Global Positioning System Reference
In-Depth Information
Using (31) for pairs
( p i ,
p j
)=( p k ,
p 1
)
and
( p l ,
p 1
)
and subtracting the second from the first,
2
2
1
2
2
2
−(
x l )
−(
y l )
−(
z l )
−(
l 1 )
=
(
k 1 )
−(
l 1 )
+ p l
x k
x
y k
y
z k
y
r
k 1
r
r 1
r
r
p k
(32)
2
x k +
y k )
=(
where
. The above formulation represents a set of linear equations with
unknowns x , y , z and r 1 for all combination of two pair of satellites having satellite 1 in common.
Let x
p k
C i represent the i th
ij , y
ij , z
represent the di
ff
erence x i
x j , y i
y j , z i
z j respectively,
ij
= {( p k i ,
)
( p l i ,
)}
combination and m represent the total number of combinations with
C i
p 1
,
p 1
.
The system of linear equations for these m combinations can be concisely written as follows:
AX = B
(33)
where,
r
l 1 1
x
y
z
k 1 l 1
k 1 1
r
k 1 l 1
k 1 l 1
r
l 2 1
x
y
z
k 2 l 2
k 2 1
r
k 2 l 2
k 2 l 2
A =
,
.
.
.
r
l m 1
x
y
z
k m l m
k m 1
r
k m l m
k m l m
2
2
2
2
(
k 1 1 )
−(
l 1 1 )
+ p l 1
r
r
p k 1
x
y
z
r 1
2
2
2
2
(
r
k 2 1 )
(
r
l 2 1 )
p k 2
+ p l 2
1
2
X =
,
B =
.
2
2
2
2
(
k m 1 )
−(
l m 1 )
+ p l m
r
r
p k m
For m
3, the system of equations can be solved. However, r 1 is related to x , y , z by (6). For
pairing and equivalence of r i
r 1
=
r 1 , observed ranges are always used in the equations
and thus the system of equations are essentially independent of relationship between ( x , y , z )
and r 1 . This is also verified by the iterative refinement of r 1 where r 1 is modified by obtained r 1
in successive runs. The results show no di
r i
ff
erence in position estimates
(
x , y , z
)
for successive
iterations.
The equal noise assumption cannot be applied to any arbitrary selection of pairs while it is
quite reasonable for satellites observing near equal ranges to have equal noise components.
The selection of pairs with near equal ranges from a single reference satellite, may not be
feasible for low visibility where only a very few satellites are available for positioning. This is
the motivation for the next solution approach.
3.7 PML with refinement of the locus of positions
The linearization using one single reference satellite raises a performance issue and while it
is superior to trilateration in most of the cases, occassionally it performs worse. In search for
a positioning approach that can give consistently better estimates than basic trilateration, a
locus refinement approach is now presented.
A refined and better approximation to planar form LOP is found from two imprecise planar
form LOPs assuming equal noise presence due to receiver bias and ionospheric error in each
pair and for specific instance of measurement as follows.
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