SSC05-XI-7
The AeroAstro Fast-Angular-Rate Miniature Star Tracker:
Algorithms and Simulation Results
Bill Seng, James Stafford, Ray Zenick, Vickie Kennedy
AeroAstro, Inc.
20145 Ashbrook Place, Ashburn, VA 20145; (703) 723-9800
bill.seng@aeroastro.com, james.stafford@aeroastro.com, ray.zenick@aeroastro.com,
vickie.kennedy@aeroastro.com
ABSTRACT: AeroAstro’s Fast Angular Rate Miniature Star Tracker (FAR-MST) is an optical-based system that
uses image processing of star fields viewed from the satellite to determine ephemeris information. The FAR-MST
system offers solutions to several problems, including the determination of an initial position fix (the “lost-in-space”
problem), determination of spacecraft attitude, and measurement of spacecraft rotational velocity – even at high
angular rates. Low-cost, high-performance microsatellites cannot afford the volume or mass required by a large star
tracker. The target physical specifications of the FAR-MST make it very attractive to microsatellites. FAR-MST
weighs less than 1 kilogram and consumes fewer than 3 Watts. It provides both spacecraft attitude to within 100
arc-seconds and angular rate information over rates up to 15 degree/second. As a result of its high performance, it
can easily handle typical spacecraft tumble conditions, providing for recovery from a lost-in-space condition without
aid from any other sensors at minimal cost and impact to spacecraft resources. FAR-MST provides the necessary
information to enable accurate thrusting and slewing in specific directions, necessary for advanced microsatellite
missions that require precise station-keeping, orbital-transfer capabilities, or rendezvous missions. FAR-MST
provides a complete low-cost, low-power attitude determination solution ideal for small, rapid-response satellites.
INTRODUCTION
realize the benefits of small satellites, they cannot be
burdened with the volume or mass required by a large
star tracker, let alone redundant, large star trackers
coupled with gyroscopes. Achieving a low-cost, highperformance satellite is not feasible without the
introduction of new components more consistent with
their goals. European star tracker manufacturers have
taken strides towards producing either star trackers that
function at both low and high angular velocities or star
trackers that are small and lightweight, but none offer
all of these characteristics in a single unit, which is
what FAR-MST is striving to achieve.
While the space industry continues to emphasize the
increasing importance of small satellites, one of the
primary hurdles to their widespread acceptance is that
there is an insufficient selection of appropriate
components to realize the full extent of a small
satellite’s capabilities. One such area in which this is
apparent is that of attitude determination components,
particularly star trackers.
To address this critical industry need, AeroAstro has
developed a single, small, lightweight system that can
replace multiple, independent components, such as
gyros and star trackers, while providing all the
functionality associated with traditional star trackers.
AeroAstro’s Fast Angular Rate Miniature Star Tracker
(FAR-MST) is an optical-based system that provides
satellite ephemeris data by processing star field images.
AeroAstro’s FAR-MST system addresses several
different problems facing small satellites. First, it
provides the ability to determine position from a lost-inspace condition (either during mission initialization or
in recovery from tumble). It also enables determination
of spacecraft attitude during normal operations. And
finally, it provides measurement of spacecraft rotational
velocity – even at high angular rates.
The goal of the FAR-MST program is to develop a lowcost, mid-range performance star tracker to provide
attitude determination and rate sensing capability to
small satellites. A small satellite would be able to use
FAR-MST both for initial detumbling and subsequent
three-axis pointing operations, thus eliminating the
need for additional gyros or magnetometers. This
would provide savings to the satellite in terms of cost,
development time, and operational complexity.
The FAR-MST hardware design is based largely on the
AeroAstro Miniature Star Tracker (MST), which is
being developed under a separate Small Business
Technology Transfer (STTR) program contract in
conjunction with the Massachusetts Institute of
Technology (MIT) Space Systems Laboratory (SSL).
Unlike MST, however, the FAR-MST star tracker will
Replacing redundant components saves volume, mass,
and power, as well as their associated costs. To truly
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have autonomous lost-in-space capability. This means
that it will autonomously determine the inertial threeaxis attitude of the satellite without prior attitude
knowledge. The addition of this capability will involve
minimal hardware changes, and is largely an upgrade to
the MST star tracker software.
low-power attitude determination solution ideal for
small, rapid-response low-earth orbit (LEO) satellites.
In addition to adding lost-in-space capability to the
existing MST hardware design through the FAR-MST
program, the team has also studied adding rate sensing
capability, with a goal of designing a method to
determine spin rates optically. Ideally, such an
algorithm would require no additional hardware to the
MST baseline design. This would give it an advantage
over even inexpensive MEMS gyros in terms of cost,
interface complexity, and power consumption.
The target physical specifications of the FAR-MST
make it very attractive to microsatellites. It weighs less
than 1 kilogram and consumes fewer than 3 Watts of
power. It provides both spacecraft attitude to within
100 arc-seconds and angular rate information over rates
up to 15 degree/second. As a result of its high
performance, it can easily handle typical spacecraft
tumble conditions, providing for recovery from a lostin-space condition without aid from any other sensors at
minimal cost and impact to spacecraft resources. Since
FAR-MST can function at high angular rates, it
eliminates the need for gyroscopes, offering further
mass savings. It provides the necessary information to
enable accurate thrusting and slewing in specific
directions, necessary for advanced microsatellite
missions that require precise station-keeping or precise
orbital-transfer capabilities, and rendezvous missions.
In recognition of the trend towards more modular and
standardized spacecraft systems, the FAR-MST has a
standard USB interface for command and telemetry.
The FAR-MST system provides a complete low-cost,
FAR-MST is not intended to compete with existing
high-performance star trackers,1,2 nor will it necessarily
provide the rate sensing accuracy or update rate
provided by inexpensive MEMS gyros. Instead, it
attempts to provide sufficient capabilities to
economically fulfill the specific mission requirements
of small satellites without burdening them with undue
mass, volume, and power consumption.
SIMULATION ENVIRONMENT
Introduction
Beginning with an internal AeroAstro simulation
environment called SIMK3 (Figure 1), which provided
Figure 1. Simulation Environment (SIMK).
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spacecraft dynamics, vector math, visualization, and
general data analysis capabilities, the team then
developed object models for each of the major
components of the FAR-MST system: the CMOS
(complementary metal oxide semiconductor) detector,
lost-in-space algorithms and star catalog, and FARMST algorithms. In addition, each object also had a
separate scripting object whose main purpose was to
interface between the simulation scripts and the
underlying C/C++ code of the algorithms themselves.
The algorithms were developed in C/C++ for later
porting to embedded systems.
star centroid position. Pixels in the imaging array
accumulate signal count through a combination of the
optics and detector electronics modeled in the following
equation:
E(x, y) = (a)(e)( flux)PSF(x ! x c , y ! y c )dt (2)
where E is the signal count in a pixel, a is the aperture
area of the optics in square meters, e is the quantum
efficiency of the detector in converting photons to
electrons, flux is the number of photoelectrons emitted
from the star per m2/s, and dt is the integration time in
seconds.
CMOS Camera Model
Lastly, several noise sources are added to better
simulate real-world performance. Noise is inherent not
only from the detector but also from the quantized
nature of the photons. Two dominant noise sources can
be modeled as Poisson processes: photon arrival, and
the detector noise from thermal sources, also known as
dark current. Noise is also added to simulate read-out
noise.
The camera model was designed with an eye toward
fabricating an optical chain through which the
spacecraft dynamics model and star catalog model
could be used to generate star map images. The
spacecraft dynamics model enabled star field views
from arbitrary attitudes and angular velocities to be
simulated.
The camera model consists of two image arrays: the
first contains the number of photoelectrons in each
pixel, the second contains the integration time for each
pixel (necessary for dark current calculations). In
SIMK, the algorithm to generate a simulated image
utilizes a sequence of time steps. In each step, stars are
extracted from the catalog and projected in the focal
plane. Then, the spacecraft dynamics model is queried
and the attitude of the star tracker is updated, causing a
new set of stars to be extracted from the catalog and
rendered onto the focal plane. Lastly, once the total
preset integration time has been reached, the
accumulated image is read out, with noise added based
on the converter electronics, the accumulated signal at
each pixel, and the dark current.
Lost-In-Space Algorithm and Star Catalog
A so-called lost-in-space algorithm is able to determine
the attitude of the star tracker from no knowledge other
than the current star image. Typically this class of
algorithms attempts to work much like the early human
astronomers did when looking at the heavens – by
looking for patterns.
In fact, star pattern recognition algorithms are arguably
the most time-consuming and computationally intensive
routines in a typical star tracker. One can generally
group these pattern recognition algorithms into three
groups:4
In the current rendering of stars in the CMOS camera
model, the spectral nature of the star, optics, and
detector is ignored. As a simplifying assumption, the
model assumes a fixed constant relating Johnson Vband star magnitude to the photon flux (photons/s/m2)
received at the focal plane of the detector. The
equation governing this, relative to the known flux F0
for a magnitude 0 star is:
Fm = F0!0.4 m
Inter-star angular separation-based
2.
Grid-based
3.
Neural networks-based
In the first class of algorithms, the stars are viewed as
vertices of a graph (triangle, pyramid, etc.) whose edges
define the angular separation between neighboring
stars.5,6 The second class of algorithms uses the
specific, well-defined pattern of stars associated with
any given star.7,8 The last class is perhaps most like the
one early astronomers used – patterns are recognized
directly.9,10
(1)
The center point of stars is positioned in the focal plane
using the stars’ three-dimensional direction cosine
vectors. To spatially distribute the star’s signal, the
point spread function (PSF) of the star is modeled by a
two-dimensional Gaussian normal curve located at the
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The first class of algorithms has been selected, based on
the fact that it is more mature and has wellcharacterized performance. The key, however, to this
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approach is to devise an efficient search method for
pattern matches against a star catalog. The search
method first must define the way patterns are generated,
then a metric to determine the best match. Specifically,
the “Pyramid” algorithm by Junkins and Mortari has
been chosen, which uses an efficient technique called
the k-vector approach, to search the star catalog and
solve the lost-in-space problem.
them, knowing the angular uncertainty and quickly
listing all possible pairs by extracting the range of
possible matching Y indexes.
The pyramid algorithm uses the identification of the
star pairs to find the star’s location in space. Given a
separation, the algorithm uses the k-vector method to
determine possible star pairs and then the algorithm
intersects those possible pairs to obtain common
matches, thereby extracting the six pairs in the pyramid.
The algorithm contains logic to reject false stars in its
search, and additionally defines an angular separation
probability distribution function in order to estimate the
uniqueness or match frequency of a given star pattern
match.
The problem solved by searching the star catalog using
the k-vector technique is an example of the more
general class of solutions known as hash tables. Hash
tables are used to shorten search times; they use the
hash function to translate the key to a data item into an
index for a table where the data is actually stored. The
search algorithm locates the data quickly without
having to compare its key to all the keys in the table.
The attitude of the star tracker can be found knowing
the stars currently in view and their orientation relative
to a star catalog. The attitude is calculated by
determining the orthonormal matrix (rotation) best
describing the relation between the observed star
positions (found from the pyramid algorithm) and the
reference star positions.
As shown in Figure 2, the distribution of sin2(!) is
nearly uniform, which means an efficient hash table can
be made by using the angular separation between the
stars as the key. The angle is then linked to the stars
defining the angle through their indices, stored in
separate arrays.
Simulation Results: Lost-In-Space Algorithms
An adaptation of the Pyramid algorithm written in C,
suitable for porting to embedded processors, was tested
against star fields generated from 10,000 randomly
oriented attitudes. Images were thresholded, centroided
to discover the stars’ locations, and run through the
lost-in-space algorithm, generating a list of stars within
the field of view. This resulting list was passed to our
implementation of the q-method to calculate the attitude
quaternion.
The simulation was based on a detector model intended
to represent a family of detectors. The detector was
modeled as a 512 x 512 array, 25µm pitch, F/#1.2
optics, and a 30-degree field of view. The detector
efficiency was modeled as 35%, with the noise set to a
dark current of 700 e-/s and read noise of 400 e-/s.
Figure 2. Histogram of sin2(q) of stars’ angular
separation; q, is nearly flat, leading to an efficient
search algorithm.
Given stars at positions ri and rj, the sorted array Y of
angular separations uij is calculated from:
uij = sin 2 (! ij ) = ri " r j
2
Results of the lost-in-space algorithm testing are shown
in Figure 3. Both the cross boresight distribution peaks
(Figure 3, top left) and the about boresight distribution
(Figure 3, top right) show that the most probable error
is fairly low: about 5 arcsec for the cross boresight error
and 10 arcsec for the about boresight error. The shapes
of the error distributions are similar, but the cross
boresight distribution becomes negligible at 30 arcsec,
whereas the about boresight tails off about 120 arcsec.
What this tells us is that there should not be any
significant error above 30 arcsec in the cross boresight,
nor above 120 arcsec in the about boresight.
(3)
The complementary arrays I and J hold the star indices
for each uij in Y. A straight-line approximation uij maps
the ij indices in Y back to their uij values and is used to
form the k-vector.
In practical terms, identifying a star pair becomes a
matter of determining the angular separation between
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Figure 3. Lost-In-Space algorithm testing: Cross Boresight Error (bottom left) and Error Distribution (top
left), and About Boresight Error (bottom right) and Error Distribution (top right).
FAR-MST Algorithm – Velocity and Attitude
typically be a small fraction of a second. A star will
then be projected into the focal plane s of the imaging
array using the perspective equations:
Fast angular rotation rates pose significant problems for
star trackers in general. A stationary tracker would
record stars as nearly point sources, but a rotating
platform tends to spread or streak the image across
several pixels in a line or curve. These effects can be
seen as a function of increasing rotational motion in
Figure 4. Because the motion of the sensor is large
compared to star parallax or star motion, these
perturbation effects are neglected in our treatment.
sx = f
•
where f is the camera focal length (in an ideal camera
model) and rz is the distance along the camera boresight
direction. Differentiating this and using the expression
for the motion of the axis of a torque-free rigid body,
the expression for the motion of the stars in the plane of
the focal array due to the angular velocity is derived:
(4)
!• $ !
#sx & = # rx ry / f
#• & "( f + ry2 ) / f
"sy %
A simplifying assumption has been made: that during
any given frame capture the angular rate would remain
nearly constant, because the exposure time would
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(5)
r
sy = f y
rz
The apparent motion of the star at position r, measured
in the reference plane of the detector rotating with
angular speed ", is:
r =! "r
rx
rz
5
'( f + rx2 ) / f
'(rx ry ) / f
!( $
ry $# x &
&(
rx %# y &
#"( z &%
(6)
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Figure 4. Star streak patterns caused by rotation of imager, shown with increasing rotation rate left to right.
•
The solution proceeds from the fact that the change in
The crux of the issue lies at estimating or measuring s
•
•
the motion, r , is by definition orthogonal to " (Eqn 4).
at several pixels in the image. Estimating s, by
definition, involves knowing the magnitude and
direction of velocity at any given pixel in the image.
The magnitude of the velocity is proportional to the
length of a star streak in the image – but high noise may
cause dim streaks to disappear, making an accurate
determination of length difficult. Also, in considering
only a single image, there is an inherent ambiguity in
the direction of the rotation.
!
If this vector is normalized and called v , we can take a
!
set of measurements of v and find the spin-axis
direction that is most orthogonal to the set of
measurements according to:
#
Our fast angular rate algorithm solves this problem in
three steps. First, raw star field images are processed
using two-dimensional, steerable Gaussian filters to
detect star streaks caused by rotation of the star tracker
and determine their orientation. Second, from these
streaks, an estimate of the direction of rotation is made,
noting that there is still an ambiguity in the sign of the
direction. Lastly, two subsequent exposures are utilized
in order to: a) remove the ambiguity in the sign of the
direction, and b) determine the magnitude of the spin.
" "
!,v i
2
$
$% min
(7)
Knowing the spin axis defines the attitude of the star
tracker.
Simulation Results: FAR-MST Algorithm
The FAR-MST algorithm was tested using 10,000
random angular velocities between 5 and 20
degrees/sec. The results are shown below in Figure 5
and Figure 6. Figure 5 shows the spin axis error
distribution, which demonstrates that the error is very
Figure 5. Spin Axis Error Distribution.
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Figure 6. Spin Magnitude Error Distribution.
small (within about 2.5%) 65% of the time. Above
30% error, the error distribution is essentially constant,
corresponding to a failure of the algorithm and the
ensuing random result.
simulation environment has been developed to test
algorithms demonstrating lost-in-space and fast angular
rate capabilities. The low signal-to-noise ratio typical
of star field snapshots taken while rotating at high
angular rates poses a significant challenge which these
algorithms overcome to a great extent.
The Spin Magnitude Error Distribution (Figure 6) is
essentially Gaussian with the addition of a spike at
–100%. This corresponds to a correct determination of
spin axis and magnitude, but with the wrong polarity.
The extreme high error cases likely correspond to
occasions where section of the algorithm determining
the spin axis failed. Spikes at ±200% represent the sum
of the rest of the errors outside the graph.
The successful development of FAR-MST will provide
microsatellites with a more viable alternative for
attitude determination, providing the level of capability
required for most microsatellite missions in a package
that is more consistent with their overall goals, in terms
of mass, volume, power consumption, and cost.
In general, it has been found that the spin axis section
of the algorithm fails where the visible star field
contains one or more short streaks grouped tightly
together. Since this is a known condition, there are
several possible approaches to rectifying this behavior:
1) increase integration time, generating longer streaks;
2) curve fit, creating longer streaks; or 3) collect images
from two orthogonal fields of view. These approaches
are actively being pursued.
The team is currently engaged in two Phase II efforts –
a Small Business Technology Transfer (STTR) contract
with the Missile Defense Agency (MDA) for
development of the basic Miniature Star Tracker
(MST), and a Small Business Innovation Research
(SBIR) contract with the Air Force Research
Laboratory (AFRL) for development of the additional
features to enable the fast angular rate version (FARMST). At the completion of these two contracts,
AeroAstro expects a fully functional flight prototype
star-tracker capable of providing microsatellites with
accurate position, attitude, and rotational velocity
determination at a cost low enough to enable
widespread incorporation. AeroAstro is actively
looking for flight opportunities to perform an on-orbit
demonstration of the technology.
CONCLUSION
AeroAstro’s Fast Angular Rate Miniature Star Tracker
(FAR-MST) is an optical-based system that uses image
processing of star fields viewed from the satellite to
determine ephemeris information. A star tracker
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ACKNOWLEDGEMENTS
5.
AeroAstro wishes to thank AFRL and the Program
Manger at AFRL, Mr. Jeff Ganley, for sponsorship and
funding under contract FA9453-04-M-0331. Also
gratefully acknowledged is the effort of our team
members at the MIT Space Systems Laboratory, Dr.
Raymond J. Sedwick and graduate research assistant
Kara Huffman.
Accardo, D. Rufino, G. “Brightness-Independent
Start-Up Routine For Star Trackers.” IEEE
Aerospace Electronic Systems, vol. 38, No. 3, pp.
813-823, 2002.
6.
Motari, D., Romoli, A., Difesa, A., 2002. “StarNav
III: A Three Fields of View Star Tracker.” In:
IEEE on Aerospace Conf. Proc., pp. 47–57, 2002.
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Birnbaum, M. “Spacecraft Attitude Control Using
Star Field Trackers.” Acta Astronautica, Vol. 39,
No. 9-12, pp. 763-773, 1996.
Padgett, C., Kreutz-Delgado, K.. “Grid Algorithm
for Autonomous Star Identification.” IEEE
Aerospace Electron. Systems 33 (1), 202–213,
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Eisenmann, A. and Liebe, C.C. “Operation and
Performance of a Second Generation, Solid State,
Star Tracker, The ASC.” Acta Astronautica, Vol
39, No. 9-12, pp. 697-705, 1996.
Clouse, D.S., and Padgett, C.W. “Small Field-ofView Star Identification Using Bayesian Decision
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Accardo, D. and Rufino, G.,. “Star Field Feature
Characterization for Initial Acquisition by Neural
Networks.” In: IEEE on Aerospace Conf. Proc., pp.
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