## 2 Preliminaries and AUV Mathematical Model
### Mathematical Model of an AUV
The added mass and damping coefficients can be considered constant for maneuvering applications [47]. From the kinematic Equation (6), we can obtain  
\[\dot{\mathbf{\eta}}=\mathbf{J}(\mathbf{\eta})\mathbf{J}(\mathbf{\eta})^{-1}\dot{\mathbf{\eta}}+\mathbf{J}( \mathbf{\eta})\dot{\mathbf{v}}. \tag{14}\]  
Then, the underwater vehicle dynamics in Equation (8) expressed in the body-fixed frame can be expressed in the inertial frame as  
\[\mathbf{M}_{\mathbf{\eta}}(\mathbf{\eta})\dot{\mathbf{\eta}}+\mathbf{C}_{\mathbf{\eta}}(\mathbf{\eta},\dot {\mathbf{\eta}})\dot{\mathbf{\eta}}+\mathbf{D}_{\mathbf{\eta}}(\mathbf{\eta},\dot{\mathbf{\eta}})\dot{ \mathbf{\eta}}+\mathbf{G}_{\mathbf{\eta}}(\mathbf{\eta})=\mathbf{\tau}_{\mathbf{\eta}}+\mathbf{\tau}_{d\bm {\eta}}. \tag{15}\]where  
\[ \begin{split} M_{\eta}(\eta)&=J(\eta)^{-1}MJ(\eta)^{-1}\\ C_{\eta}(\eta,\dot{\eta})&=J(\eta)^{-1}\Big{[}C\Big{(} I(\eta)^{-1}\dot{\eta}\Big{)}-MJ(\eta)^{-1}\dot{J}(\eta)^{-1}\Big{]}J(\eta)^{-1}\\ D_{\eta}(\eta,\dot{\eta})&=J(\eta)^{-1}D\Big{(}I( \eta)^{-1}\dot{\eta}\Big{)}J(\eta)^{-1}\\ G_{\eta}(\eta)&=J(\eta)^{-1}G(\eta)\\ \tau_{\eta}&=J(\eta)^{-1}\tau\\ \tau_{dq}(t)&=J(\eta)^{-1}\tau_{d}(t)\end{split}  \tag{16}\]  
It is important to mention that the model presented in Equation (15) is the most commonly used representation for an AUV in the specialized literature. From Equation (15) and Assumptions 1-3, we can derive input-affine nonlinear equations that governs the 4 DOF AUV, as follows:  
\[\dot{x}= c\psi f_{1}(\dot{x},\dot{y},\psi,\dot{y})+c\psi f_{3}(\dot{x},\dot{y},\psi)+s\psi f_{2}(\dot{x},\dot{y},\psi,\dot{y})+s\psi f_{4}(\dot{x},\dot{y}, \psi)(s\psi\dot{x}-c\psi\dot{y})-\dot{y}\dot{y}+\] \[c\psi\frac{\tau_{x}+\tau_{x_{d}}}{m-X_{\dot{x}}}-s\psi\frac{\tau _{y}+\tau_{y_{d}}}{m-Y_{\dot{y}}} \tag{17}\] \[\dot{y}= s\psi f_{1}(\dot{x},\dot{y},\psi,\dot{y})+s\psi f_{3}(\dot{x},\dot{y},\psi)-c\psi f_{2}(\dot{x},\dot{y},\psi,\dot{y})+c\psi f_{4}(\dot{x},\dot{y}, \psi)(-s\psi\dot{x}+c\psi\dot{y})+\dot{x}\dot{y}+\] \[s\psi\frac{\tau_{x}+\tau_{x_{d}}}{m-X_{\dot{x}}}+c\psi\frac{\tau _{y}+\tau_{y_{d}}}{m-Y_{\dot{y}}}\] (18) \[\dot{z}= f_{5}(z)+\frac{\tau_{z}+\tau_{zd}}{m-Z_{\dot{x}}}\] (19) \[\dot{\psi}= f_{6}(\dot{x},\dot{y},\psi)+f_{7}(\psi)+\frac{\tau_{\psi}+\tau_ {\psi_{d}}}{I_{zz}-N_{t}} \tag{20}\]  
where  
\[f_{1}(\dot{x},\dot{y},\psi,\dot{y})= \frac{(m-Y_{\dot{y}})(\dot{y}c\psi-\dot{x}s\psi)\dot{y}}{m-X_{\dot {x}}}\] \[f_{2}(\dot{x},\dot{y},\psi,\dot{y})= \frac{(m-X_{\dot{x}})(\dot{x}c\psi+\dot{y}s\psi)\dot{y}}{m-Y_{\dot {y}}}\] \[f_{3}(\dot{x},\dot{y},\psi)= \frac{X_{u}+X_{u|u|}|\dot{x}c\psi+\dot{y}s\psi|}{m-X_{\dot{u}}}\] \[f_{4}(\dot{x},\dot{y},\psi)= \frac{Y_{\psi}+Y_{\psi|v|}|-\dot{x}s\psi+\dot{y}c\psi|}{m-Y_{\dot {y}}}\] \[c\psi\frac{\tau_{x}+\tau_{x_{d}}}{m-X_{\dot{x}}}-s\psi\frac{\tau _{y}+\tau_{y_{d}}}{m-Y_{\dot{y}}} \tag{17}\]0 \[c\psi\frac{\tau_{x}+\tau_{x_{d}}}{m-X_{\dot{x}}}-s\psi\frac{\tau _{y}+\tau_{y_{d}}}{m-Y_{\dot{y}}} \tag{17}\]1 \[c\psi\frac{\tau_{x}+\tau_{x_{d}}}{m-X_{\dot{x}}}-s\psi\frac{\tau _{y}+\tau_{y_{d}}}{m-Y_{\dot{y}}} \tag{17}\]2  
and \(s\psi:=sin\psi\) and \(c\psi:=cos\psi\). Equations (17)-(20) represent the nonlinear equations of a completely coupled 4 DOF AUV system. We can notice that the Equations (19) and (20) for \(z\) and heading dynamics are decoupled from the \(x\) and \(y\) dynamics, so it is possible to control them separately. In this sense, for control purposes, first, the yaw angle dynamics Equation (20) is stabilized at zero, i.e., \(\psi=0\), \(\dot{\psi}=0\). It is worth mentioning that the procedure involving stabilization of the orientation of the vehicle as a first step in order to reduce the complexity of the control design is an approach that has been used for several studies related to autonomous quadrotor ([48; 49]) and underwater ([50]) vehicles.