Global Positioning System Reference
In-Depth Information
specific operation. For example, for LPV-200, the whole Integrity Risk is allocated to the
vertical domain, since this is the most demanding requirement.
4.3 RAIM tst statistic
It is not possible to obtain a direct measurement of the position error. Therefore, the overall
consistency of the solution has to be investigated (Walter & Enge, 1995). As long as there are
more than four measurements, the system is overdetermined and cannot be solved
accurately. This is why a least squares solution is performed in the first place. Since all of the
conditions cannot be met realistically and exactly, there is always an error residual to the fit.
Therefore, we need to be able to estimate the fit and assume that, if there is a good fit, the
position error is most likely small.
An estimate of the ranging errors from the least squares fit and the basic measurement
equation is given by:
⋅ =− ⋅
(26)
ε = −⋅
y
Gx
=−⋅
(
I GK y
)
(
I P y
)
wls
wls
where:
T
−
1
T
(27)
PGKGG WG
= ⋅ =
(
⋅
⋅
)
⋅
G W
⋅
From these error estimates it is possible to define a scalar measure, defined as the Weighted
Sum of the Squared Errors (WSSE):
T
T
−⋅
(28)
WSSE
=
ε
⋅
W
⋅
ε
=−⋅
[(
I
P
)
y
]
⋅
WI
⋅
[(
Py
)
]
wls
wls
which is equivalent to:
T
⋅ −⋅
(29)
WSSE
=
y
⋅
WIPy
(
)
The square root of WSSE plays the role of the basic observable, because it yields a linear
relationship between a satellite bias error and the associated induced test statistic. The test
statistic can be defined in both the horizontal and vertical planes.
Typically, a certain threshold, which depends on the required probability of false alarm, is
selected. If the statistic exceeds that threshold, then the position fix is assumed to be unsafe.
On the other hand, if the statistic is below the threshold, then the position fix is assumed to
be valid.
The statistic-vertical error plane is thus broken up into four regions consisting of: normal
operation points, missed detections, successful detections and false alarms. Ideally, there
would never be any missed detections or false alarms. In reality, a certain number of missed
detections and false alarms are allowed, based on the P
md
and P
fa
requirements, respectively.
5. Multisystem integrity
With the advent of Galileo, users will be provided with multiple signals coming from
different satellite systems. This will improve position accuracy, because the number of
satellites in view per user will be almost doubled. Moreover, the higher measurements
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