## 4 Thruster Configuration and Propulsion System  
A linear relation [18] holds between forces and moments acting on the vehicle \(\mathbf{\tau}\in\mathbb{R}^{6}\) and the motor thrusts \(\mathbf{t}(\mathbf{v},\mathbf{u})\in\mathbb{R}^{m}\) as shown in Equation (17). \[\mathbf{\tau}(\mathbf{v},\mathbf{u},\mathbf{q})=B(\mathbf{q})\mathbf{t}(\mathbf{v},\mathbf{u})\quad, \tag{17}\]  
where \(\mathbf{u}\in\mathbb{R}^{m}\) is the vector that collects the rotational speed of each motor and \(m\) is the number of motors, \(B\) is the thrust allocation matrix (TAM), which depends upon the thruster poses with respect to the vehicle center of gravity, as defined in Equation (18), and Equation (19). \[B(\mathbf{q})=\left[ \begin{array}{c}B_{1}(\mathbf{q})\\ B_{2}(\mathbf{q})\end{array} \right]\quad, \tag{18}\]  
with  
\[ \begin{array}{l}B_{1}(\mathbf{q})=\left[\,\cdots^{b}\mathbf{n}_{i}(\mathbf{q})\cdots \,\right]\\ B_{2}(\mathbf{q})=\left[\,\cdots^{b}\mathbf{r}_{i}(\mathbf{q})\times^{b}\mathbf{n}_{i }(\mathbf{q})\cdots\,\right]\quad,\end{array}  \tag{19}\]  
where \({}^{b}\mathbf{n}_{i}\) represents, in the body frame, the unit vector of the axis of the \(i\)th thruster and \({}^{b}\mathbf{r}_{i}\) its center with respect to the body-fixed frame. As suggested in [13], a more intuitive representation of the \(i\)th column of Equation (19), in different AURV configurations, can be observed in spherical coordinates. In particular, the axial unit vector of the \(i\)th thruster has been modeled as  
\[{{}^{b}\mathbf{n}_{i}(\mathbf{q})}=\left[ \begin{array}{c}\sin(\theta_{i}(\mathbf{q }))\cos(\theta_{i}(\mathbf{q}))\\ \sin(\theta_{i}(\mathbf{q}))\sin(\theta_{i}(\mathbf{q}))\\ \cos(\theta_{i}(\mathbf{q}))\end{array} \right]\quad, \tag{20}\]  
where \(\theta_{i}\) is the inclination and \(\varphi_{i}\) the azimuth angle of the longitudinal axis of the \(i\)th thruster in the \(\mathbf{q}\) configuration of the AURV. Concerning the propulsion system, the four-quadrant motor characteristic is approximated, according to [2], with the purpose of modeling the relationship between the thrust value \(t_{i}\) and the rotational speed \(u_{i}\) of the thruster:  
\[t_{i}(\
u,u_{i})=d(u_{i},\mathcal{V})\big{[}\text{sgn}(u_{i})\big{(}ku_{i}^{2}- f(p_{p},V_{a,i})\big{)}\quad\, \tag{21}\]  
where \(k\) is a coefficient that relates motor thrust and propeller speed at bollard conditions (usually divided in the forward and backward bollard coefficients, depending on the propeller motion), \(f(p_{p},V_{a,i})\) is a function depending on the \(i\)th motor advance speed \(V_{a,i}\) and the propeller pitch \(p_{p}\) (a construction parameter), and \(d(u_{i},\mathcal{V})\) is the term including in the model the dead-zone boundary values at the voltage supply level \(\mathcal{V}\). In the context of this work, a preliminary assumption has been made by neglecting the dead-zone boundary values and the term depending upon the motor advance speed. Additionally, we define a physical maximum value \(u_{i}^{max}\) as the rotational speed of each thruster, and Equation (21) can be simplified as follows:  
\[t_{i}(\hat{u}_{i}) = \text{sgn}(\hat{u}_{i})k(u_{i}^{max}\hat{u}_{i})^{2} \tag{22}\] \[u_{i} = u_{i}^{max}\hat{u}_{i}\quad, \tag{23}\]  
where \(\hat{u}_{i}\in[-1,1]\) represents the signed ratio of the thruster speed with respect to its maximum value.