## 3 Mathematical Model
### Wind Loads  
The external force due to wind is less of a concern compared to water resistance, and is often neglected during DP. However, it should not be neglected if the vessel has a bigger superstructure. This particular Quadmaran vessel had four hulls and a superstructure on which the control board and batteries were loaded. In addition, it was also found from the actual sea sailing tests that the wind effect was evident. Hence, we modeled the forces acting on the Quadmaran vessel (including the four hulls and the superstructure) generated from wind using the database of wind load tests [21]. The moment acting on the superstructure was ignored, as the simple shape was previously found from the wind load test results. Although, for hulls, the wind load test was conducted in a single hull because of the equipment limitations. The following model simulated the forces, including the moment, considering the interaction between hulls.  
Figure 14 shows the coordinate system for modelling wind loads. For simplicity, each hull was assumed to be a flat board, and the origin of the vessel coincided with origin of the space-fixed coordinate system (\(o_{0}\)-\(x_{0}\)\(y_{0}\)). The orientation of each hull facing the wind had a significant effect on the wind load received by each hull, and could not be modelled in a straightforward way. Therefore, in this study, we developed a model such that the wind load received by each hull could be considered. The wind direction angle, \(\psi_{w}\) (\(w\) means wind) is defined as the relative angle between each hull of the vessel and the streamline of the wind based on a clockwise convention. Further, \(\psi_{w,j}\) (\(i\) = 1, 2, 3, 4) for each hull can be defined as shown in Figure 14. Correspondingly, \(F_{w,j}\) (\(i\) = 1, 2, 3, 4) are the forces that each hull receives from the wind in the out of plane direction (positive force as per left hand rule). The in-plane wind force acting on the hull was small and negligible. Symbols 1-4 and 1\({}^{\prime}\)-4\({}^{\prime}\) represent the position of the stern and bow of the No. 1-4 hulls, respectively.  
The force experienced by each hull is proportional to the effective length (\(l_{w,j}\), \(i\) = 1, 2, 3, 4) of the hull facing the wind stream. Therefore, it is necessary to obtain these effective lengths based on the relative orientation of each hull and wind direction, including the wind blocked by the upstream hulls. The wind force is considered to act at the center of the effective length for each hull. As Figure 14 shows, it can be understood that the effective  
Figure 13: \(K_{T}\) of four quadrants.  
The relative hull orientation \(\psi_{w\_i}\) is defined by the formula below:  
\[\psi_{w\_i}=\begin{bmatrix}\psi_{w}+180^{\circ}-\psi\\ \psi_{w}+90^{\circ}-\psi\\ \psi_{w}-\psi\\ \psi_{w}-90^{\circ}-\psi\end{bmatrix} \tag{13}\]  
Equation (14) expresses the force generated on each hull perpendicular to the hull based on the above model and wind load test results [22].  
\[F_{w\_i}=0.64\frac{l_{w\_i}}{L}\sin\psi_{w\_i} \tag{14}\]  
Therefore, for the coordinate system fixed to the vessel body, the forces, including hulls and the superstructure (\(F_{w\_s}\), \(F_{w\_s}\),) on the \(x\) and \(y\)-axis, become  
\[X_{E}=F_{w\_4}-F_{w\_2}+F_{w\_s} \tag{15}\]  
\[Y_{E}=F_{w\_1}-F_{w\_3}+F_{w\_s} \tag{16}\]  
As mentioned earlier, the force acts at the center of the effective length for each hull. If lever arm length, \(l_{w\_arm\_i}\) of each hull is defined, yaw moment due to wind can be obtained as shown in Equation (17).  
\[N_{E}=\sum_{i=1}^{4}F_{w\_i}l_{w\_arm\_i} \tag{17}\]