Earth-Centered, Earth-Fixed Coordinate System - Fundamentals of Global Positioning System Receivers: A Software Approach - page 65

where idot can be obtained from the ephemeris data. The last term to be found is

e + ˙ (t

− ˙ ie t

er =

−

t oe ) ( 4 . 46 )

where the earth rotation rate ˙ ie is a constant, e , ˙ ,and t oe are obtained from
the ephemeris data. It should be noted that the corrected GPS time t is used in
the above two equations.
Once all the necessary parameters are obtained, the position of the satellite
can be found from Equation (4.19) as

x
y
z

r cos er cos φ − r sin er cos i sin φ

=

r sin er cos φ r cos er cos i sin φ

r sin i sin φ

+ ( 4 . 47 )

The satellite position calculated in this equation is in the ECEF frame. There-
fore the satellite position is a function of time. From the x , y , z ,andthe
pseudorange of more than four satellites the user's position can be found from
results in Chapter 2. The actual calculation of the pseudorange is discussed in
Section 9.9.

4.10 COORDINATE ADJUSTMENT FOR SATELLITES

Using the earth-centered, earth-fixed coordinate system implies that the earth's
rotation is taken into consideration. The satellite position calculated from
Section 4.9 is based on the GPS time of transmission t corrected for transit
time. However, the user position is calculated at the time of receiving. Since the
satellite and user positions are calculated at different times, they are in different
coordinate systems. This will cause an error in the user position. As discussed in
Section 3.3., if the user is on the equator of the earth and an approximate signal
traveling time of 76 ms is assumed, the user position is moved about 36 m
(2 π 10 3 ) ) due to the rotation of the earth.

In order to obtain the correct user position, a single coordinate system should
be used. Since the user position is measured at the time of receiving, it is appro-
priate to use this time in the coordinate system. The satellite position calculated
should be referenced to this time. This correction does not mean to change the
satellite position, but only changes the coordinate system of the satellites.
In order to reference the GPS time at the time of receiving, the coordinate
system of each satellite must be separately modified. Using the time of receiving
as reference, the time of transmission of satellite i in the new coordinate system is
the time of receiving minus the transition time as shown in Equation (4.31). The
transit time cannot be determined before the user position is calculated because
of the unknown user clock bias. Only when the user position is obtained can the
pseudorange be found. Once the user position is found, the pseudorange can be
found from the user position to the satellite position. The earth rotation appears
only in Equation (4.46). Equations (4.46) and (4.47) are used in the operation.

10 3 × 6368 × × 76 /( 24 × 3600 ×