A consistent, debiased method for converting between polar and Cartesian coordinate systems

A significant problem in tracking and estimation is the consistent transformation of uncertain state estimates between Cartesian and spherical coordinate systems. For example, a radar system generates measurements in its own local spherical coordinate system. In order to combine those measurements with those from other radars, however, a tracking system typically transforms all measurements to a common Cartesian coordinate system. The most common approach is to approximate the transformation through linearization. However, this approximation can lead to biases and inconsistencies, especially when the uncertainties on the measurements are large. A number of approaches have been proposed for using higher order transformation modes, but these approaches have found only limited use due to the often enormous implementation burdens incurred by the need to derive Jacobians and Hessians. This paper expands a method for nonlinear propagation which is described in a companion paper. A discrete set of samples are used to capture the first four moments of the untransformed measurement. The transformation is then applied to each of the samples, and the mean and covariance are calculated from the result. It is shown that the performance of the algorithm is comparable to that of fourth order filters, thus ensuring consistency even when the uncertainty is large. It is not necessary to calculate any derivatives, and the algorithm can be extended to incorporate higher order information. The benefits of this algorithm are illustrated in the contexts of autonomous vehicle navigation and missile tracking.