## 3 Simulation Studies
### Numerical Simulation Results in an Urban Environment
That is, although the repelling force is produced near the goal position by the repulsive potential field, the quadcopter smoothly arrives at the goal position. Unlike in Case 1, a few altitude changes were observed due to the smooth direction changes required. Figure 12 shows the relative distance to the nearest obstacle. The relative distance continuously decreases within the obstacle threshold until the quadcopter reaches the goal position. Once the quadcopter arrives at the goal position, the relative distance is not measured to perform the landing. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Description** & **Start Position (m)** & **Goal Position (m)** \\ \hline Case 1 & Local minima & \([360,130,60]^{\mathrm{T}}\) & \([250,130,60]^{\mathrm{T}}\) \\ Case 2 & GNRON & \([380,110,70]^{\mathrm{T}}\) & \([350,208,60]^{\mathrm{T}}\) \\ Case 3 & Complex environment & \([370,190,78]^{\mathrm{T}}\) & \([200,50,115]^{\mathrm{T}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Start and goal positions of the quadcopter for each case. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Parameter** & **Value** \\ \hline Quacopter’s diameter & \(d_{q}\) & 0.6 (m) \\ Quacopter’s nominal speed & \(v_{n}\) & 3 (m/s) \\ Quacopter’s mass & \(m\) & 0.65 (kg) \\ Quadcopter inertia & \(I\) & diag(\([7.5,\,7.5,\,130]\times 10^{-3}\) (kg m\({}^{2}\)) \\ Gravity constant & \(g\) & 9.81 (m/s\({}^{2}\)) \\ Rotor inertia & \(I\) & \(6\times 10^{-5}\) (kg m\({}^{2}\)) \\ Distance from center to rotors 1 and 2 & \(l\) & 0.23 (m) \\ Aerodynamic force constant & \(k_{f}\) & \(3.23\times 10^{-5}\) (N s\({}^{2}\)) \\ Aerodynamic torque constant & \(k_{\tau}\) & \(7.5\times 10^{-5}\) (N ms\({}^{2}\)) \\ Sensor’s sensing range & \(d_{s}\) & 20 (m) \\ Sensor’s horizontal FOV & \(x_{h}\) & 210 (deg) \\ Sensor’s vertical FOV & \(x_{b}\) & 70 (deg) \\  & \(k_{P,pos}\) & 2 \\ Position control gains & \(k_{l,pos}\) & 0.001 \\  & \(k_{D,pos}\) & 1 \\ Attitude control gains & \(k_{P,att}\) & 0.28 \\  & \(k_{D,att}\) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters for the quadcopter system. Thus, there is no time history after 70 s. The closest distance 7.62 m was measured at the goal position, and the quadcopter complied with the collision threshold of 2 m.  
Figure 10: Case 1: Relative distance. Figure 9: Case 1: Quadcopter’s trajectory. The last case is that of a complex environment with several static obstacles. In this scenario, the quadcopter must avoid three buildings to achieve the mission. Figure 13 shows the quadcopter's 3D and 2D trajectories. The quadcopter mainly performs the horizontal avoidance maneuver near the first and third buildings and the vertical avoidance maneuver for the second building. Since the proposed approach combines the horizontal and vertical maneuvers, the quadcopter can successfully avoid several obstacles without collision. Figure 14 illustrates the relative distance from the quadcopter to the nearest obstacle between the sensing range of 20 m and the collision threshold of 2 m. The smallest relative distance can be observed at 8.5 m so that the quadcopter safely reaches the goal  
Figure 11: Case 2: Quadcopter’s trajectory. Figure 12: Case 2: Relative distance. position without collision--even in the presence of multiple static obstacles within the limited 3D urban environment. Figure 13: Case 3: Quadcopter’s trajectory in (**a**) 3D view and (**b**) top view.