# Development and Performance Measurement of an Affordable Unmanned Surface Vehicle (USV)
## 2 Equation of Motion of the Proposed USV
### Dynamics of USV  
To model the dynamics equation, it is necessary to analyze the force of the USV. USV's dynamics model is based on [PERSON]'s model [22]. In simplifying the analysis, the USV is assumed as a rigid body. The dynamics equation of the USV is modelled in the form of a matrix, which is represented as  
\[\mathrm{M}\dot{\mathrm{V}}+\mathrm{C}(\mathrm{V})\mathrm{V}+\mathrm{D}( \mathrm{V})\mathrm{V}=\tau+\tau_{\mathrm{wind}}+\tau_{\mathrm{wave}} \tag{8}\]  
M is defined as mass matrix, C(V) represents centripetal and Coriolis matrix, and D(V) is linear drag matrix. The effect of non-linear drag is ignored as the velocity is relatively slow [17]. Meanwhile, \(\tau\) is input forces and moments generated by the thrusters. \(\tau_{\mathrm{wind}}\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**DOF** & **Motions** & \begin{tabular}{c} **Forces and** \\ **Moments** \\ \end{tabular} & \
\begin{tabular}{c} **Linear and Angular** \\ **Velocities** \\ \end{tabular} &
\begin{tabular}{c} **Position and** \\ **Euler Angle** \\ \end{tabular} \\ \hline
1 & Surge & X & u & x \\
2 & Sway & Y & v & y \\
3 & Yaw & N & r & \(\mathrm{\dot{\psi}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard notation of USV motion. \(\tau_{\rm wave}\) are external forces or moments caused by the wind and the wave, respectively. The matrix \(\rm M\), \(\rm C(V)\), \(\rm D(V)\), and \(\tau\) are expressed as Equations (9)-(12). \[\rm M=\left[ \begin{array}{ccc}\rm m-X_{\dot{u}}&0&-my_{g}\\ 0&\rm m-Y_{\dot{v}}&\rm mx_{g}-Y_{\dot{r}}\\ -my_{g}&\rm mx_{g}&\rm I_{Z}-N_{\dot{r}}\end{array} \right] \tag{9}\]  
\[\rm C(V)=\left[ \begin{array}{ccc}0&0&-m(x_{g}r+v)+Y_{\dot{v}}v+\frac{Y_{\dot {r}}+N_{\dot{v}}}{2}r\\ 0&0&(m-X_{\dot{u}})u\\ \rm m(x_{g}r+v)-Y_{\dot{v}}v-\frac{Y_{\dot{v}}+N_{\dot{v}}}{2}&-(m-X_{\dot{u}}) u&0\end{array} \right] \tag{10}\]  
\[\rm D(V)=\left[ \begin{array}{ccc}X_{u}&0&0\\ 0&Y_{v}&Y_{r}\\ 0&N_{v}&N_{r}\end{array} \right] \tag{11}\]  
\[\rm\tau=\left[ \begin{array}{ccc}X_{T1}+X_{T2}\\ 0\\ (X_{T1}-X_{T2})c\end{array} \right] \tag{12}\]  
\(\rm\tau_{wind}\) and \(\rm\tau_{wave}\) effect could be neglected because USV is sailing in calm water and a breeze environment. As a result, the effects of waves and wind (environmental disturbance) may be ignored (research was conducted at the Military Camp's Swimming Pool in Semarang City, Indonesia). The sway force emerging from yaw rotation (\(\rm y_{\dot{r}}\)) and the yaw moment caused by the acceleration in the sway direction (\(\rm N_{\dot{v}}\)) are remarkably smaller than the inertial and additional mass terms [19]. The USV center of gravity is parallel to the USV frame of reference coordinate in the horizontal direction, thus \(\rm y_{g}=0\). Then, \(\rm X_{T1}\), thrust force by thruster 1, and \(\rm X_{T2}\), thruster 2 force, are outlined as  
\[\rm X_{T1}=\frac{F_{0}}{2}+\frac{\delta}{2} \tag{13}\]  
\[\rm X_{T2}=\frac{F_{0}}{2}-\frac{\delta}{2} \tag{14}\]  
where, \(\rm F_{0}\) is defined as the total thrust force, and \(\delta\) is defined as the differential thrust force between thrusters where the magnitude of \(\delta\) affects the rotational speed of the yaw and the heading of USV. The dynamics model is reduced to Equations (15)-(18) when these assumptions are applied. \[\rm M=\left[ \begin{array}{ccc}\rm m-X_{\dot{u}}&0&0\\ 0&\rm m-Y_{\dot{v}}&\rm mx_{g}\\ 0&\rm mx_{g}&\rm I_{Z}-N_{\dot{r}}\end{array} \right] \tag{15}\]  
\[\rm C(V)=\left[ \begin{array}{ccc}0&0&-m(x_{g}r+v)+Y_{\dot{v}}v\\ 0&0&(m-X_{\dot{u}})u\\ \rm m(x_{g}r+v)-Y_{\dot{v}}v&-(m-X_{\dot{u}})u&0\end{array} \right] \tag{16}\]  
\[\rm D(V)=\left[ \begin{array}{ccc}X_{u}&0&0\\ 0&Y_{v}&0\\ 0&0&N_{r}\end{array} \right] \tag{17}\]  
\[\rm\tau=\left[ \begin{array}{ccc}F_{0}\\ 0\\ \delta c\end{array} \right] \tag{18}\]  
\(\rm X_{\dot{u}}\), \(\rm Y_{\dot{v}}\), and \(\rm N_{\dot{r}}\) are additional mass when USV moves in direction u, v, and r. \(\rm I_{Z}\) is the moment of inertia towards the z-axis.