Global Positioning System Reference
In-Depth Information
Anchor node
Regular node
Observed range
Actual range
Linear Form LOP
Hyperbolic LOP
35
Anchor node
Regular node
Observed range
Actual range
Linear Form LOP
Hyperbolic LOP
35
30
30
25
25
20
20
15
15
10
10
5
5
0
5
10
15
20
25
30
35
5
10
15
20
25
30
X
X
(a)
(b)
Fig. 2. The hyperbolic and linear form LOP of a receiver from range estimates by a pair of
satellites under equal noise assumption. (a) The general case when two observed circular
LOPs physically intersect. (b) The case when circular LOPs do not intersect due to noise and
underestimation of the ranges.
At first, we present the following observation that identifies the case when the conventional
trilateration works in consideration of noise.
Observation 1.
Assuming a receiver uses range estimates from two satellites that are located at the
same distance from the receiver and have equal noise components, it is shown below that the locus of
positions for that receiver (as the error components vary) is a straight line whose equation is independent
of range estimates.
p
3
(
)
Assume that due to noise, the range measurements for
p
1
(
x
1
,
y
1
),
p
2
(
x
2
,
y
2
) and
x
3
,
y
3
are
+ξ
3
, where
r
i
,
r
i
represent the observed and actual distance (pseudorange and actual range respectively)
between the
i
th
=
+ξ
1
,
r
2
=
+ξ
2
and
r
3
=
corrupted to give respective LOPs of radii
r
1
r
1
r
2
r
3
ξ
i
is the measurement noise at the
receiver corresponding to the measurement. The circular LOP can then be expressed as:
satellite and receiver respectively and
2
2
(
+ξ
i
)
=
p
i
−ρ
r
i
(2)
ρ=(
)
where
is the receiver position to be determined.
Equating the circular LOPs for
x
,
y
p
1
and
p
2
using (2),
L
1
becomes:
(
)
+ (
)
=
x
2
−
x
1
x
y
2
−
y
1
y
2
p
2
2
(3)
2
2
2
−
p
1
+ (
r
1
+
ξ
1
)
−
(
r
2
+
ξ
2
)
where the right hand side becomes independent of range parameters,
i.e.
, measurement values
r
1
and
r
2
whenever
r
1
=
r
2
+
ξ
2
. One particular case is equidistant satellites
and equal noise presence when the above condition is fulfilled.
r
2
⇒
r
1
+
ξ
1
=
The importance of this observation lies in the fact that it eliminates the signal propagation
dependent parameters and receiver clock bias under assumed conditions completely.
GPS measurements are mostly susceptible to these errors which are both device and
environmentally dependent.
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