Global Positioning System Reference
In-Depth Information
x
p
x
k
−
x
p
x
m
−
x
q
x
k
−
x
q
x
m
+
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
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