## 3 Kinematic and Dynamic Modeling for AURVs
With regard to the complete inverse dynamic model of the AURV, the well-known equation describing the balance of the forces acting on the center of mass of the vehicle [16; 17] has been extended to take into account several configurations of the robot by introducing the joint vector \(\mathbf{q}\):  
\[M(\mathbf{q})\dot{\mathbf{v}}+C(\mathbf{v},\mathbf{q})\mathbf{v}+D(\mathbf{v},\mathbf{q})\mathbf{v}+\mathbf{g }(\mathbf{\eta},\mathbf{q})=\mathbf{\tau}(\mathbf{u},\mathbf{v},\mathbf{q})\quad, \tag{4}\]  
where \(M(\mathbf{q})\) describes the mass matrix, \(C(\mathbf{v},\mathbf{q})\) outlines the centrifugal and Coriolis matrix, \(D(\mathbf{v},\mathbf{q})\) reports the damping matrix, \(\mathbf{g}(\mathbf{\eta},\mathbf{q})\) are the effects of gravity and buoyancy, \(\mathbf{\tau}(\mathbf{u},\mathbf{v},\mathbf{q})\) indicates the forces and torques provided by the rotational speeds \(\mathbf{u}\in\mathbb{R}^{m}\) of \(m\) thrusters, \(\mathbf{\eta}\) and \(\mathbf{v}\) as defined in Equation (3). As already mentioned, a preliminary modelhas been developed for the two configurations of the UNIFI DIEF AURV, by assuming each of them as an AUV with a different shape. Turning to the quantities on the left-hand side of Equation (4), it can be observed that there are the sum of multiple contributions due to the motion of the vehicle and to its interaction with the surrounding fluid; hereafter, these terms are comprehensively analyzed while listing the assumptions considered in the proposed research activity. In greater detail, the matrix \(M(\mathbf{q})\) comprised of the sum of two terms: \(M=M_{RB}+M_{A}\), where \(M_{RB}\) takes into account the physical properties of the vehicle, while \(M_{A}\) is usually denoted with the term added mass matrix, and describes the fact that due to the higher density (with respect to air) of the fluid where the vehicle moves in, larger accelerations are required to move not only the vehicle, but also the surrounding fluid itself. Under the assumption that the added mass matrix can be neglected, being the body-fixed frame centered in the center of gravity of the vehicle, the matrix and inertia matrix can be represented as  
\[M(\mathbf{q})=M_{RB}(\mathbf{q})=\left[ \begin{array}{cc}mI_{3\times 3}&0_{3\times 3} \\ 0_{3\times 3}&I_{O_{b}}(\mathbf{q})\end{array} \right]\quad, \tag{5}\]  
where \(m\) is the mass of the vehicle, \(I_{3\times 3}\) is the identity \(3\times 3\) matrix, and \(I_{O_{b}}\) is the inertia tensor expressed in the body-fixed frame. A further simplification is obtained by assuming that the body-fixed reference axes coincide with the principal axes of inertia. This implies that \(I_{O_{b}}\) becomes diagonal, and Equation (5) simplifies as  
\[M(\mathbf{q})=\left[ \begin{array}{ccccc}m&0&0&0&0&0\\ 0&m&0&0&0&0\\ 0&0&m&0&0&0\\ 0&0&0&I_{x}(\mathbf{q})&0&0\\ 0&0&0&0&I_{y}(\mathbf{q})&0\\ 0&0&0&0&0&I_{z}(\mathbf{q})\end{array} \right] \tag{6}\]  
where \(I_{i}\) is the moment of inertia about the \(i\)th axes of the body-fixed frame. Within the same hypothesis, the centrifugal and Coriolis matrix \(C(\mathbf{\
u},\mathbf{q})\) can be simplified as  
\[C(\mathbf{\
u},\mathbf{q})=C_{RB}(\mathbf{\
u},\mathbf{q})=\left[ \begin{array}{cc}0_{3\times 3 }&-mS(\mathbf{\
u}_{1})\\ -mS(\mathbf{\
u}_{1})&-S(I_{O_{b}}(\mathbf{q})\cdot\mathbf{\
u}_{2})\end{array} \right]\quad, \tag{7}\]  
where \(S(\cdot)\) is the operator that builds a 3-by-3 skew-symmetric matrix from a vector \(\mathbf{\bar{a}}\in\mathbb{R}^{3}\) such as  
\[S(\mathbf{\bar{a}})=\left[ \begin{array}{ccc}0&-\bar{a}_{z}&\bar{a}_{y}\\ \bar{a}_{z}&0&-\bar{a}_{x}\\ -\bar{a}_{y}&\bar{a}_{x}&0\end{array} \right]\quad. \tag{8}\]  
Figure 3: The Society of Naval Architects and Marine Engineer (SNAME) notation. NED and body-fixed frames. Furthermore, the matrix \(D(\mathbf{\
u},\mathbf{q})\) models the dissipative effects due to the motion within the fluid; the main damping contributions are given by the nonlinear skin friction due to turbulent boundary layers, and by the viscous damping force due to vortex shedding.