## 2 AUV Modeling and Problem Statements
### AUV Modeling  
The AUV motions can be modeled by two sets of differential equations: kinematics and kinetics. Kinematics deal with the geometrical relationships of the AUV motion. Kinetics follow [PERSON]'s second law and analyze how the forces and moments that are generated by the thrusters cause the motion. Notably, AUVs maneuvering underwater inevitably experience ocean currents that also influence the system motions. The AUV model used in this paper is modified from the REMUS AUV introduced in [25] where the values of all the mechanical design, hydrostatic and hydrodynamic parameters of the system have been made public. For simplicity, assumptions are made for the considered AUV dynamics that  
1. the origin \(O_{\text{b}}\) of the body-fixed coordinate system \(\{b\}\) coincides with the center of gravity of the AUV;
2. the AUV moves at a constant depth which can be measured accurately with a pressure sensor;
3. the AUV is port-starboard symmetric;
4. the AUV center of buoyancy coincides with the center of gravity. The 6 Degrees Of Freedom (6 DOFs) motions are reduced to the horizontal 3-DOF motions neglecting the roll, pitch, and behave dynamics, as shown in Figure 1. Then, the 3-DOF AUV dynamics considering the influences of ocean currents are modeled following the vectorial format [14]. For notational simplicity, time dependence for system states and control inputs are left out in this section:  
\[\hat{\mathbf{\eta}} =\mathbf{R}(\mathbf{\psi})\mathbf{\
u}, \tag{1}\] \[\hat{\mathbf{v}} =(\mathbf{M}_{\text{RB}}+\mathbf{M}_{\text{A}})^{-1}(\mathbf{\tau}-(\mathbf{D}_{ L}+\mathbf{D}_{\text{NL}}+\mathbf{C}_{\text{A}})\mathbf{\
u}_{\text{r}}-\mathbf{C}_{\text{RB}} \mathbf{\
u}+\mathbf{M}_{\text{A}}\mathbf{\tau}\mathbf{S}^{\text{T}}\mathbf{\
u}_{\text{c}}), \tag{2}\]  
where \(\mathbf{\eta}=\left[ \begin{array}{ccc}x&y&\mathbf{\psi}\end{array} \right]^{\text{T}}\) indicates the pose states in the inertial coordinate system \(\{n\}\), and \(\mathbf{\
u}=\left[ \begin{array}{ccc}u&v&r\end{array} \right]^{\text{T}}\) indicates the velocity states in the body-fixed coordinate system \(\{b\}\). The more precise definition of the body-fixed AUV velocity is the velocity of the point \(O_{b}\) with respect to \(\{n\}\) expressed in \(\{b\}\). The ocean current velocity vector \(\mathbf{v}_{\text{c}}\) is also defined in the body-fixed coordinate system \(\{b\}\), as to be specified later. The controlled forces and moments are \(\mathbf{\tau}=\left[ \begin{array}{cc}\xi&Y_{uu\delta}u^{2}\delta&N_{uu\delta}u^{ 2}\delta\end{array} \right]^{\text{T}}\) where \(\xi\) and \(\delta\), i.e., the system inputs, are the propeller thrust and rudder angle, respectively, and \(Y_{uu\delta}\) and \(N_{uu\delta}\) are the corresponding hydrodynamic parameters. The rotation matrix \(R(\psi)\) relates the motions in \(\{n\}\) and \(\{b\}\):  
\[\mathbf{R}(\psi)=\left[ \begin{array}{ccc}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{array} \right],\]  
where \(\psi\) is the heading angle, and \(\mathbf{R}\) satisfies \(\frac{\text{d}}{\text{d}t}\left\{\mathbf{R}(\psi)\right\}=\dot{\psi}\mathbf{R}(\psi) \mathbf{S}\) with  
\[\mathbf{S}=\left[ \begin{array}{ccc}0&-1&0\\ 1&0&0\\ 0&0&0\end{array} \right].\]  
In Equation (2), \(\mathbf{\
u}_{\text{r}}=\mathbf{\
u}-\mathbf{v}_{\text{c}}=\left[ \begin{array}{cc}u_{ \text{r}}&v_{\text{r}}&r_{\text{r}}\end{array} \right]^{\text{T}}\) is the relative velocity in \(\{b\}\) between the AUV hull and the ocean current. For simplification, we consider constant irrotational ocean current which is denoted as \(\mathbf{b}=[V_{\text{c}},\beta_{\text{c}}]^{\text{T}}\) where \(V_{\text{c}}\) and \(\beta_{\text{c}}\) are the current speed and angle in the inertial coordinate system \(\{n\}\), respectively. Therefore, \(\beta_{\text{c}}=0\) means that the direction of the ocean current is aligned with the \(x_{\text{n}}\) axis. Then  
\[\mathbf{v}_{\text{c}}=\mathbf{R}(\psi)^{\text{T}}\left[ \begin{array}{c}V_{\text{c} }\cos\beta_{\text{c}}\\ V_{\text{c}}\sin\beta_{\text{c}}\\ 0\end{array}