Three-Degree-of-Freedom Estimation of Agile Space ObjectsUsing Marginalized Particle Filters

Ryan D. Coder∗ and Marcus J. Holzinger†

Georgia Institute of Technology, Atlanta, Georgia 30332

and

Richard Linares‡

University of Minnesota, Minneapolis, Minnesota 55455

DOI: 10.2514/1.G001980

Several innovations are introduced for space object attitude estimation using light-curve measurements. A

radiometric measurement noise model is developed to define the observation uncertainty in terms of optical,

environmental, space object, and sensor parameters and is validated using experimental data. Additionally, a

correlated process noise model is introduced to represent the angular acceleration dynamics. This model is used to

account for the unknown inertia and body torques of agile space objects. This linear dynamics model enables the

implementation of marginalized particle filters, affording computationally tractable three-degree-of-freedom

Bayesian estimation. The synthesis of these novel approaches enables the estimation of attitude and angular velocity

states of maneuvering space objects without a priori knowledge of initial attitude while maintaining computational

tractability. Simulated results are presented for the full three-degree-of-freedomagile space object attitude estimation

problem.

I. Introduction

IMPROVEMENTS in space domain awareness (SDA) areidentified in multiple national policy documents as a top priority

to protect the United States and its allies as well as maintain itseconomic and diplomatic objectives [1]. The high-level activities ofSDA include the detection, tracking, characterization, and analysis ofspace objects (SOs), as defined in [2]. Space objects are typicallydefined as active and inactive satellites, rocket bodies, and orbitaldebris [3]. To fully characterize a space object, it is necessary toobtain knowledge about both the SO’s shape and attitude, which caninform payload capability or mission purpose [4]. For SOs in lowEarth orbit, shape and attitude estimation is performed using radar-based methods pioneered in the early 1980s [5]. The shape andattitude of large SOs can also be estimated from resolved imagerytaken by ground-based optical sensors. However, when SOs are toodistant to be imaged by radar facilities or too small to be adequatelyresolved by ground-based optical sensors, the primary source of dataprocessed is unresolved imagery [4].Each unresolved image can be analyzed to determine the power

reflected by the SO. A typical observation campaign of severalimages can then be used to create a light curve: a temporally resolvedsequence of power measured over specified wavelengths. Becausethe total amount of flux reflected by the SO is dependent on the SOshape and attitude, estimating either the attitude or shape of the SO ispossible using the observed light curve [6]. This process is referred toas light-curve inversion and was initially developed to characterizeasteroids [7]. Past efforts to characterize asteroids have used batchestimation methods, where attitude, angular rates, moments ofinertia, and shape model are all simultaneously estimated [8–11].

Although the light-curve inversion process is similar, there areseveral important differences between asteroids and man-made SOs.The first significant difference is that, unlike asteroids, many SOshave highly angular surfaces composed of several materials.Therefore, the specular reflection models required for SO light-curveinversion are more complex and nonlinear than that of asteroids.Consequently, the estimated SO attitude is often separated frommaterials and shape properties, collectively referred to as the SO“shape model” [4]. Another difference is that active satellites employactuators to change and maintain mission specific orientations,requiring dynamics models that account for this motion withoutknowledge of the external torques or inertia tensor of the satellite.This work develops an estimation approach that can deal with both ofthese challenges.The first published work outlining the potential application of

light-curve inversion to SO characterization was given by Hall et al.in 2005 [6]. Wetterer and Jah [12] provided simulated results fornonmaneuvering SOs using a single unscented Kalman filter (UKF)to estimate the SO attitude, the shape model, or both simultaneously.The synthetically generated measurements are corrupted by time-invariant, zero-mean Gaussian white noise whose covariance isrepresented in the visual magnitude scale based on historicalobservations. The authors of that work concluded that more accuratemeasurement models could alleviate discrepancies betweenobservational and simulated data [12]. To mitigate these issues, thispaper expands upon [13] to define a new radiometric measurementnoise model, based on SO and environmental parameters. Thiscontribution increases the likelihood that estimators that workwell insimulation will also perform well with operational data.More recently, Holzinger et al. [14] recognized that bidirectional

reflectivity distribution function (BRDF) measurement models arenonlinear functions of attitude states, resulting in potentially non-Gaussian state distributions. Recognizing the limitations of UKFs fordistributions not accurately summarized by amean and variance, thispast effort uses a particle filter (PF) to estimate the attitude states ofagile SOs. It is also shown how shape model bias can be included inthe estimated states. The simulated results presented use a visualmagnitude measurement noise based on historically collectedmeasurement data, like work before it. To account for the unknownSO torques and inertia properties, SO angular rates are modeled as awhite noise process. The number of particles necessary for thisapproach presents a computational challenge [14].To reduce the number of particles required, a more sophisticated

dynamics model is proposed in this investigation. Many tracking

Presented as Paper 2015 at the 25th AAS/AIAA Space Flight MechanicsMeeting, Williamsburg, VA, 11–15 January 2015; received 13 January 2016;revision received 28 July 2017; accepted for publication 28 July 2017;published online XX epubMonthXXXX. Copyright © 2017 by theAmericanInstitute ofAeronautics andAstronautics, Inc.All rights reserved.All requestsfor copying and permission to reprint should be submitted to CCC atwww.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884(online) to initiate your request. See also AIAA Rights and Permissionswww.aiaa.org/randp.

*Ph.D. Candidate, Guggenheim School of Aerospace Engineering;rcoder@gatech.edu.

†Assistant Professor, Guggenheim School of Aerospace Engineering;holzinger@gatech.edu. Senior Member AIAA.

‡Assistant Professor, Department of Aerospace Engineering andMechanics; rlinares@umn.edu. Member AIAA.

1

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. , No. ,

methods have addressed the problem of unknown motion byassuming that the unknown acceleration can bemodeled as aMarkovprocess, where the acceleration is correlated exponentially over shortperiods of time [15–19]. Thus, the second contribution of this work isto adapt this idea, such that SO angular accelerations are correlatedexponentially over short periods of time. This exponentiallycorrelated model makes the body angular velocities continuous withtime, as they are in true kinematic motion. This continuity is notreflected in the previous white noise model.Defining the attitude, attitude rates, and attitude accelerations

for three-degree-of-freedom motion requires a minimum of ninestates. Including bias terms for shape uncertainty increases thedimensionality. Because particle filter computation scales exponen-tially with state-space dimension, estimating large numbers of states iscomputational intractable. To ameliorate this curse of dimensionality,the final contribution of this work is to apply marginalized particlefilters (MPFs), also known as a Rao–Blackwellized filters [20]. Thecore concept of the marginalized particle filter is to exploit any linearsubstructure in the model that is subject to Gaussian noise [21].Because the exponentially correlated dynamics model proposed aredescribed by a linear set of equations, MPFs reduce the number ofnonlinear states to three. Critically, this enables the estimation ofattitude and angular velocity states of maneuvering space objectswithout a priori knowledge of initial attitude, while maintainingcomputational tractability.This work is organized as follows. The first contribution of this

work, the radiometric measurement noise model, is presented inSec. III.A. The second contribution, the exponentially correlatedangular acceleration dynamics model, is presented in Sec. III.B. Thethird contribution, the marginalized particle filter, is presented inSec. III.C. A; simulation framework, results, and discussion arepresented in Sec. IV.

II. Radiometric and Electrooptical Sensor

Developing a radiometric measurement noise model requires twoelements. The first is an accurate accounting of the number ofphotons emitted by the sun, reflected off the SO, and ultimatelymeasured the electrooptical (EO) sensor of a telescope. This first partis covered in Sec. II.A. The second is a description of how thesephotons are recorded by an electrooptical sensor and howmuch noiseis added to the total SO signal by the sensor itself. This second portionis covered in Sec. II.B.

A. Space Object Signatures

By convention, SO brightness is quantified using the apparentvisual magnitude system, first developed by early astronomers. Thesystem is unitless, logarithmic, and references the brightness of Vegaas the scale’s zero point. The resulting SO signature represented in theapparent visual magnitude system,mv;SO, is found using Eq. (1) [22]:

mv;SO � mv;Sun − 2.5 log10

�MSO

MSun

�(1)

The visual magnitude of the sun is typically given as −26.73, andMSO is the total radiant excitance of the SO,which is given byEq. (2):

MSO � 1

R2

ZλUL

λLL

MSun�λ�Fr

�θBI ; s; R; λ

�dλ (2)

In this equation, R is the distance from the SO to the observer,MSun�λ� is the spectral excitance of the sun,Fr is the BRDF, and the ⋅operator denotes the unit vector of the quantity. The BRDFdetermines the number of photons reflected from the SO toward theobserver and is a function of shape model, material composition, andSO attitude. The quantity s is the unit vector from the sun to the SO,and the rotation from the inertial frame to the body frame of the SO isdenoted by θBI . The integrand is evaluated between the wavelengthlimits λLL and λUL, which correspond to the sensitive wavelengths ofthe sensor or the limits of the “color” filter in the case of radiometric

measurements. The spectral excitance of the sun can be modeled

using a black-body radiator assumption [22]. The solar spectral

excitance is then converted to a photon flux density, ΦSO, inphotons∕s∕m2, assuming a weighted average for the wavelengths of

light reflected. The light-gathering capabilities of a ground-based

sensing application can then be calculated as the photon flux captured

by the optical system, qSO, measured in e−∕s, is given by Eq. (3) [23]:

qSO � ΦSOτatmτopt

�πD2

4

�QE (3)

In Eq. (3), the aperture diameter of the telescope isD, whereas τatmand τopt are the transmittance of the atmosphere and optics assembly,

respectively. The quantum efficiency of the EO sensor is defined as

QE. The value of these two transmittances and the QE are

wavelength-dependent. In lieu ofmore detailedmodeling, these threevariables are defined to have values ranging from τ ∈ �0; 1� andQE ∈ �0; 1� using a weighted average value for the wavelength of

incident light.To accurately characterize noise due to background light, the local

background radiant intensity Isky, whosemajor sources aremoonlight

and local light pollution, must be determined. In relatively light-

polluted areas, it is suggested that a sky sensor is used to directly

measure this quantity. Typical values for radiant intensity vary fromIsky ∈ �15; 22� mv∕arcsec2 for urban and rural skies, respectively

[24]. The total photon radiance at the telescope aperture due to

background sky pollution, Lsky, in photons∕s∕m2∕sr, is given by

Eq. (4) [23]:

Lsky � Φ010−0.4Isky

�180

π

�2

36002 (4)

Here, the variable Φ0 is the radiant intensity of the reference zero

point, which traditionally is the star Vega. A final expression for the

photon flux per pixel resulting from background radiant intensity,qp;sky, is expressed in e−∕s∕pixel, as shown in Eq. (5). The total

incidence on the focal plane transmitted from the radiance at the

telescope aperture is found using the “camera equation” [25]. This

work uses the simplified form of the camera equation for a singletlens, valid for all focal ratios N, yielding the photon flux incident on

the EO sensor in Eq. (5):

qp;sky �LskyτoptπQEp

2

1� 4N2(5)

In Eq. (5), the EO sensor is assumed to have square pixels. For

nonsquare pixels, p2 can be replaced by the appropriate pixel unit

of area.The radiometric model presented here is a summary of previous

work, and additional details are available to the interested reader [13].

The model defines the photon flux of SOs, in Eq. (3), and the

background sky brightness, in Eq. (5), as a function of variousenvironmental variables and optics system parameters. Using these

two quantities, it is then possible to define both the mean signal and

variance from important noise sources in terms of electrons. These

equations are valid for EO sensors capturing unresolved imagesof SOs.

B. Noise Sources in Electrooptical Sensors

An excellent discussion on EO sensor noise sources, which

includes the development of the “CCD equation” and outlines acomputer simulation of CCD noise, is presented by Merline and

Howell [26]. The goal of this subsection is to summarize the largest

noise sources outlined in that work andmathematically relate them to

the equations of Sec. II.A. Typically, the largest types of noiseinherent in unresolved images of SOs are Poisson or “shot” noise

from the SO and background radiant sky intensity (i.e., light

pollution). To quantify the largest noise sources, let the total variance

in the integrated signal be defined as Eq. (6) [26]:

2 CODER, HOLZINGER, AND LINARES

σ2S � m

�1� m

z

�σ2r �

Xmi�1

h�Ci − C∘

i

�Gi� m2

z2

Xzj�1

h�Cj − C∘

j

�Gi

(6)

It is emphasized that Eq. (6) is written in units of electrons and not

analog-to-digital units (ADUs), which are commonly referred to as

“counts”. Therefore, the variance in an SOmeasurement is calculated

from the total counts C converted to electrons by the CCD gain G.

The superscript ∘ in the second and third terms indicates that these

counts are due to direct current bias. The first term is due to readout

noise σ2r , the second term is the photon noise of the SO, and the third

term is the photon noise from the background. Each of these

components is described in more detail in the following equations.

This work neglects the variance due to the conversion from

electronics to ADUs as well as the variance in digitization offset

because these are typically less than half an ADU per pixel, which is

small compared to other sources for typical ground-based sensors

[26]. Furthermore, this derivation has assumed that thevariance of the

total signal and background are uncorrelated and zero mean.

Therefore, the subtraction of the direct current bias in Eq. (6) can be

approximated as shown in Eqs. (7) and (8), thus linking the noise in

collected images with their physical origins [26]:

Xmi�1

��Ci − C∘i�G� ≃ qSOtI �m�qp;sky � qp;dark� (7)

Xzj�1

��Cj − C∘j�G� ≃ z�qp;sky � qp;dark� (8)

In Eqs. (7) and (8), qSO is the photon flux reflected by the SO as

defined in Eq. (3); qp;sky is the photon flux per pixel from the

background sky irradiance as defined in Eq. (5); andqp;dark is the darkcurrent per pixel. Additionally, tI is the integration time (i.e.,

exposure time) of the observation. To accurately calculate the signal

received by the sensor, one must subtract the background signal

present. Althoughmanymethods are available, thiswork assumes the

simplest technique, which estimates the mean background signal

from pixels that are free of stars, SOs, and other objects of interest

[27]. The subscript j is used to denote this random sample of z object-free pixels used to determine themean background noise,whereas the

subscript i is used to denote a pixel that lies in the array of pixels

containing the SO,m. The arrival process of photons incident on the

CCD plane is described by a Poisson process. Consequently, the

variance in the number of electrons generated in an EO sensor can be

defined by Eq. (9), by combining Eqs. (7) and (8) [28]:

σ2S ≈ qSOtI �m

�1� m

z

�h�qp;sky � qp;dark�tI � σ2r

i(9)

Equation (9) shows how the noise present in images can be

rudimentarily modeled. Critically, this model assumes that the

atmospheric transmittance τatm is constant. Using the foundational

knowledge of this section, Sec. III.A demonstrates how one can

model the atmospheric transmittance as a random variable and how

this increases the measurement variance.

III. Methodology

Using the brief summary of radiometry and EO noise literature of

the previous section, a radiometric noise model is presented in

Sec. III.A. Following this, Sec. III.B summarizes the how the true

dynamics of an agile SO can be modeled with exponentially

correlated angular accelerations (i.e., the Singer model). Finally,

Sec. III.C details how the proposed dynamics model greatly reduces

computational burden when coupled with a marginalized particle

filter.

A. Measurement Noise Model

The radiometric measurement function proposed in this sectiondefines the noise present in an image as a function of optical andenvironmental parameters. This stands in contrast to previous light-curve inversion studies for estimating satellite attitude, which usevisual magnitudes and a time-invariant measurement variance[12,14,29]. It should be noted that these previous studies, whichimplemented various sequential filters, could also implement themeasurement model proposed in this work. Because the number ofphotons incident on anEO sensor are typically on the order of 103 andhigher, this Poisson process is well approximated by a Gaussiandistribution, leading to the definition of measurement noise inEq. (11):

yk�x; tk� � qSO�x; tk� � vk�x; tk� (10)

vk�x; tk� ∼N �0; Rk�x; tk�� (11)

Here, the notation for a random sample drawn from adistribution is∼, where N �⋅; ⋅� denotes a Gaussian distribution with mean andcovariance input. In Sec. II, the variance of a SO signature capturedby an EO sensor is defined by Eq. (9). In this equation, the values ofatmospheric losses and sky brightness are deterministic. However,the phase and position of themoon, time of night, presence of clouds,and scintillation effects all cause the atmospheric transmittanceand radiant sky intensity to be random variables. Atmospherictransmittance and radiant sky intensity vary temporally and spatially,and both typically increase near the local horizon. Because this worksimulates the SO light curve over a short time period, the radiantsky intensity is treated as constant, whereas the atmospherictransmittance is treated as a random variable. Depending on thepower of the scintillation, for stars near, the zenith atmospherictransmittance is best described as a Gaussian, log-normal, orF distribution [30]. For simplicity of implementation, this work willimplement the atmospheric transmittance as a Gaussian distributedrandom variable.Consequently, this work proposes amixed Poisson distribution for

the measurement model. In general, two elements are necessary todescribe a mixture distribution: the conditional distribution of arandom variable on another, and the distribution of the mixingparameter itself. Thus, the number of photons incident on the EOsensor, ζ, is a mixed Poisson random variable whose rate parameter λis conditional on the normally distributed randommixing parameter,the atmospheric transmittance. This can bewrittenmathematically asshown in Eq. (12) [31]:

ζ ∼ P�λ� ∧λN �μatm; σ2atm� (12)

As is standard, λ defines the typical number of photons perpredefined interval. In Eq. (3), the number of photons per second,qSO, is defined. Thus, λ � qSOtI , where tI is the length in seconds ofthe image exposure. To define the measurement uncertainty, oneneeds to define the variance of the mixed Poisson process. Thevariance of a mixed Poisson process has been analytically derived tobe greater than that of a homogenous Poisson process, as shown inEq. (13) [32–34]:

Var�ζ� � E�ζ2� − �E�ζ��2 � E�λ� � Var�λ� (13)

Here, the expected value of the random variables is denoted usingthe standard notation E�⋅�. To evaluate the mean and variance of therate parameter, note that the only random variable in Eq. (3) is theatmospheric transmittance. Because it has been assumed in this workthat the atmospheric transmittance may be represented as a Gaussianrandom variable, and hence has finite first and second moments, themean and variance of the shot noise can be expressed as

E�λ� ��ΦSOτopt

�πD2

4

�QEtI

�μatm (14)

CODER, HOLZINGER, AND LINARES 3

Var�λ� ��ΦSOτopt

�πD2

4

�QEtI

�2

σ2atm (15)

One can define a time-dependent zero-mean Gaussian white noiseusing the covariance defined by Eq. (16). Comparing thedeterministic noise of Eqs. (9–16) reveals that the only differenceis the addition of the second term in Eq. (16), which includes theeffect of the varying atmosphere Eq. (16):

Rk�x; tI� ≈ E�λ� � Var�λ� �m

�1�m

z

���qp;sky � qp;dark�t�

σ2rn2

�

(16)

As a demonstration of the features of the radiometric model, and tofurther illustrate the error of using constant magnitude measurementerror, refer to Fig. 1. These 2829 observations of GALAXY 15 wereexperimentally collected in May and June of 2015 using a Raven-class telescope located in Kihei, Hawaii. The 16 in. f∕5.628 Raventelescope used an Apogee U47 CCD, a Johnson R filter, and a 5 sexposure time.In Fig. 1, the gray dots are the experimentally calculated

observations, where the data are reduced using traditional “all-sky,”also known as “fixed” photometry techniques. These methodologiesassume that the atmospheric transmittance τatm is constant. To showhow the proposed measurement noise model can differ, the solid lineis the radiometric model with σatm � .001, the dashed line is theradiometric model with σatm � .01, and the dash-dotted line is thehistorical, constant measurement noise of 0.1mv. For all the linesrepresentative of the model, the atmospheric transmittance isassumed to be τatm ∼N �.6; σ2atm� (unitless), and the radiant skyintensity is assumed to be Lsky � 18 mv∕arcsec2 [23].For this specific Raven-class telescope, the historically assumed

magnitude noise is greater than actual system noise, but this trendshould not be construed to be true for all systems. Examining Eq. (9),for example, indicates that the historically assumed 0.1mv noisecould underrepresent the noise of a system observing dim objectsunder very bright skies. Further, the difference between the σatm �.001 and σatm � .01 lines illustrates the potential shortcomings of“all-sky” photometry techniques when data are collected on nightswhere the atmospheric conditions are rapidly changing. As thevariance in atmospheric extinction increases, the measurementuncertainty is consistently underestimated for bright objects.Inspection of Fig. 1 reveals that the measured data has a greater

spread than the model trends. To explain this, it is important to noteadditional sources of noise that are not captured in this model. Onesource of error present in experimentally obtained data not capturedby the model are uncertainties in the star catalog. These referencestars are used in the conversion from instrumental counts to visualmagnitude, and any uncertainties in the brightness of individual starspropagate into themeasured values of the light curve.Another sourceof error is the assumption of a constant sky brightness, where thephase of the moon and elevation of the telescope at the time of

observation both ensure the sky brightness is varying both spatiallyand temporally. Finally, local weather conditions, such as cloudcover, can cause abrupt changes in atmospheric transmittance, whichare not captured by the model.However, using the radiometric model developed in this work, it is

possible to simulate much of the experimentally observed noise. Bymodeling the stochastic process of photon arrival on the EO sensorimage plane, the measurement variance changes with objectbrightness, as opposed to constant-variance models. It should benoted that alternative measurement models using radiance as definedin the SI system (i.e., W∕m2∕sr) would also be statistically correct.However, dark current and read noise are typically defined inmanufactures’data sheets in terms of electrons, making the presentedmodel easier to implement in practice. It is hoped that, if morerealistic measurement noise models (like the one outlined in thissubsection) are implemented, the performance demonstrated instudies with simulated results will be representative of theperformance achieved with real data.

B. Exponentially Correlated Angular Acceleration Model

The light-curve attitude estimation problem involves estimatingboth attitude and angular velocity stateswhere attitude kinematics arewell known but the angular velocity dynamics may be unknown dueto unknown control torques, unknown inertia tensors, and unknowndisturbance torques. This work seeks to model the resultingmotion using an exponentially correlated process noise model. Thegeneral continuous-time dynamics of such a problem with scalarmeasurements can be described by Eq. (17):

_x�t� � f�x; t� � G�x; t�w�t� (17)

It is emphasized that the measurement is defined by Eq. (10) forthe light-curve inversion problem. The system state is defined as

xT � �θBTI ωT �, where θBI are the 3-2-1 Euler angles defining therotation between the SO body frame (B) to the inertial frame (I), andthe angular velocity of the SO is denoted byω. Assuming that the SOcan be modeled as a rigid body, the rotational motion is given byEuler’s rotational equation of motion and the Euler angles kinematicrelationship:

�_θBI_ω

��

�B�θBI �ω

−J−1�ω × Jω� � J−1T

�(18)

where

B�θBI � �1

cos θ2

24 0 sin θ3 cos θ30 cos θ2 cos θ3 − cos θ3 sin θ3cos θ2 sin θ2 sin θ3 sin θ2 cos θ3

35 (19)

Here, J and T represents the true inertia tensor matrix and the sumof all applied control and external torques, respectively. For agileSOs, the true inertia and torques acting on a SO are typicallyunknown. When the inertia properties or maneuver capabilities of atracked object are unknown, it is very common to model the resultanttime-varying accelerations of maneuvering objects as a randomprocess [35–38]. Random processes described in the literature can beclassified in three general groups: white noise models, Markovprocess models, and semi-Markov jump process models [19]. Whitenoise models were first applied to the SO attitude estimation problemby Wetterer et al. [12]. Holzinger et al. overcame these issues by

modeling both J−1T and −J−1�ω × Jω� with process noisedescribed by Eq. (20) [14]:

� _θBI � �hB�θBI ��ωμ � δω0�

i(20)

δω0 ∼N �0;Qω� (21)

Using this model, which is not statistically rigorous, the meanmotion of the SO can be defined asωμ, andmotion about this nominalFig. 1 Measurement noise of GALAXY 15 observations.

4 CODER, HOLZINGER, AND LINARES

trajectory is modeled by the process noise with appropriately sizedδω0. If the mean motion is unknown, then ωμ � 0 and Qω can be

sized such that δω0 is representative of SO maneuver capability.This work adapts a Markov process model, first proposed in the

1970s to track maneuvering aircraft. Known by its inventor, the“Singer model” defines the target accelerations as correlated in timeduring a maneuver [15]. In this work, the angular accelerations_ω�t� � α�t� are assumed to be correlated in time with theautocorrelation defined by Eq. (22):

Rα�τ� � Ehα�t�αT�t� τ�

i� σ2me

−βjτjI3×3 (22)

In Eq. (22), σ2m is the resulting variance of the maneuvering targetbody angular acceleration, and β is the inverse of the maneuveracceleration time constant τ. The angular acceleration can thereforebe expressed as the second-order Markov process shown next:

_α�t� � −βα�t� �w�t� (23)

The process noise of the angular acceleration,w�t�, is driven by thepower spectral density of the angular acceleration, as given byEq. (24):

Qα�τ� � 2βσ2mδ�τ�I3×3 (24)

Here, δ�τ� is the Dirac delta. Without loss of generality, one candefine different time constants and acceleration variances for eachaxis of the SO body-fixed frame. Letting the subscripts 1 through 3denote each orthogonal axis yields

Qα�τ� � 2δ�τ�� β1 β2 β3 �Thσ2m;1 σ2m;2 σ2m;3

i(25)

The continuous dynamics model for agile SOs proposed in thiswork is given by Eq. (26). Similar to Eq. (18), θBI defines therelationship between the body angular rates and the attitudecoordinates used to represent SO(3) in the inertial frame:

264_θBI_ω_α

375 �

24 03×3 B�θBI � 03×303×3 03×3 I3×303×3 03×3 diag�−β�

3524 θBI

ωα

35�

24 03×303×3I3×3

35w

(26)

To implement the model in discrete time, the spectral densitymatrix must be related to the discrete time exponentially correlatedprocess noise. The resultant discrete process noise for the attitude andangular velocity is given as shown in Eq. (27) [39]:

Qk�tk�1; tk; x� �Z

tk�1

tk

Φ�tk�1; s; β�GQα�τ�GTΦT�tk�1; s; β� ds

(27)

where Φ�tk�1; s; β�, the state transition matrix, is computed usingnumerical integration of the continuous-time state matrix given inEq. (26). The term G, sometimes referred to as the input matrix,maps the process noise to the appropriate states. It is equal to�03×3; 03×3; I3×3�T .The success of this model lies on the successful determination of β

and σ2m. Like previous models,Qα (by careful selection of β and σ2m)

must be sized appropriately to represent SO maneuver capability.Because β is the inverse of τ, it is selected by matching τ to the lengthof the expected SOmaneuver. It is important to note that, when τ ≠ 0,the angular velocities are continuous in time, as they are in the truekinematic motion of the SO. In the special case where τ � 0, thedynamics reduce to a white noise process model very similar to thatproposed by Holzinger et al. [14].Singer proposes modeling the variance of target acceleration as a

ternary uniform distribution, where probabilities of success andfailure of the maneuver are assigned. This work avoids assuming

these unknown probabilities by modeling the variance in theacceleration as a uniform distribution having a maximumacceleration amplitude αmax, as defined by Eq. (28):

σ2m � α2max

3(28)

Because αmax will seldom be known, one has two primarymethodologies for establishing this maximum value. The first is topick values representative of the hypothesized SO class, whereknowledge of SO size could come from auxiliary characterization.The second methodology is to hypothesize multiple targets the SOcould be tracking and selecting the maximum acceleration of thosehypotheses.

C. Marginalized Particle Filter

Although several correlation functions are available to model thedynamics of an agile SO, the exponentially correlated angularacceleration model is chosen purposefully. In Eq. (26), the Eulerangles appear nonlinearly in the dynamics and measurement models,whereas the angular velocity and angular acceleration states appearlinearly in the dynamics. The linear structure of the angular velocityand acceleration states can be exploited to reduce the computationalburden of a traditional particle filter (PF), by marginalizing out thelinear state variables. These linear states can then be estimated using aKalman filter (KF), which improves state estimates because it is theoptimal estimator of these linear states. This concept is sometimesreferred to as Rao–Blackwellization and as a marginalized particlefilter (MPF). In other words, the marginalized particle filter could bedescribed as using a particle representation of the nonlinear states,and each particle has an associatedmultivariate Gaussian distributionfor the remaining linear states.Using the notation of Schon, the total state space xk is segregated

into a nonlinear portion xnk and a linear portion xlk , as illustrated inEq. (29). Referring back to Eq. (26), one can see that the attitudestates θBI are indeed the only states that appear nonlinearly in thesystem dynamics:

xk �24 θBI

ωα

35

t�tk

��xnkxlk

�(29)

The goal of a nonlinear non-Gaussian filter, like the MPF, is todetermine the posterior probability density function, p�xkjYk�,recursively, where Yk is the output of the measurement model foreach state xk. The MPF approach analytically marginalizes out thelinear state by solving for the posterior probability density function(PDF) at time step k as

p�xnk ; xlk jYk� � p�xlk jxnk ;Yk�|�������{z�������}KF

p�xnk jYk�|���{z���}PF

(30)

where p�xlk jxnk ;Yk� is the marginalized posterior probability densityfor the linear states and can be optimally solved with the Kalmanfilter. However, the marginalized posterior probability density for thenonlinear states, p�xnk jYk�, has no closed-form solution, andtherefore it is approximated with a particle filter as shown in

p�xnk jYk� �XNi�1

wikδ�xnk − xi;nk � (31)

Here, N is the total number of particles, wik is the normalized

importance weight, and δ�xnk − xi;nk � is the Dirac delta located at xi;nk .The posterior for the linear portion of the state space is given by

p�xlk jxnk;Yk� � N �xlk ;Pk� (32)

where xlk andPk are the linear mean and covariance determined froma KF with the nonlinear states marginalized. Combining the KF

CODER, HOLZINGER, AND LINARES 5

posteriorwith the PF posterior gives the overall state posterior PDFas

p�xnk ; xlk jYk� �XNi�1

p�xlk jxi;nk ;Yk�wikδ�xnk − xi;nk � (33)

Therefore, the linear states are assumed to evolve linearly, and

because their initial distribution is Gaussian, the linear state

distribution will remain Gaussian. However, the distribution of the

nonlinear states is free to become both non-Gaussian and

multimodal. This general outline of the MPF can also found in

Schon, whereas the specific algorithm used in this work is given by

Algorithm 1 [21].This segregation allows the system dynamics given by Eq. (26) to

be defined by Eq. (34), where again it is noted the general form of the

system has been replaced with one specific to the SO light-curve

inversion problem.

xnk�1 � Fnn;k�xnk� �An

k�xnk�xlk � Gnk�xnk�wn

k

xlk�1 � Alk �xnk�xlk � Gl

k �xnk�wl

kyk � hnk�xnk� � vk (34)

The system of equations defined by Eq. (34) is necessary to

implement the MPF in practice. Here, the F and A submatrices

denote the portions of the state transition matrix that pertain to either

the nonlinear or linear states, respectively. To further illustrate the

relationship between the marginalized particle filter notation of

Eq. (34) and the state transition matrix, let the following equality be

defined:

In Eq. (35), Φ is the state transition matrix Φ�tk�1; tk; β� with

inputs suppressed for brevity. The subscripts (e.g., θ) indicate whichrows and columns of the state transition matrix correspond to the Fand A matrices, respectively. The subscript θ represents the Euler

angles, ω is the body angular velocities, and α is the body angular

accelerations. Just as the state transition matrix is segregated into

nonlinear and linear portions, the input matrix G, which appears in

Eq. (34), can be partitioned in a similar fashion:

�Gn

k

Glk

��

24 I3×3

I3×3I3×3

35 (36)

Thus, the discrete time process noise can be sampled using thediscrete time process noise covariance matrixes defined by Eq. (27):

wk ∼N �0;Qk�x�� (37)

The MPF algorithm begins by drawing random samples from anassumed distribution. The next step, the PF update equation defined

in Eq. (39), is used to calculate the likelihood of each particle, ~cik,conditioned on the true measurement, yk:

zi1;k � yk − hnk�xi;nk ; tI� (38)

~cik � p�z1;kjxk� �1�2πRk�

p exp

�−�zi1;k�22Rk

�(39)

cik �~cikPNi�1 ~cik

(40)

These likelihoods are often referred to as the “importanceweights”of each particle and are used in the PF resampling algorithm after

normalizing the weights according to Eq. (40). The fourth step,particle resampling, solves the much discussed shortcoming of the

PF, which is sample impoverishment. This approach uses residualresampling, although other methods such as Metropolis or stratified

resampling have been offered as equally effective alternatives [40].After resampling the particles, they can be propagated to the next time

step using a numerical ordinary differential equation solver. Thisleads to the final step of this specialized MPF algorithm, the KF time

update. Examining Eq. (34) reveals that the measurement equation isa nonlinear function of the body attitude only. However, the

Algorithm 1 Marginalized particle filter

xik�1 � MPF�xik; yk�1) Initialize particles2) PF measurement update [Eqs. (38) and (39)]3) Evaluate PF weights [Eq. (40)]4) PF resampling5) PF time update6) KF time update [Eqs. (41–47)]

Table 1 Assumed shape

model parameters

Facet A, m2 ξ a m

�X 2 0.5 0.25 0.3�Y 2 0.5 0.5 0.3�Z 2 0.5 0.75 0.3−X 2 0.5 0.9 0.3−Y 2 0.5 0.4 0.3−Z 2 0.5 0.1 0.3

Fig. 2. True agile SO angular acceleration.

Table 2 Radiometric model parameters

Parameter Value

Telescope aperture diameter D 0.5 mEffective focal length f 8Pixel size p 9e − 6 mMean atmospheric transmittance μatm 0.6Variance atmospheric transmittance σ2atm 2.5e − 5Optical transmittance τopt .7Quantum efficiency (QE) .5Radiant sky intensity Isky 18 mv∕arcsec2Exposure time tI 1 sDark current qp;dark 0.5 e∕pixel ⋅ sRead noise σ2r 10 e (rms)Pixels occupied by SO (m) 10 pixelsPixel sample estimate background level (z) 200 pixels

6 CODER, HOLZINGER, AND LINARES

“innovation” term zi2;k defines the coupling between the linear and

nonlinear states, as given by Eq. (34):

zi2;k � xi;nk�1 − Fnn;k�xi;nk � (41)

It is emphasized that a linearized state-observation mapping

relationship is not used here. Additionally, this is the only means for

information from the light-curve measurement to be used to update

the linear states. Using this innovation leads to the linear KF

propagation and update equations are given by [21]

xlk�1 � �Flkx

lk � Gl

k �Qlnk �T�Gn

kQnk�−1zi2;k � Lk�zi2;k − Fn

kxlk �(42)

Plk�1 � �Fl

kPlk � �Fl

k �T � Glk�Qlk �Gl

k �T − LkNkLTk (43)

where it is noted that some parameters have been dropped from thefollowing equations to make them specific to the light-curveinversion problem. The equations necessary for these computationsare given by [21]

Nk � FnkP

lk �Fn

k�T � GnkQ

nk�Gn

k�T (44)

Lk � �FlkP

lk �Fn

k�TN−1k (45)

�Flk � Fl

k −Glk �Qln

k �T�GnkQ

nk�−1Fn

k (46)

�Qlk � Ql

k − �Qlnk �T�Qn

k�−1Qlnk (47)

These equations complete the MPF algorithm. The MPF affordsthe estimation of both attitude and angular velocity states ofmaneuvering space objects, while maintaining computational

Fig. 3 Test cases at t � 0 s.

CODER, HOLZINGER, AND LINARES 7

tractability. It is also possible that the algorithm could work withoutknowledge of the initial orientation of the SO. As alluded to earlier,the initial distribution of the attitude states can be any arbitrarydistribution. It is theoretically possible for the initial distribution to bemade uniform over the attitude states. Although not a trivial task,previous work has shown how uniform distributions can be createdfor several parameterizations of the SO(3) rotation group including:quaternions, Rodrigues parameters, axis and angle of rotation, andboth symmetric and axisymmetric sequences of Euler angles [41].Complicating this proposition is the fact that different angularvelocities lead to the same object brightness. Because the linear,marginalized state is assumed to be Gaussian-distributed, the MPFcould not represent these multiple modes for the body angularvelocities. The next section provides some simulated results thathighlight these benefits compared to previous developments.

IV. Simulation Results

The emphasis of thiswork is to present novelmodels for improvingcurrent attitude estimation algorithms when applied to agile SOs. If

the algorithms presented are to one day be used operationally, thesimulation must match observational data as closely as possible.Accordingly, every effort is made to create a realistic physics-basedsimulation, as outlined later.

A. Data Flow

The first component of this simulator is the Simplified GeneralPerturbations Propagator. This software calculates the position andvelocity of a SO by propagating the information from a two-lineelement (TLE) file. A MATLAB implementation is available fromVallado et al. [42]. The next piece of software critical to the simulatoris the 1987 implementation of Variations Séculaires des OrbitesPlanétaires (VSOP87) [43]. The adaptation ofVSOP87byBretagnonand Francou combined with the coordinate transformations providedby Meeus [44] enables the ephemerides of the sun and Earth to becalculated with less than 1 in. error until 6000 A.D. The geometrynecessary to define the reflectance of light can be defined usingthe positions of the sun, observer, and SO. This geometry is usedin the final part of the simulator, a bidirectional reflectancedistribution function (BRDF) model. This particular work uses the

Fig. 4 Test cases at t � 75 s.

8 CODER, HOLZINGER, AND LINARES

Cook–Torrance BRDF model for specular contributions [45] and

Lambertian reflectance for diffuse contributions to the total radiant

flux of the SO at various attitudes [14].

B. Test Cases

All test cases presented are derived from a TLE for the NOAA 18

satellite downloaded from [46]. Each TLE comprises two lines of 80-

column ASCII text identifying the satellite by its catalog ID number

and defining its orbital elements at the given epoch time. The

simulation represents observations collected on 6 June 2014 from

10:19:21 to 10:23:19 UTC from the Fenton Hill Observatory, whose

location was modeled as 35.879998°N, 106.293058°W.

NOAA 18 [B]1 28654U 05018A 15024.53745670. 00000182 00000-012433-3 0 41472 28654 99.1733 13.7659 0014261 7.4244 99.161314.12137281498803

The NOAA 18 spacecraft orbits at a � 851 km, e � 1.4 × 10−3,i � 99.2 deg,Ω � 13.76 deg,ω � 7.42 deg, and f � 114.8 deg.The problem is simplified such that NOAA 18 is represented by asimple cube,with the shapemodel parameters presented inTable 1. Inthe shape model, A is the facet area, ξ is the affine transformationweighting parameter, a is the diffuse albedo, andm is the microfacetslope parameter, where ξ, a, and m ∈ �0; 1�. Because previous workhas addressed shapemodel uncertainty, it is assumed the shapemodelis known perfectly for the purposes of generating these simulatedresults [14]. However, the methodology described in this paper doesnot require that the shape model is known perfectly. Indeed, anypractical implementation of these methods on experimental data willrequire that the algorithm be capable of handling uncertainties inthese shape model parameters. It is also noted that the shape modelmay be so complex that any claims of computational tractabilitymade in this work would be compromised.To simulate an agile SO, the problem is modeled such that the −Z

facet of NOAA 18 is assumed to remain perpendicular to the zenithoriginating from Colorado Springs, Colrado, for the duration of the

Fig. 5 Test cases at t � 155 s.

CODER, HOLZINGER, AND LINARES 9

pass. This results in the acceleration profile depicted in Fig. 2. Thetelescope parameters used, shown in Table 2, are those of the 0.5 mf∕8 GT-SORT telescope and are representative of a typical Raven-class telescope [47]. Ideally, each improvement offered in this workcould be introduced as a separate test case. However, a comparisonbetween the time-invariant, limiting magnitude measurement modeland the radiometric measurement model is not included here.Because the benefit afforded by each improvement is judged by itsreduction in the error of estimated states, it seemed illogical tocompare estimates arrived at using measurement model thatproduced fundamentally different uncertainties. Additionally, it wascomputationally infeasible to include a comparison between thestandard PF using the exponentially correlated acceleration (ECA)dynamics model and the MPF. Indeed, computational tractability isone of the main benefits of the MPF.As a result, three test cases (TCs) are implemented to demonstrate

the improvements outlined in this paper over the current “state of theart”. TC1 represents one version of the current state of the art, using aPF and awhite noise process for the unknown angular velocities. TC1also implements the time-varying radiometric noisemodel, where the

total photon count is given using Eq. (10). Importantly, TC1 does notassume any knowledge of the mean angular velocity states. Incontrast, TC2 showcases the performance of the current state of theart even when knowledge of the true mean angular velocities isintroduced. Not only is this information unlikely to be available in anoperational setting, the introduction of this knowledge does notenable the current state-of-the-art methods to track the maneuver, asshown later. TC3 represents the culmination of all the novelcontributions of this work and is the only test case where themaneuver is successfully tracked. Like TC1, it does not assume any apriori knowledge of SO mean angular velocity. TC3 keeps theradiometric noise model and adds the ECA dynamics model,enabling the use of the MPF.All TCs are presented using the same number of particles: 10,000.

Additionally, the following levels of process noise were used to keepcomparisons between the PF and MPF consistent. For the PF,Qω�τ� � diag�1−4 ⋅ �0.0113; 0.7243; 0.0025�� rad2∕s2. Meanwhile,the MPF usedQα�τ� � diag�1−5 ⋅ �0.0029; 0.1196; 0.0003�� rad2∕s2and β1 � β2 � β3 � 1∕15 with units of 1∕s. For an initial attitudeof θBI � 0, these settings resulted in 1σ noise levels of

Fig. 6 Test cases at t � 240 s.

10 CODER, HOLZINGER, AND LINARES

δω0 � �0.061; .488; .029�T deg ∕s. The values for the process noisewere computed using Eqs. (25), (27), and (28), where the values of βare user-selected. The value of αmax was set equal to 10 times the true

accelerations required for the -Z facet of NOAA 18 to remain

perpendicular to the zenith originating from Colorado Springs,

Colorado.

Figure 3–6 illustrate the particle clouds of the three test cases at

four instances during the simulation time. TC1 is always presented on

the top row, and the middle and bottom rows are TC2 and TC3,

respectively. The particles themselves are shaded such that the

particles with the highest likelihoods (i.e., weights) are shown in

black, whereas particles that represent less likely states are

represented with lighter shades of gray. In all subfigures, the true

simulated state is shown with a red star.

Figure 3 illustrates the initialized particles for all test cases. In all

test cases, the particles are uniformly distributed in Euler angle space

in a window around the true state. Extending the results such that the

attitude space, and not the Euler angles, are uniformly sampled is an

area of future work.

Figure 4 shows both filters 75 s into the simulation. One can see

that the particles of the MPF (TC3) are distributed in smaller volume

of state space than either PF test case. Additionally, the large swaths

of gray particles in TC1 andTC2highlight that theMPF ismuchmore

computationally efficient, with each resampled particle having a

higher likelihood according to Eq. (39).

Figure 5 shows both filters 155 s into the simulation and is an

excellent illustration of the highly non-Gaussian distributions typical

of the SO attitude estimation problem. It is also evident by examining

TC1 that, without knowledge of the SO’s mean angular velocity, the

state of the art PF begins to diverge, with the true SO state no longer

bounded by the particle cloud.

Figure 6 shows all test cases at the final time step in the simulation,

at 240 s. Only in TC3 is the SO successfully tracked, with the true

Table 3 Test case descriptions

Testcase

NL/Lstates

Measurementmodel

Dynamicsmodel ωμ Filter

1 3∕0 Radiometric White-noise 0 PF2 3∕0 Radiometric White-noise Truth PF3 3∕6 Radiometric ECA 0 MPF

Fig. 7 Comparison of attitude estimates.

CODER, HOLZINGER, AND LINARES 11

state contained within the particle cloud. This same information can

be seen for all time steps in Fig. 7. The left-hand side of Fig. 7 shows

the true SO attitude, denoted by the solid black line, compared to the

first moment of the particle cloud, shown by black circles, along with

the 5 and 95 percentile bounds of the particle cloud illustrated by the

shaded gray area. The right-hand side of Fig. 7 shows the residuals of

the attitude state, denoted by black X, along with the 5 and 95

percentile residual bounds.It is cautioned that using such plots can be misleading because the

nonlinear states can become multimodal, and therefore the first

moment of the particle cloud may be a poor measure for comparison

with the true state. However, for this problem, the particle cloud is

unimodal, and Figure 7 shows that the true attitude states for θ1 and θ3are not containedwithin the 5 and 95 percentile bounds of the particle

cloud for TC1 and TC2. This is especially striking given that, for

TC2, the current state-of-the-art algorithm used explicit knowledge

of the true mean angular velocities. Therefore, for this particular

simulation, the SO maneuver is only successfully tracked by the

totality advancements outlined in this paper.It is important to grasp the physical reason why the percentile

bounds grow as observations of the SO are collected. This is directly

due to the interaction between the measurement noise and process

noise. The observing environment described in Table 2 is shot-noise

limited, and in this anecdotal example, the SO being observed

becomes brighter with time. Consequently, the shot noise increases

the measurement noise during the simulation, as defined in Eq. (9).

This increase in measurement noise causes more particles around the

truth to represent brightness values consistent with the measurement

statistics. As a result, the particles that have been randomly subjected

to higher levels of process noise are not eliminated, and the cloud of

particles grows as the simulation progresses. Not only are the

nonlinear states only successfully tracked in TC3, a key advancement

in the exploitation of light curves where no knowledge is assumed

known for the shapemodel, inertia, or torques is illustrated by Fig. 8b.

By adopting the ECA and the MPF, the angular velocity states are

able to be simultaneously estimated in TC2 only. Figure 8b gives the

body angular velocities provided by the KF update portion of the

MPF. The estimation error is given by the solid black line, whereas

the 5 and 95 percentiles are given by the shaded gray area.The first benefit of this ability is that knowledge of SO body

angular velocity can be used to better estimate the attitude states via

the KF update step of the MPF. This benefit is evident in Figs. 3–6,

where the volume of state space occupied by the particle cloud is less

for the MPF than the PF. The second benefit afforded by knowledge

of SO angular velocities manifests itself in SO operational mode

classification. Although determining the current attitude of a SO is a

critical aspect of characterization, it is not an immediately actionable

piece of information. Indeed, SSA stakeholders desire information on

the operationalmode of a SO (e.g., if the SO is currently sun-pointing,

nadir-pointing, or tracking another SO). Information on SO angular

velocity could be an important discriminator in operational mode

classification if multiple operational classifications are representedby similar SO attitude states alone.

V. Conclusions

Physics-based measurement models can be used to more closelyreflect measurement noise present in observational data. Theimplementation of a correlated angular rate dynamicsmodel, adaptedfrom the Singer Markov process model, provided a framework fordefining a space object maneuver model. This novel dynamics modelenables the implementation of marginalized particle filters, enablingestimation of attitude and angular velocity states of maneuveringspace objects without a prior knowledge of initial attitude, whilemaintaining computational tractability. These three contributionsenhance the quality of information gleaned from scarce observationassets and further improve the state estimation of agile space objects.The increased knowledge afforded by these methods directly aidsspace object characterization and the overallmission of space domainawareness.

Acknowledgments

This work was started at the Los Alamos Space Weather SummerSchool, funded by the Institute ofGeophysics, Planetary Physics, andSignatures at Los Alamos National Laboratory (LANL) andcontinued while a U.S. Air Force Research Laboratory (AFRL)Directed Energy Scholar at the U.S. Air Force Maui Optical andSupercomputing (AMOS) site. The work was completed thanks tothe support of the AFRLAMOS site and Integrity Applications Inc.–Pacific Defense Solutions (IAI-PDS) under contract FA6451-13-C-0281. IAI-PDS also provided the experimental Raven data includedin this work. A special thanks is due to David Palmer of LANL, KrisHamada of IAI-PDS, andChris Sabol andKimLuu ofAFRL for theirinsightful and thought provoking feedback.

References

[1] Rumsfeld, D. H., “Commission to Assess United States NationalSecurity SpaceManagement andOrganization,”U.S.HouseCommitteeon Armed Services, TR, Washington, D.C., Jan. 2001.

[2] “Space Operations,” U.S. Department of Defense, TR JP 3-14,Arlington, VA, May 2013.

[3] Stokes, G., Von Braun, C., Sridharan, R., Harrison, D., and Sharma, J.,“The Space-Based Visible Program,” Lincoln Laboratory Journal,Vol. 11, No. 2, 1998, pp. 205–238.doi:10.2514/6.2000-5334

[4] Hall, D., Calef, B., Knox, K., Bolden, M., and Kervin, P., “SeparatingAttitude and Shape Effects for Non-Resolved Objects,” Proceedings oftheAdvancedMauiOptical and Space Surveillance TechnologiesConf.,2007, pp. 464–475.

[5] Walker, J. L., “Range-Doppler Imaging of Rotating Objects,” IEEE

Transactions on Aerospace and Electronic Systems, No. 1, 1980,pp. 23–52.doi:10.1109/TAES.1980.308875

Fig. 8 MPF estimate of angular velocity.

12 CODER, HOLZINGER, AND LINARES

[6] Hall, D., Africano, J., Kervin, P., and Birge, B., “Non-Imaging Attitudeand Shape Determination,” Proceedings of the Advanced Maui Optical

and Space Surveillance Technologies Conf., Sept. 2005.[7] Dunlap, J., “Lightcurves and the Axis of Rotation of 433 Eros,” Icarus,

Vol. 28, No. 1, 1976, pp. 69–78.doi:10.1016/0019-1035(76)90087-7

[8] Magnusson, P., “Distribution of SpinAxes and Senses ofRotation for 20Large Asteroids,” Icarus, Vol. 68, No. 1, 1986, pp. 1–39.doi:10.1016/0019-1035(86)90072-2

[9] Kaasalainen, M., Lamberg, L., Lumme, K., and Bowell, E.,“Interpretation of Lightcurves of Atmosphereless Bodies. 1—GeneralTheory and New Inversion Schemes,” Astronomy and Astrophysics,Vol. 259, June 1992, pp. 318–332.

[10] Kaasalainen, M., Lamberg, L., and Lumme, K., “Interpretation ofLightcurves of Atmosphereless Bodies. 2—Practical Aspects ofInversion,” Astronomy and Astrophysics, Vol. 259, June 1992,pp. 333–340.

[11] Torppa, J., Kaasalainen, M., Michalowski, T., Kwiatkowski, T.,Kryszczynska, A., Denchey, P., and Kowalski, R., “Shapes andRotational Properties of Thirty Asteroids from Photometric Data,”Icarus, Vol. 164, No. 1, 2003, pp. 364–383.doi:10.1016/S0019-1035(03)00146-5

[12] Wetterer, C. J., and Jah, M. K., “Attitude Determination from LightCurves,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 5,Sept.–Oct. 2009, pp. 1648–1651.doi:10.2514/1.44254

[13] Coder, R. D., and Holzinger, M. J., “Multi-Objective Design of OpticalSystems for Space Situational Awareness,”Acta Astronautica, Vol. 128,2016, pp. 669–684.doi:10.1016/j.actaastro.2016.07.008

[14] Holzinger, M. J., Alfriend, K. T., Wetterer, C. J., Luu, K. K., Sabol, C.,and Hamada, K., “Photometric Attitude Estimation for Agile SpaceObjects with Shape Uncertainty,” Journal of Guidance, Control, and

Dynamics, Vol. 37, No. 3, 2014, pp. 921–932.doi:10.2514/1.58002

[15] Singer, R., “Estimating Optimal Tracking Filter Performance forManned Maneuvering Targets,” IEEE Transactions on Aerospace and

Electronic Systems, Vol. 6, No. 4, July 1970, pp. 473–483.doi:10.1109/TAES.1970.310128

[16] Fitts, J., “Aided Tracking as Applied to High Accuracy PointingSystems,” IEEE Transactions on Aerospace and Electronic Systems,Vol. AES-9, No. 3, 1973, pp. 350–368.doi:10.1109/TAES.1973.309721

[17] Pearson, J. B., and Stear, E. B., “Kalman Filter Applications in AirborneRadar Tracking,” IEEE Transactions on Aerospace and Electronic

Systems, No. 3, 1974, pp. 319–329.doi:10.1109/TAES.1974.307824

[18] Blair,W.,Watson,G., andRice, T., “TrackingManeuveringTargetswithan Interacting Multiple Model Filter Containing Exponentially-CorrelatedAccelerationModels,”Proceedings of the 23rd SoutheasternSymposium on System Theory, IEEE Publ., Piscataway, NJ, 1991,pp. 224–228.

[19] Li, X. R., and Jilkov, V. P., “Survey of Maneuvering Target Tracking.Part 1. Dynamic Models,” IEEE Transactions on Aerospace and

Electronic Systems, Vol. 39, No. 4, 2003, pp. 1333–1364.doi:10.1109/TAES.2003.1261132

[20] Doucet, A., Gordon, N. J., and Krishnamurthy, V., “Particle Filters forState Estimation of JumpMarkov Linear Systems,” IEEE Transactions

on Signal Processing, Vol. 49, No. 3, 2001, pp. 613–624.doi:10.1109/78.905890

[21] Schon, T., Gustafsson, F., and Nordlund, P.-J., “Marginalized ParticleFilters for Mixed Linear/Nonlinear State-Space Models,” IEEE

Transactions on Signal Processing, Vol. 53, No. 7, 2005, pp. 2279–2289.doi:10.1109/TSP.2005.849151

[22] Budding, E., and Demircan, O., Introduction to Astronomical

Photometry, Cambridge Observing Handbooks for Research Astron-omers, 2nd ed., Cambridge Univ. Press, 2007.doi:10.1017/CBO9780511536175

[23] Shell, J. R., “Optimizing Orbital Debris Monitoring with OpticalTelescopes,” Proceedings of the Advanced Maui Optical and Space

Surveillance Technologies Conf., Space Innovation and DevelopmentCenter, Sept. 2010.

[24] Falchi, F., Cinzano, P., Duriscoe, D.,Kyba, C. C., Elvidge,C.D., Baugh,K., Portnov, B. A., Rybnikova, N. A., and Furgoni, R., “TheNewWorld

Atlas of Artificial Night Sky Brightness,” Science Advances, Vol. 2,No. 6, 2016, Paper e1600377.doi:10.1126/sciadv.1600377

[25] Schott, J. R., Remote Sensing: The Image Chain Approach, OxfordUniv. Press, 1997.

[26] Merline, W., and Howell, S. B., “A Realistic Model for Point-SourcesImaged on Array Detectors: The Model and Initial Results,”Experimental Astronomy, Vol. 6, Nos. 1–2, 1995, pp. 163–210.doi:10.1007/BF00421131

[27] Schildknecht, T., “Optical Astrometry of Fast Moving Objects UsingCCD Detectors,” Geodätisch-Geophysikalische Arbeiten in der

Schweiz, Vol. 49, 1994.[28] Hines,W.W.,Montgomery, D. C., Borror, C.M., andGoldsman, D.M.,

Probability and Statistics in Engineering, Wiley, 2008.[29] Linares, R., Jah, M. K., Crassidis, J. L., and Nebelecky, C. K., “Space

Object Shape Characterization and Tracking Using Light Curve andAngles Data,” Journal of Guidance, Control, and Dynamics, Vol. 37,No. 1, 2014, pp. 13–25.doi:10.2514/1.62986

[30] Dravins, D., Lindegren, L., Mezey, E., and Young, A. T., “AtmosphericIntensity Scintillation of Stars. I. Statistical Distributions and TemporalProperties,” Publications of the Astronomical Society of the Pacific,Vol. 109, 1997, pp. 173–207.doi:10.1086/133872

[31] Rose, C., and Smith, M. D., “MathStatica: Mathematical Statistics withMathematica,” Compstat, Springer, 2002, pp. 437–442.

[32] Roggemann, M. C., Welsh, B. M., and Hunt, B. R., Imaging Through

Turbulence, CRC Press, Boca Raton, FL, 1996.[33] Feigelson, E. D., and Babu, G. J., Modern Statistical Methods

for Astronomy: With R Applications, Cambridge Univ. Press, 2012,pp. 48–52.doi:10.1017/CBO9781139015653

[34] Karlis, D., and Xekalaki, E., “Mixed Poisson Distributions,”International Statistical Review, Vol. 73, No. 1, 2005, pp. 35–58.doi:10.1111/j.1751-5823.2005.tb00250.x

[35] Berg, R., “Estimation and Prediction for Maneuvering TargetTrajectories,” IEEE Transactions on Automatic Control, Vol. 28,No. 3, 1983, pp. 294–304.doi:10.1109/TAC.1983.1103231

[36] Kumar, K., and Zhou, H., “A ‘Current’ Statistical Model and AdaptiveAlgorithm for Estimating Maneuvering Targets,” Journal of Guidance,Control, and Dynamics, Vol. 7, No. 5, 1984, pp. 596–602.doi:10.2514/3.19900

[37] Blackman, S., and Popoli, R.,Design and Analysis of Modern Tracking

Systems, Artech House, Norwood, MA, 1999.[38] Nabaa, N., and Bishop, R. H., “Validation and Comparison of

Coordinated Turn Aircraft Maneuver Models,” IEEE Transactions

on Aerospace and Electronic Systems, Vol. 36, No. 1, 2000,pp. 250–259.doi:10.1109/7.826327

[39] Jazwinski, A. H., Stochastic Processes and Filtering Theory, Dover,Mineola, NY, 2007.

[40] Douc, R., and Cappé, O., “Comparison of Resampling Schemes forParticle Filtering,” Proceedings of the 4th International Symposium on

Image and Signal Processing andAnalysis, IEEEPubl., Piscataway,NJ,2005, pp. 64–69.

[41] Shuster,M.D., “UniformAttitude ProbabilityDistributions,” Journal ofAstronautical Sciences, Vol. 51, No. 4, 2003, pp. 451–475.

[42] Vallado, D. A., Crawford, P., Hujsak, R., and Kelso, T., “RevisitingSpacetrack Report #3,” AIAA/AAS Astrodynamics Specialist Conf. andExhibit, AIAA Paper 2006-6753, Aug. 2006.

[43] Bretagnon, P., and Francou, G., “Planetary Theories in Rectangular andSpherical Variables—VSOP 87 Solutions,” Astronomy and Astrophys-ics, Vol. 202, 1988, pp. 309–315.

[44] Meeus, J. H., Astronomical Algorithms, Willmann–Bell, 1991.[45] Cook, R. L., and Torrance, K. E., “A Reflectance Model for

Computer Graphics,” Computer Graphics, Vol. 15, No. 3, Aug. 1981,pp. 307–316.doi:10.1145/965161

[46] “Space-Track.org,” June 2015, http://www.space-track.org [retrieved___________].

[47] Sabol, C., Luu, K. K., Kervin, P., Nishimoto, D., Hamada, K., andSydney, P., “Recent Developments of the Raven Small TelescopeProgram,” AAS/AIAA Space Flight Mechanics Meeting, AmericanAstronautical Society Paper AAS 02-131, Springfield, VA, 2002.

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