Global Positioning System Reference
In-Depth Information
atmosphere propagation or the major part of clock drifts. This new differential mode is
also susceptible to increasing the positioning accuracy.
Let us deal briefly with the mathematics of the first approach based on clock bias analysis.
The method is based on the use of the clock bias coordinates. As described above, once one
has carried out four receiver location computation (one for each repeater), a new vector is
available, where the ct
i
are the calculated fourth coordinates, the ct
r
(i) are the real clock bias
of the receiver at each transmission times and the d
i
the distances separating the repeaters
from the receiver.
ct
ct
()
()
()
()
t
+
d
⎡ ⎤⎡
⎤
1
r
r
1
1
⎢ ⎥⎢
⎥
ct
ct
t
d
+
⎢ ⎥⎢
2
2
2
⎥
=
⎢ ⎥⎢
(1)
⎥
ct
ct
t
+
d
3
r
r
3
3
⎢ ⎥⎢
⎥
ct
ct
t
+
d
⎣ ⎦⎣
⎦
4
4
4
The unknown variables are now the d
i
, but the problem appears to be the real clock bias of
the receiver which is naturally not a constant. Thus, in (1), one has not only the four d
i
unknowns, but also the four clock biases. The technique consists indeed in estimating the
clock bias difference between instant t
2
and instant t
1
by the way of the clock drift
computation carried out through Doppler measurements by the receiver. Thus, the idea is to
consider that:
j
+
∑
ct
()
t
=
ct
()
t
cdt
(2)
r
j
ri
k
ki
=+
1
Where cdt
k
is the clock bias rate (called the clock drift) at time t
k
. The various ct
r
(t
i
) of (1) are
now reduced to a single unknown, ct
r
(t
1
). In addition, one knows that the four distances d
i
are characterised by only three spatial coordinates, x, y and z of the receiver once the
coordinates of the repeaters are known: this is a system requirement to provide the receiver
with these coordinates. The indoor location computation is then carried out typically
through hyperboloid intersection, as soon as the receiver is able to determine which repeater
is transmitting at any given time. This is achieved through synchronisation which is made
possible since the signals transmitted by all the repeaters are identical (thus, there is just the
need for an initial calibration of the wire delays between the signal generator, or the outdoor
antenna, and the repeaters).
On the other hand, the need to estimate the clock drift is somehow a constraint since the
final performances will greatly depend on the quality of the receiver clock. Thus, another
approach was proposed, based simply on classical measurements carried out by all current
receivers: the raw pseudo-ranges. When one draws the difference of pseudo-ranges from
one instant to the next in a repeater like system, the curve of figure 12 is obtained (note that
in this example only three repeaters are deployed, leading to a 2D positioning).
Clear skips, called ”transitions” in the figure, can be seen: they correspond to the difference
of distances, d
j
-d
i
, that characterises the increase or decrease of the distance from repeaters
to the receiver when the transmitted signal switches from repeater i to repeater j. It is
positive when the distance increases, and negative otherwise. Note also that two additional
phenomena are present: 1/ a slow constant increase in the equilibrium value (which
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