An Alternate Derivation of Polar/Spherical to Cartesian Measurement Conversion Using Conditional Density

Common methods for calculating the converted Cartesian position measurement and associated covariance from polar or spherical measurements are: standard conversion, debiased conversion, unbiased conversion, and modified unbiased conversion (MUC). In this paper we present an alternate derivation of the converted Cartesian position measurement and associated covariance from polar or spherical measurements. First we obtain the conditional density of the true polar or spherical variables conditioned on the measurement. Then we derive all moments of the Cartesian position using this conditional density. We show that these two moments of the Cartesian position can be calculated exactly assuming that measurement errors are zero-mean Gaussian and independent. We propose that “conditional mean” and “conditional covariance” would be better terminology than “MUC measurement.”