## 2 System Configuration and Modeling
### System Modeling
\begin{split}\mathbf{v}_{\textit{wx}}=\big{[}\dot{r}_{\textit{wx}} \quad 0\quad 0\big{]}^{\top}+\mathbf{v}_{\textit{b}}+\mathbf{r}_{\textit{wx}}\times\mathbf{ \omega}_{\mathbf{b}},~{}~{}\mathbf{r}_{\textit{wx}}=\big{[}r_{\textit{wx}}\quad 0\quad h_{\textit{ wx}}\big{]}^{\top},\\ \mathbf{v}_{\textit{wy}}=\big{[}0\quad\dot{r}_{\textit{wy}}\quad 0\big{]}^{\top}+\mathbf{v}_{\textit{b}}+\mathbf{r}_{ \textit{wy}}\times\mathbf{\omega}_{\mathbf{b}},~{}~{}\mathbf{r}_{\textit{wy}}=\big{[}0 \quad r_{\textit{wy}}\quad h_{\textit{ wy}}\big{]}^{\top}\end{split}  \tag{6}\]  
where \(\dot{r}_{\textit{wx}}\) and \(\dot{r}_{\textit{wy}}\) are their respective relative velocities, and \(h_{\textit{wx}}\) and \(h_{\textit{wy}}\) are their corresponding vertical distances from the coordinate origin. On the order hand, the weights rotate with the rotation of the head part carrying them. Their moments of inertia taken at the body coordinate vary with their relative displacements. In other words:\[\mathbf{J}_{wx}=m_{ux} \begin{bmatrix}h_{wx}^{2}&0&-h_{wx}r_{wx}\\ 0&h_{wx}^{2}+r_{wx}^{2}&0\\ -h_{wx}r_{wx}&0&r_{wx}^{2}\end{bmatrix} ,\ \ \mathbf{J}_{wy}=m_{wy} \begin{bmatrix}h_{wy}^{2}+r_{wy}^{2}&0&0\\ 0&h_{wy}^{2}&-h_{wy}r_{wy}\\ 0&-h_{wy}r_{wy}&r_{wy}^{2}\end{bmatrix}  \tag{7}\]  
Moreover, consider that four nozzles are perfectly symmetric, and because the head is kept stabilized in operation, neglect the nozzles' elevation difference. The continuity equation indicates that the water flows through the nozzles are equal:  
\[\dot{m}_{1}=\dot{m}_{2}=\dot{m}_{3}=\dot{m}_{4}=\frac{1}{4}\dot{m}_{0} \tag{8}\]  
The thrust force of the system is generated by these flows, so the different behaviors of the nozzle--for example, when the head tilts or rolls--are always able to be considered as input disturbances. In summary, by substituting \(\mathbf{F_{t}}\) and \(\mathbf{\tau_{t}}\) from Equation (4), \(\mathbf{F_{wx}}\), \(\mathbf{F_{wy}}\), \(\mathbf{\tau_{wx}}\), and \(\mathbf{\tau_{wy}}\) from Equations (5)-(7), and \(\dot{m}_{i}\) from Equation (8), the equations of motion in all directions are obtained as follows:  
\[ \begin{array}{c}\ddot{x}_{b}=g\sin\theta+\frac{1}{a_{1}}\dot{m}_{0}\omega_{ yb}+d_{1},\\ \ddot{y}_{b}=-g\cos\theta\sin\varphi-\frac{1}{a_{1}}\dot{m}_{0}\omega_{xb}+d_{ 2},\\ \ddot{z}_{b}=\frac{1}{\rho a_{1}}\left(\frac{1}{A}+\frac{1}{4a}\right)\dot{m}_ {0}^{2}-g\cos\theta\cos\phi+d_{3},\\ \dot{\omega}_{xb}=\frac{1}{\rho_{1}}\left[\left(m_{wx}h_{x}+m_{wy}h_{y}\right) g\cos\theta\sin\phi-m_{wy}r_{y}g\cos\theta\cos\phi-\omega_{xb}\left(h^{2}+l^{2} \right)\dot{m}_{0}\right]+w_{1},\\ \dot{\omega}_{yb}=\frac{1}{\rho_{2}}\Big{[}m_{wx}r_{x}g\cos\theta\cos\phi+ \left(m_{wx}h_{x}+m_{wy}h_{y}\right)g\sin\theta-\omega_{yb}\left(h^{2}+l^{2} \right)\dot{m}_{0}\Big{]}+w_{2},\\ \dot{\omega}_{zb}=\frac{1}{\rho_{3}}\big{[}-2l^{2}\omega_{zb}\dot{m}_{0}\big{]} +w_{3}\end{array}  \tag{9}\]  
where:  
\[ \begin{array}{c}a_{1}=M+m_{wx}+m_{wy},\ \sigma_{1}=J_{xx}+m_{wx}h_{x}^{2}+m_{ wy}\Big{(}r_{y}^{2}+h_{y}^{2}\Big{)},\\ \sigma_{2}=J_{yy}+m_{wy}h_{y}^{2}+m_{wx}\big{(}h_{x}^{2}+r_{x}^{2}\big{)},\ \sigma_{3}=J_{zz}+m_{wx}r_{x}^{2}+m_{wy}r_{y}^{2}\end{array}  \tag{10}\]  
and:  
\[