Global Positioning System Reference
In-Depth Information
I
ijk
I
i jk
L
k
I
i
j k
I
L
i
L
j
I
i
jk
I
ij k
L
j
O
k
O
i
L
i
I
ij k
O
j
L
k
I
ijk
Fig. 5. Refining the locus of the receiver position under noisy measurement conditions.
Fig. 5 shows the ideal scenario where the position of the receiver to be determined,
, and the
three respective planar form LOPs
O
i
,
O
j
and
O
k
are obtained from any three arbitrary satellite
pairs
ρ
P
k
.
The equation for
L
i
,
L
j
,
L
k
can be found using (8).
P
i
,
P
j
and
ξ
is constant due to identical receiver clock bias and exposure to similar atmospheric noise.
Hence,
L
i
,
L
j
,
L
k
vary from the ideal noise free LOPs
O
i
,
O
j
,
O
k
by the extra constant terms of
2
For specific measurement instance
r
2
respectively. Crucially their slopes remain unchanged
(Left hand side of (8)), and these are shown by the solid planes
L
i
,
L
j
,
L
k
parallel to
O
i
,
O
j
and
O
k
in Fig. 5. For non co-planar satellite pairs,
L
i
,
L
j
and
L
k
will have a physical intersection
point
r
i
1
−
r
i
2
,2
r
j
r
2
and 2
r
1
−
ξ
ξ
1
−
ξ
x
ijk
,
y
ijk
,
z
ijk
.
I
ijk
=
r
i
1
−
r
i
2
with
Another plane
L
i
parallel to
L
i
can be found as follows by modifying the term 2
ξ
q
r
i
1
−
r
i
2
, where
q
is an arbitrary positive constant.
x
i
2
−
−
x
i
1
x
y
i
2
−
y
i
1
y
z
i
2
−
z
i
1
z
+
+
=
2
r
i
2
r
i
1
2
−
r
i
2
2
q
r
i
1
−
(34)
i
2
2
i
1
2
p
−
p
+
−
The original LOP
O
i
will then pass between the planes
L
i
and
L
i
as the constants have opposite
signs. A similar argument applies to
L
j
,
L
k
so that the parallelopiped bounded by the planes
O
i
,
L
i
,
O
j
,
L
j
,
O
k
,
L
k
will have an aspect ratio
AR
r
i
1
−
r
2
as
L
i
,
L
j
,
L
k
are
:
r
j
r
j
2
r
i
2
:
r
1
−
=
1
−
r
i
1
−
r
i
2
,2
r
j
2
and 2
r
1
−
r
2
distances away from
O
i
,
O
j
and
O
k
respectively as they
r
j
2
ξ
ξ
1
−
ξ
di
er only by the constant terms in (8). The
AR
of the parallelopiped bounded by the planes
O
i
,
L
i
,
O
j
,
L
j
,
O
k
,
L
k
will have exactly the same aspect ratio so indicating
ff
ijk
and
I
ijk
,
I
I
to be the
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