## 4 Results and Discussion  
A simulation was performed so that a computational model of the BlueROV2 follow the spiral trajectory described before, the parameters and gains for the proposed model-free 2 nd order SMC with finite-time convergence are shown in Table 4. The initial position for the vehicle was set to \(x=2\) meters, \(y=-1\) meter, and \(z=2\) meters so it started slightly away from the desired trajectory. The effects of ocean currents were neglected for this simulation. Results for trajectory tracking and errors of the proposed controller are presented in Figure 5. The controlled trajectory smoothly converged with the desired trajectory at the time-base proposed and without overshoots, after that, the vehicle followed the desired trajectory without deviations. The errors decreased over time to become zero at the designated \(t_{b}\) except for the pitch, which, as established in Section 2.1, was not actuated and the resulting increase in its error--smaller than \(0.5^{\circ}\)--was caused by the vehicle's dynamics while moving forward. A three-dimensional trajectory of the BlueROV2 simulation along with the control trajectory can be seen in Figure 6.  
The control coefficients for each of the BlueROV2 thrusters are shown in Figure 7. These control coefficients were in a range of \(-1\leq u\leq 1\), where \(|u|=1\) represents that the controller is demanding 100% of the thruster force. The maximum force was demanded at the start of the simulation and, after reaching the time-base, the control signals remained constant and no chattering was observed. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(t_{0}\) & 0 & \(\delta\) & 0.001 \\ \hline \(t_{b}\) & 5 & \(\kappa\) & 5 \\ \hline \(\alpha_{0}\) & 1.01 & \(K_{i}\) & \(diag[0,5,5,5,5,5]\) \\ \hline \(\alpha_{c}\) & 20 & \(K_{d}\) & \(diag[800,800,800,800,0,800]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Controller parameters for simulation of a desired spiral trajectory. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Controller** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(\alpha\) & \(\kappa\) \\ \hline PID & 140 & 120 & 180 & - & - \\ \hline Feed-back linearization & 140 & 120 & 180 & - & - \\ \hline Lyapunov-based & - & - & 80 & 0.80 & - \\ \hline Model-free 2 nd order SMC with & & & & & \\ finite-time convergence & - & 5 & 800 & \(\alpha(t)\) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Gain matrices for classic controllers and the proposed model-free 2 nd order SMC with finite-time convergence. Figure 5: Results with \(t_{b}\ =\ 5\) s. (**a**) Trajectory tracking. (**b**) Position tracking errors. Figure 6: Simulated three-dimensional trajectory of the BlueROV2 with \(t_{b}\ =\ 5\) s.  
Figure 7: Control signal for each thruster. Simulation with \(t_{b}\ =\ 5\) s.  
The controllers presented and described in Section 3.3 were simulated for comparing their error convergence to zero and the control coefficient computed for the BlueROV2 thrusters. As can be seen in Figure 10, the surge and sway error of the proposed controller converges to zero at the desired time. Meanwhile, the rest of the controllers converged at different times and, for the PID and FBL, there was not a stationary error but an oscillation in its value was observed through the simulated time. For the heave error, despite the oscillations resulting in the PID and the FBL, all the controllers converged to zero. Finally, for the yaw orientation, the FBL exhibited some troubles converging to zero during the first 20 s to finally converge to it exponentially. In general, the performance of the proposed model-free 2 nd order SMC with finite-time convergence surpassed the rest of the controllers, even when FBL and Lyapunov-based were model-based.