GPS Software Receivers - Fundamentals of Global Positioning System Receivers: A Software Approach - page 206

where x , y , z and x u , y u , z u are the coordinates of the satellite and the user,
respectively, c is the speed of light. Use this transit time to modify the angle er
in Equation (4.49) as

˙ ie t t

er ⇒ er − ( 9 . 23 )

Use this new er in the first portion of Equation (9.19) to calculate the satellite
position x , y , z in the new coordinate system. From these satellite positions, the
user position x u , y u , z u will be calculated again from Equation (9.21).
These four equations (9.19), (9.21), (9.22), and (9.23) can be used in an iter-
ative way until the changes in x , y , z (or x u , y u , z u ) are below a predetermined
value. The final position will be the desired user position x u , y u , z u .

9.14 CHANGING USER POSITION TO COORDINATE SYSTEM
OF THE EARTH

Once the user position x u , y u , z u in Cartesian coordinate system is found, it
should be converted into a spherical coordinate system, because the user position
on the surface of the earth is given in geodetic latitude L , longitude l , and altitude
h as shown in Equations (2.17) - (2.19):

x u +

r = y u + z u

tan − 1 z u L c = x u + y u

tan − 1 y u
x u

=

l (9.24)

where L c is the geocentric latitude. However, the surface of the earth is not
a perfect sphere; the shape of the earth must be taken into consideration. The
geodetic latitude L is used in maps and should be calculated from L c through
Equations (2.50) or (2.51) as

L = L c + e p sin 2 L or

L i + 1 = L c + e p sin 2 L i (9.25)

where e p is the ellipticity. The second portion of the above equation is written
in iterative form. The altitude can be found from Equation (2.57) as

x u + y u + z u − a e ( 1 − e p sin 2 L)

h = ( 9 . 26 )

These last three values, latitude L , longitude l , and altitude h , are the desired
user position. The latitude and longitude are often expressed in degrees, minutes,
and seconds or in degrees and minutes.