astrometricandphotometricdatafusion for inactive
TRANSCRIPT
ASTROMETRIC AND PHOTOMETRIC DATA FUSION FOR INACTIVE
SPACE OBJECT FEATURE ESTIMATION
Richard Linares
University at Buffalo, State University of New York, Amherst, NY, 14260-4400
Moriba K. Jah
Air Force Research Laboratory, Kirtland AFB, New Mexico, 87117
John L. Crassidis
University at Buffalo, State University of New York, Amherst, NY, 14260-4400
Fred A. Leve
Air Force Research Laboratory, Kirtland AFB, New Mexico, 87117
Tom Kelecy
The Boeing Company, Boeing LTS, Colorado Springs, CO, 80919
Abstract
This paper presents a new method to determine the mass of an inactive space object from photometricand astrometric data. Typically the effect of solar radiation pressure is used to determine area-to-massratio for space objects from angles observations. The area-to-mass ratio of a space object can greatly affectits orbital dynamics; therefore angles data are sensitive to this quantity. On the other hand, photometricdata is not sensitive to mass but is a strong function of the albedo-area and the rotational dynamics of thespace object. The albedo-area can be used to determine the amount of energy reflected from solar radiation.Since these two data types are sensitive to albedo-area and area-to-mass, then by fusing photometric datawith angles data it is possible to determine the area and mass of a space object. This work employsan unscented Kalman filter to estimate rotational states, translational states, area states and mass of aninactive space object. Mass is not observerable with only angles data or only photometric data, but itis shown in this work that with the two combined data types mass can be recovered. Recovering thischaracteristic and trajectories with sufficient accuracy is shown in this paper. The performance of the newmethod is demonstrated via simulated scenarios.
1 Introduction
In recent years space situational awareness, which isconcerned with collecting and maintaining knowledgeof all objects orbiting the Earth, has gained much at-tention. The U.S. Air Force collects the necessarydata for space object catalog development and main-tenance through a global network of radars and op-tical sensors. Due to the fact that a limited num-ber of sensors are available to track a large numberof space objects (SOs), sparse data collected must beexploited to its fullest extent. Various sensors such as
radars and optical sensors exist for SO state estima-tion, which typically includes position, velocity anda non-conservative force parameter analogous to aballistic coefficient. However, the ballistic coefficientparameters may not fully describe the SO’s motion.Hence, more detailed attitude dependent models arerequired. In this case more elaborate techniques forprocessing observation data that can extract attitudeinformation are required.
Light curves (the SO temporal brightness as seenfrom the observer) have been used to estimate theshape for an object. In particular, light curve ap-
1
proaches have been studied to estimate the shapeand state of asteroids [1, 2]. Reference [3] uses lightcurves and thermal emissions to recover the three-dimensional shape of an object assuming its orien-tation with respect to the observer is known. Thebenefits of using a light curve-based approach overthe aforementioned others is that it is not limited tolarger objects in lower altitudes, and it can be appliedto small and dim objects in higher altitudes, such asgeosynchronous orbits. Here light curve data is con-sidered for mass estimation, which is also useful sinceit provides a mechanism to estimate both positionand attitude, as well as their respective rates [4, 5].
There are several aspects of using light curve data(temporal photometry) that make it particularly ad-vantageous for object detection, identification andtracking. Light curve data includes the time-varyingsensor wavelength-dependent apparent magnitude ofenergy (i.e. photons) scattered (reflected) off of anobject along the line-of-sight to an observer. Becausethe apparent magnitude of the SO is a function of itssize, orientation and surface material properties, oneor more of these characteristics should be recoverablefrom the photometric data. This can aid in the de-tection and identification of a SO after a catalog ofspacecraft data with material properties is developed,and may also prove to be powerful for never-seen-before objects.
There is a coupling between SO attitude and non-conservative force/torques. This can be exploited toassist in the estimation of the SO trajectory. Themeasurement of the apparent magnitude is a functionof several SO characteristics. These same character-istics drive certain non-conservative forces, such assolar radiation pressure (SRP). The acceleration dueto SRP is modeled as function of an object’s Sun-facing area, surface properties, mass, position, andattitude. It has a very small magnitude comparedto gravitational accelerations, and typically has anorder of magnitude around 10−7 to 10−9 m/s2, butis the dominant non-conservative acceleration for ob-jects above 1,000 km. Below 1,000 km, drag causedby the atmospheric neutral density is the dominantnon-conservative acceleration.
Deep space optical surveys of near geosynchronous(GEO) objects have identified a class of high area-to-mass ratio (HAMR) objects [6]. The exact char-acteristics of these objects are not well known andtheir motion pose a collision hazard with GEO ob-jects due the SRP induced large variations of incli-nation and eccentricity. These objects are typicallynon-resolved and difficult to track due to dim mag-nitude and dynamic mismodeling. Therefore, char-acterizing the large population of HAMR objects in
geostationary orbit is required to allow for a betterunderstanding of their origins, and the current andfuture threats they pose to the active SO population.
Estimating the dynamic characteristics of a HAMRobject using light curve and astrometric data can al-low for mass parameters to be observable. Estimatingmass for HAMR objects can help in the developmentof a detailed understanding of the origin and dynam-ics of these objects. It has been shown that the SRPalbedo area-to-mass ratio, CrA
m , is observable fromangles data [7] through the dynamic mismodeling ofSRP forces. Reference [7] conducts a study with sim-ulated and actual data to quantify the error in theestimates of CrA
m and good performance is found us-ing data spanning over a number of months. AlsoRef. [8] shows that orbital, attitude and shape pa-rameters can be recovered with sufficient accuracyusing a multiple-model adaptive estimation approachcoupled with an unscented Kalman filter. This ap-proach works reasonably well but requires that thearea-to-mass ratio is known a priori. The purposeof this work is to shown that since CrA
m is observablefrom angles data and shape/albedo properties are ob-servable from photometric data, then by fusing thesedata types mass can be extracted with reasonable ac-curacy.
Filtering algorithms for state estimation, such asthe extended Kalman filter (EKF) [9], the unscentedKalman filter (UKF) [10] and particle filters [11]are commonly used to both estimate hidden (indi-rectly observable) states and filter noisy measure-ments. The basic difference between the EKF andthe UKF results from the manner in which the statedistribution of the nonlinear models is approximated.The UKF, introduced by Julier and Uhlmann [10],uses a nonlinear transformation called the unscentedtransform, in which the state probability densityfunction (pdf) is represented by a set of weightedsigma points (state vectors deterministically sampledabout a mean). These are used to parameterize thetrue mean and covariance of the state distribution.When the sigma points are propagated through thenonlinear system, the posterior mean and covarianceare obtained up to the second order for any nonlin-earity. The EKF and UKF assume that the processnoise terms are represented by zero-mean Gaussianwhite-noise processes and the measurement noise isrepresented by zero-mean Gaussian random variable.Furthermore both approaches assume that the a pos-
teriori and a priori pdf is Gaussian in a linear do-main. This is true given the previous assumptionsbut under the effect of nonlinear measurement func-tions and system dynamics the initial Gaussian stateuncertainty may quickly become non-Gaussian. Both
2
filters only provide approximate solutions to the non-linear filtering problem, since the a posteriori and a
priori pdf are most often non-Gaussian due to nonlin-ear effects. The EKF typically works well only in theregion where the first-order Taylor-series linearizationadequately approximates the non-Gaussian pdf. TheUKF provides higher-order moments for the compu-tation of the a posteriori pdf without the need tocalculate Jacobian matrices as required in the EKF.The light curve measurement model is highly nonlin-ear, and Jacobian calculations are non-trivial; thus,the UKF is used to provide a numerical means of es-timating the states of the SO using light curve mea-surement models.
Attitude estimation using light curve data has beendemonstrated in Ref. [12]. The main goal of this cur-rent work is to use light curve data to, autonomouslyand in near realtime, determine the mass of a SOalong with its attitude (rotational) and translationalstates. In order to accomplish this task, a UKFis designed for state estimation of these quantities.The translational dynamics includes both conserva-tive gravitational and non-conservative SRP. The ro-tational dynamics includes classic Euler dynamicscoupled with SRP torque. Light curve and anglesdata are employed in the UKF structure to estimatethe states.
The organization of this paper is as follows. First,the methods used to recover mass are discussed.Then the models used for SO shape, orbital dynamicsand attitude dynamics are discussed. Following thisa description of the measurement models used in thispaper are given. Next, a review of the UKF approachis provided. Finally, simulation results of the massand albedo-area estimation approach are provided.
2 Details of Approaches
Two approaches are considered in this work: the firstapproach estimates for albedo-area, Ar, which usesan assumed value for the albedo, r, to allow for es-timation of mass, and the second approach uses esti-mated rotational dynamics to infer information aboutthe area and therefore allow information about massto be gained. As mentioned previously, light curvedata is sensitive to Ar and angles data are sensitiveto CrA
m , where Cr is the albedo coefficient for the SRPforce [7]. The Cr coefficient is a function of the albedoof the SO; Cr = 1+r the for cannonball model [7], andtherefore, to allow for mass to be separated from area,an estimate of albedo must be determined. It will beshown in §2.3 that Cr = 1+ 2
3Cdiff for a diffuse sphere
assuming Lambert’s cosine law and Cr = 1 + 49Cdiff
for a diffuse flat plate assuming Lambert’s cosine law
with the normal direction aligned with the sun direc-tion.Air Force Maui Optical and Supercomputing site
Advance Electro-Optical System (AMOS) researchershave calibrated for nominal albedo by using observa-tions of known SOs to determine effective albedosfor different classes of objects [13]. The analysis in-dicated the best effective albedo and albedo rangethat provides a 90% confidence range for each ob-ject class. NASA and AMOS have jointly determinedthat, for payloads and rocket-bodies reflecting sun-light, an albedo of r ≈ 0.3 best reproduces knownsizes with a 90% confidence range of 0.1 ≤ r ≤ 0.7.For orbital debris, an effective albedo of r ≈ 0.15 bestreproduces the sizes estimated from radar with a 90%confidence of roughly 0.05 ≤ r ≤ 0.3. Therefore, as-suming a value for the albedo that is consistent withobservations of a typical SO can provide a means toestimate mass with estimates of albedo-area. Thestate vector for the joint attitude, position and pa-rameter estimate problem is given by
xk =
qBI
ωBB/I
rI
vI
mAr
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
tk
(1)
where position and velocity of an Earth orbiting SOare denoted by rI = [x y z]T and vI = [vx vy vz ]
T .Also, ωB
B/I is the angular velocity of the SO with re-
spect to the inertial frame, and qBI is the quaternion
that parameterizes the orientation of the SO with re-spect to the inertial frame. The mass, m, of the SOis added as a state along with the albedo-area vec-tor, Ar, containing the albedo-area of each side ofthe SO. The benefit of this approach is that since thealbedo-area vector is part of the estimated state vec-tor the orientation of the SO is not affected by theuncertainty in the assumed albedo, but rather the as-sumed albedo is used to compute the SRP force onthe SO and in turn estimate the mass of the SO. TheSRP force discussed in the following section requireseither estimates of area and albedo-area or estimatesof albedo-area and albedo. It is noted that using thisapproach will not affect the estimates of rotationaland albedo-area states since the assumed albedo willonly be used in the dynamic equations for the SRPforce. The uncertainty in the assumed albedo willaffect the estimates for mass and translational statessince the assumed albedo will be used in the SRPforce calculation. In this approach no estimate of thearea is directly needed, but the assumed albedo isused in the dynamic state propagation.
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Buu
Bvu
Bnu
obsIu
sunIu
Ihu
Figure 1: Geometry of Reflection
The second estimation approach used in this workmake uses of the fact that the observation and dy-namics are both sensitive to the desired quantities butare not by themselves sensitive to all desired quan-tities. For example photometric data is sensitive toattitude, angular velocity and the albedo-area of theSO, whereas astrometric data is sensitive to the posi-tion, velocity and area-to-mass ratio. The sensitivityof area-to-mass ratio is due to dynamic mismodelingdue to the SRP force, but this force is also a func-tion of orientation and albedo-area. By considering astate vector that consists of parameters that are ob-servable from astrometric data and parameters thatare observable from photometric data, then it is pos-sible separate area from mass.The state vector for the joint attitude, position and
parameter estimate problem with area states is givenby
xk =
qBI
ωBB/I
rI
vI
mArA
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
tk
(2)
where the states of areas, A, are added and con-tain the areas of each side of the SO. The areas andalbedo-areas are added as separate states due to thefact that each data type is sensitive to one but notto both. These states have a dependency since thealbedo-areas are just products of the albedo r for eachside with the respective area for that side.
2.1 Shape Model
The shape model considered in this work consists ofa finite number a flat facets. For curved surfaces thismodel becomes more accurate as the number of facetsis increased. Each facet has a set of three basis vec-
x
y
z
d
a
l
Figure 2: Cuboid Shape Model
tors (uBn , u
Bu , uB
v ) associated with it as defined inFigure 1. The unit vector uB
n points in the directionof the outward normal to the facet, and the vectorsuBu and uB
v are in the plane of the facet. The notationsuperscript B denotes that the vector is expressed inbody coordinates. The SOs are assumed to be rigidbodies, and the unit vectors uB
n , uBu and uB
v thereforedo not change in time since they are expressed in thebody frame.
The light curve and the SRP models discussed inthe next sections require that these vectors be ex-pressed in inertial coordinates. Since the SO bodyis rotating, these vectors will change inertially. Thebody vectors can be rotated to the inertial frame bythe standard attitude mapping given by
uBk = A(qB
I )uIk, k = u, v, n (3)
where A(qBI ) is the attitude matrix mapping the in-
ertial frame to the body frame using the quaternionparameterization. The superscript I denotes that avector is expressed in inertial coordinates. The unitvector uI
sun points from the SO to the Sun direction,and the unit vector uI
obs points from the SO to theobserver as shown in Figure 2. The vector uI
h is thenormalized half vector between uI
sun and uIobs also
shown in Figure 2. This vector is also known as theSun-SO-Observer bisector. Each facet has an areaA(i) associated with it. Once the number of facetshas been defined and their basis vectors are known,the areas A(i) define the size and shape of the SO.
For the development of the measured light curvedata, faceted SO rectangular cuboid shape models asshown in Figure 2 are used. The rectangular cuboidmodel is described by three parameters, l, a, and d,which are the length, width, and height, respectively.
4
2.2 Rotational and Translational
Models
The two-body equations of motion with SRP accel-erations are given by
rI = − µ
r3rI + aIsrp (4)
where µ is the gravitational parameter of the Earth,r = ‖rI‖, and aIsrp represents the acceleration pertur-bation due to SRP, which will be discussed in detailin the following section.
A number of parameterizations exist to specifyattitude, including Euler angles, quaternions andRodriguez parameters [14]. This paper uses thequaternion, which is based on the Euler angle/axisparametrization. The quaternion is defined as q ≡[T q4]
T with = e sin(ν/2), and q4 = cos(ν/2),where e and ν are the Euler axis of rotation and rota-tion angle, respectively. The quaternion must satisfya unit norm constraint, qTq = 1. In terms of thequaternion, the attitude matrix is given by
A(q) = ΞT (q)Ψ(q) (5)
where
Ξ(q) ≡[
q4I3×3 + [×]−T
]
(6a)
Ψ(q) ≡[
q4I3×3 − [×]−T
]
(6b)
with
[a×] ≡
0 −a3 a2a3 0 −a1−a2 a1 0
(7)
for any general 3 × 1 vector a defined such that[a×]b = a× b.
The rotational dynamics are given by the coupledfirst-order differential equations:
qBI =
1
2Ξ(qB
I )ωBB/I (8a)
ωBB/I = J−1
SO
(
TBsrp −
[
ωBB/I×
]
JSOωBB/I
)
(8b)
where ωBB/I is the angular velocity of the SO with
respect to the inertial frame, expressed in body co-ordinates, JSO is the inertia matrix of the SO andTB
srp is the net torque acting on the SO due to SRPexpressed in body coordinates. The SO is assumedto be of uniform density, and therefore the principalcomponents of the inertia tensor for the shape model
discussed in §2.1 and given in Figure 2 are given by
J1 = ma2 + b2
12(9a)
J2 = ma2 + l2
12(9b)
J3 = ml2 + b2
12(9c)
where m is the mass of the SO and a, l, and d arethe shape parameters defined in Figure 2.
2.3 Solar Radiation Force Model
For higher altitude objects (≥ 1,000 km) SRP repre-sents the primary non-conservative perturbation act-ing on SOs. Because SRP depends upon the SO’sposition and orientation, the position and attitudedynamics are thus coupled. The acceleration due toSRP is computed as a function of the total solar en-ergy impressed upon exposed SO surfaces that are re-flected, absorbed and reradiated. The rate at whichradiant energy is incident on an element of area dAis a function of angle between the normal to dA, un,and the Sun direction usun. The power of incidentradiant energy is given by
PI =Φsun,tot
(d/d0)2(un · usun) dA (10)
where Φsun,tot is the average incident radiant fluxdensity from the Sun at 1 AU, given by Φsun,tot =1, 367 W/m2. Therefore the energy flux at any dis-tance d is given by Φsun,tot/(d/d0)
2 where d0 = 1 AU.The reflected radiation will have the following diffuseand specular power:
PD = Cdiff
Φsun,tot
(d/d0)2(un · usun) dA (11a)
PS = Cspec
Φsun,tot
(d/d0)2(un · usun) dA (11b)
where the incident solar radiant energy is accountedfor in three terms: the absorbed energy, Cabs, thespecularly reflected energy Cspec, and the diffuselyreflected energy, Cdiff, which yields
Cabs + Cspec + Cdiff = 1 (12)
The elemental force on dA can be written in threeterms: incident force, dFI , specular reflection force,dFS , and diffuse reflection force, dFD. The incidentforce accounts for force due to the three terms Cabs,Cspec, and Cdiff, since for each term the radiant par-ticle is at least brought to rest before being absorbedor reflected. Therefore, dFI accounts for the trans-fer in momentum to bring a radiated particle to rest.
5
The force term for diffuse and specular reflectance ac-counts for the momentum transfer due to reflection.The momentum contribution due to incident energyis in the opposite direction of the incoming energy,given by
dFI = −PI
cun (13)
The force exerted by specularly reflected energy is inthe direction of specular reflection which is given byreflecting the vector usun about an axis defined bythe direction un. Then the force exerted by specularreflection is given by
dFS =PS
c[2 (un · usun)un − usun] (14)
Diffusely reflected energy will reflect equally in alldirections and the resulting force will be in the nor-mal direction due to symmetric components cancel-ing out. For surfaces obeying Lambert’s cosine lawof diffuse emission the diffuse term will be [15]
dFD =2
3
PD
cun (15)
where the factor 23accounts for the portion of energy
that is reflected in the normal direction. Then theforce on an element of area is given by
dF = dFI + dFS + dFD (16)
The force acting on a body due to solar radiationpressure can be determined by integrating over theSun exposed surface area, given by
F =
∫
sun
(dFI + dFS + dFD) (17)
For a spherical body this integral is calculated overthe Sun exposed area. The result is given by
F = − Φsun,tot
c(d/d0)2A[
1 +2
3Cdiff
]
usun (18)
This equation can be rewritten in terms of albedo
F = − Φsun,tot
c(d/d0)2ACrusun (19)
where Cr = 1 + 23Cdiff. Similarly the forces can be
calculated for a flat plate:
F = − Φsun,tot
c(d/d0)2A (un · usun)
[
(1− Cspec)un
+
(
4
9Cdiff + 2Cspec (un · usun)
)
usun
] (20)
Under the assumption of a perfectly diffuse flat plate,Eq. (20) becomes
F = − Φsun,tot
c(d/d0)2(un · usun)
[
Aun +4
9ACdiff usun
]
(21)We can note that, if un remains aligned with usun, asimilar expression to Eq. (19) can be written for theflat plate model, where Cr = 1+ 4
9Cdiff in the case of
the flat plate. To calculate the SRP force for the facetmodel discussed in §2.1, the SRP force is calculatedfor each facet using the SRP force equation for a flatplate, Eq. (21), and then summed over all facets toobtain the total SRP force on the body:
FIsrp =
NF∑
j=1
FIsrp,j (22)
where NF is the total number of facets. The sum isperformed over all sides of the SO. If for any side theangle between the surface normal and the Sun’s di-rection is greater than π/2, then this side is not facingthe Sun and receives no energy from the Sun. There-fore, the solar radiation pressure for these sides is setto zero, FI
srp,j = 0 if θinc > π/2. The acceleration
due to SRP is then simply given by aIsrp = FIsrp/m.
The solar radiation pressure moments can be cal-culated by assuming that the SRP force acts throughthe center of each facet. Then the SRP moments canbe written as
TBsrp =
NF∑
j=1
[
rBi ×]
AFBsrp,j (23)
where rBi is the location of the geometric center ofeach facet with respect to the center of mass of theSO in body coordinates. The SRP moments are usedwith Eq. (8) to simulate the rotational dynamics ofthe SO.
2.4 Observation Model
Consider observations made by a optical site whichmeasures azimuth and elevation to a SO. The geom-etry and common terminology associated with thisobservation is shown in Figure 3, where dI is theposition vector from the observer to the SO, rI isthe position of the SO in inertial coordinates, RI isthe radius vector locating the observer, α and δ arethe right ascension and declination of the SO, respec-tively, θ is the sidereal time of the observer, λ is thelatitude of the observer, and φ is the East longitudefrom the observer to the SO. The fundamental obser-vation is given by
dI = rI −RI (24)
6
1i
observer
observer’s
meridian
plane
space object (SO)
satellite
subpoint
inertial
reference
direction
equatorial
plane
2i
3i
u
n
e
Ir
Id
IR
Figure 3: Geometry of Earth Observations of Spacecraft Motion
In non-rotating equatorial (inertial) components thevector dI is given by
dI =
x− ||RI || cos(θ) cos(λ)y − ||RI || sin(θ) cos(λ)
z − ||RI || sin(λ)
(25)
The conversion of dI from the inertial to the observercoordinate system (Up-East-North) is given by
ρuρeρn
=
cos(λ) 0 sin(λ)0 1 0
− sin(λ) 0 cos(λ)
×
cos(θ) sin(θ) 0− sin(θ) cos(θ) 0
0 0 1
dI
(26)
The angle observations consist of the azimuth, az, andelevation, el. The observation equations are given by
az = tan−1
(
ρeρn
)
(27a)
el = sin−1
(
ρu‖dI‖
)
(27b)
In addition to the azimuth and elevation, the opticalsite also records the magnitude of the brightness ofthe SO. The brightness of an object in space can bemodeled using an Phong light diffusion model [15].
This model is based on the bidirectional reflectancedistribution function (BRDF) which models light dis-tribution scattered from the surface due to the inci-dent light. The BRDF at any point on the surface isa function of two directions, the direction from whichthe light source originates and the direction fromwhich the scattered light leaves the observed surface.The model in Ref. [15] decomposes the BRDF intoa specular component and a diffuse component. Thetwo terms sum to give the total BRDF:
ρtotal(i) = ρspec(i) + ρdiff(i) (28)
The diffuse component represents light that is scat-tered equally in all directions (Lambertian) and thespecular component represents light that is concen-trated about some direction (mirror-like). Reference[15] develops a model for continuous arbitrary sur-faces but simplifies for flat surfaces. This simplifiedmodel is employed in this work as shape models thatare considered to consist of a finite number of flatfacets. Therefore the total observed brightness of anobject becomes the sum of the contribution from eachfacet.Under the flat plate assumption the specular term
of the BRDF becomes [15]
ρspec(i) = Cspec
(uobs · uspec)α
(usun · un)(29)
where the parameter α defines the width of the specu-
7
lar lobe. The vector uspec is the direction of specularreflection and is defined as uspec = 2 (un · usun)un −usun. The diffuse term of the BRDF is
ρdiff(i) =Cdiff
π(30)
The apparent magnitude of a SO is the result of sun-light reflecting off of its surfaces along the line-of-sightto an observer. First, the fraction of visible sunlightthat strikes an object (and is not absorbed) is com-puted by
Fsun(i) = Φsun,vis ρtotal(i)(
uIn(i) · uI
sun
)
(31)
where Φsun,vis = 455 W/m2 is the power per squaremeter impinging on a given object due to visible lightstriking the surface. If either the angle between thesurface normal and the observer’s direction or the an-gle between the surface normal and Sun direction isgreater than π/2, then there is no light reflected to-ward the observer. If this is the case then the fractionof visible light is set to Fsun(i) = 0.Next, the fraction of sunlight that strikes an object
that is reflected must be computed:
Fobs(i) =Fsun(i)A(i)
(
uIn(i) · uI
obs
)
‖dI‖2 (32)
The reflected light is now used to compute the ap-parent brightness magnitude, which is measured byan observer:
mapp = −26.7− 2.5log10
∣
∣
∣
∣
∣
NF∑
i=1
Fobs(i)
Φsun,vis
∣
∣
∣
∣
∣
(33)
where −26.7 is the apparent magnitude of the Sun.
3 Unscented Kalman Filter
Formulation
The unscented Kalman filter (UKF) is chosen forstate estimation because it has at least the accu-racy of a second-order filter [10] without the require-ment of computing Jacobians like the EKF. The UKFstructure is used for estimating rotational, transla-tional, and parameter states based on fusing anglesand light curve data along with their associated mod-els as discussed in §2. The attitude UKF describedin Ref. [16] is used in the same manner as the oneshown in Refs. [4] and [12].Applying the UKF structure for attitude estima-
tion has some challenges. For instance, althoughthree parameter sets are attitude minimal representa-tions, they inherently have singularities. On the other
hand the quaternion representation, which is a fourparameter set with no singularity, has a nonlinearconstraint which results in a singular covariance ma-trix, and the quaternion is not constitute by directlyadding quaternions but through quaternion multipli-cation. This does not allow use of quaternions in astraightforward UKF implementation. Reference [16]overcomes these challenges by utilizing generalizedRodrigues parameters (GRPs), a three parameter set,to define the local error and quaternions to define theglobal attitude.
The global parameterization for attitude UKF ap-proach used for this work is the quaternion whilea minimal parametrization involving the generalizedRodriguez parameters (GRPs) is used to define thelocal error. Quaternions are useful because their kine-matics are free of singularities. The representation ofthe attitude error as a GRP is useful for the propaga-tion and update stages of the attitude covariance be-cause the structure of the UKF can be used directly.Complete explanations of the quaternion and its map-ping to GRPs is provided in Refs. [14] and [17]. Inthe UKF implementation described in Ref. [16], thecovariance matrix is interpreted as the covariance ofthe error GRP because for small angle errors, theerror GRP is additive and the UKF structure canbe used directly to compute sigma-points. The er-ror GRP sigma points are converted to error quater-nions and then to global quaternions for the propa-gation stage. To compute the propagated covariancethe global quaternions are converted to error quater-nions and then back to error GRPs. The process isthen as follows: error GRP→ error quaternion, errorquaternion → global quaternion, global quaternion→ error quaternion, and finally error quaternion →error GRP.
3.1 Model and Measurement Uncer-
tainty
A UKF is now summarized for estimating the stateof a SO’s position, velocity, orientation, rotationrate, mass, albedo-areas and areas given by x =
[qBT
I ωBT
B/I rIT
vIT
m rA]T for method I and x =
[qBT
I ωBT
B/I rIT
vIT
m rA A]T for method II . The
dynamic models from Eqs. (4) and (8) can be writtenin the general state equation which gives the deter-ministic part of the stochastic model:
x = f (x, t) +G (x, t)Γ(t) (34)
where Γ(t) is a Gaussian white noise process termwith correlation function Qδ(t1 − t2). The func-tion f (x, t) is a general nonlinear function. To solve
8
the general nonlinear filtering problem the UKF uti-lizes the unscented transformation to determine themean and covariance propagation though the func-tion f (x, t). The dynamic function used in thiswork consists of rotational and translational dynam-ics given by the sigma points, which are propagatedthrough the system dynamics:
f ([χ, q]) =
1
2Ξ(q)ωB
B/I
J−1SO
(
TBsrp −
[
ωBB/I×
]
JSOωBB/I
)
vI
− µ
r3rI + aISRP
(35)If the initial pdf p(xo) that describes the associatedstate uncertainty is given, the solution for the timeevolution of p(x, t) constitutes the nonlinear filteringproblem.
Given a system model with initial state and co-variance values, the UKF propagates the state vectorand the error-covariance matrix recursively. At dis-crete observation times, the UKF updates the stateand covariance matrix conditioned on the informa-tion gained from the measurements. The predictionphase is important for overall filter performance. Ingeneral, the discrete measurement equation can beexpressed for the filter as
yk = h (xk, tk) + vk (36)
where yk is a measurement vector and vk is the mea-surement noise, which is assumed to be a zero-meanGaussian process with covariance Rk.
All random variables in the UKF are assumed to beGaussian random variables and their distribution areapproximated by the deterministically selected sigmapoints. The sigma points are selected to be along theprincipal axis directions of the state error-covariance.Given an L×L error-covariance matrix Pk, the sigmapoints are constructed by
σk ← 2L columns from ±√
(L + λ)Pk (37a)
χk(0) = µk (37b)
χk(i) = σk(i) + µk (37c)
where√M is shorthand notation for a matrix Z such
that M = Z ZT . Given that these points are selectedto represent the distribution of the state vector, eachsigma point is given a weight that preserves the in-
formation contained in the initial distribution:
Wmean0 =
λ
L+ λ(38a)
W cov0 =
λ
L+ λ+ (1 − α2 + β) (38b)
Wmeani = W cov
i =1
2(L+ λ), i = 1, 2, . . . , 2L
(38c)
where λ = α2(L + κ) − L is a composite scaling pa-rameter.The constant α controls the spread of the sigma
point distribution and should be a small number, 0 <α ≤ 1. κ = 3−L provides an extra degree of freedomthat is used to fine-tune the higher-order moments,and β is used to incorporate prior knowledge of thedistribution by weighting the mean sigma point inthe covariance calculation.The reduced state vector, with the error GRP
states and the full state vector, with quaternion statevector, for the joint attitude and position estimateproblem are given by
xδp
k =
δp
ωBB/I
rI
vI
mrA
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
tk
xq
k =
qBI
ωBB/I
rI
vI
mrA
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
tk
(39)
where δp is the error GRP states associated with thequaternion qB
I and · is used to denote estimate. Theinitial estimate x0 is the mean sigma point and isdenoted χ0(0). The error GRP state of the initialestimate is set to zero, while the rest of the states areinitialized by their respective initial estimates.
3.2 Using Quaternions for UKF
The error quaternion, denoted by δq−
k (i), associatedwith the ith error GRP sigma point is computed by[16]
δ−
k (i) = f−1[
a+ δq−4k(i)]
χδpk (i) (40a)
δq−4k(i) =−a||χδp
k (i)||2 + f√
f2 + (1− a2)||χδpk (i)||2
f2 + ||χδpk (i)||2
(40b)
δq−
k (i) =
[
δ−
k (i)δq−4k(i)
]
(40c)
where a is a parameter from 0 to 1 and f is a scalefactor, which is often set to f = 2(a + 1). Here itis noted that the subscript I and superscript B in
9
qBI and its estimates are omitted in this section for
brevity. The representation of the attitude estimateperturbed by the ith error quaternion is computedusing the quaternion composition:
q−
k (i) = δq−
k (i)⊗ q−
k (0) (41)
where
q′ ⊗ q =[
Ψ(q′) q′]
q (42)
This forms the global quaternion, and the er-ror quaternions corresponding to each propagatedquaternion sigma point are computed through thequaternion composition:
δq−
k+1(i) = q−
k+1(i)⊗[
q−
k+1(0)]−1
(43)
where the notation for the inverse quaternion is de-fined as:
q−1 ≡[
−q4
]
(44)
Using the result of Eq. (43), the error GRP sigmapoints are computed as
δp−
k+1(i) = fδ ˆ−k+1(i)
a+ δq−4k+1(i)
(45)
Angles data can be used to determine the unknownposition and velocity of a SO but if the position iscoupled with the attitude dynamics then angles datacan assist with attitude estimation as well. Howeverif position is known accurately, then using only lightcurve data is sufficient to determine the orientation.
3.3 Constrained UKF
The estimator discussed in this works includes con-strained states since the area states are constrainedto be positive. To account for these constraints themethod from ref. [18] is used. Reference [18] incorpo-rates the information of the constraints in the UKFalgorithm by projecting sigma points and estimatesonto the constraint boundary when sigma points orestimates are found outside the feasible region. Sincer ≥ 0 we have the following constraints
rA ≥ 0 (46a)
A ≥ 0 (46b)
Then if the constraints are violated during, sam-pling the sigma pointing from covariance matrix, af-ter propagation, or after the update step, the esti-mates and or sigma points that violate the constraintsare projected onto the constraint boundary.
3.4 Summary of UKF
The UKF algorithm is described in Table 1. Theprocess starts by first defining two initial state vec-tors, one that includes quaternion states and one thatincludes error GRP states. The error GRP statesare initially set to zero. The initial covariance ma-trix is defined as the initial error covariance for thestate vector that includes the GRP states, the trans-lational, rotational and parametric states. The co-variance matrix is then used to form the error GRPsigma points. The error GRP sigma points are con-verted to quaternion sigma points by creating errorquaternions from each error GRP and then addingthe error quaternion to the initial mean quaternionusing quaternion multiplication. If any area statessigma points are found outside the feasible region dis-cussed in §3.3 they are projecting onto the constraintboundary.Next the quaternion sigma points are propa-
gated through the system dynamics using equationEq. (35). The estimated acceleration and torque dueto SRP are calculated with Eqs. (22) and (23), re-spectively. After propagating the sigma points, theerror GRP states are computed with the propagatedquaternion sigma points. If any area states sigmapoints are found outside the feasible region they areprojecting onto the constraint boundary. The propa-gated mean sigma point quaternion, q−
k+1(0), is com-puted and stored, and error quaternions correspond-ing to each propagated quaternion sigma point arecomputed. The non-attitude sigma points are thepropagated non-attitude states.
1000 2000 3000 4000 5000 6000 7000−1000
−800
−600
−400
−200
0
200
400
600
800
1000
Mas
s er
ror
(kg)
Measurement Samples
Figure 4: Mass Error Estimated Using Method I
After setting the error GRP for the mean sigmapoint to zero, the propagated sigma points are re-combined, and the propagated mean and covarianceare calculated as a weighted sum of the sigma points,where Qk+1 is the discrete-time process noise covari-
10
Table 1: UKF for Rotational, Translational, and Parameter States
Initialize with
xδpo = E{xδp
o } xq
o = E{xo} Pδpo = E{
(
xδpo − xδp
o
) (
xδpo − xδp
o
)T }Calculate GRP Sigma Points
χδp
k =[
xδp
k xδp
k + γ√
Pk−1 xδp
k − γ√
Pk−1
]
Calculate Quaternion Sigma Points
χq
k =[
xq
k xq
k + γ√
Pk−1 xq
k − γ√
Pk−1
]
Propagate Quaternion Sigma Pointsχ
q
k = F[
χq
k−1, t]
Calculate GRP Sigma Points
χδp
k =[
xδp
k xδp
k + γ√
Pk−1 xδp
k − γ√
Pk−1
]
Time update
x−
k+1 =∑2L
i=0 Wmeani χk+1(i)
P−
k+1 =∑2L
i=0 Wcovi [χk+1(i)− x−
k+1] [χk+1(i)− x−
k+1]T +Qk+1
Measurement updateγk(i) = h
[
χk(i), q−
k
]
y−
k =∑2L
i=0 Wmeani γk(i)
P yyk =
∑2Li=0 W
covi [γk(i)− y−
k ] [γk(i)− y−
k ]T
P υυk = P yy
k +Rk
P xyk =
∑2Li=0 W
covi [χk(i)− x−
k ] [γk(i)− y−
k ]T
Kk = P xyk (P υυ
k )−1
x+k = x−
k +Kk [yk − yk]P+k = P−
k −KkPvvKT
k
Quaternion updateq+k = δq+
k ⊗ q−
k (0)Set GRP to zeroδp = [0 0 0]T
ance. As previously discussed, measurements areavailable in the form of azimuth, elevation and ap-parent brightness magnitude, y ≡ [mapp az el]T .Estimated observations are computed for each sigmapoint using the observation models discussed previ-ously. The mean estimated output are computed,and the output, innovations, and cross-correlation co-variance are computed using the sigma points. Fi-nally, the Kalman gain is calculated from the sigmapoint and is used to update the estimated state vectorthat contains the error GRPs. If any of the updatedarea states are found outside the feasible region theyare projecting onto the constraint boundary. Thequaternion update is performed by converting theerror GRP states of x+
k to a quaternion, δq+k , via
Eq. (40), and adding it to the estimated quaternionusing quaternion multiplication.
4 Results
Three simulations are presented: method I is usedto estimate mass using an assumed albedo value;method II is used to estimate mass and area; anda study is conducted to determine the effect of area-to-mass ratio on mass estimation performance. In allcases the same dynamic configuration is considered,with same initial attitude, true orbit and true angu-lar velocity. For the generation of data for the truemodel, an equatorial ground station is chosen as thesite of the observer. Also, all scenarios use the sameshape model which contains six sides where the areasand albedo-areas for these sides are estimated. TheSO is simulated to fly in a near-geosynchronous or-bit in a trajectory that is continuously lighted. Thisis accomplished by inclining the orbit by 30 degreesand choosing an appropriate time of the year, therebyavoiding the shadow cast by the Earth.
The initial inertial position and velocity are chosen
11
2000 4000 6000−0.2
0
0.2
Ar 6 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−0.2
0
0.2
Ar 5 e
rror
(m
2 )
Measurement Samples
2000 4000 6000
−0.050
0.05
Ar 4 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−10
0
10
Ar 3 e
rror
(m
2 )
Measurement Samples
2000 4000 6000−0.4−0.2
00.20.4
Ar 2 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−0.4−0.2
00.20.4
Ar 1 e
rror
(m
2 )Measurement Samples
Figure 5: Albedo-Area Estimated Using Method I
as rI = [−7.8931× 102 3.6679× 104 2.1184× 104]T
km and vI = [−3.0669 − 4.9425× 10−2 − 2.8545×10−2]T km/s. The geographic position of the groundsite is 0◦ North, 172◦ West with 0 km altitude. Thetime of the start of the simulation is May 8, 2007 at5:27.55. The initial true quaternion attitude mappingfrom the inertial frame to the body frame is chosenas qB
I = [1/2 0 0 1/2]T . A constant rotation rate,defined as the body rate with respect to the inertialframe, represented in body coordinates, is used givenby ωB
B/I = [0 0.00262 0]T rad/s.
For all simulations scenarios, measurements areproduced using zero-mean white-noise error processeswith standard deviation of 0.1 for magnitude andstandard deviation of 1 arc-seconds for azimuth andelevation. The initial errors for the states are 5 degfor all three attitudes, 1, 000 deg/hr for the rotationalrates, and 1 km and 0.001 km/s for the position andthe velocity errors, respectively. The initial condi-tion error-covariance values are set to 202 deg2 forall three attitude components, 1, 4002 (deg/hr)2 forthe rotational rates, and 12 km2 and 0.0012 (km/s)2
for the position and the velocity errors, respectively.The initial condition error-covariance values for themass, albedo-area, and area are 3002 kg2, 102 m4, and102 m4, respectively. The time interval between themeasurements is set to 30 seconds. Data is simulated
for 20 nights (about 20 orbits) where observations ofthe SO are made each night for 3 hours. The simula-tion results are plotted versus number of data samplessince there are large time gaps between each 3 hourdata arc.
For the development of the measured light curvedata a six sided facet SO shape model is used. Thethree parameters for the shape model are given, l, a,and d which are the length, width, and height respec-tively, as shown in Figure 2. For the truth model asix-facet rectangular shape model is used with dimen-sion l = 4 m, a = 2 m, and d = 8 m. The mass for thetrue SO is selected to be m = 1, 500 kg, which givesa maximum area-to-mass ratio of 0.09 for this model.For the area-to-mass ratio study this ratio is variedfrom 0.01 to 10. All facets have reflectance valuesof Cspec = 0 and Cdiff = 0.5, under the assumptionthat the SO is perfectly diffuse. For method I, theassumed albedo is set to the true value r = 0.3.
4.1 Angles and Magnitude Fusion:
Method I
The estimation errors, along with their respective 3σbounds calculated from the covariance for method I,for attitude, position, velocity, rotation rate, mass,albedo-areas, and areas, are shown in Figure 6. The
12
1000 2000 3000 4000 5000 6000 7000−0.5
0
0.5
1000 2000 3000 4000 5000 6000 7000−0.5
0
0.5
1000 2000 3000 4000 5000 6000 7000−0.5
0
0.5
Measurement Samples
Roll(D
eg)
Pitch
(Deg)
Yaw
(Deg)
(a) Attitude Estimation Error
1000 2000 3000 4000 5000 6000 7000
−0.05
0
0.05
1000 2000 3000 4000 5000 6000 7000−0.1
0
0.1
1000 2000 3000 4000 5000 6000 7000−0.04−0.02
00.020.04
Measurement Samples
x(km)
y(km)
z(km)
(b) Position Estimation Error
1000 2000 3000 4000 5000 6000 7000
−5
0
5
x 10−6
v x (km
/s)
1000 2000 3000 4000 5000 6000 7000
−202
x 10−5
v y (km
/s)
1000 2000 3000 4000 5000 6000 7000
−101
x 10−5
v z (km
/s)
Measurement Samples
(c) Velocity Estimation Error
1000 2000 3000 4000 5000 6000 7000
−0.5
0
0.5
wx (
Deg
/Hr)
1000 2000 3000 4000 5000 6000 7000−2
0
2x 10
−3
wy (
Deg
/Hr)
1000 2000 3000 4000 5000 6000 7000−0.5
0
0.5
wz (
Deg
/Hr)
Measurement Samples
(d) Angular Velocity Estimation Error
Figure 6: SO State Estimation Results for Method I
attitude is estimated to within 0.03◦ of uncertainty,and attitude rate is found to within 0.05 deg/hr forthe x-axis, 4.5 × 10−4 deg/hr for the y-axis (the y-axis is the spin axis), and 0.03 deg/hr for the z-axis.Position and velocity are estimated to within 10 mand 0.0028 m/s, respectively. The mass is estimatedto within 54.76 kg 3σ, and the albedo-areas are es-timated to 0.5 m2. For the albedo-area plot the es-timates are plotted going from +x, −x, +y, −y, +zand−z. The area states are not estimated for methodI since the albedo is assumed known. Also the albedo-areas plot shows that the third area component (+ycomponent) is not observable. This is due to the factthat this side does not face the observer because theSO is spinning about the y-axis. Although this sideis not very observable the filter can still observe massstates. This is due to the fact the SRP disturbanceis a sum of all the cross-sectional errors and overtimethrough the rotational dynamics of the space objectall the sides contribute to the translational dynamiccreating observability. Most states show proper fil-ter convergence behavior in that the residual errorssettle down and are bounded by their computed 3σbounds.
1000 2000 3000 4000 5000−1000
−800
−600
−400
−200
0
200
400
600
800
1000
Mas
s er
ror
(kg)
Measurement Samples
Figure 7: Mass Error Estimated Using Method I
4.2 Angles and Magnitude Fusion:
Method II
The estimation errors, along with their respective 3σbounds calculated from the covariance for method II,for attitude, position, velocity, rotation rate, mass,albedo-areas and areas are shown in Figure 10. The
13
estimated attitude, attitude rate, position and veloc-ity show similar performance to that of method I. Thealbedo-areas also show similar performance to that ofmethod I and are estimated to 0.5 m2.
The mass estimate for method II has larger errorthan that of method I, where the mass is estimatedto within 200.67 kg 3σ. The area state estimates areshown in Figure 9, and the area state shows small im-provement in the estimated error. Although the areauncertainty is still higher, the mass is still observ-able and the uncertainty in the area is accounted forin method II. Most states show proper filter conver-gence behavior in that the residual errors settle downand are bounded by their computed 3σ bounds.
4.3 Dependence on Area-to-Mass Ra-
tio
To study the convergence rate of the mass estimatesas a function of area-to-mass ratio, a number of sim-ulations for conducted where A/m is varied over fourorders of magnitude, A/m = [0.01 0.1 1 10]. Figure11 shows the mass estimate error covariance for thearea-to-mass ratios considered. As expected, as thearea-to-mass ratio increases the convergence rate alsoincreases. This is due to the fact that for higher area-to-mass ratio objects the effect of SRP perturbationis greater; and therefore the mismodeling effect ap-pears faster in the data making the mass and areaparameters observable earlier.
Also it can be seen from Figure 11 the steadystate error is reduced for objects with higher area-to-mass ratios. This is due to fact that, for theseclass of objects, the disturbance seen from SRP ismuch higher than the estimated error in their trans-lational states; the SRP disturbance is thus shown tohave good signal-to-ratio characteristics. Figure 11that the converged mass values for A/m of 0.01, 0.1,1, and 10 are 160.12 kg, 30.45 kg, 2.71 kg, and 0.80kg respectively.
5 Conclusion
In this paper an unscented Kalman filter estimationscheme using light curve and angles data was pre-sented and was used to estimate mass, albedo-areasand areas of a SO along with its associated rotationaland translational states. This work uses an assumedshape model of six sides and estimated the area andalbedo-areas for each side. Using an unscented filterto employ brightness magnitude and angles data, theestimator was able to determine the mass of a SO towithin 54.76 kg 3σ. Simulations were conducted to
study the dependance of convergence rate and esti-mate accuracy on the area-to-mass ratio of the SO.As expected it was shown that for higher area-to-mass ratios the mass and area states could be es-timated to higher accuracy with faster convergencerates. Higher area-to-mass ratios leads to higher ac-curacy and faster convergence rates.
6 Future Work
Future studies will consider the effect of phase an-gles on the estimates of areas and mass. Since theobservable area parameters from light curve data area projection of the observer facing areas and the ob-servable parameter from angles data are Sun facingareas, these two are in better agreement for lowerphase angles. For data gathered at lower phase it isexpected the estimation performance will improve inaccuracy.
Also in this work is was assumed that the shapemodel was that of a six sided cuboid, but this as-sumption can be relaxed by using the approach de-veloped in Ref. [8], where the shape models are hy-pothesized using a multiple-model adaptive estima-tion technique. This will allow this work to be ex-tended to estimating area and mass parameters forgeneral shaped model SOs.
7 Acknowledgement
This work was supported through multiple fundingmechanisms, one of which was via the Air Force Re-search Laboratory, Space Vehicles Directorate (AS-TRIA) Space Scholars Program.
0 1000 2000 3000 4000 5000
100
101
102
Measurement Samples
σ m
AtoM=0.01AtoM=0.1AtoM=1AtoM=10
Figure 11: Mass Variance Versus Number of DataSample for Varying Area-to-Mass Ratios
14
2000 4000 6000−0.2
0
0.2
Ar 6 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−0.2
0
0.2
Ar 5 e
rror
(m
2 )
Measurement Samples
2000 4000 6000
−0.050
0.05
Ar 4 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−10
0
10
Ar 3 e
rror
(m
2 )
Measurement Samples
2000 4000 6000−0.4−0.2
00.20.4
Ar 2 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−0.4−0.2
00.20.4
Ar 1 e
rror
(m
2 )Measurement Samples
Figure 8: Albedo-Area Estimated Using Method II
2000 4000 6000−10
0
10
A6 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−10
0
10
A5 e
rror
(m
2 )
Measurement Samples
2000 4000 6000−10
0
10
A4 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−10
0
10
A3 e
rror
(m
2 )
Measurement Samples
2000 4000 6000−10
0
10
A2 e
rror
(m
2 )
Measurement Samples2000 4000 6000
−10
0
10
A1 e
rror
(m
2 )
Measurement Samples
Figure 9: Area Estimated Using Method II
15
1000 2000 3000 4000 5000−1
0
1
1000 2000 3000 4000 5000−1
0
1
1000 2000 3000 4000 5000−1
0
1
Measurement Samples
Roll(D
eg)
Pitch
(Deg)
Yaw
(Deg)
(a) Attitude Estimation Error
1000 2000 3000 4000 5000
−0.05
0
0.05
1000 2000 3000 4000 5000−0.1
0
0.1
1000 2000 3000 4000 5000
−0.01
0
0.01
Measurement Samples
x(km)
y(km)
z(km)
(b) Position Estimation Error
1000 2000 3000 4000 5000
−505
x 10−6
v x (km
/s)
1000 2000 3000 4000 5000−4−2
024
x 10−5
v y (km
/s)
1000 2000 3000 4000 5000−2
0
2x 10
−5
v z (km
/s)
Measurement Samples
(c) Velocity Estimation Error
1000 2000 3000 4000 5000
−1
0
1
wx (
Deg
/Hr)
1000 2000 3000 4000 5000
−0.01
0
0.01
wy (
Deg
/Hr)
1000 2000 3000 4000 5000−1
0
1
wz (
Deg
/Hr)
Measurement Samples
(d) Angular Velocity Estimation Error
Figure 10: SO State Estimation Results for Method II
References
[1] Kaasalainen, M. and Torppa, J., “OptimizationMethods for Asteriod Lightcurve Inversion I:Shape Determination,” Icarus , Vol. 153, No. 4,Jan. 2001, pp. 24–36.
[2] Kaasalainen, M. and Torppa, J., “Optimiza-tion Methods for Asteriod Lightcurve Inver-sion II: The Complete Inverse Problem,” Icarus ,Vol. 153, No. 4, Jan. 2001, pp. 37–51.
[3] Calef, B., Africano, J., Birge, B., Hall, D., andKervin, P. W., “Photometric Signature Inver-sion,” Proceedings of the International Society
for Optical Engineering, Vol. 6307, San Diego,CA, Aug. 2006, Paper 11.
[4] Jah, M. and Madler, R., “Satellite Charac-terization: Angles and Light Curve Data Fu-sion for Spacecraft State and Parameter Esti-mation,” Proceedings of the Advanced Maui Op-
tical and Space Surveillance Technologies Con-
ference, Vol. 49, Wailea, Maui, HI, Sept. 2007,Paper E49.
[5] Centinello, F. J., “Six Degree of Freedom Esti-mation with Light Curve Data,” Tech. rep., AirForce Research Laboratory, Kihei, HI, 2008.
[6] Schildknecht, T., “Optical Surveys for SpaceDebris,” Astronomy and Astrophysics Review ,Vol. 14, No. 1, 2007, pp. 41–111.
[7] Kelecy, T. and Jah, M., “Analysis of High Area-to-Mass Ratio (HAMR) GEO Space Object Or-bit Determination and Prediction Performance:Initial Strategies to Recover and Predict HAMRGEO Trajectories with No a priori Information,”Acta Astronautica, Vol. 69, No. 7-8, Sept. 2011,pp. 551–558.
[8] Linares, R., Crassidis, J. L., Jah, M., and Kim,H., “Astrometric and Photometric Data Fus-sion for Orbit, Attitude, and Shape Estimation,”AIAA Guidance, Navigation and Control Con-
ference, Chicago, IL, Aug. 2010, AIAA-2009-6293.
[9] Jazwinski, A. H., Stochastic Processes and Fil-
tering Theory, chap. 8, Academic Press, SanDiego, CA, 1970.
16
[10] Julier, S. J., Uhlmann, J. K., and Durrant-Whyte, H. F., “A New Method for the Nonlin-ear Transformation of Means and Covariances inFilters and Estimators,” IEEE Transactions on
Automatic Control , Vol. AC-45, No. 3, March2000, pp. 477–482.
[11] Gordon, N. J., Salmond, D. J., and Smith,A. F. M., “Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation,” IEE Pro-
ceedings, Part F - Communications, Radar and
Signal Processing, Vol. 140, No. 2, Seattle, WA,April 1993, pp. 107–113.
[12] Wetterer, C. J. and Jah, M., “Attitude Estima-tion from Light Curves,” Journal of Guidance,
Control and Dynamics , Vol. 32, No. 5, Sept.-Oct. 2009, pp. 1648–1651.
[13] Africano, J., Kervin, P., Hall, D., Sydney, P.,Ross, J., Payne, T., Gregory, S., Jorgensen, K.,Jarvis, K., Parr-Thumm, T., Stansbery, G., andBarker, E., “Understanding Photometric PhaseAngle Corrections,” Proceedings of the 4th Euro-
pean Conference on Space Debris , Vol. SP-587,Darmstadt, Germany, April 2005, pp. 141–146.
[14] Shuster, M. D., “A Survey of Attitude Represen-tations,” Journal of the Astronautical Sciences ,Vol. 41, No. 4, Oct.-Dec. 1993, pp. 439–517.
[15] Ashikmin, M. and Shirley, P., “An AnisotropicPhong Light Reflection Model,” Tech. Rep.UUCS-00-014, University of Utah, Salt LakeCity, UT, 2000.
[16] Crassidis, J. L. and Markley, F. L., “UnscentedFiltering for Spacecraft Attitude Estimation,”Journal of Guidance, Control and Dynamics ,Vol. 26, No. 4, July-Aug. 2003, pp. 536–542.
[17] Schaub, H. and Junkins, J. L., “StereographicOrientation Parameters for Attitude Dynamics:A Generalization of the Rodrigues Parameters,”Journal of the Astronautical Sciences , Vol. 44,No. 1, Jan.-March 1996, pp. 1–20.
[18] Kandepu, R., Imsland, L., and Foss, B. a., “Con-strained state estimation using the UnscentedKalman Filter,” 2008 16th Mediterranean Con-
ference on Control and Automation, , No. i, June2008, pp. 1453–1458.
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