Remote nanosatellite formation designs with orbit perturbation corrections and attitude

Remote nanosatellite formation designs with orbit perturbation corrections and attitude
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Thesis Collection
2000-06-01
Remote nanosatellite formation designs with orbit
perturbation corrections and attitude
control/propulsion subsystem correlation
Tomlin, Stephen D.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/7781
OXUBRARV
-cT<Ey
-<ADUATE SCHOOL
93943-5101
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
REMOTE NANOSATELLITE FORMATION DESIGNS
WITH ORBIT PERTURBATION CORRECTIONS AND
ATTITUDE CONTROL/PROPULSION SUBSYSTEM
CORRELATION
by
Stephen D. Tomlin
June 21, 2000
Thesis Advisor:
Brij
Norm N.
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1.
AGENCY USE ONLY (Leave blank)
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REPORT DATE
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REPORT TYPE AND DATES COVERED
June 2001
Master's Thesis
REMOTE NANOSATELLITE FORMATION
TITLE AND SUBTITLE:
DESIGNS WITH ORBIT PERTURBATION CORRECTIONS AND ATTITUDE
CONTROL / PROPULSION SUBSYSTEM CORRELATION
6. AUTHOR(S) Tomlin, Stephen D.
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
4.
Naval Postgraduate School
9.
Monterey, CA 93943-5000
SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES)
5.
PERFORMING ORGANIZATION
REPORT NUMBER
SPONSORING / MONITORING
AGENCY REPORT NUMBER
10.
N/A
11.
SUPPLEMENTARY NOTES
The views expressed
in this thesis are
FUNDING NUMBERS
those of the author and do not reflect the official
policy or position of the Department of Defense or the U.S. Government.
12a.
DISTRIBUTION / AVAILABILITY STATEMENT
Approved
13.
for public release; distribution
ABSTRACT (maximum
is
12b.
DISTRIBUTION CODE
unlimited.
200 words)
The innovative idea of distributing
the functionality of current larger satellites
among
smaller,
cooperative satellites has been sincerely considered for assorted space missions to accomplish
goals that are not possible or very difficult to do with a single satellite.
Additionally, the
maximized within formations and clusters to conduct missions
such as interferometry and earth-sensing. This paper presents a methodology to describe, populate
and analyze numerous formation designs employing the use of Hill's equations of motion to
describe a formation's dynamics.
These equations of motion are then programmed into a
MATLAB code to produce Cartesian elements for input into a Satellite Tool Kit™ (STK)
simulation that demonstrates numerous possible cluster formation designs. These simulations are
then used to determine AV requirements for overcoming LEO-type perturbations that were
modeled within STK's High Precision Orbit Propagator (HPOP).
utilization
of smaller
Finally,
satellites is
components from two subsystems [Attitude Determination and Control (ADCS) and
Propulsion], using the
AV
calculations from the simulation analysis and current advances in
MicroElectroMechanical systems (MEMs) and nanosatellite technology, are presented based on a
mass constraint of 1 0kg for the entire satellite.
14.
SUBJECT TERMS
Nanosatellite,
15.
Orbit
Satellite Propulsion,
Dynamics,
Satellite
Formation,
Satellite
Cluster,
STK
SECURITY
CLASSIFICATION OF
CLASSIFICATION OF THIS
19. SECURITY
CLASSIFICATION OF
REPORT
PAGE
ABSTRACT
17.
Unclassified
NSN
7540-01-280-5500
18.
SECURITY
Unclassified
NUMBER OF
PAGES 136
16.
PRICE CODE
20.
LIMITATION
OF ABSTRACT
UL
Unclassified
Standard Form 298 (Rev. 2-89)
Prescribed by
ANSI
Std. 239-1
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11
Approved
for public release; distribution
is
unlimited.
Remote Nanosatellite Formation Designs
Corrections and Attitude Control/Propulsion
Perturbation
With Orbit
Subsystem Correlation
Stephen D. Tpmlin
Lieutenant, United 'States
Navy
B.S., Rensselaer Polytechnic Institute,
Submitted in
partial fulfillment
1994
of the
requirements for the degree of
MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
June 2001
* EY
ABSTRACT
The innovative idea of
among
distributing the functionality
of current larger
satellites
smaller, cooperative satellites has been sincerely considered for assorted space
missions to accomplish goals that are not possible or very difficult to do with a single
Additionally, the utilization of smaller satellites
satellite.
is
maximized within
formations and clusters to conduct missions such as interferometry and earth-sensing.
This paper presents a methodology to describe, populate and analyze numerous
formation designs employing the use of Hill's equations of motion to describe a
formation's dynamics.
MATLAB
code
to
(STK) simulation
These equations of motion are then programmed into a
produce Cartesian elements for input into a
that demonstrates
These simulations are then used
numerous possible
to determine
type perturbations that were modeled within
Satellite
Tool Kit
cluster formation designs.
A V requirements
for
overcoming LEO-
STK's High Precision Orbit Propagator
(HPOP).
Finally,
components from two subsystems [Attitude Determination and Control
(ADCS) and
current
Propulsion], using the
advances
in
A V calculations from
MicroElectroMechanical systems
the simulation analysis and
(MEMs) and
nanosatellite
technology, are presented based on a mass constraint of 1 Okg for the entire
satellite.
THIS
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VI
TABLE OF CONTENTS
I.
INTRODUCTION
NANOTECHNOLOGY PHENOMENON
B.
DEFINING THE STARTING LINE
CURRENT AND FUTURE EMPHASIS
C.
D.
II.
1
A.
1
1
3
OBSTACLES
E.
CONCEPTS OF THE AEROSPACE CORPORATION
F.
THE SCOPE
LITERATURE REVIEW: PAST AND CURRENT SYSTEMS
A.
PAST SYSTEMS
SIZE
7
9
9
1.
1958
1 990-1 995
9
10
3.
1 995-2000
2000 - Present
13
11
JA WSA T
SNAP-1
Munin
a.
b.
c.
13
15
75
16
STUDENT SATELLITE PROJECTS
1.
2.
3.
C.
6
2.
4.
B.
5
SSETI - Student Space Exploration
University nanosatellite program
& Technology Initiative
a.
Three Corner SAT.
b.
ION-F
c.
Emerald
17
18
18
19
d.
UW nanosatellite (Dawgstar)
20
20
22
22
24
24
25
Miscellaneous Projects
CURRENT SYSTEMS AND INDUSTRY LEADERS
1
Aerospace Corp
2.
NASA
3.
Foreign Universities
NANOSATELLITES FLYING TOGETHER
NANOSATELLITE FORMATION DYNAMICS
A.
INTRODUCTION
B.
INITIAL
EQUATIONS OF MOTION
FORMATIONS
D.
III.
2.
In-Plane Formation
3.
In-Track Formation
Circular Formation
Projected Circular Formation
REMOTE
in the
WITH
CLUSTERS
DISTRIBUTION
27
27
OF
SATELLITE
29
32
33
34
35
37
Formation
Basic
5.
C.
Remote Motion
1.
4.
16
CONSTANT
APPARENT
39
vii
1
2.
3.
Problem Description
Linearized Approach
Orbital Mechanics
a.
Apparent Angular Width
Correction for Inclination
Apparent Vertical Size
Phase Separation in Apparent Orbit
Geometry of the Orbit
b.
c.
d.
e.
4.
D.
IV.
52
53
Example
Populating a Remote Cluster
PERTURBATIONS AND A V REQUIREMENTS
1.
STK Perturbation Propagators
a.
Two-Body,
b.
HPOP.
Perturbation Effects
3.
Formation Keeping
Station Keeping
63
67
67
SUBSYSTEM DESIGN
A.
ADCS
2.
3.
69
70
Magnetic Control
71
a.
All-Magnetic Torquer System
Reaction wheel / Magnetic Torquerod System
b.
Propulsion Option
72
Components
73
74
c
Micromechanical gyroscope
Magnetometers
Reaction Wheels
d.
Sensors.
a.
b.
B.
75
76
77
79
80
Systems
Cold/Hot Gas
81
81
a.
b.
MEMS.
c.
Electrical.
Performance
Propulsion System Comparison (Cold gas
3.
EXAM PLE SN API (SSTL)
82
84
85
2.
C.
V.
72
74
e.
DGPS
PROPULSION
1.
54
55
61
62
OTHER TOPICS
1
LOW
60
60
60
J2 and J
2.
4.
F.
57
POPULATING AND MAINTAINING A CLUSTER IN
EARTH ORBIT (LEO)
1.
E.
40
43
45
46
48
49
:
CONCLUSION
A.
THE NANOSATELLITE PUSH
B.
FORMATION DESIGNS
C.
PERTURBATION UPKEEP
viii
/
fiPPT)
86
88
89
89
89
90
APPENDIX 2.1
93
APPENDIX
2.2
95
APPENDIX 3.1
97
APPENDIX 3.2
99
APPENDIX 3.3
103
APPENDIX 3.4
105
APPENDIX 4.1
107
LIST
OF REFERENCES
1
INITIAL DISTRIBUTION LIST
09
113
IX
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PAGE INTENTIONALLY LEFT BLANK
LIST OF FIGURES
Figure
1.1
Small
Satellite
Launch Mass
3
Figure 2.1
Vanguard-
Figure 2.2
TUB SAT-N (bottom box) / -Nl
Figure 2.3
Figure 3.1
Aerospace tethered picosatellites (artist interpretation)
SNAP-1 from SSTL (UK)
Inside a CubeSat (10cm per side)
Artist Conception of orbiting AEROSPACE Nanosatellite
Hill's reference frame for satellite relative motion
Figure 3.2
Moving Frame
22
23
30
32
Figure 3.3
Reference, relative, and apparent orbits
41
Figure 3.4
Figure 3.5
Geometry description and definitions
Orbital elements of the eccentric orbit
Figure 3.6
Apparently circular cluster of satellites
Figure 3.7
Figure 4.1
Ring of eight equally spaced satellites
Cyclic motion of Subsat Orbit
Remote Geometry as viewed along Mothersat Velocity vector
BEI GYROCHIP™ Model QRS1 1 Micromachined Angular Rate Sensor
42
43
45
53
55
56
Figure 4.2
Litton
Figure 4.3
Model 533: Miniature
Figure 4.4
Small reaction wheel developed by
Figure 4.5
EDO
Figure 4.6
Details of the structure of the micro-mirrors are
Figure 4.7
Aerospace MEMS chip compared to Penny
Marotta microthruster compared to Dime
Full-sized Pulsed Plasma Thrusters from Primex Aerospace Company
Basic diagram of a pulsed-plasma thruster
Snap-1 Propellant tube
Figure 2.4
Figure 2.5
Figure 2.6
Figure 3.8
Figure 3.9
9
G2000 gyroscope with
Barnes Model
(top plate)
11
14
15
75
75
electronics
3 Axis, Fluxgate
Magnetometer
76
77
HIT
13-500 wide-angle miniature solid-state horizon
78
79
sensor
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.1
XI
shown
83
84
84
85
;
....88
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PAGE INTENTIONALLY LEFT BLANK
XII
LIST
Table
1.1
OF TABLES
2
Satellite Classifications
Table 4.1
Assumptions made
Table 4.2
Comparison of uPPT and cold-gas propulsion systems
69
for Satellite Physical Characteristics
performance)
(single thruster
87
Xlll
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xiv
BLANK
LIST OF
a
semimajor axis
Cd
drag coefficient
Ce
effective exhaust velocity
ff
SYMBOLS
8
angle that defines shape and orientation of suborbit plane
e
eccentricity
E
eccentric
(f>
rotary angle
h
angular
rj
anomaly
momentum
dimensionless pattern generator scale factor
i
inclination
Ibit
minimum impulse bit
Isp
specific impulse
k
elevation angle
X
lateral
M
mean anomaly
//
gravitational parameter {398, 600. 441 5
v
true
angle
km 3/(solar sec) 2 )
anomaly
phase angle of satellite {measured clockwise from the cross-track(z) direction)
decay
q
orbit
r
radius of the formation
rate
xv
r
position of remote from Earth (vector)
p
scale factor for formation creation
Re
radius of Earth
Rp
radius of planet
R
position of mothersat from Earth (vector)
t
time
T
thrust
u
angle from line of nodes to radial vector (w
v
velocity vector
w
argument of perigee
co
mean motion
coe
rotation rate of the Earth
Q
right ascension
x
radial difference
v
along-track difference
z
cross-track difference
AV
velocity increment
(0.
v)
000 072 921 158 553
of the ascending node
/
+
'delta-V
xvi
rad/solar sec)
ACKNOWLEDGMENTS
Of
first
thank
enough
all
the help, support and inspiration
my
wife Kimberly
to enable
more ways than
To
and
I
my
who
I
received writing this paper,
I
wanted
has affectionately encouraged, assisted and loved
completion of
this project.
Without your help
me
would be stuck
I
to
in
one, thank you for the smiles, the laughter and the unwavering love.
my three
boys, Joshua, Daniel and
thank you guys for waiting patiently
Adam,
at the
I
am
truly blessed to
be your father
chess table, train table, and changing
table respectively!
Also
who
I
wanted
my sincere
to express
thanks to
all
my family members
helped through words of prayer and encouragement to show
possible,
and
that a true faith
and a loving family
especially to Professor Brij Agrawal, Dr.
Norm
is all
me
and friends
that anything is
that is important.
Thank you
Sorenson, Dave Cook, and
the
all
OCFers.
I
also
wanted
read and critique this paper.
evaluating
to Jeff
it
with
me
my
to express
Thank you
afterwards.
King who helped with
sincere thanks to Jon Strizzi,
It
was
certainly the best review
knowledge of
I
through His blessing and grace,
receive
my
thank
my
heavenly Father
wonderful family,
words of knowledge, wisdom and
love.
vain.
xvii
my
Without
got, thanks!
And
MATLAB. Thank
the time spent and hopefully your part of the puzzle in this field will
I
took the time to
for the time spent in reading, proofing and
his far superior
Finally and most important,
who
who
blessed
Him
fit
I
also
you
for
nicely!
made
possible,
and the
ability to
has
life
would be building
in
THIS
PAGE INTENTIONALLY LEFT BLANK
xvin
INTRODUCTION
I.
Nanotechnology provides the capability to manipulate matter at the atomic
level. In the future, we will measure the way we design and build our
systems by the atom, not by the pound. Today, we are developing material
systems, at the molecular level, that are 100 times stronger than steel at
1/6 the weight. We will also develop sensors and detectors capable of
responding to a single photon of light or the stimulus from a single
electron. Using nanotechnology, we will build systems on a scale 1000
times smaller than today ~ at true molecular level. They will be based on
concepts emerging from biology, quantum mechanics and chemistry, all of
which have no current parallel. [Ref 1]
A.
NANOTECHNOLOGY PHENOMENON
Commercially and
systems
militarily,
(MEMS), nanoscale
space systems based on MicroElectroMechanical
design and materials, low power quantum electronics, and
high bandwidth photonics are of special
interest, as are the
demonstrations of space
subsystems based on these technologies. Significant reductions in individual spacecraft
mass and
cost,
large
gains in capabilities and robustness,
and novel architectures
involving large constellations and closely coupled spacecraft are expected with the
introduction of these technologies. Novel spacecraft architecture concepts for Earth
orbiting
missions
including
communication, navigation, remote earth-sensing, and
monitoring of the local space environment that can be created with the introduction of
micro/nano-technologies will be outlined briefly below.
B.
DEFINING THE STARTING LINE
First
of all,
it
is
worth defining what
current small satellite world
Cheaper'.
Small
satellite
is
we mean by a small
encompassed by the slogan
projects
are
compared with the conventional space
months. Leading-edge technology
is
characterized
satellite.
The
spirit
of the
'Faster, Better, Smaller,
by rapid development when
industry, often ranging
from
six to thirty-six
routinely included in order to provide innovative
1
solutions, permitting lighter satellite systems to
be designed inside smaller volumes.
manned space programs,
Frequently, traditional procedures, with roots in the military and
can no longer be justified, and low cost solutions are favored to match the reducing space
budgets.
So
in
many ways
it
mass of the
the philosophy, and not the size or
is
satellite
that matters.
Many
terms are used to describe this rediscovered class of
satellites,
SmallSat, Cheapsat, MicroSat, MiniSat, NanoSat and even PicoSat.
Advanced Research
Naval Space
Projects
Command
method of
adopted (see Table
TACSat's (Tactical
medium
Satellites).
Nevertheless, in recent years a
mass has been generally
1.1).
tradeoffs are typically
classes are an indication of
made, which
Although the
satellite', this
Defense
refers to these as LightSats, the U.S.
classifying satellites in terms of deployed
The boundaries of these
'wet mass').
US
as SPINSat's (Single Purpose Inexpensive Satellite Systems), and
the U.S. Air Force as
general
Agency (DARPA)
The
including
is
also
why
the
mass
is
where launcher or cost
defined including fuel
satellites in the 5 00- 1000kg are typically
causes confusion and until a better term appears
it
designated as a 'small
will
be defined here as a
sized satellite.
B
B
E
Group name
Wet Mass
Large satellite
>1000kg
Medium sized satellite
500-1000kg
Mini
100-500kg
satellite
Micro
n
H
Nano
Pico
satellite
satellite
0.1-lkg
satellite
Femto
satellite
Table
10-100kg
l-10kg
1.1
<100g
Satellite Classifications
"(i.e.
Small
Satellites
The mass
ire
no clear mass boundaries, although there
'.00kg class.
)f
distribution for small satellites plotted in Fig. 1.1 illustrates that there
launched
The
emove
a general lack of spacecraft in the
positive sloping line in the figure depicts a gradual increase in the
satellites.
vithin the next
is
few
Although
years, the
this trend will continue as heavier satellites are
number of smaller
satellites
1
00-
mass
launched
launched should
start to
this trend.
Small
Figure
1
.
1
satellite
Small
mass
Satellite
Launch Mass
CURRENT AND FUTURE EMPHASIS
Few modern Nano and
aunched, although there
echnology
is
is
Pico-Satellites
weighing less than
considerable interest in this area as advanced microsat
applied to miniaturize satellite systems even further.
nrcrosats almost
fall
10kg have been
into this category with weights
Some
launched
of 11-1 4kg, most notably the
AMSAT
than
microsat-series [Ref 2]. These satellites are cubical in shape and measure less
150mm on
each
side.
Nanosatellites are attractive to
commonly
space, as
many
available technology
educational institutions to get involved in
now makes
most importantly affordable. Although operational
are
some time
still
nanotechnology makes
on
entire satellites
weighing
it
JAWSAT
just
recently
below).
less than
launched a
Increasingly,
1kg
of
set
micro
and
possible to fabricate entire satellite sub-systems, and possibly
a chip. Considerable effort
than
less
2A:
weighing
picosatellites
Aerospace Corporation
Chapter
(see
picosatellites
off,
type of satellite feasible and
this
with
0.1kg,
is
being spent on these Femtosatellites
applications
in
remote
inspection,
distributed
measurement and disposable sensors.
For Nanosatellites, autonomous operation using a single on-board computer
making use of technology developed
feasible,
minimize mass, active
attitude
systems.
to
The downlink data
be operated
at
The key
low data
palmtop computers. To
and orbit control are often ignored, and omni-directional
The main
antennas are employed.
for laptop and.
is
rate is limited
by the
rates, or in burst
mode.
to the successful
advanced technology. Each of the
by
limits are set
the downlink and
orbit average
power generation
power generation, and has
development of nanosatellites and constellations
many
is
enabling technologies toward nanotechnology
represents a breakthrough in performance, capability, or application in a unique way.
This technological challenge
is
formidable, since currently, the smallest "full-service"
microspacecraft weighs 100 kg (220 lb) or
of
five.
more
-
size
must be reduced by almost a
factor
This will require revolutionary advances in microelectronics and spacecraft
component technologies.
In
addition,
at
present day, almost
space missions are flown as single
all
spacecraft. This is because controlling spacecraft in flight is a very
problem of flying several spacecraft as one system
is further
communication path from a constellation of spacecraft
communication
to
behave
stations
on the ground.
"intelligently"
—
In order to
work
autonomously staying
4
complex process. The
compounded by
in flight
the
complex
high above Earth with
properly, the spacecraft will have
in constant contact
with each other,
sharing information, and re-configuring onboard instruments and systems to behave as a
single unit.
SIZE OBSTACLES
D.
A
fundamental problem in spaceflight
is
spacecraft size.
It is
very expensive to
place satellites into space. For example, using expendable cost with eight launches a year
Space Shuttle,
for the
it
payload into low-Earth
as possible.
It
costs about $13,200 per kilogram ($6,000 per
As
orbit.
pound)
to
deploy a
a result, engineers try to design spacecraft to be as small
has been estimated that satellites using nanotechnology could measure
inches wide by 2 inches thick and weigh about 2 pounds. These satellites
1
would then
require smaller and therefore less expensive launchers. [Ref. 3]
Most Earth-observing
kg (2,200
spacecraft with science payloads
lbs). Microsatellites are
much
range. Nanosatellite are even smaller
—
weigh
smaller, typically in the 100
in the
in excess
kg (220
lbs).
However, small spacecraft even today
capabilities. Typically they lack
control,
lb) or larger
sub-20 kg (44 lbs) range. Small spacecraft
are nothing new. In fact, the first satellite launched into space, Sputnik,
about 90 kg (180
of 1 ,000
weighed only
are very limited in their
any propulsion, possess only limited
ability for attitude
and carry one single-function payload.
What
service",
is
needed
meaning they
is
new
a
era of "smart", miniature spacecraft that will be "full
will carry a
wide range of spacecraft services including guidance,
and control, attitude control, propulsion, high bandwidth and complex
navigation
communication functions.
Nanosatellites not only reduce launch costs, but they also reduce the risks
associated with flying missions. Currently, several instruments and payloads are flown on
a single, large spacecraft.
disable
the
entire
A
single instrument or system failure
mission.
Constellations
complementary instruments reduce the
instrument
fails.
risk
of numerous
of an
entire
may
severely degrade or
spacecraft,
mission failing
if
each
carrying
one system or
E.
CONCEPTS OF THE AEROSPACE CORPORATION
Most of the
radical
the nanosatellite concept
new
concepts for building and using spacecraft represented by
were developed
The Aerospace Corporation and formally
at
introduced in a paper, "The Concept of 'Nanosatellite' for Revolutionary
Systems," presented
at
Low
Cost Space
the 44th International Astronautical Federation Congress in Graz,
Austria, in 1993.
— who
Authors included Robinson, Siegfried Janson, Ph.D.,
and originated the concept [Ref
nanosatellite
scientist
the
in
microtechnology
~
3]
and
center
coined the term
and Henry Helvajian, Ph.D., a senior
of the just-published
editor
book,
"Microengineering Aerospace Systems."
A
of reports written by Helvajian, Janson, and Robinson and issued
series
the 1993 conference presented the details on
nanosatellites.
national
and
how
to design, build,
These nanosatellite technologies are
international
research
now
organizations
after
power and maneuver
being explored by a number of
in
addition
to
The Aerospace
Corporation
The
First
International
Conference on Integrated Micro/Nanotechnology for
Space Applications was hosted jointly by the Aerospace Corporation and Johnson Space
Center in Houston, from October 30
conference
was
to
bring
th
together
through
November
scientists
and
3
rd
1995.
engineers
The purpose of
from
the
fields
the
of
microtechnology, nanoelectronics and space technology to explore the possibilities for
applying newly emerging capabilities in microtechnology to space operations. The
evolution of microelectronic technology coupled with the growth of
5 years has
had a significant impact
in the
commercial
MEMS in the past 4-
terrestrial sector.
This influence
can be evidenced particularly in sensor, optical switching and mass data storage
applications
that
have been inserted into major industries such as transportation,
medicine, telecommunications and computers. The focus of this conference was to
anticipate and extend the incorporation of nanoelectronics
Specific Integrated Microinstruments
of space systems.
(ASIMs)
and
MEMS
into Application
in order to revolutionize the
development
THE SCOPE
F.
This paper sets out to introduce, explore and design nanosatellite formation
designs with their required orbital dynamics. Chapter two will introduce past and current
Numerous systems
systems that are based on novel nanosatellite concepts.
will
be
covered in areas ranging from experimental military designs to innovative student-driven
ideas and future-looking commercial enterprises,
developing realm of 'smaller,
faster,
all
expecting to capture the quickly
cheaper' with regards to nanosatellite technology.
This paper presents a methodology to describe, populate and analyze numerous
formation designs employing the use of Hill's equations of motion to describe a
formation's dynamics. These equations of motion are then
code
to
that
demonstrates
Satellite
Tool Kit
formation
designs.
produce Cartesian elements for input into a
MATLAB
and
numerous possible
STK
to
create
cluster
understand the required delta- V
dimensional
MATLAB
(STK) simulation
After
utilizing
(AV)
to
(LEO)
perturbations will be analyzed
maintain the
formation within given
criteria.
Finally,
(ADCS) and
current
into a
formation simulations modeled within STK's High
Precision Orbit Propagator (HPOP), low-earth orbit
to
programmed
components from two subsystems [Attitude Determination and Control
Propulsion], using the
advances
in
AV
calculations from the simulation analysis and
MicroElectroMechanical
systems
(MEMs) and
technology, are presented based on a mass constraint of 10kg for the entire
nanosatellite
satellite.
THIS
PAGE INTENTIONALLY LEFT BLANK
LITERATURE REVIEW: PAST AND CURRENT
II.
SYSTEMS
PAST SYSTEMS
A.
1.
The
1958
first
U.S. earth
satellite,
Explorer
Launched by a modified
today's standards.
February 01, 1958
Army
diameter, and with a
I,
mass of 13.6kg and an
nanosatellite
first
Ballistic Missile
orbit
by
Agency Jupiter-C on
150mm
of 347x1,859
km
at
in
33.2 deg
developed by the Jet Propulsion Laboratory, carried the U.S.-IGY
(International Geophysical Year) experiment
of James A. Van Allen and resulted
discovery of the radiation belt around the earth [Ref.
follow would weigh
the
containing the upper stage, measured 2.03m long and
it,
inclination. Explorer
was almost
I,
more and become bigger
4].
Although most
in size, there
was one
US
in the
satellites to
that could
be
considered the 'forefather' of today's nanosatellites: Vanguard-1.
Although the
first
official
'nanosatellite'
years, the first satellite launched in the 1-1 0kg
still
orbiting Earth.
in July
of 1957
it
Originally, a simple nose
was decided
would not appear
mass range
used instead to exercise the tracking stations (see Fig.
2.1).
Vanguard-1
9
ironically the oldest satellite
cone was to be carried on Vanguard-1, but
that a small 1.47-kg (3.25
Figure 2.1
is
for another thirty
pound)
test satellite
would be
Launched March
orbit, this test satellite
aluminum
108
17,
1958 into an
would
the
Vanguard
technical and scientific results
(656x3866 km), 34.25 deg inclination
consist of a simple 16 centimeter (6.4 inch) polished
alloy sphere equipped with
MHz. Although
elliptical
is
1
two transmitters operating
satellite
must be looked
Technically
impressive.
its
at
frequencies around
at as
a test satellite
purpose was to
its
and
test
evaluate solar cells, satellite terminal design and on board instrumentation. All these test
objectives
were met. The solar
six years, (interfering the
scientific results
108
cells
MHz
worked so well
proved
to
were also a success. As Soviet
be a very useful
tool.
were active
frequency) well beyond the expected lifetime.
scoffed at the diminutive size of Vanguard
it
that its transmitters
1
and
authorities
its
The
and even some in the West
lack of sophisticated instrumentation,
Analysis of the motion of Vanguard
fact that the earth is not spherical but
for
established the
1
has a bulge, disclosing an unsuspected stress deep
within the earth. These measurements indicated that there
was
large-scale convection
taking place inside the Earth, which supported the emerging theories of continental drift
and sea floor spreading. Analysis of the drag exerted by the atmosphere on Vanguard
proved the atmosphere to be
far
more extensive and
Perturbations in Vanguard l's orbit also led to a
1
variable than previously believed.
more
refined estimate of the Earths
oblateness. [Ref. 5]
2.
1990-1995
On September
03,
1990 China launched two atmospheric balloons (1990-
081B/C), weighing 4kg each to measure the magnetosphere.
775x804
into
km
orbit at
an 833x886
99 degrees and decayed on March
1,
QQW1
1991.
km orbit at 99 degrees and decayed on July 24,
was launched
QQW2
into a
was launched
1991
Also on two occasions in mid- 1990 (February 03, 1994, February 04, 1995), a
series
of spherical objects were released from the
Orbital DEbris
RAdar
Calibration Spheres
shuttle, typically in pairs.
(ODERACS) were
a few centimeters in
diameter, and were intended to provide calibration for radar echoes.
provides the data on these objects.
10
These
Appendix
2.1
1995-2000
3.
A
the
MIR
commemorative functioning
replica of the original Sputnik
(SPUTNJX-40 - aka
space station
was deployed from
Sputnik-II, PS-2, RS-17, Sputnik Jr., 1997-
05 8C, 24958) during an extravehicular activity (EVA) by Russian cosmonauts Anatoly
Solovyov and Pavel Vinogradov on November 03, 1997.
km
orbit inclined at 51.6 degrees.
rocket on October 9
th
.
The
satellite
It
It
was
therefore in a 383x391
originally arrived via a Progress automated cargo
was
1/3 scale
and weighed only 3kg and was
built
by
French students from the l'Aeroclub of France (radio transmitter), staff from the Russian
Aeronautical Federation (structure),
industry.
It
The
and funded by various sponsors in the space
May 21,
stopped transmitting on the December 29, and decayed on the
first satellites
TUBSAT-N1
launched from a submarine were
(1998-042B). The 8kg
TUBSAT-N
(1998-042 A) and
TUBSAT-N and the 3kg TUBSAT-N1
were two nanosatellites launched on July 07, 1998 as a
1998.
satellite cluster
(see Fig. 2.2)
from a Russian
nuclear powered submarine with a Shtil-1 converted missile in the Barents Sea, and was
reported to have cost on the order of $ 100k (US$1998).
orbit via
telecommand and were placed
Figure 2.2
The
each
spacecrafts
contained
three
into a
TUBSAT-N
400x776
different
experimental
satellites
were separated
km orbit inclined at
(bottom box)
were flat-box shaped with a
The
/
-Nl (top
solar panel
payloads
in
78.9 degrees.
plate)
on the
largest face,
and
provided by the Technical
University of Berlin (TUB): reaction wheel performance, star sensor performance, and
•11
and forward communication. The
store
communication transceivers
1
200 and 2400 baud.
in the
Two
for store
payload consisted of four independent
and forward communication with a baud
transceivers
worked
70-cm frequency band with FFSK
additional downlink transmitter with
latter
in the
rate
2m-frequency band, the other two
(Fast Frequency Shift Keying) modulation.
9600 Baud
of
GMSK
(Gaussian
Minimum
An
Shift
Keying) modulation was available. The attitude control of TUBS AT-N consisted of two
magnetic
coils, a
magnetometer, a reaction wheel and a
star sensor. All attitude control
devices were also developments of the Technical University of Berlin. Electrical power
was provided by 9 NiCd-battery
cells
Ah (SANYO). The
of 5
battery cells
were
connected serially and provided an unregulated bus voltage from 9 to 13 V. The current
of the nanosatellites
utilization
mammals,
is
being used for tracking medium-sized and large
stolen cars and to collect data
from autonomous buoys for earth environmental
observation. These buoys are located in the northern Atlantic
Ocean near
the
Canary
Islands.
A
MIR
second sputnik, Sputnik-41 (aka RS-18, 1998-062C), was launched from the
spacestation into a
1998.
A
313x318 inclined
orbit at 51.7 degrees
Progress-M40 cargo rocket delivered
it
to
MIR
on the November
on the October 25, 1998.
It
10,
was
financed by the Aeroclub de France, and built by French and Russian students. Sputnik41's
200m
Russian.
from
W transmitter broadcasted pre-recorded voice greetings in French, English, and
The
orbit
on the
The
from the
1999.
A
spacecraft
1
measured
in diameter,
spacestation
RS-19 (aka
Sputnik-Jr. 3rd,
It
decayed
earlier spacecraft in this series,
it
was supposed
was launched "switched off,
company (Swatch) were
1
999-01 5C), was launched
by Jean-Pierre Haignere during a spacewalk on the 16th April
Progress-M41 cargo rocket delivered
spacecraft
and weighed 4kg.
1th January 1999.
third "junior" sputnik,
MIR
230mm
it
to
MIR
on the 2nd April 1999. Like
to transmit simple messages,
as advertising messages
carried in breach
by
however the
a commercial
of International Telecommunication Union
(ITU) regulations regarding amateur bands [Ref 6].
12
2000 -Present
4.
JAWSAT
a.
On
January 26, 2000 an
Launch Vehicle) carried
JAWS AT,
Those
several
OSPSLV
Program Space
(Orbital Suborbital
within
microsatellites
a
payload
adapter
called
developed jointly by the U.S. Air Force and Weber State University in Utah.
included FalconSat, an experimental
satellites
Academy, ASUSat
built
1,
by students
at
satellite built
by the U.S. Air Force
Arizona State University; the Optical
Calibration Sphere Experiment, an inflatable 3.5-meter (11.5-foot) balloon built
by
L'Garde for the Air Force Research Laboratory that served as a target for low-powered
ground-based
lasers;
and Opal, a Stanford University
satellite that, in turn,
deployed six
smaller "picosatellites" built by Santa Clara University, the Aerospace Corporation, and
ham
radio operators.
In addition to those satellites,
remained attached
provided by
altitudes,
to
NASA's
A
microsatellite.
Satellite
Test (PEST),
at
orbital
Control Platform tested a low-cost three-axis
satellites are
covered below in more
detail.
DARPA Picosat (OPAL-#l&2), built by
Rockwell and funded by DARPA and UCLA weighing just 0.5kg
tethered set of Picosatellites,
100x750x250mm each
(see
Fig.
They were deployed on February
established 24 hours
the pair.
The Plasma Experiment
State's Attitude
The launched
the Aerospace Corp. at
measuring
after launch.
included two other payloads that
Marshall Space Flight Center, studied plasma found
Weber
while
stabilization system.
and
it
JAWSAT
later.
Gold strands
in the tether
The spacecraft performed basic
tests
on
2.3),
6th,
launched
and
were instrumental
MEMS RF
OPAL
communications were
in radar tracking
switches.
primary batteries (lithium thionyl chloride) ran out by February 10th.
13
from the
The
spacecraft
Aerospace tethered
Figure 2.3
A
ARTEMIS
in a
Picosat
5400 hour
named
picosatellite
picosatellites (artist interpretation)
after
(OPAL-#3&4, Thelma and
effort
over a period of
1
the
Greek Goddess of the moon, the
Louise), weighed just 0.5kg and
GaAs
not attitude stabilized.
solar cells.
carried a
It
were deployed on February
MASAT,
1
the
1th,
It
was designed
Very
built
months by an exclusively female team of 7
Santa Clara University students. The spacecraft employed a
battery cells and
was
Low
for
68HC1 1
microcontroller,
AA
one week of operations and was
Frequency (VLF) receiver. The spacecraft
and were reported not
to
be operational. [Ref.
Miniature Amateur Satellite, and
7]
STENSAT, NASA's
amateur radio picosatellite built by Goddard Space Flight Center (GSFC), deployed on
February
12
picosatellite
th
from the
OPAL
weighting just 0.5kg, and
Both spacecraft were reported not
ASUS AT- was
1
University.
when
microsatellite.
It
carried
GPS,
to
MASAT
STENSAT (OPAL-#6)
(OPAL-#5, JAK) was a
weighed
in at just 0.25kg.
be operational.
a 5kg nano-satellite designed and built
a camera, and a radio amateur
at
Arizona State
FM repeater that only operated
requested on the uplink. The batteries failed to charge and therefore the satellite
operated for approximately 15 hours during which telemetry
longer operational.
14
was
received, and is
no
b.
SNAP-1
SNAP-1 (2000-033C, 26385)
Satellite
is
a 6.5kg nanosatellite developed by Surrey
Technology Limited (SSTL) and the Surrey Space Centre (SSC)
Fig. 2.4).
The spacecraft was launched
COSMOS-3M
as a piggyback ride
orbit inclined at 98.13 degrees.
It
carries a
communicate with Tshinghua-1.
nanosatellite
to
was placed
microsatellite
into a
remote inspection payload, and an
carries a
The
subsystem design.
684x707
km
intersatellite
Butane propulsion system with a 3m/s
be presented on
capability. Further information will
regards
It
It
UK (see
on June 28, 2000 on a
launcher from Plesetsk, together with the Tsinghua-1
(China) and the Nadezhda-06 (Russia) primary payload.
link to
in the
this satellite in
spacecraft
is
still
Chapter 4 with
reported to be
operational.
-r-'&cp^- J^^*^
-
SNAP-1 from SSTL (UK)
Figure 2.4
c.
Munin
Munin
(2000-0, 26621 A)
is
a Swedish
6kg
satellite
(7.5kg including
separation system) to measure the electron and ion distribution in the aurora ovals and
was launched
into a
698x1800
km
orbit.
Detector of Ions and Neutral Atoms),
The Swedish
University
Institute
is
The
satellite carries
a spectrometer (DINA,
cubic in shape and measures
200x200x250mm.
of Space Physics (IRF) and the Dept. of Space Physics of
(RYP) designed
the satellite.
Munin was launched from VAFB,
15
Umea
together with
EO-1 (Earth Observing-1), and SAC-C
last
contact with
Munin was February
('Satelite
de Aplicaciones Cientificas'-C). The
12, 2001.
STUDENT SATELLITE PROJECTS
B.
In recent years, an increased effort to design, build, and operate small satellites
has taken place in universities and laboratories
and nanosatellites provide numerous
fraction
of the cost of larger
all
over the world. These microsatellites
flight opportunities for science
traditional
missions.
experiments
find themselves
working
progress to continue. Appendix 2.2
known
to
be involved
lists
an international environment for space
the
most current educational establishments
in small spacecraft projects.
The European Space Agency (ESA) has
to involve students into the building
is to
create a
Tomorrow's
in
SSETI - Student Space Exploration
1.
of
& Technology Initiative
started this ambitious educational project
satellites: "...The
network of students, educational
main objective of this
institutions
Internet) to facilitate the distributed design, construction
initiative
and organizations (on the
and launch of (micro)-satellites
and potentially more complex projects such as a moon-lander [Ref. 8]." The
distribution
a
In addition, there has been an
increasing trend towards international cooperation on space projects.
engineers will
at
round of the sub-systems resulted
in the following distribution:
AOCS:
Institute Superior Tecnico, Portugal
Communication:
UNICAL, Cosenza,
Ground
Instituto Superior Tecnico, Portugal
stations:
first
Italy
Instruments:
University of Florence, Italy
Lander, Avionics:
Escola Politecnica Superior at Universitat de Girona, Spain
Mechanism:
EPFL(Lausanne), Switzerland
Mission Analysis:
University ofZaragoza, Spain
On-Board Data Handling (OBDH):
16
UK
University of Newcastle,
•
Power:
Euroavia-Napoli, Italy
•
Programmatic:
University
•
Propulsion:
University of Stuttgart,
•
SSETI
•
Simulation:
•
Structures/Configuration:
of Stuttgart, Germany
Infrastructure: Escola Politecnica Superior at Universitat de Girona, Spain
TU Vienna,
Austria
University of the
•
Basque Country, Spain
Structures/Structural calculations:
Kingston University,
•
Germany
Manchester University,
Thermal:
2.
University nanosatellite
The Air Force
Advanced Research
UK
program
Office of Scientific Research
Projects
grants of $50k/year over
Agency (DARPA)
two years
UK
to design
(AFOSR), NASA, and
the Defense
are jointly funding 10 universities with
and assemble 10 nanosatellites (~10kg). The
universities will conduct creative low-cost space experiments to explore the military
usefulness of nanosatellites in such areas as formation flying, miniature bus technologies,
enhanced communications, miniaturized sensors, attitude control, distributed
capabilities
and maneuvering.
The
satellites are
planned to launch mid-2002.
satellite
The 10
university nanosatellites provide a broad range of technology demonstrations in the areas
of miniature spacecraft subsystem components and formation
flying.
numerous science measurements and experiments
GPS
wind, magnetic
fields,
in
such areas as
scintillation, solar
and upper atmosphere ion density.
The Air Force Research Laboratory (AFRL)
structure, securing a launch,
is
developing
a
deployment
and providing such advanced microsatellite hardware as high
efficiency solar cells and micropropulsion units.
the universities to provide approximately
flying technologies as
There are also
NASA
$1.2M funding
Goddard has
also
teamed with
to demonstrate such formation
advanced crosslink communication and navigation hardware and
17
flight control algorithms.
Numerous
industry partners are also supporting the universities
with hardware and design and testing services.
This program has the potential to provide significant payoff for very modest
DoD
funding by
NASA
and
given the broad university resources being applied and
support by industry partners. If these flight demonstrations are successful,
it
very likely
is
government sponsorship can be secured for follow-on launches of nanosatellites
universities
built
by
and other agencies.
Three Corner SAT
a.
This project
University of Colorado
Aptly named Three
Sat (3
demonstrate
and
communications,
a joint effort between Arizona State University (ASU),
Boulder (CU), and
Comer
will
nanosatellites
at
is
innovative
New Mexico
State University
(NMSU).
A Sat), the proposed
constellation of three identical
stereo
imaging,
command and
formation
data
flying/cellular-phone
handling.
In
addition,
each
University in the 3 A Sat constellation has the opportunity to fly an individual unique
payload should
b.
it
desire. [Ref. 9]
ION-F
Utah
State
of
University
University,
Washington,
and
Virginia
Polytechnic Institute are designing and developing a system of three 10-kg spacecraft to
investigate
satellite
coordination
ionospheric measurements.
The
and
management
three will coordinate
on
technologies
satellite design,
and
distributed
formation flying
and management mission development, and science instruments and mission. Advanced
hardware for distributed space system
to
be demonstrated includes micro-pulsed plasma
thrusters (uPPT), gimbaling magnetic attitude control,
addition, an Internet
control
that
its satellite
would
and an advanced tether system. In
based operations center will be designed to enable each university to
from
its
own remote
location.
ION-F
will focus
benefit TechSat 21 and future Air Force and
18
NASA
on mission objectives
missions. In addition,
(SDL, Primex, Honeywell) support has been
industrial
students, hardware,
and
design support. [Ref.
satellite
Spacecraft formation flying
and commercial potential
EMERALD,
a
low
cost,
As
and
University
Stanford
is
an evolving technology with vast
that ranges
radical reductions in operations cost.
Program,
0]
Emerald
c.
military,
1
identified, including funding for
from enhanced mission performance
to
of the TechSat 2 1 University Nanosatellite
part
Santa
two-satellite
scientific,
Clara
University
mission
for
are
jointly
validating
developing
formation-flying
technologies.
SSDL
(Space Systems Development Lab) and Santa Clara
(Santa
Clara
Stanford's
University's
SCREEM
Remote Extreme
Environment
Mechanisms
Laboratory) will work as a unified team to develop, construct, test and eventually operate
the
EMERALD
Stanford's
spacecraft.
The formation
flying experiments will
ARL (Aerospace Robotics Lab)
The
EMERALD
from a simple single
Mission
satellite to
two
is
divided into three distinct stages that progress
free flying satellites in a coarse formation.
building block experimental strategy, the research payloads
isolation.
be coordinated through
first will
Using a
be characterized
in
Then, they will be coordinated and combined to permit simple demonstrations
of fundamental formation flying control functions such as relative position determination
and position control.
At
release, the
two spacecraft will be stacked together and
travel as a single object. This will allow initial checkout, calibration,
will
and some limited
experimentation. During the second stage of operation, the satellites will separate and a
simple tether or flexible
boom
will uncoil, linking the
two
vehicles. This tethered stage
will allow full formation flying experimentation including on-orbit relative position
determination and simple closed loop relative position control using the drag panels and
microthrusters.
During the
true
final stage
two-body formation flying
of operation, the tether will be cut
for a limited period
19
in order to permit
of time. The tether will have a simple
sub-satellite at its midpoint.
Each
will separate.
Upon ground command,
satellite will retain
The University of Washington
space.
to design, build
of
UW
1 1]
UW nanosatellite (Dawgstar)
d.
in a cluster
sub-satellite
half of the tether and half of the sub-satellite,
providing very rough gravity gradient stabilization. [Ref.
program
two halves of the
the
Nanosatellite
and launch the smallest self-propelled
flown in 2002 as one of the
satellites
program
first
is
a student run
(15kg) to be used
satellite
distributed satellite testbeds in
has university partners (Utah State and Virginia Tech), each of which
designing a
allow the "cluster" testbed. The focus on the
satellite to
UW
is
program has
been on distributed control. The sensing will be done using several technologies relating
to
GPS, and a
cross-link system designed
by APL-JHU. The actuation
will
be a
set
of
eight pulsed plasma thrusters, to be used for both attitude and position control. In
addition, there are several other important technology developments in this program,
including:
development
of uPPTs
for
both
and
attitude
orbit/formation
development of horizon and sun sensors using small, lightweight,
CMOS
control,
camera
technology and distributed ionospheric science. [Ref. 12]
Miscellaneous Projects
3.
Project Starshine.
small,
reflective
optically
Washington,
DC
assembled
The Student Tracked Atmospheric Research
spherical
it
spacecraft.
Satellite is a
The Naval Research Laboratory
from eleven hundred
machined by Utah technology students and shipped
sets
in kits
in
of aluminum mirror blanks
by
project officials to schools
around the world where students polished the blanks.
The
eye.
satellite is
48cm
Students recorded
satellite's orbit
decayed
(19in) and
their
was very
bright
and easily visible
observations online while tracking the
at a rate
proportional to
how much
heated by solar activity, thus monitoring sunspots.
to the
naked
satellite.
the upper atmosphere
Starshine
was deployed by
The
was
NASA
from a Hitchhiker canister on the Space Shuttle Discovery into a highly inclined low
20
on mission STS-96
earth orbit
atmosphere on Feb
Sunsat.
African
1
8,
satellite. It
cm
x 45
cm
of 1999, and the
satellite re-entered Earth's
2000
Graduate students
was launched
on February 23, 1999.
45
May
in
SUNSAT
University of Stellenbosch built this South
at the
as a piggyback payload
is
x 60 cm, with an
a micro
satellite,
on a Delta
from Vandenberg
II
weighing 64 kg, with dimensions of
The
polar orbit (620 by 850 km).
elliptical
satellite
holds amateur radio transponders and several other instruments that allow digital store-
and-forward capability and a voice
The
educational demonstrations.
'parrot' repeater that
unit has
two
VHF
is
being used primarily for
and two
UHF
transmit-receive
systems. In addition to amateur radio and school science payloads, Sunsat carries two
NASA experiments and an experimental push-broom imager capable of taking pictures of
Earth.
The high-resolution imager operates
computer
also can be stored in
RAM
in real
aboard the
time on S-band frequencies. Images
satellite
and then downloaded
speeds for retrieval by hams and schools. The participants have
made
at
lower
a big effort to use
the project to inspire interest in science and engineering in South African high school
students.
CubeSat. The CubeSat
(Stanford's Space System's
is
a nano-satellite design from
Development Laboratory).
Bob Twiggs of the SSDL
The motivation
is to
develop a
standard, off-the-shelf-satellite small satellite kit that can be cheaply built, easily adapted
for different missions,
CubeSat
is
and launched in clusters to lower the per
satellite
launch cost. The
about 10cm per side and weighs a kilogram (see next page
-
Fig. 2.5),
allowing student groups to be able to build and launch them for around $50k each.
Eventually, multiple CubeSats will
of a single large
formed
to
satellite.
together in formation to provide the capabilities
The company OSSS (One-Stop-Satellite-Solutions) has been
commercialize SSDL's CubeSat. Several college teams are
CubeSats for launch
CubeSat
work
at the
in 2001:
CubeSat
at
now
building
Cuesta College Amateur Radio Organization,
University of Tokyo, and Cube-sat
13]
21
at
Tokyo
Institute
of Technology. [Ref
Inside a CubeSat
Figure 2.5
PANSAT. The
and
designed
Petite
Amateur Navy
by military
built
officer
(10cm per
Satellite
students,
side)
(PANSAT)
faculty,
and
a small satellite
is
staff
at
Naval
the
Postgraduate School (NPS). The main objective was to support the Space Systems
Engineering and Space Systems Operations curricula by providing a 'hands-on' hardware
project
where exposure
can be experienced.
to the
PANSAT
many
facets
of a space system development and
Shuttle into a low-Earth orbit
spacecraft
communications using
bits
C.
PANSAT
was launched from
on the STS-95 Discovery mission as
per second and 9
itself
direct
amateur radio 70
to utilize
NPS.
part
of the
the
third
Extreme Ultraviolet Hitchhiker (IEH-3) experiment.
The
in the
cycle
further provides educational training while in orbit through
a space-based laboratory for officer students at
International
life
provides
store-and-forward
(packet
sequence spread spectrum modulation.
cm band
with center frequency
at
436.5
radio)
digital
PANSAT
operates
MHz,
a bit rate of 9842
MB of message storage. Amateur radio ground stations will be able
PANSAT via a bulletin-board type user interface.
CURRENT SYSTEMS AND INDUSTRY LEADERS
Aerospace Corp.
1.
In
an
effort
sponsored
by Defense Advanced Research Projects Agency
(DARPA), Aerospace Corporation
scientists
22
and engineers are collaborating with
Lockwell Science Center and Stanford University to develop miniature low-cost space
latforms to validate microsystems for space applications and advance the development
f mass-producible, fully functional nanosatellites (see Fig. 2.6). Aerospace supported
reflight activities for
an experiment involving two tiny picosatellites to be deployed
OPAL
rom Stanford University's
tie
new Air Force OSPSLV. The
within the
ormed the
rrays
third
in space to
MEMS
A picosat
its
launch by
mounted on a 50-meter ground antenna
MEMS
element of a rudimentary constellation.
were also tested on
/[EMS
2000-present ) after
picosats were tethered to emulate formation flying
range of low-power radios.
The mission
he
satellite (see section
radio-frequency switch
this mission.
represents one of several programs for systematic testing and use of
be designed and implemented by The Aerospace Corporation. One of
devices, designed and fabricated at
The Aerospace Corporation, comprises an
rray of 19 microthrusters that could be used to orient a nanosatellite. Each of the 19 cells
epresents a separate thruster like a solid rocket motor on a launch vehicle (see Chapter
B for propulsion components).
The mission closed out February
validate
atellites
MEMS
0,
2000 and was the
operating in constellations. Another picosat mission
a third
first
of a series designed to
technology and demonstrate the capabilities of mass-produced miniature
iboard the MightySat 2.1 satellite built
md
1
more complex
was launched July
29,
2000
by the Air Force Research Laboratory (AFRL),
"inspector" mission
is
planned for 2003.
i^^^^^^^^H
1
^j^H ^^^
P^kHt?^
^H
fV': :^>?^^^^BB
*H| j*9^z
'
fcV*r-
BBkL
I
.*
"^'ilSffifipM
Figure 2.6
Artist
Conception of orbiting
23
AEROSPACE Nanosatellite
NASA
2.
The Nanosat Constellation
Trailblazer mission
NASA's New Millennium Program. Known
mission will attempt to
spacecraft
is
fly three
is
Space Technology 5 (ST-5
as
about the size of a 'big' birthday cake;
spacecraft
communication, and
payoff from
a
in
for short), the
miniature spacecraft high above the Earth. Each of the
42cm
(17 in) across and 20
ST5
high^ and weighs about 21.5 kilograms (47 pounds).
separate
the fourth deep space mission in
performing
constellation,
scientific observations as
it
was
cm
(8 in)
will attempt to fly three
movements,
coordinated
a single, larger spacecraft.
The
technology demonstration will enable a series of multi-spacecraft
this
missions in the future. Large numbers of small spacecraft are planned in "constellation
perform in-situ measurements of space weather
class" missions in the next century to
conditions.
Space Technology 5 will
about 200x 35,790
a
as
will
single
be used
system.
to test
methods
The mission
technologies in the magnetosphere.
will
The mission
is
for operating a constellation
also
test
3.
Flight Center in Greenbelt,
innovative
eight
Maryland and
is
is
new
managed by NASA's
budgeted
at
$28 million.
Foreign Universities
The purpose of the Surrey Nanosatellite Applications Program, by Surrey
Technology Ltd (SSTL) located
multi-mission
of
planned for launch in 2003 as a
secondary payload on an expendable launch vehicle. The mission
Goddard Space
orbit at
km altitude.
The spacecraft
spacecraft
major
fly three nanosatellites in a stable
satellite
at
Surrey University,
is to
Satellite
develop practical, modular,
buses within mass ranges of 1-10 kg, and to demonstrate the use of
miniature electrical and mechanical
COTS
technologies in space.
It
also
is to
provide
vehicles for the education and training of students in spacecraft engineering at the
undergraduate and post-graduate
level.
The
first
accomplishment of
this
was by
the
development of the ultra-low-cost demonstration spacecraft ('SNAP-1' - see above 2000-
24
present ) within a year,
by a team comprising undergraduates, Masters and Ph.D.
with the supporting expertise from
MicroStructure
Additionally,
Sweden
will launch in a
the Center for
SSTL
CSER staff.
and
Technology (MST)
few years the
first
European
Advanced Micro Engineering (AME)
at
Uppsala University
nanosatellite.
D.
in
Researchers within
are contributing
on microsystems
research hoping to increase the European nanosatellite knowledge base.
project has been initiated at
students,
A
nanosatellite
AME to promote system oriented research work.
NANOSATELLITES FLYING TOGETHER
One
distinctive
nanotechnology and
concept,
which
many of the above
Many
the
small satellite designs,
next chapter's thrust will be to present
numerous formations.
demonstrates
how
to
unique
is
capabilities
formation flying.
of
The
use these smaller nanosatellites in
concepts have started to surface on what types of
formations are better at certain missions, and cluster lifetime orbit analysis simulations
are quickly narrowing
down
the required fuel expense and attitude control needed to
maintain the formation dynamics.
25
THIS
PAGE INTENTIONALLY LEFT BLANK
26
NANOSATELLITE FORMATION DYNAMICS
III.
INTRODUCTION
A.
In recent years, the innovative idea of distributing the functionality of current
among
larger satellites
smaller, cooperative satellites has been sincerely considered for
various space missions to achieve goals that are not possible or very difficult to
accomplish with a single
clusters
of
imaging.
For instance, one possible use
satellite.
satellites flying together in
In this case, groups
for nanosatellites is
formation for high-resolution, synthetic-aperture
of nanosatellites are operated cooperatively
sparse aperture with an effective dimension larger than can be achieved
larger satellite.
The system adds
flexibility since the formation
and orientation are adjustable on
orbit.
[Ref.
14] Flying
to act as a
by a
single,
and therefore aperture size
many
satellites in
formation
presents flexibility to mission designers given that the individual satellites will be capable
of repositioning themselves with respect
to each other to achieve diverse tasks.
By
accurately computing the preliminary Keplerian orbit elements, the satellites can realize
the desired close separation and cluster orientation desired to operate the necessary
missions.
In particular,
formation flying
NASA
(MSFF)
buzzword denoting
and U.S. Air Force have identified multiple spacecraft
as an enabling technology for future missions.
this division
of labor among smaller
as Distributed Satellite Systems
NASA's
(DSS)
Mission to Planet Earth and
benefits of satellite formation flying
in
New
DoD,
satellites is
NASA
referred to
sectors.
Millennium programs have acknowledged the
and have incorporated an enhanced formation
a sophisticated formation for
Earth's magnetosphere, the Orion program
in formation flying,
becoming
and the commercial
experimentation in the Earth Observing System mission. [Ref. 16]
Agency has engineered
[Ref 15] Another
its
flying
The European Space
Cluster mission to study the
intended for a low cost display of GPS uses
is
and the Laser Interferometer Space Antenna (LISA) mission
heliocentric formation-flying mission intended to identify gravity waves.
Force Research Laboratory's TechSat 21 program
27
is
The
US
is
a
Air
a technology demonstration of the
The TechSat 21 program was reviewed
'virtual satellite' concept. [Ref. 17]
in Section II
(above).
Advances
nanosatellites masses of less than
3
a mass of order 10"
may have
large
MEMS
in miniaturization using
numbers of such
1
kg
kg.
In
some
technology leads to estimation of
instances, simple satellites-on-a-chip
A particular original
(picosats).
concept
is
the use of
nanosatellites/picosatellites to shape constellations to permit the
real-time acquisition of distributed information. For example, a spherical constellation of
nanosatellites has
been proposed
to provide a real-time, three-dimensional
magnetosphere. Each nanosatellite can be thought of as a
data
To supply
set.
view of the
of a three-dimensional
'pixel'
high-quality spatial and temporal resolution for such a mission, large
numbers of nanosatellites
are necessary.
Because these ultra-low mass
can be
satellites
passive sensors without active orbit control, environmental effects such as air drag will
form the evolution of the constellation. For
of order
0.1
kg or
less, constellations
MEMS
fabricated nanosatellites, with a
could include several thousand elements.
Other concepts have envisioned large numbers (>10
deployed
from a dispenser
to
mass
3
)
of nanosatellites
provide a continuous planar ring of
to
satellites
be
for
Such constellations would be fashioned by dispensing
communication purposes.
nanosatellites over a range of orbit radii at the identical inclination to induce differential
azimuthal motion, therefore forming a homogeneous ring.
satellites in basically
locations,
communication
Furthermore, the constellation
for on-orbit failures.
eventually remove
random azimuthal
all
communications, where the constellation
is
vehicle. Analysis
[Ref.
to
1
8]
The
make up
shows
that
constellation
for losses
400
random
links
large
grow
A
orbits to
Formation flying clusters of
performance during times of
be robust
are attractive for military
related concept requires clusters
and
air
-95%
coverage of the globe.
28
nanosatellites
drag removal.
satellites also
satellite failure.
of
piggyback launches on any available
would require the continual deposition of new
failures
to
short lived because air drag will
nanosatellites can provide
from on-orbit
number of
positioned from a single launch vehicle and
formed to sustain a dedicated remote operation.
nanosatellites to be launched into
is
Such concepts
the nanosatellites.
With a
provide for graceful degradation of
If a single large satellite has a
system
failure, the entire
mission
satellites in the cluster
The
is at risk.
may
If a single satellite in a cluster fails, the
maintain mission objectives
be brought back up
cluster could then
to
at a
particular
lower performance
level.
mission design specifications or even
improved with the addition of another inexpensive replacement
The following chapter
remaining
satellite.
investigates several satellite formation-flying designs, with
emphasis on the projected circular formation.
These
cluster formations are
described with regard to their basic orbital equations of motion and then are populated
through computation from a technique derived from the Aerospace Corporation. [Ref.
A MATLAB
program was written
position and velocity vectors,
simulation provides an
discussed in
AIAA
satellite applications.
motion
for
Earth
Interferometric
imaging are
Satellite Constellations
Imaging [Ref. 20], and in the following sections four
satellite
in-track , circular , projected circular) are obtained for a variety
The
of
desired trajectories are carefully designed natural orbits so that
the energy costs to fly along these trajectories
relative
basic formation
OF MOTION OF SATELLITE FORMATIONS
paper 98-4379, Optimization of Geosynchronous
( in-plane ,
Satellite
STK. This
look into the orbital dynamics and a brief look into
formations that are optimal
for Interferometric Earth
formations
in turn entered into
LEO cluster with the A V required to maintain that cluster.
INITIAL EQUATIONS
Satellite
9]
determined both the mothersat and eight remotes'
which were
initial
perturbations that affect a
B.
that
1
is
minimized.
formation flying designs can be derived from the linearized equations of
for
two objects under the influence of a point mass
These equations are commonly known as Hill's equations.
21], a detailed derivation
of Hills's equation
is
From
gravitational field.
Vallado's text [Ref.
presented, which take the following form
for unperturbed motion:
x-2o)y-3a) 2 x =
y + 2cox =
z
(3.1)
+ co 2 z =
29
Here, x
is
the radial difference between the
is
two
objects,
the cross-track difference (see Fig. 3.1 below),
and
y
is
co
the along-track difference, z
the
is
mean motion of
the
reference object.
Relative Orbit
Reference Orbit
Hill's reference frame for satellite relative motion.
Figure 3.1
The unperturbed version of
Hill's equations
can be solved
analytically, with the
solution being:
x(t)
=
^
sm(cot)
+ (3x + -^-) cos(fi>f ) + (4x + ^_)
CO
CO
y(t)
=
2x
—
2-
cos(a>/)
4v
+ (6x + -Z±) sm{cot) - (6cox + 3y
CO
z{t)
CO
CO
= -^sin(ft)/') + z
)t
2xG
-+y
(3.2)
CO
cos(<y/)
co
In order to avoid secular, or long-term, growth in the relative motion, Eq. (3.2) needs a
constraint
on
its
y = -2x
secular term to be set to zero:
co
(3.3)
30
It
can then be shown that
this constraint results in a displaced orbit
and thus the same semi-major
By enforcing the
displacements.
x(t)
=
x
—
axis, as the reference orbit
sin(&>0
+x
through
with the same energy,
first
order in the small
become:
constraint, Hill's equations
cos(cot)
CO
y{t)
=
2x
—
-
- 2x
cos(cot)
2x
°-
sin(cot)
z(t)
=
—
+ yQ
(3 .4)
CO
CO
sin(<yf )
+z
cos(cot)
CO
where terms with
'0' subscripts refer to initial conditions.
These equations provide
for
and along-track
(y)
formation flying design.
In the above equations,
components of motion
can be seen
it
are uncoupled
that the radial (x)
from the cross-track
motion. The motion in the radial/along-track plane can be
eccentricity (e
flying designs
= 0.866) with an
must project
to
be an
ellipse
relative
of fixed
All unperturbed formation
motion into the radial/along-track plane.
In the linearized model, the cross-track (z)
Combining
simple harmonic oscillator.
shown
arbitrary along-track offset.
this elliptical
component of the
(z)
component of the
the elliptical
motion
relative
motion
is
a
in the radial/along-track
direction with the oscillatory motion in the cross-track direction yields the family of
ellipses that describe all formation-flying designs. [Ref. 14]
Six
conditions must be specified in the solutions to Hill's equations. These
initial
conditions can be thought of as Cartesian or Keplerian element differences
initial
between the two
satellites
on the formation.
from the solution
One
and allow for
constraint
was
of the
specified
to Hill's equations (see
requiring the semimajor axis of the
offset
six constraints or design parameters to
elliptical
motion
two
when
the secular terms were
Eq. (3.3) above).
satellites to
of relative motion.
be equal. Another constraint
in the radial/along-track plane,
y
.
is
the
This leaves four design
initial
location in the
These also can be thought of as the size of the
31
removed
This can be thought of as
parameters, which describe the size, eccentricity, orientation, and
ellipse
be placed
ellipse in the
radial/along-track plane, the initial location within that ellipse, the amplitude of the
and the
oscillation in the cross-track plane,
initial
location in the cross-track oscillation.
Basic Remote Motion in the Formation
1.
The remote formation,
center of the formation
a
is
as
mentioned above,
satellite flying in
which then has a moving frame attached
to
direction
x and y
orbit,
defined as 'mothersat',
directions are
shown
in Fig. 3.2
pointing upward orthogonal to both the x and y axes.
is
this coordinate system, the
The
Moving Frame
Figure 3.2
The z
derived from Hill's equations.
a circular
Its
it.
is
dynamic equations of a remote,
Under
'Sat2', are:
_2-]" /2
2
..2
x-2coy-co\r +x){l-r J [(r +xy+y-+z
} = v.
.
z
y + 2o)x-co y{\-r
3
z + 6T r z[(r + x)
Under
the
2
+y +z
moving frame,
xd (?) ~
(*o
I
yd if) - (2x
zd (/)
2
= yJ3(x
]~ Yl
}
=v
(3.5)
:
= v3
+ x cos(wt)
cos(cot)
1 co)
2
z
the desired trajectory of Sat2 in the circular formation satisfies:
<y ) sin(<2tf )
1 co)
+xy+y +z TA
z
[(r
,
- 2x
sin(cot)
sin( wt)
(3.6)
+ v3jc cos(wO
32
The desired path
is
a nonthrust trajectory for the linearized dynamics of Eq. (3.5).
x and x
free variables,
fixed frame, the path
r
are the initial value of x
and the derivative of x.
The
In the inertially
is:
= 2yjxllo) 2 +x]
(3.7)
All formation designs that follow will be constrained to these equations of
As
motion of the remotes (Sat2s).
will
be seen, most parts of these equations can be
canceled out, or simplified.
In-Plane Formation
2.
The in-plane formation
of
satellites
occupying the same
equations, setting
The solutions are
x{t)
all initial
the simplest
is
orbital plane
of all cluster designs.
It
consists of a group
and separated by mean anomaly. In
conditions, except for y
,
Hill's
to zero, represents this formation.
then:
=
y(0=y
(38)
z(t)
=
where y represents the amount of in-plane spacing between two
related to the
mean anomaly
satellites.
This can be
separation by:
AM = ^_
(3.9)
a
where
AM
is
the
mean anomaly
separation and a
is
the semimajor axis of the orbits.
Again, this formulation being based on Hill's equations assumes that the orbits are
circular, are at least nearly so.
The primary advantage of
simplicity in design, deployment and control.
33
the in-plane formation
is its
In-Track Formation
3.
The
in-track cluster design is a special case
satellites all share the
same ground
different orbital planes separated
To do
track.
by
of the in-plane formation. Here the
so, the satellites
right ascension
have
to
occupy
slightly
of the ascending node (Q), which
accounts for the rotation of the earth. The difference in the equations can be seen by the
addition of a cross-track oscillation that represents the difference in right ascension of the
ascending node. The solutions to Hill's equations are then:
x(t)
=
y(t)
=y
z(t)
= —-sin(i)y
(3-10)
cos(cot)
CO
where y represents the amount of in-plane spacing between two
motion of the
Earth.
The
orbits,
i
is
the inclination of the orbits, and coe
trailing satellite is
relate this difference in
some time
mean
the rotation rate of the
behind the lead by some difference in mean anomaly;
mean anomaly to
a difference in time
nodal difference AD. such that the second
satellite at
is
satellites, co is the
in the future (t
+
At).
satellite is
(i.e. At).
Then
calculate the
over the same point as the lead
The equations take
the following form:
AM = ^-
co
aco
(3.11)
AQ = co„At = -co
^0
aco
z
=
.
zo
a sin(z)
-sm(i)y
co
34
AM
where
the
is
mean anomaly
separation and
attractiveness of the in-track formation
the
same exact spots on
example screenshot from
the ground,
STK
is
AQ
is
the nodal
The
separation.
that each satellite in the formation passes over
which
is
An
valuable for Earth sensing missions.
for an in-track formation is
shown
in
Appendix
3.1
.
Circular Formation
4.
The
one
circular formation is
each other.
It
in
which
satellites
maintain a constant distance from
can be derived from Hill's equations analytically or geometrically.
analytic approach takes the solutions to Hill's equations and determines relations
The
between
the initial conditions given the constraint:
x
where
2
2
2
+y +z =r
(3.12)
r is the radius
fact that the relative
From
2
of the formation.
The geometric approach takes advantage of
motion in the radial/along-track planes
the
(x/y) is fixed in eccentricity.
either approach, the following relations are found:
2
.
CO
y = -2cox
(3.13)
z
= ±V3x
i
= ±V3;*:
where the
first
constraints
show
Both
two conditions
that there are
set
the along-track offset and drift to zero.
two planes
in
which the
circular formation is possible.
intersect the cross-track/along-track plane along the along-track axis but
inclined 30° to that plane and the other
is
inclined at -30°.
the following solutions to Hill's equations are found:
35
These
If the
30° case
is
one
is
chosen,
x(t)
= —sin(a>t) + x
cos(cot)
CO
y(t)
=
2x
—
-cos(cot)-2x
sin(cot)
(3.14)
CO
z(t)
= >j3—sm(cot) + y/3x
cos(cot)
CO
Note
that the formation design again has
two
free parameters:
parameters specify the radius and phasing of one
The four other
reference satellite.
along-track
x
and x
satellite in its circular
.
These
free
path about the
conditions were constrained for eliminating
initial
eliminating the along-track offset, and setting the eccentricity and
drift,
orientation of the ellipse of relative motion.
If the initial conditions
are formulated in terms
of the solutions
to Hill's equations presented in Eq. (3.14)
of the cluster radius and phasing, the following equations
arise:
xn =
^3 tan
2
(#)
+4
(3.15)
2
rco
2
where r
is
y]3t<m (0)
^3 tan
2
(0)
+4
6 is
the radius of the formation, and
the phase angle of the satellite measured
clockwise from the cross-track(z) direction.
From Equation
radius and phasing,
it
(3.14)/(3.15), given a circular-reference orbit
is
and desired
cluster
possible to convert the reference elements from Keplerian to
Cartesian, after coordinate transformations
add the Cartesian differences
to
get the
position and velocity vectors of the satellites in the circular clusters, and then convert the
Cartesian elements back to Keplerian for
The Keplerian element
on the phasing and
cluster, there will
initial
all
the satellites in the cluster.
differences for a circular formation are highly dependent
conditions of the reference orbit. For two arbitrary points in the
be differences in inclination
36
(/),
right ascension
of the ascending node
(Q), argument of perigee (w), and
will
mean anomaly
{M).
Generally, satellites in a cluster
have the same eccentricity except in the case where the reference orbit
There are conditions, however, where two
satellites in the cluster
is
not circular.
can have the same
inclination or right ascension of the ascending node.
The
circular formation has
two properties which make
attractive:
it
1) the
satellites
maintain a constant distance from each other, enabling precise measurements for use in
missions such as interferometry, and 2) unlike the in-plane and in-track formations, the
circular cluster presents a
two dimensional
array, increasing
dimensional resolution for
imaging, geolocation potential, and numerous other missions.
from
STK for a circular formation with
four remotes
is
shown
in
An example
Appendix
3.1
screenshot
.
Projected Circular Formation
5.
The projected
circular formation is very close in design to the circular formation.
This formation will be covered briefly here in this section, and in more detail with
another example of equation setup in Section
D
(satellite
formations with constant
apparent distribution - see below).
The
difference
is that
the projected circular cluster only maintains a fixed distance
in the along-track/cross-track (y/z) plane.
formation
is
to say that
track/cross-track plane,
when
it
Another way to describe the projected circular
the ellipse of relative motion
produces a
circle.
is
projected onto the along-
This results in the following constraint:
2
2
2
y +z =r
where r
is
(3.16)
the radius of the projected circle.
Like the circular
cluster, the constraint
the solutions of Hill's equations to produce:
37
can be applied to the
initial
conditions to
=—2 x
.
y
o
CO
y = -2cox
(3.17)
2
= ±2*
i
= ±2x
where the
first
two conditions
set the along-track drift
planes in which the projected circular formation
is
and offset to zero. There are two
possible.
track/along-track plane along the along-track axis but one
plane and the other
is
intersect the cross-
inclined 26.565° to that
inclined at -26.565°
26.565° case
If the
is
Both
is
chosen, the following solutions to Hill's equations are
found:
x(t)
= —sin(cot) + x
cos(cot)
CO
y(t)
=
2x
—
-cos{cot)-2x
(3.18)
s\n(cot)
CO
z(t)
=
2x
—
^-sin(6tf)
+ 2x
cos(cot)
CO
The formation design again has
the
same two
free parameters,
which represent the radius
and phasing of projected circular path.
If the initial conditions
of the solutions
to Hill's equations in Eq. (3.17) are
formulated in terms of the cluster radius and phasing, the following
arise:
xo =-cos(0)
(3-19)
x
=—
sin(^)
38
Like the circular formation design, the Keplerian element differences show up in
of the ascending node, argument of perigee, and mean
inclination, right ascension
anomaly, and the variations are dependent on the reference orbit and phasing.
primary advantage of the projected circular cluster over the circular cluster
distance
separates
the
satellites
when
formation
the
is that
The
a fixed
projected onto the along-
is
track/cross-track plane. This has applications for ground observing missions.
REMOTE CLUSTERS WITH CONSTANT APPARENT DISTRIBUTION
C.
The work presented
one
-
in this section continues the
the problem of creating a cluster of satellites that
nearly constant, shape and size
to
work
have a constant apparent
would have a
satellites
(as
when viewed from
opposed
started at the
end of the
would maintain a
the Earth.
Such a
to physical) distribution. In
clear field of view of the surface, and
cluster
constant, or
would seem
such a formation,
would remain
last
all
in sufficiently
close formation to share their information. Such formations are of interest for large
distributed-aperture sensing, for example. Another possibility is that of forming clusters
from many small, inexpensive
satellites,
computing power. Remaining
in a close formation
information and computing ability
For such missions,
positions.
It
it
each with a particular type of sensor and some
among
know
satellites to share
themselves.
might be unnecessary
might be sufficient to
would allow the
to
maintain extremely precise relative
the relative position accurately, and to remain in
close enough proximity to allow intersatellite communication. In addition, an effect of
constant apparent distribution
their
ground
targets
Since
it
is
that the angular dispositions
of the
satellites relative to
and each other would be constant.
has been mentioned that one of the desires of nanosatellite designs
is to
reduce the need for station-keeping thrust, orbits in which the natural motion of the
satellites
keeps them in a cluster are required. The need for station keeping would then be
reduced to eliminating the effects of perturbations on the array (covered below in Section
E).
These
of the
effects will
be found
satellites in the array.
in the bulk
For missions
motion of the array and
in
in the relative
motions
which precise knowledge of position
39
is
the
may be
primary goal, some cyclic perturbations
acceptable if they are sufficiently small
and well understood.
Problem Description
1.
In
what follows, the motion of a
moving reference
point and as seen
satellite will
be described with respect both to a
by an observer on
the surface of a spherical planet.
The motion of the reference point can be visualized by thinking of it
(possibly hypothetical) reference satellite
on a
circular orbit
as the position
of a
of radius R, which will be
referred to as the reference orbit. This reference point will serve as the origin of a local
coordinate system. This reference frame
work on
the motion of the
Let
moon
was
first
described by Hill,
who
derived
it
in his
about the Earth.
R be the position of the reference satellite (mothersat),
following the reference
orbit
under idealized two-body motion, and r the position of a nearby point (the remote).
Both
R
and r are expressed
Hill's coordinate frame
aligned with R.
in
an
inertial
moves with
The y-axis
is
frame centered
tangent to the reference orbit
illustrated earlier in Fig. 3.1.
moving reference
Hill's
the reference point, and
z-axis completes the orthogonal set, as
Thus, the reference point
frame allows the motion of the actual
It
will
be shown
create an orbit; this orbit will
(passing through the plane at
the relative motion as
is
The
at
is
also the origin of the
frame.
to the reference point.
position
the planetary center of mass.
the reference point, and rotates such that the x-axis is
positive in the direction of the orbital motion.
was
at
it
that the
be called the
some
satellite to
motion of the
relative orbit, as
be described with respect
satellite in this
shown
The primary
arbitrary angle).
frame will
in Fig. 3.3
interest
below
however,
is in
appears to a fictional observer on the planetary surface, whose
always on the ray connecting the center of the planet to the reference point.
This observer represents the point on the surface that
is
the subject of observation
by the
satellite cluster.
The concept of an apparent
planetary surface that
orbit
now
moves with time such
40
can be defined. Consider a point on the
that
it
is
always between the center of the
planet and the origin of Hill's frame.
relative to the reference point as seen
of sight from the viewer
orbit.
However,
of sight as
it
it
is
The apparent
from
this point.
to the satellite, there is
helpful to visualize
passes through the
y
-
it
orbit is the
As
motion of the
this is purely a matter
no physical meaning
as the trace
left
satellite
of the
line
to the apparent
by the intersection of the
z plane in the Hill's coordinate frame, as
line
shown
in
Fig. 3.3.
Relative Orbit
Reference Orbit
Figure 3.3
The
Reference, relative, and apparent orbits
analysis that follows will be primarily concerned with three angles that
describe the position of the satellite on the apparent orbit.
lateral
angle X, and the rotary angle ^are
shown, with
<f>
shown
in Fig. 3.4.
The elevation angle
The angles
are
all
increasing in right-hand rotation about the Hill's-frame x-axis.
41
k, the
positive as
Viewer
Geometry description and
Figure 3.4
The maximum value of X
the
maximum
the value
R - Rp
of the
altitude
Using
total
a:
orbit,
and
will denote the radius of the planet,
and
of the
will be referred to as the angular width
as the angular height.
will be called the altitude
Rp
of the reference
orbit.
This
is
also the
mean
satellite cluster.
this definition, a perfectly circular apparent orbit
would
result in a constant
angle between the line of sight and the line joining the viewpoint to the origin of the
The apparent
Hill's frame.
Rp.
value of
definitions
The
orbit can then
difference in this formal
be calculated by multiplying the angles by
method and the
R—
visualization suggested above
is
its
mean
semimajor axis and eccentricity of an
orbit.
extremely small for clusters in which the cluster radius
is
small with respect to
altitude.
The symbols a and
The variable
for
an
co is the
e will denote the
mean motion
elliptic orbit is
or average angular motion of a body on an orbit, and
given by the relation
co
=
yj/j I
a
3
where
,
/j.
is
the gravitational
parameter of the central body (the mass of the planet times the universal gravitational
constant).
The
periapsis, as
shown
circular orbit,
orbit planes.
this line
true
anomaly
(v)
of a point on an eccentric orbit will be measured from
in Fig. 3.5. In this case, the reference plane will be the plane
and the
line
of nodes will be taken to be the
The argument of periapsis (w) of the
of nodes. The inclination will be denoted
42
line
of intersection of the two
eccentric orbit will be
i
and
will
of the
measured from
be the angle between the two
orbit planes. Finally, the angle u
Note
vector.
=
w+v
is
the angle
from the
line
of nodes
to the radial
that this is defined for circular as well as eccentric orbits.
Periapsis
Line of
Nodes
Reference Orbit
Rising
Node
Orbital elements of the eccentric orbit.
Figure 3.5
Linearized Approach
2.
The Clossey - Wiltshire (C-W) equations
motion of a
satellite
reference frame as
Remember
is
an accepted choice for describing the
near a circular reference orbit [Ref 21]. They are defined in Hill's
in
Fig.
(from Section
B
)
3.1,
and are generally referred to as Hill's equations.
the equations are written:
x-2o)y-3co'x =
y + 2cox =
z
It
+ co
2
(3.1)
=
-
follows that the out-of-plane motion
z(t)
where
Az
slightly
= A:
is
cos(a>t
the
more
is
given by:
+ <p_)
magnitude and
difficult.
(3.20)
^
is
a constant phase
shift.
Following usual practice, integrate y to
y = -2o)x + k
Solving for x and
y
is
only
get:
(3.21)
43
where &
is
a constant of integration. Substitute this into the x equation to get:
2
x = -a) x + 2o)k
This
is
directly solved to get:
x{t)
which
(3.22)
+
cos(cot
+ 2klcQ
x)
(f>
(3.23)
substituted into Eq. (3.1) and integrated to get:
is
y{t)
where c
= Ax
= -2Ax
is
sm(cot
+
)-2>kt
(f>x
+c
(3.24)
the constant of integration.
Because a purely cyclic motion
method, k
orbit for this
=
0; c is
is
requires with the
same period
as the reference
simply an offset term and can be given the value zero
without loss of generality.
Only
the orbit as
it
appears from the surface of the planet
upon. In keeping with the assumptions
assumption
orbit
is
made
= -2AX
what
is
to linearize the equations
being focused
of motion, the
that the variation in r is negligible with respect to the altitude
The appearance of the
y{t)
made
is
sm(cot
+
orbit will then
be
its
projection on they
-
of the
z plane. This shows:
(/>
x)
(3.25)
z(t)
=
A, cos(cot
Setting
<j>
x
=
+
<f>,
the initial conditions,
Setting fa
=
fa
+ n
<f>z
it
)
results in a perfectly elliptic projection.
is
possible to set
A z = 2A X and the
The statement
projection assumes that the viewpoint
is
on the
properly specifying
apparent orbit becomes a
also results in an elliptic projection,
projected motion will be reversed.
By
however the
circle.
direction of the
that the apparent orbit is the y-z
line connecting the center
of the planet to
the center of the relative orbit.
Because the
many
initial
angle fa in Eqn. (3.25)
satellites in a circular
some constant
angle. This
is arbitrary, it is
possible to place as
apparent orbit as desired, each separated from the next by
would give a "pinwheel"
44
effect
from the planetary surface as
the satellites rotate about the center of the cluster.
The
cluster
would spin about
its
own
center once during each orbit of the planet.
The
radius of the apparent orbit
cluster of an almost arbitrary
number of
reference satellite, as in Fig. 3.6.
is
also arbitrary. Thus,
satellites,
The period of
it
is
possible to create a
by creating concentric rings about the
the apparent orbit
reference orbit, so that the entire cluster will maintain
its
is
always that of the
shape relative to the center.
Thus, separation angles such as fa in Fig. 3.6 should remain constant.
z
V
o
Figure 3.6
As
the
Apparently circular cluster of satellites.
C-W equations are linearizations,
describe the relative motion of the
orbital equations
satellite.
it
is
to be expected that they
do not
Examining the higher-order terms
can make an approximation of their
This
error.
is
fully
in the
the subject of the next
section.
The height of
Thus,
this
shown
to
be independent of the width.
technique can also be used to generate a relative orbit that
circular in actuality,
3.
the apparent orbit can be
and which appears
elliptic
when viewed from
is
very nearly
the planetary surface.
Orbital Mechanics
The motion of
the satellite described
by the
C-W
differences in the orbital parameters of the satellites.
45
The
equations
is
caused by slight
satellite in the elliptic orbit
moves more slowly than
It
the reference
when
at apoapsis,
thus seems to lag behind and then to catch up.
and more quickly near
When combined
periapsis.
with inclination, this
leads to an apparent orbit about the reference satellite.
Investigating the apparent orbit in terms of the orbital elements
forward as using the linear equations, but
limits
this
is
approach allows an investigation of the
of the linearized approach, and an estimate of the corrections due
In this section, the eccentricity
not as straight
and inclination parameters required
to
to nonlinearity.
produce a circular
apparent orbit will be investigated.
In developing the ideas of this section, attention will be confined to orbits in
which periapsis
is
90° from the line of nodes. This ensures the symmetry of the apparent
orbit about the Hill's-frame z-axis. (Recall that the line
of nodes as defined here will be
the line of intersection of the orbital plane with the reference plane.)
The
of the
description of the apparent orbit will be expressed in terms of the eccentricity
satellite orbit.
may be
These terms will be on the order of e\
thus, second-order terms in e
considered first-order corrections. In the following analysis, estimates for these
correction terms will be found.
Apparent Angular Width
a.
The needed
eccentricity is first
The width
particular width.
will
examined
to
produce an apparent orbit of a
be defined by projecting the
line
of sight to the
satellite
onto the reference plane. The angle between this projection and the line of sight to the
reference point
is
the angle
X
is
then the angular separation of the satellite and the reference point (this
in Fig. 3.4).
The width of the
If the inclination
orbit is then the
of the orbit
is
maximum
zero, the angle
is
value of this angle.
easily
computed, taking
the reference orbit and the eccentric orbit to have equal semimajor axes and thus equal
periods.
The motion of
eccentric orbit.
the reference point
The separation
at
is
any point
then equivalent to the
is
then the difference between the true
anomaly v and the mean anomaly M. Assuming the
46
mean motion of the
inclination
is
small enough that
its
of some information. The expansion as given
effects are negligible then allows extraction
in
Vallado [Ref. 21]
is:
v-M = 2esinM + ±e sm2M +
2
Finding the
maximum,
(3.26)
...
taking the expansion through second order, and taking the
derivative with respect to
M
results in:
4ecosM + 5e 2 cos2M=0
For e
—
>
0, this leads to
(3.27)
values of
M approaching
through by e allows Eq. (3.27) to be rewritten
nil. Setting
M = nl 2 + 6 and dividing
as:
-2sin£-5ecos2£ =
Again, e
8
is less
-
gives
5=
0,
(3.28)
and for very small
e,
important than the knowledge that 8
Using these
results,
have 8
is
of order
Eq. (3.26) can
approximately -5 el
is
2.
The
sign of
e.
now be
written:
2
v-7W = 2esin£-{e sin2£ +
Because 8
is
small, cos
8~\
(3.29)
...
and sin2S ~ 2b. Because 8
the right in the above equation
is
-e 2 and
is
of order
e,
on
the second term
can be ignored. Thus, to second order:
v-M = 2e
(3.30)
Thus, the angular width of the apparent orbit as seen from the center of the central body
is 4e.
From
the surface of the planet, the angular spread
is
wider. Letting
X denote
the
angle between the lines of sight to the satellite and to the reference, geometry gives:
A,
=
rsin(v-M)
tan"
(3.31)
yrcos(v-M)-R py
Expanding the radius of the
r
elliptic orbit in
M and
e,
again writing
M as nl 2 +
8, gives:
= R[l-esmS + (e 2 l2)(cos2S-\) + ...]
Using the expansions
for v
-
M and r as expanded, the
47
(3.32)
first
order approximation
is:
,
A=
IRcosS
R-R p
e~
B
and
2Re
R-R p
,
,
(3.33)
n
of the second-order term
that the coefficient
order. This gives a total angular width
is
of order
of the apparent orbit
5,
making
of:
2A*4e[R/(R-R p )]
For a low-altitude
orbit,
the term of third
(3.34)
Rp may be
nearly as large as R, and thus the angular spread of
the apparent orbit as seen from the planetary surface might be several times larger than
the value of v
-
M.
Correction for Inclination
b.
To allow
of the apparent
orbit
must be extended
between the projection of the
the eccentricity vector (that
'
(
rsinv
^
is,
,
it
_,ftanv
(3.35)
v C0Si
the apparent orbit
M of the reference
is
then the difference between this angle
orbit.
This angle will be maximized near v = nfl
argument of periapsis
is
w = njl
,
this
about this point, the small parameter s
to
v
and the projection of
shows:
= tan-'P=—
The width of
Consider the angle
the vector from the center of the planet to periapsis).
i^rcosvcos/y
and the mean anomaly
to include the inclination.
radial vector onto the reference plane
Letting this angle be denoted v
v = tan
compute the width
for orbital inclination, the expansions used to
.
Since the assumption
is
the
corresponds to the descending node. Expanding
=
v
- njl is
defined, and the angle corresponding
as:
tan£ =
=>£ =
tan
[atan^]
rcos£
48
(3.36)
where for brevity use a
of cos
in place
the projection of the radial vector.
This angle
i.
The expectation
measured from the
is
is
s » 0, so
that
line
of nodes to
that Eq. (3.36) should
be valid over the area of interest.
The expansion
2
a£+[(a + a
s =
for £is:
+0(s
3
/3]e
The angular width
is
5
found by introducing the expansion for v into
expression. At this point, the assumption
correct,
and thus that 8
is in fact
constant c at least through
Knowing
third order
combined
first
5
that
in the
(3.37)
)
made
is
that the earlier results
of about the same size as e and
that
is
two
of the order of e allows the disregard to
2
2
2
)S
-3cos£-13cos3J](a/12)£>
expansion
some
all
terms above
2
]e
3
(3.38)
3
+...
(ce) for
a and expanding
the trigonometric terms gives the
as:
£-S = 2e-S
2
2
2
e-(5 + c )Se /2-(c 2 +%)e 3
through third order combined. Through
there are
for
2
+ [\6(\-a )Scos2S-5sm2S](a/4)e +[32(l-a )cos S
= cos
& ce
variables. Thus:
2
i
were "nearly"
order.
£-S = (a-l)S + (l-a )aSy3 + 2acosS[l + (\-a
Substituting cos
i
this
first
(3.39)
order, again the
width
is
2e,
and once more
no second-order terms.
Apparent Vertical Size
c.
The apparent
between the
line
of sight
vertical size
to the satellite
of the
orbit is
determined by the elevation angle
and the reference plane. Relative
to the center
of
the planet, this angle can be expressed as:
kc =
sin "' [sin
m sin/']
(3.40)
49
where u
the angle from the line of nodes to the radial vector.
is
As a
of the
result
requirement on the argument of periapsis, the absolute value will be maximized for
u—± n/y.
maximum
kc =
to
For a circular
value of v -1
sin
there
is
maximum
requires that the
it is
value of
k
c
equal the
Thus, from Eq. (3.30):
.
v-M = 2e
(3.41)
e.
The
the lateral.
=i=
(sin/)
second order in
M
orbit,
vertical angle as seen
However,
maximum
as the
from the surface of the planet
vertical spread occurs at the
is
widened, as
is
extrema of the radius,
a first-order contribution of e to r that must be examined.
When
the satellite
is
apparent elevation angle can be
at the apses, the
written as:
(
k=
i?(l-f-,?)sinz'
'
tan
R(l +
where s is equal
at periapsis.
tan
magnitude
in
Expand about s=
-
f
k=
(3.42)
£)cosi-R p
R cos i — R p
positive at apoapsis
and negative
RRsini
(R-R p ) +R
2
.
is
to get:
\
.
R sin i
'
and
to the eccentricity,
2
sin
£+
2
(3.43)
...
i
Recalling Eq. (3.41) and making small angle approximations, the
first
term reduces
K^2R-e/(R-R p )
The second term
is
assumption
is
made
Rp approaches R,
is
that
the
coefficient
that the product
Re
I
k.
of the correction contains
(R
-
Rp)
is
The
I
and Rp.
it
is
An
when
seen that the correction term
(R- Rp).
coefficient of the correction term
at periapsis
R
small. This is of concern only
as for a low-altitude orbit. In this case,
of second order in Re
of the angle
(3.44)
the first-order correction to
Note
to:
and a contraction
at
apoapsis.
50
is
negative,
The
effect
showing an expansion
of this could be
to raise
the apparent orbit slightly with respect to the reference plane. If in fact the apparent orbit
is
elevated, there should be a corresponding elevation at the points of
maximum
lateral
spread.
To
orbit at
investigate this, a value
which the apparent
is
orbit achieves
it
required for the position of the satellite on
maximum
width. Return to Eq. (3.38) in an
attempt to find this value. If the derivative of this equation
are
found that are of
first
taken, however,
order combined. Because the value of s -
show
information. Numerical investigations
is in fact
is
6,
no terms
and thus of
i,
is
second order, first-order terms are necessary to obtain any
defined only through
width
its
that the elevation
of the point of maximum
not linear with e [Ref 21].
Phase Separation
d.
When
point, they will
there
is
in
Apparent Orbit
more than one
satellite in
apparent orbit about the reference
be separated by some angle, constant
equations. In Fig. 3.6, the angle denoted
fa
is
to the accuracy
of the
C-W
such a separation angle.
The separation angle between two
satellites in the
same apparent
orbit is a
function of the angle between the lines of nodes of their orbits. Nominally, the separation
angle equals this angle.
The
separation will vary as a result of the second-order deviations
of the apparent orbit from perfect
circularity; the actual
amount of variance
will
be
dependent upon the separation.
Although
angle, the angular rate
will
sweep
same
its
this
of the
angle faster
as that for the true
makes
it
impractical to define the change in separation
satellite in its
when
it is
apparent orbit can be considered. The
at periapsis
(f>
is
than at apoapsis; the ratio will be the
anomaly rates:
*„M=[(l + *)/(l-«)r
where
satellite
(3-45)
the phase angle in the apparent orbit and the subscripts refer to periapsis and
apoapsis.
51
An
estimate of the total variation in
fit)
from that predicted by the
equations can be had by finding the value of the true anomaly
equals x/l
v=
.
This
is
accomplished through the same expansion used
M + 2<?sinM + (5e
2
/4)sin2M +...=>
v =7u/2
can be expressed approximately
— + tan"
=
R(y-M)
Recalling that
apparent angle as
two
i
M goes from
satellites in orbit
mean anomaly
earlier:
(3.46)
orbit,
measured from the
vertical,
as:
rsinwsin/^
'
the
+ 2e
through second order. The phase angle in the apparent
<f>(M =7r/2)
when
C-W
«
*^/2 + sini
(3.47)
sweeps through Ae
2e, this implies that the satellite
n/L to
3i
^A
less
than during the other half of the orbit. Thus,
about the reference, separated by a 180-degree difference in phase,
will vary in their relative positions
by ±4e radians during the
orbit.
Geometry of the Orbit
e.
The apparent
orbit described in the previous sections is in a sense the
projection of the relative orbit onto a plane normal to the radius of the reference orbit (see
The
Fig. 3.3/3.4 above).
from the
actual relative orbit
results gained in Section
The depth of
apoapsis, which
by
the apparent orbit
4Re,
the motion
is
As
from the reference point
maximum
^
A
first
As
motion
a result of
to the satellite will vary
satellite crosses the
times this amount
distance will be (to
This follows
between periapsis and
the first-order approximation of the diameter of
this implies that the relative
apparent orbital radius (when the
approximately
elliptical.
clearly the difference
angled 60° with respect to the reference plane.
actual distance
not circular, but
C above (Linearized Approach).
definition is 2Re.
is
is
(at
lies
this,
near a plane that
the variation in the
from a minimum of the
reference plane) to a
periapsis and apoapsis).
order in e) V5(/?
52
-R
)e
is
maximum
Thus the
of
actual
Example
4.
An example
linearized solution
orbital
is
presented.
To
tie all
the chapters together, the nanosatellite design's
parameters are used in this example.
As
has been mentioned in the preceding section, the appearance of the relative
of the angles between the
orbit is a matter
reference,
lines
of sight
to the satellite
and to the
from a point on the planetary surface along the vector joining the center of the
planet to the reference.
precise definition.
by the
of the correction terms on the
to demonstrate numerically the effects
altitude
For
To speak of
this case,
of the reference
The nominal
orbit.
satellites,
such as in Fig.
orbital altitude is
-1111
km
second-order effects, set the radius of the apparent orbit
spherical Earth.
The
cluster consists
of a single
ring,
such that their nominal apparent angular separation
is
at
3.7, in
apparent orbit about a
(600 nm).
To
exaggerate the
25 km. Additionally, assume a
with eight satellites equally spaced
45 deg.
Nominal
Angular
Separation
Figure 3.7
more
simply multiply the angles describing the relative orbit
Consider a cluster of Earth
central point.
the radius of the apparent orbit requires a
Ring of eight equally spaced
53
satellites.
45°
Using the
X = 25/1111 = 0.0225023
From
rad.
6378.1363 km, and £=7489.5457 km)
0.0225023 = 2 Re/(R
The
The
have the angular spread of the desired
results derived above,
Eq. (3.34) the equation works out (with
is
\0e
0.1
=
1.84
x 10
-4
which when multiplied by the nominal
204.07 m.
to the
10" 3 rad
the spread angle, so
first-order corrections to the vertical spread
Ak =
The shape of the apparent
be
Rp =
that:
-Rp )=* e=\.67 x
inclination angle
orbit to
i
= .09563 deg
= 2e =
of the
3
3.34 x 10" rad
= 0.1913
deg.
orbit are:
= 0.0105 deg
altitude
of the orbit produces an apparent
orbit is so close to circular that
it
shift
of
would appear so
naked eye.
D.
POPULATING AND MAINTAINING A CLUSTER IN
ORBIT (LEO)
The following
analysis
was used
to generate both cartesian
LOW EARTH
and classical
orbital
elements (Keplerian) that were entered into Satellite Tool Kit® (STK). Both in-plane and
out-of-plane clusters were developed and varying plane angles of the remotes were
looked
at.
some of the
Creating a Matlab program (see
final results
satellites
3.2 ) to calculate the data points,
of the different cluster formations can be found
For a cluster constellation such as
chosen so that (1) each
Appendix
satellite
that in
Fig
Appendix
3.8, the initial orbit
3.3 ,
elements are
occupies a node in an arbitrary spatial pattern, and (2) the
undergo a cyclic motion that allows the formation
AV for maneuvering.
in
Depending on the
be occupied by a mother ship or
the suborbit plane will maintain
may be
its
application, the center position in the cluster
empty.
An
arbitrary distribution
little
may
of spacecraft
in
configuration, and the suborbit normal will stay at a
fixed angle 8 relative to the mothersat orbit normal.
can
to persist with relatively
occupy the
disconnected
nanosatellites
characteristics
of a larger (km-scale) spacecraft.
54
suborbit
Large numbers of physically
plane
and
can
imitate
the
Normal
Vector
Mothersat Orbit
Normal Vector
Subsat Orbit
of
Subsat Plane
Mothersat
Orbit
adir Vector
Cyclic motion of Subsat Orbit
Figure 3.8
Populating a Remote Cluster
1.
Each subsat (remote) undergoes a
satellite (mothersat).
cyclic
motion
in the reference
The remote completes one suborbit cycle and
frame of a center
returns to
original
its
position relative to the mothersat after one revolution about the Earth, as illustrated in
Fig.
3.8.
Therefore
in the cluster
satellites
all
semimajor axis (and hence the same
The
cluster can
must have the same value
orbital period) if perturbing forces are neglected.
be populated based on the idea that each remote
o'clock position in the suborbit exactly once per revolution.
orbital velocity vector is taken to
for the
is in
At
the 6 o'clock or 12
this instant the
remote
be parallel to the mothersat velocity vector, but the two
velocity magnitudes are different.
No
further assumptions concerning the position
and
velocity of the remote at other points in the orbit are needed to determine the orbital
elements.
The
inclination
initial
i c,
mean anomaly
mothersat orbit
epoch
M
c o-
specified by the semimajor axis
Q
a c eccentricity
,
argument of perigee
wc
Three parameters that specify the cluster geometry
(p,
right ascension
at
is
of the ascending node
55
c,
,
ec,
and
rj,
S)
to
are illustrated in Fig 3.9.
The
scale factor
assigned the same distance value for
pattern generator
arrangement
rj
all
remotes
is
25 km), while the dimensionless
The angle £ specifies
and the orientation of the suborbit plane
Choosing
(i.e.
the overall cluster size of and
can be different for each remote and determines the geometrical
for circular rings, 77=1).
(i.e.
p determines
different values for
8
in
the shape of the suborbit
relative to the orbit plane
of the mothersat.
one cluster will produce a swarm having multiple
suborbit planes.
Remote
rs (12 'clock)
r|p sin5
Earth
rs (6 o'clock)
\
Remote
Remote Geometry
Figure 3.9
The following paragraphs
Astrodynamics textbook. [Ref. 21]
choosing a
set
Ec
of
as viewed along Mothersat Velocity vector
utilize
values (e.g. 0°, 45°, 90°, etc.) where
o'clock position (either
61
=2tan -i
+ e,
when
e„ sin
Ec
orbital
is
elements begins by
the eccentric
the corresponding remote
is at
anomaly of
the 6 or 12
populate the cluster). At these 'population points'
radius and orbital speed of the mothersat are:
(E„
tan
1-e.
M=E-
may be used to
mean anomaly,
l
and equations from Vallado's
Determining the remote
the mothersat at the location in the orbit
the true anomaly,
definitions
(3.48)
v
E„
(3.49)
56
1
(3.50)
+ e,
cos 0,
r
V
c=JV
/? c
p
r.
Q cJ
(3.51)
v
where
2
= a f (1 - e: )
population point
-
vs
=
are:
8) + (rc ± rjp cos Sy
y](rjp sin
f 2
jM
l>
rs
l
s
where the ± sign
is
(3.52)
(3.53)
\
it
3.9 the radius and orbital speed of the remote at the
2
rs
Here
From Fig
.
in
assumed
J
Eq
that
(3.52) is negative for the 6 o'clock and positive at 12 o'clock.
a s = a c for
all
remotes, but this choice will be modified
Using the above expressions together with the construction
velocity vectors in the mothersat perifocal coordinates
when
later.
in Fig. 3.9, the radius
a remote
is at
and
the population
point are:
COS0.
r.
=
r.
sin^
(3.54)
-s\r\6c
v„
rs
=
=
ec
+cos#
c
(re
± rjp cos 5) cos 6C
(rc
± rjp cos 8) sin 6C
±rjp sin 8
(3.55)
V,
=
V.
57
The magnitudes of these vectors agree with
the scalar values in Eqns. (3.50)-
Equation (3.55) expresses the key idea that the remote and mothersat velocity
(3.53).
vectors are parallel
Knowing
at the
population points.
that the radius vector
and velocity vector of a remote
orbit is sufficient for calculating its orbital elements (a s , es ,
is
to
transform
is ,
Q
s,
vv s ,
at
M
one point in
s0 ).
The
first
its
step
remote radius and velocity vectors from mothersat perifocal
the
coordinates to Earth-centered inertial coordinates as follows:
r*=*-r.
(3.56)
vp =-R-vJ
where R
is
a 3x3 rotation matrix
1
defined by:
R = R(Clc )*R(ie )*R(wc )
(3.57)
and with the three components identified
*(a)=
cos(-Q c )
sin(-Q c )
-sin(-Q c )
cos(-Q.)
as:
1
R(ic )
=
cos(-ic )
sin(-ic )
-sin(-/c )
cos(-*c )
cos(-wf )
*to) = -sin(-w
f
)
(3.58)
sin(-wc )
cos(-wc )
1
The second
step
is to
apply the set of equations relating the Earth-centered
inertial
radius and velocity vectors to the classical orbital elements. This procedure yields (a s , es ,
1
Matrix found in Matlab program in Appendix 3.2
58
Q w
is ,
s,
S
[1
=2 tan
E.
s
=E
The
•
S
is
the true
anomaly
anomaly and mean anomaly
eccentric
M
6 ) where 6
s,
mothersat
s
at the
-M»fr
-e sinE
s
(3.59)
(3.60)
s
mean anomaly
M
at
s
the
is
M
same
c0
=0 o
.
Using Eq. (3.49)
point, the initial
vv s ,
M
s0
M
c
at the
initial
population point and
orbital elements.
is
M
The
j0
=
M -M
s
c
,
resulting es ,
is ,
are distributed around the mothersat values, with the difference depending
on
the magnitude of/? and on the location
The
for
and hence the
remote mean anomaly
which completes the determination of the remote
s,
The corresponding
are:
cluster is initialized with the mothersat at perigee,
Eq. (3.51)for
Q
population point.
initial
on the remote within the
cluster.
subsat orbital elements can be refined to enhance cluster stability in the
presence of the non-spherical geopotential.
This
is
accomplished by calculating the
subsat displacements (relative to their original positions) after one orbital period of the
mothersat.
is
due
The most
to the
mean-motion
significant perturbation (which will be discussed in a later section)
second zonal harmonic
vP
Jj,
which appears
in the expressions for the
mean
and the secular rates of change of Q and w.
Propagating the cluster for one revolution in low-Earth orbit under the influence
of
1/2
reveals that the remote displacements are primarily in the mothersat along-track
coordinate (y-axis), with cross-track and radial displacements being at least 50 times
smaller.
This suggests that the semimajor axis of each remote can be adjusted slightly to
cancel the along-track displacement.
the
same
for all satellites in the cluster,
semimajor axis
as
In the unperturbed solution the
that
namely a s = a c
in Eq. (3.53).
compensates for along-track displacements due
=ac+ ^-
semimajor axis was
to J2
The revised
is:
(3.61)
3/T
59
where
AY
is
one revolution.
the along-track displacement after
semimajor axis
differs
from a c by
less than
5m
1111
at
km
Typically the revised
altitude.
Of course,
adjusting
a s does not reduce the cross-track and radial displacements caused by Ji.
t
E.
PERTURBATIONS AND A V REQUIREMENTS
1.
STK Perturbation
For
real
Propagators
geopotential and other perturbing forces
analysis the non-spherical
conspire to disrupt cluster coherency. Data from
propagated
the
orbits
and
presented
STK's numerous
based
influences
on
perturbation modelers
two-body,
and
Ji
J
a,
contributions, solar/lunar gravitational effects, solar radiation pressure, and atmospheric
drag for short term (3-10 days) and long term
a.
Two-Body,
A Two-Body,
gravity from the Earth,
the
year) investigation.
J2 and J
or Keplerian motion, propagator considers only the force of
which
is
modeled as a point mass. The two-body propagator uses
same basic technique outlined
technique assumes the Earth
(1
is
in the
two-body equation of motion development. This
a perfect sphere and the only force acting on a
gravity. This propagator doesn't account for
satellite is
any perturbations.
J2 Perturbation (first-order) and
J4 Perturbation (second-order) propagators
account for secular (long-term) variations in the orbit elements due to Earth oblateness.
These propagators don't model atmospheric drag or solar or lunar gravitational
J2
and
J4
are
zonal
harmonic
coefficients
in
an
forces.
infinite
series
representation of the Earth's gravity field. J2 represents the dominant effects of Earth
oblateness.
The even zonal harmonic
coefficients of the gravity field are the only
coefficients that result in secular changes in satellite orbital elements.
The
includes the first-order secular effects of the J2 coefficient while the
and second-order
of J2 and the
includes the
first-
coefficient,
which produces long period periodic
effects
60
J2 propagator
J4
first-order effects
effects,
isn't
propagator
of J4. The J3
included
in
either
propagator.
J4
oblateness.
Since the second-order J? and the first-order
there
is little
is
difference between the orbits generated
b.
J4
and
is
a result of Earth
secular effects are very small,
by the two propagators.
HPOP
HPOP
the
is
High Precision Orbit Propagator, and was the main
propagator for formation analysis for
can handle
J2
approximately 1000 times smaller than
this thesis.
circular, elliptical, parabolic
HPOP,
included as part of
and hyperbolic orbits
Moon
the surface of the Earth to the orbit of the
at distances
and beyond. As
name
its
STK/PRO,
ranging from
implies,
it
uses
a powerful propagation technique to incorporate sophisticated orbit perturbation models.
HPOP
uses a Runge-Kutta-Fehlberg integration method of order 7-8 to propagate the
satellite state in the
J2000 reference frame.
including the Joint Gravity
Earth's oblateness) with a
A variety of high-fidelity models are utilized,
Model 2 (JGM2; a highly
maximum
precise
model (70 X 70) of the
degree/order of 21
The atmospheric density model used the Jacchia-Roberts
(similar to
Jacchia-1971 but uses analytical methods to improve performance) to model drag effects
on the
spacecraft. This
model takes
into
account daily variations in the height of the
atmosphere due to solar heating among other parameters, and was based on values of
Cd = 2.0,
of 0.020
daily/average F10.7 of 150, a geomagnetic index of 3.0 and an area/mass ratio
m /kg.
2
Solar radiation pressure
was modeled using
C
r
=1.0
and the same
area/mass ratio as above. Since sunlight produces a small force on any exposed surface.
This force varies depending on
more
how
reflective the surface is
reflective than a black surface).
been updated
to
The
(i.e.
solar radiation pressure
a mirrored surface
model
in
be consistent with other commonly used propagators such as
Finally, third
body
gravity
models (solar/lunar
HPOP
is
has
GTDS.
gravitational effects) are
based on U.S. Naval Observatory data and are accurate to within 0.03 arc seconds.
Additionally, third-body gravitational perturbations are also
planetary ephemeris from
JPL
for the
Sun and Moon.
61
computed using the
DE
405
Perturbation Effects
2.
Propagations in the presence of perturbations show the circular formation to be
The primary
highly unstable.
The
oblateness or Ji effect.
factor disrupting the formation design
Jj contributions to the relative
motion are
magnitude larger than the disturbing accelerations including
the earth's
is
at least
an order of
tesseral resonance (for short
repeat ground track cases), atmospheric drag, solar radiation pressure, and third
gravitational effects.
Earth oblateness effects are most prevalent in the secular motion of
the right ascension of the ascending
anomaly
node (Q), argument of perigee
right ascension
of the ascending node.
Since the two orbital planes have slightly
for each orbit to precess at slightly different rates.
drifting apart
An
of the ascending
This results in the orbital planes
and a cross-track error growth.
additional contribution to the error growth
of apsides of the
the
and mean
cross-track error growth can be attributed to the difference in precession rates
different inclinations, the secular Jj effect causes the right ascension
node
(w),
(A/).
The
of the
body
This
orbits.
mean anomaly and
is
is
caused by the rotation of the line
close to causing the Ji effect to be equal but opposite on
the argument of perigee.
If the orbit is not circular (i.e. all the
remotes' orbit) the orbit line of the apsides begins to rotate and disrupt the formation
design.
Large variations in eccentricity and argument of perigee that occur in near.
circular
LEO
will disrupt the formation, but they can
mothersat into a 'frozen'
orbit.
be greatly reduced by placing the
This choice also simplifies the ephemeris representation,
&fc.
The NASA/JPL
stability
and maintenance
because here are no secular or long-period variations in ec and
TOPEX
mission has successfully demonstrated the long-term
of this type of orbit. [Ref. 22]
ec that
is
A frozen orbit requires w
c
determined by the choice of inclination and
satellites in the cluster (typically the
expected that the
total
AV
= 90° and a
altitude.
particular value
Although only one of the
mothersat) can achieve frozen conditions,
of formation keeping will be minimized for
62
of
it
is
this choice.
Periodic maneuvers must be performed
by
the mothersat to follow a reference orbit
having an altitude and frozen eccentricity. Typical the mothersat will have a higher areato-mass ratio then the remotes, and
compensation
rate
is
its
altitude will therefore
decay more quickly. Drag
applied so that the mothersat reference orbit matches the average decay
of all the remotes. This minimizes the fuel requirements for each remote.
For the long-term behavior (~3 years), solar radiation pressure and atmospheric
drag will disturb the frozen orbit conditions, because the eccentricity and argument of
perigee are no longer constant.
Small oscillations in inclination are
common
to the
mothersat and remotes and are not expected to disrupt the formation.
For the short-term behavior (1-10 days), effects of atmospheric drag, solar
radiation pressure, and the Jj to J& gravitational harmonics disrupt the cluster geometry.
The position deviations
indicate that frequent formation-keeping
maintain the desired separations
position deviations at
1111
among
km
the remotes.
and
altitude,
mothersat maneuvers. The cross-track deviation
Drag
this
is
is
is
maneuvers are needed
to
the largest contributor to the
compensated
efficiently using
caused primarily by differential nodal
regression:
HS
ts.Vm
where tm
orbit
<
=3^L
is
(3. 63)
the time between maneuvers,
decay rate in m/sec,
co is
the orbit
increment of each two-burn transfer.
summing
3.
362)
L
is
mean
The
the in-track tolerance in meters, q
motion, in rad/sec, and
total
AV
AVm
is
is
the
the velocity
for the remotes are obtained
by
the velocity increments for each maneuver.
Formation Keeping
The
circular
and projected circular formations were highly unstable and require
formation-keeping maneuvers to account for earth oblateness, atmospheric drag, and
tesseral
resonance (for short repeat ground track cycle orbits)
63
effects.
Both atmospheric drag and
The
track direction.
tesseral
resonance affect the formation in the along-
along-track error growth induced by these perturbations can be
controlled via small adjustments in the semimajor axis of the satellites.
Based on the
results
of the propagation, these semimajor axis adjustments are sub-centimeter for drag
effects
and on the centimeter level for deep tesseral resonance. From Gauss' variation of
parameters
plane, the
(VOP)
equations [Ref 23] for Keplerian elements in the normal-tangential
change in semimajor axis due
—=
at
to a disturbing acceleration
is:
aA
(3.64)
fj.
where a
is
V is
the semimajor axis,
constant for the earth, and a d
If a velocity
,
impulse
is
the
magnitude of the
velocity, ju is the gravitational
the acceleration in the direction of the velocity vector.
is
assumed, Eq. (3.64) can be rearranged to determine the
velocity impulse required to produce a desired change in semimajor axis:
AV,=-^-Aa
(3.65)
2a V
where the changes
to the
axis
in velocity,
AV, and semimajor
From Eq.
nominal values.
(3.65),
it
axis, Aa, are
can be shown
by 10cm, a velocity impulse of 0.00478 cm/sec
near-circular orbit.
is
assumed small compared
that to
change the semimajor
required for an 1111
maneuvers
will
altitude
Thus, the AV, and therefore propellant requirements, to account for
the differential drag and tesseral resonance effects will be quite small.
the
km
depend on the extent of the drag and resonance
The frequency of
effects,
formation
keeping error bounds, and several control related issues, such as the accuracy with which
these maneuvers are affected.
The
effects
contributions.
of Ji were seen
In terms
in the cross-track direction
of Keplerian elements, the J2 secular effects on right ascension of
the ascending node, argument of perigee, and
[Ref. 23], the
change
with indirect along-track
in right ascension
(cross-track) acceleration
mean anomaly
are of concern.
of the ascending node due
is:
64
From
to an out
Battin
of plane
dQ.
rsin(MH-v)
dt
h sin(z')
l
where r
(3.66)
dh
w
the satellite radius magnitude,
is
anomaly, h
is
the angular
momentum,
is
the argument of perigee, v
the orbit inclination, and
momentum
of the angular
in the direction
i
is
vector.
If a velocity
a&
is
impulse
is
the true
the acceleration
is
assumed, Eq.
(3.66) can be rearranged to determine the velocity impulse required to produce a desired
change
in right ascension
AVh =
h sin(z')
of the ascending node:
AQ
(3.67)
rsin(w + v)
where the changes are assumed
small.
The
differential oblateness effects
on
right
ascension of the ascending node can be derived analytically by taking the partial
derivative of the governing equation with respect to the inclination (taken from Vallado
[Ref. 21]):
2
3
2
(£]
\P )
wcos(z')
(3.68)
>
.
\
3
= — J2
2
dt
wsin(/)3/
<p J
Combining Eqs.
(3.67)-(3.68),
required to maintain nodal spacing
is
it
is
seen that the amount of velocity impulse
proportional to the size of the formation and the
length of the mission:
A^ = 5QA/
=
h sin(i')
*K
where
(3.69)
3
wsin(i)diAt
J,
rsin(w+\
2
p
j
8/ is the inclination difference in the formation
maneuvers. If the near circular assumption
AK„ =
wW(,')
a sin( w + v)
3
is
1-a-A/
and At
is
the time between
made, Eq. (3.69) simplifies
to:
(3.70)
2
65
The
AV
total
requirement
is
approximately ~7.2m/sec per kilometer of separation per
year for an 1111-kilometer altitude circular formation.
maneuvers must be performed
It
should be noted that the
at certain points in the orbit
to avoid disturbing the
inclination.
The second
disruptive influence of the earth's oblateness
flying is the rotation of the orbit line
on
satellite
formation
of apsides. Since the effects of J2 on the argument
of perigee and mean anomaly are nearly equal and opposite:
3
'**
4
kp
w = —J
•
co(4-5sin(i))
J
(3.71)
(
M
=-J
4
The
effects
2
D
N
2
2
\^-\ yll-e G)(3sm (i)-2)'
\P)
of accelerations
in the
normal direction are also nearly equal and opposite on
these elements for near circular orbits:
~17
at
= ~17 2e + - C0S ( V ) adn
eV\
a
\
—
dM =
at
J
(3.72)
A
—
~
eaV\a
-b (r
-cos(v)
\adn
J
In Eq. (3.72), only accelerations in the normal direction (mutually perpendicular to h
and v are considered since impulses
semimajor axis maneuvers
in the tangential (velocity) direction
in the cross track direction
do not
affect the
would
affect
mean anomaly.
Since the effects of these maneuvers are nearly equal and opposite on the
mean
anomaly and argument of perigee, the formation keeping analysis can be focused on
maintaining either one of the elements with maneuvers in the normal direction.
assumes
that the other
element will be maintained by those maneuvers as well. For now,
the argument of perigee
If
is
looked
at.
impulse maneuvers are assumed, Eq (3.72) can be rewritten in terms of the
amount of velocity impulse required
AVn
This
for a given
=-^—Aw
change
in
argument of perigee:
(3.73)
cos(v)
66
where the eccentricity
is
The required change
considered to be small.
in
argument of
perigee can be derived from Eq. (3.71):
Aw = wAt =
—
6>A/
J.,
4
-
(3.74)
AV =
n
eV
6
3
—--J
cos(v) 4
coAt
2
where the amount of A V required
1111
km
year.
It
is
again a function of the length of the mission. For an
altitude circular mission, the total
Av requirement
is
approximately 2.1m/sec per
should be noted again that the maneuvers must be performed
at certain points in
the orbit to avoid disturbing the eccentricity.
4.
Station Keeping
The primary
atmospheric drag.
orbit's
keeping concerns
station
For an
1 1 1 1
semimajor axis by close
drag are dependent on the
atmospheric density, which
km
formation
all
flying
designs
altitude satellite, atmospheric drag could
to 0.2 kilometers in
satellite
is
for
altitude,
one
year.
Of course
decay the
the effects of
area to mass, drag coefficient, and the
a function of the solar cycle.
A
high-density atmosphere
near the peak of the solar cycle was used in this analysis so the drag estimates
overly conservative.
is
may
be
Station keeping a 0.2 km/year decay in semimajor axis could
require approximately .097 m/sec of velocity impulse.
F.
OTHER TOPICS
Another topic
specifically important to formation
problems
is
reconfiguration.
Formations are designed based on the optimization of the various performance metric
functions for a given mission.
the mission requirements.
These performance metric functions are defined
meet
Various primary mission requirements, such as achieving the
best image quality or gaining the highest probability of detecting
different
to
performance metric functions. As a
67
result,
moving
targets, lead to
image or interferometry
qualities for
example,
is
closely related to the formation baseline and the distribution of the satellites
in the formation,
whereas the performance of moving
on the number of satellites and the
footprint.
To meet
target indication systems
depends
multiple mission requirements,
it
is
important for the formation controller to have the capability of easy reconfiguration.
Furthermore, if one
of the
satellite
has a malfunction in the middle of a mission, the adjustment
satellite distribution to
keep the system working, or the replacement of the
with the malfunction, requires reconfiguration of the formation.
Reconfiguration could
include adjustment of the relative distance between
satellites,
of a formation, the changing of the numbers of
satellites in the
combination of two formations flying closely.
68
satellite
reassignment of the leader
formation, and the
SUBSYSTEM DESIGN
IV.
To
two
incorporate a useable and attainable design for potential nanosatellite clusters,
satellite
subsystems show up as integral
to the proposal: 1) the attitude
and control subsystem, and 2) the propulsion subsystem.
satellite
bus are mutually coupled with
all
Although
all
determination
subsystems of a
other subsystems onboard, these
two systems
allow the close formation flying needed for the numerous cluster missions so far
mentioned.
The assumptions
component
offered in the remaining chapter's
selection are
derived from given reference payload requirements, the constraint of the previous
chapter's calculations of
AV
required over a course of a year to maintain the cluster
formation, and to minimize mass,
power and physical
Table 4.1 sums up most
size.
physical assumptions of the overall spacecraft.
Value
Characteristic
Satellite
shape and composition
Cylindrical and
Mass
10 kg
0.21m
Radius
Ixx,
homogenous
m
2
m
0.25 m
0.208 m
0.104 m
2
0.16 kg
hy
0.22 kg
hz
Height
Moment arm
(z-axis)
Moment arm
(x.v-axis)
Propulsion
& ADCS
35%
(3.5kg)
Mass Allowance
Table
4.
Assumptions made
for Satellite Physical Characteristics
69
A.
ADCS
The requirements of ADCS systems
are twofold:
Support formation flight
•
Manage
maneuvers
•
attitude/rates to the extent required to support precision propulsive
if available/ required.
Maintain pointing within small angles
(i.e.
±
.5° in
and maintain rates to within precise control
pitch and
roll,
±
1° in
yaw),
±0.1° per second) of the
(i.e.
desired rates about each axis.
•
Maintain attitude knowledge and spacecraft
stability at all times
Support ground communication
• Point
communication antenna
when within
sight
Attitude Control
exception.
Until
is
of ground
for
command
uplink and telemetry downlink
station
required for any satellite system and small satellites are no
now most
small
(i.e.
rudimentary attitude control systems.
Microelectromechanical Systems
and manufacturing techniques.
micro-/nanosats) used only very simple and
This can
(MEMS) and
New
now change
other small consumer electronics devices
control systems are
nanosatellite formation flying missions
with the development of
now needed
for missions such as
and space-based interferometry.
Nanosatellites represents a flexible tool to carry out scientific and technological
research in space.
Nevertheless, obvious limitations in size, mass, onboard available
power, and costs impose several constraints on the design of nanosatellite subsystems.
The
attitude control subsystem, as
particularly affected
by these
one of the more complex subsystems of a
constraints, especially
require stringent attitude control.
Therefore
new
when
satellite, is
the considered applications
solutions in terms of components and
operating logic need to be investigated to reduce costs, volume, and power requirements.
70
The Attitude Determination and Control Subsystem measures and
spacecraft's
angular orientation
(pointing
navigation, and control system, both
The simplest spacecraft
orbit).
its
direction),
or,
in
the
case of guidance,
orientation and linear velocity (which affects
are either uncontrolled or achieve control
methods as spinning or interacting with the Earth's magnetic or gravity
may
may
or
employ
not use sensors to measure the attitude or position.
controllers
to
process
the
spacecraft
attitude,
Because of
of particular
its
in
gravity
gradient
it
attitude control
be made from those
gravity, the Earth's
results.
is
stabilized
and procession control. [Ref. 24/25]
documented the
power
fields.
These
and actuators, torquers, or
momentum.
low mass and power consumption requirements, magnetic control
interest for small satellites, since
damping
acquisition,
attitude
by passive
Magnetic Control
1.
attitude
its
More complex systems
propulsion subsystem thrusters to change attitude, velocity or angular
is
controls the
extensively adopted also for active
of microsatellites, and a case for nanosatellites can then
field,
With regard
use of gravity gradient
and aerodynamic drag, to control the spacecraft
to this, various solutions
booms with eddy
magnetic damping rods to
damp
by
torques, such as those caused
by means of passive and/or semi-passive devices allows
savings.
attitude
Several authors have studied and
The use of environmental
magnetic
despin,
satellites,' initial
substantial
mass and
have been proposed based on the
current dampers, fluid ring dampers, or soft-
the satellite residual attitude motion. Nevertheless, these
solutions achieve poor attitude control accuracy (~ 5-10deg).
When
a finer control
is
required, as in the case of remote sensing applications (~ 0.1 deg), various configurations
of low-mass, low-power momentum/reaction wheels have been proposed. In
use of magnetic torquers for
momentum dumping,
as an alternative to
this case the
more
traditional
gas jets [Ref. 26], reduces the control system complexity and mass. Components
available for use in these situations will be presented later
below).
71
on
this section
( Section
A3
All-Magnetic Torquer System
a.
To develop
3-axis attitude control given the very limited
power and
weight availability on a nanosatellite, an all-magnetic torquer system where permanent
magnets on stepper motors could be used instead of traditional torquer
coils.
The
attitude
determination would be achieved by a combination of Earth horizon and sun sensors,
giving three-axis control to approximately two to three degrees.
Although
this
concept
does not provide fine control ability for most remote sensing applications, the idea
is to
progress the knowledge base by getting these nanosatellite systems into space and
start
operational testing and evaluation.
Reaction wheel / Magnetic
b.
Torquerod System
As
a last example, the attitude control system of the Italian Scientific
Microsatellite for
Advanced Research and Technology (SMART) was reviewed. Their
microsatellite attitude control system consisted of three small reaction wheels and three
magnetic torquers (torquerods).
The wheels were used
for three-axes attitude control
during station keeping. The reaction wheel design had been performed using a technique
that
minimized mass and power consumption. As
concerned,
it
was primarily driven by
unloading and
initial
far as the
magnetic torquer design was
the requirements for onboard wheel
attitude acquisition following the separation
momentum
from the launcher.
Nevertheless, the possibility of using the magnetic torquers for attitude control during
station keeping
was considered. Their technique was
attitude control
between wheels and magnetic torquers, which minimized the
to distribute the torque required for
total
power
consumption. The result of this example [Ref. 27] presented the analytical model of the
proposed technique and demonstrated
particular, the numerical analysis
•
effectiveness
by numerical simulations.
The
In
shows the following:
Wheels and torquerods can be simultaneously used
torque with
•
its
to realize a required control
minimum power consumption.
control torque distribution between torquerods and wheels
is
governed by the
torquerod control efficiency, which strongly depends on the Earth's magnetic
72
field variation
(where
R
t
is
along the orbit and on the torquerod design parameter
When more
to the torquerod
/
scf
supply current).
efficient torquerods are used, the control torque portion transferred to
the torquerods increases so that the total
•
t
the torquerod winding resistance and scf is the ratio of the torquerod
magnetic dipole
•
R
power consumption
minimized.
is
The use of torquerods with low values of the design parameter allows
power savings with
substantial
respect to the case of optimal control with reaction wheels
only.
•
As
the required torque percentage assigned to the torquerods increases, the
attitude control accuracy reduces.
Nevertheless, substantial
power savings can be
realized while retaining attitude control accuracy adequate for remote sensing
applications (0.016 deg for a
•
The
attitude control
40% power savings).
accuracy could be improved by slightly increasing the
numerical values of the control gains
•
The proposed
at the cost
of higher power consumption.
control technique could be used in addition to
unloading and attitude acquisition considerations to design the
Finally, the
wheel momentum
satellite
proposed technique does not increase the attitude control
system complexity and mass because the torquerods are also used for
acquisition and wheel
momentum
initial attitude
unloading.
Propulsion Option
2.
The
attitude control
system
is
very closely coupled with several of the other
systems aboard the spacecraft, and influences or
spacecraft to
interaction
torquerods
some
is
is
influenced by every system on the
degree. In the remote's attitude control configuration, the
normally that with the Propulsion system.
nanosatellite designers are utilizing consists
most notable
One concept
that
many
of micro-pulsed-plasma thrusters (jiPPTs)
provide control of both attitude and translation.
The shared nature of
the
ADCS
to
and
propulsion actuators also has a profound effect on the formation flight mission, placing
73
limits
on the controller design. This design option
is
explained more in detail under the
Propulsion section below ( Section Bl below).
Components
3.
Appendix
4.
shows
the overview of
Micromachined
solid-state gyroscopes use vibrating
They have no
in this section.
modes of a mechanical
highest rotation sensitivity
is
mechanical elements
rotating parts that require bearings, so they can
miniaturized. All vibration gyroscopes are based
vibration
components covered
Micromechanical gyroscope
a.
to sense rotation.
ADCS
structure,
obtained
on the
transfer
be easily
of energy between two
caused by Coriolis acceleration [Ref. 28]. The
when
the drive and sense
modes have
the
same
resonant frequency.
Resolution, drift rate, zero-rate output, and scale factor are the most
important factors that determine the performance of a gyroscope.
inertially static, the output signal is
a random function that
is
the
When
a gyroscope
sum of white
is
noise and a
cyclic noise function of the mechanical resonant frequencies.
Based on
resolution, drift rate,
QRS-11 micromachined angular
and zero-rate output, the Systron Dormer
rate sensor (see Fig. 4.1)
was seen
as the leader
from
seven commercially available micromachined gyroscopes to complement magnetometers
for
yaw
stability
determination. This gyroscope has a resolution of
(100s
at
0.004%, a short-term bias
constant temperature) of 0.004°/s, and a zero-rate output
Hz) of 0.01°/s. The gyroscope has a mass of 60g [Ref 29].
74
(DC
to
100
Figure
BEI
4.
A
Fig. 4.2). This
GYROCHIP™ Model QRS1
close alternative to the
component
1
Micromachined Angular Rate Sensor
QRS-11, would be the Litton G2000 gyro (see
offers a two-axis gyro with a size
0.76 inches wide and weighs 25 grams [Ref. 30].
of only 0.97 inches
tall
by
This gyro system can be combined
with a star sensor system to provide attitude position information to provide a pointing
accuracy of approximately
0.
1
degree.
The system
is still in
production and testing and
has not been space flown.
**M
Figure 4.2
b.
Litton
G2000 gyroscope with
electronics
Magnetometers
Magnetometers are simple, lightweight sensors
direction and magnitude
complex software
as
compared
of the Earth's magnetic
for interpretation
to horizon, sun,
and
field.
They
that
are reliable but require
and provide relatively coarse
star sensors.
GPS
measure both the
attitude determination
position measurements are used with
a computer model of the Earth's magnetic field to approximate the field direction at the
spacecraft's current position.
Over the course of an
75
orbit, the Earth's
magnetic field
direction
usually changes
computation of the
enough
to
rapidly enough with respect
field's
magnetometer. The Earth's magnetic
star
the
spacecraft
magnetometer
is
all
three Euler angles
field also varies
using only a three-axis
with time and cannot be calculated
often used with another sensor such as a sun, horizon or
sensor or a gyroscope in order to improve the accuracy.
The Applied Physics Systems Model 533 miniature
magnetometer (see
Fig. 4.3)
laboratory environment. This
its
make
to
time derivative possible. These field variations are large
enable determination of
precisely, so a
to
three-axis fluxgate
can provide direction accuracy to better than 0.1° in a
model
extremely low mass (18g) and
is
its
well suited for use in the nanosatellite because of
small size [Ref. 31].
Yaw
attitude
knowledge
is
maintained between magnetometer readings by integration of angular rate measurements.
Figure 4.3
c.
The
of
many
Model 533: Miniature
3 Axis, Fluxgate
Reaction HTteels
reaction wheel
is
used as an actuator for the attitude control and
in
vacuum,
is
one
important key technologies of nanosatellites. Hokkaido Institute of Technology
(HIT) has developed a small reaction wheel, which
Nms
Magnetometer
maximum
is
storable angular
is
about 150g in weight and 0.015
momentum. The motor, which can be used
30g and has a torque of 28gcm/2W. The wheel mass balance
precisely and the vibrational level
is restricted
76
as
low as possible.
is
in a
adjusted
Small reaction wheel developed by HIT
Figure 4.4
Sensors
d.
(1)
Horizon sensors
Horizon sensors are infrared devices
that detect the contrast
between the
of deep space and the heat of the Earth's atmosphere. Horizon sensors can provide
:old
)itch
and
).1° to
roll attitude
knowledge
for Earth-pointing spacecraft, with an accuracy
0.25°. For the highest accuracy in
low Earth
orbit
(LEO),
it is
of
necessary to correct
he data for the Earth oblateness and seasonal changes in the apparent horizon.
Two EDO
;ensors (see Fig. 4.5)
errors.
ill
Barnes Model 13-500 wide-angle miniature solid-state horizon
can be used to provide pitch and
These sensors have been space-proven on
lave fields
of view
is
knowledge
77
±
of up
1°,
including
The sensors
to
±
1 1 °,
but
Each sensor has a mass of 0.113 kg
roughly cylindrical with a diameter of about 4.1
Ref. 32].
to
six missions to date.
sufficient to allow pointing at off-nadir angles
>eak performance is limited to angles less than 9°.
md
roll
cm
and a height of about 5.6
cm
EDO Barnes Model
Figure 4.5
13-500 wide-angle miniature solid-state horizon
sensor
Star Tracker
(2)
Current
pounds,
attain
5
state-of-the-art
to 10
arc-second
commercial
accuracy,
star
sensors
typically
weigh
and use roughly 10 watts of power.
Unfortunately, the current state-of-the-art commercial star sensors do not meet
NASA's
15
"next-generation" spacecraft and instrument needs.
Nor do they
many of
satisfy
DoD's
need for micro/nano-satellite systems. The Intelligent Star Tracker [Ref. 33], built by
AFRL,
presents a low cost, miniature star tracker for nanosatellite attitude determination
and navigation. The Intelligent Star Tracker incorporates adaptive optic catadioptric
telescopes in a single, compact, robust Silicon Carbide housing.
mechanical
(MOEMs)
The Micro-opto-electro-
micro-mirrors (see Fig. 4.6) are used to compensate for various
aberrations as well as introduce aberrations such as leveraging off of adaptive optics
research, the active pixel position sensors enable
imaging of faint and bright
stars in a single
wide dynamic range and simultaneous
image frame.
78
*N- r
]r
I
i
gi ib. &
v&m^gj^
f
:=5sJ.
rz
CI)
«
.
1
>K£
Figure 4.6
Details of the structure of the micro-mirrors are shown.
The adaptive
optics telescope, using
extremely accurate tracking, and the ability
based on algebraic coding theory
-
-
MOEMs
micro-mirrors, facilitates
when coupled with a
star-matching scheme
enables the capability to track
at
least
5
stars
simultaneously. Moreover, the massively parallel architecture enables the star tracker to
operate autonomously without burdening the spacecraft processor and
supplement the onboard processor.
Because the design
utilizes
may
be used to
technologies
that
inherently integrate well together and lend themselves to batch processing, estimations
have that the Intelligent Star Tracker will have a recurring cost less than $100k. In
addition to low cost, preliminary analysis indicates that the Intelligent Star Tracker will
have a pointing accuracy exceeding 0.20 arc-sec,
consumption
less than
e.
NEA
better than 0.10 arc-sec,
power
2W and a weight of approximately 200 g.
DGPS
Formation flying will quickly revolutionize the
way
science,
sensing and surveillance missions are performed in space, enabling a whole
applications for small satellites. Currently, there are
stages involving formation flying of a constellation
to truly achieve the goals
numerous missions
of micro- or
remote
new range of
in the planning
nanosatellites.
However,
of these formation-flying missions, an accurate means of
79
determining
ranging,
relative
time
and
position
communications, and controlling the formation
The need of
integrated capabilities
timing
among
for
measurements,
becoming
states is
future formation flying missions to
spacecraft within the constellation
attitude sensor with a
The
system.
result
critical. [Ref.
34]
have technology with
communicating, relative ranging, and exchanging precise
is fast
approaching. AeroAstro Inc.
developed a system by integrating a Carrier Phase Differential
and
inter-satellite
GPS (CDGPS)
is
navigation
low power, inexpensive, compact ranging and communications
of this integration
is
a low-cost, robust, secure
GPS
micro navigation
and communication system for micro and nanosatellite constellations called Star Ranger.
The ranging accuracy of Star Ranger
the ultimate goal
is set at
3
mm. Using CDGPS,
between spacecraft will be determinable
it
cm
is
expected to be
is
expected that the relative attitude
1
or better, and
to 0.5° or better. In addition, the overall relative
position of each spacecraft with respect to each other will also be measurable to less than
1
cm, with a goal of 5
B.
mm.
PROPULSION
There
spacecraft.
is
an increasing need for on-board propulsion systems for micro- and nano-
These include upper stage engines
as very small-scale boosters to launch
and
new
1
to boost spacecraft into final orbits as well
kg
class payloads for military,
commercial
scientific applications.
Decreasing payload sizes will increase demand
platforms, including the ability to
maneuver and change
for
orbit;
smaller,
more capable
hence the need for small
propulsion systems. Such propulsive missions could include:
•
Remote
inspector to rendezvous and
• Constellations
• De-orbiting
In addition to
low
maneuver around a host spacecraft
on the same launch vehicle requiring separation
of space junk requiring rendezvous, docking and
cost,
orbit
changing
low mass and short delivery some more specific requirements
these propulsion systems include:
80
for
•
Low power consumption
•
Low,
•
High propellant
•
High density
controllable thrust
The simplest
Isp
Isp
spacecraft do not require thrust and hence have no propulsion
But most spacecraft need some controlled
equipment.
some form of metered propulsion -
thrust, so their
design includes
a propulsion system that can be turned
on and off in
small increments.
The propulsion system has
The
three objectives.
first
objective
is to
provide the
remotes with attitude control, which involves rotational disturbance rejection and angular
positioning.
The second
To maintain
objective of the propulsion system
a formation with the mother center
is to
satellite,
enable formation flying.
and other remotes, the
propulsion system must have the ability to reject translational disturbances and reposition
the remotes
the
when
A V necessary
the satellite drifts out of the formation.
for the orbital
The
third objective is to provide
maneuvers throughout the mission as determined by any
mission plan. These maneuvers could include changing from one formation to another,
changing orbital parameters, correct velocity
(e.g., drag),
errors,
maneuver, counter disturbance forces
control attitude during thrusting, and control and correct angular
The equipment
the
in
propulsion
subsystem
(propellant, tankage, distribution system, pressurant,
includes
momentum.
a propellant
supply
and propellant controls) and thruster
or engines. Significant sizing parameters for the subsystem are the total impulse and
number, orientation, and thrust levels of the
1.
thrusters.
Systems
a.
Cold/Hot Gas
Cold-gas
is
a
more
traditional propulsion system.
expanding high-pressure gas through a nozzle.
system are tank, tubing,
filter,
It
provides thrust by
Some of the components
required for the
pressure regulator, valves, and thrusters. Currently the
81
GN&C's
NASA's Goddard Space
Propulsion Branch of the
(GSFC)
Flight Center
is
conducting a broad technology development program for propulsion devices that are
ideally
suited
for
nanosatellite
missions.
The goal of
the program
nanosatellite propulsion systems that can be flight qualified in a
is
to
develop
few years and flown
in
support of nanosatellite missions. The miniature cold gas thruster technology, the
first
GSFC's propulsion component technology development program,
will
product from the
be flown on the upcoming ST-5 mission in 2003. The ST-5 mission
validate various nanosatellite technologies in
mission
to
more
ambitious
nanosatellite
major subsystem
all
missions
such
areas.
as
is
It is
designed to
a precursor
Magnetospheric
the
Constellation mission. [Ref. 35]
MEMS
b.
Small
them
in place.
The
satellites flying in clusters require periodic
required impulse
satellites in rigid formation,
is
very small
cluster.
the goal
is
not to keep the individual
but only to keep them in well-defined orbitals with respect to
one another. The necessary impulse, therefore,
the difference in drag
-
"stationkeeping" to keep
only the amount needed to overcome
is
between the most-affected and the least-affected
satellites in the
Estimates are that the differential drag can be overcome by providing ~1
(micro-Newton second)
throughout each
satellite's
mN
sec (milli-Newton second) every 10 to 100 seconds
mission.
propulsion
Currently
miniaturized systems.
~1
to
Most notable
technology
is
MEMS
is
developing
technology. The
Micro-thruster Array (see Fig. 4.7) thrusters have very low
for ignition
(~10
m Watts,
highly reliable. [Ref. 36]
mNsec
A
-100 u Joules), and no moving
rapidly
MEMS
towards
Mega-pixel
power and energy thresholds
parts so are expected to be
single thruster array contains a quarter of a million separate
thrusters.
82
Aerospace
Figure 4.7
Marotta
Scientific
Vticroelectromechanical System
the spacecraft;
it
:ontrol systems.
MEMS chip compared to Penny
Controls
(MEMS)
designing
is
very
tiny
chip that provides fine attitude adjustments on
uses 8.5 times less power and weighs 2 times less than proven attitude
[Ref.
37]
Marotta
is
also in the process of developing a unique
microthruster (see Fig. 4.8) and electronics driver combination
power operation
a
(<1W
peak), response time of <5 msec,
which
minimum
is
capable of low
pulse rate of <1 Hz,
and has a mass of 50g. The resulting low power component provides an order of
magnitude reduction in solenoid coil heating when compared to an ordinary continuous
duty solenoid valve. Aerospace, Primex, Honeywell and
MEMS
AFRL are working separately on
based thrusters such as micro-hydrazine. These will be flown on numerous
commercial and university based nanosatellite systems when the maturity of the
technology will allow
it.
The small modular nozzles would allow many options
microthruster size. Although development time will
years, the potential for nanosatellites is
very high.
83
most
likely require
as to
more than two
Marotta microthruster compared to Dime
Figure 4.8
Electrical
c.
Washington
to scale
Company
Aerospace
Primex
down
Thrusters (see Fig. 4.9).
The
the
is
working
power requirements of
UW nanosatellite will
fly a
with
the
University
their full-sized Pulsed
of
Plasma
propulsion system, and will be
either fiPPT's, or a cold gas system.
V*
Full-sized Pulsed
Figure 4.9
A
typical
Plasma Thrusters from Primex Aerospace Company
pulsed plasma thruster consists of two electrodes, a solid
Teflon" propellant bar, an igniter (spark plug), a feed spring, a power supply, and a
capacitor (shown in Fig. 4.10).
The power supply charges the
electrodes.
When
capacitor,
which
a small plasma puff from the spark plug
84
is
connected to the two
is
released between the
electrodes, the puff creates a low-resistance arc path, discharging the energy stored in the
capacitor. This arc ablates a small
it
into plasma.
resulting
JxB
The
amount of the Teflon propellant bar and turns
part of
current flowing through the arc also creates a magnetic field, and the
force accelerates the plasma
away from
Feed
the thruster, thus generating thrust.
Eleclrock
Is»riiler
Spring
Plasrtu
Electrode
Capacitor
Figure
4.
Basic diagram of a pulsed-plasma thruster
1
Performance
2.
Three parameters determine the performance of a propulsion system. These three
parameters are thrust
(7),
minimum impulse
and specific impulse
bit {hit),
propulsion system hardware, the thrust can be measured experimentally.
(Isp ).
From
Given
these
experimental data, the average thrust can be calculated. However, for a theoretical
analysis, the thrust is calculated
from Eq.
(4.1):
T = rhC,tf
where
m
is
the
(4.1)
mass flow
Ce/f
rate at the thruster exit
and
amount of thrust
be delivered by a
is
the effective exhaust
velocity.
The
hit is the smallest
that can
thruster. This is
given in the units of an impulse (force multiplied by time), such as 70 uNs. Although the
hit
may be
experimental
calculated theoretically in
data
since
the
some
cases,
it
is
usually calculated from the
minimal thrusting capability depends highly on the
85
propulsion system hardware. The /#/
is
calculated using Eqn. (4.2) from the thrust profile
data of an experiment in which a propulsion system
is
activated for the shortest duration
possible.
= [T-dt
Ibil
(4.2)
In equation (4.2),
respectively. For the
f,
and
tf
are the time at the start
most spacecraft a low 4„
is
and the end of the
desired for fine control of the attitude,
and especially during dead-band limit cycling. The specific impulse
as
shown
thrusting,
is
generally defined
in equation (4.3):
hP = ~
™-g
(4-3)
The
where g
is
impulse
a measure of the
amount of thrust a propulsion system can provide
mass flow
The Isp diminishes
is
propellant
lifetime.
The
the acceleration due to gravity at the surface of the Earth.
rate.
specific impulse is essentially a
propulsion system, so a high Isp
3.
is
for a given
most propulsion systems over
their
measure of the mass efficiency of a
desired for the propulsion system.
Propulsion System Comparison (Cold gas
Two main
uPPT and
for
specific
/
uPPT)
propulsion systems were analyzed and compared. The results of the
cold-gas system performance analyses are summarized in Table 4.2 below.
86
Propulsion
Total
System
Mass
Type
(kg)
Propellant
T
Isp
Ibit
(s)
GiNs)
(mN)
Mass per
AV
(g-s/m)
A V Time
Energy
Duration
per
(s
2 /m)
(J-s/m)
uPPTf
3.80
500
70
0.07
2
1.43xl0
cold-gas
4.58
65
100
4.5
16
2.22x1
5
3
f The performance of the {iPPT was analyzed assuming a J
X The energy
AV requirement for
per
Peak
Power
(W)
AV
17.9xl0
l~5xl0
6
12.5
4
10.1
J
Hz firing frequency.
a cold-gas thruster depends on the firing
mode, pulsed or continuous.
Comparison of uPPT and cold-gas propulsion systems
Table 4.2
(single thruster
performance).
Both uPPT and cold-gas
maximum
translational disturbances,
has a lower mass due to
and more
A V for
its
lower
a given propellant
due mostly
Ibu
uPPT
of
less
compensate for
remotes drag. The
uPPT system
,
mass than the cold-gas system. The
an impulsive bum, or a short duration per
is
to the
thrust to
and higher Isp providing better pointing accuracy
required for attitude control can be characterized
maneuvers,
enough
thrusters provide
by
fine thrust control
the long time duration per AV. Thus,
AV burn,
which
is
optimal for most orbital
importance. The peak power and energy consumption are high for
systems.
Most importantly, uPPTs do not
have a history of
complexity
and
failing
due
problems
suffer
from propellant leakage. Cold-gas systems
to propellant leakage.
inherent
to
a
Compared
high-pressure
to the cold-gas system's
and
system
the
miniaturization capability due to the flow characteristic of gases and liquids, the
more
attractive.
The \iPPT has a simple feed system with minimal moving
to higher reliability. Also,
size
uPPT system mass can be
further reduced
of the electronics, the most massive component of the
characteristics are suited to small satellites
future
commercial nanosatellites
uPPT
limited
uPPT
is
parts, leading
by decreasing the
system.
The uPPT
and thus will become the stepping stone for
that will utilize
87
uPPTs.
EXAMPLE: SNAP-I
:.
SNAP-1
>STL),
rid
is
it
is
(SSTL)
a low-cost nanosatellite built
ingle
Satellite
Technology Ltd.
amongst other objectives a technology demonstrator for 3-axis
orbit control for a future constellation
itellite
by Surrey
uses a single miniature
of small
Y-momentum
satellites
stabilization
during formation flying. The
wheel, 3-axis magnetic torquers and a
butane gas thruster to ensure a nominal nadir-pointing attitude with
and
ontrol
in-track
AV
full
pitch
The magnetic torquers do momentum
maneuverability.
mintenance and nutation damping of the Y-wheel. The primary attitude sensor used,
A
1
1
GPS
receiver supported
by an on-board
low cost propulsion system was designed and
7 months from concept to launch
site. It utilizes
olution.
lunch
was
The
site.
stored in a
spacecraft
SNAP-1 was
was no
1),
spacecraft
was used
for the valves.
The
formed titanium tube, rather than a tank, giving a low cost
successfully launched
The
providing 65
propellant
cm
Figure 4.1
3
1
pipework assembly
was stored
in
1
.
1
it
to
Snap-1 Propellant tube
is that
meter of coiled titanium
of storage volume. [Ref. 38]
88.
prior to shipping
on 28 June 2000.
feature of the complete propulsion
propellant tank.
ibe (see Fig. 4.1
SNAP-1
butane stored as a liquid and operating
was loaded with 32.6 grams of butane
The most obvious
lere
orbit estimator.
built for the
a cold gas mode. Miniature conventional technology
ropellant
a
magnetometer. Precise orbital knowledge was obtained using a
liniature 3-axis flux gate
mall single antenna
is
CONCLUSION
V.
THE NANOSATELLITE PUSH
A.
In recent years, an increased effort to design, build, and operate small satellites
has taken place in universities and laboratories
and nanosatellites provide numerous
fraction
thrust
over the world. These microsatellites
flight opportunities for science
experiments
at a
of the cost of larger traditional missions. This paper has presented an enormous
toward innovative ways, not only of
satellite's roles
'some distant
in
all
and
abilities in the
future',
and
with
built,
opportunities to
show how
commercial
and
fields
years.
but a shift in thinking about
Nanosatellites are far from being the
and indeed are not even the end of the
space commercialization.
designed
coming
satellite design,
Picosatellites
space
line for this
new movement
and femtosatellites are currently being
engineering paving
the
way
for
many powerful
useful and cunning these systems can be to both the
to the military.
FORMATION DESIGNS
B.
Imagine
satellites
numbering
in the tens,
hundreds or even thousands
being launched off surface combatant ships, submarines, mobile
Army
vehicles or even
Air Force cargo planes. The possibility of throwing a large quantity of 'sensors' quickly
and
efficiently into
an engaged theater wherever/whenever removes the dependence on
costly, vulnerable national-asset satellites.
The preceding chapters looked
at the possibility
of creating clusters
that through
their natural
motion retain a constant shape when viewed from the planet's surface.
shown
the shape of the apparent circular orbit can be described in terms of
that
eccentricity of the orbit,
and
It
is
that terms through second order in e are sufficient to
describe the motion to a high order of accuracy.
Many
formation designs have been
presented to show that utilizing natural orbits allows formation dynamics to help reduce
fuel requirements
for formation-,
and station-keeping needs.
These natural
coupled with robust control laws and precise position knowledge and
89
orbits,
inter-satellite
communication
DoD
key
abilities, are
to the
NASA,
growing need and requirement for future
and commercial space missions.
PERTURBATION UPKEEP
C.
This paper has presented the effects of the earth's oblateness on
satellite
formation flying designs like the circular and projected circular clusters. There are two
components of motion
that
must be accounted
for:
1) differential
changes
in the right
ascension of the ascending node, and 2) secular changes in the argument of perigee and
mean anomaly. The
formation.
cost to maintain relative
For a circular cluster of 25
km
node spacing
is
dependent on the size of the
radius, the cost is approximately 7.2m/sec per
year of velocity impulse. The cost to maintain the argument of perigee, not being a
function of the cluster size,
is
roughly 2.1m/sec per year for the circular
maneuvers cannot be coupled, a cluster
1
like the
cluster.
If the
one presented here could require close
to
Om/sec per year of velocity impulse. Based on the orbit propagations and an assumed
10%
error
bounded on the formation, maneuvers would be required every 40 hours or
may
Other perturbing effects
also require
maneuvers but
It
may be
satellite in the
all
directions
in the along-track, cross-track,
maneuvers
will also vary
from
and each
and
radial directions.
satellite to satellite
formation must be able to thrust
The amount of formation keeping
within the formation.
Station keeping cost for the classes of orbits discussed here
Only atmospheric drag decay of the semimajor axis
to correct for this effect
is
is that
is
is relatively
small.
a major concern. Velocity impulse
on the order of .097m/sec per year using high drag conditions.
Another major issue
maneuvers
frequency and cost
should also be pointed out that maneuvers
than the oblateness induced maneuvers.
required in
at far less
so.
that
impacts the frequency and cost of formation keeping
of attitude control. The formation keeping maneuvers discussed above
require thrusting in the along-track, cross-track and radial directions.
dynamics are very sensitive
to acceleration in the along-track directions.
maneuvers are required
in the cross-track
has substantial errors
when
cross-track
unwanted acceleration might be applied
and
radial directions.
or radial maneuvers
The
The
satellite
largest
If the satellite pointing
are
performed, some
in the along-track direction with significant
90
consequences. This will undoubtedly drive the frequency and cost of formation-keeping
higher than what has been calculated here.
Orbit determination knowledge
control.
is
another factor that will influence formation
The STK's simulated propagations show
that centimeter level differences in
semimajor axis cause significant along-track error growth over time.
determine the orbits to this level of precision
control.
91
is vital
for precise
and
The
ability to
efficient formation
THIS
PAGE INTENTIONALLY LEFT BLANK
92
APPENDIX 2.1
International
Mass
Launched from
(g)
Initial orbit
@ 51.6 deg
Decayed
Designator
Oderacs
A
STS60 9Feb94
1994-006B
4200
225x463
km @
51.6 deg
10Feb94
Oderacs
B
STS60 9Feb94
1994-006C
4200
239x451
km @
51.6 deg
10 Apr 94
Oderacs
C
STS60 9Feb94
1994-006D
500
Oderacs
D
STS60 9Feb94
1994-006E
500
Oderacs E
STS60 9Feb94
1994-006F
5
266x272
km @ 5
Oderacs F
STS60 9Feb94
1994-006G
5
266x271
km @
51.6 deg
Oderacs 2
STS63 4Feb95
1995-004C
5000
267x277
km @
51.6 deg
13
Oderacs 2B
STS63 4Feb95
1995-004D
4200
323x349
km @
51.6 deg
29 Sep 1995
Oderacs
2C
STS63 4Feb95
1995-004E
500
Oderacs
2D
STS63 4Feb95
1995-004F
1.5
314x320
km @
51.6 deg
2
Oderacs 2E
STS63 4Feb95
1995-004G
1.5
320x334
km @
51.6 deg
27 Feb 1995
Oderacs 2F
STS63 4Feb95
3
Mar 95
6 Feb95
1
.6
deg
3
Mar 95
24 Feb 95
Mar 1995
7 Feb 1995
Mar 1995
Not catalogued by
0.5
NORAD
ODERACS Data Table
93
20 Feb 1995
THIS
PAGE INTENTIONALLY LEFT BLANK
94
APPENDIX
2.2
Educational institutions involved in small
Increasingly,
satellites.
This
satellites to
becoming possible
it is
is
largely due to
be designed and
satellites
be involved in small
affordable yet sophisticated technology. This then allows
for educational institutions to
built within the course duration, or via a series
of student
projects.
The following
list
of educational establishements are those known to be involved in small
spacecraft projects. This can be either in experiments, parts of satellites, or entire
Where known, specific spacecraft project names apear in brackets, inducing a
year when launched (or expected to be launched). For more details on the projects, please
satellites.
refer to the satellite
and future mission
lists.
All the links on this page will lead you to the individual institutions
Europe
HUniversidad Complutense, Madrid, Spain
BlJniversidad Politecnica de Madrid, Spain
BSurrey Space Centre
(UPMSAT
*95)
University of Surrey, Surrey, United
Kingdom (UoSAT
series '80
onwards)
HTechnical University of Helsinki
in Finland
(HUTS AT)
(CATS AT)
Germany (TUBS AT
HUniversity of Leicester, England
HTechnical University of Berlin,
series '90
onwards)
Buniversity of Bremen, Germany
(BREMSAT
BRoval
Stockholm Sweden. (KTHSat)
Institute
of Technology
Blnstituto Superior Tecnico
,
in
'94,
Abrixas
'97)
.
Lisbon Portugal. (PoSAT-1)
Germany (Phase 3D)
of Umea University (RYP) (Munin)
BUnitersitaet Kaiserslautern ,
EDept. of Space
Physics
North America
In the U.S.,
NASA encourages participation in small missions via the University Explorer
(UNEX) Programme.
BUniversity of Illinois
Urbana-Champain.
BStanford University, U.S. (SAPHIRE and OPAL)
BArizona State University. Tempe. Arizona. U.S. (ASUSAT)
BUni versify of Arizona Tuscon, U.S., Students Satellite Program (UASAT)
Blowa State Univerity, Ames, Iowa, U.S. (ISAT)
BUniversidad Nacional Autonoma de Mexico Mexico City, Mexico (UNAMSAT)
at
,
.
,
95
(PANSAT)
BSan Jose State University, San Jose, California, U.S. ( SPARTNIK )
BSierra Community College Rocklin, California, U.S. ( MINERVA )
BUniversitv of Alabama Huntsville, Alabama, U.S. (SEDSAT)
BUtah State University Logan, Utah, U.S. (NUSAT '85, Webersat etc CAST )
BNaval Postgraduate School Monterey,
,
California, U.S.
,
,
,
BJohns Hopkins University Applied Physics Laboratory U.S.
,
BPenn
State University U.S.
,
(SAC-B CUBIC
instrument)
BUniversitv of Colorado U.S.
BMassachusetts Institute of Technology Centre for Space Research U.S. (HETE)
BWeber State Univerity Ogden, Utah, U.S. (NUSAT *85, WEBERSAT '90 and more)
,
,
,
BUniversitv of New Hampshire U.S.
,
(CATSAT)
BBoston University U.S. (TERRIERS)
BAnahuac University, Mexico (ANISAT
,
Nano-satellite project)
BCarleton University, Canada. (SUA microsatellite)
BUniversitv of Toronto (MOST microsatellite)
Africa
BUniversitv of Stellenbosch South Africa
,
(SUNSAT)
Asia
BKorean Advanced Institute of Science /Technology (KAIST), S.Korea (KITSAT
BMahanakorn University, Thailand (TMSAT)
series)
BNational Central University, Taiwan (TUU-Sat)
BATSB,
Malaysia, (JjungSAT)
BTsinghua University, China (Tsinghua-1)
South America
BUniversidad de La Frontera, Temuco, Chile, Involved with the design and construction
of CESAR-
96
APPENDIX 3.1
STK Screen Shots
In-Track Formation(2 remotes)
97
y*
•
•
*
s
_A
ISfeta
^
^
^</
*^^;~* ^~~"'"""
**N»
F&^^^Siz'^
**
"
«•-*.
'
V***-
\
WM
HTM
*9a
*
_
"Sgj^frVi
AmL.
-^X-<""-'
.,
.
/
"Si;
j-^^L
^sJ?
6;
7
^
/
A,
^
/
/
/
/
P"
/
*^-,
•
^
/
fism
/
/
/
/
/
1
.^fen
2001
/
/
/
;
/
/
OOOOOqio Time
:
f /
/
*
.
V
/
/
/
;
'
/
/
/
/
Circular Formation (4 remotes)
98
I
J
/
\vA
xN
s^fe. X
^
2
X
APPENDIX 3.2
Matlab Code for STK data input
in inertial position/velocity parameters
%Cluster Formation
clear
clc
%Need Mother parameters
%ac
-
semimajor axis
%ec
-
eccentricity
%ic
-
inclination
%RAANc
-
right ascension of ascending node
argument of perigee
%argc
-
%Mc
-
mean anomaly at epoch
%Ec
-
eccentric anomaly
%Tc
-
true anomaly
%rc
-
radius
%vc
-
orbital speed
ac = 7489.137000;
ecdeg
ec
=
ecdeg*pi/180
%deg
63.4;
=
RAANcdeg
RAANc
=
argcdeg
argc
=
=
;
%rad
%deg
;
RAANcdeg *pi/ 180 ;%rad
=
%deg
;
argcdeg*pi/180;
pc = ac* (l-ec
=
%rad
;
icdeg* (pi/180)
ic
mu
%deg
0.00;
=
icdeg
%km
A
398600.5;
%rad
2)
%km^3
s
A
-2
%Cluster geometry parameters
%rho - scale factor;
same for all subsats
%nu - dimensionless
arrangement
determines
pattern
generator;
%delta - angle; specifies shape
suborbit plane wrt centersat orbit
99
overall
of
cluster
size/
determines
suborbit
&
assigned
geometrical
orientation
of
%rs = subsat radial speed
%vs
=
subsat orbital speed
%as = subsat semimajor axis
rho
=
deltadeg
delta
%km
100;
=
%deg
63.4;
deltadeg*pi/180;
=
%rad
%km
as = ac;
%First choose eccentric anomaly, Ec, of centersat
numsats=8
for m=0 :numsats-l
Anomaly
%Eccentric
Ec= (2*pi/numsats*m)
spacing circular
Equal
%Pattern Generator for Desired Pattern
nu=
.
%Circular Spacing, one loop
5
%Calculte Center-Sat parameters
Tc=2*atan(sqrt (1+ec)
(
Mc=Ec-ec*sin (Ec)
rc=pc./
/
(1-ec)
)
vc=sqrt (mu* (2/rc-l/ac)
)
.
*tan(Ec/2)
)
;
,-
+ec. *cos (Tc)
(1
)
%Calculte Sub-Sat parameters (use 12 o'clock insertion position)
%Assumed for all subsats
as=ac;
rs=sqrt (nu*rho*sin (delta)
(
vs=sqrt (mu* (2/rs-l/as)
A
2+ (rc+nu*rho*cos (delta)
A
2)
)
%Calculate Vectors
Rc=rc* [cos (Tc) sin(Tc)
Vc=sqrt (mu/pc)
*
0]
'
[-sin(Tc) ec+cos(Tc)
Rs= (rc+nu*rho*cos (delta) *cos
nu*rho*sin (delta)
)
[
0]
[rc+nu*rho*cos (delta)
(Tc)
]
'
Vs=vs/vc.
'
*Vc,-
100
)
*sin
(Tc)
vectors
%Transform subsat
radius/velocity
Earth-centered inertial coord
Rxx
=
3x3 rotation matrix from RSW to
Rraanc
[cos(-RAANc)
=
sin(-RAANc)
0;
UK
from
centersat
coord
frame
-sin(-RAANc)
cos(-RAANc)
(-sin(-ic))
(cos(-ic))];
1];
Ric
=[10
Rargc
R
=
[cos(-argc)
=
(sin(-ic));
(cos(-ic))
0;
sin(-argc)
0;
-sin(-argc)
Rraanc*Ric*Rargc;
Rijk=R*Rs
Vijk=R*Vs
Rsub (m+1,
1
:numsats-l) =Ri jk
'
,-
Vsub(m+1,
1
:numsats-l) =Vijk'
;
%Calculate Orbital Elements
%As should equal Ac for all subsat s
(Assumed)
end
display 'SubSatl
(
'
)
101
cos(-argc)
0;
1]
0;
to
THIS
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102
APPENDIX 3.3
Example
orbital
Orbital
5 =
Parameters
MOM
63.4
Position
Inclination
1.5
63.403457 deg
3
4.5
Parameters from Matlab to
a c = 6978.137 km
STK coordinate
alt
= 600
transformation
km
Ring radius -25
km
Arg
RAAN
Mean Anomaly
0.003
180
-0.006913 deg
0.003
225.012421 deg
180
134.987579 deg
63.406913 deg
0.003
270.024842 deg
63.403457 deg
0.003
315.012421 deg
Eccentricity
63.4
6
63.4
0.003
0.006913 deg
7.5
63.396543 deg
0.003
44.987579 deg
9
63.393087 deg
0.003
89.975158 deg
10.5
63.396543 deg
0.003
134.987579 deg
12
63.4
0.003
180
F. or
5
= 63.4
MOM
63.4
Position
Inclination
Eccentricity
63.400654
0.00328833
63.06986
0.00325885
63.400654
(all
a c = 6978. 137
89.975158 deg
44.987579 deg
Bottom
315.012421 deg
-0.000001 deg
270.024842 deg
225.012421 deg
-0.006913 deg
second outt. tr ring: e=006
Orbital
Parameters
0.000001 deg
Top
180
other parameters stay the same)
km
alt
= 600
km
Ring radius -25
km
RAAN
Mean Anomaly
359.630879 deg
359.953493 deg
270 deg
Odeg
Odeg
0.00328833
179.788017 deg
0.369121 deg
0.046908 deg
63.730337
0.00326474
90 deg
Odeg
Odeg
63.400654
0.00328833
359.630879 deg
359.953493 deg
Arg
0.21 1983
deg
Top
1.5
3
4.5
6
7.5
9
10.5
12
0.21 1983
deg
For second outer ring: e=006
103
(all
other parameters stay the same)
Bottom
Orbital
Parameters
MOM
Position
5 = 63.4
a c = 7489.1 37
km
alt
= 11 11
km
Rin9 radius
—23
km
63.4
Eccentricity
Argft
o<°)
M(°)
63.400508
0.00300701
180.170765
359.61862
180
63.058986
0.00300701
90
Odeg
180
63.400508
0.00300701
359.829235
0.38138
180
63.741014
0.00300701
270
Odeg
180
63.400508
0.00300701
180.170765
359.61862
180
Inclination
(°)
Top
1.5
3
4.5
6
7.5
9
10.5
12
For second outer ring: e-.006
104
(all
other parameters stay the same)
Bottom
APPENDIX 3.4
RAAN
over
1
year(deg)
350
300
-
ORC\
0)
0)
oc\c\
zuu
.
^
150
-
i—
O)
I
uu
50
n
2001001
2001051
2001101
2001151
Date
2001051
2001101
2001251
2001301
2001351
(YYYYMMDD.hhmmss)
Arg of Perigee over
2001001
2001201
1
2001151
year(deg)
2001201
Date
105
-
uncorrected
2001251
2001301
2001351
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106
APPENDIX 4.1
Sensor
Performance
Power
Unit
(kg)
(W)
Price
short-term bias and
Notes
Units
Use
0.0047s
Micro-
Gyroscope
Mass
0.060
0.030
$450
between
1
out
resolution
using
regularly
the
magnetometer.
Nominal use
Magnetometer
±3°
magnetometer
measurements. Must be zeroed
attitude
0.018
0.030
$2200
1
only.
is
yaw
for
Can be used
attitude
for three-axis
attitude if necessary.
8300Reaction
Wheel
8400 rpm
0.015
Built
by
0.450
2.0
0.113
0.030
$40,000
2
Pitch and roll attitude
0.200
<2.0
$100,000
1
Tracker's processor
HIT
3
Nms
Horizon sensor
Star Tracker
±1°
attitude
pointing
accuracy:
0.20arcsec
NEA:
better
to
than
O.lOarcsec
TOTALS
0.954
4.120
$182,700
8
Comparison of attitude sensors
107
may be used
supplement onboard processor
THIS
PAGE INTENTIONALLY LEFT BLANK
108
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