# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 4 Comparative Simulation Test
### Discussion of Results
However, an undoubted advantage of the algorithm is an insight into the dynamics of the vehicle even with a completely different design. The algorithm can be used to check the effectiveness of underwater vehicle control and to determine the forces necessary to implement tracking of a desired trajectory. Such results are shown in Figure 2 (Kambara) and in Figure 3 (ROPOS) for linear and sine trajectory, respectively. Effectiveness of SMC/backstepping has not been clearly confirmed. Satisfactory results, for linear trajectory, for Kambara were achieved, while worse for ROPOS (Figure 5). Unfortunately, for the sine trajectory, the tracking task is not carried out for any of the two tested vehicle models as it is presented in Figures 4 and 5. Certainly, SMC/backstepping requires more thorough testing, but current tests show that the effectiveness of this algorithm depends significantly on the task and the conditions of its implementation (e.g., shape of the trajectory, vehicle dynamics). For SMC, no acceptable results were obtained for Kambara and ROPOS, neither for linear trajectory nor sine trajectory as it is seen in Figures 6 and 7. It is difficult to say in what conditions this algorithm can be used because the only case of its correct operation was obtained for the vehicle and conditions indicated in work [17]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Kambara**} & \multicolumn{2}{c}{**ROPOS**} \\  & **L-m (std)** & **S-m (std)** & **L-m (std)** & **S-m (std)** \\ \hline C1 (Lyapunov MBC) & \(-0.112\) (0.178) & \(-0.041\) (0.126) & \(-1.114\) (1.148) & \(-0.441\) (0.858) \\ C2 (SMC/backstepping) & \(-0.868\) (0.716) & \(0.092\) (0.352) & \(-0.412\) (0.450) & \(-0.282\) (0.350) \\ C3 (SMC) & \(0.473\) (0.507) & \(0.203\) (0.873) & \(0.605\) (0.929) & \(0.611\) (0.751) \\ C4 (TSMC) & \(-0.226\) (0.066) & \(-0.128\) (0.102) & \(-0.339\) (0.436) & \(-0.179\) (0.352) \\ C5 (FTSMC) & \(-0.551\) (0.107) & \(-0.240\) (0.220) & \(-0.030\) (0.552) & \(-0.031\) (0.363) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Means (m) and standard deviations (std) of tracking errors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Kambara**} & \multicolumn{2}{c}{**ROPOS**} \\  & **L** & **S** & **L** & **S** \\ \hline C1 & + & + & + & + \\ C2 & + & \(-\) & \(-\) & \(-\) \\ C3 & \(-\) & \(-\) & \(-\) & \(-\) \\ C4 & \(-\) & \(-\) & \(+\) & + \\ C5 & \(-\) & \(-\) & \(+\) & + \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effectiveness of controllers based on simulations. With the use of TSMC for Kambara (Figure 8), convergence of the position errors was very quickly achieved, but only up to a certain non-zero value (about \(-0.1\) m) for linear trajectory. The results show that the controller does not guarantee convergence of these errors to zero. Other values of variables indicate the algorithm works effectively. For the sine trajectory tracking task (Figure 8) also some position error values occur which are not close to zero. TSMC effectiveness was confirmed for ROPOS and linear trajectory by convergence the position errors to zero with acceptable time history of other physical quantities (Figure 9). If the sine trajectory is tracked then the control algorithm gives also satisfactory results. Simulation results obtained for FTSMC, linear trajectory and Kambara vehicle show that the position errors only go to a limited value other than zero (Figure 10). At the same time, the applied force and torque have acceptable values which indicate that the algorithm is working properly. Similar results were obtained for the sine desired trajectory as it is given in Figure 10. For ROPOS, the tracking errors tend to zero for both linear and sine trajectories at acceptable values of other tested variables (Figure 11). Satisfactory results can also be obtained for sine trajectory as shown also in Figure 11.  
Comment on the results from Tables 2 and 3.