# Airspace Geofencing and Flight Planning for Low-Altitude, Urban, Small Unmanned Aircraft Systems
## 6 Simulation Results  
A total of 1010 Monte Carlo simulations were run with our Manhattan maps. For each case, start and destination points were randomly defined. Selected start/end altitudes ranged from 20 m above ground level to the highest building roof. The 20 m value represents an above-ground vertical climb to hover waypoint to ensure the multicopter is well clear of people on the ground when it begins executing its lateral flight plan. If both start and end points had altitudes less than 50 m, the cruise altitude for the turn solution was set at 50 m. Otherwise, cruise altitude was adjusted based on the following condition:  
\[z_{\text{cruise}}=max\{h_{start},h_{end}\}if\ h_{start}>50\ \text{m}\ ||\ h_{ end}>50\ \text{m}. \tag{3}\]  
Our airspace volumization algorithm used this condition to choose one of the fixed-altitude datasets described in Section 4. As \(z_{\text{cruise}}\) becomes larger, fewer obstacles were present, so fewer calculations were needed to generate and plan a flight through the visibility graph. For each case, cost values of the four planning options (\"turn\", \"constant cruise alt.\", \"terrain follower\", \"combined (constant cruise altitude + turn)\" ) were calculated using Equation (2), and the minimum cost solution was selected as the best solution. Note in the Manhattan data the \"wait\" solution was never used because buildings are permanent, resulting in static geofence obstacles only. Monte Carlo results offer an opportunity to compare our airspace volumization solutions against a manual fixed airway or \"flight corridor\" airspace design. A conventional fixed-altitude airway is permanently designated on a map to enable traffic \"queues\" to organize in a way that can be managed by human air traffic controllers. It is unclear whether UTM will benefit from this legacy design practice, motivating our comparison of path costs for our airspace volumization and fixed airway solutions. Unlike our airspace volumization, fixed airway/flight corridor maps only require a local search for the closest airway to join. The UAS then follows fixed airway routes until exiting over a short final \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{\mathcal{V}}_{\text{vehicle}}\) & \(\delta_{\text{vehicle}}\) & \(\delta_{\text{building}}\) & \(N_{\text{geofence}}\) & \(z_{\text{cruise}}\) \\ \hline
5 (m/s) & 2 (m) & 5 (m) & 5 & 50 (m) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Control parameters for geofenced flight planning case studies. Figure 11: Flow chart of pathfinding logic for different start and end locations. In the chart, V.G. abbreviates visibility graph, and \(h_{\text{bldg}}\) is the height of a geofence around a cluster of buildings. If the departure/destination is not inside the keep-out geofence ROI box, \(h_{\text{bldg}}\) at start/end point is set to street/terrain altitude. segment to the end state. We generated a pair of low-altitude horizontal and vertical airways through our Lower Manhattan map to illustrate the airways concept and support our evaluation. The designed vertical airway in Lower Manhattan follows Broadway, the north-south main thoroughface, from its origin at Bowling Green to Houston Street. The horizontal airway follows Chambers Street from River Terrance in the west to Municipal Plaza in the east, and then follows the Brooklyn Bridge until it reaches the East River. We provided two sets of the same cross airways at 150 m and 500 m to offer each UAS an altitude choice since more obstacles are present at 150 m but the climb will be more substantial to 500 m. Figure 12 shows our manually defined airway corridors. To offer a practical comparison, only randomly generated start and end points that do not lie in the same quadrants (i.e., 712 out of 1010 simulation examples) were considered. If randomly-generated start and end points were located in the same quadrant, the airways were unused, thus offering no benefit to efficiency or airspace organization.