Joshua T. Horwood

Adaptive Gaussian Sum Filters for Space Surveillance

The representation of the uncertainty of a stochastic state by a Gaussian mixture is well-suited for nonlinear tracking problems in high dimensional data-starved environments such as space surveillance. In this paper, the framework for a Gaussian sum filter is developed emphasizing how the uncertainty can be propagated accurately over extended time periods in the absence of measurement updates. To achieve this objective, a series of metrics constructed from tensors of higher-order cumulants are proposed which assess the consistency of the uncertainty and provide a tool for implementing an adaptive Gaussian sum filter. Emphasis is also placed on the algorithms potential for parallelization which is complemented by the use of higher-order unscented filters based on efficient multidimensional Gauss-Hermite quadrature schemes. The effectiveness of the proposed Gaussian sum filter is illustrated in a case study in space surveillance involving the tracking of an object in a six-dimensional state space.

Nonlinear Uncertainty Propagation in Orbital Elements and Transformation to Cartesian Space Without Loss of Realism

A number of methods for nonlinear uncertainty propagation used for space situational awareness (SSA) exploit orbital-element-based representations of orbital-state uncertainty in order to mitigate the departure from Gaussianity and thereby improve performance. However, some downstream SSA functions require that orbital-state uncertainty be represented in a Cartesian space. This paper reconciles the two practices by describing a way in which uncertainty that has been propagated in orbital elements can be transformed to Cartesian space without loss of realism via Gaussian mixtures. The efficiency of this approach is compared to an alternative approach to uncertainty propagation wherein uncertainty is both represented and propagated in Cartesian space (using Gaussian mixtures). Metrics for assessing the realism of a Gaussian mixture are also presented.

Covariance consistency for track initiation using Gauss-Hermite quadrature

The initiation of a consistent state covariance or uncertainty which accurately reflects the discrepancy from truth is a prerequisite to achieving correct data association in tracking. In this paper, the treatment of non- Gaussian states in the initial orbit determination (IOD) problem for space surveillance or more general track initiation problem is considered and the accurate and consistent computation of such non-Gaussian uncertainties is addressed. The main contribution is a framework for achieving uncertainty (covariance) consistency in the IOD problem based on efficient Gauss-Hermite quadrature methods. The formalism is applicable in general tracking settings, in particular, multisensor data fusion. Additionally, a series of realtime metrics constructed from tensors of higher-order cumulants are developed which provide a tool for assessing uncertainty consistency. The effectiveness of the proposed track initiation method is illustrated through various case studies in space surveillance tracking.

A comparative study of new non-linear uncertainty propagation methods for space surveillance

We propose a unified testing framework for assessing uncertainty realism during non-linear uncertainty propagation under the perturbed two-body problem of celestial mechanics, with an accompanying suite of metrics and benchmark test cases on which to validate different methods. We subsequently apply the testing framework to different combinations of uncertainty propagation techniques and coordinate systems for representing the uncertainty. In particular, we recommend the use of a newly-derived system of orbital element coordinates that mitigate the non-linearities in uncertainty propagation and the recently-developed Gauss von Mises filter which, when used in tandem, provide uncertainty realism over much longer periods of time compared to Gaussian representations of uncertainty in Cartesian spaces, at roughly the same computational cost.

Short arc gating in multiple hypothesis tracking for space surveillance

Multiple hypothesis tracking methods are under development for space surveillance and one challenge is the accurate and timely orbit initiation from sets of uncorrelated optical observations. This paper develops gating methods for correlation of optical observations in space surveillance. A pair gate based on the concept of an admissible region is introduced. By implementing a hierarchy from fast, but coarse, to more expensive, but accurate gates, the number of hypotheses to be considered for initial orbit determination is reduced considerably. Simulation results demonstrate the effectiveness of the gating procedure, address gate parameter determination, and study the accuracy of initial orbits.

A Gaussian sum filter framework for space surveillance

While standard Kalman-based filters, Gaussian assumptions, and covariance-weighted metrics work remarkably well in data-rich tracking environments such as air and ground, their use in the data-sparse environment of space surveillance is more limited. In order to properly characterize non-Gaussian density functions arising in the problem of long term propagation of state uncertainties in the two-body problem, a framework for a Gaussian sum filter is described which achieves uncertainty (covariance) consistency and an accurate approximation to the Fokker-Planck equation up to a prescribed accuracy. The filter is made efficient and practical by (i) using coordinate systems adapted to the physics (i.e., orbital elements), (ii) only requiring a Gaussian sum to be defined along one of the six state space dimensions, and (iii) the ability to initially select the component means, covariances, and weights by way of a lookup table generated by solving an offline nonlinear optimization problem. The efficacy of the Gaussian sum filter and the improvements over the traditional unscented Kalman filter are demonstrated within the problems of data association and maneuver detection.