## 4 Results and Discussion
To evaluate the performance of the proposed controller in terms of energy consumption from the thrusters, the Root Mean Square (RMS) value of the control coefficients computed for the thrusters was obtained and results are shown in Table 5. The mean column of the table represents the mean value of the coefficients considering all six thrusters. As it can be seen in the table, the average thruster force demanded by the proposed model-free 2 nd order SMC was in the range of 0.1004 and 0.1296--1.0 being the maximum--this was up to 50% less than the forces demanded by a traditional PID or FBL controller even when it converged faster. The Lyapunov-based controller converged to the trajectory approximately at 7 s, and its mean value for control coefficient of the thrusters was 23% bigger than the 2 nd Order SMC with t\({}_{b}\ =\ 7\) s.  
Figure 10: Error convergence for different controllers. (**a**) Surge error. (**b**) Sway error. (**c**) Heave error. (**d**) Yaw error. To test the controller response to external disturbances, the effects of ocean currents were included as described in Section 3.2. Results for trajectory tracking and errors of a simulation with the proposed model-free 2 nd order SMC with \(t_{b}=5\) s are shown in Figure 11. Despite the high ocean currents introduced, the BlueROV2 still smoothly converged to the desired trajectory at the designated \(t_{b}\), there were not overshoots at all, and the vehicle followed the designed trajectory with an error of zero after the time-base was reached. A three-dimensional trajectory of the BlueROV2 simulation along with the control trajectory when subject to ocean currents can be seen in Figure 12. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Controller** & **Thruster 1** & **Thruster 2** & **Thruster 3** & **Thruster 4** & **Thruster 5** & **Thruster 6** & **Mean** \\ \hline PID & 0.2078 & 0.1946 & 0.2010 & 0.2105 & 0.1311 & 0.2041 & 0.1915 \\ \hline Feed-back linearization & 0.1676 & 0.1876 & 0.1909 & 0.1693 & 0.2851 & 0.2760 & 0.2127 \\ \hline Lyapunov-based & 0.1278 & 0.1500 & 0.1380 & 0.1192 & 0.0678 & 0.1358 & 0.1231 \\ \hline Model-free 2 nd Order SMC with \(t_{b}\,=\,3\) s & 0.1449 & 0.1400 & 0.1333 & 0.1378 & 0.0801 & 0.1418 & 0.1296 \\ \hline Model-free 2 nd Order SMC with \(t_{b}\,=\,5\) s & 0.1264 & 0.1175 & 0.0986 & 0.1059 & 0.0740 & 0.1324 & 0.1091 \\ \hline Model-free 2 nd Order SMC with \(t_{b}\,=\,7\) s & 0.1132 & 0.1067 & 0.0880 & 0.0921 & 0.0726 & 0.1295 & 0.1004 \\ \hline \end{tabular}
\end{table} 
Table 5: Root Mean Square (RMS) values for thrusters control coefficients. Figure 11: Results for the finite-time convergence controller subject to ocean currents and \(t_{b}=5\) s. (**a**) Trajectory tracking. (**b**) Position tracking errors. To evaluate the performance of the proposed controller in terms of energy consumption from the thrusters, the Root Mean Square (RMS) value of the control coefficients computed for the thrusters was obtained when the vehicle was subject to high ocean currents. Results are shown in Table 6.  
The average thruster force demanded by the model-free 2 nd order SMC with finite-time convergence was in a range of 0.1903 and 0.2105, which resulted in up to 30% less than \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Controller** & **Thruster 1** & **Thruster 2** & **Thruster 3** & **Thruster 4** & **Thruster 5** & **Thruster 6** & **Mean** \\ \hline PID & 0.3254 & 0.3131 & 0.3534 & 0.3163 & 0.1135 & 0.1761 & 0.2663 \\ \hline Feed-back linearization & 0.3978 & 0.4008 & 0.4056 & 0.3892 & 0.7889 & 0.7749 & 0.5262 \\ \hline Lyapunov-based & 0.27321 & 0.2479 & 0.2607 & 0.2306 & 0.0578 & 0.0843 & 0.1924 \\ \hline Model-free 2 nd order SMC with \(tb\,=\,3\) s & 0.2790 & 0.2684 & 0.3093 & 0.2646 & 0.0578 & 0.0840 & 0.2105 \\ \hline Model-free 2 nd order SMC with \(tb\,=\,5\) s & 0.2909 & 0.2670 & 0.2801 & 0.2535 & 0.0500 & 0.0556 & 0.1995 \\ \hline Model-free 2 nd order SMC with \(tb\,=\,7\) s & 0.2897 & 0.2525 & 0.2625 & 0.2421 & 0.0479 & 0.0469 & 0.1903 \\ \hline \hline \end{tabular}
\end{table}