GPS Software Receivers - Fundamentals of Global Positioning System Receivers: A Software Approach - page 204

The true anomaly can be found from Equation (4.41) as
ν 1 = cos − 1 cos E

− e s

1 − e s cos E

sin − 1 1 − e s sin E
1 −

ν 2 =

e s cos E

ν = ν 1 sign (ν 2 ) (9.14)

and angle φ can be found from Equation (4.42) as

π ≡ ν + ω ( 9 . 15 )

where ω is the argument of perigee (subframe 3, bits 197 - 204 and 211 - 234)
obtained from navigation data.
The following correction terms are needed as shown in Equation (4.43):

= +

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 (9.16)

where C us (subframe 2, bits 211 - 226), C ue (subframe 2, bits 151 - 166), C rs
(subframe 2, bits 69 - 84), C rc (subframe 3, bits 181 - 196), C is (subframe 3,
bits 121 - 126), and C ic (subframe 3, bits 61 - 76) are obtained from navigation
data. These three terms are used to correct the following terms as shown in
Equations (4.44) and (4.45):

φ ⇒ φ + δφ
r ⇒ r + δr

i ⇒ i + δi + idot (t − t oe ) (9.17)

where idot (subframe 3, bits 279 - 292) is the rate of inclination angle and is
obtained from the navigation data, t is obtained from Equation (9.13).
The angle between the accenting node and the Greenwich meridian er can
be found from Equation (4.46) as

˙ (t ˙ ie t

er = e + − t oe ) − ( 9 . 18 )

The final two steps are to find the position of the satellite from Equation (4.47)
and adjust the pseudorange by the overall clock correction term 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

ρ i ⇒ ρ i + ct (9.19)