Consistent unbiased linear filtering with polar measurements

The problem of tracking objects moving in Cartesian space with sensors delivering polar measurements has been under investigation of several researchers for quite some time now. Different proposals for using measurement conversion techniques in combination with a linear Kalman filter have been made in order to reduce the range bias that shows up in the filter estimates when a Cartesian pseudo-measurement is created from the polar measurements by applying the respective conversion formulae in combination with a corresponding linearized form of the measurement error covariance matrix. It turns out that the actual behavior of these different approaches strongly depends on the specific situation under consideration where observed effects range from a truly (approximate) suppression of conversion bias up to an even increased bias. In this paper, a systematic approach to analyzing the bias effects of measurement conversion in certain typical tracking situations is presented. Starting from this approach, a new measurement conversion technique is proposed that yields consistent unbiased estimates in these cases.