Global Positioning System Reference
In-Depth Information
The following day, John repeats the experiment, but he measures
t
as the difference between
the arrival time read on the school clock and the leaving time, read on his watch. John
realizes that the estimated distance is significantly different from that estimated the
previous day. Likely, his watch and the clock at school are not synchronized. In this case,
the measured time interval can be written as follow:
tt
δ
= +
(2)
Equation (2) takes into account the bias δ
t
between John's watch and the school clock.
Considering this term, John understands that this reflects to an error δ
x
on the estimated
distance.
xvtvt t x
δ
=⋅=⋅+ =+
(
δ
)
(3)
At this point, John wants to compare the result with one of his friends. He asks Alice to do
the same measurement from her house, since John knows that his house is exactly
500 m
away from hers. Before the measurements, Alice and John synchronize their watches.
Referring the measurements taken by John and Alice to the subscripts
J
and
A
, equation (3)
becomes:
⎧
xx xvt
=− = −
δ
(
δ
t
)
⎨
J
J
J
J
J
(4)
xx xvt
=− = −
δ
(
δ
t
)
⎩
AA
AAA
where:
x
and
x
are the distances estimated by John and Alice, respectively;
•
J
A
•
x
and
x
are the unknown distances John and Alice want to measure;
J
A
t
and
A
t
are the time intervals measured by John and Alice;
•
J
•
v
and
v
are the average speeds of John and Alice read on their speedometers;
•
δ is the unknown bias between Alice and John's watches and the school clock.
Recalling that Alice's house is
500 m
away from John's, the previous system of equations can
be rewritten as:
J
⎧
x
=
v
(
t
−
δ
t
)
⎨
J
J
J
(5)
x
+= −
500
v
(
t
δ
t
)
⎩
J
A
A
This new system has two equations and two unknowns:
x
and δ . In few steps, John can
finally compute the distance between his house and the school, realizing that he obtains the
same result of the first experiment. The conclusion of this simple example is that in 1
dimension, if the clocks used to measure time intervals are not synchronized, we need an
additional equation to solve the problem.
Bringing the concept to a three-dimensional space, it is easy to understand that we need four
equations to solve the problem and determine the unknown user position respect to a
reference system. This is the case of Global Navigation Satellite System (GNSS) receivers.
Referring to the geometry sketched in Fig. 1, there are satellites in view broadcasting
ranging signals, while a user on the Earth wants to estimate his unknown coordinates
(
x
u
,y
u
,z
u
). The satellites continuously transmit their positions (i.e. (
x
k
,y
k
,z
k
) considering the
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