## 3 Results
### Simulation Cases and Results  
For validating the performance of the proposed approach, one simulates five meaningful scenarios in realistic environments as follows: (i) local minima problem, (ii) GNRON problem, (iii) only static obstacles, (iv) only dynamic obstacles, and (v) complex environment. The first and second cases describe the well-known two major issues of the path planning problems  
Figure 4: An example city of urban modeling (**a**) 3D (**b**) 2D. including the APF. The third case tests a 3D environment that contains urban structures as static obstacles, demonstrated in Section 3.1. In the fourth case, a UAV senses other non-cooperative UAVs that are considered as dynamic obstacles. The last challenging case is a complex environment with multiple UAVs (i.e., moving obstacles) in the presence of static obstacles. All of these scenarios are popular in the field of path planning. In this work, for simulating practical sensing, a range sensor mounted a UAV obtains a point cloud within the measurable space (Figure 2) based on the sensor's specification. The point cloud includes relative distance information between the UAV and obstacles. In addition, for navigating safely, it assumes that collision occurs when obstacles are inside the risk sphere with a radius of 5 m, centered at the UAV. Here, one assumes no collision at the initial state and no existence of estimation errors and sensing errors. In the processes of the proposed approach, the number of intermediate waypoint candidates is multiples of 8 so that each one is distributed about every 45 deg on the perpendicular plane of the UAV's path direction. Table 2 lists simulation parameters, and the start and goal positions for each case are tabulated in Table 3. Note that each component of the column vectors represents \(x\), \(y\), and \(z\) (altitude), respectively. The following results from the proposed approach are compared to those from the existing APF method. In case 1, a point of the static obstacle is exactly on the line connecting the start and goal positions. In that environment, the APF yields the UAV gets stuck in the local minima, so the UAV cannot go further avoiding the obstacle as shown in the blue dotted line of Figure 5. In the bird's eye view, green and pink dots represent the start and goal locations, respectively. Figure 6 depicts the fact that the APF-based UAV (i.e., blue line) stays near the obstacle while maneuvering back and forth around \(d_{\text{thd}}\). In the figure, relative distances to the nearest obstacle are measured only within the sensor range (i.e., black \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Value** \\ \hline \(\mathcal{V}\) & 2 m/s \\ Discrete-time Interval & 0.1 s \\ Sensor’s Sensing Range & 20 m \\ Sensor’s Horizontal FOV & 220 deg \\ Sensor’s Vertical FOV & 70 deg \\ Radius of the Collision Risk Sphere & 5 m \\ Number of Intermediate Waypoints & 8\(\times\) \\ Maximum Number of Intermediate Waypoints & 1000 \\ Maximum Iterations for the APF & 1000 \\ Resolution of Extracted Path-Sample Points & 0.3 s \\ \(k_{\text{att}}\) & 0.01 \\ \(k_{\text{rep}}\) & \(5\times 10^{3}\) \\ \(d_{\text{thd}}\) & 10 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters.