Global Positioning System Reference
In-Depth Information
a distance of about 6371 km from the origin. We can eventually get the receiver coordinate
using correct value of
e
1
as follows:
r
1
e
1
(29)
The Kleusberg's method is geometrically oriented and uses a minimum number of satellites.
On the other hand, it cannot utilize more number of satellites even when they are available. This
method is also dependent on the proper geometrical orientation of the satellites. Moreover,
it often gives di
ρ
=
p
1
+
ff
erent results for di
ff
erent set of satellites and depending on the order of the
satellites in solving the equations.
3.5 Paired measurement localization
In trilateration, the positioning works by simultaneous solution of three spherical LOP
equations. Similar to the 2-D steps, we can equate two spherical LOP equations to find
equation for a 2-D plane representing the planar locus of position. Analogous to 2-D case,
three planar equations can be solved to find the ultimate receiver position.
As shown in section 2, the e
ect of noise will have detrimental impact on the aforementioned
simple solution. On the other hand, instead of equating the two imprecise range equations we
can maintain an equi-distant locus of position from two satellites as formulated in equation 21
for a hyperboloid LOP. This will be more accurate than a traditional 2-D planar LOP based
positioning.
ff
Solving the nonlinear hyperbolic
cult. Moreover, existing
hyperbolic positioning methods proceed by linearizing the system of equations using either
Taylor-series approximation (Foy, 1976; Torrieri, 1984) or by linearizing with another additional
variable (Chan & Ho, 1994; Friedlander, 1987; Smith & Abel, 1987). However, while linearizing
works well for existing approaches it is not readily adaptable for the proposed paired approach
as linearizing is indeed pairing with an arbitrarily chosen hyperbolic LOP. The assumption
of equal noise cannot be held for any arbitrary selection of pairs and hence alternate ways to
solve such LOPs for paired measurement is now formulated.
/
hyperboloid equations is di
3.6 PML with single reference satellite
(Chan & Ho, 1994) provided closed form least squares solution for non-linear hyperbolic LOPs
by linearizing with reference to a single satellite. Analogous to their approach a closed form
solution is found for PML using pairs having a common reference satellite in them.
The
solution is simpler than (Chan & Ho, 1994)'s approach as the e
ff
ect of noise is considered early
in the paired measurements formulations.
Let
r
ij
represent the di
ff
erence in the observed ranges for satellite pairs
(
i
,
j
)
. In case of equal
noise presence it follows:
r
r
j
After squaring and rearranging,
r
i
=
ij
=
r
ij
=
r
i
−
(30)
r
2
r
j
ij
+
+
2
r
ij
r
j
Hence, the actual spherical LOP can be transformed as follows:
(
2
2
2
2
2
)
+(
)
+(
)
=(
ij
)
+
+(
)
x
−
x
i
y
−
y
i
z
−
z
i
r
2
r
ij
r
j
r
j
(31)
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