## 3 Simulation Studies  
### Analysis and Selection of the Key Parameters  
Among several parameters of the proposed CA approach in this paper, the effects of \(\alpha\) and \(\gamma\) were analyzed through numerical simulations to select proper values. To plainly evaluate the effect of each parameter independently, the quadcopter's dynamics are not considered in this subsection. In other words, the inclusion of dynamics makes it difficult to analyze the CA trajectories purely affected by \(\alpha\) and \(\gamma\) because the quadcopter's motion is being affected by the controller and the quadcopter's parameters. Simulation studies are performed with several values of \(\alpha\) and \(\gamma\) within the ranges mentioned in the previous section. The other simulation parameters for the proposed algorithm are listed in Table 2.  
To analyze the effect of \(\gamma\)'s changes, only the horizontal avoidance maneuver (i.e., \(\alpha=0\)) was considered. The local minima situation was selected to test, and in this situation, the conventional APF approach cannot find the proper path, so the \(\gamma=0\) case does not need to be tested. Figure 4 shows the trajectories of the quadcopter with various \(\gamma\) values, and Figure 5 displays the total path lengths of the quadcopter and the minimum relative distances between the quadcopter and the nearest obstacle in accordance with \(\gamma\). As \(\gamma\) increases, the quadcopter's trajectory tends to approach the obstacles, and this result can be confirmed in Figure 5. The shorter the total path length, the lesser the energy consumption will be, by assuming the constant velocity-based operation for example. However, since too close a relative distance may cause a collision, a safety perspective must also be considered when selecting a \(\gamma\) value. This study assumed that the quadcopter is safe when the relative distance to the nearest obstacle is greater than 8 m. The minimum relative distance (i.e., safety margin) is highlighted with the red-dotted line in Figure 5. Here, the quadcopter may approach the obstacle closer than the safety margin if the dynamics of the quadcopter is considered. Furthermore, since the safety margin is 8 m, the quadcopter may collide with the nearest obstacle in 3 s when the quadcopter flies at a nominal velocity of 3 m/s. Based on this setting and information, \(\gamma=45\) deg is the best selection in this study, and the selection of \(\gamma=45\) deg results in the minimum relative distance of 8.3 m.  
With the fixed \(\gamma=45\) deg, the next analysis is that of the effect of \(\alpha\)'s changes. To analyze the vertical avoidance maneuver, the operational altitude of the quadcopter in this simulation is selected around the top height of the building. Similar to the analysis process for \(\gamma\), Figure 6 describes the trajectories of the quadcopter with various \(\alpha\) values, and Figure 7 shows the total path lengths of the quadcopter and the minimum relative distances between the quadcopter and the nearest obstacle in accordance with \(\alpha\). As \(\alpha\) changes, the degree of the portion of horizontal and vertical maneuvers on the planned trajectory also changes. Since the quadcopter operates near the top of the obstacle, the vertical avoidance maneuver (i.e., \(\alpha=0\)) provides better performance than the others in terms of path length. However, when \(\alpha=0.5\), the result is best, as shown in Figure 7, because the minimum relative distance it is required to maintain is greater than the safety margin (8 m).  
Figure 5: Path length and minimum relative distance in accordance with \(\gamma\).  
Figure 6: 3D trajectories in accordance with \(\alpha\).