SPACE situational awareness (SSA) encompasses intelligence,

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1 JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 34, No. 6, November December 2011 Gaussian Sum Filters for Sace Surveillance: Theory and Simulations Joshua T. Horwood, Nathan D. Aragon, and Aubrey B. Poore Numerica Cororation, Loveland, Colorado DOI: / While standard Kalman-based filters, Gaussian assumtions, and covariance-weighted metrics are very effective in data-rich tracking environments, their use in the data-sarse environment of sace surveillance is more limited. To roerly characterize non-gaussian density functions arising in the roblem of long-term roagation of state uncertainties, a Gaussian sum filter adated to the two-body roblem in sace surveillance is roosed and demonstrated to achieve uncertainty consistency. The roosed filter is made efficient by using only a onedimensional Gaussian sum in equinoctial orbital elements, thereby avoiding the exensive reresentation of a full six-dimensional mixture and hence the curse of dimensionality. Additionally, an alternate set of equinoctial elements is roosed and is shown to rovide enhanced uncertainty consistently over the traditional element set. Simulation studies illustrate the imrovements in the Gaussian sum aroach over the traditional unscented Kalman filter and the imact of correct uncertainty reresentation in the roblems of data association (correlation) and anomaly (maneuver) detection. Nomenclature A = lower-triangular Cholesky factor of a covariance matrix a = semimajor axis a ert = erturbing acceleration a; h; k; ; q; = equinoctial orbital elements c = arameter controlling the accuracy of the Gaussian sum filter f = system dynamics vector G = rocess noise shae matrix h = measurement function vector k = (subscrit) time index = mean longitude N = number of mixture comonents N = Gaussian robability density function n = mean motion n; h; k; ; q; = alternate equinoctial orbital elements P = covariance of a Gaussian distribution PE = rediction error = robability density function Qt = rocess noise covariance matrix R k = measurement noise covariance matrix r = Cartesian Earth-Centered Inertial (ECI) osition coordinates _r = Cartesian ECI velocity coordinates r = Cartesian ECI acceleration coordinates t 0, t = initial and current times u = orbital element coordinates (sixdimensional) fv 1 ;...; v k g = Gaussian white noise sequence wt = Gaussian white noise rocess w ;w ;... = mixture weights x = dynamic state vector k fz 1 ;...; z k g = measurement sequence Received 25 January 2011; revision received 25 Aril 2011; acceted for ublication 25 Aril Coyright 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Coies of this aer may be made for ersonal or internal use, on condition that the coier ay the $10.00 er-coy fee to the Coyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code /11 and $10.00 in corresondence with the CCC. Research Scientist; joshua.horwood@numerica.us. Research Scientist; nate.aragon@numerica.us. Chief Scientist and CEO; aubrey.oore@numerica.us ; ;... = (subscrits) indices of mixture comonents kj = Kronecker delta symbol = mean of a Gaussian distribution = Earth gravitational constant (398600:4418 km 3 =s 2 ) = standard deviation of a univariate Gaussian distribution = inverse solution flow r x = gradient oerator with resect to x (column oriented) Introduction SPACE situational awareness (SSA) encomasses intelligence, reconnaissance of all sace objects, and the rediction of sace events, ossible collisions, threats, and activities. Fundamental to SSA are the roblems of data/track association (correlation), conjunction analysis, sensor resource management, and anomaly (e.g., maneuver, change) detection. Common amongst these tracking roblems is the requirement of uncertainty consistency, which is the accurate and truthful reresentation of the robability density function (PDF) of each resident sace object (RSO) in the sace catalog. Within the roblems of nonlinear estimation and filtering, uncertainty consistency entails comuting an accurate finitedimensional reresentation of a (ossibly non-gaussian) PDF and accurately imlementing the rediction and fusion stes of the general Bayesian state estimator [1]. The roblem of tracking RSOs and correctly managing and reresenting their uncertainties resents some unique and formidable challenges not found in other tracking environments. In contrast to air or ground tracking, the sace surveillance environment is datastarved. Tyical RSO tracking roblems require the long-term roagation of state PDFs, often on the order of several orbital eriods, using high-fidelity dynamical models in the absence of measurement or track udates. Often, the state uncertainty of an RSO is assumed to be Gaussian which, although may be aroriate initially, will inevitably become non-gaussian if roagated for a sufficiently long time san under nonlinear dynamics (i.e., gravity, drag, solar radiation, third-body erturbations, etc.). This dearture from Gaussianity lagues the standard Kalman-based filtering algorithms such as those of the extended (EKF) and unscented (UKF) variety. More accurate filters which better aroximate model nonlinearities and non-gaussian PDFs are therefore sought. The

2 1840 HORWOOD, ARAGON, AND POORE Gaussian sum filter is one such examle which has been investigated in SSA alications by the authors [2,3] and other researchers [4 8]. The Gaussian sum filter (GSF) is based on a fundamental result of Alsach and Sorenson [4], which states that any PDF can be aroximated arbitrarily close (in the L 1 sense) by a weighted sum (mixture) of Gaussian PDFs henceforth called a Gaussian sum. Thus, Gaussian sums rovide a mechanism for modeling non-gaussian densities and for more accurately aroximating the solution of the Fokker Planck Kolmogorov equation (FPKE) [9], which governs the time evolution of a PDF under a nonlinear stochastic dynamical system. Comutationally, the GSF has the added advantage of being arallelizable since filters such as the EKF or UKF act indeendently on each comonent Gaussian in the rediction and correction stes. With regard to udating the weights within the filter, such a scheme is clearly dictated from Bayes rule following a measurement or track udate (fusion). However, there is not comlete agreement on how the weights should be udated (if at all) following a rediction. Our hilosohy is to not udate the weights, which is justified because uncertainty consistency is achieved by working in coordinates adated to the dynamics (i.e., orbital elements). On the other hand, some researchers [6 8] have roosed methods for adating the weights based on various L 2 otimization criteria, for examle, using the FPKE error as feedback. The imact of using one these weight udate schemes during roagation is analyzed in this aer. It is found that the method does not imrove uncertainty consistency when alied to the roosed GSF, but rather causes the accuracy of the Gaussian sum reresentation to degrade slightly. This aer builds on the earlier works of Horwood and Poore [2] and Horwood et al. [3], which rovide an initial framework for a GSF based on efficient refinement and coarsening schemes (i.e., adatively changing the number of mixture comonents). Although the revious methodology is very general and alicable to a variety of tracking roblems, it suffers from the curse of dimensionality because the refinement oeration is erformed along all dimensions of the distribution. In the six-dimensional setting of sace surveillance, the length of the Gaussian sum generated from this algorithm would be rohibitive. In the resent aer, we develo a new and more efficient GSF by noting that, in equinoctial orbital element sace, one can accurately roagate the full state PDF using only a one-dimensional Gaussian sum. This erformance enhancement is not ossible in a system of Cartesian Earth-centered inertial (ECI) coordinates because the nonlinearities of the dynamics act strongly in all six coordinates. In contrast, in orbital element sace (equinoctial or otherwise), five of the six coordinates evolve linearly under the unerturbed two-body (Keler) roblem and evolve nearly linearly under erturbed dynamics. Consequently, the uncertainty grows mainly along one coordinate (the in-track or mean longitude direction). Deendent on the size of the initial uncertainty in the radial direction (semimajor axis) and the roagation time, a onedimensional Gaussian sum in equinoctial sace can be defined. Furthermore, when the UKF is used to roagate each comonent Gaussian in arallel, the roagated Gaussian sum is argued to be a consistent reresentation of the true PDF. The underlying GSF is made very efficient because the initial Gaussian sum is comuted by solving an L 2 otimization roblem once offline (the solution is also demonstrated to be near otimal in L 1 and L 1 ) and used thereafter in any scenario through a looku table. An additional finding communicated in this aer is that enhanced uncertainty consistency can be achieved by reresenting state PDFs in alternate equinoctial orbital elements, which use the mean motion in lace of the semimajor axis. Under unerturbed Kelerian dynamics, all six of these coordinates transform linearly. To the knowledge of the authors, this alternate set of equinoctial elements does not aear to be recognized within the sace surveillance community. To demonstrate the imact of correct uncertainty characterization of an RSO s state in the roblem of long-term roagation, the rediction error comonent of the likelihood ratio used to score the association of a track to an orbit has been evaluated using a highfidelity gravity model in conjunction with the new GSF. Because the statistically correct rediction error can be accurately comuted, the GSF is shown to maintain uncertainty consistency over many orbital eriods. This is comared with the classical covariance-weighted rediction error which, while sufficient for short-term rediction in a data-rich environment, is demonstrated to be inadequate for longterm rediction required in the data-sarse sace surveillance environment. Simulation studies are also erformed illustrating the imact of correct uncertainty management on the roblems of data association (correlation) and anomaly detection. The lan of the aer is as follows. First, the coordinate systems used in this aer are defined including the alternate system of equinoctial orbital elements. Second, the mathematical theory of nonlinear filtering is summarized and subsequently secialized to mixture PDFs. Third, the new Gaussian sum filter is develoed showing recisely how the initial Gaussian sum is defined so that uncertainty consistency is maintained when it is roagated over a long time ga. Fourth, an online metric based on the rediction error is roosed, which rovides a means to assess track state uncertainty consistency. Finally, results are resented and conclusions are made. The aendix summarizes a weight udate scheme for uncertainty roagation roosed originally by Terejanu et al. [6], which is subsequently validated against the new GSF in the simulations section. Coordinate Systems In this section, the roosed orbital element coordinate systems used in this work are briefly outlined. To begin, note that with resect to Cartesian ECI osition-velocity coordinates r; _r, the acceleration r of a resident sace object (satellite, debris) can be written in the form r r 3 r a ertr; _r;t (1) In this equation, r jrj, GM where G is the gravitational constant and M is the mass of the Earth, and a ert encasulates all erturbing accelerations of the sace object other than those due to the two-body oint mass gravitational acceleration. The equinoctial orbital elements [10] a; h; k; ; q; define a system of curvilinear coordinates in osition-velocity sace. Physical and geometric interretations of these coordinates as well as the transformation from equinoctial elements to ECI are rovided, for examle, in Montenbruck and Gill [11] and Vallado [12]. Models for the erturbing acceleration a ert are also develoed in these references. The authors roose the definition of alternate equinoctial orbital elements n; h; k; ; q;, in which the mean motion n =a 3 is used as the first coordinate in lace of the semimajor axis a. The reresentation of the dynamical model (1) in coordinate systems other than ECI is straightforward to obtain using the chain rule. Indeed, if u ur; _r denotes a coordinate transformation from ECI osition-velocity coordinates r; _r to coordinates u 2 R 6 (e.g., equinoctial or alternate equinoctial), then Eq. (1) is transformed to _u a ertr; _r;t (2) where r (3) r If u is either equinoctial or alternate equinoctial orbital elements, then Eq. (3) simlifies to _u Keler 0; 0; 0; 0; 0;n T (4) where n is the mean motion. Notice that if a ert 0, then Eq. (2) is a linear system of differential equations in the alternate equinoctial elements but not in the equinoctial elements. In the latter coordinate system, the solution of Eq. (4), though trivial, deends nonlinearly on the semimajor axis since t 0 =a 3 0t t 0. Therefore, a PDF that is initially Gaussian in alternate equinoctial elements will

3 HORWOOD, ARAGON, AND POORE 1841 Fig. 1 The redictor-corrector ste for the recursive Bayesian state estimator. remain identically Gaussian when roagated under unerturbed Kelerian dynamics. This same conclusion does not hold in the (traditional) equinoctial orbital elements. Overview of Nonlinear Filtering In this section, the general Bayesian framework for nonlinear filtering is reviewed and the filter rediction ste is subsequently secialized to general mixture robability density functions (PDFs). Further secializations to Gaussian mixtures are relegated to the next section. The roblem of nonlinear filtering requires the definition of dynamical and measurement models. It is assumed that the dynamic state xt 2R n at time t evolves according to the continuous-time stochastic model x 0 tfxt;tgxt;twt (5) where f: R n R! R n, G: R n R! R nm, and wt is an m-dimensional Gaussian white noise rocess with covariance matrix Qt. In articular in Eq. (5), the function f encodes the deterministic force comonents of the dynamics (e.g., gravity, drag, etc.) while the rocess noise term models the stochastic acceleration. In many tracking alications, it is often convenient to work with a discretetime formulation of the dynamical model which assumes the form x k1 f k x k G k x k w k (6) where x k xt k, f k : R n! R n, G k : R n! R nm, and fw k g is an m-dimensional zero-mean Gaussian white noise sequence with covariance matrix Q k. In sace surveillance, the rocess noise term is often very small and discarded. In such situations, the function f k is just the solution flow corresonding to the continuous model (5) with wt0. A sequence of measurements k fz 1 ;...; z k g is related to the corresonding kinematic states x k via measurement functions h k : R n! R according to the discrete-time measurement model z k h k x k v k (7) In this equation, fv k g is a -dimensional zero-mean Gaussian white noise sequence with covariance matrix R k. More general filter models can be formulated from measurement models with non- Gaussian or correlated (e.g., colored) noise terms [13] and sensor biases [14]. In the Bayesian aroach to dynamic state estimation, one constructs the osterior PDF of the state based on information of a rior state and received measurements. Encasulating all available statistical information, the osterior PDF x k j k may be regarded as the comlete solution to the estimation roblem and various otimal state estimates can be comuted from it. The rediction and correction filter stes are summarized in the flowchart of Fig. 1 (with additional details rovided, for examle, in Ristic et al. [1]). Analytical solutions to the filter rediction and correction stes in Fig. 1 are generally intractable and are only known in a few restrictive cases. In ractice, models are nonlinear and states can be non- Gaussian; one must be content with an aroximate or subotimal algorithm for the Bayesian state estimator. While the EKF and UKF are used extensively in air and missile tracking, they only reresent state uncertainties by a covariance matrix and this may not be adequate in the sace surveillance environment. Because of the need to roagate uncertainties over extended time intervals in the absence of measurement udates, higher-order cumulants (e.g., skewness, excess kurtosis) can become nonnegligible and must be accounted for to achieve uncertainty consistency. Secialization to Mixture Densities Given a set of PDFs 1 x;...; N x, called the mixture comonents, and weights w 1 ;...;w N such that w 0, for 1;...;N, and P N w 1, the sum x XN w x (8) is called a mixture density. In ractice, the mixture comonents are members of a arametric family of distributions (such as Gaussians) each allowed to have different arameter values. To derive the filter rediction ste for a mixture density, suose the rior density x k1 j k1 at time t k1 is a mixture of the form x k1 j k1 XN w x k1 j k1 (9) Alying the Chaman Kolmogorov equation (i.e., the rediction ste equation in Fig. 1) indeendently on each mixture comonent yields x k j k1 x k jx k1 x k1 j k1 dx k1 (10) The redicted mixture PDF is found to be x k j k1 x k jx k1 x k1 j k1 dx k1 XN x k jx k1 XN w x k1 j k1 dx k1 w x k j k1 (11) as follows from Eqs. (9) and (10). The above analysis shows that the mixture weights remain invariant under uncertainty roagation by a simle consequence of linearity. However, there is no universal agreement on how the mixture weights should be udated (if at all) following a rediction. For the Gaussian sum roagation algorithm develoed in the next section, the rior Gaussian comonents x k1 j k1 are reresented in equinoctial orbital elements and seeded in such a way that the UKF rediction ste accurately aroximates the solution of Eq. (10). As shown in the simulations section, this methodology rovides excellent uncertainty consistency without the added The new work resented in this manuscrit deals rimarily with the rediction ste of the Gaussian sum filter. Details on the correction ste can be found, for examle, in [2,5]. An analogous conclusion on the invariance of the mixture weights alies if assuming continuous-time dynamics and time evolution governed by the Fokker Planck Kolmogorov equation.

4 1842 HORWOOD, ARAGON, AND POORE exense of udating the weights. Notwithstanding these comments, it is of interest to study if there is any added benefit by alying a weight adatation rocedure to the roosed Gaussian sum filter. Details of the secific weight udate scheme evaluated in this aer (which is taken from Terejanu et al. [6]) are summarized in the aendix and the results of its alication are resented in the simulations section. Gaussian Sum Proagation In the revious section, it was shown how the Bayesian nonlinear filter secializes to general mixture densities. In this section, the methodology is further secialized to Gaussian mixtures (Gaussian sums) adated to the two-body roblem. It is convenient to define the function N : R n R n R nn! R given by 1 N x; y; P ex 1 det2p 2 x yt P 1 x y (12) Note that if the second and third arguments of N are fixed and the latter is symmetric and ositive definite, then N defines a multivariate Gaussian PDF with resect to its first argument. A Gaussian sum is a mixture density of the form (8) given by x XN w N x; ; P where and P, 1;...;N, are the resective means and covariances of the comonent Gaussians. Kalman-based filters such as the EKF or UKF are the workhorse of a GSF since they act in arallel on each comonent Gaussian during the rediction and correction filtering stes. For otimal efficiency and numerical stability, the use of the discrete-time square-root version of the UKF [15] is roosed that does not directly form any covariance matrix and instead uses the (lower-triangular) Cholesky factor (i.e., square root) of the covariance. Higher-order versions of the UKF (also called Gauss Hermite filters) that make use of efficient Gauss Hermite quadrature rules due to Genz and Keister [16] are also roosed in [2]. Further, in this aer it is assumed that the initial rior density is Gaussian. This rior density is usually obtained through a batch least-squares (differential correction) rocedure which roduces a state estimate (mean) and a Gaussian covariance from a sequence of sensor measurement data. For low- Earth-orbit (LEO) radar data (e.g., a 2 min track generated from radar measurements every ten seconds), it is observed that the initial PDF is well aroximated by a single Gaussian in equinoctial orbital elements [17] or ECI coordinates [18]. The Gaussian assumtion on the rior can be relaxed; a method for determining a Gaussian sum reresentation of the rior density from radar measurements is develoed in Horwood and Poore [2]. For the remainder of this section, a new algorithm is roosed that efficiently refines a single Gaussian in equinoctial orbital elements into a Gaussian sum. The comonent means, covariances, and weights of the mixture are chosen such that each Gaussian comonent remains Gaussian (u to a rescribed accuracy) when roagated by the (square-root version of the) UKF under the nonlinear dynamics. The refinement methodology is illustrated in Fig. 2. Under a nonlinear transformation, a Gaussian (reresented by the thick black ellise) need not be maed to a Gaussian (e.g., the nonlinear transformation Fig. 2 Deiction of a single Gaussian and its Gaussian sum aroximation undergoing a nonlinear transformation. level surfaces of the transformed distribution could look crescentshaed). However, in a sufficiently small neighborhood, any (smooth) nonlinear ma will be aroximately linear. Consequently, Gaussians with smaller covariances (reresented by the colored ellitic disks) remain more Gaussian than those with larger covariances under the nonlinear maing. Therefore, a Gaussian sum refined by aroximating each constituent Gaussian by a finer Gaussian sum will exhibit better behavior through nonlinear transformations. The roagation of uncertainty under the unerturbed two-body roblem [i.e., Kelerian dynamics with a ert 0 in Eq. (1)] rovides the motivation for the new Gaussian sum refinement method. The time evolution of the equinoctial orbital elements a; h; k; ; q; under the assumed (unerturbed) dynamics is ata 0 ; hth 0 ; ktk 0 ; t 0 qtq 0 ; t 0 n 0 t t 0 (13) where n 0 =a 3 0 is the mean motion at the initial eoch. Suose an initial Gaussian at time t 0 has mean 0 and covariance P 0. The first key observation is that at any future time t>t 0, the covariance between any two of the first five elements remains constant; that is P ij tp ij 0 for all t>t 0 and i, j 2f1;...; 5g. To leading order, the other comonents of the roagated covariance matrix are n 0 P i6 tp i6 0 3 P i1 0 t; i 2f1;...; 5g 2 a 0 P 66 tp n 0 3 P a t n 0 2P 11 t a 0 where t t t 0. Note that the covariance between the mean longitude (the sixth equinoctial element) and the other five elements grows linearly in time while the variance in the mean longitude grows quadratically. Therefore, the variance in the mean longitude P 66 t is the dominant comonent in the roagated covariance for large t. Any dearture from Gaussianity would be manifested most strongly in this coordinate. With the inclusion of higher-order erturbative forces in the dynamics, a similar conclusion is conjectured. The key idea of the new refinement method is to mitigate the growth of the dominant covariance comonent P 66 t through efficient refinement of the initial Gaussian density. The roosed refinement criterion is now described. For t sufficiently large P 66 tp n 2 0 a 2 0 P 11 0 t2 (14) In Eq. (14), the only comonent of the initial covariance which drives the growth of P 66 t is P 11 0, the initial variance in the first equinoctial element, namely the semimajor axis. In other words, it is the uncertainty along the radial (semimajor axis) direction which causes the uncertainty along the in-track (mean longitude) direction to grow. Therefore, it is roosed to refine the initial Gaussian at time t 0 only along the semimajor axis a coordinate. This roosed refinement scheme also mitigates the growth of the cross-terms P i6 t. To mitigate the growth of the uncertainty along the dominant intrack (mean longitude) direction, the initial Gaussian N x; 0 ; P 0 is refined along only the first dimension (semimajor axis) such that each mixture comonent has a variance in a of ~P 11 0 <P By Eq. (14), the variance in the mean longitude of each comonent at time t, ~P 66 t,is aroximately ~P 66 tp ~n 2 0 ~P 11 4 ~a 2 0 t 2 (15) 0

5 HORWOOD, ARAGON, AND POORE 1843 The new variance ~P 11 0 is chosen such that ~P 66 t cp 66 0 (16) for some constant c>1. From Eq. (15), it follows that ~P c 1 ~a2 0 9 ~n 2 P66 0 (17) 0t2 The arameter c in the criterion Eq. (16) is a tuning arameter of the filter that controls the growth of the comonent covariances and ultimately the accuracy of the underlying Gaussian sum. For examle, if c 2, then Eq. (16) in conjunction with Eq. (17) says that the comonent Gaussians are chosen so that the variance in the mean longitude never exceeds twice the original variance in over the roagation time t. It is noted from Eq. (17) that for a fixed choice of c, the required variance in the semimajor axis of each comonent is roortional to the inverse square of the roagation time; longer roagations therefore require finer Gaussian sums. What remains to be shown in this section is how to refine a single Gaussian into a Gaussian sum where each comonent has a rescribed mean and covariance. Precisely how this is done is a delicate matter. The theory of Gaussian sums is based on the L 1 norm [4]; however, any otimization criterion based on this norm would likely be comutationally imractical due to the need to solve a nonsmooth nonlinear otimization roblem. A subotimal solution based on moment matching is roosed in [2], but the method tends to lace too many of the comonent Gaussians far out on the tails, thereby sacrificing resolution around the center. In what follows, an alternative algorithm for Gaussian sum refinement is roosed by working in L 2 because an analytic exression for the L 2 norm of a Gaussian sum is well known. In what follows, it is first shown how to refine the one-dimensional unit Gaussian N x;0; 1. The refinement method is then extended to the multivariate case with an arbitrary covariance. Although L 2 is the assumed norm, it is shown that the new algorithm exhibits nice convergence roerties not only in L 2 but also in L 1 and L 1. The general refinement method is summarized as Algorithm 1 at the end of this section. Refinement of the One-Dimensional Unit Gaussian A Gaussian sum aroximation of the unit one-dimensional Gaussian N x;0; 1 is now derived. A key feature of the ensuing algorithm is that it can be alied offline and subsequently used online to refine any univariate or multivariate Gaussian with an arbitrary covariance. The roblem has an obvious trivial solution which erfectly aroximates the target, namely the Gaussian sum with a single comonent. However, as motivated earlier, the idea is to construct a Gaussian sum aroximation whose comonent standard deviations are smaller than some fixed value <1 so that the Gaussian sum more accurately reresents the true distribution under a nonlinear transformation. This requirement leads to a constrained nonlinear otimization roblem. That said, it is assumed that the target Gaussian sum has the form GS x XN w N x; ; 2 for some redetermined length N>1. Nowdefine the objective function E 1 2 kn x;0; 1 GSxk N x;0; 1 L 2 2 XN 1 w N x; ; 2 2 dx (18) The weights w, means, and standard deviations of each comonent are determined from the solution of the following L 2 otimization roblem: Minimize w1 ;...;w E 1 ;...; N 1 ;...; N Subject to XN 1 2 N w 1; w 0; 1;...;N <1; 1;...;N (19) Noting the identity N x; 1 ; P 1 N x; 2 ; P 2 dx N 1 2 ; 0; P 1 P 2 it follows that Eq. (18) reduces to where (20) E 1 2 wt Mw w T n 1 4 (21) w w ; n N ;0; 2 1 M N ;0; 2 2 the subscrits above denote vector and matrix comonents. The minimization of Eq. (21) over the individual weights, means, and standard deviations subject to the constraints in Eq. (19) is a difficult nonlinear otimization roblem. (It is not a quadratic rogramming roblem since M deends on the unknown means and standard deviations.) Further insight into this general roblem is rovided at the end of this subsection. In what follows, a subotimal yet comutationally tractable algorithm is roosed which aroximates the solution of Eq. (19). This rocedure is summarized in Algorithm 1 at the end of the section. Secifically, each of the comonent Gaussians is constrained to have a common standard deviation <1 and the means are fixed according to m 1 (22) for 1;...;N, where m>0and N d12m=e. Note that the means are evenly distributed with the left-most and right-most comonents located at x m. To ensure adequate accuracy around the tails, it is suggested to set m 4 if 1, otherwise m 6. With 2 these additional constraints, the otimization roblem (19) reduces to a quadratic rogramming roblem which is straightforward to solve using elementary methods. Figure 3 shows lots of the L 1, L 2, and L 1 error between the unit one-dimensional Gaussian and its Gaussian sum aroximation comuted using the subotimal aroach described above. U to around N 100, the three L errors all decrease, but they lateau to L Error Number of Gaussians Fig. 3 Plots of the L error between the unit one-dimensional Gaussian and its Gaussian sum aroximation obtained using Algorithm 1. L 1 L 2 L

6 1844 HORWOOD, ARAGON, AND POORE around 10 8 for N 100. This lateau is believed not to arise because of numerical stability issues [the matrix M in Eq. (21) has a condition number on the order of 10 5 ] but simly because the standard deviations and means are fixed via Eq. (22) and the minimization is erformed only over the unknown weights; this need not be otimal. Notwithstanding these comments, the simulation studies considered in this aer use this refinement scheme with N as high as 1000 and no adverse effects have been observed. If one insists on comuting a Gaussian sum aroximation of N x;0; 1 with an L error much less than 10 8, one could relace the comonent mean locations (22) by m for 1;...;N, where N d1 4m=e. If, for examle, one takes 1 5 and m 6, then N 121 and the L1, L 2, and L 1 errors are 1: , 7: , and 8: , resectively. If 1 10 and m 8, then N 321 and the corresonding L errors diminish to 4: , 2: , and 4: In both examles, the corresonding condition number of the matrix M in Eq. (21) is on the order of ; thus quadrule recision or software floating-oint arithmetic is required. To conclude this subsection, some reliminary solutions of the full otimization roblem (19) are resented. It is observed that the numerical conditioning of this roblem worsens as the target length N of the Gaussian sum increases thereby rendering double recision floating-oint arithmetic largely inadequate. To facilitate an accurate solution, the MAPLE Otimization ackage has been used which takes advantage of built-in library routines rovided by the Numerical Algorithms Grou with the ability to call them within an arbitrary-recision software floating-oint environment. Figure 4 shows various scatter lots of the comonent weights and means comuted using two different methods. The first method uses the subotimal aroach which fixes the means according to Eq. (22) and subsequently otimizes over the weights only. The second method solves the full otimization roblem (19). Each sublot assumes a different target length N and standard deviation. One interesting observation is that the otimal comonent standard deviations lie on the constraint boundary; i.e., for all. Most imortantly, by solving the full otimization roblem, the locations of the means are no longer uniformly distributed and the resulting L 2 error is smaller than that obtained by solving the subotimal roblem. In articular, to get the L 2 error down to 10 7 requires about N 28 Gaussians using the subotimal method but only about N 9 Gaussians when solving the full otimization roblem. Thus, with the otimal solution of Eq. (19), the error in the Gaussian sum aroximation can be substantially reduced or, for the same error, the number of Gaussians needed to achieve a rescribed accuracy can be reduced. It must be emhasized that the full roblem, although very exensive to solve, can nevertheless be done offline. One can generate a library of Gaussian sum aroximations to N x;0; 1 for various values of N and. Such comutations are in rogress. Future work will also attemt to solve the analog of Eq. (19) in L 1, the norm of which the theory of Gaussian sums is based [4]. Refinement of a Multivariate Gaussian Using the scheme of the revious subsection, assume a Gaussian sum aroximation of the unit one-dimensional Gaussian of the form N x;0; 1 XN w N x; ; 2 (23) where 2 < 1. By linearity, Eq. (23) induces a Gaussian sum aroximation of the Gaussian with mean and variance 2 according to N x; ; 2 XN w N x; ; 2 2 (24) 10 0 N = 7, σ = 4/5 Subotimal Otimal 10 0 N = 11, σ = 2/3 Subotimal Otimal Weight Weight 10 2 L 2 error (subotimal): L 2 error (otimal): L 2 error (subotimal): L 2 error (otimal): x x 10 0 N = 17, σ = 1/2 Subotimal Otimal 10 0 N = 37, σ = 1/3 Subotimal Otimal Weight 10 4 L 2 error (subotimal): L 2 error (otimal): Weight L 2 error (subotimal): L 2 error (otimal): x x Fig. 4 Scatter lots of the comonent weights and means comuted using the subotimal algorithm (i.e., minimizing only over the weights) and by solving the full (otimal) otimization roblem (i.e., minimizing over the weights, means, and standard deviations).

7 HORWOOD, ARAGON, AND POORE 1845 To extend Eq. (24) to the multivariate Gaussian N x; ; Q, assume without loss of generality that refinement is done along the first coordinate x 1 of x (this is the semimajor axis if x denotes an equinoctial orbital element state). To roceed, consider a Gaussian sum aroximation of the quotient N x;0; 1 XN N x;0; 1 N x; w ; 2 XN ~w N x;0; 1 N x; ~ ; ~ 2 (25) where 2 < 1 and the weights w and means are the solution of Eq. (19), as detailed in the revious subsection. It follows that 2 2 ~w w 1 2 ex 21 2 ~ 1 2 ; ~ (26) U to a multilicative constant, the quotient (25) defines a Gaussian sum aroximation of the uniform distribution on the interval (m, m). Therefore, by the linear transformation x! 1 x, the quotient Q 11 N x 1 ; 1 ;Q 11 XN ^w N x 1 ; 1 ;Q 11 N x 1 ; ^ ; ^ 2 (27) defines (u to a multilicative constant) a Gaussian sum aroximation of the uniform distribution on the interval ( 1 m Q 11, 1 m Q 11 ), where ^w ~w ; ^ 1 Q 11 ~ ; ^ 2 ~ 2 Q 11 Thus, to obtain a Gaussian sum aroximation of N x; ; Q where refinement is done along the first coordinate, the Gaussian PDF N x; ; Q is multilied by the Gaussian sum (27) aroximating the uniform PDF. This leads to where N x; ; Q XN XN ^w N x; ; QN x 1 ; ^ ; ^ 2 w N x; ; Q (28) where n = 1 3, t is an uer bound for the total roagation time, and c>1is a growth control arameter as exlained following Eq. (17). 2) Comute the number of terms N in the Gaussian sum according to N d12m=e, where m 4 if 1 or m 6 if 2 <1. 2 3) For 1;...;N, comute m. 4) Comute the matrix M and vector n with comonents M N ;0; 2 2 ; n N ;0; 2 1 5) Minimize the quadratic function P 1 2 wt Mw w T n over w subject to the constraints N w 1 and w 0, for 1;...;N. 6) For 1;...;N, comute ~w, ~, and ~ 2 from Eq. (26) using the means from ste 3 and the weights w from Ste 5. 7) For 1;...;N, comute ^ 1 Q 1 ~ and set ^w ~w and ^ 2 ~ 2 Q 11. 8) Using the data from the revious ste, comute Q,, and w from Eq. (29). 9) For 1;...;N, comute the renormalized weight w! w = P N w. Metric for Uncertainty Consistency The metric derived in this section is motivated from the rediction error term arising in the correction ste of the Bayesian state estimator. This term also aears in the likelihood ratios for scoring the association of one reort (e.g., track, measurement) to another [19]. Its accurate evaluation is thus critical in the roblem of data/ track association (correlation). It will be shown and later demonstrated that the the rediction error rovides an offline metric to validate the accuracy of the filter rediction ste and assess uncertainty consistency when truth is not available. Referring to Fig. 1, for measurement-to-track association, the rediction error is recisely the denominator aearing in the correction (fusion) ste: PE z k j k1 z k jx k x k j k1 dx k (30) For track-to-track association, the redicted (rior) PDF x k j k1 is udated with a new track x new k (which is a random variable) to form the corrected (fused, osterior) PDF x k jx new k ; k1 using Bayes rule: Q ^ 2 e 1 e T 1 Q1 1 ; Q ^ 2 ^ e 1 Q 1 w ^w N ^ e T 1 ;0; ^2 e T 1 Qe 1 (29) x k jx new k ; k1 xnew k jx k ; k1 x k j k1 x new k j k1 (31) and e 1 ; 0;...; 0 T 2 R 6. The final ste of the calculation is the renormalization of the weights w in Eq. (28) via w! w = P N w. The refinement algorithm is summarized below. Algorithm 1. Refinement of a Gaussian into a Gaussian Sum Given a Gaussian N x; ; Q, where x denotes an equinoctial orbital element state, a Gaussian sum aroximation N x; ; Q XN w N x; ; Q is comuted as follows. 1) Select the refinement arameter 20; 1 either manually or according to 2 c 1 3nt 1 s Q 66 Q 11 Algorithm 1 is also alicable if x is an alternate equinoctial orbital element state. In such a case, the equation in Ste 1 becomes c 1 Q 66 =Q 11 =t. Note that Eq. (31) is recisely the correction ste in Fig. 1 with the identifications z k $ x new k and k $x new k ; k1. Therefore, the corresonding rediction error is the denominator in Eq. (31): PE x new k j k1 x new k jx k ; k1 x k j k1 dx k (32) If x k and x new k are indeendent, then Eq. (32) can be written in the form PE new xj k1 rior xj k1 dx (33) where new j k1 and rior j k1 are the PDFs (each conditioned on the rior reorts k1 ) of the new track and rior (redicted) tracks, resectively. If, in addition, x k and x new k are Gaussian with resective means 1 and 2 and resective covariances P 1 and P 2, then Eq. (33) reduces to Alternatively, stes 3 5 of Algorithm 1 can be substituted with the solution of the full otimization roblem (19).

8 1846 HORWOOD, ARAGON, AND POORE ln PE T P 1 P ln det2p 2 1 P 2 (34) The first term in Eq. (34) involves the familiar Mahalanobis distance. However, for non-gaussian uncertainties which commonly arise in sace surveillance tracking due to the need to roagate over long time sans, higher fidelity filters and reresentations of the uncertainty must be used to account for ossible higher-order statistics (cumulants). In other words, the rediction error is filter deendent and must be evaluated accordingly. The association roblem rovides a natural alication of the rediction error PE in Eq. (33) as a metric for uncertainty consistency. Referring to Fig. 5, suose there are two satellite states at times t 1 and t 2 (with t 1 t 2 without loss of generality) reresented by the PDFs 1 x;t 1 and 2 x;t 2, resectively, and one wishes to assess whether the two objects associate. The likelihood score for comuting the robability of association deends on the rediction error (33), which for this scenario is given by PE t 1 x;t 2 x;tdx (35) It is also convenient to the define the cost metric from Eq. (35) according to ctln PEt (36) Evaluation of the integral Eq. (35) requires reresentations of both PDFs 1 and 2 at some common time t. Thus, one could either 1) roagate 1 to time t 2 and evaluate PEt 2, or 2) roagate 1 from time t 1 to some intermediate time t, then back roagate 2 to t, and finally evaluate PEt. In fact, the value of the rediction error is indeendent of the choice of t under the assumtion that the dynamics are governed by a conservative deterministic model and the PDFs are roagated consistently. This result is formally stated and roved below. Proosition 1. Let 1 x;t and 2 x;t denote the PDFs of two indeendent states at time t. Suose the state x is governed by the conservative dynamical model x 0 tfxt;t; r T x f 0 and the time evolution of each PDF satisfies the noiseless Fokker Planck rt x f (37) where r x is the gradient with resect to x viewed as a column oerator. Then, the rediction error (35) is time-indeendent. Proof:If 1 and 2 both satisfy Eq. (37), then the same can be said for their roduct. Indeed, by the @ rx T 2 f 2 rx T 1 f rx T 1 2 f Therefore d dt PE dt dx rx T 1 2 f dx 0 as follows from the Gauss divergence theorem from elementary vector calculus. The metric (35) rovides a necessary condition for uncertainty consistency. If the evolution of the rediction error is not constant over time, then the roagated uncertainties of the two states are not consistent. The use of this metric for quantifying uncertainty consistency over long time gas is illustrated in the next section. In closing, the roof of roosition 1 can be relicated for other distance metrics such as the Kullback Leibler divergence (KLD), 1 x;t KL t 1 x;tln dx 2 x;t The KLD can thus serve as an alternative metric for uncertainty consistency. While the KLD is used in sensor resource management as a measure of mutual information gain [20], it is not what is commonly used in tracking for scoring likelihoods of association. A rigorous derivation of the likelihood ratio used to score the association of a track or measurement to an orbit is rovided, for examle, in Poore [19]; it is recisely the rediction error z k j k1 or Eq. (32) which aears in the ensuing exression and not the KLD. Simulation Studies In this section, results demonstrating the roosed GSF are resented. For a low-earth-orbit (LEO) scenario, it is first shown how the standard UKF does not necessarily maintain uncertainty consistency during long-term roagation and how higher fidelity GSFs can imrove this situation. Next, the imact of correct uncertainty management on the roblems of association and anomaly detection is illustrated. Finally, a weight adatation scheme for uncertainty roagation (full details of which are exlained in the aendix) is alied to the GSF and is shown not to rovide any additional accuracy of the forecast PDF. The simulation scenario considers a canonical test roblem involving two closely saced objects (CSOs) in LEO whose initial uncertainties are Gaussian with resect to equinoctial orbital elements. The initial states of the objects at time t 0 are: a km; h 1 k 1 1 q Object 1 a km; h 2 k 2 2 q Object 2 2 (x, t 2 ) 2 (x, t 2 ) 2 (x, t * ) 1 (x, t 2 ) 1 (x, t * ) 1 (x, t 1 ) Fig. 5 The evaluation of the rediction error (35) used to comute the robability of association between two states at times t 1 and t 2 (left) requires reresentations of their PDFs at a common time t [e.g., t t 2 (middle) or t 1 < t < t 2 (right)]. The rediction error is indeendent of the choice of t.

9 HORWOOD, ARAGON, AND POORE UKF (GH order 3) GHF order 5 GHF order 7.5 UKF (N = 1) GSF (N = 10) GSF (N = 38) GSF (N = 112) GSF (N = 347) Time (orbital eriods) Fig. 6.5 Time (orbital eriods) Evolution of the cost metric (36) comuted using the UKF, various GHFs, and various GSFs. The initial covariances of the objects are P 1 P 2 AA T, where A diag 20 km; 10 3 ; 10 3 ; 10 3 ; ; From Eq. (35), it follows that the rediction error between the two objects at t 0 is PE 3: with corresonding cost ln PE 33: (38) Fixing a articular filter, the initial Gaussians are roagated for a total time of h (which is about 20 comlete orbital eriods) and the rediction error (35) and the cost (36) are evaluated at intermediate times. By roosition 1, the cost must retain the constant value given by Eq. (38) to assert that the state uncertainties are roagated consistently (i.e., a constant cost is a necessary condition for uncertainty consistency). Thus, any dearture from a constant cost signals a degradation in the comuted uncertainty. The left half of Fig. 6 lots the evolution of the cost metric using the Gauss Hermite filters (GHFs) of order three, five, and seven (the former is the standard UKF). Indeed, there is negligible difference in the comuted costs. Therefore, the standard (third-order) UKF achieves covariance consistency. It must be emhasized that the GHFs comute the mean and covariance (of a ossibly non-gaussian PDF) directly from their resective definitions using Gauss Hermite quadrature. By increasing the order the quadrature method (i.e., increasing the number of sigma oints), the accuracy of the comuted mean and covariance does not change. However, covariance consistency is only a rerequisite to the more general uncertainty consistency. Although the UKF correctly resolves the mean and covariance (first two cumulants) of the true PDF, it fails to cature information about the cumulants of third-order and higher. Because the cost diverges from the initial value, the true state uncertainties of the CSOs are evidently non-gaussian and contain nonnegligible higher-order cumulants which imact the robability of association (POA). After 20 orbital eriods, this divergence is nearly four units of cost or about a factor of 50 in the rediction error. Consequently, the resulting POA could have an error uwards of a full order of magnitude. A value of 20 km for the uncertainty in the semimajor axis is reresentative of real data the authors have rocessed describing a breaku scenario in LEO. Later in this section, the rate of uncertainty degradation on the initial uncertainty in the semimajor axis is studied (see Fig. 7). The rediction error (or cost) deends on units. In this aer, the assumed length unit is Earth radii (R ); 1R 6378:137 km. Unless stated otherwise, the discrete-time square-root version of the UKF [15] is used to roagate Gaussian PDFs and comonent Gaussians within a Gaussian mixture. The UKF sigma oints are roagated using the Astrodynamic Standards Secial Perturbations (SP) rogram (Version 7) using a degree and order 70 gravity model. The right half of Fig. 6 shows the imrovement in uncertainty consistency when using the GSF with various numbers of mixture comonents. In generating these results, the initial Gaussian uncertainties of the CSOs were refined into Gaussian sums using Algorithm 1 and the comonent Gaussians roagated using the (third-order) UKF. For comarison uroses, the to curve is once again the cost obtained when the initial Gaussian uncertainties are roagated using the UKF. The remaining curves show the cost obtained using GSFs with 10, 38, 112, and 347 Gaussian comonents. By accounting for higher-order cumulants in the uncertainties, the GSF rovides imroved uncertainty consistency as seen by the mitigation of the divergence in the time evolution of the cost function. As the number of Gaussians increases, the uncertainty consistency imroves. The highest fidelity filter (N 347) shows no noticeable uncertainty degradation over the entire 20 orbital eriod roagation. While the GSF can give high-accuracy results, it can be exensive as a otentially large number of Gaussians must be roagated. However, in some situations (e.g., short roagation times or small initial uncertainties in the semimajor axis a), such a fine level of accuracy may not be required. For examle, Fig. 7 shows how the rate of uncertainty degradation for the UKF deends on the magnitude of the initial uncertainty (standard deviation) in the semimajor axis of the two CSOs. The dark blue (to) curve corresonds to an initial standard deviation in the semimajor axis of 20 km, which is the same value used to generate the lots in Fig. 6. The remaining curves are generated using a semimajor axis standard deviation of A 11 a, where a varies as shown in the figure. The initial means of the semimajor axes of the two CSOs are set to a km a and a km a. Because the initial states and uncertainties differ, the rediction error or cost at time zero, as determined from Eq. (35), also differs for each a case. Of greater interest is how these costs evolve in time. While the uncertainty consistency raidly degrades for larger a, with a small uncertainty of a 1km, uncertainty consistency is maintained for aroximately 20 orbits. This result is consistent with Sabol et al. [17]. However, even for this case of small initial uncertainty in the semimajor axis, uncertainty consistency cannot be maintained indefinitely; within 50 orbital eriods the degradation is significant. This indicates that for some situations (e.g., small a, short roagation times) comutationally chea low-fidelity methods can be used, while more accurate methods like the Gaussian sum filter will be needed in other cases. Figure 8 rovides insight into the source of the eriodic oscillations in the rediction error (cost). Here evolution of the cost metric is lotted using the UKF and various GSFs assuming roagation under either a full (SP) gravity model (thin curves) or unerturbed Kelerian dynamics (thick curves). The former set of curves are those from Fig. 6. For the Kelerian case, uncertainty This choice of initial conditions ensures that the objects always start with a searation of 2 a, though qualitatively the results are not deendent on this sacing.

10 1848 HORWOOD, ARAGON, AND POORE UKF (σ a = 20km) UKF (σ = 15km) a UKF (σ a = 10km) UKF (σ = 5km) a UKF (σ = 1km) a Time (orbital eriods) Fig. 7 Evolution of the cost metric (35) comuted using the UKF comaring the effect of the size of the initial standard deviation a in the semimajor axis of the two closely saced objects UKF (full) GSF N=10 (full) GSF N=38 (full) GSF N=112 (full) UKF (Keler) GSF N=10 (Keler) GSF N=38 (Keler) GSF N=112 (Keler).5 Time (orbital eriods) Fig. 8 Evolution of the cost metric (35) comuted using the UKF and various GSFs where roagation is done assuming either a full gravity model (thin curves) or unerturbed Kelerian dynamics (thick curves). consistency still degrades over time, but the degradation is essentially monotonic with a smoothly varying rate. The oscillations that aear when using the full gravity model are not resent and it is conjectured that these eriodic oscillations are due to the effects of the erturbations (and not a numerical stability issue). Thus, even under Kelerian dynamics, the initial Gaussian uncertainties evolve to something non-gaussian (and are better described by considering higher-order reresentations of the uncertainty such as Gaussian sums). Assuming use of the full gravity model for the remainder of this section, Fig. 9 illustrates the imact of uncertainty consistency when state uncertainties are reresented in Cartesian ECI coordinates versus equinoctial orbital elements and alternate equinoctial orbital elements. Here we can see that the cost metric degrades much more raidly in Cartesian sace; while there is an error of about 5.5 units of cost over 50 orbital eriods in equinoctial orbital elements, this same error is surassed in less than a single orbital eriod in ECI. It is also imortant to note that both the ECI and orbital element imlementations of the UKF used here assume the square-root version of the UKF [15]. This greatly imroves numerical conditioning and stability, allowing a Gaussian to be roagated over a long time eriod in ECI (and orbital elements) without suffering from covariance collase. Thus, in defense of ECI, the standard UKF For this comarison, the initial uncertainty is assumed to be Gaussian in all coordinate systems. For the ECI and alternate equinoctial cases, the initial uncertainty was comuted by erforming an unscented transform on the initial Gaussian (in equinoctial elements) defined at the beginning of this section. still maintains covariance consistency (since there is little change when using higher-order Gauss Hermite filters). Figures 9 and 10 also show the additional benefit of reresenting and roagating uncertainties in the alternate set of equinoctial orbital elements over the traditional set. For both the traditional and alternate element sets, the use of Gaussian sums mitigates the degradation in the evolved rediction error (cost) and hence imroves the consistency (accuracy) of the reresented state uncertainty. For long roagation times, the degradation in cost is slightly less when using the alternate set over the traditional set. However, around t 0, the cost curves in the alternate set are significantly flatter than those in the traditional set. Thus, for short roagation times, the use of alternate equinoctial orbital elements rovides a more accurate reresentation of the state PDF comared with reresentations in traditional equinoctial orbital elements. Imact on Association and Anomaly Detection The examle in Fig. 11 demonstrates the imact of correct uncertainty management on the roblems of association (correlation) and anomaly (maneuver, change) detection. The left half of the figure deicts the non-gaussianity of the uncertainty in Object 1 after only four orbital eriods (about 6.5 h). A two-dimensional slice of its PDF along the semimajor axis a and mean longitude coordinates is lotted. The true uncertainty (grayscale colorma) was comuted using a high-fidelity Gaussian sum filter with N 1127 and is contrasted with the Gaussian uncertainty (reddish colorma) obtained from the UKF. Suose one wishes to comute the rediction error (or cost) associated with a new Gaussian track state (blueish colorma) located between the tails of the true distribution. In this articular examle, a cost of 32:675 is obtained from the UKF aroximation versus a cost of 3:5595 from the N 1127 GSF. Therefore, the UKF estimates a smaller cost leading to a higher robability of association and ossibly a misassociation and ossibly a failure to detect an anomaly. Note that even a medium fidelity GSF (e.g., 112 or 347 terms) can estimate the true uncertainty and hence the cost much more accurately than the UKF, and this in turn mitigates misassociations and undetected anomalies. Effect of Weight Udate Scheme Figure 12 shows the evolution of the cost metric (36) using various GSFs and the imact of using a weight adatation method for uncertainty roagation. Unerturbed Kelerian dynamics are assumed in this figure. The solid curves corresond to the GSF of this aer, which does not use any weight udate method by default (these curves are the same as the thick curves in Fig. 8). The dashed curves in Fig. 6 show the evolved cost when using a GSF in conjunction with the weight udate scheme described in the aendix with the udates occurring at a frequency of every one-tenth of an orbital eriod (i.e., 200 times over the length of the

11 HORWOOD, ARAGON, AND POORE ECI: UKF (order 3) ECI: UKF (order 5) EqOE: UKF (order 3) EqOE: UKF (order 5) AEqOE: UKF (order 3) AEqOE: UKF (order 5) Time (orbital eriods) Fig. 9 Evolution of the cost metric (35) comuted using the UKF where the uncertainty is reresented in Cartesian ECI coordinates versus equinoctial orbital elements (EqOE) and alternate equinoctial orbital elements (AEqOE). EqOE: UKF AEqOE: UKF EqOE: GSF (N=10) AEqOE: GSF (N=10) Time (orbital eriods).5 Time (orbital eriods) EqOE: GSF (N=38) AEqOE: GSF (N=38) EqOE: GSF (N=112) AEqOE: GSF (N=112) Time (orbital eriods).5 Time (orbital eriods) Fig. 10 Evolution of the cost metric (35) comuted using the UKF and various GSFs where the uncertainty is reresented in (traditional) EqOE versus AEqOE. Fig. 11 (Left) The PDF of Object 1 in the semimajor axis a and mean longitude coordinates after four orbital eriods comuted using the UKF (reddish colorma) and a GSF with N 1127 (grayscale colorma). The blueish Gaussian reresents a track udate. The rediction error for the udate against various fidelity GSFs are shown in the table (right).