## 6 Results and Discussion  
### Simulation of Cable Disturbing Force  
The spatial changing rule of the cable disturbing force is analyzed and discussed, and a simplified model of the cable disturbing force is attempted for use in the design of the observer to reduce the influence of the cable disturbing force on the observer. In the numerical calculation model, the cable disturbing force is calculated using the lumped mass method iteratively and is regarded as the true value of the cable force in a real circumstance. A simplified model of the cable disturbing force also needs to be built for the observer as the disturbing compensation. The cable disturbing force is generated by the relative motion of the cable in the water and is influenced by various parameters, such as the current velocity, displacement and velocity boundary conditions of the cable ends, cable length, and the cable material properties. The simulation results are given below for different operating conditions, and the simplified model structure of the cable disturbing force is discussed. Case one: The top end and the bottom end of cable are static at the initial time; the top end coordinates of the cable are (0 m, 0 m, 0 m), and the bottom end coordinates of the cable are (0 m, 0 m, 100 m). Further, the current velocity is (\(u_{c}\), 0 m/s, 0 m/s). Tension results under different conditions of cable length and working depth are shown in Table 3.  
Case two: The top end of the cable is static with the coordinates (0 m, 0 m, 0 m), and the bottom end of the cable moves with the absolute velocity (\(u\), 0 m/s, 0 m/s); the relative current velocity is (\(u_{r}\), 0 m/s, 0 m/s), and the current velocity is (\(u_{c}\), 0 m/s, 0 m/s). Tension results under different relative velocity conditions are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cable Length (m)** & **Working Depth (m)** & \(u_{c}\) **(m/s)** & \(F_{cablex}\) **(N)** & \(F_{cabley}\) **(N)** & \(F_{cablez}\) **(N)** \\ \hline \multirow{8}{*}{120} & \multirow{8}{*}{100} & 0 & 0 & 0 & 0 \\  & & \(-\)0.25 & \(-\)55.45 & 0 & \(-\)45.58 \\  & & \(-\)0.3 & \(-\)79.44 & 0 & \(-\)68.01 \\  & & \(-\)0.5 & \(-\)220.21 & 0 & \(-\)190.68 \\  & & \(-\)0.75 & \(-\)495.29 & 0 & \(-\)428.54 \\  & & \(-\)1 & \(-\)880.45 & 0 & \(-\)761.07 \\  & & \(-\)1.25 & \(-\)1375.20 & 0 & \(-\)1186.70 \\  & & \(-\)1.5 & \(-\)1979.45 & 0 & \(-\)1704.94 \\ \hline \multirow{8}{*}{150} & \multirow{8}{*}{100} & 0 & 0 & 0 & 0 \\  & & \(-\)0.25 & \(-\)44.96 & 0 & \(-\)19.66 \\  & & \(-\)0.5 & \(-\)179.66 & 0 & \(-\)82.12 \\  & & \(-\)0.75 & \(-\)404.16 & 0 & \(-\)184.76 \\  & & \(-\)1 & \(-\)719.00 & 0 & \(-\)329.74 \\  & & \(-\)1.25 & \(-\)1123.27 & 0 & \(-\)514.93 \\  & & \(-\)1.5 & \(-\)1617.16 & 0 & \(-\)740.76 \\ \hline \multirow{8}{*}{150} & \multirow{8}{*}{125} & 0 & 0 & 0 & 0 \\  & & \(-\)0.25 & \(-\)68.47 & 0 & \(-\)58.53 \\  & & \(-\)0.5 & \(-\)275.12 & 0 & \(-\)237.54 \\ \cline{1-1}  & & \(-\)0.75 & \(-\)619.05 & 0 & \(-\)535.34 \\ \cline{1-1}  & & \(-\)1 & \(-\)1100.30 & 0 & \(-\)950.30 \\ \cline{1-1}  & & \(-\)1.25 & \(-\)1718.30 & 0 & \(-\)1480.80 \\ \cline{1-1}  & & \(-\)1.5 & \(-\)2472.94 & 0 & \(-\)2125.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tension on the ROV at the end of the cable under different conditions. It can be seen that when the top end of the cable is static and the bottom end moves, the tension components \(F_{cablex}\) and \(F_{cablez}\) at the end of the cable on the ROV have a linear relationship with the relative velocity with respect to the current, and have the opposite direction of the relative velocity. To conclude, the tension at the end of the cable on the ROV is a complex function of the relative velocity to the current, displacement, and velocity boundary conditions of the cable ends, the cable length, and so on.