Global Positioning System Reference
In-Depth Information
Let
t
0
be the time the satellite passes perigee, so that
µ(
t
)
=
n
(
t
−
t
0
)
.Kepler's
famous equation relates the mean anomaly
µ
and the eccentric anomaly
E
:
E
=
µ
+
e
sin
E
.
(8.4)
From Equation (8.1) we finally get
arctan
√
1
e
2
sin
E
cos
E
arctan
η
ξ
−
f
=
=
.
(8.5)
−
e
By this we have connected the true anomaly
f
, the eccentric anomaly
E
,and
the mean anomaly
µ
. These relations are basic for every calculation of a satellite
position.
It is important to realize that the orbital plane remains fairly stable in relation
to the geocentric
X
Z
-system. In other words, seen from space, the orbital
plane remains fairly fixed in relation to the equator. The Greenwich meridian
plane rotates around the Earth spin axis in accordance with Greenwich apparent
sidereal time (GAST), that is, with a speed of approximately 24 h
,
Y
,
day. A GPS
satellite performs two revolutions a day in its orbit having a speed of 3
/
.
87 km
/
s.
In the orbital plane the Cartesian coordinates of satellite
S
are given as
⎡
⎣
⎤
⎦
,
r
j
cos
f
j
r
j
sin
f
j
0
where
r
j
=
r
(
t
j
)
comes from (8.2) with
a
,
e
,and
E
evaluated for
t
=
t
j
; refer
to Figure 8.5.
This vector is rotated into the
X
,
Y
,
Z
-coordinate system by the following se-
quence of 3D rotations:
k
i
j
)
k
R
3
(
−
j
)
R
1
(
−
R
3
(
−
ω
j
).
The matrix that rotates the
XY
-plane by
ϕ
, and leaves the
Z
-direction alone, is
⎡
⎤
cos
ϕ
sin
ϕ
0
⎣
⎦
R
3
(ϕ)
=
−
sin
ϕ
cos
ϕ
0
(8.6)
0
0
1
and similarly for a rotation about the
X
-axis:
⎡
⎤
1
0
0
⎣
⎦
.
R
1
(ϕ)
=
0 s
ϕ
sin
ϕ
(8.7)
0
−
sin
ϕ
cos
ϕ
Finally, the geocentric coordinates of satellite
k
at time
t
j
are given as
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
.
X
k
r
j
cos
f
j
(
t
j
)
Y
k
k
i
j
)
k
r
j
sin
f
j
0
(
t
j
)
R
3
(
−
j
)
R
1
(
−
R
3
(
−
ω
j
)
(8.8)
Z
k
(
t
j
)
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