Bayesian Estimation and Tracking: A Practical Guide

18.2 Tracking a Target in Spherical Coordinates

In this section, we discuss tracking an object using a spherical state vector defined as

(18.28)

with the spherical velocity components {} to be defined below. As we did for the Cartesian case, where we assumed that the components of the Cartesian velocity were constant resulting in a Cartesian dynamic motion model given by (18.1), our objective will be to develop a three-dimensional dynamic motion model for an object with constant spherical velocity components.

18.2.1 State Vector Position and Velocity Components in Spherical Coordinates

Referring to Figure 18.25, the position of an object can be written as

(18.29)

or

(18.30)

where {e_x, e_y, e_z} and {e_R, e_θ, e_ϕ} are unit vector sets in Cartesian and spherical coordinates, respectively.

The spherical-to-Cartesian transformation of the position components are defined by (18.193), so we can rewrite (18.29) as

(18.31)

By definition, the spherical unit vectors are given by

(18.32)

(18.33)