GPS Receiver-Related Subjects - Fundamentals of Global Positioning System Receivers: A Software Approach - page 344

The distance from satellite to the center of the earth r can be found as

r = a s ( 1 − e s cos E) ( 4 . 40 )

The true anomaly ν can be found from the following equations as

cos − 1 cos E − e s
1 −

ν 1 =

e s cos E

sin − 1 1 − e s sin E
1 −

ν 2 = (4.41)

e s cos E

ν = ν 1 sign(ν 2 )

The definition of φ is

φ ≡ ν + ω ( 4 . 42 )

The er can be found as

ž (t GPS − ž ie t GPS

er = e + t oe ) − ( 4 . 46 )

where t is replaced by t GPS .
The position of the satellite can be found 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 obtained is in earth-center, earth-fixed (ECEF) coordination.
This approach is used to calculate the positions of all the satellites.

Step3. Coordinatetransform . In this operation, three coordinate systems are
included. The user position is usually given in latitude ( L c ), longitude ( l ), and
altitude ( h ). The satellite position is in ECEF system. The final satellite position is
referenced to the user position in east, north, and height. The following coordinate
transform can achieve this goal:
The user position is changed from L c , l ,and h into ECEF system to have the
same coordinate with the satellite position through

a e sin l

x 0 = 1 +

e e ) tan L c + h × cos l × cos L c

( 1 −

a e cos l
1 + ( 1 − e e ) tan L c + y 0 = h × sin l × sin L c (12.25)

e e ) sin L c

a e ( 1 −

z 0 = 1 − + h × sin L c

e e sin 2 L c