## 3 Mathematical Model  
### Motion Equations  
Figure 6 shows the defined coordinate system of the vessel. The space-fixed coordinate system \(o_{0}\)-\(x_{0}y_{0}\) was firstly defined. The \(x_{0}\)-\(y_{0}\) plane was aligned with the hydrostatic surface, and the yaw angle \(\psi\) concerning the \(x_{0}\) axis was defined. The coordinate system \(o\)-\(xy\) was then defined for the vessel body. On the horizontal plane, the midship and center of vessel gravity overlap with the geometric features of the vessel. In this study, since only the horizontal movement of the vessel was considered, it was feasible to align the origin of the coordinate system, as it was fixed to the vessel body along with the position of the center of gravity of the vessel. The \(x\)-axis direction on the No.1 hull side was set, and the \(y\)-axis was placed laterally. The \(x\)-\(y\) plane was also parallel to the hydrostatic surface. Table 2 shows the significance of the symbols used in this coordinate system. \begin{table}
\begin{tabular}{c c} \hline \(X\) & Hydrodynamic force in the \(x\)-axis [N] \\ \(Y\) & Hydrodynamic force in the \(y\)-axis [N] \\ \(N\) & Hydrodynamic moment around the \(z\)-axis \\ \(U\) & Forward movement speed [m/s] \\ \(u\) & Velocity in the \(x\)-axis [m/s] \\ \(v\) & Velocity in the \(y\)-axis [m/s] \\ \(r\) & Yaw rate [rad/s] \\ \(\beta\) & Drift angle [deg.] \\ \(\psi\) & Heading angle [deg.] \\ \hline \end{tabular}
\end{table} 
Table 2: Significance of symbols. Figure 5: Principal dimensions of the 1/3 scale model. This study adopts the MMG (Maneuvering Modeling Groupe) as the mathematical model [18]. Considering the coordinate system that moves with the vessel body (_o-xy_), the motion equations are as shown below:  
\[ \begin{cases}\big{(}m+m_{x}\big{)}\dot{u}-\big{(}m+m_{y}\big{)}vr=X\\ \big{(}m+m_{y}\big{)}\dot{v}+\big{(}m+m_{x}\big{)}ur=Y\\ \big{(}I_{\text{zz}}+I_{\text{zz}}\big{)}\dot{r}=N\end{cases}  \tag{1}\]  
where \(m\): mass of vessel; \(m_{x}\): added mass in the \(x\)-axis; \(m_{y}\): added mass in the \(y\)-axis; \(I_{\text{zz}}\): the moment of inertia around the center of gravity; \(J_{\text{zz}}\): the added moment of inertia around the center of gravity; \(r\): yaw rate; \(u\): velocity in the \(x\)-axis; and \(v\): velocity in the \(y\)-axis. \(X\), \(Y\), and \(N\) are the forces acting on the vessel body other than the additional mass component. Specifically, the forces acting on the vessel body can be written as shown in Equation (2). \[ \begin{cases}X=X_{H}+X_{P}+X_{E}\\ Y=Y_{H}+Y_{P}+Y_{E}\\ N=N_{H}+N_{E}\end{cases}  \tag{2}\]  
\(X_{H}\), \(Y_{H}\), and \(N_{H}\) are forces acting on the four hulls generated from water. \(X_{P}\) and \(Y_{P}\) are the thrusts of the propellers. \(X_{E}\), \(Y_{E}\), and \(N_{E}\) are other external force components, mainly wind load. This paper ignores the effect of waves and the water current since aquaculture farms are usually in calm waters. According to the test results conducted in calm water, we discovered that the parallel motion of vessel drifting during DP is slow (generally about 0.25 m/s), and the yawing rate is consistently less than 0.026 rad/s. Hence, as shown in Equation (3), the forces acting on the hulls due to parallel and static turning motions are only considered based on the  
Figure 6: The coordinate system. quasi-static assumption in this study. We will investigate \(X_{H(u,v,0)}\), \(Y_{H(u,v,0)}\) and \(N_{H(u,v,0)}\) by conducting the oblique towing tests, and \(N_{H(0,0,r)}\) by the static turning tests. \[\left\{ \begin{aligned} X_{H}&=X_{H(u,\,v,\,0)}\\ Y_{H}&=Y_{H(u,\,v,\,0)}\\ N_{H}&=N_{H(u,\,v,\,0)}+N_{H(0,\,0,\,r)}\end{aligned} \right. \tag{3}\]  
In addition, the Quadmaran vessel maintains front-back and left-right symmetry while in DP. The principal terms of additional mass and moment inertia (\(m_{x}\), \(m_{y}\), \(J_{zz}\) in Equation (1)) are considered sufficient in this study. Here, \(m_{x}=m_{y}\) since the movements in the \(x\) and \(y\) directions are happening to similar extents while in DP.