Global Positioning System Reference
In-Depth Information
RAIM algorithms require a minimum of five visible satellites in order to detect
the presence of an unacceptably large position error for a given mode of flight. If a
failure is detected, the pilot receives a warning flag in the cockpit that indicates that
GPS should not be used for navigation. Certified GPS receivers that contain FDE, an
extension of RAIM that uses a minimum of six visible satellites, can not only detect
the faulty satellite, but can exclude it from the navigation solution so the operation
can continue without interruption.
The inputs to the RAIM algorithm are the standard deviation of the measure-
ment noise, the measurement geometry, and the maximum allowable probabilities
for a false alert and a missed detection. The output of the algorithm is the
horizontal
protection level
(HPL), which is the radius of a circle, centered at the true aircraft
position that is assured to contain the indicated horizontal position with the given
probability of false alert and missed detection discussed next. If the aircraft is con-
ducting phases of flight that require vertical guidance, a
vertical protection level
(VPL) is output as well; however, VPL is generally associated with differen-
tial-based systems. This section concentrates on the generation of HPL using a
snap-
shot
RAIM algorithm that has been developed in support of RTCA SC-159 [43].
The linearized GPS measurement equation is given as
yHx
=+
(7.69)
where
x
is the 4
1 vector whose elements are incremental deviations from the nom-
inal state about which the linearization takes place. The first three elements are the
east, north, and up position components, and the fourth element is the receiver
clock bias.
y
is the
n
×
1 vector whose elements are the differences between the noisy
measured pseudoranges and the predicted ones based on the nominal position and
clock bias (i.e., the linearization point). The value
n
is the number of visible satellites
(number of measurements).
H
is the
n
×
4 linear connection matrix between
x
and
y
.
It consists of three columns of direction cosines and a fourth column containing the
value 1, which corresponds to the receiver clock state.
×
1 measurement
error vector. It may contain both random and deterministic (bias) terms.
GPS RAIM is based on the self-consistency of measurements, where the number
of measurements,
n
, is greater than or equal to 5. One measure of consistency is to
work out the least squares estimate for
x
, substitute it into the right-hand side of
(7.69), and then compare the result with the empirical measurements in
y
. The dif-
ference between them is called the range residual vector,
w
. In mathematical terms,
is the
n
×
(
)
−
1
$
x HHHy
=
T
T
(least square estimate)
LS
$
$
y x
=
LS
LS
(7.70)
(
)
(
)
−
1
−
1
$
y yHHHHyI HHHHy
wy
=−
=−
T
T
=
−
T
T
LS
n
(
)
(
)
−
1
−
1
(
)
T
T
T
T
=−
I HHHHH
x
+= −
I
H H H H
n
n
Since
is not known to the user aircraft, (7.70) is only used in simulations.
Let
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