Global Positioning System Reference
In-Depth Information
The only “receiver time” used is the relative time of reception from each of the
satellites and which makes the individual pseudorange.
A consequence of this time definition is that the computed satellite coordinates
immediately refer to the ECEF system, and therefore satellite coordinates are not
to be rotated about the Z -axis by an angle equal to the travel time times the Earth's
rotation rate.
8.5.2 Linearization of the Observation Equation
The most commonly used algorithm for position computations from pseudoranges
is based on the least-squares method. This method is used when there are more
observations than unknowns. This section describes how the least-squares method
is used to find the receiver position from pseudoranges to four or more satellites.
Let the geometrical range between satellite k and receiver i be denoted
k
i ,let c
denote the speed of light, let dt i be the receiver clock offset, let dt k be the satellite
clock offset, let T i be the tropospheric delay, let I i be the ionospheric delay, and
let e i be the observational error of the pseudorange. Then the basic observation
equation for the pseudorange P i
ρ
is
P i
k
i
dt k
T i
I i
e i .
= ρ
+
(
dt i
) +
+
+
c
(8.15)
k
i
The geometrical range
ρ
between the satellite and the receiver is computed as
k
i
X k
2
Y k
2
Z k
2
ρ
=
(
X i )
+ (
Y i )
+ (
Z i )
.
(8.16)
Inserting (8.15) into (8.16) yields
e i .
(8.17)
From the ephemerides—which include information on the satellite clock offset
dt k —the position of the satellite ( X k
P i
dt k
T i
I i
X k
2
Y k
2
Z k
2
=
(
X i )
+ (
Y i )
+ (
Z i )
+
c
(
dt i
) +
+
+
Y k
Z k ) can be computed. (The M -file sat-
,
,
pos does the job.)
The tropospheric delay T i is computed from an a priori model that is coded
as tropo ; the ionospheric delay I i may be estimated from another a priori model,
the coefficients of which are part of the broadcast ephemerides. The equation
contains four unknowns X i , Y i , Z i ,and dt i ; the error term e i is minimized by
using the least-squares method. To compute the position of the receiver, at least
four pseudoranges are needed.
Equation (8.17) is nonlinear with respect to the receiver position
(
X i ,
Y i ,
Z i )
,
so the equation has to be linearized before using the least-squares method.
We analyze the nonlinear term in (8.17)
(
X i ,
Y i ,
Z i ) =
(
X k
X i )
2
+ (
Y k
Y i )
2
+ (
Z k
Z i )
2
.
f
(8.18)
Linearization starts by finding an initial position for the receiver: ( X i , 0 ,
Y i , 0 ,
Z i , 0 ).
This is often chosen as the center of the Earth (0,0,0).
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