Global Positioning System Reference
In-Depth Information
From this
n
value the mean anomaly can be found from Equation (4.23) as
M
=
M
0
+
n(t
c
−
t
oe
)
(
4
.
34
)
where
M
0
is in the ephemeris data. In this equation
t
c
is used instead of
t
as
t
is
not derived yet.
The eccentric anomaly
E
can be found from Equations (3.29) or (3.30) through
iteration as
=
+
E
M
e
s
sin
E
(
4
.
35
)
where
e
s
is eccentricity of the satellite orbit, which can be obtained from the
ephemeris data. Let us define a constant
F
as
=
−
2
√
µ
c
2
10
−
10
sec
/(
meter
)
1
/
2
F
=−
4
.
442807633
×
(
4
.
36
)
where
µ
is the earth's universal gravitational parameter and
c
is the speed of
light. The relativistic correction term is
Fe
s
√
a
s
sin
E
t
r
=
(
4
.
37
)
The overall time correction term is
t
oc
)
2
t
=
a
f
0
+
a
f
1
(t
c
−
t
oc
)
+
a
f
2
(t
c
−
+
t
r
−
T
GD
(
4
.
38
)
where
T
GD
,
t
oc
,
a
f
0
,
a
f
1
,
a
f
2
are clock correction terms and
T
GD
is to account
for the effect of satellite group delay differential. They can be obtained in the
ephemeris data. The GPS time of transmission
t
corrected for transit time can be
corrected as
t
=
t
c
−
t
(
4
.
39
)
This is the time
t
that will be used for the following calculations.
4.9 CALCULATION OF SATELLITE POSITION
(
5,6
)
This section uses all the information from the ephemeris data to obtain a satellite
position in the earth-centered, earth-fixed system. These calculations require the
information obtained from both Chapters 3 and 4; therefore, this section can be
considered as a summary of the two chapters.
Equation (4.19) is required to calculate the position of the satellite. In this
equation there are five known quantities:
r
,
ν
+
ω
,
i
,and
er
. These quantities
appear on the right side of the equation and the results represent the satellite
position. Let us find these five quantities.
First let us find the value of
r
from Equation (4.10) as
r
=
a
s
(
1
−
e
s
cos
E)
(
4
.
40
)
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