## 4 Simulation Results and Discussion
### Results and Discussion  
In the first scenario, the firefighting robot takes off, up to a height of 10 [m], and then moves horizontally to a length of 2 [m] in each of the X- and Y-directions, respectively. The geometry of the desired path is shown in Figure 7, along with the tracking performances provided by the two considered controllers: the proposed LQI-GA and the cascade PID. Subsequently, Figures 8 and 9 depict the system response in every direction. The corresponding control inputs are given in Figure 10. For all of them, the left panels show the results when there is no disturbance and vice versa for the ones on the right. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Value** & **Unit** \\ \hline \(M_{h}\) & Mass of the head part & 3 & kg \\ \(J\) & Moment of inertia of the head part & \(diag\{0.0071,\ 0.0085,\ 0.0133\}\) & kg-m\({}^{2}\) \\ \(A\) & Inner diameter of the water hose & 0.05 & m \\ \(a\) & Inner diameter of the nozzle outlets & 0.01 & m \\ \(k\) & Correction factor of the water hose & 1.5 & - \\ \(l\), \(l\), \(h\) & Dimensions of the head part & 0.25, 0.1 & m \\ \(m_{ux}\), \(h_{x}\) & Weight and vertical location of WX & 0.5, 0.13 & kg, m \\ \(m_{uy}\), \(h_{y}\) & Weight and vertical location of WY & 0.5, 0.08 & kg, m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the proposed firefighting robot. Figure 6: Convergence of the cost function. Figure 8: Time responses of the robot position with the step-type trajectory. (**a**) Without disturbances; (**b**) With disturbances. Figure 7: Tracking performances with the step-type trajectory. (**a**) Without disturbances; (**b**) With disturbances. Figure 9: Time responses of the robot orientation with the step-type trajectory. (**a**) Without disturbances; (**b**) With disturbances. In particular, Figure 7 shows that the designed robot can perform flying tasks as required. Moreover, the proposed LQI-GA manages the robot to closely follow the trajectory. The proposed controller is also robust against the disturbances, as the tracking performance is insignificantly affected in Figure 7b. The time responses in the positioning of the LQI-GA control system are also smooth, without any overshoot. The zoomed figures in Figure 8a prove that the non-minimum phase phenomena are smaller than those in the PID control system; even the PID controllers provide much slower performances. The external disturbances also heavily influence the PID-controlled system, as seen in the figures on the right. Figure 9 indicates that the PID controllers give the robot's head less tilting and rolling. Nevertheless, the orientations are smoother with the LQI-GA controller, either with or without the presence of the disturbances. The yaw rotation of the robot is generally stable, without any control effort needed. The smooth control inputs of the proposed controller explain the smooth response and the reduction of the non-minimum phase phenomena in the system. The smooth displacements \(r_{ux}\) and \(r_{uy}\) from the LQI-GA controller indicate the small acceleration of the weight blocks, thus resulting in smaller non-minimum phase phenomena. In contrast, the rough movements of the blocks in the PID control system lead to the fluctuation in orientations and the significant non-minimum phase phenomena. In the second scenario, the firefighting robot follows a continuous trajectory while also maintaining system stabilization. The complex trajectory, in this case, is given by:  
\[r= \begin{bmatrix}10e^{\frac{t}{100}}\sin(0.1u)\cos(0.01u)\\ 0.1t\sin(0.05t)\\ 0.1t\end{bmatrix}  \tag{27}\]  
The simulation results are presented in Figures 11-14 in the same manner as the previous scenario. The tracking paths in Figure 11 show that the proposed control system remains the better one. Its resulting motion is closed to the reference, even with the presence of disturbances. Meanwhile, the PID controllers give a lot of overshoots and are easily influenced by external forces and torques.