## 4 Simulation Study and Results Discussion  
In this section, the effectiveness and superiority of the designed controller is verified by a numerical simulation using MATLAB/Simulink R2021b software. The simulation results are given for the FSMC-FDO and CSMC based on the ERL [31]. The ROVs model parameters used in the simulation are provided in Table 1.  
To further compare the performance of the different schemes, the following three measurements are introduced: the first, the overshoot, also called the maximum deviation, is a measure of the control system's dynamic performance in a step response. The average of the absolute deviations of all individual values from the arithmetic mean is the mean absolute error (MAE). The mean absolute error solves the problem of errors canceling each other out, and thus accurately reflects the actual error magnitude. A measure of the average deviation from the true value is the root mean square error (RMSE). The three measures are defined in the following:  
\[\textit{overshoot}=(\eta_{max}-\eta(\infty))\times 100\% \tag{49}\]  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}|\eta_{i}-\eta_{di}| \tag{50}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\eta_{i}-\eta_{di}\right)^{2}} \tag{51}\]  
The fast sliding control scheme parameters were selected as follows:  
\(\alpha=diag[0.04,0.01,0.05,0.01,0.01,0.01]\), \(\beta=diag[4\times 10^{5},4\times 10^{5},5\times 10^{5},4\times 10^{5},5 \times 10^{5},5\times 10^{5}]\), \(\mu=diag[0.001,0.001,0.001,0.001,0.001,0.001,0.001]\), \(k=diag[2,2,1,4,1,1]\), \(\Gamma=diag[26,5,1,15,27,16]\). The fixed-time disturbance observer parameters were set as follows:  
\(Z_{1}=diag[4,3,5]\), \(Z_{2}=diag[0.1,0.1,0.1]\), \(Z_{3}=diag[10,10,10]\), \(Z_{4}=diag[2,1,2]\), \(\delta=0.1\), \(\zeta=2\). Three cases of simulation will be considered here. Case 1: For the verification of the superiority of the proposed FSMC-FDO scheme, a comparison with the CSMC scheme is performed. Step-response reference trajectory tracking of six degrees of freedom (DOF) in a no-external-disturbance scenario was per formed for the FSMC-FDO scheme and the CSMC scheme. The original state of the ROVs is set to the following:  
\(x(0)=y(0)=z(0)=0\,\text{m},\phi(0)=\theta(0)=\psi(0)=0\,\text{rad},u(0)=v(0)=w(0)=0 \,\text{m}/\text{s}\), \(p(0)=q(0)=r(0)=0\,\text{rad}/\text{s}\). The simulation time is 20 s and the final position and attitude of the ROVs are as follows:  
\(x(0)=y(0)=z(0)=0.5\,\text{m},\phi(0)=\theta(0)=\psi(0)=0.5\,\text{rad}\). The response curves of the two control schemes at different DOF are shown in Figure 3.  
Figure 3 shows the step response capabilities of the two control schemes for position and attitude in each of the six DOF. The FSMC-FDO scheme has faster convergence and less chatter than the CSMC scheme, as shown in the figure. Table 2 summarizes the overshoot degree for the two schemes under the step response, and it is clearly seen that the overshoot of the FSMC-FDO scheme is lower than that of the CSMC scheme by between 1.83% and 40.63%. This phenomenon highlights the significant superiority of the FSMC-FDO in terms of fixed-time convergence and the precision of convergence. Figure 3: The results of the trajectory tracking of the ROVs under different schemes in the case 1. (**a**) surge trajectory. (**b**) sway trajectory. (**c**) heave trajectory. (**d**) roll trajectory. (**e**) pitch trajectory. (**f**) yaw trajectory. Case 2: To evaluate the robustness of the presented FSMC-FDO scheme, trajectory tracking simulations were performed with time-varying external disturbances using both control schemes. The initial state of the ROVs was the same as in Case 1, and the desired trajectory of the motion of the six DOF was as follows:  
\[ \begin{cases}x=2 sin(0.1t)\text{ m}\\ y=2 sin(0.1t)\text{ m}\\ z=-0.1t\text{ m}\\ \phi=2 sin(0.1t)\text{ rad}\\ \theta=2 sin(0.1t)\text{ rad}\\ \psi=2 sin(0.1t)\text{ rad}\\ \end{cases}  \tag{52}\]  
In this case, the simulation time was set to 100 s and external disturbances between 30 s and 70 s were added as follows:  
\[