## 2 System Configuration and Modeling
### System Modeling
That is, \(\mathbf{J}=diag\big{(}I_{xx},~{}I_{yy},~{}I_{zz}\big{)}\). In Equation (2), \(\big{(}M_{h}+M_{p}\big{)}\dot{\mathbf{v}}_{\mathbf{b}}\) and \(\dot{\mathbf{\omega}}_{\mathbf{b}}\) are the differentiation of the system linear momentum and the moment-of-momentum, respectively. The cross-product terms \(\mathbf{\omega}_{\mathbf{b}}\times\mathbf{v}_{\mathbf{b}}\) and \(\mathbf{\omega}_{\mathbf{b}}\times\mathbf{J}\mathbf{\omega}_{\mathbf{b}}\) represent the Coriolis phenomenon. On the right-hand side, the total forces and torques acting on the robot's head are given by \(\mathbf{F}_{\mathbf{b}}\) and \(\mathbf{\tau}_{\mathbf{b}}\), respectively. They include the gravitation force, the water thrust (\(\mathbf{F}_{t}\), \(\mathbf{\tau}_{\mathbf{t}}\)), actuating forces (\(-\mathbf{F}_{\textit{wx}}\), \(-\mathbf{F}_{\textit{wy}}\)) and torques (\(-\mathbf{\tau}_{\textit{wx}}\), \(-\mathbf{\tau}_{\textit{wy}}\)) from the weight-shifting mechanism, reaction forces from the hose \(-\mathbf{F}_{\mathbf{h}}\) and the water sprinkler \(-\mathbf{F}_{\mathbf{s}}\), and other external disturbances (\(\mathbf{F}_{\mathbf{d}}\), \(\mathbf{\tau}_{\mathbf{d}}\)), such as the wind gust, ceiling, and ground effect. That is:  
\[\mathbf{F}_{\mathbf{b}}=\mathbf{F}_{\mathbf{t}}-\mathbf{F}_{\textit{wx}}-\mathbf{F}_{\textit{wy}}+ \big{(}M_{h}+M_{p}\big{)}\big{(}\mathbf{F}_{\mathbf{b}}^{\mathbf{E}}\big{)}^{\top}\big{[} 0\quad 0\quad g\big{]}^{\top}-\mathbf{F}_{\mathbf{h}}-\mathbf{F}_{\mathbf{s}}+\mathbf{F}_{\mathbf{d}} \tag{3}\]  
The water thrust can be obtained by applying [PERSON]'s second law on the water flowing through the head part. That is, the time rate of change of the linear momentum and moment-of-momentum of the water equals the summation of external forces and torques, respectively, acting on the water [24]. To obtain simple but sufficiently reliable dynamics of the fluid, assume that the water flow through the head is frictionless and steady [27]. The water thrust force and torque are written as follows:  
\[\mathbf{F}_{\mathbf{t}}= \begin{bmatrix}0\\ 0\\ \frac{1}{\rho A}\ddot{m}_{0}^{2}+\frac{1}{\rho a}\sum\limits_{i=1}^{4}\ddot{m} _{i}^{2}+p_{p}A\\ \tau_{\mathbf{t}}=-\sum\limits_{i=1}^{4}\ddot{m}_{i}\mathbf{r}_{i}\times(\mathbf{\omega}_{ \mathbf{b}}\times\mathbf{r}_{i})\end{bmatrix} +\sum\limits_{i=1}^{4}\ddot{m}_{i}\mathbf{ \omega}_{\mathbf{b}}\times\mathbf{r}_{i} \tag{4}\]  
In which \(\ddot{m}_{0}\) and \(\dot{m}_{i}\) are the mass flowrates at the water inlet port and the \(i\)th nozzle outlet, respectively. \(a\) is the cross-sectional of each nozzle outlet. The term \(p_{p}A\) represents the water force acting at the water inlet due to the gauge pressure \(p_{p}\) at this area. Furthermore, the vector \(r_{i}\) determines the position of the \(i\)th nozzle outlet in the robot's body frame. On the other hand, the weight-shifting mechanism affects the robot's head by the reaction forces and torques resulting from the motion of its two weight blocks. Equations of their motion are written as follows:  
\[ \begin{split} m_{\textit{wj}}(\dot{\mathbf{v}}_{\textit{wj}}+\mathbf{ \omega}_{\mathbf{b}}\times\mathbf{v}_{\textit{wj}})=\mathbf{F}_{\textit{wj}}+m_{\textit{ wj}}\big{(}\mathbf{F}_{\mathbf{b}}^{\mathbf{E}}\big{)}^{\top}\big{[}0\quad 0\quad g\big{]}^{\top},\\ \mathbf{I}_{\textit{wj}}\mathbf{\omega}_{\mathbf{b}}+\mathbf{\omega}_{\mathbf{b}} \times\mathbf{I}_{\textit{wj}}\mathbf{\omega}_{\mathbf{b}}=\mathbf{\tau}_{\textit{wj}}+m_{ \textit{wj}}\mathbf{r}_{\textit{wj}}\times\big{(}\mathbf{F}_{\mathbf{b}}^{\mathbf{E}}\big{)}^{ \top}\big{[}0\quad 0\quad g\big{]}^{\top},\quad j=x,y\end{split}  \tag{5}\]  
The translational motion of each weight consists of its displacement (\(r_{\textit{wx}}\) and \(r_{\textit{wy}}\)) observed from the robot's head and the motion of the head itself. Thereby, the translational velocities of the weight WX and WY are given by:  
\[