Extension to Account for Nominal Range Biases

Least Squares Residuals RAIM 67

These equations contain both probabilities 𝑃_𝑚𝑐 and 𝑃(𝑃_𝐸>𝑃𝑃). The following results are based on an example with given satellite coordinates from a total number of 6 satellites and a given user position. All possible values for 𝑃_𝑚𝑐 and 𝑃(𝑃_𝐸>𝑃𝑃) between 0 and 1 are covered.

Figure 5-5: HPL as Function of PMD and P(PE>PL)

The underlying integrity risk is 2E-7. Figure 5-5 shows the sensitivity of the results for HPL as function of the ratio between 𝑃𝑚𝑐 and 𝑃(𝑃𝐸>𝑃𝑃). Hereby, three different scenarios are depicted based on different assump-tions regarding pseudorange variances for the satellites (1 m, 10 m, 100 m). The red line represents the ap-proach of “mapping of 𝑃𝑃𝑒𝑇𝑇” based on 𝑃(𝑃_𝐸>𝑃𝑃)=1. Green line gives the ratio corresponding to a 𝑃_𝑚𝑐 of 0.35 that refers to the “mapping of threshold” approach based on 𝑃_𝑓𝑓 of 0.01 and a number of 6 satellites. The grey vertical line at the value of 8.08E-3 indicates the worst performance with the largest HPL which is identical for all three scenarios based on different pseudorange variances. The grey area indicates a not allowed range of values that is constrained through the 𝑃_𝑓𝑓 requirement: the threshold T in the test statistic domain is the mini-mum allowable bias still satisfying the 𝑃_𝑓𝑓 requirement. Thus, 𝑃𝑃𝑒𝑇𝑇 must not get below T. This defines a maxi-mum value for 𝑃𝑚𝑐 by setting 𝑃𝑃𝑒𝑇𝑇 equal to T. It is obvious that HPL performance depend on the allocation of the probabilities where the differences in the HPL results themselves strongly depend on the pseudorange variances. Based on this example, the conclusion can be drawn that the optimum results can be achieved when the threshold T is directly used to derive a PL in the position domain.

68 Integrity Algorithms

biases must not be neglected in an adequate fault-free error model. Nominal biases are assumed to exist even under nominal conditions due to antenna phase center variations, multipath or signal deformations for example.

In the following, it will be assumed that the nominal biases on the pseudoranges are bounded by 𝑃𝑐𝑐𝑚 and are assumed to be present on all satellite measurements respectively. For further details, it is referred to section 6.7.

The theory is orientated to [Martineau 2008].

A common (additive) bias on all pseudoranges would directly translate into the user receiver clock estimate and no impact on the position solution will be observed. However, biases could be present with different signs and magnitudes on the pseudoranges. Thus, the approach that is followed here is such that the worst case impact on the position solution is considered by using the norm of the corresponding elements of the projection matrix 𝑺.

Thus, the impact on the position can be expressed as:

𝑃𝐸=𝑃𝑐𝑐𝑚∙ �|𝑊1𝑖|

𝑁

𝑖=1

; 𝑃𝑁=𝑃𝑐𝑐𝑚∙ �|𝑊2𝑖|

𝑁

𝑖=1

; 𝑃𝑉=𝑃𝑐𝑐𝑚∙ �|𝑊3𝑖|

𝑁

𝑖=1

5.34 The presence of nominal biases on the pseudorange measurements will cause the fault-free 𝜒²-distribution to be non-central (as opposed to the case where no nominal biases are considered). Analogously to the RAIM approach without considering nominal biases, the non-centrality parameter is derived in the following.

From equation 5.24, the minimum bias that can be detected in the test statistic domain 𝜆 (with a given 𝑃_𝑚𝑐) can be expressed by the 𝑊𝑊𝑊𝐼 and the relation between the bias on every pseudorange 𝑃𝑖 normalized by the pseu-dorange variance 𝜎𝑖:

𝑊𝑊𝑊𝐼=𝜆=𝑃_𝑖²

𝜎_𝑖²(1− 𝑃_𝑖𝑖)

5.35 (1− 𝑃𝑖𝑖) is the projection of satellite i into detection space. It is assumed that the magnitude of the nominal biases 𝑃𝑐𝑐𝑚 is identical for all pseudoranges N. Now, assuming a nominal bias on all pseudoranges with random sign leads to the following expression:

𝑃𝑐𝑐𝑚,𝑐𝑑𝑡𝑑𝑐𝑡𝑖𝑐𝑐 𝑠𝑠𝑓𝑐𝑑,𝑐𝑐 𝑐𝑐𝑛𝑛𝑑𝑐𝑓𝑡𝑖𝑐𝑐2=𝑃𝑐𝑐𝑚2∙ �(1− 𝑃𝑖𝑖)

𝑁

𝑖=1

5.36

Least Squares Residuals RAIM 69

This expression takes into account the projection of the nominal biases of each satellite into detection space but without considering correlations between the satellites (𝑃𝑐𝑐𝑚,𝑐𝑑𝑡𝑑𝑐𝑡𝑖𝑐𝑐 𝑠𝑠𝑓𝑐𝑑,𝑐𝑐 𝑐𝑐𝑛𝑛𝑑𝑐𝑓𝑡𝑖𝑐𝑐). The nominal biases on the satellites might influence each other. Therefore, the following expression will be used to derive 𝑃𝑐𝑐𝑚,𝑐𝑑𝑡𝑑𝑐𝑡𝑖𝑐𝑐 𝑠𝑠𝑓𝑐𝑑 which describes the projection of the nominal biases into detection space taking into account their correlations:

𝑃𝑐𝑐𝑚,𝑐𝑑𝑡𝑑𝑐𝑡𝑖𝑐𝑐 𝑠𝑠𝑓𝑐𝑑2=𝑃𝑐𝑐𝑚2∙ � ���1− 𝑃𝑖𝑖��

𝑁

𝑖=1 𝑁

𝑖=1

5.37 For the projection, the absolute values (�1− 𝑃𝑖𝑖�) are used to represent the worst case respectively. The fault-free non-centrality parameter 𝜆(𝑃) of the 𝜒²-distribution is then derived as follows:

𝜆(𝑃) =𝑃𝑐𝑐𝑚,𝑐𝑑𝑡𝑑𝑐𝑡𝑖𝑐𝑐 𝑠𝑠𝑓𝑐𝑑2

𝑚𝑇𝑥(𝜎𝑖2)

5.38 Consequently, the threshold 𝑇_{𝑊𝑊𝑊𝑊}^′ is derived based on a normalized non-central 𝜒²-distribution with non-centrality parameter 𝜆(𝑃):

𝑃_𝑓𝑓= 1− 𝛼= 1− � 𝑒_𝜒²_{𝑊𝑊𝑊𝑊}�𝑥, 1,𝑁 −4,𝜆(𝑃)�𝑑𝑥

𝑇_{𝑊𝑊𝑊𝑊}

0

5.39 This leads to the threshold T’

𝑇^′=�𝑇^{𝑊𝑊𝑊𝑊}∙ 𝑚𝑇𝑥(𝜎_𝑖²) 𝑁 −4

5.40 For the derivation of the minimal detectable bias 𝑃𝑃𝑒𝑇𝑇^′ in the test statistic domain taking into account a re-quirement for 𝑃𝑚𝑐 and the presence of nominal biases, an additional bias on a single pseudorange is assumed (analogously to the bias-free RAIM approach). The non-centrality parameter 𝜆(𝑃𝑃) is derived by solving the following equation:

𝑃𝑚𝑐= � 𝑒_𝜒′2�𝑥, 1,𝑁 −4,𝜆(𝑃𝑃)�𝑑𝑥

𝑇𝑊𝑊𝑊𝑊

0

5.41

70 Integrity Algorithms

Figure 5-6 highlights the LSR RAIM approach taking into account nominal biases on the pseudo-ranges: the presence of nominal biases leads to a fault-free non-central 𝜒²-distribution (shown in blue). Analogously to the

“bias-free” LSR RAIM, the decision threshold T is set according to the 𝑃_𝑓𝑓 requirement. The faulty non-central 𝜒² -distribution (shown in red) is set according to a respective 𝑃_𝑚𝑐.

PFA

PMDT test statistic

position error

SLOPE

PL

Figure 5-6: LSR RAIM (accounting for nominal biases on pseudoranges)

Figure 5-7 shows the non-centrality parameter �𝜆(𝑃) and �𝜆(𝑃𝑃) of the 𝜒²-distribution as function of 𝑃𝑐𝑐𝑚. The example is based on a snapshot geometry based on 6 satellites assuming a pseudo-range noise of 1 m. The prob-abilities are set to 𝑃𝑚𝑐= 1𝐸 −3 and 𝑃𝑓𝑓= 1𝐸 −2. The horizontal line in blue indicates the level for the �𝜆(𝑃𝑃) without assuming nominal biases and therefore corresponds to √𝜆 in the LSR RAIM approach previously de-scribed. Based on this example, in can be seen for example that assuming nominal biases on the pseudoranges in the order of 1m would lead to an increase in the �𝜆(𝑃𝑃) parameter in the test statistic domain from 6 m to 8 m.

Least Squares Residuals RAIM 71

Figure 5-7: Non-centrality parameters 𝜆(𝑃) and 𝜆(𝑃𝑃) as function of 𝑃_𝑐𝑐𝑚

In order to derive the PL equation, equation 5.35 is solved for the parameter 𝑃𝑖 leading to the minimum pseudor-ange bias that can be detected with at least the specified 𝑃𝑚𝑐 in the “bias-free” case:

𝑃_𝑖=𝜎_𝑖 √𝜆

�1− 𝑃_𝑖𝑖

5.42 The smallest detectable bias on the pseudorange 𝑃 needs to be expressed as the sum of the nominal bias 𝑃𝑐𝑐𝑚

(with different signs) and an additive unknown bias part 𝑃𝑖. Now, equation 5.42 is adapted in order to account for nominal biases leading to the following equation:

(𝑃_𝑖±𝑃𝑐𝑐𝑚) =𝜎𝑖 �𝜆(𝑃𝑃)

�1− 𝑃𝑖𝑖

5.43 In the worst case, the nominal bias 𝑃_𝑐𝑐𝑚 is assumed being additive to the smallest detectable bias 𝑃_𝑖 (leading to the most conservative PLs):

𝑃𝑖=𝜎𝑖 �𝜆(𝑃𝑃)

�1− 𝑃𝑖𝑖+𝑃𝑐𝑐𝑚

5.44

72 Integrity Algorithms

In the following, the derivation is done for the horizontal case. However, it can be analogously derived also for the vertical position component. The HPL is computed by deriving the impact of the minimum bias 𝑃𝑖 in the position domain using the following equation:

𝐻𝑃𝑃𝑖=�𝑊_𝐸,𝑖² +𝑊_𝑁,𝑖² ∙ 𝑃𝑖

5.45 Combining equation 5.44 and 5.45 leads to

𝐻𝑃𝑃𝑖=�𝑊𝐸,𝑖2 +𝑊_𝑁,𝑖² ∙ �𝜎𝑖�𝜆(𝑃𝑃)

�1− 𝑃𝑖𝑖+𝑃𝑐𝑐𝑚�

5.46 Previous equation can also be written as:

𝐻𝑃𝑃_𝑖=�𝑊𝐸,𝑖2 +𝑊_𝑁,𝑖²

�1− 𝑃_𝑖𝑖 �𝜆(𝑃𝑃)∙ 𝜎_𝑖+�𝑊𝐸,𝑖2 +𝑊_𝑁,𝑖²

�1− 𝑃_𝑖𝑖 ∙ �1− 𝑃_𝑖𝑖∙ 𝑃_𝑐𝑐𝑚

5.47 Further arrangements lead to the following to equations:

𝐻𝑃𝑃𝑖=�𝑊_𝐸,𝑖² +𝑊_𝑁,𝑖²

1− 𝑃𝑖𝑖 ∙ ��𝜆(𝑃𝑃)∙ 𝜎𝑖+�1− 𝑃𝑖𝑖∙ 𝑃𝑐𝑐𝑚�

5.48 The first term from equation 5.48 is denoted as 𝑊𝑃𝑆𝑃𝐸𝐻,𝑖 hereafter to highlight the analogies of the slope factor derived in section 5.3.4. This is used to perform the mapping between the test statistic and the horizontal posi-tion domain:

𝐻𝑃𝑃𝑖=𝑊𝑃𝑆𝑃𝐸𝐻,𝑖∙ ��𝜆(𝑃𝑃)∙ 𝜎𝑖+�1− 𝑃𝑖𝑖∙ 𝑃𝑐𝑐𝑚�

5.49 The final HPL (and VPL respectively) is derived by taking the maximum out of the 𝐻𝑃𝑃𝑖:

𝐻𝑃𝑃=𝑚𝑇𝑥_𝑖�𝑊𝑃𝑆𝑃𝐸_𝐻,𝑖∙ �𝜎_𝑖�𝜆(𝑃𝑃) +�1− 𝑃_𝑖𝑖∙ 𝑃_𝑐𝑐𝑚��

5.50 𝑉𝑃𝑃=𝑚𝑇𝑥𝑖�𝑊𝑃𝑆𝑃𝐸𝑉,𝑖∙ �𝜎𝑖�𝜆(𝑃𝑃) +�1− 𝑃𝑖𝑖∙ 𝑃𝑐𝑐𝑚��

5.51

Novel Maritime RAIM 73

Im Dokument of Receiver Autonomous Integrity Monitoring (RAIM) for Maritime Operations (Seite 69-75)