# A Performance-Based Airspace Model for Unmanned Aircraft Systems Traffic Management
## 3 Urban Airspace Model
### Performance-Based Elementary Cell Dimensions  
The proposed performance-based urban airspace model is intended to enhance not only safety and efficiency but also ease UTM operator's interpretability by using CNS performance as the main driver in airspace structure design, determining spacing, sector dimensions and capacity altogether, while also supporting an intuitive visualisation. Therefore, as previously indicated, the elementary cell dimensions shall be parameterized as functions of the CNS performance for a given airspace region. The reference UAS equipage is assumed as follows:  
* Communication-datalink;
* Navigation-GNSS/INS [43];  
Figure 8: Overall SA/CA process for autonomous system. * Surveillance-Automatic Dependent Surveillance Broadcast (ADS-B)  
Within the scope of this study, the elementary cell dimensions are defined as a function of GNSS performance. GNSS is in fact not only the most widely employed source of absolute positioning for UAS operations but, as recognised by ICAO, GNSS is also the key element of all Communications, Navigation, and Surveillance/ATM (CNS/ATM) systems [44]. The elementary cell dimension is set as a 3\(\sigma\) (99.7%) bound on GNSS positioning errors, accounting for errors in pseudorange and unfavourable satellite geometry over the intended operational airspace region. The uncertainty in horizontal and vertical positioning is related to the pseudorange error and DOP as:  
\[\sigma_{\mathrm{x,y}}=3\cdot\sigma_{\mathrm{UEEE}}\cdot\mathrm{HDOP} \tag{12}\]  
\[\sigma_{\mathrm{z}}=3\cdot\sigma_{\mathrm{UEEE}}\cdot\mathrm{VDOP} \tag{13}\]  
where HDOP is the Horizontal Dilution of Precision; VDOP is the Vertical Dilution of Precision; \(\sigma_{\mathrm{UEEE}}\) is the User Equivalent Range Error (m) as described by Equation (2) The elementary cell dimensions are then set as:  
\[\mathrm{r_{x}}=\mathrm{r_{y}}=\sigma_{\mathrm{x,y}} \tag{14}\]  
\[\mathrm{r_{z}}=\sigma_{\mathrm{z}} \tag{15}\]  
In order to calculate the DOP factor, the line-of-sight vector between the receiver and the satellites in the Earth-Centered Earth-Fixed (ECEF) coordinate system needs to be determined. This is readily obtained from the ephemera of the GNSS satellites. The elevation (\(\theta_{\mathrm{mu}}\)) and azimuth (\(\varphi_{\mathrm{mu}}\)) of each satellite in the NED coordinate system is then:  
\[\theta_{\mathrm{mu}}=-\mathrm{arcsinu_{as,D}} \tag{16}\]  
\[\varphi_{\mathrm{mu}}=-\mathrm{arctan}\frac{\mathrm{u_{as,N}}}{\mathrm{u_{as, E}}} \tag{17}\]  
where \(\mathrm{u_{as,D}},\mathrm{u_{as,E}}\), \(\mathrm{u_{as,N}}\) are the Down, East, and North components of the line-of-sight unit vector respectively. The geometric matrix is a concatenation of each of the visible satellite elevations and azimuths:  
\[\mathrm{H_{G}^{n}}=\left[ \begin{array}{cccc}-\cos\theta_{\mathrm{mu}}^{ \mathrm{i}}\cos\varphi_{\mathrm{mu}}^{\mathrm{i}}&-\cos\theta_{\mathrm{mu}}^{ \mathrm{i}}\sin\varphi_{\mathrm{mu}}^{\mathrm{i}}&-\sin\varphi_{\mathrm{mu}}^ {\mathrm{i}}&1\\ -\cos\theta_{\mathrm{mu}}^{\mathrm{i+1}}\cos\varphi_{\mathrm{mu}}^{\mathrm{i+1 }}&-\cos\theta_{\mathrm{mu}}^{\mathrm{i+1}}\sin\varphi_{\mathrm{mu}}^{\mathrm{ i+1}}&-\sin\varphi_{\mathrm{mu}}^{\mathrm{i+1}}&1\\ \vdots&\vdots&\vdots&\vdots\\ -\cos\theta_{\mathrm{mu}}^{\mathrm{n}}\cos\varphi_{\mathrm{mu}}^{\mathrm{n}}& -\cos\theta_{\mathrm{mu}}^{\mathrm{n}}\sin\varphi_{\mathrm{mu}}^{\mathrm{n}}& -\sin\varphi_{\mathrm{mu}}^{\mathrm{n}}&1\end{array} \right] \tag{18}\]  
The Local-navigation-frame Cofactor Matrix is then:  
\[\Pi^{n}=\left(\mathrm{H_{G}^{n}}^{T}\mathrm{H_{G}^{n}}\right)^{-1}=\left[ \begin{array}{cccc}\mathrm{D_{11}}&\mathrm{D_{12}}&\mathrm{D_{13}}&\mathrm{ D_{14}}\\ \mathrm{D_{21}}&\mathrm{D_{22}}&\mathrm{D_{23}}&\mathrm{D_{24}}\\ \mathrm{D_{31}}&\mathrm{D_{32}}&\mathrm{D_{33}}&\mathrm{D_{34}}\\ \mathrm{D_{41}}&\mathrm{D_{42}}&\mathrm{D_{43}}&\mathrm{D_{44}}\end{array}