Global Positioning System Reference
In-Depth Information
tem time. Corrections are generated by the ground-control/monitoring network
periodically to correct for this offset. These corrections are available in the naviga-
tion message and are applied within the receiver to obtain the actual satellite
transmitted frequency. Hence,
f
=+
0
f
∆
f
(2.39)
Tj
TJ
where
f
0
is the nominal transmitted satellite frequency (i.e., L1), and
∆
f
Tj
is the cor-
rection determined from the navigation message update.
The measured estimate of the received signal frequency is denoted
f
j
for the sig-
nal from the
j
th satellite. These measured values are in error and differ from the
f
Rj
values by a frequency bias offset. This offset can be related to the drift rate
t
u
of the
user clock relative to GPS system time. The value
t
u
has the units seconds/second and
essentially gives the rate at which the user's clock is running fast or slow relative to
GPS system time. The clock drift error,
f
j
, and
f
Rj
,
are related by the formula
(
)
&
f
=
f
1
+
t
(2.40)
u
Rj
j
where
t
u
is considered positive if the user clock is running fast. Substitution of (2.40)
into (2.38), after algebraic manipulation, yields
(
)
&
cf
−
f
cf t
f
j
Tj
j
u
&
+⋅
va ua
=⋅
−
j
j
j
f
Tj
Tj
Expanding the dot products in terms of the vector components yields
(
)
&
cf
−
f
cf t
f
j
Tj
u
j
+
va
+
va
+
va
=
xa
&
+
ya
&
+
&
za
−
(2.41)
xj
xj
yj
yj
zj
zj
u
xj
u
yj
u
zj
f
Tj
Tj
where
v
j
=
uuu
. All of the variables on the
left side of (2.41) are either calculated or derived from measured values. The compo-
nents of
a
j
are obtained during the solution for the user location (which is assumed to
precede the velocity computation). The components of
v
j
are determined from the
ephemeris data and the satellite orbital model. The
f
Tj
can be estimated using (2.39)
and the frequency corrections derived from the navigation updates. (This correction,
however, is usually negligible, and
f
Tj
can normally be replaced by
f
0
.) The
f
j
can be
expressed in terms of receiver measurements of delta range (see Chapter 5 for a more
detailed description of receiver processing). To simplify (2.41), we introduce the
new variable
d
j
, defined by
(
(
v
xj
,
v
yj
,
v
zj
),
a
j
=
(
a
xj
,
a
yj
,
a
zj
), and
u
&
=
(
xyz
&
,
&
,
&
)
)
cf
−
f
j
Tj
d
+
va
+
va
+
va
(2.42)
j
xj
xj
yj
yj
zj
zj
=
f
Tj
The term
f
j
/
f
Tj
on the right side in (2.41) is numerically very close to 1, typically
within several parts per million. Little error results by setting this ratio to 1. With
these simplifications, (2.41) can be rewritten as
Search WWH ::
Custom Search