DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYFORMATION FLIGHT
OF SATELLITESAROUND THE EARTH(CELESTIAL MECHANICS SEMINAR)

PhD. Candidate: Jorge Alberto Soliz Torrico

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGY

Concepts

CONTENTS

Advantages
Formation flight - missions
Recent Advances in Formation
FlightNonlinear Methodologies applied
to the formation flight
References
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYCONCEPTS
Formation Versus Constellation*
As defined by the NASA Goddard Space Flight
Centre, a constellation is composed of two or
more spacecraft in similar orbits with no active
control by either to maintain their relative
position. Is only necessary that spacecrafts
maintain themselves within their own prespecified boxes without collisioning or changing
the overall coverage of the Earth significantly.
So, the consideration of relative positions ororientations, real-time coordination and a very
high level of autonomy is not required.
In contrast, a formation is two or more
spacecraft that use an active control scheme tomaintain the relative positions between
spacecrafts. It has to be a direct control on therelative position and orientation between one
spacecraft and another one (typically its
neighbour) or many other. This means that
active, real-time and closed-loop control isrequired.
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYADVANTAGES**
Many potential applications of this enabling technology exist, one of wich is
to improve the performance of Earth observation. A cluster of satellites will
be able to synthesize a much larger aperture than can be achieved with asingleplatform,thus providing
increases
in imagin resolution
The cluster
approach
has many signicant
advantages
over a single
throughinterferometry.
large satellite:- Each spacecraft is smaller, lighter and simpler tomanufacture;
for these reasons a cluster of many satellites is lessexpensiveand less complex than a single large satellite.
- The failure of a component of a single satellite is notcritical ina cluster because failed satellites can be incrementallyreplaced
and the cluster is easily adapted to the failure.
- Lower consume of fuel used in transportation.
- Higher angular and spatial resolution imagery andinterferometry
as well as the sensitivity of scientic instruments.
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYFORMATION FLIGHT MISSIONS
Nowadays the list of missions that implement the formation flight
configuration is quite large, taking into account that the concept offormation flight is relatively new. The following list includes some of thesemissions. Some of them are briefly described later:
1. EO-1 (Earth Orbiter-1)
2. GRACE (Gravity Recovery and Climate Experiment)
3. TechSat21 (Technology Satellite of the 21st Century)
4. XEUS (X-Ray Evolving Universe Spectometrer)
5. MAXIM (Micro-Arcsecond X-ray Imaging Mission)
6. LISA (Laser Interferometer Space Antenna)
7. PRISMA
8. PROBA-3
9. Constellation-X
10. TPF (Terrestrial Planet Finder)
11. Planet Imager
12. DS3 (Deep Space 3)
13. DARWIN
14. SIMBOL X
15. FIMOS (Fire Monitoring Satellite Swarm)
16. ION-F (Ionospheric Observation Nanosatellite Formation)
17. SMARTSAT-1
18. MAGNAS (Magnetic NanoProbe Swarm)
19. Gemini (GPS-based Orbit Estimation and Laser Metrology forIntersatellite Navigation)
20. AMSAT
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGY

Darwin and TPF

FORMATION FLIGHT –MISSIONS I

Darwin (ESA) and Terrestrial Planet Finder (NASA) are
two formation
projects which were planned to find planets with theability to support life.
The goal of the projects is to measure the size,
temperature, and placementof planets similar to Earth in the habitable zones of
distant solar systems.
For this purpose, the two projects are based on aformation of spacecraftwith a virtual
apertureof(TPF)
thousands of meters
Terrestrial
PlanetFinder
The Terrestrial Planet Finder concept sets a large
baseline of a hundred of meters, composed by fourtelescope spacecraft with a diameter of 3 or 4
meters. These four spacecraft are combined with acollector, in an equilateraltriangle with the two inner spacecraft of the baseline.
The five spacecraft form a rotating formation, and there
are two target orbits
for the formation: one is a Halo orbit about the L2 Sun Earth+Moon
system, and the other one is a heliocentric orbit similar
to the SIRTF orbit
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYFORMATION FLIGHT –MISSIONS II
ST5 (Space Technology 5)
Objective: To test and validate new technologies and aid
scientists inunderstanding the harsh environment of the Earth's
magnetosphere. Using
data collected from the ST5 constellation, scientists canbegin to understandand map the intensity and direction of the magnetic field, its
relation to space
LISA
weather events, and the affects on our planet. Is part ofNASA’s New MilleniumThe
Laser(NMP)
Interferometer Space Antenna (LISA) is a joint
Programproject of ESA and NASA to study the mergers of supermassive black holes, tests Einstein’s
Theory of General Relativity, probes the early Universe, andsearches forgravitational waves the primary objective. The LISA
formation is a large formation. It has three spacecraft in thevertices of an equilateral triangle, witha distance of five million kilometers between eachspacecraft
The plane of the triangle formed by the spacecraft isinclined at an angle of 60◦ to the plane of the ecliptic. This
position is chosen to minimize the gravitational
disturbances from the Earth-Moon system and to admit thecommunication
with Earth
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT

Sedwick, Kong and Miller [1] used the Clohessy-Wilshire equations as theirstarting model to find relative orbits about a reference spacecraft. A
circular reference orbit and a spherical Earth (without J2 effect) was
assumed in their study. They used these linear equations of motion toestablish a large family of relative orbits.
/ Schaub and Alfriend [2] built on the work of Brouwer [3] and found J2
invariant relative orbits. Working with mean orbit elements, the seculardrift of the longitude of the ascending node and the sum of the argumentof perigee and mean anomaly were set equal between two neighboring
orbits. By having both orbits drift at equal rates on the average, theywould not pull apart over time due to the J2 influence.
Dong-Woo Gim and Kyle T. Alfriend [4] used a precise analytic solution for
the relative motion of satellites. Based on the relationship between therelative states and the differential orbital elements, the state transition
matrix for the linearized relative motion that includes the effects due tothe reference orbit eccentricity and the gravitational perturbations isderived. This method is called the Geometric Method.

In last years, Sabatini M, D. Izzo and R. Bevilacqua [5] see the possibility ofobtaining a natural periodic relative motion of formation flying Earthsatellites (numerically and analytically). The numerical algorithm is basedon a genetic strategy. First, They tested their algorithm using a point mass
gravitational model. In this case the period matching between theconsidered orbits is a necessary and sufcient condition to obtain invariant
relative trajectories. Then, the J2 perturbed case is considered.
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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODS
Clohessy-Wiltshire (CW) Equations [8]PAG. 9

DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODS
Clohessy-Wiltshire (CW) Equations [1]PAG. 10

DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODSFormationfor ellipticalorbit- Peter
M. Bainum[6]
The aim of this flightwork consists
on maintaining
constant
the separation
distance
between the satellites. This is trivial in a circular orbit, but in an elliptical orbit it is
not as this because the inter-satellite distance is not constant, due to the differencesin velocity along it. If two satellites are in the same orbit and a nominal distance d isfixed between them, with a simple simulation can be demonstrated that the distancecan vary about a 25% depending on the chosen orbit.
For example, In this scheme, two satellites are orbiting around the Earth in the same
orbit. The perigee is at an altitude of 600 km and the apogee is at 8000 km. Bothsatellites are pretended to be separated 500 km during the entire orbit, but thisdistancedoes
notremain constant.
The motheranddaughtersatellites in two equal elliptical orbits but one displaced
respect to the other. The orbit of daughter satellite is rotated an angle.

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODS
Elliptical orbit in formation flight - Peter M. Bainum [6]It is possible to reduce more the separation distance between satellites
error by doing another rotation to the daughter’s orbit. In this case, thesemi-major axis of the daughter satellite orbit will be rotated an angle beta
, as is illustrated in the following figure

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODS

The possibility of obtaining a natural periodic relative motion of formation flying
Earth satellites is investigated
Magic
Inclination[5] The numerical algorithm is based on a genetic
both
numericallyand analytically.
strategy, refined by means of
nonlinear programming, that rewards periodic relative trajectories. First, we test ouralgorithm using a point mass
gravitational model. In this case the period matching between the considered orbits
is a necessary and sufficientcondition to obtain invariant relative trajectories.
Then, the J2 perturbed case is considered. For this case, the conditions to obtain an
invariant relative motion are known only in approximated closed forms whichguarantee a minimal orbit drift, not a motion periodicity. Using the proposed
numerical approach, we improved those results and found two couples of inclinations(63.4 and 116.6 deg, the critical inclinations, and 49 and 131 deg, two new “special”
inclinations) that seemed to be favored by the dynamic system for obtaining periodic
relative motion at small eccentricities. ****

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYRECENT ADVANCES IN FORMATION FLIGHT - METHODS
AMagicsurprising
behaviour is foundperforming the simulation for given semi-major axis
Inclination[5]and eccentricity, but varying the inclination. Instead of the expected two peaks offitness value (corresponding to the critical inclinations), we found other two
inclinations, about 50° and its supplementary 180°-50°, which give high values forthe fitness.
This means that for these particular inclinations, the effects of the un-periodicity
due to the J2 perturbation are weaker and easier to compensate it’s confirmed by
the plot in the Figure where it’s shown how these inclinations are not sensitive tovariations of semi-major axis.
This behaviour is probably due to the fact that for the 50° and 130° inclination case,the relative orbits are never really closed, even if the causes of the drifting apart of
the formation seem to be weaker. But for large formations these benefits slowly
disappear because of the major effects of differential J2 and non linearities. *****

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT

Analysis of dynamical systems techniques
appropriate to the near
Earth case and found a family of candidatereference orbits whose
nearby orbits support formation flight.
Using Routh reduction and Poincare's section
techniques, we canfound a fixed stable point for which passes thedesired orbit.

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT
ROUTH REDUCTION - THE REDUCED EQUATIONS I

We will use the Routh reduction technique to rewrite the equations of motion. This
procedure will enable us to study rst the reduced dynamics in the meridian plane of
the satellite before dealing with the dynamics in the longitudinal direction
ρ distance from the origin to a given point (satellite), θ colatitude, Ø longitude. The
potential energy including the J2 effect is givne by: (The potential function is themain problem in articial satellite theory).

In order to find the Routhian, the equation is
given by

The potential energy equation will be:

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT
ROUTH REDUCTION - THE REDUCED EQUATIONS II
So, Routhian function becomes:

The reduced equations are given by:

So, the equations are:

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT
POINCARÉ MAP
After performing Routh reduction, We use the method of Poincare section infinding the initial conditions for orbits that are dynamically favorable to formationflight.

By studying this Poincare section and looking for the stable fixed point, We canfind the pseudo-circularorbit (which corresponds to the fixed point in the middle of Figure) whose nearby
orbits can be used for formation flight. Clearly, this fixed point corresponds to aperiodic orbit in the reduced system.
Poincare Section of (r, dr) at z=0 , E=-0.45 Hz2=0.3
z = 0:3

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT

Work Done
Analysis of Stable Fixed point depending on energy and zcomponent of theangular momentum (Hz^2 )Initial conditions- Triangular Cluster near the Pseudo-Circular Orbit
Methods of monitoring the formation of satellites- Area- Relative distances- Time of formation- Triangular relationshipAnalysis of results depending on z-component of the angularmomemtum
Orbital elements- Relation among Orbital elements, Energy and Hz^2
Useful range for positioning satellite around stable fixed point
R=1.11133496883- Analysis of three formations- Analysis of three formations in yz-plane
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PhD. Candidate Jorge Alberto Soliz
Torrico

DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT

Analysis of Stable Fixed point depending on energy and z-component of theangular momentum (Hz^2 )Thanks to Routh reduction; We can calculate the stable fixed point knowing values of E
and H2z , then We will
make an analysis of the fixed point based on the energy and the H2z . This analysis isreferred to see how the
distance of fixed point changes in relation to energy and z-component of the angularmomentum.

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT
Triangular cluster -Initial conditions

By using the stable fixed point and the points nearby as well as making slightchanges in the longitudinal angle(and possibly in the time t), we can construct different kinds of cluster which willremain together after many
years (corresponding to thousands of revolutions around the Earth). i.e. if we fix E =-0.45, Hz^2= 0.3 (for example), the fixed point for the Poincare section at z = 0 willbe (rf ; 0) where r=1:11133496883 (about 710 km above the Earth). The following
initial conditions will give a triangular cluster (with each side close to 100 meters,
approximately).
The evolution of these three satellites in a triangular cluster were plotted in a frame
whose origin is at their instantaneous barycenter, with the yz-plane orthogonal to
the line of sight, the x-axis pointing towards the center of the Earth, and the y-axis
and the z-axis pointing towards the (instantaneous) west and north respectively.

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGY

Analysis of the formation of satellites
Area: The triangle's area formed by the cluster of satellites will be calculated forknowing how the area changes along time.

Relative distances:

Time of formation: It is going to calculate the time that the satellites remain
within a hypothetical sphere of certain radius. The hypothetical sphere has its
center in the barycentre of the triangle formed by the satellites

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT

Orbital elements

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT
Useful range for positioning satellite around stable fixed point
R=1.11133496883

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGYNONLINEAR METHODOLOGIES APPLIED TO THE FORMATION FLIGHT

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DOCTORAL PROGRAMME IN AEROSPACE SCIENCE ANDTECHNOLOGY[1] Sedwick, R.J., E.M.C. Kong and D.W. Miller. "Exploring orbital dynamics andmicropropulsion for aperture synthesis using distributed satellites systems:
applications to techsat21". AIAA 52-89; 1998.

REFERENCES[2] Schaub, H. and K.T. Alfriend. "J2 invariant reference orbits for spacecraft
formations". Flight Mechanics
Symposium, Goddard Space Flight Center, May 18-20; 1999.
[3] Brouwer, D. "Solution of the problem of articial satellite theory without drag". The
Astronautical Journal
64, No. 1274, 478-397; 1959.
[4]Dong-Woo Gim and Kyle T. Alfriend "Satellite relative motion using dierential
equinoctial elements". Celest.
Mechanics and Dynamical Astronomy 92:295-336; 2005.
[5] Sabatini M, D. Izzo and R. Bevilacqua "Special Inclinations Allowing Minimal Drift
Orbits for Formation
Flying Satellites". Journal of guidance, control and dynamics 31, No. 1; 2008.
[6] Tan, Z., Bainum, P.M., “An improved Strategy for Maintaining Distance Between
Satellites in an Elliptically Orbiting Constellation” Paper AAS 01-219, AAS/AIAA Space
Flight Mechanics Meeting, Santa Barbara (California), 11-14, February 2001.
NOTE: (Figures)
TFC: Estratègies per mantenir separació constant entre satèl·lits
situats en constel·lacionsd’òrbites el·líptiques.
Autor: Meritxell Viñas Tió; Data: 27 de febrer de 2006[7]Koon, W.S., J.E. Marsden, J.J. Masdemont and R.M. Murray. "J2 Dynamics andformation ight". AIAAPAG. 26
2001-4090; 2001