Discussion

In this section we tried to distinguish the behaviour of the vertical position error during the two different phases of the approach already identified previously:

• Before the FAF: as explained, the FAF is reached 𝑇𝐹𝐴𝐹= 160 𝑖 after the beginning of the

VERTICAL POSITION ERROR STEP AT CONSTELLATION CHANGE

METERS

PERCENTAGE

APPCH: 2894 samples, min=-1.96 m, max=1.8m)

-8 -6 -4 -2 0 2 4 6 8

APPCH: 2894 samples, min=-4.37 m, max=4.32m)

• Final approach: this part takes place from the FAF to the end of the approach at 𝑡 = 270 𝑖.

The aim here is to highlight the quantity of constellation changes occurring during the final approach since these are the most critical for autoland functions due to the proximity to the ground.

Figure 38: Comparison of the histograms of the estimated mean of VPE over all simulation runs before and after the FAF in nominal configuration

The same figures as previously have been represented. Figure 38 presents the two histograms obtained by collecting the estimated mean of the Vertical Position Error during all simulations and by distinguishing the samples before and after the FAF. The previous histogram over the complete approaches is reminded as a reference. Since FAF is passed by the aircraft at 𝑡_𝐹𝐴𝐹 = 160 𝑖 to compute each value of the estimated VPE we used 𝑇𝐹𝐴𝐹× 𝑓𝑝𝑜𝑠 = 160 × 5 = 800 samples. For the histogram corresponding to the events occurring after the FAF we used for each estimate for the mean VPE (𝑇𝐴𝑃𝑃− 𝑇𝐹𝐴𝐹) × 𝑓𝑝𝑜𝑠 = 110 × 5 = 550 samples. As previously we thus obtained 𝑛𝑎𝑝𝑝× 𝑛𝑎𝑖𝑣= 433 × 16 = 6928 estimated mean VPE to build each histogram.

Minimum estimated mean

VPE

Maximum estimated mean

VPE

Complete approach -4.39 4.19

Before FAF -4.5 4.24

After FAF -4.22 4.1

Table 18: Minimum and maximum estimated mean VPE observed in nominal configuration

This figure shows that the different histograms obtained have Gaussian shapes. The maximum and minimum mean values observed which are gathered in the previous table are nearly the same which is quite logical since the position of the a/c in the approach has a limited impact on the mean VPE. We can however see here that the maximum estimated mean values of the VPE are observed before the FAF with 𝑚𝑖𝑛_𝑉𝑃𝐸= −4.5 𝑚 and 𝑚𝑎𝑥_𝑉𝑃𝐸 = 4.24 𝑚.

-6 -4 -2 0 2 4 6

0 0.005 0.01 0.015 0.02 0.025

METERS

PERCENTAGE

HISTOGRAM OF ESTIMATED MEAN OF VERTICAL POSITION ERROR

APPCH: MEAN VERT (min=-4.39 m, max=4.19 m) Before FAF: MEAN VERT (min=-4.5 m, max=4.24 m) After FAF: MEAN VERT (min=-4.22 m, max=4.1 m)

Figure 39: Comparison of the histograms of estimated standard deviation of VPE over all simulation runs before and after the FAF in nominal configuration

On the contrary, the results are slightly different when we take a look at the estimated standard deviation of the VPE represented in Figure 39. The minimum and maximum estimated standard deviations of VPE observed are gathered in the following table:

Minimum estimated std VPE

Maximum estimated std

VPE

Complete approach 0.02 1.21

Before FAF 0.02 1.52

After FAF 0.0 0.91

Table 19: Minimum and maximum estimated VPE standard deviation observed in nominal configuration

In fact, we have already seen that if we make no distinction between the first part of the approach and the final approach, the maximum observed standard deviation of VPE was about 1.21 𝑚 and we also observed that the standard deviation was lower than 0.4 𝑚 more than 95 % of the time. However, if we distinguish the standard deviation before and after the FAF, we can see that the obtained standard deviation histograms are different. On the one hand, before the FAF the observed standard deviation has a higher maximum of about 1.52 𝑚. On the other hand, after the FAF the observed standard deviation has a maximum value of about 0.91 𝑚 and the corresponding histogram is more concentrated in the lower VPE standard deviation magnitude. So, we can see that globally, the VPE standard deviation is lower during the Final approach than before the FAF.

-0.20 0 0.2 0.4 0.6 0.8 1 1.2

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

METERS

PERCENTAGE

HISTOGRAM OF ESTIMATED STANDARD DEVIATION OF VERTICAL POSITION ERROR OVER EACH APPROACH

APPCH: STD VERT (min=0.02 m, max=1.21 m) Before FAF: STD VERT (min=0.02 m, max=1.52 m) After FAF: STD VERT (min=0 m, max=0.91 m)

Figure 40: Comparison of the histograms of VPE steps at constellation change over all simulation runs before and after the FAF in nominal configuration

We then look at the distribution and the magnitude of the VPE steps due to a constellation change during the approach thanks to Figure 40. A total number of 𝑛_{𝑠𝑣𝑣𝑝𝑠} = 2894 constellation changes have been detected during our simulations resulting in VPE steps of minimum and maximum amplitude which are gathered in the following table:

Minimum VPE steps

Maximum VPE steps

Complete approach -1.96 1.8

Before FAF -1.92 1.77

After FAF -1.96 1.8

Table 20: Minimum and maximum estimated VPE steps observed in nominal configuration

The most interesting result is that we have observed the same number of VPE steps caused by constellation changes before and after the FAF. In fact, we observed 1447 steps before the FAF and 1447 steps after the FAF. IF we express the probability of occurrence of a step per approach we thus obtain:

𝑃𝑠𝑣𝑣𝑝𝑠_𝑏𝑣𝑓𝑜𝑣𝑣_𝐹𝐴𝐹 = 𝑃𝑠𝑣𝑣𝑝𝑠_𝑎𝑓𝑣𝑣𝑣_𝐹𝐴𝐹 = 1447

𝑛_𝑎𝑝𝑝× 𝑛_𝑎𝑖𝑣 ≈ 0.21

As we can see, the probability that a constellation change occurs during the approach is quite high.

In terms of magnitude, we can see that the distributions of the VPE steps magnitude have Gaussian shapes with the minimum and maximum observed values being around −1.9 𝑚 and 1.8 𝑚 equivalently before and after the FAF.

-2.50 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

0.02 0.04 0.06 0.08 0.1 0.12

VERTICAL POSITION ERROR STEP AT CONSTELLATION CHANGE

METERS

PERCENTAGE

APPCH: 2894 samples, min=-1.96 m, max=1.8m) Before FAF: 1447 samples, min=-1.92 m, max=1.77m) After FAF: 1447 samples, min=-1.96 m, max=1.8m)

Figure 41: Histogram of VPL steps at constellation change over all simulation runs before and after the FAF in nominal configuration

Figure 41 represents the steps induced by the recorded constellation changes on the RAIM vertical protection levels. The different minimum and maximum values of each histogram are gathered in the following table:

Minimum VPL steps

Maximum VPL steps

Complete approach -4.37 4.32

Before FAF -4.37 4.32

After FAF -3.43 4.3

Table 21: Minimum and maximum estimated VPL steps observed in nominal configuration

We can see in this distribution that the magnitudes of the RAIM vertical protection levels are nearly the same before and after the FAF. We observe steps having an absolute magnitude higher than 4.0 𝑚.

The algorithm proposed previously is intended to prevent such jumps in the horizontal and vertical position error as well as in the associated RAIM protection levels.