Content uploaded by John Crassidis
Author content
All content in this area was uploaded by John Crassidis on Jan 28, 2023
Content may be subject to copyright.
Spacecraft Design to Augment
Attitude Estimation from Light Curves
Stephen R. Gagnon∗and John L. Crassidis†
University at Buffalo, State University of New York, Amherst, NY 14260-4400
This paper investigates the effect of an object’s design on its suitability for attitude estimation
via light curve measurements. First a 2D analysis of the light curve model is conducted, and
the effects of various facet properties, such as spectral-dependant parameters, are observed.
Then strategies for designing objects to be more observable are investigated and simulations,
including 3D attitude optimization, are conducted to demonstrate the advantages of these
approaches. It is shown that multi-spectral measurements, in combination with sufficiently
diverse spectral-dependant material properties, can be used to make the objects attitude more
easily estimated, and even reduce the number of observers required to obtain static attitude
observability.
I. Introduction
Over the last half century, humanity have rapidly expanded into space, and they have brought a huge array of
satellites and vehicles with them. In addition to the thousands of active satellites and vehicles in orbit, there are many
times more pieces of debris and naturally occurring objects such as meteoroids. Space domain awareness (SDA) is the
process of identifying and tracking all of these resident space objects (RSOs) in real time. SDA is critical to maintaining
the safety and security of humanity in space. The rapid pace of human expansion in space poses huge challenges to
this task. Any failures in collision detection or avoidance can result in the creation of new debris. Events such as
the Iridium-Cosmos satellite collision in 2009 have demonstrated this problem; the collision between the operational
Iridium 33 satellite and the decommissioned Cosmos 2251 satellite created upwards of 500 new pieces of debris larger
than 10 cm, each with the potential to collide with other satellites in the future. There are already many technological
challenges to SDA in the geostationary (GEO) regime, including limited resolution and coverage of sensors and the
sheer scope of the data association problems that need to be solved. These problems will only become worse as human
activities extend more and more beyond GEO into the ‘XGEO’ regime encompassing a volume in excess of 1000 times
that of GEO. Cislunar activities have been a major focus of expansion recently, and these XGEO activities will only
become more prevalent with time [1].
While the significance of an objects orbit for collision avoidance is clear, the objects attitude must also be tracked to
account for orbital perturbations such as solar radiation pressure and atmospheric drag. In addition, attitude information
is critical to almost any spacecraft maneuver from station keeping to rendezvous, and determining attitude can even
give insight into the function and task of non-cooperative objects. Conventional techniques for attitude determination
typically rely on onboard sensors such as star trackers, or on resolved imagery; however, this dependence on onboard
sensors, or expensive, ground-based, high-resolution telescopes limits the applicability of these techniques. Recently
there has been a significant amount of attention and research focused on utilizing non-resolved imagery of space objects.
These techniques attempt to extract attitude information from the apparent magnitude or intensity of visible light
reflected by the object. However, there are significant obstacles that must be overcome before these measurements can
be utilized. In particular, light reflection models, such as the modified Phong anisotropic model used in this paper [
2
],
are characterized by a high degree of non-linearity, which limits the performance of typical attitude tracking techniques
such as Kalman filtering [
3
,
4
]. In practice, this means that filtering techniques are not viable without an accurate
initial guess for attitude, which limits the applicability of these techniques. Previous work [
5
,
6
] has demonstrated that
an object’s attitude can be estimated from a set of simultaneous measurements from multiple observers via heuristic
optimization.
This paper focuses on investigating the effect of the object design on attitude observability and considering specific
design choices that can make an object more conducive to attitude estimation from light curves. In particular, the
∗Ph.D. Candidate and NSTGRO Fellow, Department of Mechanical & Aerospace Engineering. Email: srgagnon@buffalo.edu.
†
SUNY Distinguished Professor and Moog Endowed Chaired Professor of Innovation, Department of Mechanical & Aerospace Engineering.
Email: johnc@buffalo.edu. Fellow, AAS, Fellow AIAA.
1
effects of spectral-varying material properties along with multi-spectral measurements are considered. These effects are
evaluated in the context of the static attitude determination problem, which is solved via optimization. Previous work
has developed various optimization approaches to this problem [
5
,
6
], and has demonstrated that the results can be used
to initialize conventional filtering approaches to attitude determination in order to avoid convergence issues with those
filtering approaches [5].
The outline of the paper is as follows. First, the object and light curve models will be presented. Then the
methodology for analysis in 2D “flatland” will be presented. Next, a brief outline of the optimization procedures used to
evaluate the static attitude estimation performance will be given. Finally, the results will be presented and conclusions
will be drawn.
II. Methodology
A. Object Geometry
(a) Simple Satellite Geometry (b) Facet Geometry
Fig. 1 Geometry
Table 1 Geometry and Properties
Part Dimensions (m) Surface Properties
Bus Height: 1.7, Width: 1.75, Depth: 1.8 𝑅spec: 0.26, 𝑅diff: 0.6
Solar Panels Length: 4 Width: 1.6 𝑅spec: 0.04, 𝑅diff: .05
Dish Diameter: 1.872 𝑅spec: 0.275/0.26 𝑅diff : 0.275/0.6
Object geometries, particularly man-made objects, can generally be approximated with a set of flat facets. Common
structures such as solar panels and satellite busses can be approximated as rectangular prisms composed of flat facets,
and more complex geometries can be represented using paneling methods. For this reason, it will be assumed that the
observed light magnitude or intensity of an object can be approximated as the sum of reflected light from a set of simple
flat facets.
The parameterization of a single flat facet is shown in Figure 1(b). Each facet is defined by a unit normal vector
𝑢𝑛
as well as two in-plane unit vectors
𝑢𝑣
and
𝑢𝑢
, which are all orthogonal to one another. In general, these facet-specific
unit vectors are known with respect to the body frame of the object. In addition, it is assumed that the locations of
the light source and observer(s) are known. More specifically, we assume we have unit vectors
𝑢sun
and
𝑢obs
pointing
towards the sun and the observer respectively. The half angle vector between the sun and observer vectors is given by
𝒖𝐼
ℎ, 𝑗 =
𝒖𝐼
sun +𝒖𝐼
obs, 𝑗
||𝒖𝐼
sun +𝒖𝐼
obs, 𝑗 || (1)
2
The superscripts
𝐼
and
𝐵
denote the inertial and body frames, respectively. The transformation of a vector between
these frames is given by
𝒗𝐼=𝐴𝑇𝒗𝐵(2)
where
𝐴
is the attitude matrix. Since some vectors are known in the inertial frame (Sun and observer vectors) and the
facet basis vectors are defined in the body frame, vectors must be transformed into a single frame for computations,
creating a dependence on the attitude. In this work, the Galaxy 15 satellite is used as a reference for geometry and
dimensions. The simple facet-based satellite model used is shown in Figure 1(a). This model is comprised of a
rectangular bus with size facets, two solar panels each with a front and back facet, and a dish with a front and back facet.
The specific surface properties of these facets will be varied to study their effect on the the static attitude determination
problem.
B. Light Intensity Model
Light curves are the temporal, wavelength dependent apparent magnitudes of energy (photons) that can be observed
passively and utilized to determine shape and state information about an object. A key component of the light curve
model is the Bi-Directional Reflectance Distribution Function (BRDF), which determines the fraction of incident light
that is reflected in any given direction. The BRDF used in this paper is derived from the Phong anisotropic light
reflection model[
2
]. Under the flat facet assumption, the total reflectance from facet
𝑖
in the direction of observer
𝑗
is
given by
𝜌total,𝑖 , 𝑗 =𝜌spec,𝑖 , 𝑗 +𝜌diff,𝑖, 𝑗 (3a)
𝜌spec,𝑖 , 𝑗 =𝐾
(𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
ℎ, 𝑗 )(𝑛𝑢(𝒖𝐼
ℎ, 𝑗 ·𝒖𝐼
𝑢,𝑖 )2+𝑛𝑣(𝒖𝐼
ℎ, 𝑗 ·𝒖𝐼
𝑣,𝑖 )2
1−( 𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
ℎ, 𝑗 )2)
(𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
sun)+(𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
obs, 𝑗 )−(𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
sun) ( 𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
obs, 𝑗 )𝐹ref (3b)
𝜌diff,𝑖 , 𝑗 =28𝑅diff, 𝑖
23𝜋(1−𝑅spec,𝑖 )
1− 1−𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
sun
2!5
1− 1−
𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
obs,j
2!5
(3c)
𝐹ref =𝑅spec,𝑖 + (1−𝑅spec, 𝑖 )(1−𝒖𝐼
ℎ, 𝑗 ·𝒖𝐼
sun)(3d)
𝐾=(𝑛𝑢+1)(𝑛𝑣+1)
8𝜋(3e)
These equations are a function of the unit vectors associated with each facet, the unit vectors corresponding the Sun
and observer positions,
𝑛𝑢
and
𝑛𝑣
, which are a function of the material properties, and
𝑅diff
and
𝑅spec
which are the
diffuse and specular reflectance of the material, respectively. In particular,
𝑛𝑢
and
𝑛𝑣
determine the size and shape of
the specular reflection cone, and
𝑅diff
and
𝑅spec
, determine the proportion of light reflected diffusely (scattered in all
directions), or reflected specularly (reflected a single direction). If the material is perfectly smooth, the reflection would
be entirely along the reflection direction, given by
𝒖𝐵
reflect =𝒖𝐵
sun −2(𝒖𝐵
sun ·𝒖𝐵
𝑛)𝒖𝐵
𝑛(4)
However, if the surface is rough, then the specular reflection will be distributed in a cone around the reflection direction.
The distribution of the cone in the 𝑢and 𝑣directions is given by 𝑛𝑢and 𝑛𝑣, respectively.
Once the reflectance
𝜌total,𝑖 , 𝑗
is calculated, the light intensity due to the Sun and visible to the observer can be
determined by first calculating the fraction of sunlight that is reflected by each facet, given by
𝐹sun,𝑖 =𝐶sun,𝑖 𝜌total, 𝑖, 𝑗 (𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
sun)(5)
and then calculating the intensity of the sunlight that reaches the observer from each facet, given by
𝐹obs,𝑖, 𝑗 =
𝐹sun,𝑖 A𝑖(𝒖𝐼
𝑛,𝑖 ·𝒖𝐼
obs, 𝑗 )
𝑑2
𝑗
(6)
These depend on the solar power per square meter impinging on the facet,
𝐶sun,𝑖 =455 𝑊
𝑚2
, the area of the facet,
A𝑖
, and
the distance from observer
𝑗
to the object,
𝑑𝑗
. The sum of the intensities from each facet gives the total intensity which
3
would be seen by the observer:
𝐹total, 𝑗 =
𝑖
𝐹obs,𝑖, 𝑗 (7)
Wave l en g th ( n m)
300 400 500 600 700 800 900 1000 1100 1200
Transmission
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
B
V
R
I
Fig. 2 Frequency Transmission for Johnson UVBRI Filters
Wavelength (nm)
400 500 600 700 800 900 1000
Reflectance
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
GaAs Diffuse
GaAs Specular
Polymide Diffuse
Polymide Specular
Al (Specular)
BK7 Glass (Specular)
Fig. 3 Spectral Dependence of Material Properties
4
Wavelength (nm)
300 400 500 600 700 800 900 1000 1100 1200
Intensity (W/(m2*nm))
0
0.5
1
1.5
2
2.5
Extraterrestrial Spectrum
Air Mass 1.5 Spectrum
Fig. 4 Solar Radiation Spectrum
C. Multi-Spectral Measurements
The above light curve model was given without any consideration for the spectral dependence of surface properties.
Following the work of Dianetti and Crassidis [
7
,
8
] we can consider a full multi-spectral light curve model. In general the
parameters
𝑅diff
and
𝑅spec
are functions of wavelength of the incident electromagnetic radiation, meaning that different
intensities will be observed at different frequencies. In practice, multi-spectral measurements are made by dividing the
frequency spectrum into discrete ranges or “bins” and measuring the intensity of light over that range of frequencies. In
Figure 2, the frequency transmission common Johnson UBVRI (ultraviolet, blue, visible, red, infrared) filters are shown.
These filters can allow the decomposition of the incoming light into frequency bins using a mono-chromatic camera.
Spectrometers can also be used that refract or diffract the incoming light such that it is spread across the sensor, which
can give higher frequency resolution as well. Figure 3 also shows values of
𝑅spec
for several common materials. These
are used to determine a reasonable range of material properties for analysis. In addition, the solar irradiance
𝐶sun,𝑖 ,𝜆
is
also now a function of wavelength, and is generated by integrating the solar radiation spectrum over the frequency range
𝜆. The solar radiation spectrum is shown in Figure 4.
The purpose of this work is investigate the fundamental potential for multi-spectral measurements to contribute to
attitude estimation via optimization, so the details of the particular materials and sensors are not considered rigorously.
Instead, the approach is to consider if multi-spectral measurements are valuable under any reasonable conditions, and to
leave the specific material and sensor choices that could achieve good practical results as the subject of future work.
The hope is to demonstrate the fundamental value of multi-spectral measurements so that materials and sensors can
be selected, or even designed, to maximize this value. It’s also apparent from Figure 3 that even for commonly used
materials in satellites, there can be a large range of spectral reflectance values over the frequency spectrum, so if more
exotic materials or coatings are considered, it should be reasonable to assume a wide variety of possible reflectance
curves could be found. For this reason, we assume to have
𝑅diff,𝜆
and
𝑅spec,𝜆
which are the surface parameters for the
frequency range
𝜆
. Then we have measurements
𝐹total, 𝑗,𝜆
for a range of frequency values
𝜆∈Λ
/ based on these material
properties. The exact frequency ranges can be taken to be UBVRI or any other common division of the frequency
spectrum, however, these details are not important to the goal of this study.
5
D. Attitude Parameterizations
1. Three Dimensional Attitude
The transformation from the body to the inertial frame shown in Eq.
(2)
depends on attitude through the attitude
matrix
𝐴
. In three dimensions, this is conventionally the direction cosine matrix (DCM) with nine elements, however,
the DCM can be uniquely represented by a set of three parameters, and it is also computationally advantageous to use
fewer parameters when encoding attitude. In this work quaternions are used to represent attitude in three dimensions,
which add a fourth parameter in order to avoid singularities.
Quaternions are related to the attitude matrix through
𝒒="®
𝒒
𝑞4#(8a)
||𝒒|| =1(8b)
𝐴=(𝑞2
4−®
𝒒𝑇®
𝒒)𝐼3×3+2®
𝒒®
𝒒𝑇−2𝑞4[®
𝒒×] (8c)
where [𝒗×] is the standard cross product matrix of the vector 𝒗.
2. Two Dimensional Attitude
In addition to full three dimensional attitude determination, we consider attitude in 2D “flatland.” In this case,
objects are exist in the 2D plane and attitude can be parameterized as a single angle of rotation about the out of plane
axis, which is advantageous for visualization and understanding. The attitude matrix is now given by
𝐴="cos 𝜃sin 𝜃
−sin 𝜃cos 𝜃#(9)
where 𝜃is the single-axis rotation angle. Consider two vectors given by
𝒖="𝑢1
𝑢2#,𝒗="𝑣1
𝑣2#
In flatland the dot product of two vectors takes the usual form:
𝒖·𝒗=𝑢1𝑣1+𝑢2𝑣2(10)
but the cross product is a scalar [9], given by
𝒖×𝒗=[𝒖⊗]𝑇𝒗=𝑢1𝑣2−𝑢2𝑣1=det[𝒖 𝒗 ]=𝒖𝑇J𝒗=−𝒗×𝒖(11)
where the flatland “cross-product matrix” is now a 2×1vector, defined by
[𝒖⊗] ≜"−𝑢2
𝑢1#=−J 𝒖(12)
and where Jis defined by
J≜"0 1
−1 0#(13)
which satisfies
J2=−𝐼2×2
, where
𝐼2×2
is a
2×2
identity matrix. This can be interpreted as
J
being the “square root”
of −𝐼2×2[9]. The attitude matrix can now be written as
𝐴=cos 𝜃𝐼2×2+sin 𝜃J(14)
The matrix
J
is the two-dimensional representation of the Levi-Civita symbol [
9
]. The flatland cross-product definition
means that the resultant of any 2×2matrix times the flatland cross-product matrix is a 2×1vector.
6
E. Optimization
In order to evaluate the effectiveness of various object design choices in making the object conducive to attitude
determination, an optimization approach is used. This follows previous work which used optimization to solve the
static attitude determination problem. For further details on the particle swarm optimization approached used here, one
should refer to references [6].
The cost function is given by:
min
𝜽
𝑗˜
𝐹total, 𝑗 −ˆ
𝐹total, 𝑗 (𝜽)2
˜
𝐹total, 𝑗 +𝛿(15)
This cost function is a sum of squared errors over the observers, where the error from each observer is normalized by
the measured value of intensity for that observer in order to equally weight the measurements from all observers. In
addition, a small factor
𝛿
is added to the denominator to avoid numerical issues when measurements are close to zero.
To extend this to multi-spectral measurements, we simply consider an additional summation over the frequency bins:
min
𝜽
𝑗
𝜆˜
𝐹total, 𝑗,𝜆 −ˆ
𝐹total, 𝑗 ,𝜆 (𝜽)2
˜
𝐹total, 𝑗,𝜆 +𝛿(16)
III. Results
A. 2D Cost Function Analysis
In this section, we consider the above cost function in 2D flatland in order to visualize the effect of various object
characteristics on the difficulty of the optimization problem. The advantage of treating the problem in 2D is that the cost
function can be plotted vs attitude easily, allowing for qualitative understanding of the effects of certain design decisions.
We will consider two simple geometries, a triangle and a rectangle, both shown in Figure 5. The exact dimensions
and properties are given in Table 2. The triangle is chosen with different length sides so that there are no geometric
symmetries of rotation, while the rectangle is chosen since it exhibits 180 degree rotational symmetry.
(a) Triangle (b) Rectangle
Fig. 5 2D Geometry
First we will consider the triangle. Cost function plots for a single observer, and for two observers are shown below.
In addition, plots for diffuse only reflections and for multi spectral measurements are included. From Figure 6(a), it is
clear that a single observer is insufficient for estimation. There are multiple attitudes where the cost function value is
zero, meaning the cost function is ambiguous, and it is impossible to identify the correct attitude from one measurement,
despite one measurement being sufficient for local observability. To solve this issue, a second observer can be added,
which does make the problem solvable, however, the cost function (shown in Figure 6(b)) is still highly non-linear
7
Table 2 2D Geometry and Properties
Object Side Lengths Surface Properties
Triangle
Side 1: 3
Side 2: 4
Side 3: 5
𝑅spec: 0.2, 𝑅diff: 0.9
Rectangle Side 1: 2 Side 2: 4
Side 3: 2 Side 4: 4 𝑅spec: 0.1, 𝑅diff: 0.9
(a) One Observer (b) Two Observers
(c) One Observer, Multi-Spectral (d) Two Observers, Diffuse Reflection
Fig. 6 Cost Function for Triangle
making the optimization or estimation problem very difficult to solve. The extreme peaks in the cost function are
products of the spectral reflection. This is illustrated in Figure 6(d), where it can be seen that removing the spectral term
(leaving only diffuse reflections) eliminates many of the extreme contours of the cost function, and significantly reduces
the number of local minima and maxima that might interfere with optimization. However, diffuse only reflections are
not necessarily realistic for most common spacecraft materials. A simpler approach to making the cost function more
tractable is to consider multi-spectral measurements. To simulate this, each side of the triangular object was assumed to
reflect more strongly in one frequency range, so the model consisted of three frequency bins to accomodate this. In
Figure 6(c), it can be seen that only 3 frequency bins from a single observer are required to create a non-ambiguous cost
function without the need for a second observer.
8
(a) Two Observers (b) One Observer, Multi-Spectral
Fig. 7 Cost Function for Rectangle
However, other more complex geometries can pose additional problems. In particular, some geometries such as the
rectangle show in Figure 5(b), can have inherent symmetries that even additional observers cannot resolve. This is
illustrated in Figure 7(a), where it can be seen that the rectangular object’s symmetries result in ambiguities in the cost
function, even when two observers are used. This is apparent in the repeated structures of the cost function, which each
individually contain points where the cost goes to zero. On the other hand, the use of multi-spectral measurements creates
a non-ambiguous cost function, even with only one observer. This indicates that the multi-spectral measurements are
able to distinguish between orientations that would be otherwise indistinguishable for mono-chromatic measurements.
B. 3D Optimization
In order to test if the effects observed in 2D translate to 3D, we consider the performance of optimization on the
3D cost function. We utilize the Multiplicative Particle Swarm Optimization (MPSO) algorithm to optimize the cost
function described in Equation 16. For details on the optimization routine and associated optimization parameters,
the reader should refer to our previous work developing the approach [
6
]. The most interesting result from the 2D
analysis was the potential for multi-spectral observations to reduce the number of required observers, so that will be the
focus of this section. In order to evaluate this, the MPSO optimizer is run 5 times for 1000 random true attitudes, with
the number of observers varied from 1 to 5, and the fraction of runs that the optimizer successfully identifies the true
attitude is recorded. All other parameters of the optimization are the same between the the simulations. Two cases are
considered, a multi-spectral case and a standard monochromatic case. The satellite dimensions and parameters are
given in Table 1. For the multi-spectral case, the spectral and diffuse coefficients are the same as the monochromatic
case, except for the bus where they are assigned such that each facet reflects strongly in one frequency and weakly in the
others. This could be viewed as having a sufficiently unique color or material on each facet of the bus. The results can
be seen in Figure 8(a). Clearly the multi-spectral measurements allow for significantly improved performance with
fewer measurements. What is even more interesting is that optimization using multi-spectral measurements performed
well with only two measurements which is less than the minimum of 3 required for observability with monochromatic
measurements [
10
], so the addition of multi-spectral measurements is adding observability in addition to eliminating
ambiguities as observed in the 2D analysis. Furthermore, it seems as though there is little improvement when adding
additional observers beyond two. This needs to be investigated further.
C. Real Materials
Next, some real materials are considered to see if these results hold up under more realistic conditions, and to
consider what design decisions could be made to make an object more conducive to attitude estimation from light
curves. To obtain the values of
𝑅Spec,𝜆
and
𝑅Diff,𝜆
, the curves in Figure 3 are divided into bins of 100 Hz, and the values
in these bins are averaged. Three materials are considered: polymide, aluminum and glass. These are chosen because
polymide has the largest variation in reflectance with frequency, while aluminum exhibits minor variation and glass
9
(a) Mono-Chrom. vs. Multi-Spectral (b) Single Material Comparison
(c) Material Combination Comparison (d) Random Reflectance Comparison
Fig. 8 Optimization Performance Comparisons
exhibits almost no variation. The facets of the satellite bus are assigned values from these materials. A first test is
consider the entire satellite bus being made of one material. These results are shown in Figure 8(b). All three materials
with mutli-spectral measurements perform better than the monochromatic case, however, the range of reflectance values
of the materials also corresponds to performance, with polymide performing best and glass performing worst. This
demonstrates that materials with high variation of reflectance over the spectrum are better for attitude estimation.
In addition, we consider combinations of materials. In Figure 8(c), results are shown for several selected
combinations. It can be seen that once again, for all the considered combinations, the performance is still far superior
to the monochromatic case, indicating that the the advantage of reducing the number of observers seems to still exist
in more realistic cases. The results also show that a large amount of polymide with some aluminum mixed in gives
the best results, but pure polymide and a mixture of polymide, aluminum and glass both give worse results. This
indicates that having diversity of materials between facets is also valuable, and substituting a material with lower range
of reflectance (aluminum) can still improve the performance by eliminating symmetries of the bus. To further highlight
this, randomized reflectance parameters are used. Figure 8(d) shows that randomized reflectance coefficients give better
performance than the realistic cases, and that a larger range of reflectance coefficients improves performance in this case
as well. While these results are not realistic, they give an indication of how materials/coatings might be designed to
maximize attitude estimation performance. Maximizing the diversity of reflectance, both within the spectral responses
of the individual materials and across the different facets seems to give the best results.
10
IV. Conclusions
We have shown that the use of multi-spectral measurements can overcome ambiguities in the light curve model and
also reduce the number of measurements required for static observability. We have demonstrated these effects both
qualitatively in the structure of the cost function for the 2D case, and also in the performance of attitude estimation
via optimization in the 3D case. In addition, we have shown that these effects exist with realistic materials found in
satellites. Finally, we have determined that increased diversity of the spectral reflectance parameters can further improve
optimization performance. Future work will focus on analytical demonstration of the reduced observability requirements
as well as further investigation into possible material or coating selection to maximally leverage these phenomenon.
Acknowledgments
This work was funded by a NASA Science and technology research fellowship. Many thanks to James McCabe for
his advice and insight.
References
[1]
Holzinger, M. J., Chow, C. C., and Garretson, P., “A Primer on Cislunar Space,” Tech. Rep. 1271, Air Force Research Laboratory,
2021.
[2]
Ashikmin, M., and Shirley, P., “An Anisotropic Phong Light Reflection Model,” Tech. Rep. UUCS-00-014, University of Utah,
Salt Lake City, UT, 2000.
[3]
Linares, R., Jah, M. K., Crassidis, J. L., and Nebelecky, C. K., “Space Object Shape Characterization and Tracking
Using Light Curve and Angles Data,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 1, 2014, p. 13–25.
https://doi.org/doi:10.2514/1.62986.
[4] Crassidis, J. L., and Junkins, J. L., Optimal Estimation of Dynamic Systems, CRC Press, Baco Raton, FL, 2012.
[5]
Gagnon, S. R., and Crassidis, J. L., “Static Attitude Estimation from Light Intensity Measurements: A Heuristic Approach,”
Astrodynamics Specialist Conference, 2020. AAS 20-735.
[6]
Gagnon, S. R., and Crassidis, J. L., “An Improved Particle Swarm Optimization Approach for Static Attitude Estimation from
Light Intensity Measurements,” Astrodynamics Specialist Conference, 2021. AAS 21-727.
[7]
Dianetti, A. D., and Crassidis, J. L., “Space Object Material Determination from Polarized Light Curves,” AIAA SciTech Forum,
2019. AIAA 2019-0377.
[8]
Dianetti, A. D., and Crassidis, J. L., “Space Object Attitude Determination from Multispectral Light Curves,” AIAA SciTech
Forum, Orlando, FL, 2019. AIAA 2020-1098.
[9]
Shuster, M. D., “Attitude Analysis in Flatland: The Plane Truth,” Journal of the Astronautical Sciences, Vol. 52, No. 1-2, 2004,
pp. 195–209.
[10]
Linares, R., and Crassidis, J. L., “Dynamic Observability Analysis for Attitude, Angular Velocity, Shape, and Surface
Parameters,” AAS/AIAA Space Flight Mechanics Meeting, Napa, CA, 2016. AAS 16-515.
11
