## 4 Simulation Study and Results Discussion
\begin{cases}w_{x}=60[cos(5t-120)+cos(2t+90)]\text{ N}\\ w_{y}=50[cos(5t+90)+sin(3t+90)]\text{ N}\\ w_{z}=60[cos(5t+120)+sin(3t+90)]\text{ N}\\ w_{\phi}=40 cos(2t)\text{ N}\cdot\text{m}\\ w_{\theta}=50 cos(2t)\text{ N}\cdot\text{m}\\ w_{\phi}=50 cos(2t)\text{N}\cdot\text{m}\\ \end{cases}  \tag{53}\]  
Trajectory tracking of each DOF of the ROVs and the trajectory tracking error can be found in Figures 4 and 5. From the figures, both the proposed FSMC-FDO scheme and the CSMC scheme can converge the tracking error of each DOF of the ROVs to close to zero in the period without external disturbances. However, the FSMC-FDO scheme has a smaller convergence error and less chattering compared to the CSMC scheme. After external disturbances were added to the ROVs, it is clear that both schemes are affected by them, but the FSMC-FDO scheme has notably smaller error fluctuations. It is demonstrated \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(X_{u}\) & \(-\) 36.27 N/s & \(N_{t}\) & \(-\)6.23 kg \\ \(Y_{\text{p}}\) & \(-\)43.72 N/s & \(X_{u|u}\) & \(-\)9.44 N\({}^{2}\)/m\({}^{2}\) \\ \(Z_{u}\) & \(-\)105.29 N/s & \(Y_{v|v}\) & \(-\)13.53 N\({}^{2}\)/m\({}^{2}\) \\ \(K_{p}\) & \(-\)7.86 N/s & \(Z_{u|v}\) & \(-\)35.62 N\({}^{2}\)/m\({}^{2}\) \\ \(M_{q}\) & \(-\)8.16 N/s & \(K_{p|P}\) & \(-\)10.65 N\({}^{2}\)/m\({}^{2}\) \\ \(N_{t}\) & \(-\)10.21 N/s & \(M_{q|t}\) & \(-\)9.23 N\({}^{2}\)/m\({}^{2}\) \\ \(X_{ii}\) & \(-\)17.25 kg & \(N_{r|t}\) & \(-\)102.32 N\({}^{2}\)/m\({}^{2}\) \\ \(Y_{p}\) & \(-\)20.22 kg & \(I_{x}\) & 10.13 Nms\({}^{2}\) \\ \(Z_{u}\) & \(-\)22.73 kg & \(I_{y}\) & 16.48 Nms\({}^{2}\) \\ \(K_{p}\) & \(-\)2.65 kg & \(I_{z}\) & 17.52 Nms\({}^{2}\) \\ \(M_{q}\) & \(-\)1.23 kg & \(m\) & 23.5 kg \\ \hline \end{tabular}
\end{table} 
Table 1: ROVs system parameters. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(x\) & 14.81\% & \(x\) & 23.91\% \\ \(y\) & 11.08\% & \(y\) & 27.01\% \\ \(z\) & 2.75\% & \(z\) & 4.58\% \\ \(\phi\) & 11.19\% & \(\phi\) & 26.22\% \\ \(\theta\) & 7.72\% & \(\theta\) & 30.88\% \\ \(\psi\) & 2.40\% & \(\psi\) &the presented FSMC-FDO scheme has superior robustness, allowing it to accurately and quickly estimate and compensate for the environmental disturbances. Case 3: During ROVs operation, the trajectory corresponds to the trajectory of movement in 3D space. The 3D movement trajectory tracking capability of ROVs is an important performance index for an ROVs control system. In this section, to evaluate the 3D motion trajectory tracking ability of the presented FSMC-FDO scheme, simulations were performed and this scheme was compared with the CSMC scheme. The initial state of the ROVs is the same as in Case 1, and the desired motion trajectory each DOF is as follows:  
\[ \begin{cases}x=sin(0.05t)\text{ m}\\ y=cos(0.05t)-1\text{ m}\\ z=-0.1t-0\text{ m}\end{cases}  \tag{54}\]  
Figure 4: The results of the trajectory tracking of the ROVs under different schemes in the case 2. (**a**) surge trajectory. (**b**) sway trajectory. (**c**) heave trajectory. (**d**) roll trajectory. (**e**) pitch trajectory. (**f**) yaw trajectory. External disturbances are considered as follows:  
\[ \begin{cases}w_{x}=60[cos(5t-120)+cos(2t+90)]\text{ N\textminus}\text{ m}\\ w_{y}=50[cos(5t+90)+sin(3t+90)]\text{ N\textminus}\text{ m}\\ w_{z}=60[cos(5t+120)+sin(3t+90)]\text{ N\textminus}\text{ m}\end{cases}  \tag{55}\]  
The 3D motion trajectory of the ROVs under the two control schemes is shown in Figures 6-9. In the presence of environmental disturbances, it is observed that both controllers can achieve precise trajectory tracking. However, it is clear that the FSMC-FDO scheme has smoother tracking trajectories and exhibits a good robustness to time-varying external perturbations. The CSMC scheme's tracking trajectory is also close to the desired trajectory, but it exhibits chattering throughout the simulation.