## 3 Kinematic and Dynamic Modeling for AURVs  
The kinematic and dynamic description of the UNIFI DIEF AURV is briefly discussed in this Section; for the sake of completeness, the two different and extreme vehicle configurations have been studied in a decoupled manner, without the development of a multibody model [14; 15] (whose development is, nonetheless, planned as a strategic solution for future research activities). In light of these considerations, the kinematic models, commonly exploited for AUVs by the scientific community [16; 17], have been employed for both the hovering and survey configurations. To be more specific, in the following, we assume that once one of the two extreme configurations, described by the joint coordinate vector \(\mathbf{q}\), has been reached, it does not change dynamically and the reconfigurable vehicle can be considered as a unique rigid body. Within the context of this work, the following notation is employed: a generic vector \(\mathbf{p}\in\mathbb{R}^{3}\) expressed in a particular \(\{O^{0}x^{0}y^{0}z^{0}\}\) frame is denoted with \({}^{0}\mathbf{p}\), whereas, if it is convenient to define vectors without an explicit reference to a specific coordinate frame (coordinate free vector), it will be simply indicated with \(\mathbf{p}\). A generic rotation matrix \(R\in\mathbb{SO}(3)\) is indicated with two indices \(R^{i}_{i}\), where the notation represents the unit vectors of the frame \(i\) with respect to frame \(j\). Following the Society of Naval Architects and Marine Engineers (SNAME) notation [16], the state of an underwater vehicle (considered as a rigid body) is represented by using two reference frames. A local Earth-fixed reference frame (hypothesized as inertial) with axes pointing north, east, and down (NED frame) \(\{O^{N}x^{N}y^{N}z^{N}\}\), and a right-handed body reference frame \(\left\{O^{b}x^{b}y^{b}z^{b}\right\}\) whose origin is the center of mass of the vehicle (see Figure 3) with its \(x\)-axis pointing in the forward motion direction, its \(z\)-axis pointing down, and its \(y\)-axis completing a right-handed reference frame. The pose of a vehicle is therefore represented with  
\[\mathbf{\eta}=\left[{}^{N}\mathbf{\eta}_{1}\ \mathbf{\eta}_{2}\right]^{\top}\quad, \tag{1}\]  
where \({}^{N}\mathbf{\eta}_{1}\) is the position of \(O^{b}\) with respect to the NED frame, and \(\mathbf{\eta}_{2}\) is the orientation the body-fixed frame w.r.t. the NED frame, where roll (\(\phi\)), pitch (\(\theta\)), and yaw (\(\psi\)) (RPY) angles are employed to describe the orientation. Additionally, AUV linear and angular velocities along the axis of the body-fixed reference (surge, sway, and heave motion) can be denoted as  
\[\mathbf{v}=^{b}\mathbf{v}=\left[{}^{b}\mathbf{v}_{1}\ {}^{b}\mathbf{v}_{2}\right]^{\top}\quad. \tag{2}\]  
The differential kinematic model is shown in Equation (3), and further information can be found in [17]. \[\dot{\mathbf{\eta}}=J(\mathbf{\eta})\mathbf{v} \tag{3}\] \[\left( \begin{array}{cc}{}^{N}\dot{\mathbf{\eta}}_{1}\\ \dot{\mathbf{\eta}}_{2}\end{array} \right)=\left[ \begin{array}{cc}{}^{R}_{b}(\bm {\eta}_{2})&0_{3\times 3}\\ 0_{3\times 3}&T^{N}_{b}(\mathbf{\eta}_{2})\end{array} \right]\left( \begin{array}{ cc}{}^{b}\mathbf{v}_{1}\\ {}^{b}\mathbf{v}_{2}\end{array} \right)\quad,\]  
where \(R^{N}_{b}\) is the rotation matrix between the body and the NED frame, and \(T^{N}_{b}(\mathbf{\eta}_{2})\) is the matrix mapping the angular velocity \({}^{b}\mathbf{v}_{2}\) onto the derivatives of the orientation angles.