Global Positioning System Reference
In-Depth Information
The hyperbolic LOP represents the actual LOP for a pair of satellites under the equal noise
assumption. The linear form LOP does not truly represent the locus of the receiver in presence
of noise unless both ranges to the satellites are equal as clarified in Fig. 2. Two possible cases
could arise due to equal noise presence:
a
) the circular ranges have a physical intersection
and
b
) the circular ranges do not have any physical intersection. In both cases, the hyperbolic
LOP is able to represent the original receiver position whereas linear form LOP deviates from
receiver position significantly. As establishing the LOP is the first step in positioning, any error
present at this step could aggravate the result significantly and hence finding a LOP closer to
the original receiver position is fundamental to achieving high accuracy positioning.
It is also crucial to compare the hyperbolic and linear form LOPs for unequal noise components
in individual measurements as in reality this assumption can be void. In these general
situations three possible cases could arise.
a
) the observed circular ranges have a physical
intersection;
b
) the observed circular ranges do not have any common intersection region; and
c
) One of the observed circular ranges overlap completely within the other circular region.
These three cases are shown in Fig. 3 where Fig. 3(a), (b) shows the hyperbolic and linear
form LOPs for noise ratio
(ξ
1
/
)
of 2 while Fig. 3(c) shows the LOPs for noise ratio of
4. Fig. 3(c) also shows that for completely overlapped ranges the hyperbolic formulation
turns into elliptic formulation. This is the case when coe
ξ
2
cient of
y
2
in (5) changes sign as
ff
>
the range di
a
). The noise
presence generally attenuates the signal more than that of ideal propagation scenario causing
overestimation of the range. However, it is theoretically possible to imagine the case where
range is underestimated due to noise. The simultaneous overestimation and underestimation
of ranges is supposed to be the most detrimental for LOP estimation and hence this case is
shown in Fig. 3(d). It is evident from the figures that for all the three cases of unequal noise
presence as well as for noise having di
erence becomes greater than distance between the satellites (
c
erent signs, hyperbolic formulation is better suited
than linear form and the impact of noise is less detrimental on hyperbolic LOPs than it is on
linear form LOPs.
ff
3. Analytical approaches for global positioning
We have discussed about the mathematical basis for positioning and presented the problems
of regular trilateration from the viewpoint of noisy measurements. The positioning algorithms
for GPS need greater care for noise and often augmented by filtering process to mitigate the
e
ff
ect of noise. However, they still largely depend on basic analytical positioning both for initial
estimation and for error correcting
/
ff
filtering phase.
In this chapter, we present the di
erent
analytical algorithms for GPS.
We begin with the 3-D analogous formula for equation 2 which represents a sphere.
2
2
r
i
2
2
2
2
= (
+ξ
i
)
=
p
i
−ρ
=
(
)
+(
)
+(
)
r
i
x
−
x
i
y
−
y
i
z
−
z
i
(6)
A generally acceptable modeling of the ranging error
ξ
i
is described by the following equation
(Strang & Borre, 1997).
e
i
(7)
where
I
i
is the ionospheric error,
T
i
is the tropospheric error,
c
is the speed of light,
dt
i
is the
satellite clock o
ξ
i
=
I
i
+
T
i
+
c
(
dt
i
(
t
−τ
i
)
−
dt
(
t
))
−
ff
set,
dt
is the receiver clock o
ff
set,
t
is the receiver time and
τ
i
is the signal
propagation time and
e
i
represents all other unmodelled error terms.
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