatenessoftheEarthisneglectedthecoecients J k aresetto0andthe potentialfunctionissimply V r ; andEquation2{6reducestothesphericallysymmetricresult g p r q r 3 r ; {9 asinEquation2{5.HigherdelitymodelsoftheEarthhavebeendevelopedtoaccount foramassdistributionwhichdoesnotpossessazimuthalsymmetry.Inthesemodels, therotationoftheEarthmustbeaccountedforbyfurtheraugmentingthegravitational 22

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potentialfunction.Thesetesseralandsectoralmodelsaccountforthemassdistribution usinganexpansioninsphericalharmonicsandarenotaddressedinthisresearch. 2.4TwoBodyMotion MotiondescribedbyEquation2{5isreferredtoasKeplerianmotion,orthetwo bodyproblem;itiscoveredextensivelyinliterature[18,20,23{26].Thescalarquantity r || r || isgovernedby r p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e cos f ; {10 where e and f aretheeccentricityandtrueanomaly,respectively, p a p 1 e 2 q isthe orbitalparameterorsemi-latusrectum,and a isthesemi-majoraxis.Equation2{10is thepolarequationdescribingconicsections.Theeccentricitydeterminesthefourtypesof possibleorbits,aslistedinTable2-1. Table2-1.ConicSections e 0circular 0 e 1elliptical e 1parabolic e 1hyperbolic Inmanyapplicationsofastrodynamics,itistheclosedorbitswhichareofprimary interest,particularlyforsatelliteswhichorbitthesamecentralbodyforthedurationof theirfunctionallifetime.Torelatethissolutiontothetimedomain,thetrueanomalyis relatedtotheeccentricanomaly E forellipticorbitsthroughGauss'equation: tan f 2 d 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e 1 e tan E 2 ; {11 andtheeccentricanomalyisrelatedtotimethroughKepler'sequation: n p t q E e sin E: {12 where n and arethemeanmotionandtimeoflastperiapsispassageminimumorbital radius,respectively.AppendixBprovidessomeadditionalcommentsregardingthe 23

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relationbetween f and E .Theorbitalperiodisgivenby T 2 n 2 d a 3 : Equations2{10,2{11,and2{12describetheorbitalradiusofbody m intheorbitalplane, orperifocalplane.Theperifocalplane F pqw isdepictedinFigure2-3,withbasisvectors ^ p ; ^ q ; ^ w .The ^ p vectorpointsfromthecentralbody,orfocus,inthedirectionofperiapsis. The ^ w vectorpointsinthedirectionofspecicorbitalangularmomentum h r v whichisaninertiallyxeddirectionforKeplerianmotion.Thevector ^ q ^ w ^ p completes theorthonormal,dextralset. Figure2-3.PerifocalReferenceFrame Figure2-4.EarthCenteredInertialReferenceFrame 24

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AnotherusefulreferenceframeistheEarthcenteredinertialECIframe F eci .The originoftheECIframeislocatedatthecenteroftheEarth,andthebasisvectorsare ^ e 1 ; ^ e 2 ; ^ e 3 .The ^ e 3 vectorpointsinthedirectionoftheEarth'sspinaxis.The ^ e 1 isparallel tothelineofintersectionbetweentheequatorialandeclipticplanes,pointingfromthe EarthtotheSunatthevernalequinox.Thevector ^ e 2 ^ e 3 ^ e 1 completestheorthogonal, dextralset,asdepictedinFigure2-4.AnalternatedepictionisprovidedinAppendixA. Thecomponentsofvectorsresolvedin F eci canbetransformedto F pqw viathe directioncosinematrixDCM R c p q c p q c p i q s p q s p q c p q s p q)]TJ/F21 11.9552 Tf 18.967 0 Td [(s p q c p i q c p q s p q s p i q s p q c p q c p q c p i q s p q s p q s p q)]TJ/F21 11.9552 Tf 18.967 0 Td [(c p q c p q c p i q c p q s p i q s p i q s p q s p i q c p q c p i q ; {13 where c pq cos pq and s pq sin pq ,and ;i;! aretherightascensionofascending node,inclination,andargumentofperiapsis,respectively.Equation2{13istheresult ofasymmetric3-1-3rotationsequencethroughtheEulerangles ;i;! .FiguresA-1and A-2inAppendixAdepictthegeometry.Both F eci and F pqw arebydenitioninertial referenceframesforthisresearch,andtherefore 9 9 i 9 0.Thisconditionisaresult oftheKeplerianmotion.Theevolutionofatwobodysystemmaybesemi-analytically obtainedthroughtheuseofEquations2{10,2{11,2{12,and2{13,orthroughtheuseof Lagrangiancoecients[20,27{29],whichrelatethepresentpositionandvelocityvectors tothepositionandvelocityvectorsataninitialepoch: r v FG 9 F 9 G r 0 v 0 ; 25

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wheretheLagrangecoecientscanbewrittenintermsofeithertrueoreccentric anomaly: F 1 a r 0 p 1 cos E q r p p r 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(cos f G a 0 ? r 1 cos E s)]TJ/F21 11.9552 Tf 18.447 0 Td [(r 0 d a sin E rr 0 ? p sin f 9 F ? a rr 0 sin E ? r 0 p r 0 p 1 cos f q ? p sin f s 9 G 1 a r p 1 cos E q 1 r 0 p r 1 cos f s ; and 0 r v ? .Numericalintegrationisalsousedforpropagation,usingthedierential Equation2{5oralternatively,throughthedierentialequationsfortheanglesand orbitalradius, 9 f h r 2 9 E na r n 1 e cos E 9 r d p e sin f ane sin E 1 e cos E : Thetwobodysolutionpresentedhereformsthebasisforrelativemotiondynamics,which arediscussedinChapter3.Forallrelativemotiondiscussedinthisdissertation,both satellitesareassumedtobeinKeplerianorbits. 26

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CHAPTER3 RELATIVEMOTION Aninterestinspacemissionsbasedontherelativeposition,orientation,and/or dynamicsofmultiplespacecraftwasinitiatedwithintheUnitedStatesbeginningwith theGeminiprogram.RendezvousandproximityoperationsRPOsconsistofactions oroperationsinvolvingtwoormorespacecraft.Suchoperationsincludeinspection, interception,rendezvous,docking,andon-orbitservicing.Rendezvoushasplayedacentral roleintheGemini,Apollo,Hubble,andInternationalSpaceStationmissionswithinlow EarthorbitLEO. Insection1.2someeventsandaspectsofthespaceenvironmentwerediscussedas motivationalreasonsforexaminingrelativemotiondynamics.Theincreasingresident spaceobjectRSOpopulation,multi-agentmissionarchitectures,anddesireformorerisk tolerantsystemsarebroadmotivatingfactors.Someofthespecicaspectsoftheseitems arenowdiscussedasameansoffurthermotivatingrelativemotiondynamicsresearch. GlobalNavigationSatelliteSystemsGNSSsuchastheGlobalPositioningSystem GPS[30]andtheRussianGLONASSconstellationhavedemonstratedsomeofthe utilityattainablefromspaceassetsbasedonmultiplespacecraftwhichisnotpossible withasinglesatellite.SyntheticApertureRadarSARutilizingmultiplesatellitesisan emergingtechnologyunderinvestigationforsatelliteapplications.Stereoimagingofthe Earthcanalsobeperformedwithmultiplesatellites. Spaceassetdevelopmentprogramshavetypicallyrequiredmillionsofdollarsand adecadeormoretobringaconcepttooperationalstatusonorbit.Anadditional motivatingfactorformulti-bodysystemsisthatthefunctionalitymaybescaledover aperiodoftimeassatellitesareaddedtoaformation.Distributingfunctionalityover multiplesatellitespermitsriskreductioninthatthelossofonesatellitedoesnotproduce missionfailure,asinthecaseofGPS.AllnotionsofRPOs,constellations,andcontrolled formationsarerootedinrelativemotiondynamics. 27

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3.1PhysicalDescription Thescenariounderinvestigationisthemotionofonebodyinorbitrelativetoa referenceorbit,whichmayormaynotbeoccupied.Themathematicsapplyequallytoa multitudeofscenarios;theterminologyforwhichislistedinTable3-1. Table3-1.RelativeMotionScenarios ReferenceorbitSecondorbit chiefdeputy targetinterceptor disabledspacecraftservicingspacecraft residentspaceobjectRSOinspectingsatellite Henceforththetermschief"anddeputy"willbeusedtodenotethereferenceand secondaryorbits,respectively.Thesub/superscriptsc"andd"areusedtodenote quantitiesofthechiefanddeputyrespectively.Thepositionofthedeputyrelativetothe chiefis r r d r c ; {1 whichisafreevector,asdepictedinFigure3-1. Figure3-1.RelativeMotionGeometry Thedierentialequationgoverningtheevolutionofthisvectorquantitycanbe derivedbyapplicationofEquation2{5forboththechiefanddeputy.Takingthe 28

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dierenceofthetwobodyequationforthetwoorbitsyields : r g p r c )]TJ/F21 11.9552 Tf 11.759 0 Td [( r q g p r c q ; {2 where g p r q isgivenbyEquation2{9.Equation3{2representsthreecouplednonlinear secondorderdierentialequations.ManyapproximationsofthesolutionofEquation3{2 involveaTaylorseriesexpansionofthedierentialequationabout r c oftheform : r B g p r q B r r c r )]TJ/F21 11.9552 Tf 11.76 0 Td [(:::: {3 Closedformexpressionsexist,howevertheyutilizeindepedentvariablesotherthantime, andoftenrequiresolvingatranscendentalequationnamelyKepler'sequation.Onesuch solution[31]is LVLH r x y z p cf c T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(sf c T 21 q a d p cos E d e d q)-222(p cf c T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(sf c T 22 q b d sin E d r c p sf c T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(cf c T 21 q a d p cos E d e d q)-222(p sf c T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(cf c T 22 q b d sin E d T 31 a d p cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(T 32 b d sin E d : {4 Thiscolumnmatrixconsistsoftherelativepositionvectorexpressedintherotatinglocal vertical,localhorizontalLVLHreferenceframe,denotedas LVLH r .Theindependent variablesinEquation3{4arethechieftrueanomaly f c andthedeputyeccentric anomaly E d ,whichinteractwiththedirectioncosinematrixelements T ij toyieldthe cartesianrelativepositioncomponents x;y;z .TheDCM T transformsthecomponents ofavectorinthedeputyperifocalframetothechiefperifocalframe.TheLVLHreference frameisdescribedinsection4.1.Thisanalyticsolutionwillformthebasisforthree formationightmetricspresentedinChapter4. 3.2LiteratureReview Researchinrelativemotiondynamicscanbegroupedintothreecategories.Therst categoryisthemodelingofthedynamics,withoutanycontrolapplied.Thisrepresentsthe bulkoftheliteraturepublishedfrom1960to2000.Theequationsofmotiondescribingthe 29

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motionofonesatelliterelativetoanotherarenonlineardierentialequationsforwhich analyticsolutionsinthetimedomainhavenotyetbeenfound.Manyapproacheshave beentakeninformulatingapproximatesolutionstothisnonlinearsystem.Thesecond categoryconsistsoftheapplicationofcontroltheoryforpurposesofeectinginterception, rendezvous,orinspection.Thethirdcategoryconsistsofeortstocharacterizeordesign spacecraftformationsorconstellations.Thiscategoryhasemergedoverthepastdecade. Eortstodesignconstellationsormaintainformationgeometryinthepresenceof disturbancesand/ornon-Kepleriandynamicsrepresentadeparturefromthetraditional approachofmanipulatingdierentialequations. ThemostwellknownrelativemotionformulationisthatofClohessyandWiltshire CW[32],orHill's[33]equations,whichconsistsofalineartimeinvariantmodel, obtainedfromtherstorderTaylorseriesapproximationoftheKepleriandynamicsand assumptionsofsmallseparationdistancesbetweenthespacecraftrelativetotheirdistance fromthecentralbodycenterofmass,andtherestrictionofthechiefsatellitetocircular orbits.TheCWformulationhasservedasthebasisformanyanalysesandguidance systems[1,34].Unfortunatelyitpredictsconditionsforboundedseparationwhichare notphysical,andfailstoproperlycapturethetruemotion.Theunderlyingassumptions restricttheutilityofthismodel. London[35]formulatedasecondorderapproximationusingsuccessiveapproximations. DeVries[36]recastthelinearizationusingtrueanomalyastheindependentvariableand transformedthedependentvariablestoobtainexpressionsforaslightlyeccentricorbit. AnthonyandSasaki[37]alsoextendedtheCWformulationtoincludesecondorderterms inEquation3{3alongwithsmalleccentricitiesinthereferenceorbit. Shulmanetal.[38]presentedaformulationforterminalrendezvousinareference orbitofarbitraryeccentricityforsmallseparationdistances,concludingthathigher orderexpansionswerenecessaryforpracticalapplications.EulerandShulman[39]then extendedthesecondorderexpansionstoincludeellipticalorbits.Manyformulations 30

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utilizedtransformationsoftheindependentvariableordependentstatevariablessuchas theTschauner-Hempelequations[40,41],asameansofobtainingapproximatesolutions tothenonlineardierentialequationsgoverningrelativemotiondynamics.Tschauner appliedpreviouswork[42]toarendezvousapplicationinellipticalorbitswiththrust limitedactuators[40].Schneider,Prussing,andTimin[43]formulatedanalgorithm intendedforastronautstoeectrendezvous.Schneideretal.[44]producedasimplemodel formoderatelyeccentricreferenceorbits e 0 : 05andcomparedtheaccuracyofthree rotatingcartesianreferenceframes. Lancaster[45]formulatedanexactsolutiontoEquation3{1usingtheLagrange coecientsF&G[20],howeverthismodelrequiressolvingatranscendentalequation ateachtimestepandthus,itisnotamenabletoon-orbitimplementation.Nacozy andSzebehely[46]publishedaformulationforrelativemotionfromthestandpointof maintainingnumericalaccuracy.Theindependentvariableusedistrueanomaly,andwhile noapproximationsareused,thealgorithmisnotintendedforon-orbitimplementation. BerreenandCrisp[47]analyzedtheseparationdynamicsofaparticleejectedfroma satellite.Curvilinearcoordinateswereusedtoobtainexpressionsfreefromrestrictionson theseparationdistance.Inafollowuppaper[48],Berreenetal.highlightedtheneedfor simpliedsolutionsofcoplanarsatellitesinellipticorbits.Jezewski[49]exploredoptimal impulsebasedtrajectoriesbasedonaCWmodelforrendezvous.Carterexploredfuel optimalrendezvousinnoncircularorbits[50,51],inadditiontoprovidinganexcellent sourceofreferenceonthissubject[52],comparingmanyofthevariousformulation techniques,includingquadraticdragperturbations[53]. MorerecentdevelopmentsincludeMelton's[54]timeexplicitsolutioninvolvingrst ordertermsusingaseriesexpansionofthestatetransitionmatrixforellipticorbits. YamanakaandAnkersen[55]providedananalyticsolutiontotheTschauner-Hempel equations,whichconstitutearstorderlinearizationaboutanellipticchieforbit,using transformationsoftheindependentanddependentvariables.Wiesel[56]appliedFloquet 31

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theorytonearlycircularorbitstoaccountforzonalharmonicsinEarth'sgravitational eld.KarlgaardandLutzealsoextendedthelinearizationtoincludesecondorderterms insphericalcoordinates[57]aswellasaformulationinthecontextofLambert'sproblem [58].Alfriendetal.[59]haveaccountedforsomeoblatenesseectsbyincluding J 2 terms andnoncircularreferenceorbits,aswellasproposedamethodforcomparingtheutilityof relativemotionmodels[60].Additionalnotablerelativemotionmodelsinclude[61{66]. Mucheorthasbeenspentonproximityoperationssuchasformationight.The CWformlationhasservedasthebasisforanumberofformationightanalyses,namely YehandSparks[67],Saboletal.[68],thePRISMAmission[69],Sabatinietal.[70],Reali andPalmerini[71],andBandoetal.[72].Formationightanalyseshavebeenderived usingapproachesnotbasedontheCWmodel.SchweighartandSedwick[73]formulated aconstantcoecientlineardierentialequationtocapture J 2 eectsforformationight scenarios.Vaddietal.[74]focusedoncorrectinginitialconditionsforformationsby combiningthenonlineardynamicsforcircularorbitswiththelinearizeddynamicsfor eccentricorbitsinatimeexplicitmodel.Additionalformationightanalysesinclude Schaubetal.[75{77],Inalhanetal.[78],GurlandKasdin[79],Laneetal.[80],Guibout [81],Sengupta[65],Chungetal.[82],Yanetal.[83],andTragresser[84]. Rendezvousisanotherkeyproximityoperationforsomespacemissions[85,86]. Rendezvousspecicresearchincludes[1,34,87{96].Somerendezvousalgorithms andcontrollersarebasedonCWmodelse.g.,Shitbataetal.[95]whileothersutilize nonlineardynamicmodels[97].Inparallelmanycontrolschemesforeectingrendezvous weredevised,typicallybasedonimpulsivethrusting,withminimumfuelasatopicofhigh interest[34,43,49,50,98,99]. Thesurveypresentedherehighlightsthemostsalientresearchtodate.Manyofthe citationsstraddletheboundarybetweenthethreefocusareas,presentingadvancesin morethanonetopic.Itisclearfromthenumberofformulationspresentedoverftyyears ofresearchthattherearemanyapproachestoexaminingtherelativemotionproblem. 32

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Asstatedpreviouslythereisatradeobetweencomputationalcostandaccuracywith correspondingexchangesbetweentheoreticalandpracticalapplicability.Despitethe factthattheequationsofmotioncanbeexpressedcompactly,arelativemotionmodel appropriateforonemissionisnotnecessarilysuitableforothers. Someeortshavefocusedongeometricmethods[59,100{102]insteadofdierential equations.AlfriendandYan[60]providedarigorouscomparisonofsomeofthemore moderndynamicsformulations.VeryrecentlytheworkbyKholshevnikovandVassiliev [103,104],andGurlandKholshevnikof[31]haveprovidedboundsforthemetric ofrelativeseparationdistancebetweentwosatellitesinboundorbitsthatinvolveno approximationsoftheKepleriandynamicsorrestrictionsontheorbitofeithersatellite,as wellasanapproximationfortheaverageofthisdistance.Theseresultsarerepresentative ofthepushtodevelopmoreaccurateanalysistoolsforformationcharacterization.The resultsobtainedareusefulinmissionplanningforformationightmissionsbecause theyprovidedesignconstraintsforspacecrafthardware.Thisgeometricapproachfor characterizingsatelliteformationsthroughextremalvaluesofquantitiesofinterest formsthebasisforthecontributionspresentedhere.Propertiesoftherelativemotionof satellitesinKeplerianorbitsareobtainedthroughthedeterminationoftheglobalextrema ofrelevantmetricsofinterest. 33

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CHAPTER4 FORMATIONFLIGHT InadditiontotheRPOsdescribedinChapter3,itisdesirabletocharacterize aformationofsatelliteswhichmaybeuncontrolled,byplacingboundsonphysical descriptorsoftheformation.Suchacharacterizationisusefulinplanningrendezvousand proximityoperations,specifyingighthardwarerequirements,andunderstandingmission constraints.Someofthephysicalquantitiesofinterestarenowpresented. 4.1EquationsofMotion Relativemotiondynamicsareoftencoordinatizedintherotating,localvertical-local horizontalLVLHreferenceframe,where ^ i x pointsinthezenithofthechief, ^ i z pointsin thedirectionoforbitalangularmomentumofthechief,and ^ i y ^ i z ^ i x completesthe orthonormal,dextralset,asshowninFigure4-1.ThedirectioncosinematrixDCM whichtransformsthecomponentsofavectorfromtheperifocalframeofthedeputyorbit PQW d totheperifocalframeofthechieforbitPQW c ,is T R p c ;i c ;! c q R T p d ;i d ;! d q ; {1 where R wasgiveninEquation2{13.Thepositionofthedeputy r d ,resolvedinto componentsalongitsperifocalbasisvectorsindicatedasPQW d canbeparameterizedin termsofitseccentricanomaly E d as PQW d r d a d p cos E d e d q b d sin E d 0 ; {2 where a d ;e d ;b d arethesemi-majoraxis,eccentricity,andsemi-minoraxisofthedeputy, respectively.TheDCMwhichtransformsthecomponentsofavectorrepresentedinthe 34

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Figure4-1.LocalVerticalLocalHorizontalReferenceFrame chiefperifocalframetotheLVLHframeis C p f c q cos f c sin f c 0 sin f c cos f c 0 001 ; {3 where f c isthetrueanomalyofthechief.Thepositionofthechief,expressedintheLVLH referenceframeis LVLH r c r c 0 0 ; {4 where r c a c p 1 e 2 c q 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e c cos f c : {5 35

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UsingEquations4{1through4{5,therelativeseparationofthedeputywithrespecttothe chief r r d r c canbeexpressedintheLVLHframeas LVLH r x y z C p f c q T a d p cos E d e d q b d sin E d 0 r c 0 0 ; {6 whichreducesto x y z a d p T 11 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 21 sin f c qp cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d p T 12 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 sin f c q sin E d r c a d p T 21 cos f c T 11 sin f c qp cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d p T 22 cos f c T 12 sin f c q sin E d a d T 31 p cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d T 32 sin E d : {7 Inthisformulationthepositionofthedeputyrelativetothechiefisexpressedinthe LVLHframe,andasolutiontotherelativemotionproblemhasbeenobtainedwithout solvingdierentialequations[31,102].Thetrueanomalyofthechief,andtheeccentric anomalyofthedeputyserveastheindependentvariables.Theperiodicpropertiesof satelliteformationsdependsontheperiodsofthechiefanddeputy.Twoorbitsaresaidto becommensurateiftheratiooftheirorbitalperiodsisarationalnumber.Thatisif, T c T d P Q ; thenthesystemiscommensurateandafteratime T equaltotheleastcommonmultiple of T c and T d ,thesatelliteswillreturntotheirsamerelativepositionsandvelocities. Sincetheperiodisafunctionofsemi-majoraxis,itisthesizesoftheorbitsandnottheir shapes,whichgovernthecommensurabilityproperty. 4.2FormationFlightMetrics Theseparationdistance,orrange || r || ,wasformulatedinreferences[31,103,104] asafunctionoftheeccentricanomalies E c and E d ofthechiefanddeputy.Thedistance functionwasthendierentiatedwithrespectto E c and E d toobtaintwoextremal equations.Thetwotrigonometricequationswerethenconvertedtoanalgebraicsystemof 36

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threemultivariatepolynomials.AreducedGrobnerbasiswascalculatedforthissystem. Oneofthebasispolynomialsisaunivariatepolynomialin E d asaresultoftheterm orderingofthepolynomials.Therootsofthisunivariatepolynomialcanthenbefound throughnumericalmethodsinordertoobtainthevaluesof E d whichsatisfytheextremal equations.Theserootswerethenusedtoobtainthecorrespondingvaluesof E c .In thismanner,theextremalvaluesof || r || canbeobtained,allowingonetoboundthe separationdistanceoftwosatellitesinellipticalorbits.Theutilityofthismetricincludes 1.Estimatingrequiredsignalstrengthforinter-satellitecommunications. 2.Determiningtheclosestapproachbetweensatelliteconjunctions. 3.Predictingmaximumseparationdistanceforspacebasedobservationofassets. Asimilarapproachwillbeappliedtoothermetricsofinterestasameansofdeveloping furtherinsighttorelativemotiondynamics.Adiscussionofalgebraicsettheory,followed bypolynomialalgebra,andthentheGrobnerbasisisnowprovidedtoclarifythecontext ofeldsandpolynomialsoveraringandservestodenetheterminology. 4.2.1AlgebraicTheory Asetisdenedasacollectionofelementswithnonrepeatingmemberse.g,the complexnumbers C ,thereals R ,ortheintegers Z .Giventwosets, X and Y ,amapor functionisarulewhichassignstoeachelement x P X oneandonlyoneelement y P Y Suchamap f isdenotedas f : X Y: Theset X isreferredtoasthedomain D f of f andtheset Y isreferredtoasthe codomainof f .Anelement y P Y whichisassignedtoan x P X through f isknown astheimageof x under f .Theset t y P Y | f p x q y u isknownastherange R f of f .Withtheidenticationofadomainandcodomain,amap canbedeterminedasinjective,surjective,orbijective[105].Let a;b;c beelementsofan 37

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arbitraryset A .Analgebraiccompositionlaw[105]isaruleanoperatordenotedby l whichassignstoanyorderedpair p a;b q anelement c ,writtenas a l b c .Aninternal compositionlaw k actsuponelementsofthesameset k : A A A; whileanexternalcompositionlawactsuponelementsofdierentsets k : A B A; whereinthiscontext denotestheCartesianproductoftwosets,yieldinganordered pairwithoneelementfromtherstsetandoneelementfromthesecondset.Anexample ofaninternalcompositionlawisscalaradditionofrealnumbers.Anexampleofan externalcompositionlawisthemultiplicationofavectorandascalar.Aset A forwhich analgebraiccompositionlawisgivenisknownasanalgebraicstructure,denotedas p A; l q .Acompositionlawmayexhibitsomeofthefollowingproperties: associative: a l p b l c qp a l b q l c ,forall a;b;c P A commutative: a l b b l a ,forall a;b P A leftdistributive: a 4 p b l c qp a 4 b q l p a 4 c q rightdistributive: p a l b q 4 c p a 4 c q l p b 4 c q Inthelasttwoproperties,thenotionofdistributivityappliesonlytosetsforwhich twocompositionlawshavebeendened.If 4 isbothleftandrightdistributiveover l ,itissaidthat 4 isdistributiveover l .Analgebraicstructurewithasingleinternal associativecompositionlawisdenedasasemigroup.Aunitelement e P A foraninternal compositionlawobeys e l a a and a l e a forall a P A; andifitexists,itisunique[105].Anexampleisthenumber1forthecompositionlawof multiplicationoverthecomplexnubmers C ,orthenumber0forthecompositionlawof 38

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additionover C .Aregularelement r P A obeysthestatements r l a r l b a b a l r b l r a b: Ifthestructure p A; l q hasaunitelement e ,then a P A hasaninverseifthereexistsa unique a 1 P A suchthat a 1 l a e a l a 1 e: Analgebraicstructure p G; l q wheretheinternalcompositionlawisassociative,aunit elementexists,andeveryelementpossessesaninverseisdenedasagroup.Ifthe compositionlawiscommutative,then G isanAbeliangroup. Ofinteresttothepresentdiscussionisthenotionofaring,whichisanalgebraic structurecontainingtwointernalcompositionlaws p R ; l ; q .Therstcompositionlaw l iscommutativeandthesecondlaw isassociativeanddistributiveover l .Inmost connotationstherstlawistermedadditionandisdenotedby+"andthesecondlawis termedmultiplication.Furthermore,aring R satisestheaxioms 1.azeroelement0existssuchthat0 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a a )]TJ/F15 11.9552 Tf 11.759 0 Td [(0 a forall a P R 2. a )-222(p b )]TJ/F21 11.9552 Tf 11.759 0 Td [(c qp a )]TJ/F21 11.9552 Tf 11.759 0 Td [(b q)]TJ/F21 11.9552 Tf 18.968 0 Td [(c forall a;b;c P R 3.forevery a P R thereexistsanelement a P R suchthat a )-222(p a qp a q)]TJ/F21 11.9552 Tf 18.967 0 Td [(a 0 4. a )]TJ/F21 11.9552 Tf 11.759 0 Td [(b b )]TJ/F21 11.9552 Tf 11.759 0 Td [(a forall a;b P R 5. a p bc qp ab q c forall a;b;c P R 6. a p b )]TJ/F21 11.9552 Tf 11.759 0 Td [(c q ab )]TJ/F21 11.9552 Tf 11.759 0 Td [(ac and p a )]TJ/F21 11.9552 Tf 11.759 0 Td [(b q c ac )]TJ/F21 11.9552 Tf 11.759 0 Td [(bc forall a;b;c P R If ab ba forall a;b P R thentheringiscommutative,andifthereisaunitelement formultiplicationthen R isaringwithaunit.Aeld F isacommutativeringinwhich everyelementhasamultiplicativeinverse.Anideal I isasubsetofacommutativering R forwhich I isnotempty. I isclosedundersubtraction,sothat a )-222(p b qP I forall a;b P I I isclosedundermultiplicationbyanyelementof R .Stateexplicitly,if a P I then ra P I forall r P R 39

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Idealsplayanimportantroleinthesolutionofalgebraicsystems,connectingasystemof polynomialswiththeGrobnerbasis. 4.2.1.1Polynomialalgebra Polynomialsconsistofmonomialsandcoecientsselectedfromaring K .Thering fromwhichcoecientsareselectedmayalsobeaeld.Amonomialin n indeterminatesis aproductoftheform[106] x 1 1 x 2 2 :::x n n ; wherethe i arenon-negativenumbersin N .Theorderoftheindeterminatesisreferred toasthetermorderingdenotedby .Thetotaldegreeofamonomialisthesum 1 )-281()]TJ/F21 11.9552 Tf 42.025 0 Td [( n .When n 1thepolynomialisunivariate;when n 2thepolynomial ismultivariate.Acompactnotationwhichisoftenusedformonomialsistodenethe n-tuple p 1 ; 2 ;:::; n q andthenwrite x x 1 1 x 2 2 :::x n n : Then | | 1 )-239()]TJ/F21 11.9552 Tf 40.509 0 Td [( n denotesthetotaldegreeofamonomial.Apolynomial f isthen denedasasumofnitetermsconsistingofacoecientandmonomial 1 : f a x ;a P K : Thesetofallpolynomialsin x 1 :::x n withcoecientsin K isdenotedas K r x 1 ;:::;x n s Thetotaldegreeofapolynomial f isthemaximum | | suchthatitscoecient a is nonzero.Adivisionalgorithmforunivariatepolynomialsstatesthatforevery f P K r x s thereexisttwopolynomials q;r P K r x s suchthat f qg )]TJ/F21 11.9552 Tf 11.759 0 Td [(r 1 Thisnotationismoreakintothatofanindexedset,inthat isanindexbutnotan integer. 40

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withdeg p r q deg g ,where q isthequotientand r istheremainder.Formultivariate polynomialsthedivisionalgorithmcanbegeneralizedwiththeconceptofamonomial ordering.Let Z n 0 denotethesubsetof Z containing N and0.Atotalorderingmeansthat foreverypairofmonomials x and x oneofthefollowingthreeconditionswillhold: x x or x x or x x : Amonomialorderingon K r x 1 :::x n s isanyrelation onthesetofmonomials x P Z n 0 satisfying 1. isatotalorderingon Z n 0 2.if and P Z n 0 ,then )]TJ/F21 11.9552 Tf 11.759 0 Td [( )]TJ/F21 11.9552 Tf 11.759 0 Td [( 3. isawell-orderingon Z n 0 ,meaningthateverynonemptysubsetof Z n 0 hasa smallestelementunder Thestipulationsabovepermittheconstructionofmanytypesofmonomialorderings.Two prevalentorderingswhichhavewidespreadutilityarethelexicographicallexanddegree reverselexicographicalgrevlex[106].Inthelexicographicalordering,giventwon-tuples ; therelation lex ifinthedierence P Z n theleftmostnon-zeroentryis positive.Inthedegreereverselexicographicalordering, | | n i 1 i grevlex | | n i 1 i or, if | || | ; thenin P Z n therightmostentryisnegative. Onceamonomialorderinghasbeenselected,apolynomial f a x canbe characterizedintermsofits multidegree:themultidegreeof f ismultideg f =max P Z n 0 : a 0. leadingcoecient:theleadingcoecientof f islc f = a multideg p f q P K leadingmonomialof f islm f = x multideg p f q leadingtermof f isLT f =lc f lm f Dividingamultivariatepolynomial f byasecondpolynomial g proceedsmuchinthesame waythatitdoesfortheunivariatecase,inthat g ismultipliedbyapolynomial q and 41

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subtractedfrom f soastocanceltheleadingtermof f ,yielding f gq )]TJ/F21 11.9552 Tf 11.759 0 Td [(r: Intermsofnotation,apolynomial f maybedividedbyanotherpolynomial f 1 toobtain thepolynomial h ,writtenas f f 1 h; andreadas f reducesto h modulo f 1 ".Inordertogeneralizethedivisionalgorithmto multivariatepolynomials,amonomialorderingmustbeselected.Intheunivariatecasea monomialorderingbasedondegreeisimplicitlychosen. Givenacollection F ,consistingof s polynomials F t f 1 ;:::;f s u ,multiplereductions of f arepossible,dependingontheorderofthe f i dividedinto f .Theresultof f F h isnotuniquelydetermined[107],i.e., k F f F h; where k;h P K r x 1 ;:::;x n s .Thedivisionalgorithmformultivariatepolynomialspermitsa polynomial f P K r x 1 ;x 2 ;:::;x n s tobewrittenintheform[106] f a 1 f 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(:::a s f s )]TJ/F21 11.9552 Tf 11.759 0 Td [(r; {8 where a i ;r P K r x 1 ;:::;x n s andeither r 0or r isalinearcombinationofmonomials, noneofwhichisdivisiblebyanyLT f 1 q ;:::; LT f s .Thisformistheresultofthe reduction f F r .If a i f i 0thenmultideg f q multideg a i f i .Given s polynomials, apolynomial f canbewrittenintheformofEquation4{8withthecoecients a i determinedusingthedivisionalgorithm.Thedivisionalgorithmiseectedbycancelling theleadingtermof f viasubtractingamultipleofeach f i .Theleadingterm,andhence thecoecients a i andremainder r aredependentonthemonomialorderingchosen,aswell 42

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astheorderofthes-tuple p f 1 ;:::;f s q .Asanexample,consider f x 2 y )]TJ/F21 11.9552 Tf 12.16 0 Td [(xy 2 )]TJ/F15 11.9552 Tf 12.16 0 Td [(3 x )]TJ/F21 11.9552 Tf 12.16 0 Td [(y with F t f 1 ;f 2 ut xy )]TJ/F15 11.9552 Tf 11.759 0 Td [(1 ;y 2 )]TJ/F15 11.9552 Tf 11.759 0 Td [(1 u usinglexordering.Then f F 2 x and f p x )]TJ/F21 11.9552 Tf 11.759 0 Td [(y q f 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(0 f 2 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 x: Iftheorderofthe2-tupleisreversedsothat F t f 1 ;f 2 ut y 2 )]TJ/F15 11.9552 Tf 12.395 0 Td [(1 ;xy )]TJ/F15 11.9552 Tf 12.395 0 Td [(1 u thenthe reductionyields f F x )]TJ/F21 11.9552 Tf 11.759 0 Td [(y and f xf 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(xf 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x )]TJ/F21 11.9552 Tf 11.759 0 Td [(y: Regardlessofthemonomialorderingors-tupleordering,multivariatepolynomialdivision willalwaysterminateinanitenumberofoperationsasshownin[107]andanalgorithm isprovidedonpage63ofCox[106]. 4.2.1.2Grobnerbasis AGrobnerbasis G isasetofpolynomialswiththespecialpropertyamongothers thatthereductionofapolynomialby G isunique[107].TheGrobnerbasishasmany applicationsinsystemstheory[108,109]andisdiscussedinfurtherdetailintheliterature [106,107,110{113].Itschiefrelevancetothepresentresearchisasatoolforsolvingane varieties.Ananevariety V isthesetofpointswhichsimultaneouslysolve s multivariate polynomialsin n indeterminates,or V p f 1 ;:::;f 2 qtp z 1 ;:::;z n qP R n | f i p z 1 ;:::;z n q 0 u for i 1 ;:::;s: Thedenitionofanidealonpage39canbespeciedtopolynomials.Asubset I K r x 1 ; ;x n s isanidealintheringofpolynomialsif 1. I containsazeroelement. 2.if f;g P I ,then f )]TJ/F21 11.9552 Tf 11.76 0 Td [(g P I 3.if f P I and h P K r x 1 ; ;x n s ,then hf P I Givenasetofpolynomials t f 1 ;:::;f s u then f 1 ;:::;f s # s i 1 h i f k : h 1 ;:::;h s P K r x 1 ;:::;x n s + 43

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iscalledtheidealgeneratedby t f 1 ;:::;f s u andisanidealin K r x 1 ;:::;x n s .TheHilbert basistheorem[106]statesthateverynitedimensionalidealcanbewrittenasinthis form,usinganitenumberofbasiselements.Furthermore,if f 1 ; ;f s and g 1 ;:::;g t arebasesofthesameidealin K r x 1 ;:::;x n s sothat f 1 ;:::;f s g 1 ;:::;g t ,then V p f 1 ;:::;f s q V p g 1 ;:::;g t q [106].Thesolutionofananevarietyisgovernedbyan ideal,andcanbeexpressedintermsofdierentbases,orsetsofpolynomials,sincemore thanonesetofequationsmayhavethesamepointsassolutions.Let V K n with n P N beananevariety.Thentheset I p V qt f P K r x 1 ;:::;x n s : f p a 1 ;:::;a n q 0 ; @p a 1 ;:::;a n qP V u isanideal,asubsetof K r x 1 ;:::;x n s ,andiscalledtheidealof V .Theidealofavarietyis relatedtoanidealgeneratedbyasetofpolynomialsthroughtherelation f 1 ;:::;f s I p V p f 1 ;:::;f s qq ; whereequalitycan,butneednotalwaysoccur[106].Thesolutionsofsomeanevarieties canbeexpressedasanexplicitparameterization,involvingrationalfunctionsforsome problems.AmorepowerfultoolistochangethepolynomialbasistoaGrobnerbasis consistingofpolynomialswhicharemembersofthesameidealastheoriginalpolynomials. TherstalgorithmforcomputingaGrobnerbasiswasputforthbyBuchberger[107]. TheprocessforconstructingaGrobnerbasisistoutilizepolynomialdivisionalongwith theconceptofS-polynomials.TheS-polynomialoftwopolynomials f 1 ;f 2 istheremainder whenbothpolynomialsaremultipliedbymonomialfactorssothatthedierenceofthe resultingproductsforcestheleadingtermofbothexpressionstocancel.Buchberger's algorithm[107]consistsofthefollowingstepsforconstructingaGrobnerbasis G ofaset ofpolynomials F : 1.Set G F 2.Startwith f 1 ;f 2 P F andcomputeS-polynoimal p f 1 ;f 2 q 44

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aReduceS-polynomial p f 1 ;f 2 q to h withrespectto G ,i.e.,S-polynomial p f 1 ;f 2 q G h bif h 0repeatforthenextpair cif h 0,add h totheset G andrepeat 3.ContinueuntilallS-polynomialscanbereducedto0 Inthecaseofalinearanevariety,whereall i 1,forall f i P F ,theBuchberger algorithmisequivalenttoGaussianeliminationofasystemoflinearequations.Inthe caseofasetofunivariatepolynomials,thealgorithmisequivalenttotheEuclidean algorithm,whichprovidesthegreatestcommondivisoroftwopolynomials.TheGrobner basisofanidealwhoseanevarietycontainsanitenumberofsolutionswillcontain aunivariatepolynomialinthelowestrankedindeterminate,baseduponthemonomial ordering[107].Thisunivariatepolynomialcanbesolvedforitsrootstoobtainthevalues ofoneofthevariablesinthesolutionspaceoftheanevariety.Thesevaluescanbe usedwiththeremainingGrobnerbasispolynomialstoobtainthevaluesoftheremaining n 1indeterminates.Whilethelexorderingistypicallythemostdesirablemonomial orderingduetothesemi-triangularformoftheresultingbasisequations,ittendstobe themostcomputationallyintensive.Thegradedreverselexorderingiscomputationally cheaper,althoughitisnotasamenabletosolvingvarieties.Itispossibleinsomecases toconvertaGrobnerbasisinonemonomialorderingtoanother[114].Theringalgebra topicsandGrobnerbasistheorydiscussedwillnowbeappliedtothedeterminationof motionextremaintheformationightproblem. Metricsrelevanttoformationightwillbepresentedandtheirextremalequations arederived.Theextremalequationsarethencastinmultivariatepolynomialformusing achangeofvariables.Thesolutionoftheresultinganevarietyisfacilitatedusinga Grobnerbasis.Eightnewmetricsareintroducedinthisresearch.Therstsetofmetrics describethepositionofthedeputyrelativetothechief,expressedintheLVLHreference frame.Theradial,alongtrack,andcrosstrackfunctionsderivedinEquation4{7are analyzedandtheirextremaaredetermined.Nexttherelativespeed || d dt r || isformulated andamethodologyfordeterminingitsextremaispresented.Thevelocityofthedeputy 45

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relativetothechief,expressedintheLVLHreferenceframeisthenanalyzed.Finally,the rangeratefunction d dt || r || isderived.Rangerateextremafortwospecialcases,namely circularcoplanarandcircularnon-coplanararepresented.Numericalsimulationsare providedinsections5.4and6.6. 46

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CHAPTER5 POSITIONMETRICS Equation4{7providedananalyticsolutionfortherelativepositionintheLVLH frame.Itisreproducedhere: x y z a d p T 11 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 21 sin f c qp cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d p T 12 cos f c )]TJ/F21 11.9552 Tf 11.76 0 Td [(T 22 sin f c q sin E d r c a d p T 21 cos f c T 11 sin f c qp cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d p T 22 cos f c T 12 sin f c q sin E d a d T 31 p cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d T 32 sin E d : {1 Thethreefunctionswhichprovidethecomponentsof LVLH r x : R 2 R ; where p f c ;E d q x y : R 2 R ; where p f c ;E d q y z : R R ; where E d z eachgeneratea2-surface,examplesofwhichareshowninFigure5-1.Ananalysisofthese threefunctionsandtheirextremaisnowdiscussed. Thecombinedlocusofpointsin F lvlh isshowninFigure5-2foranincommensurate system.ThistoruslikesurfacewaspublishedbyGurletal.[31]andrepresentsall possiblepointsofthestatespacegeneratedbythesurfacesinFigure5-1.Although nottrulyparametrizableasatorus,thisrelativepositionmanifoldwillbereferredto henceforthassuchforbrevity.Asthesystemevolvesthroughthestatespace p f c ;E d q the relativepositionvectorwillwanderoverthesurfaceofthismanifold.Acommensurate systemwillexhibitatrajectoryintheLVLHframe,whichisaclosedcurvein R 3 ,as showninFigure5-3.InFigures5-2and5-3chiefsatelliteoccupiestheorigin,with instantaneousinertialvelocityinthe ^ i x ; ^ i y plane.Theverticalaxisintheguresrepresents thezenithdirectionofthechief. 47

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ARadialSeparation BAlongTrackSeparation CCrossTrackSeparation Figure5-1.RelativePositionManifolds 48

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Figure5-2.RangeUpperandLowerBoundsIncommensurateSystem 49

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Figure5-3.RangeUpperandLowerBounds:3CommensurateSystem Inordertocharacterizethetorusingreaterdetail,itisrecognizedthattheboundsfor therangeprovidedbyGurletal.[31]representtwospherescenteredonthechief.The sphericalupperboundenclosesthetorus,andthelowerboundingsphererepresentsthe closestapproachbetweenthesatellites.Sinceincommensuratesystemswillrealizeevery pointinthestatespace,themanifoldwillcontacttheoutersphereatsomepoint.The tworangeboundsareincludedinFigures5-2and5-3,whichinvolveachiefinLEOand adeputyingeosynchronousGEOandmidEarthorbitMEO,respectively.Sincethe outersphereprovidesanupperboundon r b x 2 )]TJ/F21 11.9552 Tf 11.76 0 Td [(y 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(z 2 ,itwouldbebenecial tobeabletoobtainmorestringentboundsonthemanifold,throughtheboundingof thepositioncomponents t x;y;z u ,whichamountstodeterminingboundsforthe surfacesdepictedinFigure5-1.Suchacharacterizationprovidesinformationregardingthe 50

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directionallocationofthedeputyrelativetothechiefanditsexcursionsin F lvlh ,which hasutilitytoobservationeortsandrendezvousandproximityoperations.Amethodology fordeterminingsuchextremaisnowpresented,beginningwiththeradialcomponentof thelocalvertical,localhorizontalframe. 5.1LocalVerticalPositionMetric UsingEquation5{1,theextremaof x : R 2 R where p f c ;E d q x aresought.Dierentiating x withrespecttoitsindependentvariablesyields B x B f c a d p cos E d e d qp T 21 cos f c T 11 sin f c q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d sin E d p T 22 cos f c T 12 sin f c q r 1 c 0 B x B E d a d p T 11 cos f c )]TJ/F21 11.9552 Tf 11.76 0 Td [(T 21 sin f c q sin E d )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d cos E d p T 12 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 sin f c q 0 : where r 1 c B r c B f c r 2 c p c e c sin f c .Thecommonsolutionofthetwoextremalequationsis facilitatedbyuseofaGrobnerbasis,providedthattheycanbeexpressedasasystem ofmultivariatepolynomials.Therstextremalequationcontainsdenominatorswith trigonometricexpressions.Multipyingthisequationby p 1 )]TJ/F21 11.9552 Tf 11.157 0 Td [(e c cos f c q 2 leadstoaformmore amenabletotheGrobnerbasisanddoesnotintroduceanynewsolutionssince r c 0. Dening x 1 sin f c x 3 sin E d x 2 cos f c x 4 cos E d ; {2 51

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andtheauxiliaryvariables A 1 a d T 11 p e d x 4 q e c p c b d T 12 x 3 A 2 a d T 21 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d T 22 x 3 A 3 2 e c p a d T 11 p e d x 4 q b d T 12 x 3 q A 4 2 e c p a d T 21 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d T 22 x 3 q A 5 e 2 c p a d T 21 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d T 22 x 3 q A 6 e 2 c p a d T 11 p e d x 4 q b d T 12 x 3 q A 7 b d T 22 x 4 a d T 21 x 3 A 8 b d T 12 x 4 a d T 11 x 3 ; permitsthealgebraicsystemtobewrittenastheanevariety V p f 1 ;f 2 ;f 3 q ,with f 1 A 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 3 x 1 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 4 x 2 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 5 x 3 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 6 x 1 x 2 2 f 2 A 7 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 x 2 f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 : {3 5.1.1Grobnerbasisfordeputyeccentricanomaly Let F x denotethealgebraicsysteminEquation5{3,or F x t f 1 ;f 2 ;f 3 u .Thesolution totheradialextremalequationsistheanevariety V x ofthealgebraicsystemdenedin Equation5{3,or V x p x 1 ;x 2 ;A 1 ;:::;A 8 qP R 10 | f i p x 1 ;x 2 ;A 1 ;:::;A 8 q 0for i 1 ; 2 ; 3 : Anevarietiesaregovernedbytheirideals.Sincemorethanonesetofpolynomialscan representavariety,anysetwhichgeneratesthesameidealwillyieldthesamesolutions. Let R r x 1 ;x 2 ;A 1 ;A 2 ;:::;A 8 s designatethesetofpolynomialsintheindeterminates x 1 ;x 2 ;A i withcoecientsin theideal R .TheGrobnerbasis[106,107,112,115] G isasetofpolynomialswhich 52

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aremembersoftheidealofthevarietyof F ,and G hasthesamesolutionsas F TheGrobnerbasisisformulatedbyrepeatedapplicationofthedivisionalgorithm formultivariatepolynomials,afteratermorderingoftheindeterminateshasbeen established.Itcanbeshown[106]thataGrobnerbasiscanbeconstructedusinga nitenumberofapplicationsofthedivisionalgorithm.Severalsymbolicsoftwarepackages forcomputationalcommutativealgebraarecapableofreducingasetofpolynomialstoa Grobnerbasis.Mathematica,forexample,canbeusedtoconstructaGrobnerbasisfor thealgebraicsystem F x Usingamonomialorderingof x 1 x 2 A 1 A 8 ,bothlexicographicaland degreereverselexicographicalbasescanbeconstructed.Thelexbasiscontainstwenty sixelementswiththerstelementbeingunivariatein E d .TheGrobnerbasisindicates thatthereexistanitenumberofsolutionstothesystem F x [115].Thegrevlexcontains twentythreeelementswiththefteenthelementunivariatein E d .Thebasiselements areomittedsaveforthemembersrelevanttotheanalysis.Theunivariateelementsare identicaluptoasign.Theunivariatepolynomialis g x A 2 5 A 6 7 2 A 1 A 5 A 5 7 A 8 2 A 5 A 6 A 5 7 A 8 )-222(p A 2 1 A 2 3 q A 4 7 A 2 8 )-222(p 2 A 1 A 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 6 q A 4 7 A 2 8 2 A 1 A 5 A 3 7 A 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 2 1 A 2 7 A 4 8 A 2 3 A 2 7 A 4 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 1 A 6 A 2 7 A 4 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 1 A 6 8 A 2 4 A 4 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 A 3 A 4 A 3 7 A 8 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(A 2 2 A 2 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q 2 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 A 2 A 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 qp A 5 A 3 7 A 8 p A 6 A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 p A 2 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 2 8 qqq : Thisfunctionisaneighthordertrigonometricpolynomialandassuch,hasatmostsixteen zeros[116].Numericaltechniquescanbeusedtodeterminetherealzerosofthisfunction. Dependingontheorbitalgeometrysomeoftherootsmayhavemultiplicitygreaterthan one.Rootndingtechniquestypicallybenetfromgradientinformationofthefunction. 53

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Thegradientis B g x B E d 2 A 5 A 1 5 A 6 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 A 2 5 A 5 7 A 1 7 2 p A 1 1 A 5 A 5 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 A 1 5 A 5 7 A 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(5 A 1 A 5 A 4 7 A 1 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 A 5 A 5 7 A 1 8 q 2 p A 1 5 A 6 A 5 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 5 A 1 6 A 5 7 A 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(5 A 5 A 6 A 4 7 A 1 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 5 A 6 A 5 7 A 1 8 q p 4 A 3 7 A 1 7 A 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 4 7 A 8 A 1 8 qp A 2 1 A 2 3 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 1 A 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 6 q )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 p A 4 7 A 2 8 qp A 1 A 1 1 A 3 A 1 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 1 A 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 A 1 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 6 A 1 6 q 2 p A 1 1 A 5 A 3 7 A 3 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 A 1 5 A 3 7 A 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 A 1 A 5 A 2 7 A 1 7 A 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 A 1 A 5 A 3 7 A 2 8 A 1 8 q )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 p A 1 A 1 1 A 2 7 A 4 8 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 2 1 A 7 A 1 7 A 4 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 2 1 A 2 7 A 3 8 A 1 8 q 2 A 3 A 1 3 A 2 7 A 4 8 2 A 2 3 A 7 A 1 7 A 4 8 4 A 2 3 A 2 7 A 3 8 A 1 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 p A 1 1 A 6 A 2 7 A 4 8 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 1 A 1 6 A 2 7 A 4 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 1 A 6 A 7 A 1 7 A 4 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 A 1 A 6 A 2 7 A 3 8 A 1 8 q )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 A 1 A 1 1 A 6 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 A 2 1 A 5 8 A 1 8 p 2 A 4 A 1 4 A 4 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 A 2 4 A 3 7 A 1 7 qp A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q 2 p A 2 4 A 4 7 qp A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 A 1 8 q )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 qp A 1 3 A 4 A 3 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 3 A 1 4 A 3 7 A 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 A 3 A 4 A 2 7 A 1 7 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 3 A 4 A 3 7 A 1 8 q )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 A 3 A 4 A 3 7 A 8 p A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 A 1 8 q )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 A 2 2 A 2 7 p A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 A 1 8 qp A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q 2 p A 2 A 1 2 A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 2 A 7 A 1 7 q )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 p A 5 A 3 7 A 8 p A 6 A 2 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 1 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 qqq r A 1 2 A 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q)]TJ/F21 11.9552 Tf 18.968 0 Td [(A 2 A 1 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 2 8 q)]TJ/F15 11.9552 Tf 18.967 0 Td [(2 A 2 A 7 p A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 A 1 8 qs )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 A 2 A 7 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 q r A 1 5 A 3 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 A 5 A 2 7 A 1 7 A 1 8 p A 6 A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 p A 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 8 qq A 8 p A 1 6 A 2 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 6 A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 1 p A 2 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 2 8 q)]TJ/F15 11.9552 Tf 18.967 0 Td [(2 A 1 p A 7 A 1 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 8 A 1 8 qqs : 54

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with A 1 1 a d T 11 x 3 b d T 12 x 4 A 1 2 a d T 21 x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d T 22 x 4 A 1 3 2 p a d e c T 11 x 3 b d e c T 12 x 4 q A 1 4 2 p a d e c T 21 x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d e c T 22 x 4 q A 1 5 a d e 2 c T 21 x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d e 2 c T 22 x 4 A 1 6 a d e 2 c T 11 x 3 b d e 2 c T 12 x 4 A 1 7 b d T 22 x 3 a d T 21 x 4 A 1 8 b d T 12 x 3 a d T 11 x 4 : Oncetherootsof g x havebeenobtained,thevaluesof E d whichsatisfytheextremal equationsareknown,andthevaluesof t A 1 ;:::;A 8 u canbecomputed. 5.1.2Backsolvingforchieftrueanomaly Thecorrespondingvaluesof f c whichsatisfytheextremalequationsremaintobe determined.ThisbacksolvingisaccomplishedusingtheremainingelementsoftheGrobner basis.Theeleventhelementofthelexicographicalbasiscanbewrittenas x 6 2 x 5 2 x 4 2 x 3 2 x 2 2 x 2 1 A 2 5 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 6 2 p A 3 A 6 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 4 A 5 q 2 A 1 A 6 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 A 2 A 5 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 4 A 2 6 2 p A 1 A 3 )]TJ/F21 11.9552 Tf 11.76 0 Td [(A 2 A 4 A 3 A 6 q A 2 1 2 A 1 A 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 2 A 2 3 2 A 1 A 3 A 2 1 0 : Thispolynomialyieldsatmostsixrealrootsin x 2 .Selectingonlythoserootsforwhich x 2 Pr 1 ; 1 s yieldsvaluesof x 2 whichrepresentphysicalanglesfor f c ,since x 2 cos f c Thevariable x 1 maybecalculatedfromanyofthelexicographicalbasiselementsnumbers thirteenthroughtwentyve.Theexpressionsinelementsthirteenandfteenarethemost 55

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compact.Thethirteenthelementofthelexicographicalbasisis x 1 A 1 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 2 p A 5 A 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 6 A 7 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(x 2 2 p A 3 A 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 4 A 8 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(x 2 p A 2 A 8 A 6 A 7 q A 3 A 7 0 ; andcanbeusedtoobtainavalueof x 1 correspondingtoagivensetofvaluesfor x 2 and A i .Thenafourquadrantinversetangentfunctioncanbeappliedto x 1 and x 2 toobtain thevaluesof f c correspondingtoaparticularvalueof E d .Exploitingthesemi-triangular formofthelexicographicalbasisinthismannerguaranteesthatallsolutionsofthe extremalequationswillbefound.Notethat A 1 vanishesforonlytwovaluesof E d whichinterfereswiththecalculationof x 1 .Theextremalvaluesof E d canbecheckedto determineiftheycause A 1 tovanishpriortousingthisbacksolvingmethodology.The coecient A 8 vanishesonlyforvaluesof E d satisfying tan E d T 12 b 1 e 2 d T 11 ; whichcanbecheckedaprioriaswell.Shouldeitherofthesetwovariablesvanishthen analternateexpressionfromelementsnumberthirteenthroughtwentyvecanbeused insteadforthecalculationof x 1 .Theeleventhlexicographicalelementisasixthorder polynomialin x 2 oncethe A i havebeendetermined.Thispolynomialwillnecessarily containrootswhicharenotsolutionsofsomeoftheotherGrobnerbasisequations.Since theextremalsolutionsmustsatisfyalloftheGrobnerequationssimultaneously,any erroneoussolutionsobtainedfromusingthesubsetdescribedherecanbeeliminatedby discardingsuchsolutionswhichdonotsatisfyalloftheGrobnerbasisequations,and hencetheextremalequations.Inthismannerthevarietyisdeterminedbywhittlingdown thesetofcandidatesolutions.Theglobalextremaof x providetheheightdiameterof thetorusintheLVLHframeforanincommensuratesystem.Foracommensuratesystem thepositioncurvein R 3 mayrealizethisextremadependingontheinitialconditions p f c0 ;E d0 q .The x functioncouldtheoreticallyhaveasmanyassixteenextremalpoints, butinpracticeithasfarfewer,typicallyaroundeight. 56

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5.2LocalHorizontalPositionMetric UsingEquation5{1theextremaof y : R 2 R where p f c ;E d q y aresought.Dierentiating y withrespecttoitsindependentvariablesyields B y B f c a d p cos E d e d qp T 11 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 21 sin f c q b d sin E d p T 12 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 sin f c q 0 B y B E d a d sin E d p T 11 sin f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 21 cos f c q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d cos E d p T 12 sin f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 cos f c q 0 : Itisworthnotingthat B y B f c p f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(;E d q B y B f c p f c ;E d q B y B f c p f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(;E d q B y B E d p f c ;E d q : Solutionsoftheextremalequationsoccurwith periodicityinthechieftrueanomaly. ThevariablesinEquation5{2permittheextremalequationstobewrittenastheane variety V p f 1 ;f 2 ;f 3 ;f 4 q ,with f 1 a d p x 4 e d qp x 2 T 11 x 1 T 21 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d x 3 p x 2 T 12 x 1 T 22 q f 2 a d x 3 p x 1 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 T 21 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b d x 4 p x 1 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 T 22 q f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 f 4 x 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 4 1 : Althoughnotnecessary,morecompactexpressionscanbeobtainedfortheGrobnerbasis iftheauxiliaryvariables B 1 a d T 21 p e d x 4 q b d T 22 x 3 B 2 a d T 11 p e d x 4 q b d T 12 x 3 B 3 a d T 11 x 3 b d T 12 x 4 B 4 b d T 22 x 4 a d T 21 x 3 57

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aredened,sothatthealgebraicsystem F y t f 1 ;f 2 ;f 3 u canbeexpressedas f 1 B 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 x 2 f 2 B 3 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 4 x 2 f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 2 2 1 : Asinthelocalverticalcase,theauxiliaryvariables B i arenotindependent,butrelated through E d 5.2.1Grobnerbasisfordeputyeccentricanomaly Thesolutiontotheextremalequationsforthealongtrackdisplacementisgivenby theanevariety V y p x 1 ;x 2 ;B 1 ;:::;B 4 qP R 6 | f i p x 1 ;x 2 ;B 1 ;:::;B 4 q 0for i 1 ; 2 ; 3 : AGrobnerbasisforthisvarietycanbecomputedusingthetermordering x 1 x 2 B 1 B 2 B 3 B 4 andeitheralexicographicalorgrevlexmonomialordering.The lexicographicalbasiscontainsnineelementswiththesixthelementbeingunivariatein E d Thegrevlexbasiscontainssixelementswiththerstelementbeingunivariate.Thetwo univariatepolynomialsareequivalentuptoasign.Theunivariatepolynomialis g y B 1 B 4 B 2 B 3 ; anditsexistencearmsthatthereareanitenumberofsolutionstothesystem F y .The rootsofthisequationprovidetheextremalvaluesof E d .Thistrigonometricpolynomial isofdegreetwoandthereforehasatmostfourzeros[116].Therootsof g y areusedto obtainvaluesfor B 1 B 4 .Thegradientof g y is B g y B E d B 1 1 B 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 1 B 1 4 B 1 2 B 3 B 2 B 1 3 ; 58

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with B 1 1 a d T 21 x 3 b d T 22 x 4 B 1 2 a d T 11 x 3 b d T 12 x 4 B 1 3 a d T 11 x 4 b d T 12 x 3 B 1 4 b d T 22 x 3 a d T 21 x 4 : Oncethe B i coecientsareknown,theremainingelementsoftheGrobnerbasiscanbe usedtosolvefor f c 5.2.2Backsolvingforchieftrueanomaly Inordertoobtainthevaluesof f c correspondingtoavalueof E d ,theeight remainingelementsoftheGrobnerbasismustbesimultaneouslysolvedfor x 1 and x 2 ThreeofthelexicographicGrobnerbasiselementsareappropriateforobtainingvaluesfor x 2 : lex2 x 2 2 p B 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 4 q B 2 3 0 lex3 x 2 2 p B 1 B 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 B 4 q B 1 B 3 0 lex4 x 2 2 p B 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 2 q B 2 1 0 : {4 Anyofthesethreeequationscanbeusedtoobtainpossiblevaluesof x 2 ,aslongas x 2 Pr 1 ; 1 s .TheoptionofutilizingoneofthethreeexpressionsinEquation5{4 providessomerobustness,intheeventthatthequantities p B 2 3 )]TJ/F21 11.9552 Tf 12.084 0 Td [(B 2 4 q p B 1 B 3 )]TJ/F21 11.9552 Tf 12.084 0 Td [(B 2 B 4 q ,or p B 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 2 q vanish.Thevariable x 1 canbeobtainedfromanyoftheexpressions lex5 B 3 x 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(B 4 x 2 0 lex6 B 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 x 2 0 lex7 B 4 x 1 x 2 B 3 x 2 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 3 0 lex8 B 2 x 1 x 2 B 1 x 2 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 1 0 : {5 Eachofthetwosolutionsfor x 2 cannowbeusedtoobtainacorrespondingvaluefor x 1 ,providedthat x 1 Pr 1 ; 1 s .Thenafourquadrantinversetangentfunctioncan beusedtoobtainthevaluesof f c whichsatisfytheextremalequations.Ifwhileusing oneoftheexpressionsinEquation5{5shouldanyofthe B i vanish,thenoneofthe 59

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otherexpressionsmaybeusedasanalternativeforcalculating x 1 .Inthismanner,all solutionsoftheextremalequationsfor y canbecalculated.Earlieritwasnotedthat y isantisymmetricunderthetransformation f c f c )]TJ/F21 11.9552 Tf 12.544 0 Td [( .Thispropertyisreected intheGrobnerbasiselements.Undersuchatransformationtheindeterminateschange as x 1 x 1 and x 2 x 2 .TheGrobnerbasispolynomialsfor y areinvariantunder suchatransformationandelementstwothroughfourarequadraticin x 2 oncethe B i are known,indicatingtwosolutionsforagivenvalueof E d .Theglobalextremaprovidethe lengthdiameterofthetorusintheLVLHframe.Theoreticallythe y functioncanhavea maximumofeightextremalpoints. Thepositionofthedeputyrelativetothechiefhasbeenboundedwithinthechief orbitalplanethroughthecharacterizationof x and y throughtheuseofGrobner bases.Thechecksthat x 1 and x 2 arerealandliein r 1 ; 1 s areanecessaryresultof thesevariablesrepresentingtrigonometricquantities.Therearesomesolutionstothe anevarietieswhichdonotrepresentphysicalangles.Suchsolutionsarediscarded, astheyareartifactsofthevariables x i usedtoconvertthetrigonometricpolynomials toaneexpressions.Thedeviationofthedeputyfromthechieforbitalplaneisnow analyzed.Since z isafunctionof E d only,thesingleextremalequationismuchmore straightforwardtosolvethanaretheexpressionsfor x and y 5.3CrossTrackPositionMetric Itisofinteresttocalculatethemaximumcrosstrackoutofplaneseparationof thedeputysatellitewithrespecttotheorbitalplaneofthechiefsatellite,forEarth observationorsyntheticaperturearraymissions.Theinplaneseparationcanbebounded, andforsomesystemsthecrosstrackseparationmayrepresentthelargestdeviation inposition.ThegeometryisdepictedinFigure5-4.Thecrosstrackcomponentfrom Equation5{1, z T 31 a d p cos E d e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(T 32 b d sin E d ; {6 60

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is2 periodicin E d ,orthedeputy'sorbitalperiodinthetimedomain.Thismetricis periodicandharmonicwithanamplitude z amp a d b T 2 31 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 32 p 1 e 2 d q ,andaconstant biasof T 31 a d e d ,governedbytherelativeorbitalgeometry.Similarly,ifthecrosstrack separationofthechiefwithrespecttothedeputywereofconcern,thentheperiodof crosstrackseparationwouldbethatofthechief.Theextremalvaluesofthecrosstrack separationoccurwhen B z B E d a d T 31 sin E d )]TJ/F21 11.9552 Tf 11.76 0 Td [(b d T 32 cos E d 0 ; or g z tan E d T 32 T 31 b 1 e 2 d 0 ; withvaluesof z min { max T 31 a d e d a d b T 2 31 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 32 p 1 e 2 d q : Since B z B E d p E d )]TJ/F21 11.9552 Tf 11.759 0 Td [( q B z B E d p E d q ; therootsof g z are periodicin E d Figure5-4.CrossTrackSeparation 61

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5.3.1Crosstrackpositionmeanvalue Inadditiontothecrosstrackamplitude,period,andextrema,themeancrosstrack valuecanbederived,andmaybeofinterestwhendesigningaformationorconstellation. Ingeneraltwosatelliteswillnothavethesameorbitalplane,anditisdesirabletoknow whattheaverageoutofplanedisplacementofthedeputyisforcertainmissions,suchas stereoimaging.Deningtheconstants A T 31 a d B T 32 b d C T 31 a d e d {7 forconvenienceallowstheoutofplaneseparationinEquation5{6tobeexpressedas z A cos E d )]TJ/F21 11.9552 Tf 11.759 0 Td [(B sin E d )]TJ/F21 11.9552 Tf 11.759 0 Td [(C: {8 Therootmeansquareoutofplaneseparationdistanceis z rms d 1 T b a z 2 d; where isthedomainthemetricisbeingaveragedover.WhenEquation5{8isaveraged overtheeccentricanomalyofthedeputy, T 2 ,andthemeanvalueis z rmsE d d 1 2 p A 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(C 2 : {9 Amoreusefulaverageisinthetimedomain.Sincethemeananomaly M n p t q is relatedtotimebyaconstant,averagingovertimeisequivalenttoaveragingover M .The rootmeansquarevalueis z rmsM d 1 2 2 0 z 2 p M d q dM d : {10 ConvertingtheintegrandinEquation5{10toeccentricanomalyyields z rmsM d 1 2 2 0 z 2 p 1 e d cos E d q dE d : {11 62

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CalculatingtheintegralinEquation5{11yields z rmsM d A 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 e d AC: {12 Theintegrandisalwaysnon-negative,whichcanbeprovenbysubstitutingEquation 5{7intoEquation5{12.Thisistherootmeansquareofthedistanceofthedeputy fromthechieforbitalplane.Ingeneralthechiefwillhaveadierentaveragedistance fromthedeputyorbitalplane,aswellasdistinctvaluesforthebiasandamplitudeterms. Calculatingthemeanvalueof z withrespecttotimeanomalyyields z mean 3 T 31 a d e d ; whichvanishesas e d 0,asexpected. 5.3.2ComparisonwithClohessy-Wiltshireformulation ItisofinteresttocomparetheamplitudederivedfromEquation5{6totheoutof planeamplitudedictatedbytheCWformulation,asmanyanalyseshavebeenbasedon CWdynamics,butfewclearstatementshavebeenmadeabouttheirdomainofvalidity. TheClohessy-Wiltshireformulationyieldsasolutionforthecrosstrackmotionas[24,32] z p t q z 0 cos n c t )]TJ/F38 11.9552 Tf 15.505 8.089 Td [(9 z 0 n c sin n c t; {13 where n c isthemeanmotionofthechief,and z 0 ; 9 z 0 aretheinitialcrosstrackseparation andspeed,respectively.Thecrosstrackamplitudeinthisformulationis z amp:cw g f f e z 2 0 )]TJ/F38 11.9552 Tf 15.473 8.089 Td [(9 z 2 0 n 2 c : {14 ThisexpressioncanbeequatedtoorbitalelementsbyevaluatingEquation5{6for z 0 ,andthenconstructinganexpressionfor 9 z 0 .Thequantity 9 z canbeobtainedby constructingtherelativevelocityas LVLH v r )]TJ/F15 11.9552 Tf 13.392 0.166 Td [(~ r .TheangularvelocityoftheLVLH framerelativetotheECIframe,expressedintheLVLHframeis 00 9 f c T ,and r isthelocalderivativeoftherelativepositionvectorintheLVLHframe.Theoperator 63

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p ~ q isaskewsymmetricmatrix,whereif a a 1 a 2 a 3 T ,then ~ a 0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 : Thevelocityofthechief v c anddeputy v d relativetotheEarthcanberesolvedinto componentsalongtheirrespectiveperifocalbasisvectorsandparameterizedintermsof eccentricanomaliesas PQW v a sin E b cos E 0 n 1 e cos E : Therelativevelocityis v v d v c .ExpressingthelocalderivativeintheLVLHframe andrearrangingyields LVLH r 9 x 9 y 9 z v ~ r C p f c q T PQW d v d PQW c v c ~ r : Multiplyingoutthematricesandretainingthethirdequationyields 9 z n d p T 31 a d sin E d )]TJ/F21 11.9552 Tf 11.76 0 Td [(T 32 b d cos E d q 1 e d cos E d : {15 EvaluatingEquation5{15forsomeinitialconditionsandsubstitutingintoEquation 5{14yieldsanamplitudeof z 2 amp:cw p T 31 a d p cos E d0 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(T 32 b d sin E d0 q 2 )]TJ/F21 11.9552 Tf 12.955 8.087 Td [(n 2 d n 2 c p T 31 a d sin E d0 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 32 b d cos E d0 q 2 p 1 e d cos E d0 q 2 ; {16 where0subscriptsindicateinitialconditions.When i c i d and c d ,theorbital planesofthechiefanddeputycoincide,and T 31 T 32 0,sothatboththeCWandtrue 64

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crosstrackamplitudesare0,asexpected.If a d a c and e d 0,thenthetwoamplitude expressionsareidentical,reducingto z amp a d b T 2 31 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 32 : {17 Itisseenthatonlyinthelimitingcaseofadeputysatelliteinexactlyacircularorbit ofthesamesemi-majoraxisasthechiefsatellitedotheexpressionsagreeexactly.Note thattheCWformulationyieldsacrosstracksolutionwhichisperiodicwiththechief's orbitalperiod,insteadofthatofthedeputy.TheCWformulationalsofailstocapturethe biasterminthecrosstrackmotion,exceptforthecase e d 0,forwhichthebiastermis identicallyzero.TheCWformulationiscapablehoweverofcapturingapureinclination dierencebetweenthetwosatellitesinthecrosstracksolution.Numericalsimulationsof Equation5{14and z amp revealthattheCWcrosstrackamplitudecanbegreaterthan orlessthanthetruecrosstrackamplitude,dependingontheorbitalgeometry. 5.4PositionMetricSimulations Todemonstratetheutilityofthepositionbasedmetricsderivedabove,considerthe caseoftwosatelliteswithorbitsdenedbytheorbitalparameterslistedinTable5-1;the twoorbitsare1:1commensurateintheirorbitalperiods[102]andthereforetheirrelative positionandvelocityisperiodicintheircommonorbitalperiod.Thiscommensurability conditionisnotnecessaryandwaschosentohighlightthebehavioroftheposition functionsdescribedbelow.Theinertialtrajectoriesofthechiefanddeputyareshownin Figure5-5.Theunivariatefunctions g x ;g y ,and g z areshowninFigures5-6A,5-6B,and 5-6CinEarthcanonicalunitswiththeirzerosidentied. Themanifold x : R R R isshowninFigure5-7.Thesurfacerepresentsall possiblevaluesthatthelocalverticalpositioncomponentcanattain.Theorbitalelements t a c ;e c ; c ;i c ;! c ;a d ;e d ; d ;i d ;! d u determinetheshapeofthesurface.Thefunction x is2 periodicinboth f c and E d andthereforeplottingoverthisintervalissucient forvisualizationpurposes.Thetrajectoryproducedbytheinitialconditionshasbeen 65

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overlaidonthemanifoldinFigure5-7.Thetrajectoryof x onthissurfacewillcycleo ofoneaxis,wheneither f c or E d reach2 andreappearat0ontheotherside.Sincethe orbitsarecommensurate,thetrajectoryrepeatsitself,andsincethecommensurability is1:1,asinglecontourispresent.Thevaluesof E d obtainedasthezerosof g x ,and theircorrespondingpairsin f c havebeenidentiedonthemanifoldaswell.Thesepoints constitutethemaxima,minima,andinectionpoints.Similarstatementsholdforthe functions y and z ,themanifoldsofwhichareshowninFigures5-8and5-9,respectively. TherelativepositionintheLVLHframeisdepictedinFigure5-10,alongwiththe boundingboxobtainedfromtheglobalextremaof t x ; y ; z u .Itisimportanttonotethat theinitialconditions p f c0 ;E d0 q determinewhetheracommensuratesystemwillrealize theglobalextremaofthemanifoldsdepictedabove.Sincethetrajectoryencountersonly asubsetofthestatespace,theboundingboxdepictedinFigure5-10willingeneral, overestimatetheextremafor x and y inincommensuratesystems,unlesstheinitial conditions p f c0 ;E d0 q aresuchthatthetrajectorypassesthroughtheglobalextrema. Table5-1.OrbitalParametersforPositionSimulations:1Commensurate ChiefDeputy a8,0508,050km e0.10.2 1550deg i 6010deg 55deg f 0 15deg 66

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Figure5-5.EarthCenteredInertialTrajectories 67

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ALocalVerticalPositionUnivariatePolynomial BLocalHorizontalPositionUnivariatePolynomial CCrossTrackPositionUnivariatePolynomial Figure5-6.PositionUnivariatePolynomials:1Commensurate 68

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Figure5-7.LocalVerticalPositionManifold:1Commensurate 69

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Figure5-8.LocalHorizontalPositionManifold:1Commensurate 70

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Figure5-9.CrossTrackPositionManifold:1Commensurate 71

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Figure5-10.RelativePositionTrajectory:1Commensurate 72

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Ingeneral,theorbitalperiodsaredistinctandoneofthehorizontalaxesinFigures 5-7,5-8,and5-9willcyclefasterthantheother,leadingtointerestingcontours.Ifthe semi-majoraxisofthedeputyischangedto a d a c 3 ? 2aslistedinTable5-2,thenthe orbitalperiodshavearatioof ? 2andareincommensurate.Themanifoldsfor x ; y ,and z areshowninFigures5-11,5-12,and5-13respectively.Intheincommensuratecase,the trajectorywillwanderovertheentiremanifold,eventuallycoveringeverypoint,without everrepeatinganyofthem. Withthisobservationcomestherealizationthatthetrajectoriesonthemanifold donotintersectoneanother.Theapproachpresentedhereisparticularlyusefulinthe caseofincommensurateorbits.Sincenopointsinthestatespace f c E d arerepeated, anumericalsearchofaparticularsetoforbitswouldtheoreticallyrequireaninnite searchdomaintoensurethateverycombinationof p f c ;E d q hasbeenevaluated.Using theGrobnerbasisapproachpresentedherepermitsarapidglobalboundingforallinitial conditions.Themethodologypresentedhereguaranteeslocatingtheglobalmaximaand minimaforasetoforbitsspeciedby t a c ;e c ; c ;i c ;! c ;a d ;e d ; d ;i d ;! d u ,butdoesnotnd extremaoveraparticularcontour,i.e.,whentheinitialvaluesof p f c ;E d q arespecied. TherelativepositionintheLVLHframeisdepictedinFigure5-14,alongwiththe boundingboxobtainedfromtheglobalextremaof t x ; y ; z u .Sincetheincommensurate systemencounterseverypointinthestatespace,theglobalextremaofallthreeposition componentswillberealizedeventually,andtheboundingboxistangenttothemanifold oneachofitssixfaces.Furthermore,therangemetricintroducedin[103]providesa globalminimaandmaximafortherange,whichisalsorealizedforanincommensurate system.ThemanifoldtouchestheinnerandouterboundingspheresinFigure5-14.The globalextremafortheorbitalparameterslistedinTable5-2arelistedinTable5-3. Commensurateorbitscanmimicincommensuratesystemsiftheleastcommon multipleoftheorbitalperiodsislarge.Ifoneorbithasasemi-majoraxiswhichis considerablysmallerthantheother,thenthisorbitwillhaveagreatermeanmotion. 73

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Althoughsuchasystemwillnotrealizeeverypointinthestatespace,itmaycoveragreat dealofthestatespacebeforereplicatingitself.Anexampleofsuchasystemisspeciedin Table5-4.ThegeometryisdepictedinFigure5-15andthepositionmanifoldisshownin Figure5-16.FurtherdiscussionofthestatespaceisincludedinSection7.1. Table5-2.OrbitalParametersforPositionSimulationsIncommensurate ChiefDeputy a8,05010,142.3645km e0.10.2 1550deg i 6010deg 55deg f 0 15deg Table5-3.GlobalExtremaforPositionSimulationsIncommensurate MetricMinimumkmMaximumkm r 1,239.1318,995.78 x -18,812.682,714.16 y -11,565.0811,565.08 z -6,797.979,021.84 74

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Figure5-11.LocalVerticalPositionManifoldIncommensurateSystem 75

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Figure5-12.LocalHorizontalPositionManifoldIncommensurateSystem 76

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Figure5-13.CrossTrackPositionManifoldIncommensurateSystem 77

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Figure5-14.RelativePositionTrajectoryIncommensurateSystem 78

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Table5-4.OrbitalParametersforPositionSimulationsDisparateSizes ChiefDeputy a10,05042,050km e00.4 2700deg i 890deg 00deg f 0 11deg Figure5-15.DisparateOrbitGeometry 79

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Figure5-16.RelativePositionTrajectoryDisparateOrbitSizes 80

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5.5PositionManifoldDescription NowthatthecomponentsoftherelativepositionvectorintheLVLHframehavebeen bounded,theregionofpossibletrajectoriescanberestrictedfurtherfromthespherical regionsdepictedinFigures5-2and5-3.Themanifoldcanbeconnedwithabounding boxbasedontheradial,alongtrack,andcrosstrackextrema.Thecrosstrackextrema z max and z min canbeusedtocalculatethethicknessofthetorusas z max z min .The biasterminthecrosstrackmetricistheamountthatthecenterplaneofthetorusis displacedfromtheoriginthelocationofthechief.Themaximumvaluesoftheradial andalongtrackdisplacementsdenetheheightandlengthofthetorus.Commensurate systemswillnotnecessarilyrealizetheglobalmaximumandminimumofthe x and y functions,dependingontheinitialconditions.Incommensuratesystemswillcontactthe outersphereatsomepoint,andthelocationofthispointdependsontheorientationof theorbits.Allsystems,whethercommensurateornot,willrealizethecrosstrackextrema; thecrosstrackboundrepresentsasignicantreductionoftheregionofaccessiblespaceby reducingtheouterspheretoasectionalsliceofitsinterior.Themostrestrictivebounds forthemanifoldconsistoftheintersectionofthesphericalboundsandtheboundingbox. AsnotedbyGurl[31]anddepictedinFigure5-2,therelativepositionmanifold canhavegeometryresemblingatorus.Figure5-16providesacontrastingexample,where themanifoldlacksaholeandacquiresashaperesemblinganellipsoid.InFigure5-14, themanifoldexhibitsthehollowcenter,yetlackstheclosedsurfacepropertyofthetorus example.TherelativepositionmanifoldoftheLVLHframedoesnotalwaysresemblea torus,andamoregeneraldescriptionisasourceofcontinuingeort. Inanalogywiththefourpositionmetricsdiscussedinsections5.1-5.3,velocity metricsarenowderivedandanalyzed. 81

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CHAPTER6 VELOCITYMETRICS 6.1RelativeSpeedMetric Therelativespeedbetweensatellites, || v |||| v d v c || mayalsobeofinterestwhen designingformationmissions.Inadditiontocalculatingthisquantity,itmaybedesirable toboundthisquantitybetweenitsextremaforpurposesof 1.Estimatingconjunctionspeedwithconjunctionbeingdenedascloseapproach betweenthespacecraftinpositionwithoutmatchingvelocities. 2.Boundingapproachspeedsduringarendezvousmaneuver. 3.Boundingdopplershiftininter-satellitecommunicationslinks 4.Boundingtherelativespeedwhichanopticallysensormustaccommodatewhen inspectingaspacecraft. ConstellationssuchasIridium[117]requireextensiveplanningandthesemetricsmay servetofacilitateconstellationdesign.IncontrastwithSections5.1-5.3,wherematrix notationwasusedindescribingcomponentsofvectors,thissectionusesvectornotation forcompactness.Thevelocityofasatelliteexpressedinitsperifocalunitvectorsis v d p r sin f ^ p )-222(p cos f )]TJ/F21 11.9552 Tf 11.759 0 Td [(e q ^ q s : Thespeedofthedeputyrelativetothechief || v || isdependentontheeccentricities, trueanomalies,parameters,andperifocalunitvectorsofthedeputyandchief;i.e., v || v || b p v d v c qp v d v c q : {1 Followingadevelopmentsimilartothatusedinreferences[31,103],thetermsofEquation 6{1arecollectedas || v || 2 W W 0 )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 1 cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 2 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 3 cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 4 sin f c )]TJ/F21 11.9552 Tf 9.103 0 Td [(W 5 cos f d cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 6 cos f d sin f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 7 sin f d cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 8 sin f d sin f c ; wheretheconstants W i are 82

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W 0 p d p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e 2 d q)]TJ/F21 11.9552 Tf 21.709 8.088 Td [( p c p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e 2 c q 2 ? p c p d e d e c ^ q d ^ q c W 1 2 e d p d 2 e c ? p c p d ^ q d ^ q c W 5 2 ? p c p d ^ q d ^ q c W 2 2 e c ? p c p d ^ p d ^ q c W 6 2 ? p c p d ^ q d ^ p c W 3 2 e c p c 2 e d ? p c p d ^ q d ^ q c W 7 2 ? p c p d ^ p d ^ q c W 4 2 e d ? p c p d ^ q d ^ p c W 8 2 ? p c p d ^ p d ^ p c : Amethodologyfordeterminingtheextremaoftherelativespeedmetricisnowpresented. 6.1.1Extremalequations Therelativespeed || v || isanon-negativequantity.Let z representeither f c or f d Dierentiating || v || withrespectto z yields B|| v || B z v || v || B v B z ; whichisproblematicforsituationswheretheglobalminimumis || v || 0.Insteadit isadvantageoustodierentiate || v || 2 W withrespectto f c and f d ,whichyieldsthe expressions: B W B f c W 3 sin f c )]TJ/F21 11.9552 Tf 11.76 0 Td [(W 4 cos f c W 5 cos f d sin f c )]TJ/F21 11.9552 Tf 9.103 0 Td [(W 6 cos f d cos f c W 7 sin f d sin f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 8 sin f d cos f c 0 B W B f d W 1 sin f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(W 2 cos f d W 5 sin f d cos f c W 6 sin f d sin f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 7 cos f d cos f c )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 8 cos f d sin f c 0 Theseextremalequationsdonotpossessanyundesirablebehaviorforformationswhere v 0 .Dening x sin f c ,and y cos f c ,theextremalequationscanbeformulatedas analgebraicsystem F v t f 1 ;f 2 ;f 3 u ,with f 1 C 1 x )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 y C 3 0 f 2 C 4 x )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 5 y 0 f 3 x 2 )]TJ/F21 11.9552 Tf 11.76 0 Td [(y 2 1 0 ; 83

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and C 1 W 8 cos f d W 6 sin f d C 2 W 7 cos f d W 5 sin f d C 3 W 1 sin f d W 2 cos f d C 4 W 3 W 5 cos f d W 7 sin f d C 5 W 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 6 cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 8 sin f d : 6.1.2Grobnerbasisforrelativespeed Thesolutiontotherelativespeedextremalequationsisgivenbytheanevariety V v p x 1 ;y;C 1 ;:::;C 5 qP R 7 | f i p x 1 ;y;C 1 ;:::;C 5 q 0for i 1 ; 2 ; 3 : Ifthemonomialorderingisselectedaslexicographicwith x y C 1 C 2 C 3 C 4 C 5 ,thentheGrobnerbasisconsistsofthirteenpolynomials.Oneofthepolynomials isindependentof x;y andisunivariatein f d through t C 1 ;:::;C 5 u .TheGrobnerbasis indicatesthatthereexistanitenumberofsolutionstothesystem F v .Thispolynomial denotedas g speed is g speed C 2 4 p C 2 2 C 2 3 q 2 C 1 C 2 C 4 C 5 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 p C 2 1 C 2 3 q : {2 Therootsof g speed canbefoundusingnumericalrootndingtechniques,andyieldthe trueanomaliesofthedeputywhichsatisfythesysteminSection6.1.1.Equation6{2is afourthdegreetrigonometricpolynomialandhasamaximumofeightroots[116],which determinethevaluesof C 1 C 5 .Numericalrootndingtechniquescanbeusedtondthe valuesof f d forwhich g speed p C 1 ;C 2 ;C 3 ;C 4 ;C 5 q 0,andthegradientof g speed : B g speed B f d 2 C 4 C 1 4 p C 2 2 C 2 3 q)]TJ/F15 11.9552 Tf 18.967 0 Td [(2 C 2 4 p C 2 C 1 2 C 3 C 1 3 q 2 p C 1 1 C 2 C 4 C 5 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 1 C 1 2 C 4 C 5 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 1 C 2 C 1 4 C 5 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 1 C 2 C 4 C 1 5 q )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 5 C 1 5 p C 2 1 C 2 3 q)]TJ/F15 11.9552 Tf 18.967 0 Td [(2 C 2 5 p C 1 C 1 1 C 3 C 1 3 q ; 84

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with C 1 1 p W 8 sin f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(W 6 cos f d q C 1 2 p W 7 sin f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(W 5 cos f d q C 1 3 W 1 cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(W 2 sin f d C 1 4 W 5 sin f d W 7 cos f d C 1 5 W 8 cos f d W 6 sin f d ; canbeusedtofacilitatetherootndingprocessinmanyalgorithms. 6.1.3Backsolvingforchieftrueanomaly ThelexicographicalmonomialorderingproducesaGrobnerbasiswithasemi-triangular formwhichfacilitatesthebacksolvingprocessfortheremainingindeterminates.Foreach valueof f d whichsatisesEquation6{2,thecorrespondingvaluesfor t C 1 ;:::;C 5 u are computed.TheninthandtenthelementsoftheGrobnerbasisare lex9 C 3 C 5 x C 3 C 4 y )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 4 C 1 C 5 0 lex10 C 1 x )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 y C 3 0 : {3 ThelinearsysteminEquation6{3canbeusedtosolvefor x and y .Finally,afour quadrantinversetangentfunctionisusedtoobtainthevalueof f c correspondingtoa particularvalueof f d ,andtheextremalpointshavebeendetermined.Sincethespeed metrichasbeenformulatedintermsoftrueanomalies,thestatespaceconsistsof p f c ;f d q incontrastwiththepositionmetricsabove,whichutilizedtheeccentricanomalyof thedeputy.Theconstraint f 3 x 2 )]TJ/F21 11.9552 Tf 12.687 0 Td [(y 2 1 0isnotautomaticallyenforcedand appearsasoneoftheGrobnerbasispolynomialstobesatised.Solutionsoftheninthand tenthpolynomialsinEquation6{3mustalsosatisfythistrigonometricconstraint.Itis possibletoobtainvaluesfor x and y whichdonotcorrespondtophysicalangles.Such erroneoussolutionscanbeidentiedbyevaluatingtheextremalequationstoensurethat theyvanishtoanacceptablenumericaltolerance.Inordertorepresentaphysicalangle, thevariablesmustsatisfy x Pr 1 ; 1 s y Pr 1 ; 1 s ,and x;y P R .Inthismannerallofthe extremalvaluesoftherelativespeedcanbedetermined. 85

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Therelativespeedhasbeenanalyzedbutnoeortshavebeenmadetocharacterize thevelocitycomponents.Thisanalysisproceedsalonganapproachparalleltothatofthe positionmanifoldabove.Switchingbacktomatrixnotation,therelativevelocity v can beexpressedintheLVLHframeas LVLH v v x v y v z C p f c q T sin f d cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d 0 d p d e c sin f c 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(e c cos f c 0 d p c ; whichreducesto v x v y v z d p d r cos f c p T 11 sin f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(T 12 p cos f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(e d qq )]TJ/F15 11.9552 Tf 11.095 0 Td [(sin f c p T 21 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qqs d p c e c sin f c d p d r sin f c p T 11 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 12 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qq )]TJ/F15 11.9552 Tf 11.095 0 Td [(cos f c p T 21 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qqs d p c p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e c cos f c q d p d r T 32 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d q T 31 sin f d s : {4 Thequantity LVLH v representstheinertialvelocityvectorofthedeputyrelativetothe chief,resolvedintheLVLHframe.Thethreefunctions v x : R 2 R ; where p f c ;f d q v x v y : R 2 R ; where p f c ;f d q v y v z : R R ; where p f d q v z ; eachgeneratea2-surfaceinthestatespace p f c ;f d q .Boundsforeachofthesesurfacesare soughtinananalogousmannertothepositionmetricsformulations. 86

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6.2LocalVerticalVelocityMetric Theextremalequationsfor v x areobtainedbydierentiatingthelocalvertical componentofEquation6{4. B v x B f c d p d r sin f c p T 11 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 12 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qq)]TJ/F15 11.9552 Tf 23.519 0 Td [(cos f c p T 21 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qqs d p c e c cos f c 0 B v x B f d d p d r cos f c p T 11 cos f d T 12 sin f d q)]TJ/F15 11.9552 Tf 18.967 0 Td [(sin f c p T 21 cos f d T 22 sin f d qs 0 : Theextremalequationsareanti-symmetricunderthetransformation f c f c )]TJ/F21 11.9552 Tf 12.325 0 Td [( ,and thereforesolutionsoftheextremalequationsoccurwith periodicityin f c .Deningthe variables x 1 sin f c x 3 sin f d x 2 cos f c x 4 cos f d ; {5 andtheauxiliaryvariables A 1 d p d p x 3 T 11 x 4 T 12 e d T 12 q A 2 d p d p T 22 p x 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d q x 3 T 21 q e c d p c A 3 d p d p x 3 T 22 x 4 T 21 q A 4 d p d p x 3 T 12 x 4 T 11 q ; permitstheextremalequationstobeexpressedasthealgebraicsystem F v x t f 1 ;f 2 ;f 3 u with f 1 A 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 x 2 0 f 2 A 3 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 4 x 2 0 f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 0 : AGrobnerbasisisnowsoughtforthismultivariatesystem. 87

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6.2.1Grobnerbasisfordeputytrueanomaly Thesolutionof F v x isgivenbytheanevariety V v x p x 1 ;x 2 ;A 1 ;:::;A 4 qP R 6 | f i p x 1 ;x 2 ;A 1 ;:::;A 4 q 0for i 1 ; 2 ; 3 : Usingatermorderingof x 1 x 2 A 1 A 2 A 3 A 4 andlexicographicalmonomialorderingresultsinaGrobnerbasiswithnineelements.The rstelementisunivariatein f d : g v x A 1 A 4 A 2 A 3 0 : Thistrigonometricpolynomialissecondorderandhasatmostfourrealroots.The gradientof g v x is B g v x B f d A 1 1 A 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 A 1 4 A 1 2 A 3 A 2 A 1 3 ; with A 1 1 d p d p x 4 T 11 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 3 T 12 q A 1 2 d p d p x 3 T 22 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 4 T 21 q A 1 3 d p d p x 3 T 21 x 4 T 22 q A 1 4 d p d p x 3 T 11 x 4 T 12 q : Oncetherootsof g v x havebeendetermined,eachvalueof f d canbeusedtocalculatethe correspondingvaluesof f c .ThisisperformedbybacksolvingwiththeGrobnerbasis elements. 88

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6.2.2Backsolvingforchieftrueanomaly ThreeoftheGrobnerbasiselementscanbeusedtosolvefor x 2 : lex2 x 2 2 p A 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 4 q A 2 3 0 lex3 x 2 2 p A 1 A 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 A 4 q A 1 A 3 0 lex4 x 2 2 p A 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 2 2 q A 2 1 0 : Havingdetermined x 2 2 cos 2 f c ,fourvaluesof f c arepossible.Ofthesefour,the candidatesareselectedbasedontheirconsistencywiththefollowingGrobnerelements: lex5 x 1 A 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 A 4 0 lex6 x 1 A 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 A 2 0 lex7 x 1 x 2 A 4 x 2 2 A 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 3 0 lex8 x 1 x 2 A 2 x 2 2 A 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(A 1 0 : Thetrigonometricconstraint f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.113 0 Td [(x 2 2 1 0mustbesatisedaswell.Inthisfashion, alltheextremalvaluesof v x havebeendeterminedandthelocalverticalcomponentof therelativevelocityhasbeenbounded. 6.3LocalHorizontalVelocityMetric Theextremalequationsfor v y areobtainedbydierentiatingthelocalhorizontal componentofEquation6{4: B v y B f c d p d r cos f c p T 11 sin f d )]TJ/F21 11.9552 Tf 11.76 0 Td [(T 12 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qq sin f c p T 21 sin f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 22 p cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d qqs )]TJ/F41 9.9626 Tf 9.103 10.623 Td [(b p c e c sin f c 0 B v y B f d d p d r sin f c p T 11 cos f d T 12 sin f d q)]TJ/F15 11.9552 Tf 18.967 0 Td [(cos f c p T 21 cos f d T 22 sin f d qs 0 : Theextremalequationsareanti-symmetricunderthetransformation f c f c )]TJ/F21 11.9552 Tf 12.864 0 Td [( Solutionsoftheextremalequationsoccurwith periodicityin f c .Usingthesame 89

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variables x i asinEquation6{5,anddeningtheauxiliaryvariables B 1 d p d p x 3 T 21 x 4 T 22 e d T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(e c d p c B 2 d p d p x 3 T 11 x 4 T 12 e d T 12 q B 3 d p d p x 3 T 12 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 4 T 11 q B 4 d p d p x 3 T 22 x 4 T 21 q permitstheextremalequationstobewrittenasthealgebraicsystem F v y t f 1 ;f 2 ;f 3 u with f 1 B 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 x 2 0 f 2 B 3 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 4 x 2 0 f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 0 : 6.3.1Grobnerbasisfordeputytrueanomaly Thesolutionof F v y isgivenbytheanevariety V v y p x 1 ;x 2 ;B 1 ;:::;B 4 qP R 6 | f i p x 1 ;x 2 ;B 1 ;:::;B 4 q 0for i 1 ; 2 ; 3 : Usingatermorderingof x 1 x 2 B 1 B 2 B 3 B 4 ; andlexicographicalmonomialorderingresultsinaGrobnerbasiswithnineelements.The rstelementisunivariatein f d : g v y B 1 B 4 B 2 B 3 0 : Thistrigonometricpolynomialissecondorderandhasatmostfourrealroots.The gradientof g v y is B g v y B f d B 1 1 B 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 1 B 1 4 B 1 2 B 3 B 2 B 1 3 ; 90

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with B 1 1 d p d p x 4 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 T 22 q B 1 2 d p d p x 4 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 T 12 q B 1 3 d p d p x 4 T 12 x 3 T 11 q B 1 4 d p d p x 4 T 22 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 3 T 21 q : Oncetherootsof g v y havebeendetermined,eachvalueof f d canbeusedtocalculatethe correspondingvaluesof f c .ThisisperformedbybacksolvingwiththeGrobnerbasis elements. 6.3.2Backsolvingforchieftrueanomaly ThreeoftheGrobnerbasiselementscanbeusedtosolvefor x 2 : lex2 x 2 2 p B 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 4 q B 2 3 0 lex3 x 2 2 p B 1 B 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 B 4 q B 1 B 3 0 lex4 x 2 2 p B 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 2 2 q B 2 1 0 : Havingdetermined x 2 ,fourvaluesof f c arepossible.Ofthesefour,thecandidatesare selectedbasedontheirconsistencywiththefollowingGrobnerelements: lex5 x 1 B 3 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 2 B 4 0 lex6 x 1 B 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 2 B 2 0 lex7 x 1 x 2 B 4 x 2 2 B 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 3 0 lex8 x 1 x 2 B 2 x 2 2 B 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(B 1 0 ; alongwiththetrigonometricconstraint x 2 1 )]TJ/F21 11.9552 Tf 12.05 0 Td [(x 2 2 1 0.Inthisfashion,alltheextremal valuesof v y canbedeterminedandthevelocityofthedeputyrelativetothechiefinthe localhorizontaldirectionhasbeenbounded. 91

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6.4CrossTrackVelocityMetric Theextremalequationfor v z isobtainedbydierentiatingthecrosstrackcomponent ofEquation6{4withrespectto f d : B v z B f d d p d p T 31 cos f d )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 32 sin f d q : Theextremaoccurwhen g v z tan f d )]TJ/F21 11.9552 Tf 12.954 8.087 Td [(T 31 T 32 0 : Thefunction v z isaharmonicfunctionwithabiasterm.Thebiasis b p d T 32 e d andthe amplitudeis b p d p T 2 32 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 31 q .Theextremalvaluesof v z are v z min { max d p d T 32 e d d p d p T 2 32 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 31 q : Since B v z B f d p f d )]TJ/F21 11.9552 Tf 11.759 0 Td [( q B z B E d p f d q ; therootsof g z are periodicin f d .Asillustratedinthesimulationsofsection6.6, thecrosstrackvelocityextremawillalwaysberealized,forbothcommensurateand incommensuratesystems.Nowthattherelativespeedandvelocitycomponentshave beenanalyzed,aspecicprojectionoftherelativevelocity,namelytherangerate,willbe discussed. 6.5RangeRateMetric Anadditionalrelevantvelocitymetricistherangerateofthedeputyrelativetothe chief,denotedby D .Therangerateisgivenby D d dt || r || v r || r || : Therangeratecanbeinterpretedaseithertherateofchangeoftheseparationdistanceor thecomponentofrelativevelocityalongthelineofseparation.Therangeratehasutility tothemissionplannerinthatitprovidesmorestringentboundsontheDopplershiftthat 92

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inter-satellitecommunicationssystemsmustaccommodatethantherelativespeedmetric [118].Furthermore,ithasutilityintheplanningofrendezvousandproximityoperations withregardtodockingmaneuvers,whereitisusefultoknowtheapproachvelocityof satelliterelativetoanother.Thenecessaryquantitiesarederivedusingvectornotationas inSection6.1.Thefunction D canbeformulatedintermsofeccentricanomalies,withthe relativevelocityandrelativepositiongivenby v p E d ;E c q n d r a d sin E d ^ p d )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d cos E d ^ q d s 1 e d cos E d n c r a c sin E c ^ p c )]TJ/F21 11.9552 Tf 11.759 0 Td [(b c cos E c ^ q c s 1 e c cos E c and r p E d ;E c qr a d p cos E d e d q ^ p d )]TJ/F21 11.9552 Tf 11.759 0 Td [(b d sin E d ^ q d sr a c p cos E c e c q ^ p c )]TJ/F21 11.9552 Tf 11.759 0 Td [(b c sin E c ^ q c s ; respectively,sothat || r || 2 a 2 d p 1 e d cos E d q 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 c p 1 e c cos E c q 2 2 r a d a c p cos E d e d qp cos E c e c q ^ p d ^ p c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c b d sin E d p cos E c e c q ^ q d ^ p c )]TJ/F21 11.9552 Tf 9.102 0 Td [(b c a d sin E c p cos E d e d q ^ p d ^ q c )]TJ/F21 11.9552 Tf 11.759 0 Td [(b c b d sin E d sin E c ^ q d ^ q c s : Thequantityofinterestistherangeofvaluesthat D mayacquireforallpossiblevaluesof E c and E d .Theextremalequationsaregeneratedfromtheexpression B D B E || r || B B E p v r qp v r q B B E p|| r ||q || r || 2 0 ; {6 where E representsboth E c and E d .Itisthezerosofthenumeratorwhichareofinterest. Intheeventthat || r || 0,thedenominatorofEquation6{6vanishesandthesatellites havecollided.As || r || 0,thedirectionofseparation ^ r becomesundenedand D isnot wellbehaved,exhibitingrapidchanges.Consequently,onlysituationswheretheorbitsdo notintersectareconsidered.FurtherexaminationofEquation6{6revealsthatitcanbe expressedas B D B E || r || 2 B B E v r )]TJ/F21 11.9552 Tf 11.759 0 Td [( v B B E r p v r qp r B B E r q || r || 3 : 93

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Theproblemhasnowbeenreducedtondingthevaluesof E forwhichthequantity N || r || 2 B B E v r )]TJ/F21 11.9552 Tf 11.759 0 Td [( v B B E r p v r q r B B E r vanishes.Constructingtheappropriatepartialsyields B r B E c a c sin E c ^ p c b c cos E c ^ q c B r B E d a d sin E d ^ p d )]TJ/F21 11.9552 Tf 11.76 0 Td [(b d cos E d ^ q d B v B E c n c p 1 e c cos E c q 2 r c B v B E d n d p 1 e d cos E d q 2 r d : Thedotproductbetweentherelativepositionandvelocityis v r n d e d a 2 d sin E d )]TJ/F21 11.9552 Tf 11.76 0 Td [(n c e c a 2 c sin E c )]TJ/F15 11.9552 Tf 9.912 0.166 Td [(^ p d ^ p c a c a d n d sin E d p cos E c e c q 1 e d cos E d )]TJ/F21 11.9552 Tf 12.955 8.087 Td [(a c a d n c sin E c p cos E d e d q 1 e c cos E c )]TJ/F15 11.9552 Tf 9.912 0.166 Td [(^ p d ^ q c a d b c n d sin E c sin E d 1 e d cos E d a d b c n c cos E c p cos E d e d q 1 e c cos E c )]TJ/F15 11.9552 Tf 9.725 0.166 Td [(^ q d ^ p c a c b d n c sin E c sin E d 1 e c cos E c a c b d n d cos E d p cos E c e c q 1 e d cos E d )]TJ/F15 11.9552 Tf 9.725 0.166 Td [(^ q d ^ q c b c b d n d sin E c cos E d 1 e d cos E d b c b d n c sin E d cos E c 1 e c cos E c : 94

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Theremainingrelevantdotproductsare: B v B E c r n c p 1 e c cos E c q 2 p r c r d r 2 c q B v B E d r n d p 1 e d cos E d q 2 p r d r c r 2 d q v B r B E c n d a d a c sin E c sin E d 1 e d cos E d ^ p d ^ p c )]TJ/F21 11.9552 Tf 12.954 8.087 Td [(n d a d b c sin E d cos E c 1 e d cos E d ^ p d ^ q c )]TJ/F21 11.9552 Tf 10.298 8.088 Td [(n d b d a c cos E d sin E c 1 e d cos E d ^ q d ^ p c n d b c b d cos E c cos E d 1 e d cos E d ^ q d ^ q c )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c a 2 c p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e c cos E c q v B r B E d n c a c a d sin E c sin E d 1 e c cos E c ^ p c ^ p d )]TJ/F21 11.9552 Tf 12.954 8.088 Td [(n c a c b d sin E c cos E d 1 e c cos E c ^ p c ^ q d )]TJ/F21 11.9552 Tf 10.298 8.087 Td [(n c b c a d cos E c sin E d 1 e c cos E c ^ q c ^ p d n c b c b d cos E c cos E d 1 e c cos E c ^ q c ^ q d )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d a 2 d p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d cos E d q B r B E c r a c a d sin E c p cos E d e d q ^ p c ^ p d )]TJ/F21 11.9552 Tf 11.76 0 Td [(a c b d sin E c sin E d ^ p c ^ q d a d b c cos E c p cos E d e d q ^ q c ^ p d b c b d cos E c sin E d ^ q c ^ q d )]TJ/F21 11.9552 Tf 9.102 0 Td [(a 2 c e c sin E c p 1 e c cos E c q B r B E d r a d a c sin E d p cos E c e c q ^ p d ^ p c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a d b c sin E d sin E c ^ p d ^ q c a c b d cos E d p cos E c e c q ^ q d ^ p c b c b d sin E c cos E d ^ q d ^ q c )]TJ/F21 11.9552 Tf 9.102 0 Td [(a 2 d e d sin E d p 1 e d cos E d q 6.5.1PolynomialFormulation Theextremalequationscanbeexpressedasanalgebraicsystembywritingthemina multivariatepolynomialform.Dening x 1 sin E c x 2 cos E c x 3 sin E d x 4 cos E d {7 95

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andwritingthedotproductsoftheperifocalbasisvectorsintermsofthedirectioncosine matrix T as T 11 ^ p c ^ p d T 21 ^ q c ^ p d T 12 ^ p c ^ q d T 22 ^ q c ^ q d : permitstheextremalequationstobewritteninamorecompactform,andthedot productsbecome: B v B E c r n c a 2 c r 2 c r a d a c p x 4 e d qp x 2 e c q T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c b d T 12 x 3 p x 2 e c q )]TJ/F21 11.9552 Tf 9.103 0 Td [(b c a d T 21 x 1 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b c b d T 22 x 1 x 3 a 2 c p 1 e c x 2 q 2 s B v B E d r n d a 2 d r 2 d r a c a d p x 4 e d qp x 2 e c q T 11 )]TJ/F21 11.9552 Tf 11.76 0 Td [(a c b d T 12 x 3 p x 2 e c q )]TJ/F21 11.9552 Tf 9.103 0 Td [(b c a d T 21 x 1 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b c b d T 22 x 1 x 3 a 2 d p 1 e d x 4 q 2 s v B r B E c 1 r d n d a 2 d a c x 1 x 3 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d a 2 d b c x 3 x 2 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d b d a c a d x 4 x 1 T 12 n d b c b d a d x 2 x 4 T 22 )]TJ/F21 11.9552 Tf 9.103 0 Td [(n c a 2 c p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e c x 2 q v B r B E d 1 r c n c a 2 c a d x 1 x 3 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c a 2 c b d x 1 x 4 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c b c a c a d x 2 x 3 T 21 n c a c b c b d x 2 x 4 T 22 )]TJ/F21 11.9552 Tf 9.102 0 Td [(n d a 2 d p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e d x 4 q || r || 2 a 2 d p 1 e d x 4 q 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 c p 1 e c x 2 q 2 2 r a d a c p x 4 e d qp x 2 e c q T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c b d T 12 x 3 p x 2 e c q )]TJ/F21 11.9552 Tf 9.102 0 Td [(b c a d T 21 x 1 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(b c b d T 22 x 1 x 3 s 96

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and v r )]TJ/F21 11.9552 Tf 28.167 0 Td [(n d a 2 d e d x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c a 2 c e c x 1 )]TJ/F21 11.9552 Tf 9.102 0 Td [(T 11 a c a d n d x 3 p x 2 e c q 1 e d x 4 )]TJ/F21 11.9552 Tf 12.955 8.088 Td [(a c a d n c x 1 p x 4 e d q 1 e c x 2 )]TJ/F21 11.9552 Tf 9.102 0 Td [(T 21 a d b c n d x 1 x 3 1 e d x 4 a d b c n c x 2 p x 4 e d q 1 e c x 2 )]TJ/F21 11.9552 Tf 9.102 0 Td [(T 12 a c b d n c x 1 x 3 1 e c x 2 a c b d n d x 4 p x 2 e c q 1 e d x 4 )]TJ/F21 11.9552 Tf 9.102 0 Td [(T 22 b c b d n d x 1 x 4 1 e d x 4 b c b d n c x 3 x 2 1 e c x 2 B r B E c r a c a d T 11 x 1 p x 4 e d q)]TJ/F21 11.9552 Tf 18.967 0 Td [(a c b d T 12 x 1 x 3 a d b c T 21 x 2 p x 4 e d q b c b d T 22 x 2 x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 c e c x 1 p 1 e c x 2 q B r B E d r a d a c T 11 x 3 p x 2 e c q a c b d T 12 x 4 p x 2 e c q )]TJ/F21 11.9552 Tf 9.102 0 Td [(a d b c T 21 x 1 x 3 b c b d T 22 x 1 x 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d e d x 3 p 1 e d x 4 q Thealgebraicsystemnowconsistsof t f 1 ;f 2 ;f 3 ;f 4 u : f 1 || r || 2 B v B E c r )]TJ/F21 11.9552 Tf 11.759 0 Td [( v B r B E c p v r q r B r B E c f 2 || r || 2 B v B E d r )]TJ/F21 11.9552 Tf 11.759 0 Td [( v B r B E d p v r q r B r B E d f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 f 4 x 2 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 4 1 : {8 Thegeneralrangerateextremalequationsarenowspecializedtotwospeciccases ofinterest,forwhichtherangerateextremalexpressionsaregreatlysimplied.Case1is thesimpliedscenarioofboththechiefanddeputyincircularcoplanarorbits.Case2is thescenariooftwocircularorbitswhicharenotcoplanar.Theeccentricanomalybecomes equivalenttothetrueandmeananomalyinthecaseofcircularorbits;theeccentric 97

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anomalynotationisretainedforconsistencyinanticipationofmoregeneral,continuing eorts. 6.5.2Case1 ThegeometryisshowninFigure6-1.Thesymmetryofthisscenariocanbeexploited todeterminetherangerateextremabyrecognizingthatifthepositionoftheouter satelliteisfrozen,thentherangeratefunctionwillrepeatitselfwiththesynodicperiodof thesystem.Insteadofparametrizingtheproblemintermsofbotheccentricanomalies,it issucienttospecifythedierenceineccentricanomalies E E d E c byassumingthat thetwoorbitsshareacommonlineofapsides.Therangeratecanthenbeexpressedas d dt || r || a c a d p n d n c q sin E b a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d 2 a c a d cos E : Dierentiatingwithrespectto E yieldstheextremalequation B D B E a c a d p n d n c q || r || cos E a c a d sin 2 E a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d 2 a c a d cos E 0 : Thebracketedtermistheonlyportionwhichcanvanish;theextremaoccurwhen cos E a c a d sin 2 E a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d 2 a c a d cos E 0 ; or g rr1 a c a d cos 2 E p a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d q cos E )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c a d 0 : Dening z cos E permitstheextremalconstrainttobewritteninthequadraticform a c a d z 2 p a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d q z )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c a d 0 ; sothattherootsare z p a 2 c )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 d q b p a 2 c a 2 d q 2 2 a c a d t a c a d ; a d a c u : Withoutlossofgenerality, z 1 isdesignatedastheminimumofthetworoots.Thesecond rootis 1andyieldsacomplexangle.Thisrootisnotphysical.Theextremalvalues 98

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are E extremal t cos 1 p z 1 q ; cos 1 p z 1 qu ,whichcorrespondtothedierencesinanomalies producingtheminimumandmaximumvaluesoftherangerate.Theextremalvaluesof E willoccurrepeatedlywiththesynodicperiodofthesystemasthegeometryreplicates itself.Forcommensuratesystemstheangularlocationofthisreplicationwillbexedwith respecttothelineofnodes.Forincommensuratesystemsthereplicationwilldriftaround intheorbitalplane. Figure6-1.CircularCoplanarGeometry 6.5.3Case2 Whenbothorbitsarecircularnotnecessarilycoplanar,theextremalequations naturallyfallintoapolynomialformwhentheauxiliaryvariablesinEquation6{7are 99

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used.Thedotproductsare: B v B E c r n c p r c r d r 2 c q B v B E d r n d p r c r d r 2 d q r c r d a c a d p x 2 x 4 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 x 3 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 1 x 4 T 21 )]TJ/F21 11.9552 Tf 11.76 0 Td [(x 1 x 3 T 22 q v B r B E c n d a d a c p x 1 x 3 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 x 2 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 1 x 4 T 12 x 2 x 4 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n c a 2 c v B r B E d n c a d a c p x 1 x 3 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 1 x 4 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 x 3 T 21 x 2 x 4 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n d a 2 d || r || 2 a 2 d )]TJ/F21 11.9552 Tf 11.759 0 Td [(a 2 c 2 a d a c p x 4 x 2 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 x 2 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 1 x 4 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 x 1 T 22 q v r a c a d rp n d x 3 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c x 1 x 4 q T 11 )-222(p n c x 1 x 3 n d x 2 x 4 q T 12 p n d x 1 x 3 n c x 2 x 4 q T 21 p n c x 2 x 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d x 1 x 4 q T 22 s B r B E c r a c a d p x 1 x 4 T 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 1 x 3 T 12 x 2 x 4 T 21 x 2 x 3 T 22 q B r B E d r a c a d p x 3 x 2 T 11 x 4 x 2 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 3 x 1 T 21 x 1 x 4 T 22 q Fornotationalconvenience,theconstants a 1 a 2 c a 2 d a 2 a 3 c a d )]TJ/F21 11.9552 Tf 11.759 0 Td [(a c a 3 d ; 100

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aredenedalongwiththeauxiliaryvariables C 1 x 3 p n c T 22 n d T 11 q a 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 4 p n d T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c T 21 q a 2 C 2 x 3 p T 12 n c )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 21 n d q a 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 4 p n c T 11 n d T 22 q a 2 C 3 x 2 3 p n c T 2 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d T 12 T 21 2 p n d T 11 T 22 n c T 2 22 qq a 1 x 2 4 p n c T 2 11 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 n d T 12 T 21 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 n c T 2 21 n d T 11 T 22 q a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 3 x 4 p n d p T 11 T 21 T 12 T 22 q n c p 2 T 11 T 12 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 T 21 T 22 qq a 1 C 4 x 2 3 p n d T 11 T 12 2 n c T 12 T 22 n d T 21 T 22 q a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 3 x 4 p 2 n c p T 12 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 11 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n d p T 2 11 T 2 12 T 2 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 22 qq a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 2 4 p n d T 21 T 22 2 n c T 11 T 21 n d T 11 T 12 q a 1 C 5 x 2 3 p 2 p n c T 2 12 )]TJ/F21 11.9552 Tf 11.76 0 Td [(n d T 12 T 21 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n d T 11 T 22 n c T 2 22 q a 1 x 3 x 4 p 4 n c T 11 T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d T 11 T 21 n d T 12 T 22 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 n c T 21 T 22 q a 1 x 2 4 p 2 n c T 2 11 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d T 12 T 21 )]TJ/F21 11.9552 Tf 11.76 0 Td [(n c T 2 21 2 n d T 11 T 22 q a 1 C 6 x 3 p n d T 22 n c T 11 q a 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 4 p n c T 12 )]TJ/F21 11.9552 Tf 11.76 0 Td [(n d T 21 q a 2 C 7 x 3 p n d T 12 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n c T 21 q a 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 4 p n d T 11 n c T 22 q a 2 C 8 x 2 3 p n c T 12 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(n d T 2 21 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 p n d T 2 22 n c T 11 T 22 qq a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 3 x 4 p n c p T 11 T 21 T 12 T 22 q 2 n d T 21 T 22 q a 1 x 2 4 p n c p 2 T 12 T 21 T 11 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n d p 2 T 2 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 22 qq a 1 C 9 x 2 3 p n c p T 11 T 12 T 21 T 22 q n d p 2 T 11 T 21 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 T 12 T 22 qq a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 3 x 4 p 2 n d p T 12 T 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 11 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n c p T 2 11 T 2 12 T 2 21 )]TJ/F21 11.9552 Tf 11.759 0 Td [(T 2 22 qq a 1 x 2 4 p n c p T 11 T 12 T 21 T 22 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n d p 4 T 11 T 21 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 T 12 T 22 qq a 1 C 10 x 2 3 p n d p T 2 11 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 T 2 12 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n c p 2 T 12 T 21 T 11 T 22 qq a 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(x 3 x 4 p 2 n d T 11 T 12 )]TJ/F21 11.9552 Tf 11.76 0 Td [(n c p T 12 T 22 T 11 T 21 qq a 1 x 2 4 p n d p T 2 12 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 T 2 11 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(n c p T 12 T 21 2 T 11 T 22 qq a 1 101

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whichpermitsthealgebraicsystemtobewrittenas F D t f 1 ;f 2 ;f 3 u with f 1 C 1 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 4 x 1 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 5 x 2 2 f 2 C 6 x 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 7 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 8 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 9 x 1 x 2 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 10 x 2 2 f 3 x 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(x 2 2 1 : {9 6.5.4Grobnerbasis Thesolutiontothecircularrangerateextremalequationsaregivenbytheane variety V rr tp x 1 ;x 2 ;C 1 ;:::;C 10 qP R 12 | f i p x 1 ;x 2 ;C 1 ; :::;C 10 q 0 ; for i 1 ; 2 ; 3 u : Ifthetermorderingisselectedas x 1 x 2 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 ; thenadegreereverselexicographicalgrevlexGrobnerbasiscanbeconstructedforthis system,consistingoftwentyeightpolynomials.Thetwentyeighthpolynomialconsists of568termsinvolving C i andisthereforeafunctionof E d only,denotedas g rr2 .Itis presentedinAppendixC.Thepolynomial g rr2 hasunitsofkm 32 s 8 .Thetermsin g rr2 are oftheorder C 8 i andeach C i isoftheorderof a 4 .Thislargedynamicrangecanintroduce dicultieswhennumericalrootndingtechniquesareapplied.Rootndingtechniques arefacilitatedbytheuseofEarthcanonicalunits 1,1 DU C R C .Thefunction g rr2 isatrigonometricpolynomialofdegreesixteenandcanhaveatmostthirtytwozeros intheinterval r 0 ; 2 s [116],althoughitwilltypicallyhavefarfewer.Numericaltoolscan beappliedto g rr2 todeterminethevaluesof E d whichsatisfytheextremalequations.For eachvalueof E d obtained,theremainingGrobnerbasisequationscanbeusedtosolve forthecorrespondingvaluesof E c whichsatisfytheaneequations.Selectingthebasis 102

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elementnumberselevenandfourteenpermitsalinearsystemoftheform A x 1 x 2 )]TJ/F21 11.9552 Tf 11.76 0 Td [(b 0 ; whichcanbesolvedfor x 1 sin E c and x 2 cos E c ,with A 11 C 2 C 3 C 6 C 7 C 2 C 5 C 6 C 7 C 2 2 C 6 C 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 6 C 8 C 4 C 5 C 7 C 8 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 1 p C 2 10 C 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 7 p C 3 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 5 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 8 q C 10 p C 2 C 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 5 C 8 q)]TJ/F21 11.9552 Tf 18.967 0 Td [(C 3 C 5 C 7 C 9 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 10 p C 2 2 C 6 C 3 C 5 C 6 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 C 4 C 8 C 2 C 3 C 9 q A 12 C 2 10 p C 2 C 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 1 C 4 q C 1 C 4 C 2 7 C 2 5 C 7 C 8 C 2 2 C 6 C 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 6 C 9 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 10 C 5 p C 4 C 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 3 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 8 C 1 C 9 q)]TJ/F21 11.9552 Tf 18.968 0 Td [(C 2 C 7 p C 4 C 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 1 C 9 q A 21 C 2 10 C 2 3 C 4 C 5 C 6 C 7 C 2 3 C 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 3 C 5 C 2 7 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 3 C 7 C 8 C 2 C 5 C 7 C 8 C 2 2 C 2 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 2 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 1 C 5 C 7 C 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 10 p C 2 2 C 8 2 C 3 C 5 C 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 p C 4 C 6 C 3 C 7 C 1 C 9 qq A 22 C 2 10 C 3 C 4 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 3 C 5 C 6 C 7 C 2 5 C 6 C 7 C 3 C 4 C 2 7 C 1 p C 10 C 2 C 5 C 7 qp C 10 C 8 q )]TJ/F21 11.9552 Tf 9.102 0 Td [(C 2 C 4 C 7 C 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 3 C 7 C 9 C 2 2 C 8 C 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 8 C 9 C 10 p C 2 C 3 C 6 C 2 C 5 C 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 4 C 5 C 8 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 3 C 5 C 9 q b 11 C 2 2 C 2 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 2 6 C 4 C 5 C 6 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 3 C 5 C 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 1 p C 2 10 C 2 7 q C 2 C 5 C 7 C 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 10 C 2 p C 4 C 6 C 3 C 7 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 C 8 q )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 1 p 2 C 10 C 5 C 6 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 6 C 7 C 10 C 2 C 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 5 C 7 C 9 q b 21 C 2 C 3 C 6 C 7 C 2 C 5 C 6 C 7 C 2 2 C 6 C 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 6 C 8 C 4 C 5 C 7 C 8 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 1 p C 2 10 C 3 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 7 p C 3 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 5 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 8 q C 10 p C 2 C 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 5 C 8 qq )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 3 C 5 C 7 C 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 10 p C 2 2 C 6 C 3 C 5 C 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 C 4 C 8 C 2 C 3 C 9 q : Solutionsofthislinearsystemmustsatisfythethetrigonometricidentity x 2 1 )]TJ/F21 11.9552 Tf 11.882 0 Td [(x 2 2 1 0. Usingthisapproachensuresthatallpossiblesolutionsoftheextremalequationswill befound.Asstatedabove,thefunction D isnotwellbehavedas || r || 0.Since therestrictiontocircularorbitshasbeenmade,thisprecludestheanalysisof1:1 commensuratesystems,where a c a d .Furthermore,theuseofcanonicalunitsis mosteectiveinrescaling g rr2 whentheorbitsdonotdiertoomuchinsize,e.g.,a 103

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systemconsistingofaLEOchiefandgeosynchronousdeputyisnotwellsuitedforthis technique.Numericalexamplesarenowpresentedtoshowtheecacyofthevelocity methodsderivedabove. 6.6VelocityMetricSimulations Todemonstratetheutilityoftheresultsoftheprecedingsections,thecaseoftwo satellitesfollowingorbitsdenedbytheorbitalparameterslistedpreviouslyinTable Table5-1isconsidered.Thevelocitymetricsarenowdemonstratedintheordertheywere derivedabove.Therelativespeedunivariate g speed isplottedinFigure6-2andtherelative speedmanifoldisshowninFigure6-3.Theunivariatepolynomials g v x g v y ,and g v z are plottedinFigures6-4A,6-4B,and6-4C.Themanifoldsfor v x v y ,and v z areshown inFigures6-5,6-6,and6-7,respectively.ThevelocitymanifoldisshowninFigure6-8in theLVLHframe.Sincethesystemiscommensurate,themanifolddoesnotcontactthe boundingboxandconsistsofareplicatingcurveinvelocityspace. Figure6-2.RelativeSpeedUnivariatePolynomial 104

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Figure6-3.RelativeSpeedManifold:1Commensurate 105

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ALocalVerticalVelocityUnivariatePolynomial BLocalHorizontalVelocityUnivariatePolynomial CCrossTrackVelocityUnivariatePolynomial Figure6-4.VelocityUnivariatePolynomials:1Commensurate 106

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Figure6-5.LocalVerticalVelocityExtrema:1Commensurate 107

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Figure6-6.LocalHorizontalVelocityExtrema:1Commensurate 108

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Figure6-7.CrossTrackVelocityExtrema:1Commensurate 109

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Figure6-8.RelativeVelocityTrajectory:1Commensurate 110

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Ifthesemi-majoraxisofthedeputyischangedto a d a c 3 b 2 p 0 : 95 q 2 9 ; 801 : 404 km,thentheresultingincommensuratesystemwillhavemetricswhicharealmostperiodic [119{122]intime.ThetimehistoryoftherelativespeedisshowninFigure6-9.Since T d T c 0 : 95 ? 2 ; theorbitalperiodsincommensurateandevenaftertwentychieforbits,theglobalextrema ofthespeedfunctionhavenotyetbeenrealized.Thedirectnessofthemethodology presentedhereisapparent;theGrobnerbasispermitsadirectcalculationoftheextrema withouttheneedforlongnumericalsimulations.Theglobalextremaforthissystemare listedinTable6-1. Table6-1.GlobalExtremaforVelocitySimulationsIncommensurate MetricMinimumkm/sMaximumkm/s || v || 0.15614.364 v x -7.2657.265 v y -14.3370.193 v z -5.8664.392 111

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Figure6-9.TimeHistoryofRelativeSpeedIncommensurateSystem 112

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Asnotedabove,therangerateformulationscannotbeusedtoanalyze1:1commensurate systems,ascase1collapsestoidenticalorbits,andcase2willcontainapointwherethe orbitsintersectandtherangerateisundened.Specifyingtwocircularorbitswiththe orbitalelementslistedinTable6-2,thecircularrangeratemetricsarenowdemonstrated. Forcase1above,thersttworowsofTable6-2areapplicable.Thequadratic function g rr1 forcos E isplottedinFigure6-10A.ThemanifoldisshowninFigure6-10B. ThegeometryforthisscenarioisillustratedinFigure6-11.Theextremalvaluesofthe rangerateare d dt || r || 2 : 0993km/s,whichoccurwhen E 40 : 26 .Forcase2above, theremainingrowsofTable6-2areapplicable.TheEarthcenteredinertialtrajectories areshowninFigure6-12.Theunivariatepolynomial g rr2 isplottedinFigure6-13and therangeratemanifoldisshowninFigure6-14.Forthisscenariotheglobalminimumis -5.934km/sandtheglobalmaximumis5.834km/s. Table6-2.OrbitalParametersforRangeRateSimulations:3Commensurateand Circular ChiefDeputy a10,05013,169.23km e00 3040deg i 40110deg 00deg f 0 512deg 113

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ARangeRateUnivariatePolynomial BRangeRateCircularCoplanarManifold Figure6-10.RangeRateCircularCoplanarExtrema:3Commensurate 114

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Figure6-11.RangeRateCircularCoplanarGeometry:3Commensurate 115

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Figure6-12.EarthCenteredInertialTrajectoriesforRangeRateSimulation 116

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Figure6-13.RangeRateUnivariatePolynomial 117

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Figure6-14.RangeRateCircularManifold 118

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Atangibleapplicationforthemetricsdemonstratedisinthedesignofconstellations. Iridiumwasmentionedasanexamplewhichrequiresinter-satellitecommunications capabilities.AnIridiumsimulationisnowpresentedasameansofapplyingthemetrics toarealisticexample.ThetwolineelementsTLEsforIridiumsatellites98and91were obtainedfromhttps://www.space-track.org.Theorbitalelementswereextractedfrom theseTLEsandsatellites98and91weredesignatedasthechiefanddeputy,respectively. Thestartingpointofthedeputyatepochwasadjustedtoaccountforthedierencein thetimestampoftheTLEs.Table6-3liststheorbitalelementsandepochinformation forthetwosatellites.TheinertialtrajectoriesofthetwosatellitesaredepictedinFigure 6-15.TheglobalextremaofthepositionandvelocitymetricsarelistedinTable6-4. Thesatellitesareinverynearlycircularorbits.Inordertoapplytherangeratemetric, theeccentricitiesweresettozero.TheabilitytocalculatetheparametersinTable6-4 representsanassettothehardwaredesigner,missionplanner,andconstellationmanager. Themetricsmayinuencemaneuverplanningaswellastheevaluationofthepost maneuverstateoftheconstellation. Table6-3.IridiumExample Iridium98Iridium91 Catalog#27,45127,371 TLE#86157 TLEEpochday297.7104day297.79972011 LaunchdateJune20February112002 a 7,126.99857,155.9363km e 0.00054750.0005103 164.429469.5172deg i 86.452386.394deg 113.7459121.0981deg M 0 246.4317239.0718deg TLEdataobtainedfromhttps://www.space-track.org 119

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Figure6-15.IridiumSimulationGeometry Table6-4.GlobalExtremaforIridiumSimulation MetricMinimumkmMaximumkm r 147.9314,282.41 x -14,283.07229.076 y -7,159.1837,159.183 z -7,130.5217,133.852 Minimumkm/sMaximumkm/s || d dt r || 0.01514.942 v x -7.4647.464 v y -14.942-0.015 v z -7.4427.435 d dt || r || -9.9749.974 120

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6.7EvaluationofMotionExtremaMethodology Theobjectiveoftheresearchpresentedherewastodevelopamethodologyfor calculatingtheextremalvaluesofcertainquantitiesrelevanttoformationight.Extremal valuesofthesequantitiescanbedeterminedusingpurelynumericaltechniques,however therearetwodrawbackstousingthesetechniques: 1.Solutionsfoundmaynotincludetheglobalextrema. 2.Thesolutionfoundmaynotbetheclosestsolutiontotheinitialguess. Allofthemetricsdiscussedherearefunctionsoftwovariables,andmaybeeasilyplotted andvisuallyexaminedasameansofqualitativelyevaluatingtheaccuracyofacquired solutions.Aninitialguessmayberepeatedlyadjustedasameansofseekingaparticular extremalpoint,althoughobtainingitisnotguaranteed.Themethodologyderivedabove, usingtheGrobnerbasisasameansofobtainingananevarietyaddressesbothofthe shortcomingsabove.Sinceallsolutionswillbefoundusingthemethodologyhere,the globalextremawillbeincludedinthesolutionset.Withallsolutionsknown,aninitial guessisnotknownandthereisnoneedtodetermineamethodforcalculatingone. AlthoughthemethodologyderivedinChapters5and6aboveaddressesthenumerical concernsabove,useofthistechniquerequiresarobustimplementationwithrespectto numericalaccuracy.Thetwoundesirablesituationswhichcanoccurare: 1.Anextremalsolutioncanbemissed. 2.Falsepositivesolutionswhichdonotsatisfytheextremalequationscanbereturned. Inbothcasestheseeectsarenumericalartifacts.Therstconcernistheaccuracy numericalrootndingoftheunivariatepolynomials g x ;g y ;g v x ;g v y ;g speed ,and g rr2 .Ifa rootofoneofthesepolynomialsismissed,thencase#1willoccur;anextremalsolution orinsomecases,multiplesolutions,willnotbecalculated.Ifafalserootoftheunivariate polynomialisreturned,thencase#2willoccur.Inthissituation,apointinthestate spaceiscalculatedwhichisneitheraminimum,maximum,orinectionpoint.Ifoneis 121

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onlyinterestedintheglobalextrema,thenthiscaseisnotofconcern,asitwillnotaect theoutcome,whereascase#1may. Thesecondconcernisintheaccuracyofsolvingforthevaluesofthechiefanomaly afterthedeputy'sanomalyhasbeenfound.Thisbacksolvingprocessvariesforeach Grobnerbasis,butinallcasestheprocessamountstostartingwithanitenumberof solutionsandremovingelementswhicharenotphysical.Roundoerrorinthesineand cosineofthechiefanomalycanshifttheoutputoftheinversetangentfunction.Ifthesine orcosineofananomalyiscalculatedtobeoutsideoftheinterval r 1 ; 1 s duetoroundo error,thensuchasolutionwillbeneglectedandcase#1willoccur. Ifthemethodologypresentedhereisimplementedinanumericallyrobustmannerso astoguardagainstthesepotentialconcerns,thenthismethodrepresentsapowerfultool forcharacterizingcertainaspectsofformationight.Anyfalsepositivesolutionsincase #2canbeguardedagainstbysimplyevaluatingtheextremalequationsforeachsolution andcomparingthefunctionevaluationswithasucientlysmallthresholdvalue.When careistakenintheunivariaterootnding,andbacksolvingmethods,bothconcernslisted abovecanbenegated. 122

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CHAPTER7 STATESPACEGEOMETRY 7.1StateSpaceEvolution Allofthemetricsdiscussedaboveinvolvedfunctionsoftrueand/oreccentric anomaliesofthechiefanddeputy.Themetricsarefunctionsoftwoindependentvariables p q 1 ;q 2 q whichdenethestateofthesystem,withtheexceptionofthecrosstrackmetrics whichdependonthestateofthedeputyonly.Variouscombinationsofthesevariables wereusedtorepresentthestatespaceofthesystemforthemetrics,aslistedinTable7-1. Table7-1.StateSpaceVariables q 1 q 2 Metric E c E d || r || f c f d || v || f c E d x;y;v x ;v y E d z;v z E c E d d dt || r || Givenapoint p inthestatespacedenedas p p q 1 ;q 2 qP R 2 ; theevolutionofthesystemisgovernedbythedynamicsofthestatevariables.The evolutionof p isgivenby 9 p whichfollowsfrom 9 f h p 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e cos f q 2 a 2 p 1 e 2 q 2 9 E n 1 e cos E ; dependingonwhichofthestatevariablesinTable7-1areused.Thestatespacevelocity 9 p isaboundvectoreldin R R .Thetrajectoryisgivenbyintegralcurvesofthisvector eldforprescribedinitialconditions.Anexampleofsuchanintegralcurveisshownin Figure7-1.Atrajectoryinthestatespaceisdeterminedbyitsinitialconditions p q 1 0 ;q 2 0 q andtheelements t a c ;e c ;a d ;e d u .ThecontoursinFigure7-1arelevelsetsoftherange 123

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function,givenby tp E c ;E d q| r p E c ;E d q c u ; forsome c P R .Eachmetrichasaboundvectoreldassociatedwithit,givenbyits gradient.Let X representametricofinterest.Thenormalvectoreldisgivenby N r X B X B q 1 ; B X B q 2 ; whicharepreciselytheextremalequations.Thegradientatapoint p isalwaysorthogonal tothesurfaceproducedbyalevelsetpassingthrough p .Anexampleofthegradient eldfortherangefunctionisshowninFigure7-2.Givenapoint p onalevelset r c thenormalvectoreldcanbeusedtodetermineifthesystemismovingtowardsgreater orlesservaluesoftherangeoranyothermetric.Let V denotetheboundvectoreld producedby 9 p .Then V N 0themetricisincreasing V N 0themetricisdecreasing V N 0themetricisinstantaneouslyunchanging. Inthismanner,agivepointinthestatespacecanbeanalyzedtodeterminehowvarious metricsarechanging.Thisstatespacevisualizationprovidesalucidmethodforexamining thesystemevolution,andisasourceofcontinuingresearchforpurposesofmission planningandmaneuverdesign. 124

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Figure7-1.IntegralCurveofEccentricAnomalyStateSpace:2Commensurate 125

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Figure7-2.GradientofRangeFunction 126

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7.2LineofSightObstruction Theformationightmotionextremadiscussedabovewereboundedglobally,overall possiblepositioncombinations.Somevaluesofthesemetricsarenotphysicallymeasurable whenthelineofsightbetweensatellitesisobstructedbytheEarth.Acomputationally ecientmethodologyfordeterminingEarthobstructionisnowpresented.Itisequivalent toVallado'salgorithm[23]fordetermininggroundstationlineofsightavailability.The convexvectortransformation[123] V r d )-222(p 1 q r c ; with Pr 0 ; 1 s representsamappingfrom r c to r d as :0 1.Foranyvalue Pr 0 ; 1 s V isavectorpointingfromthecenteroftheEarthtoapointalongthelineconnectingthe chiefanddeputy.Thenormof V is || V || b 2 || r d || 2 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 p 1 q r c r d )-222(p 1 q 2 || r c || 2 : Theminimumof || V || canbefoundbydierentiatingtoobtain d d || V |||| V || 1 { 2 || r d || 2 )-222(|| r c || 2 p 1 q)]TJ/F45 11.9552 Tf 18.967 0 Td [(r c r d p 1 2 q 0 : Since || V || 0exceptinthesingularcaseofradiallyopposedsatellies,thebracketedterm vanisheswhen || r d || 2 )-222(|| r c || 2 p 1 q)]TJ/F45 11.9552 Tf 18.967 0 Td [(r c r d p 1 2 q 0 ; or m r c p r c r d q p r d r c qp r d r c q r c r || r || 2 : Thisvalueof m yieldsmin || V || .Twogeometricsituationsarepossible,asdepictedin Figure7-3.If m Pr 0 ; 1 s thenthepointofclosestapproachliesbetween r c and r d as showninFigure7-3A.If m Rr 0 ; 1 s thenmin p|| r c || ; || r d ||q isthepointofclosestapproach asshowninFigure7-3B.Theextremalvalue m canbecalculatedandthepointofclosest approachcanbecomparedto R C .If || V m || R C ,thenthelineofsightisobstructed 127

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bytheEarth.Thisiscomputationallyecientanddenitive.Thistechniqueassumesa sphericallysymmetricEarth.Theimportanceofdetermininglineofsightavailabilityis germanetodiscussionsofspacesituationalawarenessandmissionplanning.Boundingthe positionandvelocitymetricsderivedaboveishelpfulindesigningsensorsandactuators forighthardware.Ifaparticularextremumoccurswhenthelineofsightisobscuredby Earth,theninsomecasesalesscapablepieceofhardwaremaybeacceptableforamission sincetheextremuminquestionwillneverberealizable. A:0 m 1 B: m 0or m 1 Figure7-3.PointsofMinimumDistanceBetweenEarthandLineofSightVector 128

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CHAPTER8 CONCLUSION Themajorityofproblemsintheeldofastrodynamicsdonothavepurelyanalytical solutionsavailablefortheirdescription.Thedierentialequationsdescribingthemotion ofonespacecraftrelativetoasecondhaveproventobeparticularlyelusivewithregards toanalytictechniques,asevidencedbythemanyformulationsintheliterature.Alternate descriptiontechniquessuchasgeometricformulationsarebeingusedtofurtherelucidate thebehaviorofsatellitesinformationight.Althoughthemacroscopicnatureofsuch geometrictechniquesdonotprovidethedetailthatananalyticsolutiontothedierential equationwould,suchtechniquesrevealcomplementaryinformation.Thecontributionof thisresearchwastofurtherthisapproach. Dierentialequationsareprimarilyusedformodelingthetrajectoryoftheformation throughthestatespaceandforcontrollersynthesis.Characterizationofcertainquantities ofinterestandtheirextremalvaluesinparticularisnecessarytopresentacomplete descriptionofasatelliteformation.Severalmetricsofsatelliteformationightwere derivedandboundedthroughthedeterminationoftheirextremalvalues.Theposition basedmetricsdiscussedweretherelativepositioninthelocalvertical,localhorizontal, andcrosstrackdirections.Amethodologyforcomputingalloftheextremalvaluesof theseexcursionswasdetailed.Thecrosstrackseparationfunctionwaspresented,and itsmeanvaluewasdeterminedanalytically.Thecrosstrackseparationisrelevantto Earthobservationmissionssuchasstereoimagingandsyntheticapertureradar.Such coordinatedmissionsrelyonknowledgeofhowmuchseparationthesystemmusttolerate. Thecombinationofthethreepositionmetricsprovidesamoredetaileddescriptionofthe relativemotionpositionmanifoldthanwaspreviouslyavailable.Thepositionofadeputy satelliterelativetoachiefsatellitecannowbedescribedaslyingbetweentwoconcentric spheres,aswellaswithinaprismaticregioninthelocalvertical,localhorizontalreference 129

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frame.Thelimitationsofthemetricboundswasalsodiscussedwithregardtowhetheror notasystemiscommensurate. Thevelocitybasedmetricsdescribethesizeofthevelocitymanifoldinthelocal vertical,localhorizontal,andcrossdirections.Inadditiontherelativespeedwasalso presented.Finally,therangeratewasderived,andamethodologyforcomputingits boundswasdetailedforthespecialcaseoftwocircularorbits.Therelativevelocityof adeputysatelliterelativetoachiefsatellitecannowbedescribedaslyingbetweentwo concentricspheresandwithinaprismaticregioninthelocalvertical,localhorizontal referenceframe. Theextremalvaluesforsixofthemetricsabovewerefoundbytransformingthe extremalequationsintoanalgebraicsystem,andusingaGrobnerbasistosolvetheane varietyusingdierentequationsfromthesameideal.Numericaltechniques,specically rootndingwereavoideduntilthepenultimatestep,extendingtheaccuracyofthis approachovertraditionalnumericalsearches.Furthermore,theapproachpresentedhereis particularlyusefulinsituationsinvolvingincommensurateorbits,wherenumericalsearches wouldotherwiseberequiredoveraninnitetimedomainandcannotguaranteethatthe globalextremawillbelocated.Themethodologyispredicatedontheassumptionthat thetwosatellitesareinboundKeplerianorbitsandnoapproximationswereutilized.The contextusedinthisresearchhasbeenthatofEarthorbitingsatelliteswithrelevance tocommunicationsconstellations,Earthobservation,spacesituationalawareness,and proximityoperations.ThemethodologyisnotrestrictedtoEarthbasedorbitshowever, andmaybeappliedequallywelltoothergravitationalbodies. Acontinuationoftheseeortsisongoing.Amethodologyforboundingthegeneral noncircularrangerateextremaisbeingsought.Furtherinsighttothebehaviorthe relativemotionpositionandvelocitymanifoldsmaybeobtainedthroughtheuseoftools fromdierentialgeometry,andpossiblyatopologicalanalysis.Inadditiontofurther renementofthedescriptionofthemanifolds,itisdesirabletoexploitthedynamicsfor 130

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purposesofsatellitemaneuvers.Levelsetshavebeenusedinsomepreliminaryanalysis formissionplanningofrendezvousandproximityoperations.Itmayalsobepossibleto conductmaneuverplanningfromageometricperspective,usingthemanifoldspresented above. Thestatespacediscussionofsection7.1servesasthebasisofadditionalongoing eorts.Thelevelsetsoftherangefunctionindicatewhenthedeputyiswithinaballof speciedradius,centeredonthechief.Sincethenormalvectoreldprovidesamechanism forclassifyingpointsonthelevelsetaseitherincreasing,decreasing,orconstantrange, pointsinthestatespacecanbemarkedtoindicatewherethesatelliteareapproachingone anotherandwhentheyareseparating.Anoptimizationproblemisbeingformulatedto determinewhichpointsmaximizecontacttimetimespentbythedeputyinthespecied ballforpurposesofproximityoperations.Theconverseproblem,thatofndingwhich pointsmaximizethetimethedeputyspendsoutsideoftheballwouldbebenecial forsatelliteevasionapplications.Similaroptimizationproblemscanbeformulatedto minimizeormaximizeagivenmetricthroughmaneuvers. 131

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APPENDIXA REFERENCEFRAMES TheEarthcenteredinertialECIandorbitalorperifocalPQWreferenceframes areillustratedinFigureA-1.Thedirectioncosinematrixwhichrelatestheorbitalplane totheequatorialplanewasgiveninEquation2{13.AnedgeviewofFigureA-1isshown inFigureA-2. FigureA-1.OrbitalElements 132

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FigureA-2.OrbitalPlaneGeometryProgradeExample 133

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APPENDIXB ASTRODYNAMICRELATIONS Thetrueanomaly f inFigureA-1canberelatedtotheeccentricanomaly E in FigureB-1viaGauss'equation tan f 2 d 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(e 1 e tan E 2 ; whichisvalidfororbitswitheccentricity e 1.Theeccentricanomalycanberelatedto timethroughKepler'sequation, M n p t q E e sin E: Themeananomaly M isafunctionof p t q ,theelapsedtimesincelastperiapsepassage. Kepler'sequationcanbeobtainedfromGauss'equation,andcanalsobederivedusing geometricarguments,orviadirectintegrationoftermsinvolvingtrigonometricfunctions of f and E 134

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FigureB-1.EccentricAnomalyGeometry 135

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APPENDIXC TRIGONOMETRICPOLYNOMIALFORCIRCULARRANGERATE Theunivariatepolynomialwhoserootsyieldthedeputyeccentricanomalyforthe rangerateextremaoftwocircularorbitsis g rr2 p C 4 10 q C 4 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 10 C 2 7 C 4 1 C 2 10 C 2 8 C 4 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 7 C 2 8 C 4 1 C 2 10 C 2 9 C 4 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 7 C 2 9 C 4 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 3 10 C 8 C 4 1 2 C 10 C 2 7 C 8 C 4 1 2 C 10 C 4 C 3 7 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 3 9 C 3 1 2 C 5 C 7 C 3 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 3 C 6 C 2 7 C 3 1 2 C 10 C 5 C 6 C 2 7 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 5 C 6 C 2 8 C 3 1 2 C 2 C 6 C 7 C 2 8 C 3 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 5 C 6 C 2 9 C 3 1 2 C 2 C 6 C 7 C 2 9 C 3 1 2 C 3 10 C 3 C 6 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 10 C 5 C 6 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 10 C 4 C 7 C 3 1 2 C 2 10 C 2 C 6 C 7 C 3 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 4 C 3 7 C 8 C 3 1 2 C 3 C 6 C 2 7 C 8 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 5 C 6 C 2 7 C 8 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 3 C 6 C 8 C 3 1 6 C 2 10 C 5 C 6 C 8 C 3 1 2 C 2 10 C 4 C 7 C 8 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 6 C 7 C 8 C 3 1 2 C 3 C 3 7 C 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 5 C 3 7 C 9 C 3 1 2 C 4 C 6 C 2 7 C 9 C 3 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 10 C 2 C 2 8 C 9 C 3 1 2 C 5 C 7 C 2 8 C 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 10 C 2 C 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 4 C 6 C 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 3 C 7 C 9 C 3 1 4 C 2 10 C 5 C 7 C 9 C 3 1 4 C 2 10 C 2 C 8 C 9 C 3 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 5 C 7 C 8 C 9 C 3 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 3 C 4 7 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 4 C 4 7 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 4 7 C 2 1 2 C 3 C 5 C 4 7 C 2 1 C 2 2 C 4 9 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 4 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 3 C 3 7 C 2 1 2 C 10 C 2 C 5 C 3 7 C 2 1 2 C 10 C 2 2 C 3 8 C 2 1 2 C 10 C 2 5 C 3 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 5 C 7 C 3 8 C 2 1 2 C 10 C 4 C 5 C 3 9 C 2 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 4 C 7 C 3 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 4 10 C 2 3 C 2 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 2 10 C 2 2 C 2 6 C 2 1 C 2 10 C 2 3 C 2 6 C 2 1 C 2 10 C 2 4 C 2 6 C 2 1 6 C 2 10 C 2 5 C 2 6 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 3 C 5 C 2 6 C 2 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 2 10 C 2 2 C 2 7 C 2 1 3 C 2 10 C 2 3 C 2 7 C 2 1 C 2 10 C 2 4 C 2 7 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 3 C 2 6 C 2 7 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 4 C 2 6 C 2 7 C 2 1 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 2 5 C 2 6 C 2 7 C 2 1 2 C 3 C 5 C 2 6 C 2 7 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 3 C 5 C 2 7 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 C 4 C 6 C 2 7 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 10 C 2 2 C 2 8 C 2 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 2 10 C 2 4 C 2 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 5 C 2 8 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 6 C 2 8 C 2 1 C 2 5 C 2 6 C 2 8 C 2 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 2 C 2 7 C 2 8 C 2 1 2 C 2 4 C 2 7 C 2 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 C 2 5 C 2 7 C 2 8 C 2 1 4 C 3 C 5 C 2 7 C 2 8 C 2 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 2 10 C 3 C 5 C 2 8 C 2 1 2 C 10 C 2 C 4 C 6 C 2 8 C 2 1 10 C 10 C 2 C 5 C 7 C 2 8 C 2 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 4 C 5 C 6 C 7 C 2 8 C 2 1 C 2 10 C 2 2 C 2 9 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 10 C 2 3 C 2 9 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 10 C 2 4 C 2 9 C 2 1 )]TJ/F21 11.9552 Tf 9.102 0 Td [(C 2 2 C 2 6 C 2 9 C 2 1 C 2 5 C 2 6 C 2 9 C 2 1 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 2 C 2 7 C 2 9 C 2 1 C 2 3 C 2 7 C 2 9 C 2 1 C 2 4 C 2 7 C 2 9 C 2 1 2 C 2 5 C 2 7 C 2 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 C 5 C 2 7 C 2 9 C 2 1 C 2 2 C 2 8 C 2 9 C 2 1 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 5 C 2 8 C 2 9 C 2 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 10 C 3 C 5 C 2 9 C 2 1 6 C 10 C 2 C 4 C 6 C 2 9 C 2 1 4 C 10 C 2 C 3 C 7 C 2 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 5 C 7 C 2 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 4 C 5 C 6 C 7 C 2 9 C 2 1 4 C 10 C 2 5 C 8 C 2 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 5 C 7 C 8 C 2 9 C 2 1 4 C 3 10 C 2 C 4 C 6 C 2 1 4 C 10 C 2 C 3 C 2 6 C 7 C 2 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 2 C 5 C 2 6 C 7 C 2 1 2 C 3 10 C 2 C 3 C 7 C 2 1 2 C 2 10 C 4 C 5 C 6 C 7 C 2 1 2 C 2 C 3 C 3 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 2 C 5 C 3 7 C 8 C 2 1 2 C 3 10 C 2 2 C 8 C 2 1 2 C 3 10 C 2 3 C 8 C 2 1 2 C 3 10 C 2 4 C 8 C 2 1 2 C 10 C 2 2 C 2 6 C 8 C 2 1 136

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)]TJ/F15 11.9552 Tf 9.102 0 Td [(6 C 10 C 2 5 C 2 6 C 8 C 2 1 4 C 10 C 3 C 5 C 2 6 C 8 C 2 1 2 C 10 C 2 2 C 2 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 3 C 2 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 10 C 2 4 C 2 7 C 8 C 2 1 4 C 10 C 2 5 C 2 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 3 C 5 C 2 7 C 8 C 2 1 6 C 2 C 4 C 6 C 2 7 C 8 C 2 1 4 C 3 10 C 3 C 5 C 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 2 C 4 C 6 C 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 3 C 2 6 C 7 C 8 C 2 1 4 C 2 C 5 C 2 6 C 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 2 10 C 2 C 3 C 7 C 8 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 2 C 5 C 7 C 8 C 2 1 2 C 2 C 4 C 3 7 C 9 C 2 1 4 C 10 C 4 C 5 C 2 6 C 9 C 2 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 10 C 3 C 4 C 2 7 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 4 C 5 C 2 7 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 C 3 C 6 C 2 7 C 9 C 2 1 6 C 2 C 5 C 6 C 2 7 C 9 C 2 1 2 C 10 C 4 C 5 C 2 8 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 4 C 7 C 2 8 C 9 C 2 1 2 C 3 10 C 3 C 4 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 C 3 C 6 C 9 C 2 1 2 C 2 10 C 2 C 5 C 6 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 4 C 2 6 C 7 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 10 C 2 5 C 6 C 7 C 9 C 2 1 8 C 10 C 3 C 5 C 6 C 7 C 9 C 2 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 3 C 4 C 2 7 C 8 C 9 C 2 1 6 C 4 C 5 C 2 7 C 8 C 9 C 2 1 2 C 2 10 C 3 C 4 C 8 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 4 C 5 C 8 C 9 C 2 1 4 C 10 C 2 C 3 C 6 C 8 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 5 C 6 C 8 C 9 C 2 1 4 C 2 5 C 6 C 7 C 8 C 9 C 2 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 C 5 C 6 C 7 C 8 C 9 C 2 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 10 C 3 5 C 3 6 C 1 6 C 10 C 3 C 2 5 C 3 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 2 C 3 C 3 6 C 1 2 C 10 C 2 2 C 5 C 3 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 3 C 5 C 3 6 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 10 C 2 4 C 5 C 3 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 3 C 4 C 3 7 C 1 2 C 2 C 2 3 C 6 C 3 7 C 1 2 C 2 C 2 4 C 6 C 3 7 C 1 2 C 2 C 2 5 C 6 C 3 7 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 2 C 3 C 5 C 6 C 3 7 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 10 C 2 C 4 C 5 C 3 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 5 C 6 C 3 8 C 1 2 C 2 2 C 5 C 6 C 3 8 C 1 4 C 4 C 2 5 C 7 C 3 8 C 1 4 C 2 2 C 4 C 7 C 3 8 C 1 2 C 10 C 2 C 2 3 C 3 9 C 1 2 C 4 C 2 5 C 6 C 3 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 4 C 6 C 3 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 3 C 5 C 7 C 3 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 3 2 C 8 C 3 9 C 1 2 C 2 C 2 5 C 8 C 3 9 C 1 4 C 2 10 C 2 C 3 C 4 C 2 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 2 10 C 2 C 4 C 5 C 2 6 C 1 2 C 10 C 3 3 C 6 C 2 7 C 1 4 C 10 C 3 C 2 4 C 6 C 2 7 C 1 4 C 10 C 2 2 C 3 C 6 C 2 7 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 2 C 5 C 6 C 2 7 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 3 C 5 C 6 C 2 7 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(8 C 2 C 3 C 4 C 2 7 C 2 8 C 1 4 C 2 C 4 C 5 C 2 7 C 2 8 C 1 8 C 2 10 C 2 C 3 C 4 C 2 8 C 1 8 C 2 10 C 2 C 4 C 5 C 2 8 C 1 4 C 10 C 3 5 C 6 C 2 8 C 1 2 C 10 C 3 C 2 5 C 6 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(6 C 10 C 2 2 C 3 C 6 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 2 C 5 C 6 C 2 8 C 1 2 C 10 C 2 4 C 5 C 6 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 4 C 2 5 C 7 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 2 C 4 C 7 C 2 8 C 1 2 C 3 2 C 6 C 7 C 2 8 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 C 2 4 C 6 C 7 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 2 C 2 5 C 6 C 7 C 2 8 C 1 4 C 2 C 3 C 5 C 6 C 7 C 2 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 2 2 C 3 C 6 C 2 9 C 1 2 C 10 C 2 3 C 5 C 6 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 4 C 5 C 6 C 2 9 C 1 2 C 3 2 C 6 C 7 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 2 3 C 6 C 7 C 2 9 C 1 4 C 2 C 2 4 C 6 C 7 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 2 5 C 6 C 7 C 2 9 C 1 4 C 2 C 3 C 5 C 6 C 7 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 2 C 3 C 4 C 8 C 2 9 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 2 C 4 C 5 C 8 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 5 C 6 C 8 C 2 9 C 1 2 C 3 C 2 5 C 6 C 8 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 3 C 6 C 8 C 2 9 C 1 4 C 2 2 C 5 C 6 C 8 C 2 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 4 C 2 5 C 7 C 8 C 2 9 C 1 6 C 2 2 C 4 C 7 C 8 C 2 9 C 1 4 C 3 C 4 C 5 C 7 C 8 C 2 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 3 10 C 3 3 C 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 10 C 3 C 2 4 C 6 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 10 C 2 2 C 3 C 6 C 1 4 C 3 10 C 2 3 C 5 C 6 C 1 2 C 2 C 2 3 C 3 6 C 7 C 1 137

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2 C 2 C 2 4 C 3 6 C 7 C 1 2 C 2 C 2 5 C 3 6 C 7 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 3 C 5 C 3 6 C 7 C 1 2 C 10 C 4 C 2 5 C 2 6 C 7 C 1 6 C 10 C 2 2 C 4 C 2 6 C 7 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 3 C 4 C 5 C 2 6 C 7 C 1 2 C 3 10 C 2 3 C 4 C 7 C 1 2 C 2 10 C 3 2 C 6 C 7 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 2 10 C 2 C 2 3 C 6 C 7 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 C 2 4 C 6 C 7 C 1 4 C 2 10 C 2 C 3 C 5 C 6 C 7 C 1 2 C 3 5 C 3 6 C 8 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 3 C 2 5 C 3 6 C 8 C 1 2 C 2 2 C 3 C 3 6 C 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 2 C 5 C 3 6 C 8 C 1 2 C 3 4 C 3 7 C 8 C 1 2 C 4 C 2 5 C 3 7 C 8 C 1 4 C 2 3 C 4 C 3 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 C 4 C 5 C 3 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 3 C 4 C 2 6 C 8 C 1 8 C 10 C 2 C 4 C 5 C 2 6 C 8 C 1 8 C 10 C 2 C 3 C 4 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 4 C 5 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 5 C 6 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 C 2 4 C 6 C 2 7 C 8 C 1 10 C 3 C 2 5 C 6 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 2 C 3 C 6 C 2 7 C 8 C 1 4 C 2 2 C 5 C 6 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 3 C 5 C 6 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 2 4 C 5 C 6 C 2 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 3 10 C 2 C 3 C 4 C 8 C 1 2 C 2 10 C 3 C 2 4 C 6 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 2 10 C 3 C 2 5 C 6 C 8 C 1 10 C 2 10 C 2 2 C 3 C 6 C 8 C 1 2 C 2 10 C 2 3 C 5 C 6 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 3 4 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 4 C 2 5 C 2 6 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(6 C 2 2 C 4 C 2 6 C 7 C 8 C 1 4 C 3 C 4 C 5 C 2 6 C 7 C 8 C 1 2 C 2 10 C 2 2 C 4 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 10 C 2 3 C 4 C 7 C 8 C 1 4 C 2 10 C 3 C 4 C 5 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 3 2 C 6 C 7 C 8 C 1 4 C 10 C 2 C 2 3 C 6 C 7 C 8 C 1 4 C 10 C 2 C 2 4 C 6 C 7 C 8 C 1 4 C 10 C 2 C 2 5 C 6 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 10 C 2 C 3 C 5 C 6 C 7 C 8 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 4 C 2 5 C 3 6 C 9 C 1 2 C 2 2 C 4 C 3 6 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 3 3 C 3 7 C 9 C 1 2 C 2 3 C 5 C 3 7 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 3 8 C 9 C 1 2 C 2 C 2 5 C 3 8 C 9 C 1 2 C 3 10 C 2 C 2 3 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 10 C 2 C 2 3 C 2 6 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 C 2 4 C 2 6 C 9 C 1 2 C 10 C 2 C 2 5 C 2 6 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 4 C 6 C 2 7 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 4 C 2 5 C 6 C 2 7 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 2 C 4 C 6 C 2 7 C 9 C 1 4 C 10 C 3 2 C 2 8 C 9 C 1 2 C 10 C 2 C 2 4 C 2 8 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(6 C 10 C 2 C 2 5 C 2 8 C 9 C 1 4 C 10 C 2 C 3 C 5 C 2 8 C 9 C 1 4 C 3 5 C 7 C 2 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 3 C 2 5 C 7 C 2 8 C 9 C 1 2 C 2 2 C 3 C 7 C 2 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 5 C 7 C 2 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 4 C 5 C 7 C 2 8 C 9 C 1 2 C 2 10 C 3 4 C 6 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 10 C 2 2 C 4 C 6 C 9 C 1 4 C 2 10 C 3 C 4 C 5 C 6 C 9 C 1 2 C 2 10 C 3 3 C 7 C 9 C 1 4 C 3 5 C 2 6 C 7 C 9 C 1 )]TJ/F15 11.9552 Tf 9.103 0 Td [(6 C 3 C 2 5 C 2 6 C 7 C 9 C 1 6 C 2 2 C 3 C 2 6 C 7 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 2 C 5 C 2 6 C 7 C 9 C 1 2 C 2 3 C 5 C 2 6 C 7 C 9 C 1 6 C 2 4 C 5 C 2 6 C 7 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 10 C 2 3 C 5 C 7 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 3 C 4 C 6 C 7 C 9 C 1 8 C 10 C 2 C 4 C 5 C 6 C 7 C 9 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 2 10 C 3 2 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 2 C 2 3 C 8 C 9 C 1 2 C 2 10 C 2 C 2 4 C 8 C 9 C 1 2 C 2 C 2 3 C 2 7 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(6 C 2 C 2 4 C 2 7 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 2 5 C 2 7 C 8 C 9 C 1 4 C 2 10 C 2 C 3 C 5 C 8 C 9 C 1 4 C 10 C 4 C 2 5 C 6 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 9.102 0 Td [(8 C 10 C 3 C 4 C 5 C 6 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 2 C 3 C 7 C 8 C 9 C 1 4 C 10 C 2 2 C 5 C 7 C 8 C 9 C 1 4 C 10 C 2 3 C 5 C 7 C 8 C 9 C 1 4 C 10 C 2 4 C 5 C 7 C 8 C 9 C 1 8 C 2 C 3 C 4 C 6 C 7 C 8 C 9 C 1 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 2 C 4 C 5 C 6 C 7 C 8 C 9 C 1 C 4 10 C 4 3 C 4 5 C 4 6 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 3 C 3 5 C 4 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 3 C 4 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 4 C 4 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 5 C 4 6 C 2 3 C 2 5 C 4 6 C 2 4 C 2 5 C 4 6 2 C 2 2 C 3 C 5 C 4 6 C 4 3 C 4 7 C 2 3 C 2 4 C 4 7 C 2 3 C 2 5 C 4 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 3 C 5 C 4 7 C 4 2 C 4 8 C 4 5 C 4 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 2 5 C 4 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 3 C 4 9 138

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C 2 3 C 2 5 C 4 9 2 C 10 C 2 C 3 4 C 3 6 4 C 10 C 2 C 4 C 2 5 C 3 6 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 3 2 C 4 C 3 6 2 C 10 C 2 C 2 3 C 4 C 3 6 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 2 C 3 C 4 C 5 C 3 6 2 C 10 C 2 C 3 3 C 3 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 2 3 C 5 C 3 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 C 3 4 C 6 C 3 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 C 4 C 2 5 C 6 C 3 7 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 3 3 C 4 C 6 C 3 7 4 C 2 3 C 4 C 5 C 6 C 3 7 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 4 2 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 3 C 3 5 C 3 8 2 C 10 C 2 2 C 2 4 C 3 8 2 C 10 C 2 2 C 2 5 C 3 8 2 C 10 C 2 4 C 2 5 C 3 8 4 C 10 C 2 2 C 3 C 5 C 3 8 2 C 2 C 4 C 2 5 C 6 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 4 C 6 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 3 5 C 7 C 3 8 4 C 2 C 3 C 2 5 C 7 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 2 C 3 C 7 C 3 8 2 C 3 2 C 5 C 7 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 2 4 C 5 C 7 C 3 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 3 C 4 C 5 C 3 9 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 C 3 C 2 5 C 6 C 3 9 2 C 3 2 C 3 C 6 C 3 9 2 C 2 C 2 3 C 4 C 7 C 3 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 C 4 C 2 5 C 8 C 3 9 2 C 2 2 C 3 C 4 C 8 C 3 9 )]TJ/F21 11.9552 Tf 9.102 0 Td [(C 2 10 C 4 2 C 2 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 10 C 4 4 C 2 6 )]TJ/F15 11.9552 Tf 11.759 0 Td [(3 C 2 10 C 2 2 C 2 3 C 2 6 2 C 2 10 C 2 2 C 2 4 C 2 6 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 10 C 2 3 C 2 4 C 2 6 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 3 C 2 5 C 2 6 2 C 2 10 C 3 3 C 5 C 2 6 4 C 2 10 C 3 C 2 4 C 5 C 2 6 4 C 2 10 C 2 2 C 3 C 5 C 2 6 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 4 3 C 2 7 C 2 10 C 2 2 C 2 3 C 2 7 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 2 10 C 2 3 C 2 4 C 2 7 C 4 4 C 2 6 C 2 7 2 C 3 C 3 5 C 2 6 C 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 3 C 2 6 C 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 4 C 2 6 C 2 7 C 2 3 C 2 4 C 2 6 C 2 7 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 5 C 2 6 C 2 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 3 C 2 5 C 2 6 C 2 7 C 2 4 C 2 5 C 2 6 C 2 7 2 C 3 3 C 5 C 2 6 C 2 7 2 C 2 2 C 3 C 5 C 2 6 C 2 7 2 C 2 10 C 3 3 C 5 C 2 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 2 3 C 4 C 6 C 2 7 4 C 10 C 2 C 3 C 4 C 5 C 6 C 2 7 C 2 10 C 4 2 C 2 8 C 2 10 C 4 4 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 2 C 2 3 C 2 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 10 C 2 2 C 2 4 C 2 8 6 C 2 10 C 2 3 C 2 5 C 2 8 )]TJ/F21 11.9552 Tf 9.102 0 Td [(C 4 2 C 2 6 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 4 5 C 2 6 C 2 8 2 C 3 C 3 5 C 2 6 C 2 8 C 2 2 C 2 4 C 2 6 C 2 8 3 C 2 2 C 2 5 C 2 6 C 2 8 )]TJ/F21 11.9552 Tf 9.102 0 Td [(C 2 4 C 2 5 C 2 6 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 3 C 5 C 2 6 C 2 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 4 4 C 2 7 C 2 8 2 C 3 C 3 5 C 2 7 C 2 8 6 C 2 2 C 2 3 C 2 7 C 2 8 C 2 2 C 2 4 C 2 7 C 2 8 C 2 2 C 2 5 C 2 7 C 2 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 3 C 2 5 C 2 7 C 2 8 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 4 C 2 5 C 2 7 C 2 8 4 C 3 C 2 4 C 5 C 2 7 C 2 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(6 C 2 2 C 3 C 5 C 2 7 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 10 C 3 C 2 4 C 5 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 10 C 2 2 C 3 C 5 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 3 4 C 6 C 2 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 10 C 2 C 4 C 2 5 C 6 C 2 8 2 C 10 C 3 2 C 4 C 6 C 2 8 4 C 10 C 2 C 3 C 4 C 5 C 6 C 2 8 2 C 10 C 2 C 3 C 2 5 C 7 C 2 8 6 C 10 C 3 2 C 3 C 7 C 2 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 3 2 C 5 C 7 C 2 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 10 C 2 C 2 3 C 5 C 7 C 2 8 2 C 10 C 2 C 2 4 C 5 C 7 C 2 8 2 C 4 C 3 5 C 6 C 7 C 2 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(6 C 3 C 4 C 2 5 C 6 C 7 C 2 8 2 C 2 2 C 3 C 4 C 6 C 7 C 2 8 2 C 3 4 C 5 C 6 C 7 C 2 8 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 10 C 2 2 C 2 3 C 2 9 C 2 10 C 2 3 C 2 4 C 2 9 )]TJ/F21 11.9552 Tf 9.103 0 Td [(C 4 2 C 2 6 C 2 9 2 C 3 C 3 5 C 2 6 C 2 9 2 C 2 2 C 2 3 C 2 6 C 2 9 C 2 2 C 2 4 C 2 6 C 2 9 C 2 2 C 2 5 C 2 6 C 2 9 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 3 C 2 5 C 2 6 C 2 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 4 C 2 5 C 2 6 C 2 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 2 C 3 C 5 C 2 6 C 2 9 C 2 2 C 2 3 C 2 7 C 2 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 3 C 2 4 C 2 7 C 2 9 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 3 C 2 5 C 2 7 C 2 9 2 C 3 3 C 5 C 2 7 C 2 9 C 4 2 C 2 8 C 2 9 2 C 3 C 3 5 C 2 8 C 2 9 )]TJ/F21 11.9552 Tf 11.759 0 Td [(C 2 2 C 2 4 C 2 8 C 2 9 )]TJ/F21 11.9552 Tf 11.76 0 Td [(C 2 2 C 2 5 C 2 8 C 2 9 C 2 4 C 2 5 C 2 8 C 2 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 2 C 3 C 5 C 2 8 C 2 9 2 C 2 10 C 3 3 C 5 C 2 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 2 C 2 3 C 4 C 6 C 2 9 4 C 10 C 2 C 3 C 4 C 5 C 6 C 2 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 2 C 3 3 C 7 C 2 9 2 C 10 C 2 C 2 3 C 5 C 7 C 2 9 2 C 3 C 4 C 2 5 C 6 C 7 C 2 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 2 C 3 C 4 C 6 C 7 C 2 9 2 C 2 3 C 4 C 5 C 6 C 7 C 2 9 4 C 10 C 2 2 C 2 3 C 8 C 2 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 2 3 C 2 5 C 8 C 2 9 4 C 10 C 3 C 2 4 C 5 C 8 C 2 9 139

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)]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 2 C 3 C 2 4 C 7 C 8 C 2 9 2 C 2 C 3 C 2 5 C 7 C 8 C 2 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 3 2 C 3 C 7 C 8 C 2 9 4 C 3 10 C 2 C 2 3 C 4 C 6 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 3 10 C 2 C 3 3 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 4 C 3 5 C 3 6 C 7 4 C 3 C 4 C 2 5 C 3 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 4 C 5 C 3 6 C 7 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 3 C 4 C 5 C 3 6 C 7 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 2 C 3 3 C 2 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 C 3 C 2 4 C 2 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 C 3 C 2 5 C 2 6 C 7 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 3 2 C 3 C 2 6 C 7 2 C 10 C 3 2 C 5 C 2 6 C 7 10 C 10 C 2 C 2 3 C 5 C 2 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 C 2 4 C 5 C 2 6 C 7 2 C 2 10 C 3 C 3 4 C 6 C 7 2 C 2 10 C 3 3 C 4 C 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 2 2 C 3 C 4 C 6 C 7 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 2 3 C 4 C 5 C 6 C 7 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 4 C 2 5 C 3 6 C 8 2 C 3 2 C 4 C 3 6 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 3 3 C 3 7 C 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 3 C 2 4 C 3 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 3 C 2 5 C 3 7 C 8 6 C 2 C 2 3 C 5 C 3 7 C 8 2 C 3 10 C 2 2 C 2 3 C 8 2 C 3 10 C 2 3 C 2 4 C 8 2 C 10 C 4 2 C 2 6 C 8 4 C 10 C 3 C 3 5 C 2 6 C 8 4 C 10 C 2 2 C 2 3 C 2 6 C 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 10 C 2 2 C 2 4 C 2 6 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 2 C 2 5 C 2 6 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 3 C 2 5 C 2 6 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 4 C 2 5 C 2 6 C 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 10 C 2 2 C 3 C 5 C 2 6 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 2 C 2 3 C 2 7 C 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 10 C 2 3 C 2 4 C 2 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 10 C 2 3 C 2 5 C 2 7 C 8 4 C 10 C 3 3 C 5 C 2 7 C 8 4 C 10 C 2 2 C 3 C 5 C 2 7 C 8 2 C 2 C 2 3 C 4 C 6 C 2 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 3 C 4 C 5 C 6 C 2 7 C 8 )]TJ/F15 11.9552 Tf 9.103 0 Td [(4 C 3 10 C 3 3 C 5 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 10 C 2 C 2 3 C 4 C 6 C 8 4 C 2 10 C 2 C 3 3 C 7 C 8 2 C 2 10 C 2 C 3 C 2 4 C 7 C 8 2 C 2 C 3 5 C 2 6 C 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 3 C 2 5 C 2 6 C 7 C 8 2 C 3 2 C 3 C 2 6 C 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 5 C 2 6 C 7 C 8 4 C 2 C 2 4 C 5 C 2 6 C 7 C 8 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 10 C 3 2 C 3 C 7 C 8 2 C 2 10 C 2 C 2 3 C 5 C 7 C 8 4 C 10 C 3 C 4 C 2 5 C 6 C 7 C 8 4 C 10 C 2 3 C 4 C 5 C 6 C 7 C 8 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 C 3 5 C 3 6 C 9 2 C 2 C 3 C 2 5 C 3 6 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 3 C 3 6 C 9 2 C 3 2 C 5 C 3 6 C 9 )]TJ/F15 11.9552 Tf 9.103 0 Td [(2 C 2 C 2 3 C 4 C 3 7 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 4 C 3 5 C 3 8 C 9 2 C 2 2 C 4 C 5 C 3 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 3 C 4 C 2 5 C 2 6 C 9 6 C 10 C 2 2 C 3 C 4 C 2 6 C 9 2 C 10 C 3 4 C 5 C 2 6 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 10 C 2 2 C 4 C 5 C 2 6 C 9 2 C 10 C 2 3 C 4 C 5 C 2 6 C 9 2 C 10 C 3 3 C 4 C 2 7 C 9 2 C 10 C 2 3 C 4 C 5 C 2 7 C 9 4 C 2 C 3 3 C 6 C 2 7 C 9 6 C 2 C 3 C 2 4 C 6 C 2 7 C 9 2 C 2 C 3 C 2 5 C 6 C 2 7 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 C 2 3 C 5 C 6 C 2 7 C 9 2 C 10 C 3 C 4 C 2 5 C 2 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 10 C 2 2 C 3 C 4 C 2 8 C 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 10 C 3 4 C 5 C 2 8 C 9 2 C 10 C 2 2 C 4 C 5 C 2 8 C 9 2 C 2 C 3 5 C 6 C 2 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 2 C 3 C 2 5 C 6 C 2 8 C 9 4 C 3 2 C 3 C 6 C 2 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 5 C 6 C 2 8 C 9 2 C 2 C 3 4 C 7 C 2 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 3 2 C 4 C 7 C 2 8 C 9 4 C 2 C 3 C 4 C 5 C 7 C 2 8 C 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 3 10 C 3 3 C 4 C 9 4 C 2 10 C 2 C 3 3 C 6 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 10 C 2 C 3 C 2 4 C 6 C 9 2 C 2 10 C 3 2 C 3 C 6 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(6 C 2 10 C 2 C 2 3 C 5 C 6 C 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(2 C 2 C 3 4 C 2 6 C 7 C 9 2 C 3 2 C 4 C 2 6 C 7 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(2 C 2 C 2 3 C 4 C 2 6 C 7 C 9 4 C 10 C 2 2 C 2 3 C 6 C 7 C 9 8 C 10 C 2 3 C 2 5 C 6 C 7 C 9 )]TJ/F15 11.9552 Tf 9.103 0 Td [(8 C 10 C 3 3 C 5 C 6 C 7 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 3 C 2 4 C 5 C 6 C 7 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(4 C 10 C 2 2 C 3 C 5 C 6 C 7 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 10 C 3 C 3 4 C 8 C 9 2 C 4 C 3 5 C 2 6 C 8 C 9 2 C 3 C 4 C 2 5 C 2 6 C 8 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 2 C 3 C 4 C 2 6 C 8 C 9 )]TJ/F15 11.9552 Tf 11.76 0 Td [(2 C 2 2 C 4 C 5 C 2 6 C 8 C 9 2 C 3 C 3 4 C 2 7 C 8 C 9 2 C 3 C 4 C 2 5 C 2 7 C 8 C 9 4 C 2 2 C 3 C 4 C 2 7 C 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(6 C 2 3 C 4 C 5 C 2 7 C 8 C 9 2 C 2 10 C 2 2 C 3 C 4 C 8 C 9 2 C 2 10 C 2 3 C 4 C 5 C 8 C 9 4 C 10 C 2 C 3 C 2 4 C 6 C 8 C 9 4 C 10 C 2 C 3 C 2 5 C 6 C 8 C 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(4 C 10 C 3 2 C 3 C 6 C 8 C 9 4 C 10 C 2 C 2 3 C 4 C 7 C 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 10 C 2 C 3 C 4 C 5 C 7 C 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(8 C 3 C 3 5 C 6 C 7 C 8 C 9 )]TJ/F15 11.9552 Tf 9.102 0 Td [(8 C 2 2 C 2 3 C 6 C 7 C 8 C 9 8 C 2 3 C 2 5 C 6 C 7 C 8 C 9 )]TJ/F15 11.9552 Tf 11.759 0 Td [(4 C 3 C 2 4 C 5 C 6 C 7 C 8 C 9 8 C 2 2 C 3 C 5 C 6 C 7 C 8 C 9 : 140

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BIOGRAPHICALSKETCH ShawnwasborninAugustof1980,theChineseyearofthemonkey.Hespenthis formativeyearsgrowingupinNewJersey.Althoughhedisplayedapredilectionfor Americanmusclecarsatanearlyage,heacquiescedtoconventionalwisdomwhenhewas presentedwiththeopportunitytoattendcollege,andenrolledinBucknellUniversity.He completedaBachelorofScienceinphysicswithaminorinmathematicsin2002.Wearyof thenorthernclimate,hepursuedgraduatestudiesattheUniversityofFlorida,completing aMasterofScienceinphysics,in2004. AfterhavinganepiphanyonabalmyJuneevening,herealizedthathedidnot wanttospendtherestofhislifesquintinginadarklaserlaboratory,andenrolledinthe DepartmentofMechanical&AerospaceEngineering,withanintenttostudystressand strain,andtherivetinginterplaybetweenthetwo.Uponlearningwhatstressandstrain were,Shawnfoundhisinterestslaymorewithdynamicalsystems.ShawnsoughtoutDr. Fitz-Coyafterreadinginthedepartmentbrochurethathisresearchportfolioincluded spacecraftdesign,andjoinedtheSpaceSystemsGroupinthefallof2005. Shawnwasdevastatedwhenheconcludedthathisgraduatestudieswouldnotentail buildinganyspacecraft,oratripintospace,butrathercopiousnightsspentderiving equationsandwritingcomputerprograms.Shawnspearheadedacontributiontothe bettermentofhumanitywhenheleadarenovationofthelaboratory.Theresulting600 ft 2 facilityhasbeenarguedbynooneinparticularasthenestspacecraftprocessing facilityonEarth.AfterspendingmanyyearsworkingontheSwampSatightprogram andhoninghisengineeringskills,ShawncompletedasecondMasterofSciencein aerospaceengineeringinthefallof2010,whileinterningattheAirForceResearch Laboratory.Eventuallythecallofthewildbecametoocompellingandheacceptedthat hemustgrowupandleavetheacademicnest.Theculminationofhisformationight eortsresultedinthisdissertation. 150

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Shawnintendstopursueopportunitiesinthespaceindustrywhichwillpermithim toadvancethestateoftheartofsmall,fractionatedspaceassets.Inhissparetime, hewillpursuehisothertwopassions,namelybruisinghisknucklesunderneathvarious automobiles,andsailingonasmanybodiesofwateraspossible.Astimepermits,hewill resumemakingmusicandrenewhisinterestinphotography. 151