# A Performance-Based Airspace Model for Unmanned Aircraft Systems Traffic Management
## 3 Urban Airspace Model
### Airspace Sectorisation  
A sectorization scheme based on the previously defined elementary cells is proposed here, which optimally balances computational complexity and communication overheads.  
A UAS operation spanning multiple sectors requires a handover to be performed at each sector boundary. The number of cells allocated to a given sector is determined so as to optimize two key factors: the communication overhead required to perform all the required handovers for all involved traffic (which increase proportionally with the density of sectors), and the computational complexity associated with computing an avoidance volume and avoidance trajectory (which increases proportionally with sector dimensions).  
The communication overhead is parameterized as a dimensionless factor which is a function of the time required to perform a 'handshake' or to establish a link between the UAS and the UTM operator, and the average time spent in each sector:  
\[\mathrm{OH}=\ t_{\mathrm{HS}}\Bigg{(}\frac{\overline{\mathrm{v}}_{\mathrm{x} \mathrm{y}}}{\mathrm{d}_{\mathrm{x}\mathrm{y}}}+\frac{\overline{\mathrm{v}}_{ \mathrm{z}}}{\mathrm{d}_{\mathrm{z}}}\Bigg{)} \tag{22}\]  
where \(\ t_{\mathrm{HS}}\) is the handshake time (s); \(\overline{\mathrm{v}}_{\mathrm{x}\mathrm{y}}\) is the average horizontal velocity of the vehicle over a sector (m/s); \(\overline{\mathrm{v}}_{\mathrm{z}}\) is the average vertical velocity of the vehicle (m/s); \(\mathrm{d}_{\mathrm{x}\mathrm{y}}\) is the average horizontal distance covered in a sector (m); \(\mathrm{d}_{\mathrm{z}}\) is the average horizontal distance covered in a sector (m).  
Figure 9: UAS protection volume (**left**) and relative occupancy grid (**right**).  
Equation (22) essentially represents the proportion of time spent in establishing a handover communication link between sectors. The sector horizontal and vertical dimensions are by definition set as multiples of the elementary cell dimensions. Thus, \(\mathrm{d}_{x,y}\) and \(\mathrm{d}_{z}\) are calculated from the multiplication of elementary cell numbers (\(\mathrm{k}_{x,y},\ \mathrm{k}_{\mathrm{z}}\)) and elementary cell dimension (\(\mathrm{r}_{x,y},\ \mathrm{r}_{z}\)):  
\[\mathrm{d}_{x,y}=\ \mathrm{k}_{x,y}\cdot\mathrm{r}_{x,y} \tag{23}\]  
\[\mathrm{d}_{z}=\ \mathrm{k}_{z}\cdot\mathrm{r}_{z} \tag{24}\]  
where:  
\[\mathrm{k}_{z}=\frac{\mathrm{HDOP}}{\mathrm{VDOP}}\mathrm{k}_{x,y}\]  
The computational complexity is approximated as a factor that captures the proportion of time spent in generating avoidance volumes and avoidance trajectories as well as the maximum potential number of aircraft simultaneously in the sector, which as a worst case can be assumed equal to the number of elementary cells in each sector:  
\[\mathrm{CP}=\ \frac{\mathrm{CD}}{\mathrm{CoP}}=\frac{(\mathrm{n}-1)[\mathrm{t}_{ \mathrm{AVG}}+\mathrm{t}_{\mathrm{ATG}}]}{\mathrm{CoP}} \tag{25}\]  
where CD is the computational demand (s); CoP is the available computing capacity (s); \(\mathrm{t}_{\mathrm{AVG}}\) is time to generate avoidance volume; \(\mathrm{t}_{\mathrm{ATG}}\) is time to generate trajectory; \(\mathrm{n}\) is the number of elementary cells in a sector.  
\[\mathrm{n}\ =\ \mathrm{k}_{x,y}^{2}\cdot\mathrm{k}_{z} \tag{26}\]