## 6 Results and Discussion  
In order to evaluate the maneuverability of each designed system, the above system models and control law are taken into comparative simulations. In particular, the numerical solution of the partial differential equation of the water base can be rewritten as follows:  
\[ \begin{gathered}(m_{f}+m_{p})\Delta\ddot{r}_{s_{n}}=-2m_{f}V_{h} \big{(}\ddot{r}_{s_{n}}-\dot{r}_{s_{n-1}}\big{)}-(\tau_{s_{n}+1}-2\tau_{s_{n} }+\tau_{s_{n-1}})/\Delta s\\ -\Big{(}m_{p}+m_{f}\Big{)}\Delta s\mathbf{g}+\mathbf{r}_{s}\Delta s( _{s_{n}}-\sqrt{\lambda_{k}/\pi})-m_{f}V_{h}^{2}(\tau_{s_{n+1}}-2\tau_{s_{n}}+ \tau_{s_{n-1}})/\Delta s,\\ \tau_{s_{n}}=k\mathbf{\Phi}_{s_{n}}+c\dot{\mathbf{\Phi}}_{s_{n}},\;\;\mathbf{r}_{s_{n}}= \mathbf{r}_{s_{n-1}}+\Delta s\big{[}\;\;s_{\beta\gamma}\;\;c_{\beta\gamma}s_{\alpha _{i}}^{\alpha}\;\;c_{\beta\gamma}\;c_{\alpha_{i}}^{\alpha}\;\;\big{]}^{T}\end{gathered}  \tag{27}\]  
where the bending moment \(\mathbf{\tau}\), according to the elementary beam theory, is \(\mathbf{\tau}=EI\mathbf{r}_{s}{}^{\prime\prime}\). Additionally, \(k=EI/L\) is the spring stiffness, \(c\) is the damping coefficient, and \(\mathbf{\Phi}\) is the vector of rotation. Hence, the approximation of a water base contains a collection of rigid bodies with mass \((m_{f}+m_{p})\Delta s\) coupled with springs and dampers, as seen in the first and last lines of Equation (27). This is similar to the lumped parameter method for modeling a flexible body [40]. Moreover, each body element is also influenced by the water flow through it which is given by the second line of Equation (27). The system is assumed to operate in an onshore mission. That is, the interactive force \(\mathbf{F}_{r}\) includes the normal contact force with the ground and the frictional force generated by the motion of the hose on the ground surface. They are modeled as in [41]. The specifications of the designed systems and the parameters used in the simulation are listed in Table 1. It is also worth noting that the actuators have their own dynamics in actual systems. Simply put, there is a time required for the response of the actuators to reach the values given by the controller. In the simulation studies, we integrated their complete CAD models and implemented first-order filters to mimic these characteristics. We use MathWorks Simulink and Simscape Multibody to solve the models and conduct the simulation tests. Two scenarios were considered, simply put, one with and one without the firefighting action. Three systems were initially land on the ground and take off and up the operating altitude at 03 [m]. The beginning of the hose is simply fixed in the simulations. The systems then perform positioning at the desired set points on the horizontal plane. At the same time, the heading angles of the head parts vary following a sinusoidal trajectory. These motions mimic the maneuver that a firefighting system has to perform while on duty, including approaching several fire sites and spaying the water widely to cover the fires. Figure 5 and Supplementary Video S1 show the motions of the three considered systems without the firefighting action. This simulation is to validate the maneuverability of these designs. On the other hand, Figure 6 and Supplementary Video S2 present the results in the other scenario, i.e., with the firefighting action. The action is represented by an additional water flow of water through the fire extinguishing nozzle at a flow rate of 01 [kg/s], lasting 10 [s] every 10 [s] interval. This flow is depicted in Figure 6.  
In the first scenario, all of the systems' heads are able to follow the required references.