## 2 Preliminaries and AUV Mathematical Model
### Mathematical Model of an AUV  
In this section, the mathematical model of the 4 DOF AUV object of our analysis is addressed. Following the underwater modeling methodology proposed by [PERSON] [46], two reference frames are defined: the inertial coordinate frame (\(O_{I},X_{I},Y_{I},Z_{I}\)) and the body-fixed coordinate frame (\(O_{B},X_{B},Y_{B},Z_{B}\)), as presented in Figure 1. In order to conduct the modeling, the following assumptions are considered  
**Assumption 1**. : _By adequate placement of the buoyancy and gravity centers of the AUV in Figure 1, the roll and pitch angles are kept small, i.e., \(\phi\approx 0\) and \(\theta\approx 0\). This gives the vehicle only four degrees of freedom (4 DOF)._  
**Assumption 2**. : _The vehicle is completely actuated in the remaining 4 DOF, i.e., the translational displacements in the \(X\), \(Y\), and \(Z\) axes, and the angular movement around the \(Z\) axis._  
**Assumption 3**. : _The vehicle will operate at relatively low velocity (less than 2 m/s)._  
For the 4-DOF vehicle, the linear and angular velocities are defined in the body-fixed frame by the vector \(\mathbf{\
u}=\left[u,\,v,\,w,\,r\right]^{\top}\), whereas the three-dimensional position and the heading angle are defined in the inertial frame by the vector \(\mathbf{\eta}=\left[x,\,y,\,z,\,\psi\right]\). The kinematic equation, which relates the velocities between inertial and body frame, is given as  
\[\dot{\mathbf{\eta}}=J(\mathbf{\eta})\mathbf{\upsilon}, \tag{6}\]  
Figure 1: Reference frames: inertial (red) and body-fixed (blue). where, under Assumption 1, \(\mathbf{J}(\mathbf{\eta})\) is defined as  
\[\mathbf{J}(\mathbf{\eta})= \begin{bmatrix}\cos\psi&-\sin\psi&0&0\\ \sin\psi&\cos\psi&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{7}\]  
The underwater vehicle dynamics expressed in the body-fixed frame is obtained by using the Newton-Euler formulation and are written in matrix form as follows [46]:  
\[\mathbf{M}\dot{\mathbf{v}}+\mathbf{C}(\mathbf{\
u})\mathbf{\
u}+\mathbf{D}(\mathbf{\
u})\mathbf{\
u}+\mathbf{G}(\bm {\eta})=\mathbf{\tau}+\mathbf{\tau}_{d}, \tag{8}\]  
where \(\mathbf{M}\) is the inertia matrix, \(\mathbf{C}(\mathbf{v})\) is the Coriolis and centripetal matrix, \(\mathbf{D}(\mathbf{v})\) is the hydrodynamic damping matrix, and \(\mathbf{G}(\mathbf{v})\) is the vector of restoring forces and moments. The control input vector \(\mathbf{\tau}\) represents the vector of forces and moments exerted by the thrusters, and the vector \(\mathbf{\tau}_{d}\) denotes the environmental disturbances produced by wave- or current-induced effects. These matrices and vectors have the following form. \[\mathbf{M}= \begin{bmatrix}m-X_{\dot{u}}&0&0&0\\ 0&m-Y_{\dot{v}}&0&0\\ 0&0&m-Z_{\dot{u}}&0\\ 0&0&0&I_{zz}-N_{r}\end{bmatrix} , \tag{9}\]  
\[\mathbf{C}(\mathbf{\
u})= \begin{bmatrix}0&0&0&-(m-Y_{\dot{v}})\mathbf{\tau}\\ 0&0&0&(m-X_{\dot{u}})u\\ 0&0&0&0\\ (m-Y_{\dot{v}})\mathbf{v}&-(m-X_{\dot{u}})u&0&0\end{bmatrix} , \tag{10}\]  
\[\mathbf{D}(\mathbf{\
u})=- \begin{bmatrix}X_{u}+X_{u|u|}|u|&0&0&0\\ 0&Y_{\dot{v}}+Y_{\dot{v}|\dot{v}|}|\mathbf{v}&0&0\\ 0&0&Z_{\dot{w}}+Z_{\dot{w}|\dot{w}|}|\mathbf{w}&0\\ 0&0&0&N_{r}+N_{r|r|}|\mathbf{r}|\end{bmatrix} , \tag{11}\]  
\[\mathbf{G}(\mathbf{\eta})= \begin{bmatrix}0&0&-(W-B)&0\end{bmatrix} ^{\top}, \tag{12}\]  
\[\mathbf{\tau}= \begin{bmatrix}\tau_{x},\,\tau_{y},\,\tau_{z},\,\tau_{\dot{v}}\end{bmatrix}  ^{\top},\,\,\,\,\,\tau_{\dot{u}}= \begin{bmatrix}\tau_{x_{d}},\,\tau_{y_{d}}, \,\tau_{z_{d}},\,\tau_{\dot{v}_{d}}\end{bmatrix} ^{\top} \tag{13}\]  
The parameter \(m\) is the mass of the underwater vehicle. The parameters \(W\) and \(B\) are the weight and buoyancy forces acting on the vehicle. The hydrodynamic added mass effects are associated to the coefficients \(X_{\dot{u}}\), \(Y_{\dot{v}}\), \(Z_{\dot{w}}\), and \(N_{r}\), and the linear and nonlinear hydrodynamic damping phenomena are characterized by the coefficients \(X_{u}\), \(Y_{\dot{v}}\), \(Z_{\dot{w}}\), \(N_{r}\) and \(X_{u|u|^{\prime}}\), \(Y_{v|\dot{v}}\), \(Z_{w|\dot{w}}\), \(N_{r|r}\), respectively.