# 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
Substituting each coefficient into an external force component, Equation (11) results in:  
\[ \begin{split}\text{m}\big{[}\dot{u}-vr+wq&-x_{G} \big{(}q^{2}+r^{2}\big{)}+y_{G}\big{(}pq-\dot{r}\big{)}+z_{G}\big{(}pr+\dot{q} \big{)}\big{]}\\ &=X_{HS}+X_{u|u|}\,u|\,u|+X_{\dot{u}}\dot{u}+X_{uq}wq+X_{qq}qq+X_ {vv}vr+X_{rr}rr\\ \text{m}\big{[}\dot{v}-wp+ur&-y_{G}\big{(}r^{2}+p^{2 }\big{)}+z_{G}\big{(}qr-\dot{p}\big{)}+x_{G}\big{(}qp+\dot{r}\big{)}\big{]}\\ &=Y_{HS}+Y_{v|v|}\,v|+Y_{r|r}\,r|\,r|+Y_{\dot{v}}\dot{v}+Y_{r} \dot{r}+Y_{ur}ur+Y_{wp}wp+Y_{pq}pq+Y_{wp}uv\\ \text{m}\big{[}\dot{w}-uq+vp&-z_{G}\big{(}p^{2}+q^{ 2}\big{)}+x_{G}\big{(}pq-\dot{r}\big{)}+y_{G}\big{(}pr+\dot{q}\big{)}\big{]}\\ &=Z_{HS}+Z_{w|w|}\,w|+Z_{q|q|}\,q|\,q|+Z_{\dot{w}}\dot{w}+Z_{uq} \dot{q}+Z_{up}vp+Z_{rp}rp+Z_{uw}uw\\ \text{l}_{x}\dot{p}+\big{(}I_{z}-I_{y}\big{)}qr&+ m\big{[}y_{G}\big{(}\dot{w}-uq+vp\big{)}-z_{G}\big{(}\dot{v}-wp+u\big{)} \big{]}=K_{HS}+K_{p|p|}\,p|\,p|+K_{\dot{p}}\dot{p}\\ \text{l}_{y}\dot{q}+(I_{x}-I_{z})rp&+m\big{[}z_{G} \big{(}\dot{u}-vr+wq\big{)}-x_{G}\big{(}\dot{w}-uq+vp\big{)}\big{]}\\ &=M_{HS}+M_{w|w|}\,w|\,w|+M_{q|q|}\,q|\,q|+M_{\dot{w}}\dot{w}+M_ {\dot{q}}\dot{q}+M_{uq}uq+M_{wp}vp+M_{rp}rp+M_{uw}uw\\ &+M_{uw\dot{s}}u^{2}\dot{\delta}_{s}\\ \text{l}_{z}\dot{r}+\big{(}I_{y}-I_{x}\big{)}pq&+ m\big{[}x_{G}\big{(}\dot{v}-wp+up\big{)}-y_{G}\big{(}\dot{u}-vr+wq\big{)} \big{]}\\ &=N_{HS}+N_{v|v|}\,v|+N_{r|r|}\,r|+N_{\dot{v}}\dot{v}+N_{r}\dot{r }+N_{ur}ur+N_{wp}wp+N_{pq}pq+N_{uw}uv+N_{uu\dot{s}}u^{2}\dot{\delta}_{r}\end{split}  \tag{11}\]  
The center of mass (\(r_{G}\)), mass moment of inertia (\(l_{t}\)), and center of buoyancy (\(r_{B}\)) change in real time, owing to the movement of the internal mass and the buoyancy engine. Due to the movement of the internal moving mass and the movement of the buoyancy engine, the center of mass, mass moment of inertia, and center of buoyancy and buoyancy force of the UG change in real time. Since the speed of the motion of the moving mass and the buoyancy engine is very slow, the dynamic effect of these two masses on the activity of the entire UG can be ignored. Therefore, in this study, the dynamic coupling between the movement of the UG and the movement of the internal moving mass and buoyancy engine was excluded, where each was considered as an independent movement, and a dynamic modeling of the entire system was performed, including only of the kinematic effects. \[r_{G}(t)=\frac{m_{h}r_{h}+m_{s}r_{s}+m_{m}r_{m}(t)}{m_{t}} \tag{12}\]  
Here, \(m_{t}\) is the total mass of the UG, \(m_{h}\) is the mass of the external hull of the UG, \(m_{s}\) is the mass of the outer hull of the UG and fixed components such as the internal control board and sensor module, excluding the inner variable mass, and \(m_{m}\) represents the internal variable mass, respectively. \(r_{h}\), \(r_{s}\), and \(r_{m}\) are the center of mass of the hull from the origin of the hull's fixed coordinate system, the position vectors from the center of mass of the fixed mass, and the center of mass of the internal moving mass, respectively. In addition, the total mass moment of inertia of the underwater glider can be obtained as Equation (13):  
\[I_{t}(t)=(I_{h}-m_{h}\hat{r}_{h}\hat{r}_{h})+(I_{s}-m_{s}\hat{r}_{s}\hat{r}_{s})+( I_{m}-m_{m}\hat{r}_{m}(t)\hat{r}_{m}(t)) \tag{13}\]  
Here, \(I_{h}\), \(I_{s}\), and \(I_{m}\) represent the mass moment of inertia of the hull, the mass moment of inertia of the fixed mass, and the mass moment of inertia of the internal moving mass expressed at each center of mass, respectively. The operator \(\hat{\ }\) is an operator that converts a vector into a symmetric matrix to express the cross-product of vectors.