## 2 ROV Modeling
### Coordinate System  
According to Figure 3, this paper establishes a dynamic model of the ROV in two coordinate systems, namely the earth frame O-XYZ and the body-fixed frame o-xyz. The earth frame is a fixed system with O as the origin, where OX and OY are two horizontal directions perpendicular to each another, and OZ points vertically to the water surface. Meanwhile, the ROV's position and direction are described by the body-fixed frame, which  
Figure 1: This is an ROV 3D model that has an electrical machine, regulating device, electronic module, and buoyancy material.  
Figure 2: The ROV is designed to operate on the mooring line.  
is a local coordinate system fixed on the moving rigid body. The body-fixed frame has o as the origin, where ox represents the front of the ROV movement, oz represents the bottom, and oy and the other two axes form a right-handed coordinate system.  
The ROV has six-degree-of-freedom (DOF) motions: three translational motions, surge (\(X\)), sway (\(Y\)), and heave (\(Z\)). There are three Euler angles, roll (\(\varphi\)), pitch (\(\theta\)), and yaw (\(\psi\)), which represent the position and attitude of the ROV relative to the inertial system. The linear velocity and angular velocity of the ROV are denoted by \((u,v,w)\) and \((p,q,r)\), respectively.  
The motion state of the ROV on the anchor chain is constrained according to the working state specified in Section 2.1, and the self-stabilizing roll and pitch motions make it unnecessary to consider the roll and pitch angles. Therefore, the ROV motion model's 4-DOF are simplified, and the motion parameters are defined as presented in Table 1[27].  
In the inertial coordinate system, the attitude vector is \(\eta=[x\ y\ z\ \psi]^{\mathrm{T}}\), where \(xyz\) is the three-axis translation displacement, and \(\psi\) is the heading angle. The motion coordinate system's velocity vector is \(V=[u\ v\ w\ r]^{\mathrm{T}}\), where \(uvw\) is the linear velocity, and \(r\) is the angular velocity around the \(z\)-axis. A few assumptions are made as follows:  
* The ROV density is distributed uniformly;
* There is low-speed movement;
* The center of buoyancy and gravity are congruent.  
The space conversion equation is given by  
\[\dot{\eta}=J(\eta)V, \tag{1}\]  
where \(J(\eta)\) is Jacobian matrix  
\[J(\eta)=\begin{bmatrix}I_{1}(\eta)&0_{3\times 3}\\ 0_{3\times 3}&J_{2}(\eta)\end{bmatrix} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**DOF** & **Motions** & **Forces and Moments** & \
\begin{tabular}{c} **Linear and Angular** \\ **Velocities** \\ \end{tabular} &
\begin{tabular}{c} **Position and Euler** \\ **Angles** \\ \end{tabular} \\ \hline
1 & Surge & \(X\) & \(u\) & \(x\) \\
2 & Sway & \(Y\) & \(v\) & \(y\) \\
3 & Heave & \(Z\) & \(w\) & \(z\) \\
4 & Yaw & \(N\) & \(r\) & \(\psi\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definition of motion parameters.  
Figure 3: ROV coordinate system.  
\(J_{1}(\eta)\) and \(J_{2}(\eta)\) are given by  
\[J_{1}(\eta)=\begin{bmatrix}cos\psi&-sin\psi cos\varphi+cos\psi sin\theta sin\varphi& sin\psi sin\varphi+cos\psi sin\theta sin\varphi\\ sin\psi cos\theta&cos\psi cos\varphi+sin\varphi sin\theta sin\psi&-cos\psi sin\varphi +sin\theta sin\psi cos\varphi\\ -sin\theta&cos\theta sin\varphi&cos\theta cos\varphi\end{bmatrix}, \tag{3}\]  
\[J_{2}(\eta)=\begin{bmatrix}1&sin\varphi tan\theta&cos\varphi tan\theta\\ 0&cos\varphi&-sin\varphi\\ 0&sin\psi sec\theta&cos\varphi sec\theta\end{bmatrix}, \tag{4}\]