Bayesian Estimation and Tracking: A Practical Guide

Appendix 18.B Three-Dimensional Coordinate Transformations

When using a tracking filter that defines the state vector in Cartesian coordinates and the observation vector in spherical polar coordinates, both Cartesian-to-spherical and a spherical-to-Cartesian transformations are required. These transformations must be applied to the vector under transformation as well as the covariance matrix. In this section, we will derive these transformations and show how to develop a subroutine to achieve these transformation. As a side benefit, the Jacobian of the transformation is created and becomes part of the output.

First, let us present some general concepts about multidimensional coordinate transformations. A vector transformation from one coordinate system to another can be written as

(18.152)

where x is the vector in its original coordinate system and r is the vector in the transformed system.

To develop a transformation of the covariance matrix, examine the definition of the covariance matrix in the transformed system, given by

(18.153)

where represents the expected value and . From ...