## 4 System Modeling  
### Coordinate Systems and Kinematic Relationships  
In order to analyze the motion characteristics of each design, mathematical models are formulated in this Section. For the sake of modeling, two Cartesian coordinates are defined and considered: the Earth reference and the head part fixed coordinates. A curvilinear coordinate is placed along the centerline of the water-conveying hose, with the beginning at the outlet of the pump and the ending at the head inlet. The coordinate systems are illustrated in Figure 3.  
Figure 3: Coordinate systems definition. The kinematic relationship between two Cartesian coordinates is given by the following transformation matrix:  
\[ \begin{bmatrix}\mathbf{v}\\ \dot{\mathbf{\varphi}}\end{bmatrix} = \begin{bmatrix}\mathbf{R}_{E}^{b}&\mathbf{O}_{3}\\ \mathbf{O}_{3}&\mathbf{R}_{Euler}^{b}\end{bmatrix}\begin{bmatrix}\mathbf{v}_{b}\\ \mathbf{\omega}_{b}\end{bmatrix}  \tag{1}\]  
where the dot stands for the differential with respect to time (the time variable is neglected in the remaining for simplicity). Hence, \(\mathbf{v}= \begin{bmatrix}\dot{X}&\dot{Y}&\dot{Z}\end{bmatrix} ^{T}\) and \(\mathbf{v}_{b}= \begin{bmatrix}\dot{X}_{b}&\dot{Y}_{b}&\dot{Z}_{b}\end{bmatrix} ^{T}\) are the translational velocity vectors of the head in the reference frame and body frame, respectively. \(\dot{\mathbf{\varphi}}= \begin{bmatrix}\dot{\mathbf{\varphi}}&\dot{\mathbf{\theta}}&\dot{ \mathbf{\psi}}\end{bmatrix} ^{T}\) is the vector of Euler angle rates in \(XYZ\) rotation order, and \(\mathbf{\omega}_{b}= \begin{bmatrix}\omega_{xb}&\omega_{yb}&\omega_{zb}\end{bmatrix}  ^{T}\) is the vector of the corresponding angular velocities. \(\mathbf{R}_{E}^{b}\) and \(\mathbf{R}_{Euler}^{b}\) are the rotation matrices from the body frame to the reference frame and Euler angles, respectively. \(\mathbf{O}_{n}\) is the \(n\)-by-\(n\) zero matrix. For the sake of clarity, the subscript \(b\) in the remaining of the paper is for the body-fixed frame. Subsequently, the velocity of the water flow at an inlet or outlet \(j\), which is positioned by the vector \(\mathbf{r}_{jb}\) in the body-fixed coordinate, namely \(\mathbf{V}_{jb}\), is given by:  
\[\mathbf{V}_{jb}=V_{j}\overset{\wedge}{\mathbf{n}}_{jb}+\mathbf{\omega}_{b}\times\mathbf{r}_{jb }+\mathbf{v}_{b}, \tag{2}\]  
where \(j=h,\ l,\ i\ (i=1\sim 4)\) indicate the head inlet section, the outlet of the fire extinguishing nozzle, and the \(i\)th thrust nozzle's outlet, respectively. \(V_{j}\) is the axial velocity of the water flowing through the \(j\) section and \(\overset{\wedge}{\mathbf{n}}_{jb}\) is the normal vector of the section. The direction of \(\overset{\wedge}{\mathbf{n}}_{jb}\) follows the direction of the water flow going through the area. Especially, the normal vector of the head inlet, i.e., \(\overset{\wedge}{\mathbf{n}}_{h}\) observed in the reference frame, is also the tangent vector of the water-conveying hose at this end. It is noted that the inlet port is designed such that it locates at the origin of the coordinate. Hence, \(V_{j}\overset{\wedge}{\mathbf{n}}_{jb}\) is the relative velocity of the jetted water with respect to the head center. The cross-product term \(\mathbf{\omega}_{b}\times\mathbf{r}_{jb}\) is the change in the water's velocity due to the head rotation \(\mathbf{\omega}_{b}\). Finally, \(\mathbf{v}_{b}\) is the head translation contributing to the absolute velocity of the water flow. On the other hand, let \(\mathbf{r}_{s}= \begin{bmatrix}X(s)&Y(s)&Z(s)\end{bmatrix} ^{T}\) define the location of the hose element at \(s\) curvilinear coordinate. The initial value of \(\mathbf{r}_{s}\) is denoted by \(\mathbf{r}_{s0}\). The hose has a uniform cross-sectional area so the water velocity at any point is equal to that at the head inlet, i.e., \(V_{h}\).