Gaussian Sum Filters for Space Surveillance: Theory and Simulations

Abstract

While standard Kalman-based filters, Gaussian assumptions, and covariance-weighted metrics are very effective in data-rich tracking environments, their use in the data-sparse environment of space surveillance is more limited. To properly characterize non-Gaussian density functions arising in the problem of long-term propagation of state uncertainties, a Gaussian sum filter adapted to the two-body problem in space surveillance is proposed and demonstrated to achieve uncertainty consistency. The proposed filter is made efficient by using only a onedimensional Gaussian sum in equinoctial orbital elements, thereby avoiding the expensive representation of a full six-dimensional mixture and hence the "curse of dimensionality." Additionally, an alternate set of equinoctial elements is proposed and is shown to provide enhanced uncertainty consistently over the traditional element set. Simulation studies illustrate the improvements in the Gaussian sum approach over the traditional unscented Kalman filter and the impact of correct uncertainty representation in the problems of data association (correlation) and anomaly (maneuver) detection. Nomenclature A= lower-triangular Cholesky factor of a covariance matrix a = semimajor axis a pert = perturbing acceleration a; h; k; p; q; ' = equinoctial orbital elements c = parameter controlling the accuracy of the Gaussian sum filter f = system dynamics vector G = process noise shape matrix h = measurement function vector k = (subscript) time index '= mean longitude N = number of mixture components N = Gaussian probability density function n = mean motion n; h; k; p; q; '= alternate equinoctial orbital elements P = covariance of a Gaussian distribution PE = prediction error p = probability density function Qt = process noise covariance matrix R k = measurement noise covariance matrix r = Cartesian Earth-Centered Inertial (ECI) position 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 process w ; w ; . . . = mixture weights x = dynamic state vector Z k fz 1 ; . . . ; z k g = measurement sequence ; ; . . . = (subscripts) indices of mixture components 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 operator with respect to x (column oriented)

A Gaussian sum filter framework for space surveillance

“…[8,11]). With regards to updating the weights w α within the filter, such a scheme is clearly dictated from Bayes' rule following a measurement or track update (fusion).…”

“…Some researchers [11][12][13] have proposed online methods for adapting the weights based on various L 2 optimization criteria, for example, using the FPKE error as feedback. In another paper, 8 it was found that such weight update schemes do not improve uncertainty consistency when applied to the authors' GSF for the specific two-body problem in space surveillance. Indeed, our philosophy is to not update the weights after a prediction step.…”

“…16 The latter has been shown to work well in space surveillance applications involving only short term propagations with many measurement or track updates. 8 For scenarios requiring long term propagations in the absence of updates, higher fidelity nonlinear estimation and filtering algorithms are required which account for higher-order cumulants in the state PDF beyond the standard Gaussian state and covariance. The Gaussian sum filter is one such example.…”

Impact Of Non-Gaussian Error Volumes On Conjunction Assessment Risk Analysis

“…Figure 4 shows the comparison in the nominal orbit plane for both position and velocity respectively. 4 and Horwood et al 10 For Sabol's "catalog-class scenario," converting a covariance from osculating Cartesian into osculating equinoctial restores a distribution that is consistent with a Gaussian distribution even after ten days of propagation. Horwood indicates that accurate techniques such as Gaussian sum filters may not be necessary in some situations (e.g., small initial uncertainty in semi-major axis).…”

“…Gaussian sum filters estimate any arbitrary pdf as a weighted sum of Gaussian components and have been successfully applied to determining satellite error volumes. 10,18,19 An added benefit is that Gaussian sum filters may make use of existing uncertainty mapping techniques, such as those employed by extended Kalman filters or unscented Kalman filters, for propagating the Gaussian error volume components. One can also represent the error volume pdf with other formulations, as demonstrated by Edgeworth filters 6 and polynomial chaos expansions.…”