# Integrity Analysis for GPS-Based Navigation of UAVs in Urban Environment
## 2 Integrity Monitoring
 \tag{2}\]  
Although there is no common agreement on the terminology describing the quality of navigation performance based on DOP coefficient values, DOP values are characterized in this study with respect to similar works in the literature [18; 19] and our test results as shown in Table 2.  
DOP coefficients can be calculated as follows:  
\[G=\left[ \begin{array}{cccc}-cos\theta_{1}sin\rho_{1}&-cos\theta_{1}cos\varphi _{1}&-sin\theta_{1}&1\\ -cos\theta_{2}sin\sigma_{2}&-cos\theta_{2}cos\varphi_{2}&-sin\theta_{2}&1\\ -cos\theta_{3}sin\sigma_{3}&-cos\theta_{3}cos\varphi_{3}&-sin\theta_{3}&1\\ -cos\theta_{4}sin\sigma_{4}&-cos\theta_{4}cos\varphi_{4}&-sin\theta_{4}&1 \end{array} \right] \tag{3}\]  
\[H=\left(G^{T}.G\right)^{-1} \tag{4}\]  
\[ \begin{split}\text{GDOP}=\sqrt{\text{H}_{11}\text{H}_{22}\text{H} _{33}\text{H}_{44}}\text{,}\quad\text{PDOP}=\sqrt{\text{H}_{11}\text{H}_{22} \text{H}_{33}}\text{,}\\ \text{HDOP}=\sqrt{\text{H}_{11}\text{H}_{22}}\text{,}\quad\text{VDOP}=\sqrt{ \text{H}_{33}}\text{,}\quad\text{TDOP}=\sqrt{\text{H}_{44}}\end{split}  \tag{5}\]  
where \(\theta_{(1,2,3,4)}\) and \(\varphi_{(1,2,3,4)}\) are the elevations and azimuths of the best four satellites for DOP calculations. DOP values multiply positioning error of the receiver and increase with the following anomalies:  
* Sub-optimal satellite geometry
* Satellite obscuration  
Satellite geometry has a direct impact on DOP coefficients. Even under normal conditions, because of the movement of satellites and vehicle with the antenna installed, geometry is continuously changing based on the number of visible satellites and their positions. Larger DOP values mean worse satellite geometry. The overall quality of satellite geometry also varies according to receiver locations on the Earth. While observations from the equatorial region give a high number of visible satellites and high-quality satellite geometry, a localization near the arctic circle can be very challenging based on satellite visibility and geometry [37]. Good and bad satellite geometry examples are illustrated in Figure 4.  
Satellite obscuration is another factor affecting DOP coefficients. In fact, obscuration acts as an elevation cut-off mask and has a direct effect on the number of visible satellites by corrupting user-satellite geometry. As a result, DOP coefficients are increased. Especially in urban environments, because of the high buildings and other obstructions, cut-off elevation may increase significantly. While high cut-off elevation affects satellite visibility, low satellite elevations can also cause multipath, which is beyond the scope of this paper. Thus, specifying the integrity limitations based on cut-off elevation is beneficial for detecting integrity risks caused by local error sources. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DOP Value** & **\textless{1}** & **1-2** & **2-5** & **5-10** & **10-20** & **20\textless{}** \\ \hline
**Rating** & Ideal & Excellent & Good & Moderate & Fair & Poor \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Meaning of dilution of precision (DOP) coefficient. In contrast to integrity monitoring systems aimed at aviation applications, such as one described in [27], urban environment sets different challenges that generally require a different approach for integrity monitoring. Due to smaller sizes of UAVs, the difference in flight pattern in urban areas in comparison to larger aircrafts and the complexity of the environment, one of the main challenges that need to be addressed, is related to integrity failures due to the obscuration pattern, and it can be expressed through the cut-off elevation imposed by the buildings. Figure 5 illustrates the typical urban environment and the physical impact of cut-off elevation. This challenge defines the focus of our work related to integrity analysis in urban environments through corresponding environmental parameters and observables. The integrity of the positioning and corresponding accuracy aspects directly depends on the distribution of position error.