Global Positioning System Reference
In-Depth Information
x p
x k
x p
x m x q
x k
x q
x m +
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P pq
km, 1
d pq
km, 1 ,P
ε pq
km, 1 ,P
=
+
pq
km
d pq
ε pq
km, 1 ,P
= ρ
+
km, 1 ,P +
(7.64)
available at each epoch. The hardware delay terms cancel in (7.64) and the unknown
pseudorange multipath d pq
km , while potentially large, is typically neglected. If the
system (7.64) is solved by linearization and subsequent least-squares, then the row
of the design matrix contains, respectively,
∂P pq
km
x m =
e m
e q m
(7.65)
fo r each double difference.
In relative positioning with pseudoranges, the processing is usually carried out
baseline by baseline and mathematical correlation between the double-differenced
pseudorange observations are often neglected. These correlations are typically not
neglected for carrier phase observations, which are the more accurate type of obser-
vations, and will be discussed below.
A closed solution is readily available for relative positioning with pseudoranges.
Consider the three double differences that can be formed from the observations of
four satellites,
[26
Lin
* 2 ——
Nor
PgE
k m = x p
x k x p
x m x i
x k x i
x m ,
P pi
1
i
3
(7.66)
[26
Le t p denote the base satellite, in this case p
4. Since the satellite coordinates and
th e station coordinates x k are known, we can compute the auxiliary quantity Q ,
=
x p
x k + x i
x k
Q pi
km
P pi
km
=
(7.67)
Co mparing (7.66) and (7.67), we find that Q relates to the unknown x m as
=− x p
x m + x i
x m
Q pi
km
(7.68)
Following Chaffee and Abel (1994), we translate the origin of the coordinate system
to satellite p
x i
x i
x p
=
(7.69)
N oting that in the translated coordinate system x p
=
0, we obtain from (7.68)
x i
x m
Q pi
km
+
x m =
(7.70)
Eq uations (7.70) and (7.50) are of the same form. Once x m is computed the coordi-
nates can be translated to x m using (7.69). Squaring (7.70) gives
 
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