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Intrinsic Fault Resistance for Nonlinear Filters with State-Dependent Probability of Detection

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Abstract

The probability of detection can be thought of as the probability that a given measurement scan contains a valid, target-oriented return. In most systems, the detection probability is inherently a function of the state, which can make forming closed-form filtering solutions exceedingly difficult. Oftentimes, closed-form filters will either neglect the probability of detection outright, or in some cases may approximate it as state-independent. Both assumptions simplify calculations, yet bar the filter from extracting any state-information from the probability of detection. This work seeks to reevaluate current estimation practices by testing and comparing several probability of detection models of varying fidelity. This is done by proposing a filter update with intrinsic fault resistance capable of processing multiple sensor returns contained within a single measurement scan. Three different methods of detection probability modeling are described, which are subsequently used to form three distinct Gaussian mixture filters. To test the filters, Monte Carlo results are taken from two different simulations: the first a simple falling body tracking scenario, and the second a more complex orbit determination scenario. The results from both simulations indicate that including detection probabilities, even when modeled incorrectly, can increase filter robustness, and will improve estimate uncertainty if modeled as state-dependent.

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Acknowledgements

This research is supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program under Fellowship Number F-8445592924.

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Authors and Affiliations

  1. Applied Physics Laboratory, Johns Hopkins University, 11100 Johns Hopkins Rd, Laurel, MD, 20723, USA

    Gunner S. Fritsch

  2. Aerospace Engineering, Texas A&M, 701 H.R. Bright Building, College Station, TX, 77843, USA

    Kyle J. DeMars

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Correspondence to Gunner S. Fritsch.

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This article belongs to the Topical Collection: Advanced Maui Optical and Space Surveillance Technologies (AMOS 2021) Guest Editors: Lauchie Scott, Ryan Coder, Paul Kervin, Bobby Hunt.

Appendix: Gaussian Product Identity

Appendix: Gaussian Product Identity

Given \({\varvec{h}}(\cdot )\), \({\varvec{R}}\), \({\varvec{m}}\), and \({\varvec{P}}\) are of appropriate dimensions and \({\varvec{R}}\) and \({\varvec{P}}\) are symmetric, positive definite [31, 39]

$$\begin{aligned} p_g({\varvec{z}} \vert {\varvec{h}}({\varvec{x}}),{\varvec{R}})p_g({\varvec{x}}\vert {\varvec{m}},{\varvec{P}}) = p_g({\varvec{z}} \vert {\varvec{h}}({\varvec{m}}),{\varvec{H}}({\varvec{m}}){\varvec{PH}}^T({\varvec{m}})+{\varvec{R}})p_g({\varvec{x}} \vert {\varvec{\mu }},{\varvec{\Pi }}) , \end{aligned}$$
(58a)

where

$$\begin{aligned} {\varvec{\mu }}&= {\varvec{m}} + {\varvec{K}}[{\varvec{z}} - {\varvec{h}}({\varvec{m}})], \end{aligned}$$
(58b)
$$\begin{aligned} {\varvec{\Pi }}&= {\varvec{P}} - {\varvec{K}}{\varvec{H}}({\varvec{m}}){\varvec{P}}, \end{aligned}$$
(58c)
$$\begin{aligned} {\varvec{K}}&= {\varvec{PH}}^T({\varvec{m}})[{\varvec{H}}({\varvec{m}}){\varvec{PH}}^T({\varvec{m}})+{\varvec{R}}]^{-1}. \end{aligned}$$
(58d)

Note that \({\varvec{H}}({\varvec{m}})\) is the Jacobian of \({\varvec{h}}({\varvec{x}})\) evaluated at \({\varvec{x}} = {\varvec{m}}\).

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Fritsch, G.S., DeMars, K.J. Intrinsic Fault Resistance for Nonlinear Filters with State-Dependent Probability of Detection. J Astronaut Sci 69, 1821–1854 (2022). https://doi.org/10.1007/s40295-022-00353-9

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