Global Positioning System Reference
In-Depth Information
5. Position, velocity and time (PVT) computation
This section completes the chapter and deals with the estimation of the user's PVT that
comes after the measurement of a set of pseudoranges, for at least four satellites in view.
In order to have an accurate estimate of the user's position, the receiver has to consider
additional error sources that typically affect the measured pseudorange and that have to be
compensated. These sources include atmospheric effects (e.g. ionosphere and troposhere,
that generate a delay in the signal broadcast by the satellite) and other kinds of noise related
to the presence of multipath and interference.
A valid PVT can be estimated after the receiver retrives the satellites' positions, (i.e.:
x
,
y
,
z
as stated in equation (6)) from the navigation message. To compute the satellite
position, the receiver needs the ephemeris and the time of transmission, which is usually
referred to the beginning of the subframes. All the information the receiver needs is
embedded in the navigation message. The time of transmission can be read every 6 seconds
at the beginning of a subframe in a specific word that corresponds to the HOW. From the
HOW the receiver retrieves a truncated version of the absolute GPS time (TOW). This
number is referred to as Z-count. The Z-count is the number of seconds passed since the last
GPS week rollover in units of 1.5s. The truncated Z-count in the HOW corresponds to the
time of transmission of the next navigation data subframe. To get the time of transmission of
the current subframe, the Z-count should be multiplied by 6 and 6s should be subtracted
from the result (Borre et al. 2006).
Therefore, if we assume to perform the pseudorange estimation at the beginning of a new
subframe, the time of transmission will be exactly equal to the value reported in the HOW of
the previous subframe. Otherwise, if we implement the computation of the pseudorange in
a different instant, we have to count the time elapsed between the beginning of the
subframe and that instant. The way the time of transmission is computed represents the
main difference between the two aforementioned methods (i.e. “common transmission
time” and “common reception time”). According to the first method, all the satellites
transmit the signal at the same time and, if we assume to calculate the pseudorange at the
beginning of a subframe, this correspond to the TOW. On the contrary, if we consider the
second approach, we have to keep in mind a different time of transmission for each satellite.
Practically speaking, we have to sum up the TOW with the
δ
i
delay that elapsed from the
starting point of the subframe and
u
t
. Since every satellite has a different distance with
respect to the Earth, it follows that the δ
i
delay will vary according to the satellite under
consideration.
When four satellite have been correctly tracked, the full set of equations can be rewritten
after having removed the satellite offset and atmospheric effects. According to the “common
transmission time”, the equations can be stated as in equation (6):
⎧
2
2
2
(
xx
−
)
+
(
yy
−
)
+
(
zz
−
)
=
ρ
+ ⋅
cb
Δ
⎪
⎪
1
u
1
u
1
u
1
2
2
2
(
xx
−
)
+ −
(
yy
)
+ −
(
zz
)
=
ρ
+⋅ +⋅
cbc
Δδ
⎨
⎪
2
u
2
u
2
u
1
2
(18)
2
2
2
(
xx
−
)
+− +−
(
yy
)
(
zz
)
=
ρ
+⋅
cbc
Δδ
+⋅
3
u
3
u
3
u
1
3
⎪
⎪
2
2
2
(
xx
−
)
+ −
(
yy
)
+ −
(
zz
)
=
ρ
+⋅ +⋅
cbc
Δδ
⎩
4
u
4
u
4
u
1
4
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