On the trajectory design, guidance and control for spacecraft rendezvous and proximity operations

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spacecraft rendezvous and proximity operations

Georgia Deaconu

To cite this version:

Georgia Deaconu. On the trajectory design, guidance and control for spacecraft rendezvous and

proximity operations. Automatic Control Engineering. Université Paul Sabatier - Toulouse III, 2013.

English. �tel-00919883�

THÈSE

THÈSE

En vue de l’obtention du

DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE

Délivré par : l’Université Toulouse 3 Paul Sabatier (UT3 Paul Sabatier)

Présentée et soutenue le 29/10/2013 par :

Georgia Iuliana DEACONU

On the trajectory design, guidance and control for spacecraft rendezvous

and proximity operations.

JURY

M. Ali ZOLGHADRI

_{Professeur, Université Bordeaux I}

_{Président du Jury}

M. Denis ARZELIER

_{Directeur de Recherche, CNRS}

_Examinateur

M. Jean-Claude BERGES

_{Ingénieur, CNES}

_{Membre Invité}

M. Thomas CARTER

_{Em. Prof., East. Connecticut SU}

_Examinateur

M. Patrick DANES

_{Professeur, UT3}

_Examinateur

M. Alexandre FALCOZ

_{Ingénieur, Astrium EADS}

_{Membre Invité}

M. Christophe LOUEMBET

_{Maître de conférences, UT3}

_Examinateur

M. Sorin OLARU

_{Maître de conférences, SUPELEC}

_Rapporteur

M. Andrew SINCLAIR

_{Associate Prof., Auburn Univ.}

_Rapporteur

M. Alain THERON

_{Chercheur affilié, LAAS-CNRS}

_Examinateur

École doctorale et spécialité :

EDSYS : Automatique 4200046

Unité de Recherche :

Laboratoire d’Analyse et d’Architecture des Systèmes du CNRS

Directeur(s) de Thèse :

M. Christophe LOUEMBET et M. Alain THERON

Rapporteurs :

There are many people that I would like to thank for their dire t or indire t ontribution to the

ompletion ofthis thesis. Iwould like tothank rst myadvisors,Christophe LouembetandAlain

Théron,fortheirguidan eandforthefreedomtheygrantedmein hoosingtheresear hdire tions.

ThisthesishasbeensupportedbytheFren hNationalCenterforSpa eStudiesandbyAstrium

EADSandIwouldlike tousethiso asionto thankJean-Claude BergesandAlexandreFal oz for

their help andfor their parti ipation.

IwouldliketothankDenisArzelier,thehead oftheMethods andAlgorithmsinControl whi h

hosted me at LAAS-CNRS for the opportunity he oered me a while ago for the internship and

for supporting my andidature for this subje t. Iwouldalso like to thank all themembersof the

MACgroupfor making theworkenvironment soenjoyable.

Spe ial thanksto Eri Kerrigan and to Paola Falugi and to theother members of theControl

and Powergroupat theImperialCollege ofLondon,fortheinterestingand hallenging dis ussions

and for thegood timesduring mystayinLondon.

A³vrea deasemeneasãlemulµumes pãrinµilormei,Ni oleta³iMarian, pre um³isuroriimele

Teodora pentrutotajutorul oferit peperioada studiilor.

IwouldalsoliketothankFran es oforhis areandforhissupportandmyoldandnewfriends

that have shared good and bad days with me over the years: Rãzvan, Dinu, Oana, Magda and

Re entspa emissionsrelymore andmoreonthe ooperationbetween dierentspa e raftinorder

to a hieve a desiredobje tive. Among thespa e raft proximity operations, theorbitalrendezvous

is a lassi al example that has generated a large amount of studies sin e the beginning of the

spa e exploration. However, the motivations and obje tives for the proximity operations have

onsiderably hanged. The need for higher autonomy, better se urity and lower osts prompts

for the development of new guidan e and ontrol algorithms. The presen e of dierent types of

onstraints and physi al limitations also ontributes to the in reased omplexity of the problem.

In this hallenging ontext, this dissertation represents a ontribution to the development of new

spa e raft guidan e and ontrol algorithms.

The works presented in this dissertation are based on a stru tural analysis of the spa e raft

relative dynami s. Using a simpliedmodel, a new set of parametri expressions is developed for

the relative motion. This parametrization is very well suited for the analysis of the geometri

properties of periodi relative traje tories and for handling dierent types of state onstraints. A

formal onne tion is eviden ed between the set of parameters that dene onstrained traje tories

and the oneof positive semi-denite matri es. Thisresult isexploited inthedesign of spa e raft

relative traje tories for proximity operations, in the impulsive ontrol framework. The resulting

guidan e algorithms enable the guaranteed ontinuous onstraints satisfa tion, while still relying

on semi-denite programming tools. The problem of the robustness of the omputed maneuvers

La réalisation des missions spatiales repose de plus en plus souvent sur la oopération entre

dif-férentsengins spatiaux. Parmi les opérations de proximité, lerendez-vousorbital estune pratique

aussian iennequela onquêtespatiale,qui ontinuedegénérer denombreuxtravauxdere her he.

Cependant, les motivations etles obje tifs desré entes missions de rendez-vous orbital ont

large-ment évolués. En eet, les besoins d'une autonomie a rue, d'une sé urité améliorée, d'une plus

grande exibilité etd'une rédu tion des oûts, onstituent autant d'in itations au développement

de nouvelles méthodes de guidage et ontrle. La satisfa tion de ontraintes très variées, dues à

des onsidérations de sé urité ouà deslimitations te hnologiques in ontournables desa tionneurs

ou des apteurs, ontribuent à la ri hesse du problème posé. Dans e ontexte, le développement

de nouveauxalgorithmes de ommande onstitue unvrai dés ientique que ette thèse tente de

relever.

Les travaux de ette thèse sont basées sur l'analyse stru turelle des expressions dé rivant le

mouvement relatifentredeuxvéhi ulesenorbite. Surlabasedesmodèlesdetransitiondisponibles

dans la littérature, une nouvelle paramétrisation du mouvement relatif est proposée. Celle- i,

parti ulièrement adaptée àla ara térisation destraje toires périodiques, orelapossibilitéd'une

prise en ompte de ontraintes d'état très variées. Un lien formel est mis en éviden e entre les

paramètres dénissant les traje toires ontraintes et le ne des matri es semi dénies positives.

Ces résultatssont exploités dans ledéveloppement desalgorithmes de design de traje toires pour

desopérationsde proximité,sous hypothèse de poussée impulsionnelle. Ces algorithmesont,entre

autre,lapropriétédepermettrelasatisfa tiondes ontraintessurlatraje toiredemanière ontinue

dans letemps, tout en utilisant lesoutils numériques de laprogrammation onvexe. Le problème

spé iquedelarobustessedesman÷uvresauxin ertitudesdela haînedemesureestaussiabordé

dans emanus rit. Desappro hesdetype ommandeprédi tivesontmisesenpla eandegarantir

A knowledgements i

Abstra t iii

Résumé v

Nomen lature ix

Introdu tion 1

1 Spa e raft relative motion 11

1.1 Introdu tion . . . 11

1.2 Dynami s ofa spa e raftorbiting theEarth . . . 12

1.3 Spa e raft relative motion . . . 16

1.3.1 Lo alCartesian dynami s . . . 16

1.3.2 Orbital elements dieren esdynami s . . . 19

1.4 Linearized Cartesianrelative motion . . . 20

1.4.1 State-spa e representation . . . 20

1.4.2 The state transitionmatrix . . . 21

1.5 Properties ofrelative traje tories . . . 24

1.5.1 Periodi ity onditions . . . 24

1.5.2 Inter-satellite distan e . . . 27

1.5.3 Geometry oftheperiodi spa e raftrelative motion. . . 28

1.6 Con lusions . . . 30

2 Parametri expressions for the spa e raft relative traje tory 33 2.1 Denition ofthe parameters . . . 34

2.2 Properties ofspa e raft relative traje tories . . . 35

2.2.1 Dynami s oftheve tor of parameters . . . 36

2.2.2 Properties ofperiodi traje tories . . . 38

2.3 Numeri alanalysis oftheperiodi relative motion. . . 40

2.3.1 The ee tsof thee entri ity of theleaderorbit . . . 40

2.3.2 The ee tsof thevalues of theparameters . . . 40

2.4 Con lusion. . . 43

3 Constrained spa e raft relative traje tories 45 3.1 Denition ofadmissible traje tories . . . 46

3.2 Finite des riptionof admissible traje tories . . . 47

3.2.1 Finite des riptionusing onstraints dis retization . . . 48

3.2.2 Finite des riptionusing non-negative polynomials . . . 48

3.3 Des ription of onstrained traje tories usingnon negative polynomials . . . 49

3.3.1 Rational expressionsfor thespa e raft relative motion . . . 49

3.3.2 Constrained nonperiodi traje tories . . . 52

4 Traje tory design for spa e raft rendezvous 57

4.1 Fixed-time linearizedimpulsivespa e raft rendezvous. . . 58

4.1.1 General formulation oftheguidan e problem . . . 58

4.1.2 Consumption riteria . . . 59

4.1.3 Saturation onstraints . . . 60

4.1.4 Using dire tshooting methods for theguidan e problem . . . 61

4.2 Fixed-time rendezvouswithtraje tory onstraints . . . 63

4.2.1 Guidan e towardsa onstrained periodi relative motion . . . 63

4.2.2 Passively safe traje tories for spa e raftrendezvous . . . 66

4.2.3 Spa e raft rendezvous withvisibility onstraints . . . 68

4.3 Numeri alexamples . . . 70

4.3.1 Rea hing a onstrained periodi relative traje tory . . . 70

4.3.2 Passively safe rendezvoustraje tories . . . 74

4.3.3 Constrained nonperiodi relative traje tories . . . 78

4.4 Con lusion. . . 79

5 Spa e raft rendezvous robust to navigation un ertainties 81 5.1 ModelPredi tive Control and spa e rafttraje torydesign . . . 82

5.2 The robusttraje toryplanning problem . . . 84

5.2.1 The spa e raftrelative dynami s . . . 85

5.2.2 The ee tsof navigation un ertainties . . . 86

5.2.3 The nominaltraje tory . . . 87

5.2.4 General formulation oftheguidan e problem . . . 87

5.3 Ane state-feedba kMPC. . . 89

5.3.1 Computation ofthefeedba kgains . . . 89

5.3.2 Computation ofthenominal ontrol . . . 92

5.4 Ane disturban e feedba kMPC . . . 93

5.5 Numeri alevaluation oftherobust ontrol te hniques. . . 97

5.5.1 Des ription ofthe simulation pro edure . . . 97

5.5.2 The PRISMAmission . . . 99

5.5.3 The Simbol-X mission . . . 103

5.6 Con lusion. . . 108

6 Analyti al bi-impulsive ontrolaround a desired periodi traje tory 109 6.1 Stabilityarounda periodi relative traje tory . . . 110

6.2 Analyti al bi-impulsive stabilizing ontrol for theperiodi motion . . . 111

6.2.1 Computation ofthe ontrol . . . 112

6.2.2 Domain ofvalidity . . . 113

6.2.3 Performan es inpresen eof navigation un ertainties . . . 115

6.3 Robust guidan etowards a spa e raftperiodi relative motion . . . 117

6.4 Numeri alexamples . . . 119

6.4.1 Inuen eof thee entri ityof thereferen eorbit . . . 120

6.4.2 Inuen eof theintervalbetween ontrols . . . 122

6.4.3 Inuen eof thenavigation un ertainties . . . 123

6.5 Con lusion. . . 124

B Properties of non negative polynomials 137

B.1 Che king polynomials non negativityon anite interval . . . 137

B.2 Che king polynomials non negativityon aninnite interval . . . 138

C Ellipsoidal sets 141

C.1 Representations ofellipsoidal sets . . . 141

C.2 Operations withellipsoids . . . 142

C.3 The S-pro edure . . . 142

µ

Earth'sgravitational onstant

ν

true anomaly

Ω

longitudeof theas endingnode

ω

argument ofperigee

a

semi-major axis

B

0

Earth entered inertial base

B

l

Spa e raft entered lo alCartesian base

E

e entri anomaly

e

e entri ity

i

orbit in lination

M

mean anomaly

n

mean motion

LMI Linear MatrixInequality

LP Linear Program

LTI Linear TimeInvariant

LTV Linear TimeVarying

LVLH Lo al Verti al Lo alHorizontal

MPC ModelPredi tive Control

Résumé: Le su ès des missions spatiales repose de plus en plus souvent sur la oopération entre

plusieurs véhi ules en orbite. L'approvisionnement de la Station Spatiale Internationale par

ex-emple est assuré par des opérations de rendez-vous orbital, tandis que des nombreuses missions

s ientiques utilisent des formations de satellites pour relever des mesures. Ce type d'opérations

ont des besoins spé iques en termes d'algorithmes de ontrle, vue la distan e réduite entre les

véhi ules, les ontraintes d'autonomieetde sé uritédes missionspatiales etles ressour es limitées.

Les travaux de ettethèse portent sur ledéveloppement des algorithmes deguidage pour des op

éra-tionsdeproximitéentreles satellites,oùla distan e réduitepermetla navigationrelative. L'obje tif

est de fournir des plans de man÷uvres optimisés du point de vue de la onsommation de

om-bustible, qui prennenten ompteles ontraintes opérationnellesde la missionetqui soientrobustes

à des in ertitudes. Le adre detravail hoisiest elui des méthodes dîtesdire tes, qui permettentla

formulation duproblème deguidage omme unproblème d'optimisationparamétrique.

Ba kground and motivations

Spa e raft rendezvousand do king apabilities arerequired for alarge arrayofspa e appli ations

thatinvolvemorethanonespa e raft. Itisakeyte hnologyforthein-orbitassemblyoflargeunits,

su h as the spa e stations (Mir, Skylab, ISS). The spa e stations further rely on rendezvous and

do kingmissionsinorderto re eivesupplies orto ex hange the rew. Forinstan e, theunmanned

Automated Transfer Vehi le (ATV) from the European Spa e Agen y periodi ally supplies the

International Spa eStation(ISS)withpropellant,water, air, payloadsand experiments. Re ently,

theDragon spa e raft be ame therst ommer ial spa e raft to su essfullydo kwiththeISS.

Spa e rendezvous has also been used for a variety of other purposes, in luding the servi e

missions to the Hubble Spa e Teles ope and the EURECA spa e raft retrieval. Other on-orbit

servi ing missions areunder study for existing spa e raft [7,85℄. The in reasing number of spa e debris in the Low Earth Orbit originating from mutual ollisions, motivated the study of a tive

debris removalmissions[13℄.

In the re ent years, a lot of interest hasbeen shown for spa e s ienti missions that rely on

dierent instruments distributedovera eet of spa e raft. This onguration an provide several

an usemultiple "massprodu tion" vehi les to assemble theeet. The robustness of themission

is alsoin reased bythis ongurationsin e thepayloadsaredistributedamong thespa e raft and

an eventually be repla ed in ase of failure [98℄. Formation ying oers more exibility be ause the formation an be re ongured in order to follow new mission requirements. This approa h

has been onsidered for s ienti missionswith very diverse obje tives, su h asEarth observation

(A-train),interferometryforEarth-likeplanetsdete tion(DARWIN),measurementofgravitational

waves fromsupermassivebla kholebinaries (LISA)or X-raysspa e teles ope (Simbol-X).

The su ess of spa e raft rendezvous and formation ying missions depends on the pre ise

ontrol of the spa e raft relative state, often-times in the ontext of relatively small spa e raft

separations. Inordertoensurethese urityofthemission,ahighdegreeofautonomyandrobustness

is desiredfor therelative motion ontrol pro edure. For missionssu hastheMarsSample Return

[86℄, for whi h the ommuni ation delay between the ground station and the spa e raft is very large,anautonomousguidan ealgorithmwhi hguaranteesthatno ollisionwillo urbetweenthe

spa e raft isof vitalimportan e.

The fuel- ost of the spa e raft maneuvers is also a matter of on ern. The propulsion system

ana ountforupto50%ofthespa e raftmassatthelaun htime,redu ingtheavailablepayload

massandinuen ingthe ostofthelaun h. The ontrolalgorithmsmustensurethatthe omputed

maneuversarefuel-optimal,su hthatthedesiredlifetimeforthemission an be a hieved withthe

smallest amount of propellant.

The spa e raft rendezvous

Theorbitalrendezvouspro ess onsistsinexe utingaseriesoforbitalmaneuverswiththepurpose

of bringing two spa e raft in lose vi inity of ea h other. Usually one of the spa e raft, alled

thetarget, is onsideredto be inert,while these ondspa e raft, alledthefollower or the haser,

exe utesthemaneuvers. Whentheobje tiveisto physi allyjointhetwospa e raftinvolvedinthe

rendezvous, we speakabout do king or berthing.

W. Fehse identied in[29℄several phases ofa rendezvous mission, ea h one withits own hal-lenges. For the laun h stage, the purpose is to bring the two spa e raft in vi inity by pla ing

them in the same orbital plane. The phasing stage aims at redu ing the phase angle between

the target and the follower (see the illustration inFigure 1). During the phasing maneuvers, the

follower spa e raft is ontrolled from the ground station and the navigation is based on absolute

Figure 1: Viewofthetarget's orbital planeat thebeginning of thephasing stage

a stablepositioninproximityof thetarget,using relative navigation measurements. For theATV

rendezvous s enario with the ISS for instan e, this stage starts at a range of few tens of

kilome-tres and ends at a range of few kilometres from the target spa e raft. The following lose range

rendezvous phase is usually divided in two stages: the losing maneuvers guiding the spa e raft

towards thenalapproa h orridor andthenal approa h stage leading to mating onditions.

The dierent stages of an orbital rendezvous mission aresummarized inFigure 2. The works

presentedinthisdissertationarerelatedtothehomingandthe losingphases,duringwhi hthetwo

spa e raftrelyonrelativenavigationmeasurementsinordertoa hievethedesirednal onditions.

Some of the presented examples also refer to the nal approa h phase, leading to the spa e raft

do king.

Mission onstraints and te hni al hallenges

The spa e raft rendezvous guidan e is a omplex pro ess due to thedierent types of onditions

and onstraintsthatmustberespe tedduringea hphaseofthemission. Forthephases onsidered

in this dissertation, the far range and lose range rendezvous stages, theapproa h traje tory an

be required for instan e to pass through spe ied hold points where the follower vehi le must

waitfor thepermissionto pro eed,either fromtheground ontrol station or fromthe rew ofthe

targetspa e raft[29℄. Se urity onsiderationsmightimposethe hoi eofapproa htraje toriesthat are inherently safe, meaning that they are guaranteed to avoid any ollision with target vehi le,

Figure 2: Thedierent phases ofa rendezvousmission

visibility oneofthetargetspa e raftfor ontinuousvisual onta t. Thesespe i ations orrespond

to onstraints that the rendezvous traje tory must respe t in order to ertify that the mission

requirementsaremet.

Another fa tor that must be taken into onsiderationwhen designing the approa h traje tory

is the fuel- ost of the maneuvers. Thrust maneuvers an be approximated with impulses, i.e.

instantaneous hanges of velo ity at the time of maneuver. This simplies the omputation and

the analysis of a fuel-optimal maneuvers plan for the spa e raft rendezvous [29℄. The impulsive approximation is espe ially well adapted for the liquid propellant engines whi h are used for a

wide spanofspa e raftmaneuvers, rangingfromorbitaltransfermaneuversto stationkeepingand

Orbitaldisturban es,navigationerrorsand ontrolexe utionerrors analtertheout omeofthe

omputed maneuvers. Thepresen eandtheee tsofthesedisturban esneedto beintegratedinto

therendezvoustraje torydesignphase. Navigationerrorsaredenedasthedieren ebetweenthe

state per eivedbytheonboardsystemandtherealstateofthevehi le. They anbe ausedbythe

sensors measurement performan elimitations, byerrorsin thealignment between thesensors and

spa e raft axes, by the onboard information pro essing and ltering, et . The ontrol exe ution

errors refer to deviations in magnitude, dire tion or appli ation time from the desired impulsive

thrusts. They an be due to mounting errors, to misalignments with the me hani al axes, to the

engine performan es, et . The de ision autonomy of the spa e raft annot be in reased without

providing a priori guarantees for its behaviour inperturbed onditions. Thismust be done while

using ontrolalgorithmsofredu ed omplexitysin ethe omputationalresour esavailableonboard

thespa e raft arelimitedwithrespe tto thoseavailableon ground.

Some of these hallenges are addressed in this dissertation. The main obje tive is to provide

algorithms for the omputation of robust fuel-optimal maneuversplans leading to rendezvous

tra-je tories that respe tthe onstraintsimposedby themission'srequirements, even inpresen eof a

ertain lassofun ertainties. A study ofthespa e raft onstrained naturally periodi traje tories

is arriedout inrelation tothese urityspe i ationsfor theapproa htraje tory.

Thespa e rafttraje torydesignisa hallenging problemduetothepresen eoftraje toryand

ontrol onstraints, totherobustness onsiderationsandtothelargenumberofdesignparameters.

In the most general ase, only the initial time of the mission is xed and the traje tory design

pro edure must provide a hoi e for the nal time, the number and the distribution of thrusting

instants, the amplitude and the dire tion of the thrusts. If the design algorithm is intended for

use onboard the spa e raft, then restri tions are added on its omputational omplexity. A brief

presentationofthemaintraje torydesignapproa hesisgiveninwhatfollows,withafo usontheir

abilityto handlethedierentmission requirements.

Spa e raft relative traje tory design approa hes

Thetraje torydesignforspa e raftrendezvousandproximityoperationsreferstothe omputation

ofaseriesofmaneuversthatsteerthespa e raftfromsomeknowninitialrelative onditionstosome

nal desired relative onditions. The design pro edure generally onsists insolving an open-loop

optimal ontrol problemwhose solution orresponds tothe best approa h traje tory thatrespe ts

respe ted and an in rease thelifetime of the spa e raft. The te hniques for solving this type of

onstrainedopen-loopoptimal ontrolproblemsareusuallydividedintodire tmethodsandindire t

methods [24℄.

Indire t methods are based on analyti al ne essary optimality onditions derived using the

al ulus of variations and thePontryagin maximumprin iple. The optimal solution an be found

by solving the two-point-boundary-value problem (TPBVP) resulting from these onditions [24℄. When using the indire t methods, the optimal spa e raft traje tory for the rendezvous problem

is omputed indire tly, based on the evolution of the adjoint state ve tor or the so- alled primer

ve tor [58,62℄. For impulsive traje tories, the primer ve tor indi ates the times and thepositions of the thrust impulses that minimize the total fuel ost. However, the resolution of the problem

is ompli ated in thegeneral ase, espe ially when onstraints are added to the problem. It also

requires a good guess for the initial value of the primer ve tor. Re ent works on the spa e raft

rendezvous problem have fo used on transforming the ne essary onditions for optimality into

onstru tive onditions forthe optimalsolution [3,4℄.

Dire t methods rely on the transformation of the optimal ontrol problem into a parameter

optimization problem. This isusually a hieved through ontrol parametrization and through

dis- retization [45℄. The obtained nite-dimensional optimal ontrol problem an be e iently solved using the existing algorithms [11℄. There are dierent types of dire t methods depending on the hoi eforthede isionvariablesandontheusedintegrationmethod. Amongthem,thedire t

shoot-ing methods are used in the ases where the parametrisation on erns only the ontrol variables.

The system'sdynami s areusuallylinear andare integrated analyti allyor numeri ally [49℄. Theindire tresolutionmethods ertifytheglobaloptimalityofa omputedsolutionby he king

a set of ne essary and su ient (ifavailable) onditions. Howeverthey lead to problems thatare

hard to solve numeri ally, espe ially when onstraints are onsidered. Dire t methods are ableto

deal withstate and ontrol onstraints more ee tively and to integraterobustness elementswith

respe t to dierent types of disturban es. Even if the obtained solution an only be ertied as

optimal for the parti ular parametrization and/or dis retization that has been onsidered, they

provide an attra tive alternative for therendezvousguidan e problem.

The algorithms developed in this dissertation for the design of spa e raft rendezvous

traje -tories fall into the ategory of dire t shooting methods. Other than theadvantage related to the

redu ed omplexity of theresulting optimization problem, this approa h also oersthepossibility

willbeonredu ingtheee tsofrelativenavigationun ertainties onthenalrendezvouspre ision.

Spa e raft traje tory ontrol: losing the loop

The dire t and indire t approa hes for spa e raft relative traje tory design provide a series of

fuel-optimal maneuvers that need to be exe uted at the spe ied instants in order to rea h the

desired nal obje tive. The maneuvers plan is obtained based on open-loop predi tions of the

evolution of the spa e raft relative traje tory. As previously dis ussed, the presen e of orbital

perturbations, navigation un ertainties or ontrol exe ution errorsmight alter theout ome of the

omputed maneuvers. Inorder to limit their undesiredee ts andto rea h aspe ied rendezvous

pre ision, thetraje tory ontrol needsto be implementedina losed-loop manner.

Theresolutionofa onstrainedopen-loopoptimal ontrolproblem anbeintegratedina

losed-loop setting by using the Model Predi tive Control (MPC) methodology [84℄. Model Predi tive ControlorRe edingHorizon Controlisa ontrolte hnique forwhi hthe ontrol a tionisobtained

bysolving at ea h sampling instant anite-horizon open-loop optimal ontrol problem, using the

urrent state ofthesystemasinitial state. The optimizationdeliversevery timeaseries of ontrol

a tions out of whi h only therst one is applied to thesystem. Therest of the planis dis arded

be ause a new solution, based on new measurement information, will be omputed at the next

sampling time[68℄.

ModelPredi tiveControlisapopular ontrolte hniqueforspa e raftrendezvousandproximity

operations [16,18,26,32,41,43,86℄. Itspopularity isdue totheabilityto integrate onstraints and un ertainties dire tly into the traje torydesign problem. Dierent other ontrol approa hes have

been proposed for spa e raft proximity operations and formation ying, spanning over a large

range of te hniques. A non exhaustive list in ludes adaptive ontrol [2,95℄, non-linear quadrati regulator [6℄, feedba k impulsive ontrol [89℄, Lyapunov-based nonlinear output feedba k ontrol [104℄, time-delayed feedba k ontrol [12℄andseveralothers [87,88℄. Butvery fewofthem onsider thepresen eof onstraintsorthefuel ostofthemaneuvers,andfo usonlyonrea hingthespe ied

nal onditions.

Insteadofdeterminingo-lineafeedba kpoli ythatprovidestheoptimal ontrolforallsystem

states, MPC solves an open-loop optimal ontrol problem on-line whi h takes into onsideration

the urrent state of the system. The periodi re omputation of the solution reates an impli it

losed-loop. Therobustnesspropertiesofthisimpli it losed-loopwithrespe ttodierenttypesof

The presen e of un ertainties raises questions related to the hanges indu ed in the ontrol

performan es. Inthe aseofspa e rafttraje tory ontrol, theperforman es aredened inrelation

to the fuel onsumption and to the pre ision with respe t to the desired nal obje tive of the

maneuvers. The Model Predi tive Control possesses some inherent robustness properties, dened

as the robustness of the losed-loop for the ontrol that has been omputed without expli itly

onsidering theun ertainties [36,67℄. But for problems thatin lude ontrol and state onstraints, the omputed ontrol a tions must guarantee that no transgressions of the onstraints will o ur

for allthepossiblerealizationsoftheun ertainties. Inthis ase, theinherent robustnessproperties

areno longersu ientand thepresen eofun ertainties needsto be in luded inthewriting ofthe

optimizationproblem[27,59,69,79℄. Anotherkeyaspe tisthepropertyofre ursivefeasibilityofthe ontrol problem inpresen eof un ertainties. The optimal ontrol is re omputed at ea h sampling

instant anditisimportantto providetheoreti alguarantees that,iftherst optimizationproblem

is feasible, then all the subsequent optimization problems will also be feasible. These important

properties areinvestigated for theguidan e algorithmsproposedin thisdissertation.

Obje tives and organization of the dissertation

The works presented inthis dissertation are oriented following two main axes: theanalysisof the

spa e raftrelativemotionandthedesignand ontrolofthespa e raftrelativetraje tory. Thestudy

of the relative motion on entrates on spa e raft naturally periodi relative traje tories. These

periodi traje tories,intheabsen eofperturbations,require no ontrolinorderto bemaintained.

Thisproperty ouldmakethemgood andidatesfor parkingorbitsinbetween dierent phasesofa

rendezvousmission,forautonomousinspe tiontraje toriesforon-orbitservi ingmissionsorforfail

traje tories in ase of systemmalfun tion. Chapter 1 summarizes the most ommon-used models

for representing the spa e raft relative motion. It also provides an overview of the properties

of the spa e raft relative traje tories that are of interest for the rendezvous guidan e problem,

su hasperiodi ity onditions, inter-satellite distan eand geometri properties ofperiodi relative

traje tories.

Thedierentperiodi motioninitialisationte hniquespresentedintheliteraturedonotgiveany

information about the geometri properties of the resulting traje tory. To address this problem,

a new parametrization for the spa e raft relative traje tories is developed in Chapter 2. This

parametrizationprovidesagoodframeworkforanalysingtheirpropertiesanditisinusedinChapter

The spa e raft relative traje tory ontrol on entrates around the problem of designing

fuel-optimal maneuvers plans leading the spa e raft from an arbitrary initial relative state towards a

desirednalrelativestate,followingtraje torieswhi hrespe tsdierentmission onstraints.

Chap-ter 4 details the writing of the spa e raft rendezvous guidan e problem as an impulsive optimal

ontrol problem using dire t shooting methods. It illustrates the ontribution of the results

pre-sented in Chapter 3 in obtaining approa h traje tories that respe t visibility onstraints or that

are guaranteed to be safe for a large rangeof system errors. The robustness aspe ts with respe t

to navigation un ertainties aretreatedinChapter5. Theguidan eproblemismodiedinorderto

provideasolutionwhi hguaranteesapriori onstraintssatisfa tionforalladmissiblevaluesforthe

un ertainties, without modifyingthe omplexityof the ontrol algorithm. Moreover, theproposed

ontrol strategyalsominimizestheee tsofthesensingnoise onthepre isionwithwhi hthenal

obje tive isa hieved.

Thepresen e ofperturbationsalso ae tsthespa e raft naturally periodi motion. Chapter 6

presentsalow- omplexitystabilizing ontrolstrategyforthespa e raftperiodi motioninpresen e

of sensingnoise. Thedeveloped methodisbasedontheparametrizationfor thespa e raftrelative

traje torypresentedinChapter 2.

Spa e raft relative motion

Contents

1.1 Introdu tion . . . 11

1.2 Dynami sof a spa e raft orbitingthe Earth . . . 12

1.3 Spa e raft relative motion . . . 16

1.3.1 Lo alCartesiandynami s . . . 16

1.3.2 Orbitalelementsdieren esdynami s . . . 19

1.4 Linearized Cartesianrelative motion. . . 20

1.4.1 State-spa erepresentation . . . 20

1.4.2 Thestatetransitionmatrix . . . 21

1.5 Properties of relative traje tories. . . 24

1.5.1 Periodi ity onditions . . . 24

1.5.2 Inter-satellitedistan e . . . 27

1.5.3 Geometryoftheperiodi spa e raftrelativemotion . . . 28

1.6 Con lusions . . . 30

Résumé: L'étude du mouvement relatif des satellites onsiste à analyser la dynamique d'un

satelliteappeléle hasseurparrapportàunautre satellite,appeléla ible. Diérentes hoixexistent

pour la représentation de l'état relatif, ha une ave ses avantages. Plusieurs représentations sont

passéesenrevueen e hapitre,notammentdans le adre des orbitesképlériennes. L'a ent estmis

sur lades riptionbasée surles positionsetlesvitessesrelatives,exprimées dansunrepère artésien

lo al atta hé au satellite ible. Les propriétés des traje toires relatives sont également étudiées,

omme les distan es minimale et maximale entre les satellites, l'existen e des traje toires relative

périodiques et leur propriétés géométriques.

1.1 Introdu tion

Thespa e raftrelativemotionreferstothestudyofthedynami sofaspa e raft, alledthefollower,

with respe t to the dynami s of another spa e raft, alled the leader or the target. The motion

of an individual satellite orbiting the Earth an be expressed using dierent representations for

most ommon des riptions will be presented in this hapter. The nal hoi e is usually driven

by the purpose of the study. Histori ally, models based on orbital elements and orbital elements

dieren eshavebeenusedforformationyingappli ations[16,34,60℄,whileCartesianmodelshave been preferredfor spa e raftrendezvous and ollision avoidan e problems [17,31,41,57℄.

Regardless of the representation hosen for the spa e raft relative motion, a distin tion an

be made between Keplerian models and non Keplerian models. Under Keplerian assumptions,

the Earth is represented as an homogeneous sphere and the spa e raft motion is ae ted only

byNewtonian a elerations. The non Keplerian models take into a ount theEarth's oblateness,

usually through the spheri al harmoni model for the Earth's potential, the atmospheri dragor

thesolar radiation pressure, amongother orbitaldisturban es.

TheKeplerian framework leadsto lessa urate butsimplieddynami almodels for the

spa e- raftrelative motion. Thesesimpliedmodels arewellsuitedfor ontrolsynthesispurposes,likein

the ase of maneuversplans design for spa e raft rendezvousmissions for instan e. Therelatively

small distan es between the spa e raft when ompared to thedistan e with respe t to the enter

of the Earth and the short time horizons asso iated with rendezvous missions justify the usage

of simplied relative motion models. For this reason we will fo us mainly on Keplerian models

throughout thisdissertation, while referringthe interested readerto publi ationstreating some of

theother representations.

Inwhat follows, aparti ular interestwill be paidto theperiodi solutions of theequations

de-s ribing thespa e raft relative motion. Thesesolutions enable thesatellitesto maintaina desired

onguration without external intervention and without any fuelexpenditure. This property has

beenextensivelyusedintheformationightliterature[1,5,46,55,92℄andhasre entlygained atten-tion for orbital rendezvous and ollision avoidan e appli ations [25,41,43℄. Dierent initialization methods forperiodi motionwillbepresentedalongwithsomeofthegeometri alpropertiesofthe

resulting traje tories.

1.2 Dynami s of a spa e raft orbiting the Earth

The Keplerian dynami s of a spa e raft with respe t to the Earth an be derived from Newton's

equations of motionbetween two massparti les. In this ase, themotion of a spa e raft orbiting

theEarth isdes ribed bythefollowing dierential equation[8℄:

d

2

_R

~

dt

2

!

B

0

= −

µ

k ~

Rk

3

~

R

(1.1)

where

R

~

representstheve torfromthe enteroftheEarthtothespa e raft enterofmassand

µ

is theEarth's gravitational onstant. The dynami s areexpressedwithrespe tto an Earth entered

inertial frame

R

0

= (0, ~

X, ~

Y , ~

Z)

illustrated in Figure 1.1. The fundamental plane for

R

0

is the Earth's equatorial plane, the

Z

~

axis oin ides with the rotation axisof theEarth and is oriented towards theNorth Pole, the

X

~

axis points thevernal equinoxand the

Y

~

axisis orthogonalto the

~

X ~

Z

plane.

Figure 1.1: The EarthCentered Inertialframe and thesatellite traje tory

Even thoughthedierential equation (1.1)governing therelative motionof twobodies is

non-linear,theequationadmitsageneralanalyti alsolution[8℄. The onstantsofintegrationasso iated to thesolution are alled theorbital elements of thesatellite motion and they playan important

role inthestudy ofthe properties of thespa e raft traje tory.

Let the orbital plane be the plane whi h ontains the traje tory of the orbiting spa e raft

(see Figure1.1). Theequation ofthespa e raft traje toryexpressedusing polar oordinateswith

respe tto this planeisgiven by [8℄:

R = k ~

Rk =

a(1 − e

2

₎

1 + e cos ν

(1.2)

where

a

is alled the semi-major axis of the spa e raft orbit,

e

is alled the e entri ity and

ν

is alledthetrueanomaly. Thesatellite'sorbitisboundedif

e < 1

andunboundedif

e ≥ 1

. For

e = 0

the spa e raft traje tory is a ir le of radius

a

and for

0 < e < 1

the traje tory is an ellipse. The true anomaly

ν

representstheangle between thespa e raft's urrent positionand thedire tion of theperigee(Figure 1.1).

orientation of the orbital plane are required in order to ompletely hara terize the spa e raft

traje tory. A ommon hoi e is represented bythe angles

i

,

Ω

and

ω

dened withrespe t to the Earth's equatorialplane, asindi atedinFigure 1.2.

Figure 1.2: The denitionof the lassi alorbital elements

Theline of nodes denotes theline of interse tion between thespa e raftorbital planeand the

equatorial plane. The as ending node refers to thepoint where the satellite is rossing the line of

nodesinanorthbound dire tion. Thelongitude of the as ending node,

Ω

,istheangle between the

X

axis of the

R

0

frame and theas ending node, the argument of perigee,

ω

,is the angle between theas ending node and theperigeewhilethe in lination,

i

,is theanglebetween theorbital plane and theequatorialplane.

Theset oforbital elementsis dened by:

oe =

h

a e i Ω ω ν

i

T

(1.3)

andit ompletelydes ribesthestateofasatelliteorbitingtheEarth. UnderKeplerianassumptions,

therst ve parameters are onstant andonly thetrueanomaly hangeswithtime[8℄:

˙ν =

r

_µ

a

3

_{(1 − e}

2

₎

3

(1 + e cos ν)

2

(1.4)

Sometimes, the e entri anomaly,

E

, or the mean anomaly,

M

, are used instead of

ν

as the varyingstate. Thee entri anomalyandthetrueanomalyarerelatedthrough geometri al

trans-formations (Figure 1.3):

tan

ν

2

=

r 1 + e

1 − e

tan

E

2

(1.5)

while e entri anomaly andthemean anomalyare relatedthroughKepler'sequation:

M = E − e sin E = M

0

+ n(t − t

0

)

(1.6)

As shown in (1.6), the mean anomaly an also be dened as a linear fun tion of time, where

n =

pµ/a

3

is the mean motion of the satellite,

t

0

is the referen e time and

M

0

is the mean anomaly at

t

0

.

Figure1.3: Thedenition ofthee entri anomaly

When the orbit is ir ular or near ir ular (

e ≈ 0

) or when theorbit is planar or near planar (

i ≈ 0

), some of the lassi al orbital elements

oe

are not dened. Inthose ases, the state of the spa e raft an be represented using dierent fun tionsof the lassi al orbital elements thatavoid

this problem. Among the solutions proposed in the literature, we an mention the nonsingular

orbital elements, the equino tial elements or the Delaunay anoni al elements, used for studying

thesatellite motionina Hamiltonianframework[90℄.

The hoi eofusingtheinertialpositionandvelo ityorthevarioussetsoforbitalparametersin

ordertodes ribethestateofaspa e raftorbitingtheEarthismadedependingontheappli ation.

Throughout this dissertation, the lassi al orbital elements

oe

arepreferred for therepresentation oftheleader'sstate. This hoi eismotivatedbythefa tthat, intheKeplerian ontext onsidered

here, the resulting dynami s have a very simple form (only one state that hanges over time).

To omplete the des ription of the spa e raft relative motion, the state of the follower satellite

1.3 Spa e raft relative motion

The spa e raft relative motion refers to the study of the dynami s of the leader spa e raft

om-bined with the study of the dynami s of the follower spa e raft. As previously stated, there are

dierent possible state denitions whi h an be used inthe des ription of the motion of a single

spa e raft (Cartesian positionand velo ity, dierent sets of orbital parameters). In a similarway,

dierent representations an be onsidered for the spa e raft relative state, ea h one bearing its

ownadvantages.

1.3.1 Lo al Cartesian dynami s

The spa e raft relative motionrepresented using lo al Cartesiandynami s is dened with respe t

to a lo alrotatingCartesianframe entered on theleader satellite. A ommonlyusedframeisthe

Lo al Verti al Lo al Horizontal (LVLH) frame

R

l

= (S

l

, ~x, ~y, ~z)

illustrated in Figure (1.4). The

~z

axis is radially oriented from the leader satellite towards the enter of the Earth, the

~

y

axis is orthogonal to the orbital plane, inthe oppositedire tion with respe t to the angular momentum

ve tor, andthe

~x

axislays intheleader's orbitalplane inthedire tionof thesatellite's velo ity.

PSfragrepla ements

~x

~z

_ν

a

O

_P

~

~

Q

S

l

S

2

~r

Figure 1.4: The spa e raftrelative positionand theleader's LVLHframe

The relative position between theleader spa e raft

S

l

and the follower spa e raft

S

f

is repre-sentedby

~r =

−−→

S

l

S

f

inFigure 1.4. Considering that theKeplerian dynami s of ea h satellite with respe tto the Earth an be des ribed using (1.1), therelative inertial a eleration an bewritten

as:

 d

2

~r

dt

2



B

0

= −

µ

k ~

R + ~rk

3

( ~

R + ~r) +

µ

k ~

Rk

3

~

R

3

(1.7)

where

~

R =

−−→

OS

f

representstheinertial positionof theleader spa e raft. Theterm onthelefthand side of (1.7) an be furtherdeveloped usingthederivation rulewithrespe tto arotating frame:

 d

2

~r

dt

2



B

0

=

 d

2

_~r

dt

2



B

l

+ 2 ~

Ω

_B

_l

_/B

0

×

 d ~r

dt



B

l

+

d ~

Ω

B

l

/B

0

dt

!

B

l

× ~r + ~Ω

B

l

/B

0

×

~Ω

B

l

/B

0

× ~r



(1.8)

Theterms inthesum orrespond to thespa e raftrelative a eleration inthelo alframe,the

Eulera eleration, theCoriolisa elerationandthe entrifugala elerationrespe tively. Theterm

~

Ω

B

l

/B

0

representstherotationvelo ity ofthelo al basis

B

l

withrespe tto theinertial basis

B

0

.

Assuming that the dynami s of the leader spa e raft areexpressed using the orbital elements

dened in (1.3) and that the spa e raft relative state is given by the lo al relative position and

velo ity

X =

h

x y z v

x

v

y

v

z

i

T

, the dierent terms in (1.8) an be omputed individually.

In the aseof Keplerian motion,wehave:

~

Ω

_B

_l

_/B

0

=











0

− ˙ν

0











B

l

~

R =











0

0

R











B

l

~r =











x

y

z











B

l

(1.9)

Afterintrodu ing theelements from(1.9),equation (1.8) be omes:

 d

2

_~r

dt

2



B

0

=











¨

x − 2 ˙ν ˙z − ¨ν z − ˙ν

2

x

¨

y

¨

z + 2 ˙ν ˙x + ¨

_{ν x − ˙ν}

2

z











B

l

(1.10)

Developing therighthand sideof (1.7)leadsto thefollowingnonlinearequations forthe

spa e- raft relative dynami s:

¨

x − 2 ˙ν ˙z − ¨ν z − ˙ν

2

x = −

µ x

p(x

2

_{+ y}

2

_{+ (R − z)}

2

₎

3

¨

y = −

µ y

p(x

2

_{+ y}

2

_{+ (R − z)}

2

₎

3

¨

z + 2 ˙ν ˙x + ¨

_{ν x − ˙ν}

2

_{z = −}

µ(R − z)

p(x

2

_{+ y}

2

_{+ (R − z)}

2

₎

3

+

µ

R

2

(1.11)

In the ase where the distan e between the two satellites is a lot smaller than the distan e

equations an beusedto des ribe thespa e raft relative motion[101℄:

¨

x = 2 ˙ν ˙z + ¨

ν z + ˙ν

2

_{x −}

µ

R

3

x

¨

y = −

_R

µ

₃

y

¨

z = −2 ˙ν ˙x − ¨ν x + ˙ν

2

z + 2

µ

R

3

z

(1.12)

It an be noti ed thatfor thelinearizedequations,thedynami s on the

y

axisarede oupled from thedynami s inthe

xz

planeand dene aharmoni al os illator.

Inthe asewheretheorbitoftheleader spa e raftis ir ular,asimpliedform anbeobtained

for theaboveequations. If

e = 0

then

R = a =

onst,

˙ν = n =

onst and

ν = 0

¨

. After introdu ing these values in(1.12),thewell knownHill-Clohessy-Wiltshire equations for thespa e raftrelative

motionwith respe tto a ir ular referen eorbit anbe dedu ed[23,42℄:

¨

x = 2 n ˙z

¨

y = −n

2

_y

¨

z = −2 n ˙x + 3 n

2

z

(1.13)

It an be noti ed that in this ase the spa e raft relative dynami s orrespond to a Linear Time

Invariant system.

The non Keplerian relative dynami s

Long term predi tions of thespa e raft relative traje tory arene essaryfor formation ying

mis-sions. In this ase, maintaining the assumption that there are no external perturbing for es or

nonlinear terms introdu es una eptable predi tion errors. Therefore, dierent models of

spa e- raftrelativemotiona ountingforsomeoftheee tsoforbitaldisturban eshavebeendeveloped.

For ir ular referen e orbits, S hweighart and Sedwi k presented in [91℄ a set of onstant- oe ient lineardierential equationsthatin lude theperturbationdue totheEarth'soblateness,

representedthroughthe

J

2

potential. HamelanddeLafontainedevelopedin[39℄asetoflinearized equations ofrelativemotionabouta

J

2

perturbed ellipti alreferen eorbit. Ke hi hiangavein[50℄ the expression of the rotation velo ity

Ω

~

B

l

/B

0

for the ase where disturban es due to air drag and Earth oblateness are onsidered. The resultis very general but itleads to omplex nonlinear

expressions for therelative motionthatarenot easy to useinpra ti e.

Even ifthe dynami s modelled by the Ts hauner-Hempel equations (1.12) do not in lude the

model whi h is well suited for ontrol synthesis and has been widely used for spa e raft relative

traje torydesign [6,41,47,86,93,99℄.

1.3.2 Orbital elements dieren es dynami s

The dierential orbital elements are dened asthe dieren e between the orbital elements of the

leader spa e raft

oe

l

andtheorbitalelementsof follower spa e raft

oe

f

:

X

oe

= oe

l

− oe

f

=

h

δa δe δi δΩ δω δν (

or

δM

or

δE)

i

T

(1.14)

UnderKeplerian assumptions, ve of thesixorbital elementsdening thestate ofa spa e raft

are onstant. In this ase, the relative dynami s expressed using thedierential orbital elements

exhibitsimilarproperties. Thesimplestformfortherelativedynami sisobtainedwhenthevarying

term intheorbital elementsis hosen to be themeananomaly

M

:

˙

X

oe

=





























δ ˙a

δ ˙e

δ˙i

δ ˙

Ω

δ ˙ω

δ ˙

M





























=































0

0

0

0

0

−

3

₂

r µ

_a

₅

δa































(1.15)

Variationalmethods an beusedto analysetheee tofperturbinga elerationsontheorbital

elementsdes ribingthespa e raftmotion,inthenonKeplerian ase[90℄. Theperturbing a elera-tions anmodelforinstan etheee tsoftheEarthoblatenessand/ortheee tsoftheatmospheri

drag. The well known GaussVariational Equations (GVE)represent a spe i formulationof the

orbitalelements variation problem,written fordisturban esexpressedintheleader'sLVLHframe.

The spa e raft relative dynami s represented using the orbital elements dieren es have been

su essfully usedinformation ight appli ations, espe iallyfor ongurations that require a large

separation between the spa e raft [1,16℄. In the ase of the spa e raft rendezvous, the mission's obje tivesareusuallyspe iedusingtherelativeCartesianlo al oordinates, intermsofnal

rela-tive positionandvelo ity,givensome position/velo ity onstraints. For thisreason thedes ription

1.4 Linearized Cartesian relative motion

StartingfromtheTs hauner-Hempelequations(1.12)forthelinearizedCartesianrelativedynami s,

a state spa e representation of the spa e raft relative dynami s an be obtained. Based on this

formulation, losed form solutions for the relative traje tories an be omputed. These solutions

enablethepropagationofthespa e raftrelativestatewithoutmakinguseofnumeri alintegration,

whi h makesthemvery valuable for spa eappli ations where omputationalpowerislimited.

1.4.1 State-spa e representation

Letthespa e raftrelativestateve torbedenedbytherelativepositionandvelo ityproje tedon

ea h axis ofthe leader's LVLHframe:

X =

h

x y z v

x

v

y

v

z

i

T

. Ifin (1.12) theindependent

variable time is repla ed by the true anomaly of the leader spa e raft, a simplied form an be

obtained for the equations des ribing the relative dynami s between the leader and the follower

spa e raft. The derivatives withrespe tto timearerepla ed by:

d(·)

dt

=

d(·)

dν

dν

dt

= (·)

′

_˙ν

d

2

()

dt

2

=

d

2

()

dν

2

˙ν

2

₊

d()

dν

ν

¨

(1.16)

and thefollowing variable hange isused:

˜

X(ν) =





(1 + e cos ν)I

3

0

3

−e sin νI

3

(1 + e cos ν)

˙ν

I

3





X(t)

(1.17)

where

I

3

∈ R

3×3

is theidentity matrix and

0

3

∈ R

3×3

is the zeromatrix. Thisoperationleads to

a periodi state-spa e modelfor thespa e raftrelative dynami s:

˜

X

′

(ν) = ˜

A(ν) ˜

X(ν) + ˜

B ˜

u

(1.18)

where thedynami al matrix

˜

A(ν)

isgiven by:

˜

A(ν) =































0

0

0

1

0 0

0

0

0

0

1 0

0

0

0

0

0 1

0

0

0

0

0 2

0 −1

0

0

0 0

0

0

3

1 + e cos ν

−2 0 0































(1.19)

the ontrol matrix

B

˜

is dened by

B = [0

˜

3

I

3

]

T

and

u = [˜

˜

u

x

u

˜

y

u

˜

z

]

T

represents the a eleration

generated bythespa e raft thrusters.

Closedform solutions an be omputed for theperiodi system(1.18) and thegeneral method

for obtaining themis summarizednext.

1.4.2 The state transition matrix

The statetransitionmatrixprovidesa onvenientwayto representthesolutionoftheautonomous

dynami s of a linear system. For the spa e raft relative motion, omputing the state transition

matrix would enable the propagation of the relative state starting from any initial onditions,

without relying onnumeri al integration:

˜

X(ν) = Φ(ν, ν

0

) ˜

X(ν

0

)

(1.20)

Fromthedynami s ofthesystem(1.18) ,it anbededu edthatthestate transitionmatrixveries

thefollowing dierential equation:

Φ

′

(ν, ν

0

) = A(ν)Φ(ν, ν

0

),

Φ(ν, ν) = I ∀ν

(1.21)

For Linear Time Varyingsystems su h as(1.18), there is no general analyti al expression for the

state transition matrix. Numeri al methods developed for omputing

Φ

are usually based on the resolution of the dierential equation (1.21). In the ase of the spa e raft relative motion, the

spe ialstru ture of thedynami almatrix

A(ν)

enables the omputation of ananalyti al solution. Forthelinearizedspa e raftrelativemotion,thedynami sonthe

y

axisarenotae tedbythe motion in the

xz

plane and aredes ribed bythe following homogeneous se ond order dierential equation (see(1.18)):

˜

y

′′

_{= −˜y}

(1.22)

The solutionof (1.22) an be dire tlyexpressedasa fun tion oftheinitial onditions:

˜

X

y

(ν) = Φ

y

(ν, ν

0

) ˜

X

y

(ν

0

)

(1.23)

where

ν

0

istheinitial true anomalyfor theun ontrolled motionand:

˜

X

y

(ν) =





˜

y(ν)

˜

v

y

(ν)





Φ

_y

(ν, ν

₀

) =





cos(ν − ν

0

)

sin(ν − ν

0

)

− sin(ν − ν

0

) cos(ν − ν

0

)





(1.24)

From(1.18),thehomogeneous dierential equations for the

xz

planearegiven by:

˜

x

′′

= 2˜

z

′

(1.25)

˜

z

′′

=

3

1 + e cos ν

z − 2˜x

˜

′

(1.26)

Integrating (1.25) on eleads to:

˜

x

′

= 2˜

z + K

(1.27)

where Kis a onstant ofintegration. After introdu ing(1.27)in(1.26), ase ond orderdierential

equation only in

˜

z

is obtained:

˜

z

′′

+



4 −

3

1 + e cos ν



˜

z = K

(1.28)

Asre alled byCarter in[20℄, themethod for solving this type of dierential equation onsists in nding a family of parti ular solutions

ϕ

1

,

ϕ

2

for the homogeneous dierential equation su h that :

ϕ

1

ϕ

′

2

− ϕ

2

ϕ

′

1

=

onstant (1.29)

and then applying the te hnique of variation of parameters [80℄. The hoi e of the parti ular solutions

ϕ

1

,

ϕ

2

determines thenalformof thetransitionmatrix.

A transition matrix for the periodi system (1.18) has been proposed by Carter in [20℄. A slightly dierent solution hasbeen given by Yamanaka and Ankersen in [103℄, whi h presents the advantage of having a simpler form. The Yamanaka-Ankersen transition matrix will be used for

some of thedevelopments inthisdissertation and itis reprodu ed herefor ompleteness.

Taking

˜

X

xz

(ν) =

h

˜

x(ν) ˜

z(ν) ˜

v

x

(ν) ˜

v

y

(ν)

i

T

,thepropagationoftherelativestateisgivenby:

˜

X

xz

(ν) = Φ

xz

(ν, ν

0

) ˜

X

xz

(ν

0

)

(1.30)

where thetransition matrix

Φ

xz

(ν, ν

0

)

an bewritten as:

The matrix

φ

xz

(ν)

isdened by[103℄:

φ

xz

(ν) =



















1 − cos ν(2 + e cos ν)

sin ν(2 + e cos ν)

3(1 + e cos ν)

2

_J

0

sin ν(1 + e cos ν)

cos ν(1 + e cos ν)

_{2 − 3e sin ν(1 + e cos ν)J}

0

2 sin ν(1 + e cos ν)

_{2 cos ν(1 + e cos ν) − e}

_{3 − 6e sin ν(1 + e cos ν)J}

0

cos ν + e cos 2ν

_{− sin ν − e sin 2ν}

_−3e



(cos ν + e cos 2ν)J +

sin ν

1 + e cos ν





















(1.32)

Theterm

J

isrelatedtothe hoi eoftheparti ularsolution

ϕ

2

andinthe aseofthe Yamanka-Ankersentransition matrix isgivenby:

J(ν) =

Z

ν

ν

0

dτ

(1 + e cos τ )

2

=

n(t − t

0

)

(1 − e

2

₎

3/2

(1.33)

From(1.33)it follows thatfor theinitial trueanomaly

ν

0

wehave

J(ν

0

) = 0

. Thisenables the analyti al omputation of theinverse of the

φ

xz

(ν)

matrix at

ν

0

:

φ

−1

_xz

(ν

0

) =

1

e

2

₋₁





















e

2

_{−1 −}

3e sin ν

0

(2 + e cos ν

0

)

1 + e cos ν

0

e sin ν

0

(2 + e cos ν

0

)

2−e cos ν

0

(1+e cos ν

0

)

0

3 sin ν

0

(e cos ν

0

+1+e

2

₎

1 + e cos ν

0

− sin ν

0

(2 + e cos ν

0

)

−(cos ν

0

+e cos

2

ν

0

−2e)

0

3(e + cos ν

0

)

−(2 cos ν

0

+e cos

2

ν

0

+e)

sin ν

0

(1 + e cos ν

0

)

0

_{−(3e cos ν}

0

+ e

2

+ 2)

(1 + e cos ν

0

)

2

−e sin ν

0

(1 + e cos ν

0

)





















(1.34)

The omplete transition matrix

Φ

orresponding to the state ve tor

X(ν)

˜

an be obtained by ombining theblo ks fromthe

Φ

y

and

Φ

xz

matri es inthe appropriateorder.

Overview of losed form solutions

Several works have been dedi ated to the omputation of the transition matrix for the spa e raft

relativemotion,inthe asewheretheleadersatelliteevolvesonanarbitraryellipti alorbit. Melton

provides in[70℄a solution that uses dire tlythe timeas theindependent variable, obtained using seriesexpansionsofthee entri ity. However,thisisanapproximatesolutionanditlosesa ura y

for higher values of the e entri ity. Re ently, a transition matrix obtained starting from the

Ts hauner-Hempelequationsthatalsoin ludestheee tsofthe

J

2

perturbationhasbeenproposed byYamadaandKimurain[102℄. Thegiven solutionis umbersomeandnot easyto usefor ontrol design purposes. Moreover, the obtained transition matrix is shown to be a urate only for short

predi tion horizons.

motionhavebeenpresentedin[33,39℄. Theyarebasedonthe onne tionbetween thelo al Carte-sian relative state and thedierential orbital elementsand no longer require theresolution of the

dierential equations ofmotion. Gim and Alfriend onsider in[33℄both the short-period and the long-periodee tsofthe

J

2

perturbation,leadingtoaverya uratebut omplexsolutionthatstill requirestheknowledgeoftheevolutionoftheorbitalparametersfortheleadersatellite. Hameland

de Lafontaine simplify theproblemin[39℄bynegle ting theshort-term ee ts of

J

2

. Theyobtain a solution thatguarantees abounded predi tion erroreven for longhorizonsbut thatrequiresthe

knowledgeof therelativese ular driftof themeanorbital elements.

Closed form solutions of the spa e raft relative dynami s are sought for the omputational

advantage obtained from removing the integration pro ess from thetraje tory design algorithms.

Moreover, they an also provide some insight into thegeometri al properties of theresulting

tra-je tories. Some examples of traje tory parametrizations that have been derived from su h losed

form solutions will be presented inthenextse tion.

1.5 Properties of relative traje tories

The spa e raft ability to maintain a naturally periodi relative motion has been thoroughly

in-vestigated, espe ially in the ontext of formation ight appli ations. Some of the initialisation

te hniques for obtaining periodi solutions to the equations of spa e raft relative motion will be

presentednext,along withsome ofthegeometri al propertiesof theresulting traje tories.

The onne tion between the initial onditions of the periodi motion and the dimensions of

the obtained traje tory bears a lot of importan e in the mission design pro ess. The estimation

of theminimal distan e between the spa e raft is essential for ollision avoidan e purposes while

theevaluationof themaximaldistan eplaysan important role inthe hoi e ofthesensorsfor the

relative navigation. However, su ient understandingofthis onne tionhasnot yetbeen rea hed.

The next se tions summarize some interesting results found in the literature in relation to this

topi .

1.5.1 Periodi ity onditions

The distan e between two spa e raft on Keplerian orbits annot grow unboundedly [37℄. This observationisbasedonthefa tthatintheKeplerian asethespa e raftevolveontraje toriesthat

arebounded anddonot hangeovertime. However,unlesssome parti ular onditionsaremet,the

motionbetween spa e raft evolving onorbitsthat verifythefollowing ondition:

p T

l

= q T

f

, p, q ∈ N

(1.35)

where

T

l

and

T

f

aretheorbitalperiodsoftheleaderandthefollowerspa e raftrespe tively. Sin e:

T = 2π

s

a

3

µ

,

(1.36)

the ondition(1.35) anbeeasilytransformed into a onditiononthesemi-majoraxisoftheorbits

orrespondingto thetwo spa e raft:

a

f

=

3

s

p

2

q

2

a

l

(1.37)

or in a ondition between the energy of theorbits. The restri tion in (1.37) indu es a restri tion

ontherelativetraje tory. Figure1.5illustratesthetraje toryobtainedbypropagatingtherelative

motion over 10 orbital periods for dierent ratios between the orbital periods of two spa e raft.

The relative traje toryappears to layon a losed surfa e whose shape anddimensions depend on

theratio hosenbetween theorbital periods.

−12

−10

−8

−6

−4

−2

x 10

6

−5

0

5

x 10

6

−2

−1

0

1

2

x 10

6

y [m]

x [m]

z [m]

p=2, q=3

−12

−10

−8

−6

−4

−2

x 10

6

−6

−4

−2

0

2

4

6

x 10

6

−2

−1

0

1

2

x 10

6

y [m]

x [m]

z [m]

p=5, q=7

Figure1.5: Relative traje tories obtained for dierent ratiosbetween theorbital periods

In the asewhere

p = q = 1

, onstraint (1.37) be omes:

a

f

= a

l

(1.38)

In this ase, therelative traje torybetween thetwo spa e raftis periodi (see Figure1.6).