Theoretical Insights for Bearings-Only Tracking in Log-Polar Coordinates

The choice of the coordinate system in a bearings-only tracking problem influences the methods used to observe and predict the state of a moving target. Modified polar coordinates and log-polar coordinates have some advantages over Cartesian coordinates. In this article, we derive closed-form expressions for the target state prior distribution after ownship maneuver: the mean, covariance, and higher order moments in log-polar coordinates. We explore the use of these closed-form expressions in simulation by modifying an existing bearings-only tracker that uses the unscented Kalman filter. Rather than propagating sigma points, we directly substitute the obtained expressions for the mean and covariance into the time update equations at the ownship turn. This modified algorithm, the closed-form expressions unscented Kalman filter, performs similarly to the pure unscented Kalman filter in terms of precision. However, the available closed-form third and fourth moments provide measures of non-Gaussianity of the target state when the ownship turns, giving insights about the underlying target state distribution and tracking performance. The availability of these higher order moments facilitates other extensions of the tracker not possible with a standard unscented Kalman filter.