## 2 Preliminary Information  
### System Models  
We introduce our system model briefly. This paper considers the modelling of the system dynamic under fully developed cavitating conditions, as presented in Reference [5], which used a full six-Degree-Of-Freedom (DOF) nonlinear model that captures the physics of the system. Table 1 shows the state variables, which are commonly used in six-DOF systems. This paper uses 12 state variables as presented in Table 1.  
We controlled four fins (fins 1, 2, 3, and 4) as well as the pitch of the cavitator. Therefore, our control variables were as follows. \[u=\left[\delta_{f1},\ \delta_{f2},\ \delta_{f3},\ \delta_{f4},\ \delta_{exp_{ pitch}}\right] \tag{1}\]  
In (1), \(\delta_{fi}\) denotes the deflection angle of the \(i\)-th fin. Also, \(\delta_{exp_{ pitch}}\) denotes the deflection angle of the cavitator pitch. This paper considers the case where the origin of the body-fixed coordinate system is located at the center of gravity. This paper also considers the case where four fins are oriented in the cross formation and line up with the principal axis. The arrangement of four fins is shown in Figure 1. This view is from the nose of the vehicle. In this figure, numbers 1, 2, 3, and 4 represent fin numbers. Under these settings, mass moments of inertia (J) matrix is a diagonal matrix. Then, the six-DOF system dynamics are computed as follows. \[\mathrm{m}\left[\dot{\mathrm{u}}+\mathrm{qw}-\mathrm{vr},\ \dot{\mathrm{v}}+ \mathrm{u}\mathrm{r}-\mathrm{pw},\ \dot{\mathrm{w}}+\mathrm{pv}-\mathrm{uq}\right]^{\mathrm{T}}=\mathrm{F_{ thrust}}+\mathrm{F_{plane}}+\mathrm{F_{cavitator}}+\mathrm{F_{gravity}}+ \mathrm{F_{fin}} \tag{2}\]  
\[\left[\mathrm{I_{x}}\ \dot{\mathrm{p}}+\mathrm{qr}(\mathrm{I_{z}}-\mathrm{I_{y}} ),\ \mathrm{I_{y}}\dot{\mathrm{q}}+\mathrm{pr}(\mathrm{I_{x}}-\mathrm{I_{z}}),\ \mathrm{I_{z}}\ \dot{\mathrm{r}}+\mathrm{pq}(\mathrm{I_{y}}-\mathrm{I_{x}})\right]^{ \mathrm{T}}=\mathrm{M_{thrust}}+\mathrm{M_{plane}}+\mathrm{M_{cavitator}}+ \mathrm{M_{gravity}}+\mathrm{M_{fin}} \tag{3}\]  
In (2) and (3), \(\mathrm{m}\) is the mass of the vehicle. \(\mathrm{F_{A}}\) is the force generated by \(\mathrm{A}\), and \(\mathrm{M_{A}}\) is the moment generated by \(\mathrm{A}\). \(\left[\mathrm{I_{x}},\mathrm{I_{y}},\mathrm{I_{z}}\right]\) are diagonal elements in J. See Reference [5] for the detailed explanation for six-DOF system dynamics of the vehicle. Moreover, Reference [5] presented how to derive each force \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Symbols** & **Units** \\ \hline Roll Rate & p & rad/s \\ Pitch Rate & q & rad/s \\ Yaw Rate & r & rad/s \\ Longitudinal Velocity & u & m/s \\ Lateral Velocity & v & m/s \\ Normal Velocity & w & m/s \\ Euler Roll & \(\Phi\) & Rad \\ Angle & & \\ Euler Pitch & \(\theta\) & Rad \\ Angle & & \\ Euler Yaw & & \\ Angle & & \\ North Position & X & M \\ East Position & Y & M \\ Depth & Z & M \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State Variables. Figure 1: Fin arrangement. In this figure, numbers 1, 2, 3, and 4 represent fin numbers. (cavitator force \(\rm F_{\rm cavitator}\), thruster force \(\rm F_{\rm thrust}\), plane force \(\rm F_{\rm plane}\), gravity force \(\rm F_{\rm gravity}\), and fin force \(\rm F_{\rm fin}\)) and its associated moment in (2) and (3). We omit the presentation of each force and moment due to page limit of the paper. See Reference [5] for detailed explanation of each force and its associated moment in (2) and (3). Using the six-DOF system dynamics in (2) and (3), and open loop system of the vehicle can be derived as presented in Reference [5]. Based on the open loop system, Reference [5] presented how to find the trim input so that the vehicle moves straight with a constant velocity. Note that there is no rotation about the center of gravity (CG) for a trimmed vehicle. Let \(u_{0}\) denote the trim input of the vehicle, which is given as follows.