# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 4 Comparative Simulation Test
### Discussion of Results  
Summarizing the obtained simulation results, the following observation can be made and conclusions can be drawn. Comment on manual selection of gains. The method of manual selection of control parameters, i.e., the trial and error method, can be effective as long as the relationships between these parameters are noticeable (or one can recognize such groups of parameters for which there is such a relationship) that regulate the same physical quantity, e.g., reaching a sliding surface, ensuring the position, and velocity errors convergence. One can set a few parameters and then tune others and possibly change the previous values. This is some sort of heuristic method, but sometimes it turns out to be effective. In such cases, manual selection of parameters is possible even if there are many of them. This type of observation was used, for example, in work [47] where 24 control parameters were selected. The relationship between the control parameters can also be detected using a genetic algorithm, as noted in the paper [48] (for 18 control parameters). Unfortunately, for the control algorithms that did not lead to the achievement of acceptable results, it was not possible to find relations facilitating effective control. Perhaps the application of an additional method, e.g., a genetic algorithm, method of parameter optimization, would lead to an improvement in the results. However, doing so does not fall within the scope of the proposed test. For Lyapunov MBC, the gains were tuned using the trial and error method based on the \(k_{1}=k_{2}=2\) taken from [4]. For SMC/backstepping, the values of parameters \(k_{s1},k_{s2}\), \(w_{s1},w_{s2}\) were determined (these gains can be grouped), and then attempts were made to tune the remaining gains (by trial and error method recommended in [11]). For the SMC method described in [17], some relationships between \(l_{x}\), \(l_{y}\), \(k_{x}\), and \(k_{y}\) were noted and parameters were tuned on this basis. However, it turned out that the impact of other parameters is so important that the method previously indicated could not guarantee the convergence of errors. For the control TSMC strategy given [18] after initial determination of values \(l_{x}\), \(l_{y}\), \(k_{x}\), and \(k_{y}\), it is possible to look for the value of the proportion \(q_{1}/p_{1}\), \(q_{2}/p_{2}\) (manual selection of parameters) so as to guarantee the convergence of errors in the assumed time (here \(t=30\) s or \(t=60\) s). Next, \(W_{1}\), \(W_{2}\) and \(\beta_{1}\), \(\beta_{2}\) should be chosen. Then it is necessary to tune the values of the preselected parameters \(l_{x}\), \(l_{y}\), \(k_{x}\), and \(k_{y}\). In addition, for FTSMC, the smallest possible values for \(\alpha_{1}\) and \(\alpha_{2}\) must be selected to avoid loss of stability. In order to compare the algorithms in terms of ensuring the convergence of tracking errors, the mean values and standard deviations were calculated for each controller, for each trajectory and for both vehicles. The results are shown in Table 2.  
Effectiveness of the tested control algorithms based on simulation results is presented in Table 3. It was also noted that for the algorithms that turned out to be effective, it is possible to slightly change the parameters to obtain acceptable control results. Despite the satisfactory results obtained in all tested cases, the Lyapunov MBC has significant drawbacks from a practical point of view. Due to the intended use of for a hovercraft, the mass in the longitudinal and transverse directions must be the same, which requires averaging and thus distorts the vehicle model. The vehicle must be equipped with a rudder and the tested vehicles have a different propulsion. Even if the structure of the vehicle could be changed, the tests show that the forces in motion have maximum permissible values or very large values. It is not known, therefore, whether such steering would be possible with a rudder.