Global Positioning System Reference
In-Depth Information
ijk
denotes the intersection point of planes
L
i
,
L
j
diagonal points of the parallelopiped where
I
and
L
k
.
Hence, the equation of the actual LOP
ijk
I
ijk
I
passing through
I
is found from the three
intersection points
I
i
j
k
which are available from equations (8) and (34) and
analogous equations for LOPs
L
j
,
L
k
,
L
j
and
L
k
.
As the LOPs obtained in this way are expressed by linear equations with unknowns
x
,
y
and
z
, they can be solved using simple algebraic or least squares methods.
I
ijk
,
I
ij
k
and
The locus refinement formulation assumes noise to be present in the formulae. However, if the
noise is absent the diagonal points
ijk
would be very close and during the calculation
I
ijk
and
I
process whenever pairs having distance
<
2
m
are observed the estimated location is found as
the mean of these two points.
The planar form LOP obtained from each satellite pair must be linearly independent so they
do not represent either the same or a parallel planar LOP. Such satellite pairs are referred to as
mutually independent, so a key objective is to identify such satellite pairs where each satellite
has nearly similar distance from the receiver. PML may be intuitively viewed as positioning
exploiting bearing measurements, as LOPs e
ectively denote a directional line. It is known that
angular measurements are consistently more accurate compared to TOF range measurements
and in (Chintalapudi et al., 2004) a combination of range and angular measurement has been
shown to achieve better positioning results, providing a valuable insight as to why the LOP
refinement furnishes better location estimation.
ff
3.8 Selection of satellite pairs for PML
It is apparent from observation 1 that the existence of a pair of satellites having equal distance
from the receiver position can have equal atmospheric noise exposure, with this prerequisite
being relaxed and generalized by LOP refinement approach. Observation 1 highlights the
significance of pairing the satellites for better noise cancellation and a better selection process
can result in considerable improvement. With practical range estimations there is no explicit
way to determine the best possible pairs following the observation. However, the range
estimation ratios can be used as a rough measure for adhering to observation 1 which is the
basis for the following empirically defined ranking criteria. The ranking criteria also considers
the closeness of the satellites. If the two satellites are too close to each other they might have
the best range estimation ratio while e
ectively they are like two satellites placed at the same
place and hence providing no additional redundancy to help positioning. Utilizing, the above
mentioned two principles the following empirical ranking criteria is introduced.
ff
1
p
1
p
2
r
1
r
2
=
(35)
where
r
1
and
r
2
are the observed range estimates for satellite pair (
p
1
,
p
2
) such that
r
1
≥
r
2
and
p
1
p
2
)
are preferred over ones with higher ranks. The complete satellite selection algorithm is given
as follows.
is the Euclidean distance between the two satellites. The pairs having lower ranks (
Algorithm 1 searches all available satellites for a particular receiver so its computational
complexity is
O
(
available satellites
2
) if an exhaustive search is applied. This selection process
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