Global Positioning System Reference
In-Depth Information
The assumptions made for the derivation of the user integrity equation are summarized as
follows:
•
In a Faulty Free mode, the true SISE for a satellite is zero mean Gaussian distributed
with standard deviation SISA.
•
In general, a faulty satellite will be flagged as Don't Use.
•
For each instance in time, one satellite of those flagged as OK is considered to be faulty
but not detected (Faulty Mode). The distribution for the SISE of a faulty satellite is
Gaussian with an expectation value TH and a standard deviation SISMA.
Once the distribution of the error in the reference frame is known (Gaussian overbounding
distribution with SISA and SISMA respectively), the derivation of the associate integrity risk
is straightforward.
Therefore, the error distribution for the vertical (one dimensional Gaussian distribution) and
horizontal (Chi Squared distribution with two degree of freedom) cases needs to be derived,
and the corresponding integrity risk can be easily computed by analyzing the integral for
both distributions with respect to the given alert limit. Finally, the integrity risk at the alert
limits HAL (Horizontal) and VAL (vertical) are computed by adding the vertical and
horizontal contributions (Oehler et al., 2004).
P
(
L
,
LP
)
=
+
P
=
HMI
IntRisk V
,
IntRisk H
,
2
HAL
⎛
⎞
−
VAL
2
2
ξ
=−
1
erf
⎜
⎟
+
e
+
FF
⎜
⎟
2
σ
⎝
⎠
uV FF
,,
⎛
⎛
⎞ ⎛
⎞
⎞
(2)
⎛
⎞
⎛
⎞
N
VAL
+
μ
VAL
−
μ
1
∑
⎜
uV
,
uV
,
⎟
⎜
⎟ ⎜
⎟
+
P
1
−
erf
⎜
⎟
+
1
−
erf
⎜
⎟
+
fail sat
,
⎜
⎟
⎜
⎟
⎜
⎜
⎟ ⎜
⎟
⎟
2
2
σ
2
σ
j
⎝
⎝
⎠
⎠ ⎝
⎝
⎠
⎠
j
=
1
⎝
uV FM
,,
uV FM
,,
⎠
⎛
⎞
N
⎛
2
⎞
HAL
∑
2
+
P
⎜
1
−
χ
cdf
⎟
⎜
⎟
⎜
⎟
fail sat
,
⎜
2 ,
δ
,
H
⎟
2
FM
u
ξ
⎝
⎠
⎝
⎠
j
=
1
where N is the number of satellites used for the positioning algorithm.
The Integrity Risk computed by the user represents the probability of exceeding the
specified alert limits, since the system works according to the hypothesis described above.
The Integrity Risk guaranteed by Galileo is partially allocated to user computation and
partially to the system itself. This means that a proper design and implementation of the
system must guarantee that the system have a sufficiently low probability of being in a
condition in which the performance relevant assumption is no longer valid. Only this will
ensure that the true overall integrity risk is below the required limit, in accordance with the
specified level of service when this service is declared available by the integrity system.
2.5 HMI probability computation algorithm (HPCA)
In order to better understand the P
HMI
formula (Eq. 2) and all the elements contributing to its
design, it is necessary to show the main passages leading to the construction of that
equation. These passages could be collected into an algorithm leading to the HPCA
algorithm (HMI Probability Computation Algorithm) (Luongo et al., 2004).
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