## 3 Kinematic and Dynamic Modeling for AURVs
Damping effects are highly nonlinear and coupled; for underwater vehicles, to a first approximation, coupling effects and terms higher than the second order (with respect to body-fixed velocity) can be neglected, which in turn equals to assume a diagonal structure for \(D(\mathbf{\
u})\). Finally, since quadratic terms dominate over linear terms, \(D(\mathbf{\
u},\mathbf{q})\) can be approximated with  
\[D(\mathbf{\
u},\mathbf{q})=-\operatorname{diag}_{6\times 6}\bigl{\{}\cdots X_{i}(\mathbf{q} )|\
u_{1,i}|\cdots X_{j}(\mathbf{q})|\
u_{2,j}|\cdots\bigr{\}}\quad,\quad i=j=x,y,z\quad, \tag{9}\]  
where \(\operatorname{diag}_{n\times n}(\cdot)\) is the operator which builds a \(n\)-by-\(n\) diagonal matrix from a vector \(\mathbf{\bar{a}}\in\mathbb{R}^{n}\) such as  
\[\operatorname{diag}_{n\times n}\mathbf{\bar{a}}=\left[ \begin{array}{ccc}\bar{a }_{1}&0&0\\ 0&\ddots&0\\ 0&0&\bar{a}_{n}\end{array} \right]\quad. \tag{10}\]  
For the sake of completeness, the angular damping coefficients have been neglected in this work (i.e., \(X_{j}(\mathbf{q})|\
u_{2,j}|=\mathbf{0}_{3\times 1}\)), while the first three terms in Equation (9) can be calculated as described in [1] with  
\[D_{i}(\
u_{1,i},\mathbf{q})=X_{i}(\mathbf{q})|\
u_{1,i}|=\frac{1}{2}\rho A_{f,i}(\mathbf{q })C_{D,i}(\mathbf{q})|\
u_{1,i}|\quad,\quad i=x,y,z\quad, \tag{11}\]  
where \(\rho\) represents the water density, \(A_{f,i}\) is the projection of the area of the hull of the vehicle on a plane perpendicular to the \(i\)-axis of the body-fixed frame, and \(C_{D,i}\) is the drag coefficient, which quantifies the fluid resistance against the vehicle motion. Additionally, the vector \(\mathbf{g}(\mathbf{\eta},\mathbf{q})\) of generalized forces due to the gravity \({}^{b}\mathbf{f}_{G}\) and buoyancy forces \({}^{b}\mathbf{f}_{B}\) in the body-fixed frame is represented by  
\[\mathbf{g}(\mathbf{\eta},\mathbf{q})\;=\;-\left[ \begin{array}{c}{}^{b}\mathbf{f}_{G}( \mathbf{\eta}_{2},\mathbf{q})+{}^{b}\mathbf{f}_{B}(\mathbf{\eta}_{2},\mathbf{q})\\ {}^{b}\mathbf{r}_{G}\times{}^{b}\mathbf{f}_{G}(\mathbf{\eta}_{2},\mathbf{q})+{}^{b}\mathbf{r}_{B} \times{}^{b}\mathbf{f}_{B}(\mathbf{\eta}_{2},\mathbf{q})\end{array} \right]\quad, \tag{12}\]  
where \({}^{b}\mathbf{r}_{G}\) and \({}^{b}\mathbf{r}_{B}\) represent the positions of the center of gravity and the center of buoyancy with respect to the body-fixed frame. In the case of vehicles whose center of mass and center of buoyancy are constant w.r.t. the joint configuration \(\mathbf{q}\), the expression of the gravity vector and the buoyancy vector are simply:  
\[{}^{b}\mathbf{f}_{G}(\mathbf{\eta}_{2},\mathbf{q})=^{b}\mathbf{f}_{G}(\mathbf{\eta}_{2})=R_{N}^{b }\left[ \begin{array}{c}0\\ 0\\ m\big{\|}\mathbf{g}^{I}\big{\|}\end{array} \right]\quad, \tag{13}\]  
where \(\mathbf{g}^{I}\) is the gravity acceleration vector expressed in the NED frame, \(\
abla\) is the volume of the vehicle submerged body, and \(\rho\) is the water density. In the context of this work, the center of gravity and buoyancy have been assumed as coincident with the center of mass (i.e., with the origin of the body-fixed frame), any torque due to gravity/buoyancy force will be applied to the vehicle and Equation (12) is approximated as  
\[\mathbf{g}(\mathbf{\eta},\mathbf{q})=\mathbf{g}(\mathbf{\eta})=-\left[ \begin{array}{c}{} ^{b}\mathbf{f}_{G}(\mathbf{\eta}_{2})+{}^{b}\mathbf{f}_{B}(\mathbf{\eta}_{2})\\ \mathbf{0}_{3\times 1}\end{array} \right]\quad. \tag{14}\]  
Furthermore, the study of marine vehicles cannot neglect the effects of specific disturbances such as ocean current [17]. Simplified modeling of the current effect can be obtainedby assuming the current irrotational and constant in the Earth-fixed frame; thus, its effect on the vehicle can be modeled as a constant disturbance in the Earth-fixed frame that is further projected onto the vehicle-fixed frame. Indeed the ocean current, expressed in the NED frame, \({{}^{N}\mathbf{v}_{c}}\) can be represented as,  
\[{{}^{N}\mathbf{v}_{c}}=\left[