## 4 Simulation Results and Discussion  
### Simulation  
To evaluate the dynamical characteristics of the designed firefighting robot and validate the feasibility of the proposed LQI control system, two scenarios reflecting two common motion trajectories are considered. One is a step-type trajectory, which is similar to the case where the firefighting system moves from one point to another to battle individual flames. The other is a continuous flying path replicating the way in which the system overcomes obstacles and approaches the fire site. In simulation studies, each scenario is carried out with and without the influence of external forces and torques. These external terms represent the unformulated effects in Section 2, including the effects of the water sprinkler \(-\mathbf{F_{s}}\) and the water hose \(-\mathbf{F_{h}}\) and external disturbances \(\left[\mathbf{F_{d}}^{T}~{}\mathbf{\tau_{d}}^{T}\right]^{T}\). The latter include the effects of the wind gust, ceiling, ground effect, etc. in the practical operation of the system, but they are neglected in the modeling process. This is to validate the robustness of the proposed control system when operating the fire-extinguishing sprinkler or facing disturbances. The formulations of the external forces and torques are adopted from the previous publication of [PERSON] and [PERSON] [30], where they were also used to test the robustness of an aerial system. As mentioned in the previous section, the step-type trajectory is also used in the optimization process of finding the controller's parameters.  
Moreover, a cascade proportional--integral--derivative (PID) control structure from [24] is also compared to the proposed LQI control with the GA approach, in which the outer PID controllers use the horizontal tracking errors in \(\mathbf{X}\)- and \(\mathbf{Y}\)-directions to compute the robot's required roll and tilt rotations, respectively. They are the references for the inner controllers, where the control inputs \(r_{ux}\) and \(r_{vy}\) are correspondingly obtained. In addition, another PID controller computes the required flowrate \(\dot{m}_{0}\) from the altitude control error of the system. Those control laws are mathematically represented in the following formulation:  
\[\mathbf{U_{PID}}=\mathbf{K_{p}}\mathbf{e}+\mathbf{K_{l}}\overset{\dagger}{f}\mathbf{ed}\tau+\mathbf{ K_{D}}\overset{\dd}{dt}\mathbf{e}, \tag{26}\] \[\mathbf{e}=\begin{bmatrix}x_{d}&\theta_{d}&y_{d}&\phi_{d}&z_{d}\end{bmatrix} ^{\overset{\overset{\overset{\overset{\overset{\overset{\overset{\overset{ \rightrightrightright}}}}}}{{T}}}}-\begin{bmatrix}\mathbf{\sigma}&\mathbf{\sigma}&\mathbf{y }&\mathbf{\phi}&z\end{bmatrix}^{T},\] \[\mathbf{U_{PID}}=\begin{bmatrix}\theta_{d}&r_{ux}&\phi_{d}&r_{uy}& \dot{m}_{0}\end{bmatrix}^{T}\]  
The system model given by (9)-(11) is carried into the simulation. The parameters of the designed concept are shown in Table 1. The equilibrium point for linearization is at \(\overline{z}_{b}=10~{}[\mathrm{m}]\) height with respect to an injected flowrate \(\overline{\dot{m}}_{0}=2.7~{}[\mathrm{kg/s}]\) to maintain this altitude for the system. The weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) are found by the optimization process with the GA after 331 generations. The convergence of the cost function during the process is depicted in Figure 6. The feedback control gains \(\mathbf{K_{i}}\) and \(\mathbf{K_{f}}\) are computed as in Equations (18) and (20) with the previously obtained \(\mathbf{Q}\) and \(\mathbf{R}\). They are shown in Table 2, where the PID controllers' gains are also presented.