Global Positioning System Reference
In-Depth Information
L 1
Same Linear LOP
P 2
P 1
(a) Ideal 2-D trilateration
scenario where linear form
LOPs are found from the
corresponding two circular
LOPs (Ca
(b) LOP from equidistant satellites
in presense of equal noise.
ff
ery, 2000).
Fig. 1. Depiction of observation 1.
p 1 (
)
p 2 (
)
The
circles
surrounding
the
satellites
with
known
positions
x 1 , y 1
,
x 2 , y 2
and
p 3 (
)
, denote the LOPs obtained from the individual range measurements for each
satellite. Ideally, the LOPs surrounding satellite i is given by,
x 3 , y 3
2
r i = p i −ρ
2
2
=
(
x
x i
)
+(
y
y i
)
(1)
In 2-D, it is feasible to calculate the exact receiver position using only three range
measurements. Two range measurements can result in two solutions corresponding to the
intersection of two circular LOPs. The third measurement resolves this ambiguity.
However, equating two circular LOPs will result in a straight line equation (in case of 3-D, it
will be planar equation) passing through two intersecting points of the circular LOPs. This line
does not represent the actual locus of the receiver position as it will be clarified later. However,
following (Ca
ery, 2000) this line is referred as Linear Form LOP in the subsequent discussions.
In Fig. 1, L 1 and L 2 are determined from the circular LOPs corresponding to satellite pairs (
ff
p 1 ,
p 2 ) and (
p 3 ) respectively, with the intersection point ( x , y )of L 1 and L 2 denoting the actual
position of the receiver.
p 1 ,
As shown in Fig. 1,the positioning geometry works correctly for ideal case of exact range
estimates being measured by the positioning devices. However, in reality it is quite di
cult to
measure the exact range both for external noise impact and internal errors such as receiver clock
bias and satellite clock skews.
However, we also showed the fact that accurate positioning
can be obtained if the noise e
ect is exactly the same for two satellites. However, in case of
variable noise presence for two satellite range estimates usual linear form LOP obtained from
circular LOPs deviates significantly from the true position of the receiver and leads to a bad
positioning geometry. This is further explained as follows.
ff
As it is clarified before that the range equations are mostly not accurate in practical scenario.
Though trilateration is a mathematical approach and ideally it can find the exact receiver
position, however it cannot find the position very well when the range estimates are perturbed
by noise. In this section we will specifically identify the problems of trilateration for inaccurate
range equations. For the ease of understanding we still limit this discussion for 2-dimensions
only.
Search WWH ::




Custom Search