The Flower Constellations An Overview of the Theory, Design Process, and Applications

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The Flower ConstellationsAn Overview of the
Theory, Design Process, and Applications

• Matthew Wilkins
• Daniele Mortari
• Christian Bruccoleri
• Aerospace Engineering Dept.
• Texas AM University

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Overview

• A Brief History
• Fundamental Concepts of the Theory
• Design Challenges and Solutions
• Examples
• An Inverse Design Technique
• Potential Applications
• A Discussion on Perturbations
• Conclusions and Future Work

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It all began with the Clover Constellation!
Sistema Quadrifoglio, by Luigi Broglio (1967)
John Junkins (right) meets Luigi Broglio in Italy
in 2000.
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Some similar constellation concepts based upon
repeating ground tracks

• JOCOS
• LOOPUS
• COBRA

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JOCOS Constellation (1/2)

• Juggler Orbit COnStellation (JOCOS)
  constellation is so named because it juggles 3
  3 satellites simultaneously with three up and
  three down at any given time
• Goal of JOCOS constellation design is to
  maximize Earth coverage
• 8 hr, circular, inclined, repeating orbits
• Inclination of 75 degrees chosen (apogee
  location irrelevant)
• 6 satellites are placed with nodes evenly
  arrayed and mean anomalies chosen to place 3
  satellites in the northern hemisphere and 3 in
  the southern hemisphere
• 1 extra satellite is required to fill in
  coverage gaps at high latitudes during cross-over

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JOCOS Constellation (2/2)
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LOOPUS Constellation (1/2)

• LOOPUS (quasi-geostationary Loops in Orbit
  Occupied Permanently by Unstationary Satellites)
• Constructed from circular or HEO orbits
• Focuses on solutions where loops are formed in
  the ground track.
• The satellites are arrayed such that two
  satellites will reach the intersection of the
  loop (one entering and one leaving) almost
  simultaneously where a communications hand-over
  is performed.
• For the non-circular orbits, the orbital
  inclination is chosen to be the critical
  inclination
• The goal of the LOOPUS constellation is to
  maximize Earth coverage

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LOOPUS Constellation (2/2)
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COBRA Constellation (1/2)

• The COBRA Teardrop concept involves two MIOs
  where the argument of perigee is neither 90
  degrees nor 270 degrees
• By choosing other values for the argument of
  perigee, a lean is created in the ground track
• By combining two repeat ground track orbits, one
  with a right lean and the other with a left
  lean, a teardrop intersection is created
• As in the LOOPUS concept, the intersection points
  are used to hand over the communications
  responsibilities between satellites in the
  constellation

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COBRA Constellation (2/2)
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Current Applications

• To date, most of the recent applications of
  multi-stationary inclined orbits (MIOs) have been
  focused on telecommunications
• MIOs are usually comprised of highly elliptic
  orbits (HEOs) and provide excellent coverage
  properties
• The HEO provide a much larger grazing angle
  w.r.t. the horizon for higher latitude regions
  such as countries in northern Europe
• The quasi-stationary properties provide an
  alternative to GEO satellites

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But Then Came the Flower Constellations
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The Flower Constellations
Depends on 8 parameters Np Nd Ns Fn
Fd w hp and i

• Compatible Orbits
• Phasing rule Mf(?)
• Symmetric, Restricted,
• and Non-Symmetric
• Phasing Schemes

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FC Theory in Brief (1/4)
For a repeating space track relative to an
arbitrary rotating reference frame, the period of
repetition can be written in two different ways.
One is the number of orbit revolutions it takes
the satellite to complete the space track. The
second is the number of revolutions that the
rotating reference frame makes in the same time
period. Here, the period of repetition is written
w.r.t. an Earth Centered Earth Fixed frame.
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FC Theory in Brief (2/4)
If one includes the J2 perturbation, one can
solve for the semi-major axis required to achieve
a given Flower Constellation initially defined by
Np, Nd, i, w, and hp.
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FC Theory in Brief (3/4)
The phasing is a function of two parameters, the
right ascension of the ascending node and the
initial mean anomaly. These orbit angles are
functions of the number of petals, the number of
days to repeat, the semi-major axis, the orbit
inclination, the height of perigee, and the
argument of perigee in addition to two arbitrary
phasing parameters Fn and Fd. These angles must
be specified in a very particular way in order
for all the satellites in a single Flower
Constellation to belong to the same repeating
space track.
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FC Theory in Brief (4/4)

• In general, there are three kinds of phasing
  schemes.
• Symmetric about the constellation axis of
  symmetry
• Restricted schemes where the RAAN angle is
  constrained to lie within a certain range
• Non-symmetric schemes where the change in RAAN
  between any two satellites is arbitrary but the
  relationship between RAAN and initial mean
  anomaly is maintained

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Ok, I understand the relative path
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But where does this come from???
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A New Phenomena in Phasing
Secondary Closed Paths!
These secondary closed paths (SCP) occur for
specific choices of the Flower Constellation
parameters. Even though a large number of
satellites is required to completely visualize
the SCP, any single satellite will trace out both
the relative path AND the SCP.
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Categories of FCs

• Basic Flowers
• Planar Patterns (i 0 deg)
• Planes of Satellites
• Uniformly Distributed Satellites
• Helixes
• Figure 8s
• Rings
• And more these categories are so named because
  they are our interpretation of a mathematical
  phenomena. As we continue to explore Flower
  Constellations, more categories of constellation
  types will be developed.

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Five Basic Steps to Designing a FC

• Specify a rotating reference system for the
  compatible orbits
• Specify the orbit inclination, argument of
  periapsis, and height of periapsis
• Decide upon an overall shape (select Np and Nd)
• Decide upon a phasing scheme (select Fn and Fd)
• Specify an orientation for the axis of symmetry

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Design Challenges

• An infinity of possibilities!
• Optimal selection of FC parameters based upon
  mission design criteria can be difficult.
• Specifying a final shape a priori is desirable
  but how does one solve for the required FC
  parameters?

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Fear not! There are solutions! A Simple Design
Example

• ESAs Galileo Constellation
• 27 active satellites 3 spares
• 3 orbit planes
• Circular orbits, a 23,616 km, i 56 deg

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An Inverse Design Technique
Projection of an Arbitrary Shape onto a Flower
Constellation Surface
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Example Flower Constellation Surface
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Projection from an Arbitrary Point
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Triangle Formation Example
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Potential Applications

• Earth Observation
• Deep Space Observation
• Global Navigation Systems
• Formation Flying
• Many more

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Global Navigation Systems
The GNFC
Two uniformly distributed Flower Constellations
provide global coverage with superior geometric
and attitude dilution of precision (GDOP and
ADOP). -Park, Wilkins, Mortari (AAS 04-297 Maui,
HI Feb 2004)
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Some Comments on Perturbations

• Resonance is a major concern
• Nd 1 or when Nd Np (read Nd divides Np)
• Critical inclination required to maintain
  stationary line of apsides
• FCs at arbitrary inclinations wilt
• Secular drift of node, argument of perigee, and
  mean anomaly
• Node drift causes FCs to spin
• Mean anomaly drift disrupts phasing

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Conclusions

• A novel theory for constellation design has been
  developed!
• Extensive possible applications.
• Readily duplicates currently known constellation
  concepts.

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Future Work

• Add additional phasing parameters
• Investigate genetic algorithms for finding
  optimal designs
• Expand projection technique into 3D constellation
  design
• Realistic mission study including perturbations

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http//flowerconstellations.tamu.eduThank you!