## 3 Results
### Simulation Cases and Results
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Description** & **Start Position (m)** & **Goal Position (m)** \\ \hline Case 1 & Local Minima & (350, 270, 60)T [FOOTNOTE:]Footnote : T (200, 270, 60)T [FOOTNOTE:]Footnote : T (330, 207, 65)T [FOOTNOTE:]Footnote : T (350, 110, 70)T [FOOTNOTE:]Footnote : T (350, 207, 65)T [FOOTNOTE:]Footnote : T (350, 207, 65)T [FOOTNOTE:]Footnote : T (350, 110, 70)T [FOOTNOTE:]Footnote : T (350, 207, 65)T [FOOTNOTE:]Footnote : T (350, 207, 65)T [FOOTNOTE:]Footnote : T (350, 110, 70)T [FOOTNOTE:]Footnote : T (350, 110, 70)T [FOOTNOTE:]Footnote : T (350, 30, 30)T [FOOTNOTE:]Footnote : T (350, 70, 30)T [FOOTNOTE:]Footnote : T (350, 20)T [FOOTNOTE:]Footnote : T (350, 20)T [FOOTNOTE:]Footnote : T (350, 20)T [FOOTNOTE:]Footnote : T (350, 20)T [FOOTNOTE:]Footnote : T (350, 200)T [FOOTNOTE:]Footnote : T (350, 200)T [FO  
dotted line), and both paths do not collide since they are no closer than the collision risk distance (i.e., pink dotted line). In addition, since the heading direction (i.e., sign of the APF's velocity vector) changes instantaneously as shown in the middle one of Figure 7b, the planned path is dynamically infeasible. However, the proposed approach generates the collision-free path resolving the local minima problem. That is, the proposed method completes the mission within around 82 seconds but the APF cannot since it stays in the local minima area forever (see Figure 7). The red stars illustrated in Figure 5 represent obstacle points that influence re-planning in the approach proposed. With minimal obstacle points (i.e., red stars), a smooth and dynamically feasible trajectory is created, shown in the position vector of Figure 7a. Figure 5: Bird’s eye view of the local minima problem case. Figure 6: Relative distance to the nearest obstacle in the local minima problem case. The next case is static obstacles only. Figures 11 and 12 highlight the performance of the proposed approach, which provides smooth and continuous position and velocity trajectories. Both methods reach their destination without invading the 5 m collision risk radius as shown in Figure 13, but the APF method is shown to be inefficient since it is affected by more obstacle points and required re-plans more often (i.e., the number of blue squares in Figure 11) than the proposed method experienced (i.e., the number of red stars in Figure 11). Moreover, the APF plans the dynamically impracticable velocity profile enough to require infinite acceleration as shown in the middle one of Figure 12b. In other words, like the proposed approach, the continuous change of the velocity profile is possible dynamically, but like the APF, the relatively discrete variation of the velocity profile is not dynamically executable. Figure 10: Translation of the GNRON problem case (**a**) Position (**b**) Linear velocity and speed. Figure 9: Relative distance to the nearest obstacle in the GNRON problem case. Figure 11: Bird’s eye view of the static obstacles only case. Figure 12: Translation of the static obstacles only case (**a**) Position (**b**) Linear velocity and speed. Figure 13: Relative distance to the nearest obstacle in the static obstacles only case. In case 4, three UAVs' starting points (green dots) and destinations (pink dots) are set as if they meet each other around the middle area in the open space (see Figure 14). Here, each UAV is a decentralized agent without any communication like automatic dependent surveillance-broadcast. That is, one UAV does not know the location of the other two UAVs via communication in advance, and like moving obstacles, it can instead predict their rough locations only through real-time measurements of the point cloud.