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

Abstract

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 $X$ and $Y$ , 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 $X$ and $Y$ 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.

Keywords

• Bearing
• Gaussian
• Kalman filter
• PDF