massari alenia constellations

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Constellations and Formations of SatellitesAlcatel Alenia Space Torino, 27 February 2007

Mauro Massari

Department of Aerospace Engineering

Politecnico di Milano

[email protected]
http://fullscreen/


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Constellations

Formations

Constellations and Formations of Satellites AAS Torino, 27 February 2007 - p. 2/90

Outline

ConstellationsIntroductionApplications

Design ProcessSTK Examples

FormationsIntroductionApplications

Relative DynamicsDesign Process
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Constellations

Introduction

Applications

Constellation Design

Orbit Design

Coverage

Classical Design Approach

Walker Constellations

Street of coverage

Advanced Design Approach

Performance analysisDeployment

Maintenance

Formations


Constellations


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Maintenance

Formations


Introduction

A Constallation is a group of satellites which: Collaborateto provide a serviceCan be modeled withabsolute dynamics(high relative

distances)

Objectives Globalor nearly globalcoverageof the Earth surface Improve system performanceand data collection capacityProvidenew services(e.g. global positioning and navigation,

worldwide telecommunications, new applications in the Earthobservation and scientific mission domain)

Take advantage ofscale economyfor the satellitemanufacture(reduction of cost and production times)


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Maintenance

Formations


Applications

The Major Application of Satellites Constellations are related to:

Navigation: Provide global positioning on the earth surface.GPS, GLONASS, Galileo

Communications: Provide telephone, internet andtelecomunication service for space and terrestrial application.Globastar, TDRSS, Orbcomm

Earth Observation: Provide Earth Imaging and monitoringservice.

GOES, DMC


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Maintenance

Formations


ApplicationsNavigation

GPS: Global Positioning System

US department of

Defence24 satellites

6 orbital plane

circular orbitsAltitude: 20200 km

(MEO)

Inclination: 55.5

Satellite Mass: 1665kg


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Maintenance

Formations



GLONASS

Russian Positioning System

21 satellites3 orbital plane

circular orbits

Altitude: 19000 km(MEO)Inclination: 64.8

Satellite Mass: 1300 kg


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage


Performance analysis

Deployment

Maintenance

Formations



Galileo: The European alternative

European (Civil)

30 satellites

3 orbital plane

circular orbits

Altitude= 23222 km(MEO)Inclination= 56

Satellite Mass= 700 kg

1 spare per orbit


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations


ApplicationsCommunication

Globalstar

Global Communication in Real Time, High Bandwidth

48 satellites on circular orbits in 8 orbital planesAltitude: 1410 km(LEO)Inclination: 52

Latitude band: [70, +70]


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations



Orbcomm

Global Communication, Messaging, Low Bandwidth

24+2 satellites on 3 orbital plane + 1 polar

Altitude: 780 km(LEO)

Inclination45


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations



TDRSS

Data Relay in real Time, High Bandwidth

7 satellites

Geostationary Orbit


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations


ApplicationsEarth Observation

DMC: Disaster Monitoring Constellation

Disaster Monitoring

AlSAT-1, BILSAT-1, NigeriaSat-1,UK-DMC, Beijing-1

1 orbital plane, Sun-synchronous

Altitude= 685 km(LEO)

Inclination= 98


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations


ApplicationsEarth Observation

GOES: Geostationary Operational Environmental Satellites

Earth Observation, US Weather, Ocean monitoring

2 satellites E-W

Geostationary Orbit


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

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The Constellation Design is divided in five steps:

MissionrequirementsAnalysis Orbit Design:

Identify the number of satellites and their orbital planes Performance Analysis:

Evaluate performance index of the designed configuration

Deploymentstrategy:Identify launchers, and deployment strategy

Maintenance:Identify recovery strategy and end of life procedures

The first step is related to themission objectiveand can usuallybe translated in requirements oncoverage or revisit time.


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



Deployment

Maintenance

Formations


Orbit Design

The classical way of defining a constellation is through the followingparameters:

Numberandspacingof satellites per orbit plane Numberandrelative orientationof orbit planesOrbitsemi-major axisandeccentricityOrbitinclination Argument of perigeein case of elliptic orbits

These parameters are usually identified developing algorithmswhich bind them to mission requirements.

Usually constellation mission requirements are related to:

Earth Observation and MonitoringGlobal or Regional Telecommunications

Global Navigation and Localization


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Constellations

IntroductionApplications


Orbit Design

Coverage



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Orbit DesignObjectives

The design algorithms should address:

Continuous coverage ofEarth surfaceContinuous coverage over aLatitude Band(Around equator)Continuous coverage over high Latitude Band (Poles)Continuous coverage overboxed regions(Latitude and

Longitude)Minimumrevisit timeover a certain location


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Coverage

The Earth Coverage is the part of theEarth surfacewhich asatelliteinstrument or antennacan see at an instant of time orover a certain period.

C


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CoverageDefinitions

The Earth Coverage is the part of theEarth surfacewhich asatelliteinstrument or antennacan see at an instant of time orover a certain period.

Coverage analysis is based on the following definitions: Footprint Area(FA) or Field of View (FOV):

The area that an instrument or antenna can see at any moment Instantaneous Access Area(IAA):

The area that an instrument or antenna can potentially see ifscanned over its operational range

Area Coverage Rate(ACR):The rate at which theinstrument or antennais sensing new

land Area Access Rate(AAR):

The rate at whichnew landcome into the Access Area of thesatellite

C


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CoverageInstrument Type

Definitions:

Multi instrument Satellite

Radar, optical, general sensing:

SAR:

Rotating, METEOSAT:

Co erage


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CoverageFootprint Area

TheLengthof the Footprint is:

LF =KL(FO FI) D sin / sin

whereFOand FIare expressed in degrees and:

KL = 1 for length in degrees

KL = 111.319 for length in km

Coverage


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CoverageFootprint Area

TheWidthof the Footprint is:

WF =REsin1 (D sin /RE) D sin

whereREis the Earth radius.Assuming that the Footprint projection is an ellipse theAreais:

FA= (/4) LFWF

Coverage


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CoverageArea Coverage Rate

TheInstantaneous Area Coverage Rateis:

ACRinstantaneous= FA

TwhereTis the exposure time for the instrument.

TheAverage Area Coverage Rateis:

ACRaverage = FA(1 Oavg) DCT

whereDCis the duty cycle of the instruments and Oavgis theaverage overlap between successive footprints.

Coverage


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Orbit Design

Coverage



Street of coverage



Deployment

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CoverageInstrument Geometry

Coverage


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Orbit Design

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CoverageInstantaneous Access Area

The instantaneous Access Areadepends on the instrumentsgeometry:

A B C D

KA(1 cos ) KA(cos 2 cos 1) 2KL(1 2) KL(1+ 2)

WhereKLhas been defined above and:

KA= 2 for area in steradians

KA= 20626.48 for area in deg2

KA= 2.556 108 for area in km2

Coverage


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CoverageAccess Area Rate

TheAccess Area Rateis computed depending on the instrumentgeometry:

(A) AAR= (2KAsin )P

(B) AAR= (2KAsin 1)

P

(C) AAR= 2KA(sin 1 sin 2)P

(D) AAR= 2KA(sin 1+ sin 2)

P

wherePis the orbital period.

Coverage


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Orbit Design

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CoverageFigures of Merit

Earth Coverage for a single satellite or for a constellation can beevaluated on the basis of different Figures of Merit:

Percent Coverage

It is the the time a point is covered over the total simulation time Maximum Coverage Gap

It is the maximum time in which a point is not covered Mean Coverage Gap

It is the average gap over a certain amount of time Mean Response Time

It is the mean time between a request of observation and theactual coverage of a point

Cl i l D i A h


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Constellations



Orbit Design

Coverage



Street of coverage



Deployment

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Formations



The design of a constellation is guided by the aim of identifying theorbital elementswhich characterize thesmallestpossibleconstellationthat fulfill mission requirements.

In the classical design the selection is done through anoptimizationconsidering thecoverage as performance index.

This choice is suitable for:Global NavigationCommunication

In particulartwo possible configurationshave been identifiedthat can be described by a subset of parameters describing the

overall constellation:Symmetrically inclined constellation (Walker)Street of Coverage


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A Walker constellation is:composed by satellitesevenly spaced on circular orbitsorbit planes areevenly spaced in RAANand at the same

altitudeis uniquely identified byfour parameters: i, the constellation inclination T, the total number of satellites P, the total number of orbit planes F, the relative spacing between satellites in adjacent planes

The notation T/P/F iscommonly used to describe aWalker constellation

Walker constellation aresuitable forglobal coverageoflatitude bandsaround the

equator



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Walker ConstellationsDesign

Globalstar constellation isa walker (48/8/1)

6 S/C per plane8 orbit plane

= 7.5

S=T /PNumber of satellites perplane

RAAN= 360/P degInterval

between ascending nodesOn each orbital plane satellites are

distributed at interval of360/S deg

=F(360/T)is the relative phase

between satellites in adjacent planes

Plane 1: 0, 60, 120, 180, 240, 300

Plane 2: 7.5, 67.5, 127.5 . . .

St t f


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Constellations

Introduction

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Orbit Design

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Street of coverage



Deployment

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Street of coverage

A method to design large optimal constellation forcontinuousglobal coverage

Multiple satellites are placed on acircular orbitat the same

altitude and on asingle planeto create astreet of coverage

Theobjectiveis to determineanalytically thenumber of streets of

coveragethat are required toguarantee theglobal coverageofthe desired zone

Streets of coverage can be used to

guarantee globalcoverage of polarcaps

Street of coverage


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Constellations

Introduction

Applications


Orbit Design

Coverage



Street of coverage



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Street of coveragePolar Networks

Polar networkare designed usingStreets of Coverage

In each orbital plane there aren2satellites evenly spaced every360/n2 deg

Then1polar orbital plane are

evenly distributed within 180 degwith a constant spacing of180/n1 deg

Synchronization between satellitesof adjacent planes isaccomplished with appropriatephasing

Street of coverage


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Constellations

Introduction

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Orbit Design

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Street of coverage



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gCoverage Gaps

Street of Coverageadjacent planesaresynchronizedto avoidcoverage gaps betweenco-rotational planes.

Counter-rotatingplanepresents gapsin a symmetrical

configuration.



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Constellations

Introduction

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Orbit Design

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Street of coverage



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Increasingly, satellite constellations are being designed withregional coverageand withnon-classical design criteria(failure, revisit time, cost).

The two classical constellation design algorithms have certainlimitations:

Their inability to take anything but classical,geometric designfactorsinto consideration

Their inability to consider alatitude-longitude boxof optimalcoverage

The methods used for advanced constellation design are:

Genetic algorithms, as may be inferred from the name, wereoriginally developed for biological uses using the concepts ofgenetics to arrive at the best solution

Any Stochastic Search algorithm used for global optimization



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Constellations

Introduction

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Orbit Design

Coverage



Street of coverage



Deployment

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The most commonPerformance Indexesfor Constellations are:

Max/Min/Meancoverage percentageat any grid point andglobally over a zone

Max/Min/Meanrevisit timeat any grid point and globally over azone

Max/Min/Meanresponse timeat any grid point and globallyover a zone

Zone andground stations visibilityfor different instrumentsIllumination conditions (eclipse analysis) andsun geometry

Theinstrument and payload featuresand modes of operationare the driving factors when analyzing the performance of anEarthObservationsatellite system.



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Introduction

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Orbit Design

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ForTelecommunicationConstellations, the following aspects areimportant:

Single andmultiple satelliteEarthcoverageversus time Elevationangle (on grid points or on iso-contours)Satellite and userantenna analysis(field-of-view, pointing

angles and swath angles)

For Navigation and Positioning Constellations, the following criteriaare generally considered:

Number of satellitesin visibility

Satellite elevations Dilution of Precision(DOP)Availability, positioning accuracy and integrity for navigation

systems



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Constellations

Introduction

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Orbit Design

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Street of coverage



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Ground Stations Coverage

Ground Station Coverageperformance can be evaluatedconsidering:

Number oforbits without contact(blind orbits)Max number of orbits without contactMaxtime withoutground stationcontact



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Introduction

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Dilution of Precision

TheDilution of Precision(DOP) measures the relativedegradation orreduction in the certaintyof anavigationsolutionbased on one-way range measurements.

A navigation solution consists of the position of the receiver and theoffset between the receiver and the satellite clock

Theuncertaintyin the navigation solution is function of uncertaintyon therangemeasurementson thenumber of measurementsavailableon therelative geometryof the set of transmitters



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Introduction

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It is possible to define the matrix

A= HTH

1

whereHis the matrix of the cosine directionscijto the satellitei:

H=

c11 c12 c13 1

c21 c22 c23 1

c31 c32 c33 1

c41 c42 c43 1

TheGeometric DOP(GDOP) measures the DOP of theentire

solution. GDOP combines position and clock-related components

GDOP =

A11+ A22+ A33+ A44



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ThePosition DOP(PDOP) measures only the DOP associatedwith thepositional portionof the navigation solution

PDOP = A11

+ A22

+ A33

TheHorizontal Dilution of Precision(HDOP) measureslatitude/longitudeDOP

HDOP =

A11+ A22TheVertical Dilution of precision(VDOP) measuresaltitude

DOP

V DOP = A33The Time Dilution of Precision(TDOP) measures the DOP of the

time portionof the navigation solution

TDOP = A44

Deployment


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Introduction

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Deployment

TheDeploymentof a constellation is not a trivial operation Multiple launchare needed.Deployment takes time aslaunchersarenot always available

Shared launch requirespositioning maneuvers Partial operationneed to begin before the end of deployment

Deployment


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Introduction

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Orbit Design

Coverage



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Deployment

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Deployment Strategy

Deployment Problem Formulation: reduce deployment timelimiting the number of launchesachieve asubstantial level of servicewhile deploying the

constellationPrimary Issues

Thesequence of launchesused to place the s/c into theconstellation, so as to achieve an acceptable service with a

minimum number of s/cThenumber of s/cper launch

Correlation between the launch and the deployment phasesThe number of s/c per launch depends on the launcher(s) Ground segmentsneed to be operative forintermediate

constellation

Deployment


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Constellations

Introduction

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Orbit Design

Coverage



Street of coverage



DeploymentMaintenance

Formations


Launchers Selection

Injection orbitcapabilities and performance(mass, dispersions)

Fairing capacityand dimensions, Single/multiple spacecraftcapabilities

Launch delay for adedicatedor for asharedlaunch (for a "spare"satellite)

Launcherreliability,availability

Launchcost

Deployment


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Introduction

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Orbit Design

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Set-Up

There are three possible choices for the deployment of aconstellation Direct Injectioninto final orbit

Propulsive maneuversto reach final orbit Indirect Injectioninto final orbit (J2)

In the case ofDirect injectioninto the final orbit

The launcher may require anupper stage, adding to the launchcost

Quick deployment, so as to start early system operationsFeasibility depends on final orbit altitude and inclination

Deployment


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Set-Up

In the case the s/c performs Propulsive Maneuversto reach thefinal orbit from an initial injection orbit additional propellantand structure are needed

deploymenttimeis maintainedlowIn the case ofIndirect Injectioninto final orbit it is possible:

To populate several operational orbital planes using thedifferential effect of the Earth oblatenesson the node

To launch several small s/c using a launcher with considerableinjection capability

No quickdeployment, due to the requireddrift time

= 32

nR2eJ2cos i

a2(1 e2)2

= T

DeploymentS t U


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Set-Up


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Maintenance


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It is necessary tomaintainthe constellation in a configuration thatensures thenominal level of serviceduring its whole plannedlifetime

Orbital perturbationscan compromise the performance of theconstellation varying that configuration

To avoid this it is necessary to identify:Criteria andalgorithmsthat handle thelong-term deviationof

the constellation from its objectiveDetermination of theoptimal maneuver times, so as to take

into account constraints as:Minimizing thetotal number of maneuversMinimizing theglobal cost(V) or distributing it equally

among the satellitesNecessaryfuel budgetfor orbit maintenance over the mission

lifetime

MaintenanceOrbit Pert rbation


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Introduction

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Orbit Design

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Orbit Perturbation

Relevant Perturbations are: LEOconstellations: atmospheric

drag effect and Earth gravitational

perturbations MEOconstellations: Earthgravitational perturbations, Sunradiation pressure and luni-solarperturbations (particularly from

10000 km onwards) HEEOconstellations: Earth

gravitational perturbations,luni-solar perturbations and Sun

radiation pressure

MaintenanceOrbit Control Strategy


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Orbit Control Strategy

It is necessary to analyze thelong-term evolution: Minimum altitudeof the constellation during its lifetime Orbit shapedeviation

Orbitalspatial configurationdeviation (i.e. orbital plane nodalseparation, inclination and argument of perigee)

Absolute orbit control: to maintain each satellite in a well-definedcontrol boxwith respect to its nominal reference trajectory

Applied to each satellite of the constellation independentlyRequire a lot of fuel

Relative orbit control: toguarantee the global geometryof theconstellation without trying to maintain each satellite on a referenceorbit in an absolute mannerCannot be employed if ground coverage areas are essential for

the mission


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Constellations

Formations Introduction

Applications

Relative Dynamics

Clohessy Wiltshire Equations

Eccentric Reference Orbit

General Reference Trajectory

Mission Design

Deployment

Maintenance

Rendezvous


Formations

Introduction


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Relative Dynamics




Mission Design

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A Formation is a group of satellites which: Collaborateto provide a serviceNeed to be modeled withrelative dynamics(low relative

distances)

ObjectivesSynthesize alarger aperturethan a single satellites

Providedistributed instrument(different observation points)Achieve the mission objectives morecost effectivelyTaking advantage of low-cost satellite approachesand/or

cheaper launchers

Applications


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The MajorApplicationof Formations of Satellites are related to:

Earth Observation: Provide Earth Monitoring Imaging service.

EO, interferometric SAR

Scientific: Gravity field recovery service, telescope.Grace, Lisa, Darwin, TPF

On Orbit Servicing:Autonomous On Orbit Assembly and Servicing


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ApplicationsInterferometric SAR


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Interferometric SAR

ERS-1/ERS-2

European Remote-Sensingsatellites

ERS-2 orbit on an orbit withground track near that ofERS-1

LEO sun-synchronous orbitNot a typical formation (no

real time cooperation)

ApplicationsGravity Field Recovery


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y y

Grace

Gravity field recovery

in track formation of 2satellites

Polar orbit at 300-500 km

separation along track

220 50kmAccurate measurements of

the inter-s/c range changebetween the two satellites

ApplicationsOptical Interferometry


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p y

Darwin, TPF

Optical and Infrared

interferometryL2 orbit

1.5 million km from Earth

High precision control of

relative distance andorientation required

ApplicationsGravity Waves Observation


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LISA

Laser Interferometer SpaceAntenna

Try to detect GravitationalWaves

circular orbit around sun

3 satellites in triangularformation

5 million km relative distance

Very high accuracy of relative

distance control

Relative Dynamics


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Satellites in formations fly with smallrelative distances

The dynamic is expressed with respect to anorbiting referenceframe

Theoriginof the reference frame is usually identifiable with anorbiting point mass

Theframeis defined by axis in theradial ix,normal izto the

orbital plane andbinormal iydirections

ix = ir = r0

|r0|

iy = i = ih ir

iz = ih= h

|h|

wherer0is the vector representing the position of the origin of thereference frame andhis the orbital momentum vector

Relative Dynamics


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Theposition of the spacecraftorbiting near the reference pointcan be expressed in theabsolute reference frameas:

rs= r0+ r= (r0+ x) ir+ yi+ zih

Representing with ihthe angular velocity of the relative reference

frame, it is possible to express theaccelerationof the satellite withrespect to theabsolute frame:

rs = r0+ x 2y y 2 (r0+ x) ir+

+

y+ 2 (r0+ x) + (r0+ x) 2y

i+ zih

Relative Dynamics


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Imposing that theangular momentumvector isconstantthe

following expression for can be found:

h=r

2

0+ 2r0r0

= 0 =

= 2

r0

r0

while considering theacceleration of the originof the orbitalreference frame this other relation can be found:

r0 =

r0 r02

ir = r20

= r0=r02

1 r0p

wherep=a(1 e2)is the semilatus rectum of the orbit.

Relative Dynamics


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Relative Dynamics




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Substituting the previous relations in the expression of theacceleration andequatingitwith gravity accelerationthefollowing equation are obtained:

x 2

y yr0

r0

x2

r20=

r3s(r0+ x)

y+ 2x xr0r0 y

2 =

r3sy

z =

r3sz

Describe the relative motion with respect to anunperturbed

reference orbit.The reference orbit and the satellites are subject to an ideal point

mass gravitational field.

Relative DynamicsSmall Relative Coordinates


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Therelative coordinates are smallwith respect to the relativereference frame origin position:

rs r0

1 + 2

x

r0

With that approximation it is possible to obtain the followingequations of motion:

x x

2 + 2 r30

2

y y r

0

r0

= 0

y+ 2x xr0r0 y2

r30 = 0

z+

r30z = 0

This equations are valid for any kind of orbit, but with the restriction

tosmall relative motion.



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One of the most important case is that ofcircular reference orbit

Imposing that is constant and that the eccentricity is null (p=r0)theClohessy Wiltshireequations are obtained:

x 2ny 3n2x = 0

y+ 2nx = 0

z+ n2z = 0

wheren ihis the constant angular velocity of the relative referenceorbit.

These equations are linear but approximate and are valid only if:Gravity field derived bypoint mass(no perturbations or third

body) Small relative distancewith respect to reference origin position Quasi circularreference orbit (e 0)

Clohessy Wiltshire EquationsAnalytical Solutions


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The Clohessy Wiltshire equations arelinearand have aanalyticalsolutions:

x (t) = x0n

sin nt 2y0n + 3x0 cos nt + 2y0n + 4x0y (t) = 2x0

n cos nt +

4y0n

+ 6x0

sin nt +

y0

2x0n

(3y0+ 6nx0) t

z (t) =z0cos nt + z0n sin nt

The motion in thenormal direction(zcoordiante) is independentfrom the other directions and is pureharmonic oscillation

The motion in theorbital planeiscoupled harmonic oscillationwith the presence of asecular termwhich produce a drift.

Clohessy Wiltshire EquationsAnalytical Solutions


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50 0 50 100 150 2006

4

2

0

2

Tangential

Radial

0 1 2 3 4 5 6 7

x 104

1.5

1

0.5

0

0.5

1

1.5

Time

Binormal

Clohessy Wiltshire EquationsBounded Periodic Motion


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Relative Dynamics




Mission Design

Deployment

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Rendezvous


Formation flying require that thesatellites remain in theneighborhood of the originof the reference frame (which can beidentified with the master satellite)

From an absolute point of view it is necessary to impose theequality of the energy levels(semi-major axis) which lead toconstraining the orbital periods to be equal.

as =a0

From the relative point of view the in the case of the ClohessyWiltshire Equations it is necessary tocancel the secular termofthe solution:

y0+ 2nx0 = 0

which constitutes aconstraintin the definition of theinitialrelative velocityon the tangential direction



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The solution of the Clohessy-Wiltshire equations become:

x (t) = x0n

sin nt + x0cos nt

y (t) =

2x0

n cos nt 2x0sin nt +

y0

2x0

n

z (t) =z0cos nt +z0n

sin nt

which is abounded periodic motion.

15 10 5 0 5 10 15 20 25 30 3520

15

10

5

0

5

10

15

20

Tangential

Outofplane

Flight direction



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It is possible to obtain periodic motioncentered in the originof therelative reference frame nullifying the term2x0 ny0 = 0obtaining:

x (t) = x0n

sin nt + x0cos nt

y (t) = 2x0n

cos nt 2x0sin nt

z (t) =z0cos nt +z0n

sin nt

It is possible to obtain acircular projectionimposingz0 = 2x0.

10 5 0 5 10

10

5

0

5

10

Tangential

Out

ofplane

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A second interesting case is consideringeccentricity in thereference orbit

The relative dynamic equations in the case of small relative

coordinates can be expressed in the form of alinear time varying(LTV) system:

r= Ar + Br

where:

r=

x

y

z

A=

0 2 0

2 0 0

0 0 0

B=

2 + 2 r3

0

2 r0r0 0

2 r0r0

2

r30

0

0 0 r30



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Considering the expression ofr0, r0and for keplerian orbit:

r0 = a

1 e2

1 + e cos =

n (1 + e cos )2

(1 e2)3

2

r0 = nae

(1 e2)1

2

wheren=

a3

is the natural frequency of the reference orbit.

It is possible to transform all the time derivative inderivative withrespect to the true anomaly:

() = ()

() = () 2 + ()



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Applying the relations above it is possible to write the followingLTVsystemvalid in the case of eccentric reference orbit:

r= r + r

where:

r=

x

y

z

=

2e sin 1+e cos 2 0

2 2e sin 1+e cos 0

0 0 2e sin 1+e cos

=

3+e cos 1+e cos

2e sin 1+e cos 0

2e sin 1+e cos

e cos 1+e cos 0

0 0 11+e cos

whereeis the eccentricity.

Eccentric Reference OrbitAnalytical Solutions


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Also these equations haveanalytical solutionswhich areexpressed in terms ofeccentric anomaly:

x () = sin

d1e + 2d2e2

H() cos

d2e(1 + e cos )2 + d3

y () = sin

d3e

(1 + e cos )+ d3

+ cos

d1e + 2d2e

2H()

+

+

d1+ d4

(1 + e cos )+ 2d2eH()

z () = sin d5

(1 + e cos ) + cos d6

(1 + e cos )whered1,d2,d3,d4,d5and d6are integration constants

H(),sin andcos can be expressed in terms of the eccentricanomaly.

Eccentric Reference OrbitAnalytical Solutions


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The functionH()can be expressed as:

H() = (1 e2)5

2

3eE

2 1 + e

2

sin E+e

2sin Ecos E+ dH

whereEis the eccentric anomaly anddHcan be computed fromH(0) = 0

The integration constantd1,d2,d3,d4,d5and d6can be related to

initial conditionsby:

d1

d2

d3d4

=

1e

0 0 0

0 (2+e)(1+e)2

e2(1+e)3

e2 0

0 2(1+e)

e 2(1+e)

e 02(1+e)

2

e 0 0 (1 + e)

x (0)

x (0)

y

(0)y (0)

Eccentric Reference OrbitBounded Periodic Motion


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It is possible to obtainbounded relative motionalso in the case ofeccentric reference orbit

it is necessary to impose continuity constraints after one period(only

xand

y)

x (0) =x (2)

x (0) =x (2)

y (0) =y (2)

y (0) =y (2)

These conditions bindy

(0)tox (0), which grant that the motion isbounded but not symmetrical with respect to the radial direction

In order to obtain asymmetrical projectionx (0)should becanceled

Eccentric Reference OrbitBounded Periodic Motion


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Mission Design

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20

2

2

0

26

4

2

0

2

RadialTangential

O

utofPlane

2 1 0 1 22

1

0

1

2

Radial

T

angential

2 0 26

4

2

0

2

Tangential

OutofPlane

2 1 0 1 26

4

2

0

2

Radial

OutofPlane

25 20 15 10 5 0 5 10 15 20 2520

15

10

5

0

5

10

15

Tangential

OutofPlane

Flight path

30 20 10 0

10 20 30

20

0

20

100

80

60

40

20

0

20

RadialTangential

OutofPlane



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It is possible to express equations for the relative dynamics in thecase of ageneric reference trajectory

The reference system is defined consideringTheradialdirectionThe direction of theinstant angular momentum, which is no

longer constantthe directionbinormalto those two

ix = ir = r

0|r0|

(1)

iy = i = ih ir (2)

iz =

ih=

(r0 v0)

|r0 v0| (3)

where r0and v0are the vectors representing the position andvelocity of the originof the reference frame.



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It is possible to express the relative velocity and acceleration with:

rs = r0+ rr

vs = v0+ vr+ r0

as = a0+ ar+ r0+ ( r0) + 2 vr

it is possible to express the relative acceleration in terms of theabsolute acceleration of the satellite and of the cinematic states of

the reference system r0, v0, a0,and With the reference system defined above it is possible to express

and as function of the states of the origin of the reference systemr0, v0and a0:

= r0 v0

|r0|2 =

r0 a0

|r0|2 2

(r0 v0) (r0v0)

|r0|4

http://showbookmarks/http://showbookmarks/


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Relative Dynamics




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Both the acceleration of the origin of the reference system and theabsolute acceleration of the satellites can be expressed asproduced by ageneric force field

Expressing the relative acceleration of the satellites as a function ofthe absolute acceleration of the satellites and of the states of theorigin of the reference system, it is possible to write a set ofdifferential equations whichadjoinsthe differential equations representing themotion of the originthe differential equations representing therelative motion

r0= U (r0)

rr = U (rs) a0 r0 ( r0) 2 vr

where U (r0)is ageneral gravity potential


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Mission DesignIn-Track


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Mission Design

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TheIn-Trackformation is the easier to implement and control

It can be used to design: Earth observationmissions (ground-registerd formation) Atmosphericmonitoring missions (co-phased instrument)

does not allow measures perpendicular to the velocity vector


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Mission DesignCartwheel


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TheCartwheelformation take advantage of the periodic motionin the CW equation

It is realized putting a group of satellites in aperiodic motionbehind a mastersatellite

It has been proposed forInSARmeasures (TerraSAR-L):Allows acquisition of one in track and two across track

componentsRequire active control of the cartwheel formation

Deployment


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Thedeploymentof a Formation it is not a major issue in themission design

However some point should be taken into consideration: Launcher selectionon the basis of dedicated shared launchMultiple launches solution or Single launch Deployment maneuversmust be considered in computing

propellant mass


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MaintenanceDifferential Orbital Perturbation


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Mission Design

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The problem related toDifferential Orbital perturbationcan befaced in two way: Controllingrelative and absolute positions Designing relative motion taking into account perturbation

reducing the amount of propellant needed for control

J2 differential perturbationcan be taken into account

The J2 invariance condition are

= 0

M = 0

Imposing these condition it is possible to identify appropriate initial

condition for bounded periodic motion

Rendezvous


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A GeneralRendezvous maneuveris divided into three principalphases:Orbitphasing Rendezvousphase Dockingphase

Theorbit phasingends at 10 km from targetis analyzed inabsolute coordinatesystem

Consists in a transfer on anelliptic phasing orbitand back onthe target orbitThe target and satellite are considered aspoint masses

RendezvousBOB maneuver


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TheRendezvousphase start at 10 km and ends at 500mIt is analyzed usingrelative dynamics(CW)The satellite and target are considered aspoint masses attitude is ignored

This phase is usually constituted by aBOB maneuverin which twoVare applied, to begin and finish the maneuver

in order to compute theVof the BOB maneuver the CWequations are used

Final position at the origin ofthe reference frame

Final Velocity equal to zero

Fixed Transfer time

RendezvousDocking

D ki


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TheDockingphase start at 500 m and ends at 0mIt is analyzed usingrelative dynamics(CW)The satellite and target are considered asbody with

dimensions attitude is considered

This phase is usually acontrolledphase

A control term is added to the CW Equations

x 2ny 3n2x = fx

y+ 2nx = fy

z+ n2z = fz

The problem can be solveddesigning a controllerto reduce theerror

Optimal trajectorycan be found with numerical methods

References


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T. Ely. Satellite constellation design for zonal coverage using genetic algorithms. Spaceflight mechanics 1998, pages 443460, 1998

JJ Spilker. Satellite constellation and geometric dilution of precision.Global Positioning System: Theory and applications., 1:177208,

1996

J.E. Draim. Three- and four-satellite continuous-coverage constellations.Journal of Guidance, Control, and Dynamics, 8(6):725730,

1985

C. Brochet, J.M. Enjalbert, and J.M. Garcia. A multiobjective optimization approach for the design of Walker constellation. IAF,

International Astronautical Congress, 50 th, Amsterdam, Netherlands, 1999

JG Walker. Satellite constellations.British Interplanetary Society, Journal (Space Technology)(ISSN 0007-084X),, 37:559572, 1984

D. Luders and LJ Ginsberg. Continuous zonal coverage-A generalized analysis. AIAA Mechanics and control of flight conference, 9,

1974

H. Schaub, S.R. Vadali, and K.T. Alfriend. Spacecraft formation flying control using mean orbit elements. Journal of the Astronautical

Sciences, 48(1):6987, 2000

H. Schaub and K.T. Alfriend. J2 invariant reference orbits for spacecraft formations. Celestial Mechanics and Dynamical Astronomy,

79:7795, 2001

K.T. Alfriend, H. Schaub, and D.W. Gim. Gravitational perturbations, nonlinearity and circular orbit assumption effect on formation flying

control strategies. InAAS Guidance and Control Conference, Breckenridge, CO, February 2000. Paper No. AAS 00-012

W.S. Koon, J.E. Marsden, and R.M. Murray. J2 dynamics and formation flight. InAIAA Guidance, Navigation and Control Conference,

Montreal, Canada, August 2001. paper 2001-4090

S.R. Vadali, K.T. Alfriend, J.L. Junkins, and H. Schaub. Fuel optimal control for formation flying. InAIAA Guidance, Navigation andControl Conference, Portland, OR, August 1999

M.W. Lo, W.S. Koon, J.E. Marsden, and R.M. Murray. Formation flight near libration points: Survey and recommendations. Journal of

Space Mission Architecture, pages 3548, 2000

References 2


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G. Inalhan, M. Tilerson, and J.P. How. Relative dynamics and control of spacecraft formations in eccentric orbits. Journal of Guidance,

Control and Dynamics, 25(1), January-February 2002

P. Gurfil and J.N. Kasdin. Nonlinear modeling and control of spacecraft relative motion in the configuration space. In 13th AAS/AIAA

Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 2003

J.P. How G. Inalhan, F.D. Busse. Precise formation flying control of multiple spacecraft using carrier-phase differential gps. In

AAS/AIAA Space Flight Mechanics, 2000

T. Carter. New form for the optimal rendezvous equation near a keplerian orbit. AIAA Journal of Guidance, Control, and Dynamics,

13:183186, January-February 1990

T. Carter and M. Humi. The clohessy-wiltshire equations can be modified to include quadratic drag. In 13th AAS/AIAA Space Flight

Mechanics Meeting, Ponce, Puerto Rico, February 2003

W.H. Clohessy and R.S. Wiltshire. Terminal guidance for satellite rendezvous. Journal of the Aerospace Sciences, 27(9):653658,

1960

J. Tschauner and P. Hempel. Elliptic orbit rendezvous. AIAA Journal, 5(6):11101113, 1967

W.E. Wiesel. Relative satellite motion about an oblate planet. Journal of Guidance ,Control, and dynamics, 25:776785, 2002

S.A. Schweighart and R. Sedwick. A high fidelity linearized model for satellite formation flying.Journal of Guidance Control and

Dynamics, 25(6):107380, November-Dicember 2002

D. Izzo, M. Sabatini, and C. Valente. A new linear model describing formation flying dynamics under j2 effect. InXVII AIDAA congress,

pages 493500, 2003

R.G. Melton. Time-explicit representation of relative motion between elliptical orbits. Journal of Guidance, Control, and dynamics,

23(4):604610, 2000

S.R. Vadali. An analytical solution for relative motion of satellites. In Cranfield University Press, editor, 5th International Conference

on Dynamics and Control of Structures and Systems in Space, pages 309316, 2002

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