# Analysis of a New Twin Hybrid Autonomous Underwater Vehicle
## 3 Mathematic Model  
In order to analyze the motion of the THAUV underwater, the coordinate system of the THAUV's motion was established as shown in Figure 4. The coordinate system includes a reference coordinate system E-XYZ and body coordinate system O-xyz, with the buoyancy  
Figure 3: Schematic diagram of roller rotation. center of the THAUV as the coordinate origin, in addition to the velocity coordinate system \(\mathbf{\pi}_{0}\) of the THAUV's motion. The position and attitude of the THAUV in the inertial coordinate system can be expressed as \(\textbf{b}= \begin{bmatrix}x&y&z\end{bmatrix} ^{T}\) and \(\mathbf{\Omega}= \begin{bmatrix}p&q&r\end{bmatrix} ^{T}\). The linear and angular velocities of the THAUV in the body coordinate system can be expressed as \(\textbf{V}= \begin{bmatrix}u&v&w\end{bmatrix} ^{T}\) and \(\mathbf{\omega}= \begin{bmatrix}p&q&r\end{bmatrix} ^{T}\). The rotation matrix from the body frame to the inertial frame is **R**. \[\textbf{R}= \begin{bmatrix}cos\phi\cos\theta&-sin\psi cos\phi+cos\psi sin\theta sin \phi&sin\psi sin\phi+cos\psi cos\phi sin\theta\\ sin\psi cos\theta&cos\psi cos\phi+sin\psi sin\theta sin\psi&-cos\psi sin\phi+ sin\psi cos\phi\\ -sin\theta&cos\theta sin\phi&cos\theta cos\phi\end{bmatrix}  \tag{8}\]  
Then, the kinematic equation of the THAUV can be expressed as:  
\[\dot{\mathbf{b}}=\textbf{R}\textbf{V} \tag{9}\]  
\[\dot{\textbf{R}}=\textbf{R}\hat{\mathbf{\omega}} \tag{10}\]  
A detailed derivation of the kinetic equations of the underwater glider is given in the literature [12; 13]. Referring to the above literature, the equations of motion governing a THAUV propagating in a three-dimensional space are expressed as:  
\[ \begin{bmatrix}\dot{\textbf{R}}\\ \dot{\textbf{b}}\\ \dot{\textbf{v}}\\ \dot{\textbf{u}}\\ \dot{\textbf{r}}_{p}\\ \dot{\textbf{P}}_{p}\\ \dot{\textbf{P}}_{p}\\ \dot{\textbf{P}}_{p}\\ \dot{\textbf{m}}_{0}\end{bmatrix} = \begin{bmatrix}\textbf{R}\hat{\mathbf{\omega}} \\ \textbf{R}\textbf{v}\\ \textbf{M}^{-1}\textbf{F}\\ \dot{\textbf{I}}^{-1}\mathbf{\tau}\\ \dot{\textbf{d}}_{p}\\ m_{p}\big{(}\textbf{v}+\mathbf{\omega}\times\mathbf{r}_{p}+\dot{\mathbf{r}}_{p}\big{)}\\ \textbf{u}_{p}\\ \textbf{u}_{0}\dot{\textbf{d}}_{p}\end{bmatrix}  \tag{11}\]  
The total moment and force are expressed as follows:  
\[\mathbf{\tau}=\big{(}\textbf{J}\mathbf{\omega}+\mathbf{\hat{r}}_{p}\textbf{P}_{p}\big{)} \times\mathbf{\omega}+\textbf{M}\textbf{v}\times\textbf{v}+\mathbf{\omega}\times\bm {r}_{p}\times\textbf{P}_{p}+m_{p}\hat{\mathbf{r}}_{p}\textbf{g}\textbf{R}^{T}\mathbf{ k}+\mathbf{\tau}_{ext}-\hat{\mathbf{r}}_{p}\textbf{u}_{p} \tag{12}\]  
\[F=\big{(}\textbf{M}\textbf{v}+\textbf{P}_{p}\big{)}\times\mathbf{\omega}+m_{0} \textbf{g}\textbf{R}^{T}\mathbf{k}+\mathbf{F}_{ext}-\textbf{u}_{p} \tag{13}\]  
Here, \(\mathbf{\tau}_{ext}\) includes the hydrodynamic moment and thruster torque, and \(\mathbf{F}_{ext}\) includes the hydrodynamic forces and thrust of thrusters. Figure 4: Coordinate system of the THAUV. The THAUV in this paper can not only perform pitching motion in the vertical plane, but it can also change the heading in the horizontal direction by using the thrusters and achieve gyratory motion by controlling the deflection angle of the roller. In this paper, we focus on the pitch motion attitude of the THAUV, where the thruster, roller, and elevator remain stationary during the pitch motion.