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Chapter 3 / Sensors / NAVSTAR GPS
_ _
S
O
S
( t − t ) + C ⋅ sin ( 2 ⋅ ϕ ) + C ⋅ cos(
2 ⋅ )
ι : = ι + & ι ⋅
ϕ
Argument of Perigee ( ωS )
REF
3.11-5
is
S
ic
S
Equation 3.11-2
The perigee of a satellite's orbit is the point at which the satellite is closest to
the centre of the Earth with respect to the ascending node.
Right Ascension (Longitude) of the Ascending Node ( ΩO )
The right ascension of the ascending node is the angle from X C to the
satellite's ascending node in the equatorial plane. This is the point at which
the satellite crosses the equatorial plane when travelling from south to north
measured positive eastwards at the reference time. The navigation message
contains the rate of right ascension and correction coefficients accounting
for earth wobble and polar wander.
S
E
E
( t ) : = ΩO
+ ( ΩO
− ωC,
E ) ⋅ ( t − t REF ) − ωC,
E ⋅ t REF
Ω &
Mean Anomoly ( MO )
Equation 3.11-3
The 6 th classical Keplerian parameter is the Time-of-Perigee Passage, the
time when the satellite last passed through the perigee. This is replaced in
GPS by the Mean Anomaly at the reference time to improve the stability of
the algorithms for near circular orbits. The Mean Anomaly is the phase of a
satellite in its circular orbit measured from the ascending node at the
reference time when it is travelling at a uniform angular velocity.
3.11.3.1 Satellite Position Derived from Ephemeris Data
The GPS model superimposes errors on the LOS range caused by the
extraction of satellite position from Ephemeris data. It is therefore useful to
understand how satellite position is derived. One of the simplest methods
amongst those available is successive substitution in Kepler’s transendental
equation, Carvalho [C.4] . This relates the True Anomaly to the relative orbital
position known as the Eccentric Anomaly (ES),
M : = E − e ⋅ sin
S
S
S
( E )
S
Equation 3.11-4
Since the orbital eccentricity is small, Duris [D.5] and Dailey [D.6] solve this
equation by iteration,
( E S ) : = ES
+ eS
⋅ sin ( ES
) where ( ES
) : = MO
k + 1
k
0
Equation 3.11-5