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

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 direct or indirect contribution to the

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

Théron,fortheirguidanceandforthefreedomtheygrantedmeinchoosingtheresearchdirections.

ThisthesishasbeensupportedbytheFrenchNationalCenterforSpaceStudiesandbyAstrium

EADSandIwouldlike tousethisoccasionto thankJean-Claude BergesandAlexandreFalcoz for

their help andfor their participation.

IwouldliketothankDenisArzelier,thehead oftheMethods andAlgorithmsinControl which

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

for supporting mycandidature for this subject. Iwouldalso like to thank all themembersof the

MACgroupfor making theworkenvironment soenjoyable.

Special thanksto Eric Kerrigan and to Paola Falugi and to theother members of theControl

and Powergroupat theImperialCollege ofLondon,fortheinterestingand challenging discussions

and for thegood timesduring mystayinLondon.

A³vrea deasemeneasãlemulµumesc pãrinµilormei,Nicoleta³iMarian, precum³isuroriimele

Teodora pentrutotajutorul oferit peperioada studiilor.

IwouldalsoliketothankFrancescoforhiscareandforhissupportandmyoldandnewfriends

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

Recentspacemissionsrelymore andmoreonthecooperationbetween dierentspacecraftinorder

to achieve a desiredobjective. Among thespacecraft proximity operations, theorbitalrendezvous

is a classical example that has generated a large amount of studies since the beginning of the

space exploration. However, the motivations and objectives for the proximity operations have

considerably changed. The need for higher autonomy, better security and lower costs prompts

for the development of new guidance and control algorithms. The presence of dierent types of

constraints and physical limitations also contributes to the increased complexity of the problem.

In this challenging context, this dissertation represents a contribution to the development of new

spacecraft guidance andcontrol algorithms.

The works presented in this dissertation are based on a structural analysis of the spacecraft

relative dynamics. Using a simpliedmodel, a new set of parametric expressions is developed for

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

properties of periodicrelative trajectories and for handling dierent types of state constraints. A

formal connection is evidenced between the set of parameters that dene constrained trajectories

and theconeof positive semi-denite matrices. Thisresult isexploited inthedesign of spacecraft

relative trajectories for proximity operations, in the impulsive control framework. The resulting

guidance algorithms enable the guaranteed continuous constraints satisfaction, while still relying

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

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

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

aussianciennequelaconquêtespatiale,quicontinuedegénérer denombreuxtravauxderecherche.

Cependant, les motivations etles objectifs desrécentes missions de rendez-vous orbital ont

large-ment évolués. En eet, les besoins d'une autonomie accrue, d'une sécurité améliorée, d'une plus

grande exibilité etd'une réduction descoûts, constituent autant d'incitations au développement

de nouvelles méthodes de guidage et contrôle. La satisfaction de contraintes très variées, dues à

des considérations de sécurité ouà deslimitations technologiques incontournables desactionneurs

ou descapteurs, contribuent à la richesse du problème posé. Dans ce contexte, le développement

de nouveauxalgorithmes de commande constitue unvrai déscientique quecette thèse tente de

relever.

Les travaux de cette thèse sont basées sur l'analyse structurelle des expressions décrivant le

mouvement relatifentredeuxvéhiculesenorbite. Surlabasedesmodèlesdetransitiondisponibles

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

particulièrement adaptée àlacaractérisation destrajectoires périodiques, orelapossibilitéd'une

prise en compte de contraintes d'état très variées. Un lien formel est mis en évidence entre les

paramètres dénissant les trajectoires contraintes et le cône des matrices semi dénies positives.

Ces résultatssont exploités dans ledéveloppement desalgorithmes de design de trajectoires pour

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

autre,lapropriétédepermettrelasatisfactiondescontraintessurlatrajectoiredemanièrecontinue

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

spéciquedelarobustessedesman÷uvresauxincertitudesdelachaînedemesureestaussiabordé

danscemanuscrit. Desapprochesdetypecommandeprédictivesontmisesenplaceandegarantir

Acknowledgements i

Abstract iii

Résumé v

Nomenclature ix

Introduction 1

1 Spacecraft relative motion 11

1.1 Introduction . . . 11

1.2 Dynamics ofa spacecraftorbiting theEarth . . . 12

1.3 Spacecraft relative motion . . . 16

1.3.1 LocalCartesian dynamics . . . 16

1.3.2 Orbital elements dierencesdynamics . . . 19

1.4 Linearized Cartesianrelative motion . . . 20

1.4.1 State-space representation . . . 20

1.4.2 The state transitionmatrix . . . 21

1.5 Properties ofrelative trajectories . . . 24

1.5.1 Periodicity conditions . . . 24

1.5.2 Inter-satellite distance . . . 27

1.5.3 Geometry oftheperiodicspacecraftrelative motion. . . 28

1.6 Conclusions . . . 30

2 Parametric expressions for the spacecraft relative trajectory 33 2.1 Denition ofthe parameters . . . 34

2.2 Properties ofspacecraft relative trajectories . . . 35

2.2.1 Dynamics ofthevector of parameters . . . 36

2.2.2 Properties ofperiodictrajectories . . . 38

2.3 Numericalanalysis oftheperiodic relative motion. . . 40

2.3.1 The eectsof theeccentricity of theleaderorbit . . . 40

2.3.2 The eectsof thevalues of theparameters . . . 40

2.4 Conclusion. . . 43

3 Constrained spacecraft relative trajectories 45 3.1 Denition ofadmissible trajectories . . . 46

3.2 Finite descriptionof admissible trajectories . . . 47

3.2.1 Finite descriptionusing constraints discretization . . . 48

3.2.2 Finite descriptionusing non-negative polynomials . . . 48

3.3 Description ofconstrained trajectories usingnon negative polynomials . . . 49

3.3.1 Rational expressionsfor thespacecraft relative motion . . . 49

3.3.2 Constrained nonperiodic trajectories . . . 52

4 Trajectory design for spacecraft rendezvous 57

4.1 Fixed-time linearizedimpulsivespacecraft rendezvous. . . 58

4.1.1 General formulation oftheguidance problem . . . 58

4.1.2 Consumption criteria . . . 59

4.1.3 Saturation constraints . . . 60

4.1.4 Using directshooting methods for theguidance problem . . . 61

4.2 Fixed-time rendezvouswithtrajectoryconstraints . . . 63

4.2.1 Guidance towardsa constrained periodicrelative motion . . . 63

4.2.2 Passively safe trajectories for spacecraftrendezvous . . . 66

4.2.3 Spacecraft rendezvous withvisibilityconstraints . . . 68

4.3 Numericalexamples . . . 70

4.3.1 Reaching aconstrained periodicrelative trajectory . . . 70

4.3.2 Passively safe rendezvoustrajectories . . . 74

4.3.3 Constrained nonperiodic relative trajectories . . . 78

4.4 Conclusion. . . 79

5 Spacecraft rendezvous robust to navigation uncertainties 81 5.1 ModelPredictive Control and spacecrafttrajectorydesign . . . 82

5.2 The robusttrajectoryplanning problem . . . 84

5.2.1 The spacecraftrelative dynamics . . . 85

5.2.2 The eectsof navigation uncertainties . . . 86

5.2.3 The nominaltrajectory . . . 87

5.2.4 General formulation oftheguidance problem . . . 87

5.3 Ane state-feedbackMPC. . . 89

5.3.1 Computation ofthefeedbackgains . . . 89

5.3.2 Computation ofthenominal control . . . 92

5.4 Ane disturbance feedbackMPC . . . 93

5.5 Numericalevaluation oftherobust control techniques. . . 97

5.5.1 Description ofthe simulation procedure . . . 97

5.5.2 The PRISMAmission . . . 99

5.5.3 The Simbol-X mission . . . 103

5.6 Conclusion. . . 108

6 Analytical bi-impulsive controlaround a desired periodic trajectory 109 6.1 Stabilityarounda periodicrelative trajectory . . . 110

6.2 Analytical bi-impulsive stabilizing control for theperiodicmotion . . . 111

6.2.1 Computation ofthecontrol . . . 112

6.2.2 Domain ofvalidity . . . 113

6.2.3 Performances inpresenceof navigation uncertainties . . . 115

6.3 Robust guidancetowards a spacecraftperiodicrelative motion . . . 117

6.4 Numericalexamples . . . 119

6.4.1 Inuenceof theeccentricityof thereferenceorbit . . . 120

6.4.2 Inuenceof theintervalbetween controls . . . 122

6.4.3 Inuenceof thenavigation uncertainties . . . 123

6.5 Conclusion. . . 124

B Properties of non negative polynomials 137

B.1 Checking polynomials non negativityon anite interval . . . 137

B.2 Checking 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-procedure . . . 142

µ

Earth'sgravitational constant

ν

true anomaly

Ω

longitudeof theascendingnode

ω

argument ofperigee

a

semi-major axis

B

0

Earthcentered inertial base

B

l

Spacecraft centered localCartesian base

E

eccentric anomaly

e

eccentricity

i

orbit inclination

M

mean anomaly

n

mean motion

LMI Linear MatrixInequality

LP Linear Program

LTI Linear TimeInvariant

LTV Linear TimeVarying

LVLH Local Vertical LocalHorizontal

MPC ModelPredictive Control

Résumé: Le succès des missions spatiales repose de plus en plus souvent sur la coopération entre

plusieurs véhicules 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

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

ont des besoins spéciques en termes d'algorithmes de contrôle, vue la distance réduite entre les

véhicules, les contraintes d'autonomieetde sécuritédes missionspatiales etles ressources limitées.

Les travaux decettethèse portent sur ledéveloppement des algorithmes deguidage pour des op

éra-tionsdeproximitéentreles satellites,oùla distance réduitepermetla navigationrelative. L'objectif

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

com-bustible, qui prennentencompteles contraintes opérationnellesde la missionetqui soientrobustes

à des incertitudes. Le cadre detravail choisiest celui des méthodes dîtesdirectes, qui permettentla

formulation duproblème deguidage comme unproblème d'optimisationparamétrique.

Background and motivations

Spacecraft rendezvousand docking capabilities arerequired for alarge arrayofspace applications

thatinvolvemorethanonespacecraft. Itisakeytechnologyforthein-orbitassemblyoflargeunits,

such as the space stations (Mir, Skylab, ISS). The space stations further rely on rendezvous and

dockingmissionsinorderto receivesupplies orto exchange thecrew. Forinstance, theunmanned

Automated Transfer Vehicle (ATV) from the European Space Agency periodically supplies the

International SpaceStation(ISS)withpropellant,water, air, payloadsand experiments. Recently,

theDragon spacecraft became therstcommercial spacecraft to successfullydockwiththeISS.

Space rendezvous has also been used for a variety of other purposes, including the service

missions to the Hubble Space Telescope and the EURECA spacecraft retrieval. Other on-orbit

servicing missions areunder study for existing spacecraft [7,85]. The increasing number of space debris in the Low Earth Orbit originating from mutual collisions, motivated the study of active

debris removalmissions[13].

In the recent years, a lot of interest hasbeen shown for space scientic missions that rely on

dierent instruments distributedovera eet of spacecraft. This congurationcan provide several

can usemultiple "massproduction" vehicles to assemble theeet. The robustness of themission

is alsoincreased bythis congurationsince thepayloadsaredistributedamongthespacecraft and

can eventually be replaced in case of failure [98]. Formation ying oers more exibility because the formation can be recongured in order to follow new mission requirements. This approach

has been considered for scientic missionswith very diverse objectives, such asEarth observation

(A-train),interferometryforEarth-likeplanetsdetection(DARWIN),measurementofgravitational

waves fromsupermassiveblackholebinaries (LISA)or X-raysspace telescope (Simbol-X).

The success of spacecraft rendezvous and formation ying missions depends on the precise

control of the spacecraft relative state, often-times in the context of relatively small spacecraft

separations. Inordertoensurethesecurityofthemission,ahighdegreeofautonomyandrobustness

is desiredfor therelative motioncontrol procedure. For missionssuchastheMarsSample Return

[86], for which the communication delay between the ground station and the spacecraft is very large,anautonomousguidancealgorithmwhichguaranteesthatnocollisionwilloccurbetweenthe

spacecraft isof vitalimportance.

The fuel-cost of the spacecraft maneuvers is also a matter of concern. The propulsion system

canaccountforupto50%ofthespacecraftmassatthelaunchtime,reducingtheavailablepayload

massandinuencingthecostofthelaunch. Thecontrolalgorithmsmustensurethatthecomputed

maneuversarefuel-optimal,suchthatthedesiredlifetimeforthemissioncan be achieved withthe

smallest amount of propellant.

The spacecraft rendezvous

Theorbitalrendezvousprocessconsistsinexecutingaseriesoforbitalmaneuverswiththepurpose

of bringing two spacecraft in close vicinity of each other. Usually one of the spacecraft, called

thetarget, isconsideredto be inert,while thesecondspacecraft, calledthefollower or thechaser,

executesthemaneuvers. Whentheobjectiveisto physicallyjointhetwospacecraftinvolvedinthe

rendezvous, we speakabout docking or berthing.

W. Fehse identied in[29]several phases ofa rendezvous mission, each one withits own chal-lenges. For the launch stage, the purpose is to bring the two spacecraft in vicinity by placing

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

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

follower spacecraft is controlled 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 scenario with the ISS for instance, this stage starts at a range of few tens of

kilome-tres and ends at a range of few kilometres from the target spacecraft. The following close range

rendezvous phase is usually divided in two stages: the closing maneuvers guiding the spacecraft

towards thenalapproachcorridor andthenal approach stage leading to matingconditions.

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

presentedinthisdissertationarerelatedtothehomingandtheclosingphases,duringwhichthetwo

spacecraftrelyonrelativenavigationmeasurementsinordertoachievethedesirednalconditions.

Some of the presented examples also refer to the nal approach phase, leading to the spacecraft

docking.

Mission constraints and technical challenges

The spacecraft rendezvous guidance is a complex process due to thedierent types of conditions

andconstraintsthatmustberespectedduringeachphaseofthemission. Forthephasesconsidered

in this dissertation, the far range and close range rendezvous stages, theapproach trajectory can

be required for instance to pass through specied hold points where the follower vehicle must

waitfor thepermissionto proceed,either fromtheground control station or fromthe crew ofthe

targetspacecraft[29]. Securityconsiderationsmightimposethechoiceofapproachtrajectoriesthat are inherently safe, meaning that they are guaranteed to avoid any collision with target vehicle,

Figure 2: Thedierent phases ofa rendezvousmission

visibilityconeofthetargetspacecraftforcontinuousvisualcontact. Thesespecicationscorrespond

to constraints that the rendezvous trajectory must respect in order to certify that the mission

requirementsaremet.

Another factor that must be taken into considerationwhen designing the approach trajectory

is the fuel-cost of the maneuvers. Thrust maneuvers can be approximated with impulses, i.e.

instantaneous changes of velocity at the time of maneuver. This simplies the computation and

the analysis of a fuel-optimal maneuvers plan for the spacecraft rendezvous [29]. The impulsive approximation is especially well adapted for the liquid propellant engines which are used for a

wide spanofspacecraftmaneuvers, rangingfromorbitaltransfermaneuversto stationkeepingand

Orbitaldisturbances,navigationerrorsandcontrolexecutionerrorscanaltertheoutcomeofthe

computed maneuvers. Thepresenceandtheeectsofthesedisturbancesneedto beintegratedinto

therendezvoustrajectorydesignphase. Navigationerrorsaredenedasthedierencebetweenthe

state perceivedbytheonboardsystemandtherealstateofthevehicle. Theycanbecausedbythe

sensors measurement performancelimitations, byerrorsin thealignment between thesensors and

spacecraft axes, by the onboard information processing and ltering, etc. The control execution

errors refer to deviations in magnitude, direction or application time from the desired impulsive

thrusts. They can be due to mounting errors, to misalignments with the mechanical axes, to the

engine performances, etc. The decision autonomy of the spacecraft cannot be increased without

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

usingcontrolalgorithmsofreducedcomplexitysincethecomputationalresourcesavailableonboard

thespacecraft arelimitedwithrespectto thoseavailableon ground.

Some of these challenges are addressed in this dissertation. The main objective is to provide

algorithms for thecomputation of robust fuel-optimal maneuversplans leading to rendezvous

tra-jectories that respecttheconstraintsimposedby themission'srequirements, even inpresenceof a

certain classofuncertainties. A study ofthespacecraftconstrained naturally periodictrajectories

is carriedout inrelation tothesecurityspecicationsfor theapproachtrajectory.

Thespacecrafttrajectorydesignisachallenging problemduetothepresenceoftrajectoryand

control constraints, totherobustness considerationsandtothelargenumberofdesignparameters.

In the most general case, only the initial time of the mission is xed and the trajectory design

procedure must provide a choice for the nal time, the number and the distribution of thrusting

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

use onboard the spacecraft, then restrictions are added on its computational complexity. A brief

presentationofthemaintrajectorydesignapproachesisgiveninwhatfollows,withafocusontheir

abilityto handlethedierentmission requirements.

Spacecraft relative trajectory design approaches

Thetrajectorydesignforspacecraftrendezvousandproximityoperationsreferstothecomputation

ofaseriesofmaneuversthatsteerthespacecraftfromsomeknowninitialrelativeconditionstosome

nal desired relative conditions. The design procedure generally consists insolving an open-loop

optimal control problemwhose solution corresponds tothe best approach trajectory thatrespects

respected and can increase thelifetime of the spacecraft. The techniques for solving this type of

constrainedopen-loopoptimalcontrolproblemsareusuallydividedintodirectmethodsandindirect

methods [24].

Indirect methods are based on analytical necessary optimality conditions derived using the

calculus of variations and thePontryagin maximumprinciple. The optimal solution can be found

by solving the two-point-boundary-value problem (TPBVP) resulting from these conditions [24]. When using the indirect methods, the optimal spacecraft trajectory for the rendezvous problem

is computed indirectly, based on the evolution of the adjoint state vector or the so-called primer

vector [58,62]. For impulsive trajectories, the primer vector indicates the times and thepositions of the thrust impulses that minimize the total fuel cost. However, the resolution of the problem

is complicated in thegeneral case, especially when constraints are added to the problem. It also

requires a good guess for the initial value of the primer vector. Recent works on the spacecraft

rendezvous problem have focused on transforming the necessary conditions for optimality into

constructive conditions forthe optimalsolution [3,4].

Direct methods rely on the transformation of the optimal control problem into a parameter

optimization problem. This isusually achieved through control parametrization and through

dis-cretization [45]. Theobtained nite-dimensional optimal control problemcan be eciently solved using the existing algorithms [11]. There are dierent types of direct methods depending on the choiceforthedecisionvariablesandontheusedintegrationmethod. Amongthem,thedirect

shoot-ing methods are used in the cases where the parametrisation concerns only the control variables.

The system'sdynamics areusuallylinear andare integrated analyticallyor numerically [49].

Theindirectresolutionmethodscertifytheglobaloptimalityofacomputedsolutionbychecking

a set of necessary and sucient (ifavailable) conditions. Howeverthey lead to problems thatare

hard to solve numerically, especially when constraints are considered. Direct methods are ableto

deal withstate andcontrol constraints more eectively and to integraterobustness elementswith

respect to dierent types of disturbances. Even if the obtained solution can only be certied as

optimal for the particular parametrization and/or discretization that has been considered, they

provide an attractive alternative for therendezvousguidance problem.

The algorithms developed in this dissertation for the design of spacecraft rendezvous

trajec-tories fall into the category of direct shooting methods. Other than theadvantage related to the

reduced complexity of theresulting optimization problem, this approach also oersthepossibility

willbeonreducingtheeectsofrelativenavigationuncertainties onthenalrendezvousprecision.

Spacecraft trajectory control: closing the loop

The direct and indirect approaches for spacecraft relative trajectory design provide a series of

fuel-optimal maneuvers that need to be executed at the specied instants in order to reach the

desired nal objective. The maneuvers plan is obtained based on open-loop predictions of the

evolution of the spacecraft relative trajectory. As previously discussed, the presence of orbital

perturbations, navigation uncertainties or control execution errorsmight alter theoutcome of the

computed maneuvers. Inorder to limit their undesiredeects andto reach aspecied rendezvous

precision, thetrajectory control needsto be implementedina closed-loop manner.

Theresolutionofaconstrainedopen-loopoptimalcontrolproblemcanbeintegratedina

closed-loop setting by using the Model Predictive Control (MPC) methodology [84]. Model Predictive ControlorRecedingHorizon Controlisacontroltechnique forwhichthecontrol actionisobtained

bysolving at each sampling instant anite-horizon open-loop optimal control problem, using the

current state ofthesystemasinitial state. The optimizationdeliversevery timeaseries ofcontrol

actions out of which only therst one is applied to thesystem. Therest of the planis discarded

because a new solution, based on new measurement information, will be computed at the next

sampling time[68].

ModelPredictiveControlisapopularcontroltechniqueforspacecraftrendezvousandproximity

operations [16,18,26,32,41,43,86]. Itspopularity isdue totheabilityto integrateconstraints and uncertainties directly into the trajectorydesign problem. Dierent othercontrol approaches have

been proposed for spacecraft proximity operations and formation ying, spanning over a large

range of techniques. A non exhaustive list includes adaptive control [2,95], non-linear quadratic regulator [6], feedback impulsive control [89], Lyapunov-based nonlinear output feedback control [104], time-delayed feedbackcontrol [12]andseveralothers [87,88]. Butvery fewofthem consider thepresenceofconstraintsorthefuelcostofthemaneuvers,andfocusonlyonreachingthespecied

nal conditions.

Insteadofdeterminingo-lineafeedbackpolicythatprovidestheoptimalcontrolforallsystem

states, MPC solves an open-loop optimal control problem on-line which takes into consideration

the current state of the system. The periodic recomputation of the solution creates an implicit

closed-loop. Therobustnesspropertiesofthisimplicit closed-loopwithrespecttodierenttypesof

The presence of uncertainties raises questions related to the changes induced in the control

performances. Inthecaseofspacecrafttrajectorycontrol, theperformances aredened inrelation

to the fuel consumption and to the precision with respect to the desired nal objective of the

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

as the robustness of the closed-loop for the control that has been computed without explicitly

considering theuncertainties [36,67]. But for problems thatinclude control and state constraints, the computed control actions must guarantee that no transgressions of theconstraints will occur

for allthepossiblerealizationsoftheuncertainties. Inthiscase, theinherent robustnessproperties

areno longersucientand thepresenceofuncertainties needsto be included inthewriting ofthe

optimizationproblem[27,59,69,79]. Anotherkeyaspectisthepropertyofrecursivefeasibilityofthe control problem inpresenceof uncertainties. The optimal control is recomputed at each sampling

instant anditisimportantto providetheoreticalguarantees that,iftherst optimizationproblem

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

properties areinvestigated for theguidance algorithmsproposedin thisdissertation.

Objectives and organization of the dissertation

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

spacecraftrelativemotionandthedesignandcontrolofthespacecraftrelativetrajectory. Thestudy

of the relative motion concentrates on spacecraft naturally periodic relative trajectories. These

periodictrajectories,intheabsenceofperturbations,require nocontrolinorderto bemaintained.

Thispropertycouldmakethemgoodcandidatesfor parkingorbitsinbetween dierent phasesofa

rendezvousmission,forautonomousinspectiontrajectoriesforon-orbitservicingmissionsorforfail

trajectories in case of systemmalfunction. Chapter 1 summarizes the most common-used models

for representing the spacecraft relative motion. It also provides an overview of the properties

of the spacecraft relative trajectories that are of interest for the rendezvous guidance problem,

suchasperiodicityconditions, inter-satellite distanceand geometric properties ofperiodicrelative

trajectories.

Thedierentperiodicmotioninitialisationtechniquespresentedintheliteraturedonotgiveany

information about the geometric properties of the resulting trajectory. To address this problem,

a new parametrization for the spacecraft relative trajectories is developed in Chapter 2. This

parametrizationprovidesagoodframeworkforanalysingtheirpropertiesanditisinusedinChapter

The spacecraft relative trajectory control concentrates around the problem of designing

fuel-optimal maneuvers plans leading the spacecraft from an arbitrary initial relative state towards a

desirednalrelativestate,followingtrajectorieswhichrespectsdierentmissionconstraints.

Chap-ter 4 details the writing of the spacecraft rendezvous guidance problem as an impulsive optimal

control problem using direct shooting methods. It illustrates the contribution of the results

pre-sented in Chapter 3 in obtaining approach trajectories that respect visibility constraints or that

are guaranteed to be safe for a large rangeof system errors. The robustness aspects with respect

to navigation uncertainties aretreatedinChapter5. Theguidanceproblemismodiedinorderto

provideasolutionwhichguaranteesapriori constraintssatisfactionforalladmissiblevaluesforthe

uncertainties, without modifyingthecomplexityof thecontrol algorithm. Moreover, theproposed

control strategyalsominimizestheeectsofthesensingnoise ontheprecisionwithwhichthenal

objective isachieved.

Thepresence ofperturbationsalso aectsthespacecraft naturally periodic motion. Chapter 6

presentsalow-complexitystabilizingcontrolstrategyforthespacecraftperiodicmotioninpresence

of sensingnoise. Thedeveloped methodisbasedontheparametrizationfor thespacecraftrelative

trajectorypresentedinChapter 2.

Spacecraft relative motion

Contents

1.1 Introduction . . . 11

1.2 Dynamicsof a spacecraft orbitingthe Earth . . . 12

1.3 Spacecraft relative motion . . . 16

1.3.1 LocalCartesiandynamics . . . 16

1.3.2 Orbitalelementsdierencesdynamics . . . 19

1.4 Linearized Cartesianrelative motion. . . 20

1.4.1 State-spacerepresentation . . . 20

1.4.2 Thestatetransitionmatrix . . . 21

1.5 Properties of relative trajectories. . . 24

1.5.1 Periodicityconditions . . . 24

1.5.2 Inter-satellitedistance . . . 27

1.5.3 Geometryoftheperiodicspacecraftrelativemotion . . . 28

1.6 Conclusions . . . 30

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

satelliteappelélechasseurparrapportàunautre satellite,appelélacible. Diérenteschoixexistent

pour la représentation de l'état relatif, chacune avec ses avantages. Plusieurs représentations sont

passéesenrevueence chapitre,notammentdans lecadre des orbitesképlériennes. L'accent estmis

sur ladescriptionbasée surles positionsetlesvitessesrelatives,exprimées dansunrepère cartésien

local attaché au satellite cible. Les propriétés des trajectoires relatives sont également étudiées,

comme les distances minimale et maximale entre les satellites, l'existence des trajectoires relative

périodiques et leur propriétés géométriques.

1.1 Introduction

Thespacecraftrelativemotionreferstothestudyofthedynamicsofaspacecraft,calledthefollower,

with respect to the dynamics of another spacecraft, called the leader or the target. The motion

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

most common descriptions will be presented in this chapter. The nal choice is usually driven

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

dierenceshavebeenusedforformationyingapplications[16,34,60],whileCartesianmodelshave been preferredfor spacecraftrendezvous andcollision avoidance problems [17,31,41,57].

Regardless of the representation chosen for the spacecraft relative motion, a distinction can

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

the Earth is represented as an homogeneous sphere and the spacecraft motion is aected only

byNewtonian accelerations. The non Keplerian models take into account theEarth's oblateness,

usually through the spherical harmonic model for the Earth's potential, the atmospheric dragor

thesolar radiation pressure, amongother orbitaldisturbances.

TheKeplerian framework leadsto lessaccurate butsimplieddynamicalmodels for the

space-craftrelative motion. Thesesimpliedmodels arewellsuitedfor controlsynthesispurposes,likein

thecase of maneuversplans design for spacecraft rendezvousmissions for instance. Therelatively

small distances between the spacecraft when compared to thedistance with respect to the center

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

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

throughout thisdissertation, while referringthe interested readerto publicationstreating some of

theother representations.

Inwhat follows, aparticular interestwill be paidto theperiodicsolutions of theequations

de-scribing thespacecraft relative motion. Thesesolutions enable thesatellitesto maintaina desired

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

beenextensivelyusedintheformationightliterature[1,5,46,55,92]andhasrecentlygained atten-tion for orbital rendezvous and collision avoidance applications [25,41,43]. Dierent initialization methods forperiodicmotionwillbepresentedalongwithsomeofthegeometricalpropertiesofthe

resulting trajectories.

1.2 Dynamics of a spacecraft orbiting the Earth

The Keplerian dynamics of a spacecraft with respect to the Earth can be derived from Newton's

equations of motionbetween two massparticles. In this case, themotion of a spacecraft orbiting

theEarth isdescribed bythefollowing dierential equation[8]:

d

2

_R

~

dt

2

!

B0

=

₋

µ

k ~

R

_k

3

~

R

(1.1)

where

R

~

representsthevectorfromthecenteroftheEarthtothespacecraftcenterofmassand

µ

is theEarth's gravitational constant. The dynamics areexpressedwithrespectto an Earthcentered

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 coincides 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 trajectory

Even thoughthedierential equation (1.1)governing therelative motionof two bodies is

non-linear,theequationadmitsageneralanalyticalsolution[8]. Theconstantsofintegrationassociated to thesolution arecalled theorbital elements of thesatellite motion and they playan important

role inthestudy ofthe properties of thespacecraft trajectory.

Let the orbital plane be the plane which contains the trajectory of the orbiting spacecraft

(see Figure1.1). Theequation ofthespacecraft trajectoryexpressedusing polar coordinateswith

respectto this planeisgiven by [8]:

R =

k ~

R

k =

a(1

− e

2

₎

1 + e cos ν

(1.2)

where

a

is called the semi-major axis of the spacecraft orbit,

e

is called the eccentricity and

ν

is calledthetrueanomaly. Thesatellite'sorbitisboundedif

e < 1

andunboundedif

e

≥ 1

. For

e = 0

the spacecraft trajectory is acircle of radius

a

and for

0 < e < 1

the trajectory is an ellipse. The true anomaly

ν

representstheangle between thespacecraft's current positionand thedirection of theperigee(Figure 1.1).

orientation of the orbital plane are required in order to completely characterize the spacecraft

trajectory. A common choice is represented bythe angles

i

,

Ω

and

ω

dened withrespect to the Earth's equatorialplane, asindicatedinFigure 1.2.

Figure 1.2: The denitionof theclassicalorbital elements

Theline of nodes denotes theline of intersection between thespacecraftorbital planeand the

equatorial plane. The ascending node refers to thepoint where the satellite iscrossing the line of

nodesinanorthbound direction. Thelongitude of the ascending node,

Ω

,istheangle between the

X

axis of the

R

0

frame and theascending node, the argument of perigee,

ω

,is the angle between theascending node and theperigeewhilethe inclination,

i

,is theanglebetween theorbital plane and theequatorialplane.

Theset oforbital elementsis dened by:

oe =

h

a e i Ω ω ν

i

T

(1.3)

anditcompletelydescribesthestateofasatelliteorbitingtheEarth. UnderKeplerianassumptions,

therst ve parameters areconstant andonly thetrueanomaly changeswithtime[8]:

˙ν =

r

_µ

a

3

₍₁

_{− e}

2

₎

3

(1 + e cos ν)

2

(1.4)

Sometimes, the eccentric anomaly,

E

, or the mean anomaly,

M

, are used instead of

ν

as the varyingstate. Theeccentric anomalyandthetrueanomalyarerelatedthrough geometrical

trans-formations (Figure 1.3):

tan

ν

2

=

r

1 + e

1

− e

tan

E

2

(1.5)

while eccentric 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 can also be dened as a linear function of time, where

n =

p

µ/a

3

is the mean motion of the satellite,

t

0

is the reference time and

M

0

is the mean anomaly at

t

0

.

Figure1.3: Thedenition oftheeccentric anomaly

When the orbit is circular or near circular (

e

≈ 0

) or when theorbit is planar or near planar (

i

≈ 0

), some of the classical orbital elements

oe

are not dened. Inthose cases, the state of the spacecraft can be represented using dierent functionsof the classical orbital elements thatavoid

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

orbital elements, the equinoctial elements or the Delaunay canonical elements, used for studying

thesatellite motionina Hamiltonianframework[90].

Thechoiceofusingtheinertialpositionandvelocityorthevarioussetsoforbitalparametersin

ordertodescribethestateofaspacecraftorbitingtheEarthismadedependingontheapplication.

Throughout this dissertation, theclassical orbital elements

oe

arepreferred for therepresentation oftheleader'sstate. Thischoiceismotivatedbythefactthat, intheKepleriancontext considered

here, the resulting dynamics have a very simple form (only one state that changes over time).

To complete the description of the spacecraft relative motion, the state of the follower satellite

1.3 Spacecraft relative motion

The spacecraft relative motion refers to the study of the dynamics of the leader spacecraft

com-bined with the study of the dynamics of the follower spacecraft. As previously stated, there are

dierent possible state denitions which can be used inthe description of the motion of a single

spacecraft (Cartesian positionand velocity, dierent sets of orbital parameters). In a similarway,

dierent representations can be considered for the spacecraft relative state, each one bearing its

ownadvantages.

1.3.1 Local Cartesian dynamics

The spacecraft relative motionrepresented using local Cartesiandynamics is dened with respect

to a localrotatingCartesianframe centered on theleader satellite. A commonlyusedframeisthe

Local Vertical Local 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 center of the Earth, the

~

y

axis is orthogonal to the orbital plane, inthe oppositedirection with respect to theangular momentum

vector, andthe

~x

axislays intheleader's orbitalplane inthedirectionof thesatellite's velocity.

PSfragreplacements

~x

~z

_ν

a

O

_P

~

~

Q

S

l

S

2

~r

Figure 1.4: The spacecraftrelative positionand theleader's LVLHframe

The relative position between theleader spacecraft

S

l

and the follower spacecraft

S

f

is repre-sentedby

~r =

−−→

S

l

S

f

inFigure 1.4. Considering that theKeplerian dynamics of each satellite with respectto the Earthcan be described using (1.1), therelative inertial acceleration can bewritten

as:



d

2

~r

dt

2



B0

=

₋

µ

k ~

R + ~r

_k

3

( ~

R + ~r) +

µ

k ~

R

_k

3

~

R

3

(1.7)

where

~

R =

−−→

OS

f

representstheinertial positionof theleader spacecraft. Theterm onthelefthand side of (1.7) can be furtherdeveloped usingthederivation rulewithrespectto arotating frame:



d

2

~r

dt

2



B0

=



d

2

~r

dt

2



B

l

+ 2 ~

Ω

_Bl/B0

_×



d ~r

dt



B

l

+

d ~

Ω

Bl/B0

dt

!

Bl

× ~r + ~Ω

Bl/B0

×



~

Ω

_Bl/B0

_{× ~r}



(1.8)

Theterms inthesumcorrespond to thespacecraftrelative acceleration inthelocalframe,the

Euleracceleration, theCoriolisaccelerationandthecentrifugalaccelerationrespectively. Theterm

~

Ω

B

l

/B0

representstherotationvelocity ofthelocal basis

B

l

withrespectto theinertial basis

B

0

.

Assuming that the dynamics of the leader spacecraft areexpressed using the orbital elements

dened in (1.3) and that the spacecraft relative state is given by the local relative position and

velocity

X =

h

x y z v

x

v

y

v

z

i

T

, the dierent terms in (1.8) can be computed individually.

In thecaseof Keplerian motion,wehave:

~

Ω

_Bl/B0

=











0

− ˙ν

0











Bl

~

R =











0

0

R











Bl

~r =











x

y

z











Bl

(1.9)

Afterintroducing theelements from(1.9),equation (1.8) becomes:



d

2

_~r

dt

2



B0

=











¨

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

space-craft relative dynamics:

¨

x

− 2 ˙ν ˙z − ¨ν z − ˙ν

2

_{x =}

₋

_p

µ x

(x

2

_{+ y}

2

_{+ (R}

_{− z)}

2

₎

3

¨

y =

₋

_p

µ y

(x

2

_{+ y}

2

_{+ (R}

_{− z)}

2

₎

3

¨

z + 2 ˙ν ˙x + ¨

ν x

_{− ˙ν}

2

_{z =}

₋

_p

µ(R

− z)

(x

2

_{+ y}

2

_{+ (R}

_{− z)}

2

₎

3

+

µ

R

2

(1.11)

In the case where the distance between the two satellites is a lot smaller than the distance

equations can beusedto describe thespacecraft relative motion[101]:

¨

x = 2 ˙ν ˙z + ¨

ν z + ˙ν

2

_x

₋

µ

R

3

x

¨

y =

₋

µ

R

3

y

¨

z =

−2 ˙ν ˙x − ¨ν x + ˙ν

2

_{z + 2}

µ

R

3

z

(1.12)

It can be noticed thatfor thelinearizedequations,thedynamics on the

y

axisaredecoupled from thedynamics inthe

xz

plane and dene aharmonical oscillator.

Inthecasewheretheorbitoftheleader spacecraftiscircular,asimpliedformcanbeobtained

for theaboveequations. If

e = 0

then

R = a =

const,

˙ν = n =

const and

ν = 0

¨

. After introducing these values in(1.12),thewell knownHill-Clohessy-Wiltshire equations for thespacecraftrelative

motionwith respectto a circular referenceorbit canbe deduced[23,42]:

¨

x = 2 n ˙z

¨

y =

_−n

2

_y

¨

z =

−2 n ˙x + 3 n

2

_z

(1.13)

It can be noticed that in this case the spacecraft relative dynamics correspond to a Linear Time

Invariant system.

The non Keplerian relative dynamics

Long term predictions of thespacecraft relative trajectory arenecessaryfor formation ying

mis-sions. In this case, maintaining the assumption that there are no external perturbing forces or

nonlinear terms introduces unacceptable prediction errors. Therefore, dierent models of

space-craftrelativemotionaccountingforsomeoftheeectsoforbitaldisturbanceshavebeendeveloped.

For circular reference orbits, Schweighart and Sedwick presented in [91] a set of constant-coecient lineardierential equationsthatinclude theperturbationdue totheEarth'soblateness,

representedthroughthe

J

2

potential. HamelanddeLafontainedevelopedin[39]asetoflinearized equations ofrelativemotionabouta

J

2

perturbed ellipticalreferenceorbit. Kechichiangavein[50] the expression of the rotation velocity

Ω

~

Bl/B0

for the case where disturbances due to air drag and Earth oblateness are considered. The resultis very general but itleads to complex nonlinear

expressions for therelative motionthatarenot easy to useinpractice.

Even ifthe dynamics modelled by the Tschauner-Hempel equations (1.12) do not include the

model which is well suited for control synthesis and has been widely used for spacecraft relative

trajectorydesign [6,41,47,86,93,99].

1.3.2 Orbital elements dierences dynamics

The dierential orbital elements are dened asthe dierence between the orbital elements of the

leader spacecraft

oe

l

andtheorbitalelementsof follower spacecraft

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 spacecraft

are constant. In this case, the relative dynamics expressed using thedierential orbital elements

exhibitsimilarproperties. Thesimplestformfortherelativedynamicsisobtainedwhenthevarying

term intheorbital elementsis chosen 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)

Variationalmethodscan beusedto analysetheeectofperturbingaccelerationsontheorbital

elementsdescribingthespacecraftmotion,inthenonKepleriancase[90]. Theperturbing accelera-tionscanmodelforinstancetheeectsoftheEarthoblatenessand/ortheeectsoftheatmospheric

drag. The well known GaussVariational Equations (GVE)represent a specicformulationof the

orbitalelements variation problem,written fordisturbancesexpressedintheleader'sLVLHframe.

The spacecraft relative dynamics represented using the orbital elements dierences have been

successfully usedinformation ight applications, especiallyfor congurations that require a large

separation between the spacecraft [1,16]. In the case of the spacecraft rendezvous, the mission's objectivesareusuallyspeciedusingtherelativeCartesianlocalcoordinates,intermsofnal

rela-tive positionandvelocity,givensome position/velocityconstraints. For thisreason thedescription

1.4 Linearized Cartesian relative motion

StartingfromtheTschauner-Hempelequations(1.12)forthelinearizedCartesianrelativedynamics,

a state space representation of the spacecraft relative dynamics can be obtained. Based on this

formulation, closed form solutions for the relative trajectories can be computed. These solutions

enablethepropagationofthespacecraftrelativestatewithoutmakinguseofnumericalintegration,

which makesthemvery valuable for spaceapplications where computationalpowerislimited.

1.4.1 State-space representation

Letthespacecraftrelativestatevectorbedenedbytherelativepositionandvelocityprojectedon

each 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 replaced by the true anomaly of the leader spacecraft, a simplied form can be

obtained for the equations describing the relative dynamics between the leader and the follower

spacecraft. The derivatives withrespectto timearereplaced by:

d(

_·)

dt

=

d(

_·)

dν

dν

dt

= (

·)

0

_˙ν

d

2

()

dt

2

=

d

2

()

dν

2

˙ν

2

₊

d()

dν

ν

¨

(1.16)

and thefollowing variable change 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 periodicstate-space modelfor thespacecraftrelative dynamics:

˜

X

0

(ν) = ˜

A(ν) ˜

X(ν) + ˜

B ˜

u

(1.18)

where thedynamical 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 control matrix

B

˜

is dened by

B = [0

˜

3

I

3

]

T

and

u = [˜

˜

u

x

u

˜

y

u

˜

z

]

T

represents the acceleration

generated bythespacecraft thrusters.

Closedform solutions can becomputed for theperiodicsystem(1.18) and thegeneral method

for obtaining themis summarizednext.

1.4.2 The state transition matrix

The statetransitionmatrixprovidesaconvenientwayto representthesolutionoftheautonomous

dynamics of a linear system. For the spacecraft relative motion, computing the state transition

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

without relying onnumerical integration:

˜

X(ν) = Φ(ν, ν

0

) ˜

X(ν

0

)

(1.20)

Fromthedynamics ofthesystem(1.18) ,itcanbededucedthatthestate transitionmatrixveries

thefollowing dierential equation:

Φ

0

(ν, ν

0

) = A(ν)Φ(ν, ν

0

),

Φ(ν, ν) = I

∀ν

(1.21)

For Linear Time Varyingsystems such as(1.18), there is no general analytical expression for the

state transition matrix. Numerical methods developed for computing

Φ

are usually based on the resolution of the dierential equation (1.21). In the case of the spacecraft relative motion, the

specialstructure of thedynamicalmatrix

A(ν)

enables thecomputation of ananalytical solution. Forthelinearizedspacecraftrelativemotion,thedynamicsonthe

y

axisarenotaectedbythe motion in the

xz

plane and aredescribed bythe following homogeneous second order dierential equation (see(1.18)):

˜

y

00

=

−˜y

(1.22)

The solutionof (1.22) can be directlyexpressedasa function oftheinitial conditions:

˜

X

y

(ν) = Φ

y

(ν, ν

0

) ˜

X

y

(ν

0

)

(1.23)

where

ν

0

istheinitial true anomalyfor theuncontrolled 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

00

= 2˜

z

0

(1.25)

˜

z

00

=

3

1 + e cos ν

z

˜

− 2˜x

0

(1.26)

Integrating (1.25) onceleads to:

˜

x

0

= 2˜

z + K

(1.27)

where Kis aconstant ofintegration. After introducing(1.27)in(1.26), asecond orderdierential

equation only in

˜

z

is obtained:

˜

z

00

+



4

₋

3

1 + e cos ν



˜

z = K

(1.28)

Asrecalled byCarter in[20], themethod for solving this type of dierential equationconsists in nding a family of particular solutions

ϕ

1

,

ϕ

2

for the homogeneous dierential equation such that :

ϕ

1

ϕ

0

2

− ϕ

2

ϕ

0

1

=

constant (1.29)

and then applying the technique of variation of parameters [80]. The choice of the particular solutions

ϕ

1

,

ϕ

2

determines thenalformof thetransitionmatrix.

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

some of thedevelopments inthisdissertation and itis reproduced herefor completeness.

Taking

˜

X

xz

(ν) =

h

˜

x(ν) ˜

z(ν) ˜

v

x

(ν) ˜

v

y

(ν)

i

T

,thepropagationoftherelative stateisgivenby:

˜

X

xz

(ν) = Φ

xz

(ν, ν

0

) ˜

X

xz

(ν

0

)

(1.30)

where thetransition matrix

Φ

xz

(ν, ν

0

)

can 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

isrelatedtothechoiceoftheparticularsolution

ϕ

2

andinthecaseofthe Yamanka-Ankersentransition matrix isgivenby:

J(ν) =

Z

ν

ν0

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 analytical computation 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 complete transition matrix

Φ

corresponding to the state vector

X(ν)

˜

can be obtained by combining theblocks fromthe

Φ

y

and

Φ

xz

matrices inthe appropriateorder.

Overview of closed form solutions

Several works have been dedicated to thecomputation of the transition matrix for the spacecraft

relativemotion,inthecasewheretheleadersatelliteevolvesonanarbitraryellipticalorbit. Melton

provides in[70]a solution that uses directlythe timeas theindependent variable, obtained using seriesexpansionsoftheeccentricity. However,thisisanapproximatesolutionanditlosesaccuracy

for higher values of the eccentricity. Recently, a transition matrix obtained starting from the

Tschauner-Hempelequationsthatalsoincludestheeectsofthe

J

2

perturbationhasbeenproposed byYamadaandKimurain[102]. Thegiven solutioniscumbersomeandnot easyto usefor control design purposes. Moreover, the obtained transition matrix is shown to be accurate only for short

prediction horizons.

motionhavebeenpresentedin[33,39]. Theyarebasedontheconnectionbetween thelocal Carte-sian relative state and thedierential orbital elementsand no longer require theresolution of the

dierential equations ofmotion. Gim and Alfriend consider in[33]both the short-period and the long-periodeectsofthe

J

2

perturbation,leadingtoaveryaccuratebutcomplexsolutionthatstill requirestheknowledgeoftheevolutionoftheorbitalparametersfortheleadersatellite. Hameland

de Lafontaine simplify theproblemin[39]byneglecting theshort-term eects of

J

2

. Theyobtain a solution thatguarantees abounded prediction erroreven for longhorizonsbut thatrequiresthe

knowledgeof therelativesecular driftof themeanorbital elements.

Closed form solutions of the spacecraft relative dynamics are sought for the computational

advantage obtained from removing the integration process from thetrajectory design algorithms.

Moreover, they can also provide some insight into thegeometrical properties of theresulting

tra-jectories. Some examples of trajectory parametrizations that have been derived from such closed

form solutions will be presented inthenextsection.

1.5 Properties of relative trajectories

The spacecraft ability to maintain a naturally periodic relative motion has been thoroughly

in-vestigated, especially in the context of formation ight applications. Some of the initialisation

techniques for obtaining periodic solutions to the equations of spacecraft relative motion will be

presentednext,along withsome ofthegeometrical propertiesof theresulting trajectories.

The connection between the initial conditions of the periodic motion and the dimensions of

the obtained trajectory bears a lot of importance in the mission design process. The estimation

of theminimal distance between the spacecraft is essential for collision avoidance purposes while

theevaluationof themaximaldistanceplaysan important role inthechoice ofthesensorsfor the

relative navigation. However, sucient understandingofthisconnectionhasnot yetbeen reached.

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

topic.

1.5.1 Periodicity conditions

The distance between two spacecraft on Keplerian orbits cannot grow unboundedly [37]. This observationisbasedonthefactthatintheKeplerian casethespacecraftevolveontrajectoriesthat

arebounded anddonotchangeovertime. However,unlesssome particularconditionsaremet,the

motionbetween spacecraft evolving onorbitsthat verifythefollowing condition:

p T

l

= q T

f

, p, q

∈ N

(1.35)

where

T

l

and

T

f

aretheorbitalperiodsoftheleaderandthefollowerspacecraftrespectively. Since:

T = 2π

s

a

3

µ

,

(1.36)

thecondition(1.35)canbeeasilytransformed into aconditiononthesemi-majoraxisoftheorbits

correspondingto thetwo spacecraft:

a

f

=

3

s

p

2

q

2

a

l

(1.37)

or in a condition between the energy of theorbits. The restriction in (1.37) induces a restriction

ontherelativetrajectory. Figure1.5illustratesthetrajectoryobtainedbypropagatingtherelative

motion over 10 orbital periods for dierent ratios between the orbital periods of two spacecraft.

The relative trajectoryappears to layon a closed surface whose shape anddimensions depend on

theratio chosenbetween 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 trajectories obtained for dierent ratiosbetween theorbital periods

In thecasewhere

p = q = 1

,constraint (1.37) becomes:

a

f

= a

l

(1.38)

In thiscase, therelative trajectorybetween thetwo spacecraftis periodic(see Figure1.6).