Global Positioning System Reference
In-Depth Information
where ρ corresponds to the reference channel relative to the satellite with the shortest path
to the user. In case we want to follow the second approach (i.e. “common reception time”),
equation (18) keeps the same except for the time delay δ that has to be substituted by Δ.
In case two different GNSS systems are tracked and used for the PVT calculation, equation
(18) has to be slightly modified. For example, let assume to have 4 GPS and 2 Galileo
satellites in view, respectively. By following the “common transmission time” method, we
can rewrite equation (18) as:
GPS
2
GPS
2
GPS
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)
=
ρ
+ ⋅
c
Δ
b
u
u
u
1,
GPS
GPS
1
1
1
GPS
2
GPS
2
GPS
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)
=
ρ
+ ⋅
c
Δ
b
+ ⋅
c
δ
u
u
u
1,
GPS
GPS
2,
GPS
2
2
2
GPS
2
GPS
2
GPS
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)
=
ρ
+ ⋅
c
Δ
b
+ ⋅
c
δ
u
u
u
1,
GPS
GPS
3,
GPS
3
3
3
(19)
GPS
2
GPS
2
GPS
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)=
ρ
+⋅
cb
Δ
+⋅
c
δ
u
u
u
1,
GPS
GPS
4,
GPS
4
4
4
Gal
2
Gal
2
Gal
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)
=
ρ
+ ⋅
cb
Δ
+ ⋅
cb
Δ
u
u
u
1,
Gal
GPS
GPS
/
Gal
1
1
1
Gal
2
Gal
2
Gal
2
(
x
x
)
+
(
y
y
)
+
(
z
z
)
=
ρ
+ ⋅
cb
Δ
+ ⋅
cb
Δ
+ ⋅
c
δ
2
u
u
u
1,
Gal
GPS
GPS
/
Gal
2 ,
Gal
2
2
When we work with more than one GNSS we have to keep in mind that different GNSSs
are not synchronized among each others. This fact implies to introduce additional
unknowns that take into account the time-bias between the GNSS systems. For example, if
we consider a GPS/Galileo receiver as stated in equation (19), we need a variable that
estimates the bias offset between the GPS and the Galileo time scales. Finally the receiver is
able to compute a valid position and velocity. One of the most commonly used algorithm for
the position estimation is based on the least-squares (LS) method. The description of this
technique is out of scope in this Chapter and a lot of material can be found in the scientific
literature (Bjork, 1990; Borre et al., 2006; Kaplan & Hegarty, 2006).
Another noteworthy technique that is used in most of the commercial receivers to improve
the accuracy of the PVT computed by using the LS approach, is the so-called Kalman filter
(Anderson & Moore, 1979; Brown & Hwang, 1997; Kalman, 1960). By combining a system
model with the measurements, this algorithm is able to smooth the solution calculated by
the LS as well as to provide estimation of the user's position even when less than four
satellites are tracked (e.g. this can be done by using the modeled system only).
5.1 Examples using GPS and Galileo data
This section provides an example of the evaluation of user's position, presenting the results
obtained with the LS algorithm. Most of these results have been taken from (Rao et al. 2011).
Taking as an example the GPS satellite with PRN 30, Fig. 9 shows the comparison of the
pseudoranges estimation obtained implementing the common transmission time and the
common reception time methods. The blue marks represents the pseudorange computed by
considering the “common transmission time ”, while the reds correspond to observables
calculated fixing a unique time of reception. Though these two methods are conceptually
different, as expected, no significant differences can be noticed in the pseudoranges estimates,
that are substantially similar, but shifted in time due to the different computation instant.
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