## 7 Conclusions  
Three designs of water-powered aerial systems with different actuators have been proposed and analyzed in this paper, namely the flow rate control system, nozzle rotation control system, and weight distribution control system. The systems have potential in the firefighting missions performed in water areas and can extend to every fire site with sufficient water availability. In this study, the dynamical characteristics of the three systems have been formulated. A linear control scheme has been designed and implemented for them. In the end, the comparative simulations indicate that both the flow rate control system and nozzle rotation control system provide adequate motion performance. For these two systems, the designed control law preserves the systems' stability and guarantees the input saturations. The nozzle rotation control system is the better one, with greater tracking results of the head and smaller fluctuations of the water hose. Therefore, this design is chosen to be developed further. Figure 6: Performance of the proposed designs in a firefighting mission. (**a**) Tracking performance in three-dimensional geometry; (**b**) Time responses of translations and rotations; (**c**) three-dimensional partial motion of the water-conveying hose; (**d**) Control inputs; (**e**) Fire extinguishing nozzle’s flow rate. However, the designed control law cannot deal with perturbations effectively. The linear controller is designed based on the linearization of the system about a local equilibrium point, such that nonlinearities and other effects are neglected. For the chosen actuator method, more advanced control techniques will be considered where the motion of the hose will be incorporated into the control law. At the same time, prototypes of the nozzle rotation control system will be fabricated, and experimental studies will be conducted for evaluation. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13031965/s1](https://www.mdpi.com/article/10.3390/app13031965/s1)]([https://www.mdpi.com/article/10.3390/app13031965/s1](https://www.mdpi.com/article/10.3390/app13031965/s1)), Video S1: Supplementary Video S1, Video S2: Supplementary Video S2. Conceptualization, T.H. and Y.-B.K. ; methodology, T.H. ; software, Y.-B.K. ; validation, T.H., D.-H.L., and Y.-B.K. ; investigation, T.H. and Y.-B.K. ; writing--original draft preparation, T.H. ; writing--review and editing, T.H., D.-H.L., and Y.-B.K. ; visualization, T.H. ; supervision, Y.-B.K. ; project administration, Y.-B.K. ; funding acquisition, Y.-B.K. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1A2C1003486). Not applicable. This work was also supported by the National Research Foundation (NRF), South Korea, under Project BK21 FOUR (Smart Convergence and Application Education Research Center). The authors declare no conflict of interest. Details of the linear state-space of each system given in Equation (21), and the control matrices obtained by Equations (24) and (25), are as follows:  
For the flow rate control system:  
\[\mathbf{A}=\left[ \begin{array}{cccccccc}0&0&0&0&\frac{\hat{m}_{0}^{2}z_{0}}{ \hat{m}_{0}^{2}z_{0}}&0&0&0&0&\frac{\hat{m}_{0}\hat{m}_{0}}{\hat{m}_{0}}&0\\ 0&0&-\frac{\hat{m}_{0}z_{0}}{\hat{m}_{0}^{2}z_{0}}&0&0&0&0&-\frac{\hat{m}_{0} \hat{m}_{0}}{\hat{m}_{0}}&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&-\frac{\hat{m}_{0}\hat{m}_{0}}{\hat{m}_{0}}&0&-\frac{\hat{m}_{0} \hat{m}_{0}^{2}+\hat{m}_{0}\hat{m}^{2}}{\hat{m}_{0}}&0&0\\ 0&0&0&0&0&0&\frac{\hat{m}_{0}\hat{m}_{0}}{\hat{m}_{0}}&0&0&0&-\frac{\hat{m}_{0 }\hat{m}_{0}^{2}+\hat{m}_{0}\hat{m}^{2}}{\hat{m}_{0}}&0\\ 0&0&0&0&0&0&0&0&0&0&-\frac{\hat{m}_{0}\hat{m}^{2}+\hat{m}_{0}\hat{m}^{2}}{ \hat{m}_{0}}\end{array} \right],\] (A1)  
\[\mathbf{B}=\left[