# Design and Analysis of the High-Speed Underwater Glider with a Bladder-Type Buoyancy Engine
## 3 Mathematical Model of the UG
### Mathematical Model of UG  
The hull was assumed to be a rigid body to define the mathematical model of UG. The hull-fixed coordinate system and the earth-fixed coordinate system were used, as shown in Figure 13. The right-hand coordinate system was used with the forward direction as positive x, the starboard direction as positive y, and the vertically downward direction as  
Figure 11: Relationship between maximum velocity and ballast water weight. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Value** & **Units** \\ \hline Length & 2.14 & m \\ Diameter & 0.28 & m \\ Width & 1.45 & m \\ Height & 0.348 & m \\ Weight & 108 & kg \\ Buoyancy & 108 & kg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of UG. Figure 12: Developed UG. the positive z direction. The translational and rotational motions about the UG's x, y, and z axes are expressed as six-degree-of-freedom (DOF) motions, and the coordinate axes and the motions are shown in Table 3.  
The equation of motion of the UG uses the six-DOF motion equation of a general submersible, as shown in Equation (8) [24]. The equations of motion are expressed in terms of forces and moments because of a significant number of hydrodynamic coefficients. Generally, the fluid force coefficients included in the fluid force on the correct side use values obtained through constraint model tests such as CFD analysis, planar motion mechanism (PMM), or empirical equations. \[ \begin{split}&\text{m}\left[\dot{u}-vr+wq-x_{\text{C}}\big{(}q^{2 }+r^{2}\big{)}+y_{\text{C}}\big{(}pq-\dot{r}\big{)}+z_{\text{C}}\big{(}pr+\dot{q }\big{)}\right]=X\\ &\text{m}\left[\dot{w}-wp+ur-y_{\text{C}}\big{(}r^{2}+p^{2}\big{)} +z_{\text{C}}\big{(}qr-\dot{p}\big{)}+x_{\text{C}}\big{(}qp+\dot{r}\big{)} \right]=Y\\ &\text{m}\left[\dot{w}-uq+vp-z_{\text{C}}\big{(}p^{2}+q^{2}\big{)} +x_{\text{C}}\big{(}pq-\dot{r}\big{)}+y_{\text{C}}\big{(}pr+\dot{q}\big{)} \right]=Z\\ & I_{x}\dot{p}+\big{(}I_{z}-I_{y}\big{)}qr+m\big{[}y_{\text{G}} \left(\dot{w}-uq+vp\right)-z_{\text{C}}\big{(}\dot{v}-wp+ur\big{)}\big{]}=K\\ & I_{y}\dot{q}+(I_{x}-I_{z})rp+m\big{[}z_{\text{C}}\big{(}\dot{u} -vr+wq\big{)}-x_{\text{C}}\big{(}\dot{w}-uq+vp\big{)}\big{]}=M\\ & I_{z}\dot{r}+\big{(}I_{y}-I_{x}\big{)}pq+m\big{[}x_{\text{G}} \big{(}\dot{v}-wp+up\big{)}-y_{\text{C}}\big{(}\dot{u}-vr+wq\big{)}\big{]}=N \end{split}  \tag{8}\]  
Here, _u, v, w_ and _p, q, r_ represent the UG's angular velocity of translational and rotational motion for the axes _x, y,_ and \(z\), respectively. _Iij_ represents the mass moment of inertia of the UG for the axis of each subscript where _xG_, _yG_, and _zG_ represent the location of the UG mass center. _X, Y, Z, K, M, and N represent the external forces and  
Figure 13: Coordinates frame of underwater glider. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Classification** & **Axis** & **Motion** &
\begin{tabular}{c} **Fore and** \\ **Moment** \\ \end{tabular} & **Velocity** & **Displacement** \\ \hline \multirow{3}{*}{Translational motion} & x & Surge & \(X\) & \(u\) & \(x\) \\  & y & Sway & \(Y\) & \(v\) & \(y\) \\  & z & Heave & \(Z\) & \(w\) & \(z\) \\ \hline \multirow{3}{*}{Rotational motion} & x & Roll & \(K\) & \(\mu\) & \(\varphi\) \\  & y & Pitch & \(M\) & \(q\) & \(\theta\) \\  & z & Yaw & \(N\) & \(r\) & \(\psi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: 6 DOF motion of the underwater vehicle. moment working on the UG for each motion direction, such as thrust, buoyancy, gravity, and hydrodynamic force [25]. The external force term can be divided into the contributing forces, as shown in the following Equations (9) and (10). Hydrostatic, inertial fluid, damping fluid, and control fluid forces, as well as the respective moments result in:  
\[\sum F_{0}=F_{hydrostatic}+F_{inertial}+F_{damping}+F_{control} \tag{9}\]  
\[\sum M_{0}=M_{hydrostatic}+M_{inertial}+M_{damping}+M_{control} \tag{10}\]  
The following fluid force coefficients were obtained using empirical formulas for the hull shape of the designed underwater glider [23].