Global Positioning System Reference
In-Depth Information
In order to apply least squares estimation the equations for each satellite are organized as
follows:
⎣
⎦
x
1
y
1
z
1
−
r
1
x
2
y
2
z
2
−
r
2
B
=
,
.
.
.
.
x
m
y
m
z
m
−
r
m
⎣
⎦
p
1
,
p
1
⎣
⎦
1
1
.
1
p
2
,
p
2
1
2
1
2
ρ
a
=
e
=
∧ =
,
, and
,
ρ
.
p
m
,
p
m
We can now rewrite equation 17 as:
a
−
B
ρ
+
∧
e
=
0
⇒
B
ρ
=
a
+
∧
e
(18)
For more than 4 satellites, we can have closed form least squares solution as follows:
ρ
=
B
+
a
+
∧
e
(19)
B
+
=(
B
T
B
)
−
1
T
is the pseudoinverse of Matrix
where
B
B
.
However, the solution
ρ
involves
∧
which is defined in terms of unknown
ρ
. This problem
is avoided by substituting
ρ
into the definition of the scalar
∧
and using the linearity of the
Lorentz inner product as follows:
2
1
B
+
(
a
+∧
e
)
B
+
(
a
+∧
e
)
∧ =
,
After rearranging,
1
2
B
+
e
B
+
e
+
B
+
e
B
+
a
−
+
B
+
a
B
+
a
=
∧
,
2
∧
,
,
0
(20)
B
+
e
B
+
e
(
B
+
e
B
+
a
−
This is a quadratic equation in
∧
with coe
cients
,
,2
,
1
)
,
and
B
+
a
B
+
a
,
. All these three values can be computed and we can solve for two possible values
of
∧
using the quadratic equation. If we get the two solutions to this equation
∧
1
and
∧
2
, then
we can solve for two possible solutions
ρ
2
in equation 19. One of these solutions will
make sense, it will be on the surface of the earth (which has a radius of approximately 6371
km), and one will not.
ρ
1
and
The major advantage of the Bancroft's method is to have a closed form least squares solution for
GPS equations. It has the same advantage of least squares approach of using all the available
satellites for location estimation. On the contrary, it uses the fundamental equation of spherical
ranging that in the course of solution leads to planar form LOPs which are than hyperboloid
LOPs. Therefore as discussed before, this method cannot be used for high-accuracy positioning
in presence of noise.
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