Global Positioning System Reference
In-Depth Information
When more than four observations are available we can compute the correction values
δ
z
for the preliminary estimate.
x
,
δ
y
,
δ
The least squares formulation can be concisely
written as follows.
⎣
⎦
x
0
y
1
−
y
0
z
0
x
1
−
z
1
−
−
−
−
1
⎣
⎦
r
1
r
1
r
1
δ
x
y
2
−
y
0
x
2
−
x
0
z
2
−
z
0
−
−
−
1
δ
y
r
2
r
2
r
2
Ax
=
=
b
−
(12)
.
.
.
.
δ
z
δ
cdt
y
0
x
0
r
m
−
y
m
−
z
0
x
m
−
z
m
−
−
−
1
r
m
r
m
The least squares solution is
⎣
⎦
δ
x
δ
y
A
T
Σ
−
1
A
)
−
1
A
T
Σ
−
1
=(
b
(13)
δ
z
δ
cdt
If the code observations are independent and assumed to have equal variance, then the above
can be simplified to
⎣
⎦
δ
x
δ
y
A
T
A
)
−
1
A
T
=(
b
(14)
δ
z
δ
cdt
x
0
z
T
.
xy
0
yz
0
The final position vector can be estimated by
ρ
=
+
δ
+
δ
+
δ
3.3 Bancroft's method (least squares solution)
We want to turn positioning into a linear algebra problem. Here is a clever method due to
Bancroft (1985) that does some algbraic manipulations to reduce the equations to a least-squares
problem. Multiplying things out in equation 6 and using the receiver clock bias
= ξ
i
s
as
−
b
the only noise parameter, we get
x
i
−
x
2
y
i
−
y
2
z
i
−
z
2
r
i
−
b
2
2
x
i
x
+
+
2
y
i
y
+
+
2
z
i
z
+
=
2
r
i
b
+
(15)
Rearranging,
x
i
+
r
i
x
2
r
2
y
i
+
z
i
−
y
2
z
2
(
+
+
) +
+
+
=
−
2
x
i
x
y
i
y
z
i
z
−
r
i
b
−
0
(16)
T
denote the receiver position vector and
T
denote the
i
th
satellite
Let
ρ
= [
xyzr
]
p
i
= [
x
i
y
i
z
i
r
i
]
position and range vectors.
Using
Lorentz inner product
for 4-space defined by:
u
,
v
=
u
1
v
1
+
u
2
v
2
+
u
3
v
3
−
u
4
v
4
Equation 16 can be rewritten as:
1
2
p
i
,
1
2
ρ
p
i
−
p
i
,
ρ
+
,
ρ
=
0;
(17)
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