# Integrity Analysis for GPS-Based Navigation of UAVs in Urban Environment
## 3 Simulation
### Simulation Results  
During the statistical analysis of the first simulation results, mean values, 67 th percentiles, and 95 th percentiles of absolute position error of all simulated cases were calculated as presented in Table 4. As expected, position errors steadily increase with increasing DOP coefficients. Considering assumed alert limits from Section 2 (10 m horizontally and 25 m vertically) one can see from Table 4 that position errors are within the alert limits for the first four test cases. More specifically, the 95 th percentile of the horizontal error is within alert limits when HDOP \(\leq\) 3.13, and for the vertical error, this is also true when VDOP \(\leq\) 5.13. In cases 5 and 6, navigation based on GPS only cannot be considered safe both horizontally and vertically. For the purposes of integrity analysis, 95 th percentile error cannot provide enough confidence to meet assumed integrity requirements. According to the assumed integrity requirements, the expected rate of position error measurements falling outside of the nominal operation zone in the Stanford-ESA Diagram should not exceed \(1-2\times 10^{-7}\)/h. Integrity failure occurs when PE \(>\) PL \(>\) AL during more than TTA, which is specified as 10 s. For generality and reproducibility of the results, achievable HPL and VPL are calculated based on UERE of the employed receiver, which represents the condition where HDOP and VDOP equal 1. In this case, derivation of achievable protection levels for different test configurations is a straightforward task. Achievable protection levels are calculated from position error and DOP measurements of the first four test cases since PE is higher than AL in the last two cases. For more sensitive analysis, sigma correction following approach [40] is also applied while obtaining the pdf of the position errors. The 99.99998 th percentile of UERE is accepted as a protection level to guarantee that integrity risk is below \(1-2\times 10^{-7}\)/h. The alert limits we have specified in Section 2 are based on assumptions about the urban environment, such as buildings' heights and street sizes, which may be changed, for instance, for applications with more strict requirements to positioning errors. If this is a case, requirements to acceptable HDOP and VDOP will be tightened. Additionally, since HPL and VPL are specified based on the employed receiver model performance, protection levels can slightly change for another receiver \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Simulation 1**} & \multicolumn{3}{c}{**Simulation 2**} \\ \hline
**Test No.** & **GDOP** & **Cut-off Elevation** & **Number of Visible Satellites** & **Test No.** & **Cut-off Elevation** & **Number of Visible Satellites** \\ \hline
1 & 1.69 & & & 1 & \(5^{\circ}\) & \\ \hline
2 & 2.90 & & & 2 & \(10^{\circ}\) & \\ \hline
3 & 4.85 & \(5^{\circ}\) & Fixed - 4 satellites only & 3 & \(15^{\circ}\) & Variable, depending on cut-off elevation \\ \hline
4 & 7.09 & & & 4 & \(20^{\circ}\) & \\ \hline
5 & 13.56 & & & 5 & \(25^{\circ}\) & \\ \hline
6 & 31.96 & & & 6 & \(30^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation scenarios. model based on its UERE. Calculated UERE based on Equation (1) and its 95 th, 99 th and 99.99998 th percentile values after sigma correction are presented in Table 5. Protection levels with various DOP values can be calculated according to Equation (6). \[\text{HPL}=\text{HDOP-UERE}_{99.99998},\text{ VPL}=\text{VDOP-UERE}_{99.99998}, \tag{6}\]  
where UERE\({}_{99.99998}\) is the 99.99998 th percentile of UERE. After specifying base values of HPL and VPL considering HDOP = VDOP = 1, the impact of cut-off elevation on DOP coefficients should be investigated to derive achievable protection levels under various cut-off elevation levels. Table 6 shows the distribution of DOP coefficients for different cut-off elevation levels. It can be seen that for cut-off elevations up to 5\({}^{\circ}\) HDOP is less than 1 with 95.08% probability.