## 2 Problem Formulation and Preliminaries
### Kinematic and Dynamic Modeling  
The kinematic and dynamic model of the underactuated AUV with five degrees of freedom is defined as follows:  
\[\hat{\mathbf{\eta}}=\mathbf{J}(\mathbf{\eta})\mathbf{\
u} \tag{11}\]  
\[\mathbf{M}\hat{\mathbf{v}}+\mathbf{C}(\mathbf{\
u})\mathbf{\
u}+\mathrm{D}(\mathbf{\
u})\mathbf{\
u}+\mathbf{ G}(\mathbf{\eta})+\mathbf{f}=\mathbf{\tau} \tag{12}\]  
In Figure 1, Here, \(\mathbf{\eta}=\left[\mathbf{\eta}_{1};\mathbf{\eta}_{2}\right]^{T}\) represents the position and orientation vector in the earth-fixed frame, where \(\mathbf{\eta}_{1}=\left[x,y,z\right]^{T}\) denotes the position and \(\mathbf{\eta}_{2}=\left[\theta,\psi\right]^{T}\) represents the orientation. The velocity vector \(\mathbf{v}=\left[\mathbf{v}_{1};\mathbf{v}_{2}\right]^{T}\) includes \(\mathbf{v}_{1}=\left[u,v,w\right]^{T}\), the linear velocities, and \(\mathbf{v}_{2}=\left[q,r\right]^{T}\), the angular velocities, both expressed in the body-fixed frame. The reference frames are denoted as \(\{E\}=\left(x_{E},y_{E},z_{E}\right)\) for the earth-fixed frame and \(\{B\}=\left(x_{B},y_{B},z_{B}\right)\) for the body-fixed frame. \(\mathbf{J}(\mathbf{\eta})\in\mathbb{R}^{5\times 5}\) signifies the transformation matrix,  
Figure 1: Earth-fixed frame and body-fixed frame. \(\mathbf{M}\in\mathbb{R}^{5\times 5}\) represents the inertia matrix, \(\mathbf{C}(\mathbf{v})\in\mathbb{R}^{5\times 5}\) denotes the Coriolis and centripetal forces, and \(\mathbf{D}(v)\in\mathbb{R}^{5\times 5}\) indicates the hydrodynamic drag, while \(\mathbf{G}(\mathbf{\eta})\in\mathbb{R}^{5}\) accounts for gravitational and buoyancy forces. The term \(\mathbf{f}\in\mathbb{R}^{5}\) denotes external disturbances, and \(\mathbf{\tau}\in\Re^{5}\) represents the control inputs. The detailed expressions of \(\mathbf{J}(\mathbf{\eta}),\mathbf{M},\mathbf{C}(\mathbf{v}),\mathrm{D}(\mathbf{v}),\mathbf{G},\mathbf{\tau}\) are as follows. \[\mathbf{J}= \begin{bmatrix}J_{1}&\mathbf{O}_{3\times 2}\\ \mathbf{O}_{2\times 3}&J_{2}\end{bmatrix} ,\mathbf{J}_{1}= \begin{bmatrix}\cos\psi\cos\theta&- \sin\psi&\cos\psi\sin\theta\\ \sin\psi\cos\theta&\cos\psi&\sin\psi\sin\theta\\ -\sin\theta&0&\cos\theta\end{bmatrix} ,\mathbf{J}_{2}= \begin{bmatrix}1&0\\ 0&1/\cos\theta\end{bmatrix}  \tag{13}\]  
\[\mathbf{M}=diag\Big{(}\Big{[}M-X_{i\
u},M-Y_{i\
u},M-Z_{i\
u}I_{yy}-M_{i\
u}I_{ zz}-N_{i}\Big{]}\Big{)} \tag{14}\]  
\[\mathbf{C}(\mathbf{v})= \begin{bmatrix}0&0&0&(M-Z_{i\
u})w&-(M-Y_{i\
u})v\\ 0&0&0&0&(M-X_{i\
u})u\\ 0&0&0&-(M-X_{i\
u})u&0\\ -(M-Z_{i\
u})w&0&(M-X_{i\
u})u&0&0\\ (M-Y_{\
u})v&-(M-X_{i\
u})u&0&0&0\end{bmatrix}  \tag{15}\]  
\[\mathbf{D}(\mathbf{v})=diag\Big{(}\Big{[}X_{u|u}\Big{|}u\Big{|}+X_{u},Y_{i\
u|u|}\Big{|} v\Big{|}+Y_{\
u},Z_{w|u|}\Big{|}w\Big{|}+Z_{u\
u},M_{q|q|}\Big{|}u\Big{|}+M_{q}, N_{i\
u|}\Big{|}r\Big{|}+N_{r}\Big{]}\Big{)} \tag{16}\]  
\[\mathbf{G}= \begin{bmatrix}0,0,0,Z_{B}G\sin\theta,0\end{bmatrix} ^{T} \tag{17}\]  
\[\mathbf{f}= \begin{bmatrix}f_{1},0,0,f_{2},f_{3}\end{bmatrix} ^{T} \tag{18}\]  
\[\mathbf{\tau}= \begin{bmatrix}\tau_{u},0,0,\tau_{q},\tau_{r}\end{bmatrix} ^{T} \tag{19}\]  
This paper simplifies the controller design by decoupling the underactuated AUV system into the actuating subsystem and the non-actuating subsystem. The model of the actuating subsystem is established as follows. [23]  
\[\dot{\mathbf{\eta}}_{1}=\mathbf{J}_{1}(\mathbf{\eta}_{1})\mathbf{v} \tag{20}\]  
\[\mathbf{M}_{a}\dot{\mathbf{v}}+\mathbf{C}_{a}(\mathbf{v})\mathbf{v}+\mathbf{D}_{a}(\mathbf{v})\mathbf{v}+\mathbf{G}_ {a}+\mathbf{f}_{a}=\tau_{a} \tag{21}\]  
\[\mathbf{v}= \begin{bmatrix}u,q,r\end{bmatrix} ^{T},\mathbf{M}_{a},\mathbf{C}_{a},\mathbf{D}_{a}, \mathbf{G}_{a},\mathbf{f}_{a},\mathbf{\tau}_{a}\text{ as follows:}\]  
\[\mathbf{M}_{a}=diag\Big{(}\Big{[}M-X_{i\
u},I_{yy}-M_{i\
u}I_{zz}-N_{i}\Big{]} \Big{)} \tag{22}\]  
\[\mathbf{C}_{a}= \begin{bmatrix}0&(M-Z_{ii})w&-(M-Y_{i\
u})v\\ (Z_{i\
u}-X_{i\
u})w&0&0\\ (X_{i\
u}-Y_{i\
u})v&0&0\end{bmatrix}