Global Positioning System Reference
In-Depth Information
After the first iteration, we can see that almost all of the correction from the GPS
receiver has been placed into the clock bias and clock drift. As we proceed with fur-
ther iterations, the error will be placed in the clock, position, and velocity error.
After a few hundred iterations, the filter should stabilize (if the noise parameters
have been properly set and truncation/roundoff errors have been minimized). The
errors in the position and velocity will not greatly exceed the errors in the
pseudorange and pseudorange rate measurements.
9.2.4.6 Carrier Loop Aiding
As previously mentioned, aiding a phase lock loop with inertial velocity is quite dif-
ficult, due to the small GPS wavelength (20 cm). A simplified, linear continuous time
model for an aided carrier loop can be constructed in a manner very similar to that
used for the aided code loop. In Figure 9.11, the range delay
ρ
and related quantities
(i.e.,
, respectively. The
code loop filter
K
c
is replaced by the carrier loop filter (also a gain
K
in this simple
model), and the rate of change of the range delay
d
ρ
est
and
δρ
) are replaced by their counterparts
θ
,
θ
est
and
δθ
/
dt
INS
. The
resultant model for an aided carrier loop can be used to derive (9.6), expressed in
terms of Laplace (continuous time) transforms:
ρ
/
dt
INS
is replaced by
d
θ
[
]
[
]
()
(
)
()
(
)
()
INS
δ
Θ
s
=
ssK
+
Θ
s
−
ssK
+
Θ
s
(9.6)
θ
θ
INS
represents a carrier phase estimate constructed from the INS velocity fol-
lowing initialization. Note that
where
Θ
INS
(
s
) is simply a mathematical construct intro-
duced in the equation derivation: it is not calculated in the carrier phase aiding
process. The INS constructed carrier phase estimate can be expanded as:
Θ
()
()
()
INS
INS
Θ
s
=
Θ
s
+ δ
Θ
s
(9.7)
Substituting into (9.6), we see the aided tracking loop error is independent of
Θ
(
s
), the actual carrier phase history, and dependent only upon the INS error. (We
have neglected the effects of noise and clock error in starting with (9.6) to reach a
conclusion about the required INS velocity accuracy.)
[
]
()
(
)
()
INS
δ
Θ
s
=−
s
s
+
K
δ
Θ
s
(9.8)
θ
But
δΘ
INS
(
s
) can be related to the satellite LOS component of INS velocity error
using:
()
()
δ
Θ
INS
s
=
uv
T
δ
INS
s
s
(9.9)
Finally, we can express the carrier phase error of an aided loop in terms of the INS
velocity error:
[
]
()
(
)
()
TI
S
δ
Θ
s
=−
1
s
+
K
uv
δ
s
(9.10)
θ
The carrier phase error in steady state, determined by setting
s
to 0 in (9.10), is
the LOS INS velocity error component divided by
K
θ
. Equivalently, the aided carrier
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