## 4 Simulations  
This section presents the results of numerical simulations, and MATLAB is used to elaborate upon the findings and developments of this paper. Initially, the controller conferred earlier was developed along with the model of the quadrotor UAV in order to track the computed final trajectory. The model parameters and controller gains, that have been used in the simulation environment, are listed in Tables 1 and 2, respectively. Because any path search algorithm requires a map and start and goal points, a three-dimensional environment containing the obstacles was modelled in MATLAB. The concept of real-time tall buildings is used to model the obstacles. Subsequently, the start point and the goal point were defined in the constructed obstacle-cluttered environment. Figure 4 illustrates the 2D and 3D views of the environment. Before handing this map over to the search algorithm, meshing is performed, which divides the whole map into small, cubical, interconnected blocks. Mesh size is decisive in performing the computation. A relatively smaller mesh size would compute a more optimal and shorter path but would do so at the cost of heavy computational power. Figure 5 demonstrates the map environment with meshing. This map is handed over to the A* search algorithm, which subsequently finds a shorter path between start and goal points. Figure 6 shows the progression of the search algorithm in 2D and 3D views. The shortest path found by the augmented A* algorithm is illustrated in Figure 7, which shows 2D and 3D views.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Mass (m) & 0.2 & kg \\ Arm length (\(l\)) & 0.09 & m \\ Inertia (\(lx\)) & 0.00026 & kgm\({}^{2}\) \\ Inertia (\(ly\)) & 0.00026 & kgm\({}^{2}\) \\ Inertia (\(l\)) & 0.00040 & kgm\({}^{2}\) \\ Gravity (g) & 9.81 & m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters.  
Technically, after the completion of the algorithmic search, the path found is still not the optimized version. Specifically, in a three-dimensional case, the path can further be optimized depending on the situation. Moreover, the shortest path found after the truncation process contains infinite curvatures at the corner points and traversal of the path using any quadrotor UAV is rather challenging. This requires smoothing of these corner points by incorporating any higher-ordered polynomial. The fifth-order polynomial is used to shape these corner points to reduce the curvatures to a finite extent. Consequently, the found shortest path is optimized further by smoothing the corner points. The 2D and 3D views of this process can be seen in Figure 8.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Controller** & **Parameter** & **Value** \\ \hline Outer Loop Controller & \(K_{p}(\text{x},\text{y},\text{z})\) & 2 \\ Outer Loop Controller & \(K_{d}(\text{x},\text{y},\text{z})\) & 1.4 \\ Inner Loop Controller & \(K_{p}(\phi,\theta,\psi)\) & 7 \\ Inner Loop Controller & \(K_{d}(\phi,\theta,\psi)\) & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Controller gains.  
Figure 4: Map containing obstacles and start and goal points (2D and 3D views).  
Figure 5: Meshing of the obstacle-cluttered environment (2D and 3D views).  
Figure 8: Extra node truncation followed by optimal trajectory generation.  
Figure 6: Augmented A* search in progress through the meshed environment (2D and 3D views).  
Figure 7: Path generated by conventional A* algorithm between start and goal points (2D and 3D views). This path contains extra expanded nodes.