## 3 Simulation Studies
### Numerical Simulation Results in an Urban Environment  
To test the proposed CA approach in realistic environments, a high-resolution urban area was simulated using real datasets measured by an airborne LiDAR. Since the open source urban data are available, this work utilized data obtained from Open-Topography ([[https://opentopography.org/](https://opentopography.org/)]([https://opentopography.org/](https://opentopography.org/)), accessed on 17 October 2021). Note that LiDAR data (i.e., \"*.las\" file here) of San Diego downtown were selected as an example of 3D urban environments. The data are classified into \(x\), \(y\), and \(z\) components of a point cloud, and one filters out the data so that all of \(z\) is less than 200 m. The filtered point cloud data are visualized in Figure 8.  
To validate the CA performance of the proposed approach, the numerical simulations of three scenarios in the selected urban environment were performed. The local minima and GNRON cases were chosen as the first and second scenarios, respectively, to highlight the fact that the proposed approach resolves the conventional APF's well-known issues. The third case is a complex urban environment in the presence of multiple static obstacles (e.g., buildings). In addition, a quadcopter in this study is mounted on a 3D scanner (e.g., depth camera) to simulate practical sensing. By the 3D scanner, the point cloud of objects (i.e., obstacles here) existing within the sensing range and FOV of the sensor are measured. The simulation parameters for the quadcopter, sensor's specifications, and control gains are listed in Table 3. Here, the quadcopter's dynamics and control logic are considered, unlike in Section 3.1. Furthermore, the quadcopter's diameter size was chosen based on information about the commercial drone's specifications [22]. Moreover, this work assumed that the quadcopter maintains a constant nominal speed during operation. In addition, the  
Figure 8: Visualization of the selected urban data in (**a**) top view and (**b**) 3D view. Figure 7: Path length and minimum relative distance in accordance with \(\alpha\). parameters for the proposed CA algorithm listed in Table 2, and \(\alpha\) and \(\gamma\) values selected in Section 3.1 were used. For each case, the start and goal positions are tabulated in Table 4.  
In Case 1, an obstacle point is located exactly between the start and goal positions. Figure 9 shows the quadcopter's avoidance trajectory and the nearest obstacle point sensed by the 3D scanner at each simulation discrete-time. Note that the blue trajectory represents the desired trajectory obtained from the proposed CA algorithm, while the black one indicates the quadcopter's actual trajectory. At every time step, the CA algorithm computes the desired position for avoiding obstacles, and the control logic mentioned in Section 2.2 uses the desired position calculated. Furthermore, Figure 10 depicts the relative distance between the quadcopter and the nearest obstacle point. In the trajectory figure, it is observed that there are slight changes in the quadcopter's altitude near the obstacle, and this is because of the portion of the vertical avoidance direction inclusion. In addition, the relative distance during the maneuver remains greater than the collision threshold 2 m that represents physical collision. Note that the obstacle threshold of 10 m denotes the relative distance that activates the repulsive potential field, as defined in Table 2. Since the smallest relative distance is observed at 8.42 m, one can say that the quadcopter completes the mission safely. In Case 2, to construct the GNRON case, the goal position is placed near the obstacle. The quadcopter's CA trajectory by the proposed approach is described in Figure 11. The quadcopter completes the mission without any collision, although the goal position is within the range (10 m) in which the repulsive potential field is activated.