The Probability Density Function of Bearing Obtained From a Cartesian-to-Polar Transformation

• Main Authors: Ford, K.R (Author), Haug, A.J (Author)
• Format: Article
• Language: English
• Published: Institute of Electrical and Electronics Engineers Inc. 2022
• Subjects: Arctangent; Bearing; Bivariate gaussian distributions; Cartesians; Close range; Extended Kalman filters; Gaussian; Gaussian distribution; Gaussian noise (electronic); Gaussians; Kalman filter; Mathematical transformations; Non-linear transformations; PDF; Polar transformations; Probability density function; State-variables; Two-dimensional
• Online Access: View Fulltext in Publisher

 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    and    , 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    and    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. © 2013 IEEE.