Global Positioning System Reference
In-Depth Information
The increments
X
,
Y
,
Z
are defined as
X
i
,
1
=
X
i
,
0
+
X
i
,
Y
i
,
1
=
Y
i
,
0
+
Y
i
,
(8.19)
Z
i
,
1
=
Z
i
,
0
+
Z
i
.
They update the approximate receiver coordinates. So the Taylor expansion of
f
(
X
i
,
0
+
X
i
,
Y
i
,
0
+
Y
i
,
Z
i
,
0
+
Z
i
)
is
Z
i
,
0
)
+
∂
f
(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
(
X
i
,
1
,
Y
i
,
1
,
Z
i
,
1
)
=
(
X
i
,
0
,
Y
i
,
0
,
f
f
X
i
∂
X
i
,
0
+
∂
f
(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
Y
i
+
∂(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
Z
i
.
(8.20)
∂
Y
i
,
0
∂
Z
i
,
0
Equation (8.20) includes only first-order terms; hence the function determines an
approximate position. The partial derivatives in Equation (8.20) are
X
k
∂
f
(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
−
X
i
,
0
=−
,
k
i
∂
X
i
,
0
ρ
Y
k
∂
f
(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
−
Y
i
,
0
=−
,
k
i
∂
Y
i
,
0
ρ
Z
k
∂
f
(
X
i
,
0
,
Y
i
,
0
,
Z
i
,
0
)
−
Z
i
,
0
=−
.
k
i
∂
Z
i
,
0
ρ
k
i
Let
0
be the range computed from the approximate receiver position; the first-
order linearized observation equation becomes
ρ
,
X
k
Y
k
Z
k
−
X
i
,
0
−
Y
i
,
0
−
Z
i
,
0
P
i
k
i
=
ρ
0
−
X
i
−
Y
i
−
Z
i
,
k
i
k
i
k
i
ρ
ρ
ρ
,
0
,
0
,
0
dt
k
T
i
I
i
e
i
,
+
c
(
dt
i
−
)
+
+
+
(8.21)
where we explicitly have
k
i
X
k
2
Y
k
2
Z
k
2
ρ
0
=
(
−
X
i
,
0
)
+
(
−
Y
i
,
0
)
+
(
−
Z
i
,
0
)
.
(8.22)
,
8.5.3 Using the Least-Squares Method
A least-squares problem is given as a system
A
x
=
b
with no solution.
A
has
m
rows and
n
columns, with
m
n
; there are more observations
b
1
, ...,
b
m
than free parameters
x
1
, ...,
x
n
. The best choice, we will call it
>
x
, is the one that
ˆ
minimizes the length of the error vector
e
ˆ
=
b
−
A
x
. If we measure this length in
ˆ
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