Global Positioning System Reference
In-Depth Information
The next step is to define the mean anomaly
M
and from Equation (3.26) the
result is
µ
a
s
(t
M
≡
(E
−
e
s
sin
E)
=
−
t
p
)
(
3
.
27
)
If one defines the mean motion
n
as the average angular velocity of the satellite,
then from Equation (3.18) the result is
µ
a
s
2
π
T
=
=
n
(
3
.
28
)
Substituting this result into Equation (3.27) the result is
≡
−
=
−
M
(E
e
s
sin
E)
n(t
t
p
)
(
3
.
29
)
This is referred to as Kepler's equation. From this equation one can see that
M
is linearly related to
t
; therefore, it is called the mean anomaly.
3.12 TRUE AND MEAN ANOMALY
The information obtained from a GPS satellite is the mean anomaly
M
.From
this value, the true anomaly must be obtained because the true anomaly is used to
find the position of the satellite. The first step is to obtain the eccentric anomaly
E
from the mean anomaly, Equation (3.29) relates
M
and
E
. Although this
equation appears very simple, it is a nonlinear one; therefore, it is difficult to
solve analytically. This equation can be rewritten as follows:
E
=
M
+
e
s
sin
E
(
3
.
30
)
In this equation,
e
s
is a given value representing the eccentricity of the satellite
orbit. Both
e
s
and
M
can be obtained from the navigation data of the satellite.
The only unknown is
E
. One way to solve for
E
is to use the iteration method.
Anew
E
value can be obtained from a previous one. The above equation can
be written in an iteration format as
E
i
+
1
=
M
+
e
s
sin
E
i
(
3
.
31
)
where
E
i
+
1
is the present value and
E
i
is the previous value. One common
choice of the initial value of
E
is
E
0
=
M
. This equation converges rapidly
because the orbit is very close to a circle. Either one can define an error signal
as
E
err
=
E
i
and end the iteration when
E
err
is less than a predetermined
value, or one can just iterate Equation (3.31) a fixed number of times (i.e., from
5to10).
E
i
+
1
−
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