Earth-Centered, Earth-Fixed Coordinate System - Fundamentals of Global Positioning System Receivers: A Software Approach - page 59

If the angle er is used in Equation (4.14) to replace ,the x -axis will
be rotating in the equator plane. This x -axis is the direction of the Greenwich
meridian. Using this new angle in Equation (4.14) the result is

cos er − sin er 0

C 3 =

sin er cos er 0 ( 4 . 16 )

0 0 1

In this equation the rotation of the earth is included, because time is included
in Equation (4.15). Using this time t er in the system, every time the Greenwich
meridian is aligned with the vernal equinox, t er = 0. The maximum length of this
time is a sidereal day, because the Greenwich meridian and the vernal equinox
are aligned once every sidereal day.
The time t er should be changed into the GPS time t . The GPS time t starts
at Saturday night at midnight Greenwich time. Thus, the maximum GPS time
is seven solar days. It is obvious that the time base t er and the GPS time t are
different. A simple way to change the time t er to GPS time t is a linear shift of
the time base as

t er = t + t ( 4 . 17 )

where t can be considered as the time difference between the time based on t er
and the GPS time t . Substituting this equation into Equation (4.15), the result is

er = − ˙ ie t er = − ˙ ie t − ˙ ie t ≡ − α − ˙ ie t ≡ e − ˙ ie t
where e ≡ − α and α ≡ ˙ ie t (4.18)

The reason for changing to this notation is that the angle α is considered as

one angle e , and this information is given in the GPS ephemeris data. However,
this relation will be modified again in Section 4.7 and the final result will be used
to find er in Equation (4.16). Before the modification of e , let us first find the
overall transform.

−

4.6 OVERALL TRANSFORM FROM ORBIT FRAME TO EARTH-CENTERED,
EARTH-FIXED FRAME

In order to transform the positions of the satellites from the satellite orbit frame
to the ECEF frame, there need to be two intermediate transforms. The overall
transform can be obtained from Equation (4.7). Substituting the results from
Equations (4.16), (4.13), and (4.12) into (4.7), the following result is obtained:

= C 3 C 2 C 1

x 4
y 4
z 4

r cos ν
r sin ν
0

cos er − sin er 0 10 0 cos ω − sin ω 0 r cos ν

r sin ν
0

= sin er cos er 0 0c s i − sin i sin ω cos ω 0

0 0 1 0 n i cos i 0 0 1