The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation
The problem of tracking a two-dimensional Cartesian state of a target using polar observations is well known. At a close range, a traditional extended Kalman filter (EKF) can fail owing to nonlinearity introduced by the Cartesian-to-polar transformation in the observation prediction step of the filt...
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| Format: | Article |
| Language: | English |
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IEEE
2022-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9740628/ |
| _version_ | 1811310084936957952 |
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| author | Kevin R. Ford Anton J. Haug |
| author_facet | Kevin R. Ford Anton J. Haug |
| author_sort | Kevin R. Ford |
| collection | DOAJ |
| description | The problem of tracking a two-dimensional Cartesian state of a target using polar observations is well known. At a close range, a traditional extended Kalman filter (EKF) can fail owing to nonlinearity introduced by the Cartesian-to-polar transformation in the observation prediction step of the filter. This is a byproduct of the nonlinear transformation acting on the state variables, which make up a bivariate Gaussian distribution. The nonlinear transformation in question is the arctangent of Cartesian state variables <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula>, which corresponds to the target bearing. At long range, the bearing behaves as a wrapped Gaussian random variable, and behaves well for the EKF. At close range, the bearing is shown to be non-Gaussian, converging to the wrapped uniform distribution when <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> are uncorrelated. This study provides a concise derivation of the probability density function (PDF) for bearing for the EKF observation prediction step and explores the limiting behavior for this distribution while parameterizing the target range. |
| first_indexed | 2024-04-13T09:53:24Z |
| format | Article |
| id | doaj.art-fc60e78b07684f4bbd2427c8d6387609 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-13T09:53:24Z |
| publishDate | 2022-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-fc60e78b07684f4bbd2427c8d63876092022-12-22T02:51:32ZengIEEEIEEE Access2169-35362022-01-0110328033280910.1109/ACCESS.2022.31619749740628The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar TransformationKevin R. Ford0https://orcid.org/0000-0002-8556-0888Anton J. Haug1https://orcid.org/0000-0002-6749-5576Applied Physics Laboratory, Johns Hopkins University, Laurel, MD, USAApplied Physics Laboratory, Johns Hopkins University, Laurel, MD, USA (Retired)The problem of tracking a two-dimensional Cartesian state of a target using polar observations is well known. At a close range, a traditional extended Kalman filter (EKF) can fail owing to nonlinearity introduced by the Cartesian-to-polar transformation in the observation prediction step of the filter. This is a byproduct of the nonlinear transformation acting on the state variables, which make up a bivariate Gaussian distribution. The nonlinear transformation in question is the arctangent of Cartesian state variables <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula>, which corresponds to the target bearing. At long range, the bearing behaves as a wrapped Gaussian random variable, and behaves well for the EKF. At close range, the bearing is shown to be non-Gaussian, converging to the wrapped uniform distribution when <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> are uncorrelated. This study provides a concise derivation of the probability density function (PDF) for bearing for the EKF observation prediction step and explores the limiting behavior for this distribution while parameterizing the target range.https://ieeexplore.ieee.org/document/9740628/BearingGaussianKalman filterPDF |
| spellingShingle | Kevin R. Ford Anton J. Haug The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation IEEE Access Bearing Gaussian Kalman filter |
| title | The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation |
| title_full | The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation |
| title_fullStr | The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation |
| title_full_unstemmed | The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation |
| title_short | The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation |
| title_sort | probability density function of bearing obtained from a cartesian to polar transformation |
| topic | Bearing Gaussian Kalman filter |
| url | https://ieeexplore.ieee.org/document/9740628/ |
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