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3.3 IMN based SIS integrity assessment

3.3.2 Single Difference Squared L2 norm

Performing the Single Difference between two reduced pseudorange referring to two different satellites observed by the n-th RS it is possible to observe how all the biases introduced by the RS itself can be removed.

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

,

,

i j i j i j

n n n n n n n

i j i j

n n n

k k k c t k k c t k k

k k k

ρ ρ ρ δ ζ δ ζ

ζ ζ ζ

∆ = ∆ − ∆ = + − − =

= − = (26)

Evaluate the single difference on two reduced pseudorange corresponds to perform the single difference among the pseudorange residuals associated to them. It is possible to build for each RS and for each satellite a vector ξin

[ ]

k whose elements are the single differences of the residual pseudoranges by using the n-th RS and the i-th satellite as pivot:

[ ] (

,1

[ ]

,2

[ ]

, sat

[ ] )

T

i i i i N

n n n

n k k k k

ξ = ζ ζ ⋯ ζ (27)

In case of fault on one or more satellites, all the singe differences involving that satellite will be affected by that event. In [12], authors evaluated the squared L2 norm of ξin

[ ]

k for all

the satellite in view. Authors compared the maximum of those evaluated values with a threshold; if that maximum is over the threshold, the corresponding satellite is labelled as faulty, and the process is repeated excluding it. This process is repeated until there are no more satellites to be checked or the maximum value is within the threshold. Figure 13 depicts a

38 schematic of the algorithm that is able to exclude more than one faulty satellite. The most important part is the threshold tuning. The procedure followed is based on the Neyman-Pearson criterion. According to this approach, the threshold is evaluated by imposing the desired false alarm probability that becomes a system requirement. The miss detection probability is then evaluated as a performance indicator. Let the indicator y be the squared L2 norm of the single ni difference vector i

ξn; for sake of compactness the time dependency is neglected.

( )

i T i

i

n n n

y = ξ ξ (28)

As shown in the previous paragraph, the pseudorange residual observed by the n-th Rs for the i-th satellite is a Gaussian distributed random variable with standard deviation σi n2, and expectation µ βni

( )

where β is the effective satellite position error that would have produced an error on the estimated receiver position of equal entity. If no fault occurs, such a random variable can be considered as zero mean. The single difference residuals, can be then considered again as Gaussian distributed with variance σi n2, +σ2j n, and a mean value respectively equal to zero in the fault free scenario, and equal to µ βin

( )

i if the i-th is affected by a fault and the j-th is healthy and, more generally, equal toµ βni

( )

i µ βnj

( )

j in the case of both faulty satellites.

Figure 13. Single Different Squared L2 Norm SIS integrity assessment algorithm

39 If the elements of the vector i

ξn were Gaussian distributed independent and identically distributed random variables, y would follow the chi square distribution with a number of ni degrees of freedom equal to the number of elements that would have been summed (satellites number excluding the pivot one). However, being obtained by means of differencing, the elements of i

ξn are correlated, thus means that y follows the generalized chi square ni distribution with Ny =Nsat − degrees of freedom. The distribution is centred or not centred 1 depending on whether the SIS is healthy or not. According to Neyman Pearson criterion the exclusion threshold γ can be evaluated as:

( )

2 ,

1 1

G Ny fa

Dχ P

γ = − (29)

Where:

( )

2 ,

1

G N y

Dχ • is the generalized chi square inverse cumulative distribution with N degrees of y freedom

Pfa is the desired false alarm probability

As a consequence, the miss detection probability will be retrieved as:

( )

2 ,

,

G Ny

nc

Pmd =Dχ γ λ (30)

Where:

( )

2 , G Ny

Dχnc • is the generalized chi square non centred cumulative distribution with N degrees y of freedom

λ is the parameter of non centrality

For the assessment of the performance of the of this algorithm, a GNSS simulator developed by RadioLabs in the framework of the ESA Artes-20 3InSat and H2020 ERSAT projects has been employed. Data have been provided by Sogei GRDNet (GNSS R&D

40 Network) [68]. According to [12], the single difference of residuals pseudorange in nominal conditions have a trend similar to the ones depicted in Figure 14. Data shown in Figure 14 leads to y indicators shown in Figure 15. In order to verify the algorithm, 3 scenarios are taken into ni account:

i. absence of SIS fault (i.e. nominal conditions);

ii. injection of fitting errors in the pseudoranges of a single satellite as due to wrong satellite clock parameters broadcasted by the satellite;

iii. injection of white noise to all pseudoranges of one RS.

Data used in the i) test are real data GPS only measurement acquired on field. In that occasion all the satellites were healthy. As shown in Figure 16, the algorithm correctly labelled as healthy all the satellite in view. To emulate a satellite fault used in case ii), an adulterated clock offset bias parameter for PRN23 satellite has been introduced in the ephemeris file. In this way, the receiver has been forced to estimate the satellite clock offset by means of a wrong second order polynomial. In order to verify the correct identification, a fault free window of about 1 hour has been left in the centre of the observation.

Figure 14. Single Difference pseudorange residuals trend in nominal conditions

3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96

x 105 -3

-2 -1 0 1 2 3 4 5

GPS time [s]

SD on pseudorange [m]

SD on pseudorange in nominal conditions

PRN 10 vs PRN 7 PRN 10 vs PRN 9

41

Figure 15. y indicator in nominal conditions

Figure 16. GPS constellation status in nominal conditions

Finally, in the case of scenario iii), a white Gaussian noise with standard deviation of 2.5 meters has been added to all the pseudorange measured by the RS in the observation window. As shown in Figure 18, all the satellites have been identified as faulty. In this scenario it becomes clear how is important to identify malfunctioning RS in order to exclude them from the SIS assessment procedure.

3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96

x 105 0

10 20 30 40 50 60 70

GPS time [s]

y [m2]

y indicator in nominal conditions

PRN 7 PRN 9

3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96

x 105 0

5 10 15 20 25 30

GPS time [s]

PRN

Constellation Status (green healthy, red faulty)

42

Figure 17. GPS constellation status with one injected fault

Figure 18. GPS constellation status in case of malfunctioning RS