## 6 Results and Discussion
The tracking performances in Figure 5 and the time responses in each direction in Figure 5 indicate that the flow rate control and nozzle rotation control systems perform significantly  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Part**} & \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Notation**} & \multicolumn{6}{c}{**Value**} \\ \cline{3-10}  & & & & \multicolumn{4}{c}{**Flow Rate Control**} & \multicolumn{4}{c}{**Nozzle Rotation**} \\  & & & & & & **Control** & & **Weight Distribution Control** \\ \hline \multirow{4}{*}{Robot head} & Mass [kg] & \(M_{h}\) & 3.10 & & 2.55 & & 3.79 & \\  & Inertial tensor & \(I_{h}\) & \( \begin{bmatrix}0.035&-0.001&-0.003\\ -0.001&0.034&-0.001\\ -0.003&-0.001&0.034\end{bmatrix} \) & \( \begin{bmatrix}0.032&0.002&-0.001\\ 0.002&0.038&-0.001\\ -0.001&-0.001&0.039\end{bmatrix} \) & \( \begin{bmatrix}0.052&-0.005&-0.007\\ -0.005&0.051&0.008\\ -0.007&0.008&0.062\end{bmatrix} \) \\ \hline \multirow{4}{*}{Thrust nozzles} & Outlet & & & & & & & & \\  & position vector [m] & \( \begin{bmatrix}r_{1b},\\ r_{10b},\\ d\end{bmatrix} \) & \( \begin{bmatrix}0.12&0.23&-0.03\end{bmatrix} ^{T}\) & & \(r_{10b}= \begin{bmatrix}0.15&0.15&0.05\end{bmatrix} ^{T}\), & \( \begin{bmatrix}0.15&0.22&-0.02\end{bmatrix} ^{T}\) \\  & & & & & & & & & & \\ \cline{2-10}  & Outlet area & \(A_{u}\) & & & & & & \(0.002^{2}\pi\) & \\  & [m\({}^{2}\)] & & & & & & & & \\ \cline{2-10}  & Nozzle angle & \(\delta_{b}\) & & & & & \(\pi/6\) & \\ \hline \multirow{4}{*}{Actual system} & Weight block & & & & & & & & \\  & actuator’s parameters & \(J_{\text{zw}},M_{\text{zw}},z_{\text{w}}\) & & & & & \(I_{\text{zw}}= \begin{bmatrix}0.14&0&0\\ 0&0.13&0&0\\ 0&0.15\end{bmatrix} \) & \( \begin{bmatrix}\text{kg}\cdot\text{m}^{2}\end{bmatrix} \), \\  & & & & & & & \(M_{\text{zw}}=0.5\;\left[\text{kg}\right],z_{\text{w}}^{2}=0.1\;\left[\text{m}\right]\), \\  & Control & \(U_{\text{max}}\) & \( \begin{array}{c}\dot{m}_{\text{k,max}}=7.5\;\left[\text{kg}/s\right]\), \\ u_{i,\text{max}}=1,\\ i=13,\;1,\;2\end{array} \) & \( \begin{array}{c}\dot{m}_{\text{k,max}}=7.5\;\left[\text{kg}/s\right]\), \\ \dot{u}_{i,\text{max}}=\pi/2\left[\text{rad}\right]\), \\ i=1,\;2,\;4\end{array} \) & \(\begin{array}{c}\dot{m}_{\text{k,max}}=7.5\;\left[\text{kg}/s\right]\), \\ \dot{x}_{\text{zw,max}}=y_{\text{zw,max}}=0.2\;\left[\text{m}\right]\) \\ \cline{2-10}  & Time constants of first-order filters & & Pump 0.1 [s], Valves: 0.01 [s] & Pump 0.1: [s], Nozzles: 0.01 [s] & Pump 0.1 [s], Weight block: 0.2 [s], Impeller fan: 0.01 [s] \\ \hline \multirow{4}{*}{Control system} & Linearization point & \(Z_{o}\), \(\dot{m}_{10}\) & & \(Z_{o}=3\;\left[\text{m}\right]\), \(\dot{m}_{10}=2.3\;\left[\text{kg}/s\right]\) & \\  & Bound on & \(\infty\)-norm & \(\gamma\) & 1.45 & & 3.6 \\ \hline \multirow{4}{*}{Water hose nozzle} & Cross area & & & & & & \\  & [m\({}^{2}\)] & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  & Stiffness & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  & & & & & & & & \\ \cline{2-10}  & & & & & & & \\ \cline{2-10}  &  
better than the weight distribution control approach. The two systems quickly reach the desired altitude and properly track their motion references most of the time. Meanwhile, the responses and control inputs (Figure 5d) of the weight distribution control system indicate that it struggles to perform in every direction. The inertia force due to the movement of the actuating weight affects the outputs of the system. Figure 5d also validates that the control inputs of all systems are within their respective saturations. On the other hand, the roll and pitch rotations of the heads should be ideally maintained at zeros. However, these rotations are coupled with horizontal motions. Their values vary with time in order to either move the system heads or compensate for the disturbances acting on the systems.