An insightful approach to bearings-only tracking in log-polar coordinates

TL;DR

This paper derives closed-form expressions for target state distributions in bearings-only tracking using log-polar coordinates and demonstrates their application in a modified UKF tracker, improving understanding of non-Gaussian effects.

Contribution

It introduces closed-form expressions for target state moments in LPC and applies them to enhance a UKF-based tracker with non-Gaussianity considerations.

Findings

Modified UKF (CFE-UKF) performs comparably to standard UKF.

Higher-order moments reveal non-Gaussianity during ownship turns.

Monitoring these moments improves range error estimation.

Abstract

The choice of coordinate system in a bearings-only (BO) tracking problem influences the methods used to observe and predict the state of a moving target. Modified Polar Coordinates (MPC) and Log-Polar Coordinates (LPC) have some advantages over Cartesian coordinates. In this paper, we derive closed-form expressions for the target state prior distribution after ownship manoeuvre: the mean, covariance, and higher-order moments in LPC. We explore the use of these closed-form expressions in simulation by modifying an existing BO tracker that uses the UKF. Rather than propagating sigma points, we directly substitute current values of the mean and covariance into the time update equations at the ownship turn. This modified UKF, the CFE-UKF, performs similarly to the pure UKF, verifying the closed-form expressions. The closed-form third and fourth central moments indicate non-Gaussianity of the target state when the ownship turns. By monitoring these metrics and appropriately initialising relative range error, we can achieve a desired output mean estimated range error (MRE). The availability of these higher-order moments facilitates other extensions of the tracker not possible with a standard UKF.