# Airspace Geofencing and Flight Planning for Low-Altitude, Urban, Small Unmanned Aircraft Systems
## 6 Simulation Results
Figure 13 shows a top-down route view comparing our airspace volumization and flight corridor solutions. Cost weights \(\alpha=0.6,\ \beta=0.2,\ \gamma=0.0\) were again used, so \(d_{travel}\) was prioritized in minimizing overall cost. For the illustrated case, the \"turn\" solution is best. Flight corridor solution cost was in fact typically higher than any of our airspace volumization solutions. For the same example, altitude vs. time plots for each solution are shown in Figure 14. Examples of geofencing solutions are shown in Figures 15-17, where three alternative trajectory solutions are generated, ensuring the avoidance of no-fly zones. Building keep-out geofences are shown in green. Figure 12: Example horizontal and vertical airway corridors in Manhattan. Figure 13: Top-down view of example flight paths for airspace volumization and fixed flight corridor solutions. Distances traveled are 770 m (turn), 1051 m (constant cruise), 1139 m (terrain follower), 1528 (150 m flight corridor), and 1977m (500 m flight corridor). Figure 16: Example 3-D geofencing solution for a “constant cruise altitude” flight plan solution. Polyhedra in green denote keep-out geofences around buildings near the trajectory’s keep-in geofence. The remaining 2-D polygons denote keep-out geofences around buildings that are more distant from the sUAS flight path. Figure 14: Flight altitude time histories for airspace volumization and flight corridor solutions for Figure 13 example. Figure 15: Example of a 3-D geofence wrapping a “turn” flight plan solution. Polyhedra in green denote keep-out geofences around buildings near the trajectory’s keep-in geofence. The remaining 2-D polygons denote keep-out geofences around buildings that are more distant from the sUAS flight path. For each Monte Carlo simulation, the minimum-cost \(\mathcal{C}\) solution was compared to flight corridor solution costs at 150 m and 500 m per Table 3. Since the flight corridor at 150 m was almost always better than the flight corridor at 500 m, benchmark data compare the best solution obtained using dynamic airspace volumization with the flight corridor at 150 m. The results indicate our airspace geofencing volumization solutions generally have lower cost than flight corridors at 150 m or 500 m do. The average distance and power consumption of the two-dimensional straight-line path between each start and destination location are shown in Table 4.  
The mean and standard deviation for \(d_{travel}\), \(p_{travel}\) for the minimum cost airspace volumization solution are summarized in Figure 11:0 The percent frequency distributions of the four solution options are shown in Figure 11:1 \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mu_{d_{2D~{}path}}\) & \(\mu_{P_{2D~{}path}}\) & \(min\{d_{2D~{}path}\}\) & \(max\{d_{2D~{}path}\}\) \\ \hline
1391 (m) & 91259 (J) & 189 (m) & 3003 (m) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average distance (\(d\)), power consumption (\(P\)), and minimum and maximum distances of 2D straight-line paths between start and destination states for the Monte Carlo simulations. Figure 17: Example 3-D geofencing solution for a “terrain follower” flight plan solution. Polyhedra in green denote keep-out geofences around buildings near the trajectory’s keep-in geofence. The remaining 2-D polygons denote keep-out geofences around buildings that are more distant from the sUAS flight path. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mu_{d_{travel}}\) & \(\sigma_{d_{travel}}\) & \(\mu_{P_{travel}}\) & \(\sigma_{p_{travel}}\) & \(min\{d_{travel}\}\) & \(max\{d_{travel}\}\) \\ \hline
1595 (m) & 606 (m) & 94,338 (J) & 39,609 (J) & 254 (m) & 3349 (m) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean \(\mu\) and standard deviation \(\sigma\) of the minimum-cost airspace volumization solution. A similar analysis was performed to compute travel distance and power consumption statistics for the flight corridor solutions at 150 m and 500 m, as shown in Tables 6 and 7.