Earth-Centered, Earth-Fixed Coordinate System - Fundamentals of Global Positioning System Receivers: A Software Approach - page 60

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

r sin ν
0

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

0 sin i cos i 0 0 1

cos er cos ω − sin er cos i sin ω − cos er sin ω − sin er cos i cos ω sin er sin i

= sin er cos ω + cos er cos i sin ω − sin er sin ω + cos er cos i cos ω − cos er sin i

sin i sin ω sin i cos ω cos i

r cos ν
r sin ν
0

·

r cos er cos (ν + ω) − r sin er cos i sin (ν + ω)

(4.19)

= r sin er cos (ν + ω) + r cos er cos i sin (ν + ω)

r sin i sin (ν + ω)

This equation gives the satellite position in the earth-centered, earth-fixed coor-
dinate system.
In order to calculate the results in the above equation, the following data
are needed: (1) a s : semi-major axis of the satellite orbit; (2) M : mean anomaly;
(3) e s : eccentricity of the satellite orbit; (4) i : inclination angle; (5) ω : argument
of the perigee; (6) - α : modified right ascension angle; (7) GPS time. The first
three constants are used to calculate the distance r from the satellite to the center
of the earth and the true anomaly ν as discussed in Section 3.12. The three values
i , ω ,and - α are used to transform from the satellite orbit frame to the ECEF
frame. In order to find er in the above equation the GPS time is needed.

4.7 PERTURBATIONS

The earth is not a perfect sphere and this phenomenon affects the satellite orbit.
In addition to the shape of the earth, the sun and moon also have an effect on
the satellite motion. Because of these factors the orbit of the satellite must be
modified by some constants. The satellites transmit these constants and they can
be obtained from the ephemeris data.
Equation (4.19) is derived based on the assumption that the orbit of the satellite
is elliptical; however, the orbit is not a perfect elliptic. Thus, the parameters
in the equations need to be modified. This section presents the results of the
correction terms.
In Equation (4.15) the right ascension will be modified as

+ ˙ (t ⇒ − t oe ) ( 4 . 20 )

where t is the GPS time, t oe is the reference time for the ephemeris, and ˙ is
the rate of change of the right ascension. In this equation it is implied that the
right ascension is not a constant, but changes with time. The ephemeris data
transmitted by the satellite contain t oe and

˙ . Substituting this equation into