Earth-Centered, Earth-Fixed Coordinate System - Fundamentals of Global Positioning System Receivers: A Software Approach - page 64

In this equation, the value E must be calculated first from ephemeris data. In
order to find the r value the following steps must be taken:

1. Use Equation (4.22) to calculate n where µ is a constant; a s and n can
be obtained from the ephemeris data.
2. Use Equation (4.34) to calculate M where M 0 and t oe can be obtained from
ephemeris data and t c can be obtained from the discussion in Section 9.10.
3. The value of E can be found from Equation (4.35), where e s can be
obtained from the ephemeris data. The iteration method will be used in
this operation.
4. Once E is obtained, the value of r can be found from Equation (4.40).

In the above four steps, the first three steps are to find the value of E .Once
E is calculated, Equations (4.36) - (4.39) can be used to find the corrected GPS
time t .
Now let us find the true anomaly ν . This value can be found from

Equations (3.40) and (3.41) as
ν 1 = cos − 1 cos E

−

e s

1 − e s cos E

ν 2 = sin − 1 1

− e s sin E

1 − e s cos E

ν = ν 1 sign (ν 2 ) (4.41)

The argument ω can be found from the ephemeris data. Using the definition
in Equation (4.24), the value of φ is

φ ≡ ν + ω ( 4 . 42 )

The following correction terms are needed

δφ = C us sin 2 φ + C uc cos 2 φ

δr = C rs sin 2 φ + C rc cos 2 φ

δi = C is sin 2 φ + C ic cos 2 φ (4.43)

where the C us , C uc , C rs , C rc , C is , C ir are from ephemeris data:

φ ⇒ φ + δφ

r ⇒ r + δr (4.44)

The inclination angle i can be obtained from the ephemeris data and be cor-
rected as

i ⇒ i + δi + idot (t − t oe ) ( 4 . 45 )