# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 4 Comparative Simulation Test
### Discussion of Results
Acceptable results by both tables were obtained for Kambara and both trajectories (Lyapunov MBC), for ROPOS and sine trajectory (TSMC), as well as for ROPOS and both trajectories (FTSMC). Table 2 shows that acceptable results were obtained for SMC/backstepping (sine) and TSMC (sine) for the Kambara. However, it can be seen from Figures 4d and 8d that the trajectory tracking task is not being performed correctly. On the other hand, from Figure 3c,d, it can be noticed that for the ROPOS vehicle (linear and sine trajectory), the use of the Lyapunov MBC algorithm is effective, although the mean and standard deviation values do not indicate it. A similar observation can be made by comparing Figure 9c for ROPOS (linear trajectory) and the TSMC algorithm with the values of standard deviation and mean. It can therefore be concluded that to evaluate the work of the controller, both objective criteria (here: std and mean) and subjective criteria, i.e., the evaluation of tracking errors based on figures obtained in simulations, are necessary. Based on the tests performed, it can be concluded that the trajectory tracking results for TSMC and FTSMC depend on the underwater vehicle being tested. TSMC and FTSMC proved in simulations to be the most promising. SMC/backstepping is also worth more detailed research. To sum up, however, to show the real effectiveness of the control algorithms, more detailed tests would be required to determine the scope of applicability of these algorithms. It can be supposed that unsatisfactory results are due to inadequate choice of parameters. However, no rules for the correct selection of control parameters were found in the reference works. Research has shown that obtaining satisfactory control results is difficult and in some cases not possible. Therefore, if the selection of parameters by means of an intuitive method is really difficult for a control algorithm, then in the source works additional methods should be given to facilitate this task, as in [48; 49]. Additional test for effective algorithms were performed for selected control algorithms with taking into consideration disturbances functions. For SMC/backstepping, Kambara, and linear trajectory the time-varying disturbances were assumed as in [11], namely: \(\tau_{du}=10^{-1}m_{1}+r(\sin 0.01t-1)\) N and \(\tau_{dr}=10^{-1}m_{3}+r(\sin 0.01t-1)\) Nm. For TSMC, FTSMC ROPOS and linear and also sine trajectory, the time-varying disturbances of the type as in [18] were applied: \(d_{u}(t)=0.3\cos 0.02t\) N, \(d_{r}(t)=0.2\sin 0.02t\) N.  
However, the control parameters had to be changed to obtain satisfactory control performance. The following control gains set was chosen for Kambara and C2:  
\[(K)\qquad k_{1}=0.5,k_{2}=0.5,k_{3}=2.0,c_{1}=2.9,c_{2}=2.9,k_{s1}=0.1,k_{s2}= 0.1,w_{s1}=30,w_{s2}=30.\]  
For ROPOS and C4 (linear trajectory), the following set of parameters was used:\(I_{x}=2.50,I_{y}=2.50,k_{x}=0.17,k_{y}=0.17,w_{1}=7,w_{2}=8,\beta_{1}=0.5,\beta_{2}=0. 5,q_{1}=1,p_{1}=2,\)\(q_{2}=5,p_{2}=7.\)  
For ROPOS C4 (sine trajectory), the following set of parameters was used:  
\(I_{x}=2.0,I_{y}=2.5,k_{x}=0.4,k_{y}=0.4,w_{1}=11,w_{2}=11,\beta_{1}=0.5,\beta_{ 2}=0.5,q_{1}=1,p_{1}=2,\)\(q_{2}=5,p_{2}=7.\)  
For ROPOS C5 (sine trajectory), the following set of parameters was used:  
\(I_{x}=1.0,I_{y}=1.0,k_{x}=0.15,k_{y}=0.15,w_{1}=20,w_{2}=20,\beta_{1}=0.5,\beta _{2}=0.5,q_{1}=1,p_{1}=2,\)\(q_{2}=5,p_{2}=7,\alpha_{1}=0.35,\alpha_{2}=0.5.\)  
As it can be observed from Figures 12 and 13, all controllers work correctly. For SMC/backstepping (Kambara), the results are only slightly different (cf. Figure 4). However, the algorithm reaches the steady state after a long time. For TSMC (ROPOS) and linear trajectory, the position errors are smaller than previously (cf. Figure 9) but the applied forces/torques time history is different. For TSMC and FTSMC (ROPOS) and sine trajectory, a similar observations can be made (cf. Figures 9 and 10).