## 3 Dynamic Model of ROV  
### Coordinate Systems  
The dynamic model of an underwater vehicle is established and analyzed in two orthogonal coordinate systems, as shown in Figure 3, namely the earth-fixed frame O-XYZ and the body-fixed frame o-xyz.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Mass in the air (kg) & 3450 \\ Mass in the water (kg) & 0 \\ Working depth (m) & 3500 \\ Length \(\times\) width \(\times\) height (m) & 3.17 \(\times\) 1.81 \(\times\) 1.76 \\ Coordinate of buoyancy center (m) & 0, 0, \(-\)0.4 \\ Coordinate of gravity center (m) & 0, 0, 0 \\ Moment of inertia \(I_{xx}\), \(I_{yy}\), \(I_{zz}\) (kgm\({}^{2}\)) & 2200, 710, 652 \\ \hline \end{tabular}
\end{table}
Table 1: Main parameters of the ROV.  
Figure 3: Earth-fixed and body-fixed reference frames.  
Figure 2: Thrunster configuration of ROVs.  
The earth-fixed frame O-XYZ is a global inertial coordinate system fixed at the ocean surface ship with the origin at O. The OZ axis points vertically down to the water, and the OX and OY axes are in two mutually perpendicular horizontal directions.  
The body-fixed frame o-xyz is a local coordinate system fixed on the vehicle with the origin at o. Here, the ox axis points to the front of the vehicle, the oz axis points downward, and the oy axis completes the right-hand system with the other two axes.  
Underwater remotely operated vehicles have 6-DOF motions, namely, the surge, sway, heave, roll, pitch, and yaw. Three translation displacements, namely X (surge), Y (sway), and Z (heave), and three Euler angles, namely \(\varphi\) (roll), \(\theta\) (pitch), and \(\psi\) (yaw), represent the position and the attitude of the vehicle with respect to the inertial frame, respectively. The instantaneous velocity and the angular velocity with respect to body-fixed frame are represented by \((u,v,w)\) and \((p,q,r)\), respectively.  
The transformation of the forces and motions from the global to the local coordinate system can be fulfilled by using the transformation matrix through the Euler angles \(\phi\ \theta\ \psi\). The orientation of the vehicle in the global coordinates can be specified by the vector \(r_{o}\) from \(O\) to \(o\).  
The position of the vehicle is denoted as \(\left[\begin{array}{cc}X&Y&Z\end{array}\right]^{T}\) in the inertial coordinates (earth-fixed). Linear velocities \(\left[\begin{array}{cc}u&v&w\end{array}\right]^{T}\) and angular velocities \(\left[\begin{array}{cc}p&q&r\end{array}\right]^{T}\) are expressed in the body-fixed coordinates. There exists  
\[\left[\begin{array}{cc}\dot{X}&\dot{Y}&\dot{Z}\end{array}\right]^{T}=J_{1} \left[\begin{array}{cc}u&v&w\end{array}\right]^{T} \tag{1}\]  
where  
\[J_{1}=\left[\begin{array}{cccc}\cos\theta\cos\psi&\sin\varphi\sin\theta\cos \psi-\cos\varphi\sin\psi&\cos\varphi\sin\theta\cos\psi+\sin\varphi\sin\psi\\ \cos\theta\sin\psi&\sin\varphi\sin\theta\sin\psi+\cos\varphi\cos\psi&\cos \varphi\sin\theta\sin\psi-\sin\varphi\cos\psi\\ -\sin\theta&\sin\varphi\cos\theta&\cos\varphi\cos\theta\end{array}\right] \tag{2}\]  
\(J_{1}\) is the coordinate transformation matrix from the body-fixed to the earth-fixed frame, \({J_{1}}^{-1}={J_{1}}^{T}\), and \({J_{1}}^{-1}\) are both units of an orthogonal array.  
Similarly, the relationship between the angular velocities and the attitude angles can be obtained as follows:  
\[\left[\begin{array}{cc}\dot{\varphi}&\dot{\theta}&\dot{\psi}\end{array} \right]^{T}=J_{2}\left[\begin{array}{cc}p&q&r\end{array}\right]^{T} \tag{3}\]  
where  
\[J_{2}=\left[\begin{array}{cccc}1&\sin\varphi\tan\theta&\cos\varphi\tan \theta\\ 0&\cos\varphi&-\sin\varphi\\ 0&\sin\varphi\sec\theta&\cos\varphi\sec\theta\end{array}\right] \tag{4}\]