# Numerical Test of Several Controllers for Underactuated Underwater Vehicles
## 4 Comparative Simulation Test
### Performance of Controllers for Linear and Sine Trajectory  
The software for the Lyapunov MBC coming from [4] was prepared in [43]. In this test, identification of dynamic parameters and disturbances arising from friction is omitted. This simplification allows for comparison of results in preliminary studies. Performance of SMC/backstepping [11] was carried out using software from [44], SMC [17] based on the software from [45], TSMC, and FTSMC developed in [18] using the software from [46]. Results of simulation tests are presented in Figures 2-11. Figure 3: Simulation results for ROPOS (Lyapunov MBC)—linear and sine trajectory. (**a,b**) Desired and realized trajectory; (**c,d**) position errors; (**e,f**) applied forces/torques. Figure 2: Simulation results for Kambara (Lyapunov method-based controller (MBC))—linear and sine trajectory. (**a,b**) Desired and realized trajectory; (**c,d**) position errors; (**e,f**) applied forces/torques. Figure 4: Simulation results for Kambara (SMC/backstepping)—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 5: Simulation results for ROPOS (SMC/backstepping)—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 6: Simulation results for Kambara (SMC)—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 7: Simulation results for ROPOS (sliding mode controller (SMC))—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 8: Simulation results for Kambara (terminal sliding mode control (TSMC))—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 9: Simulation results for ROPOS (TSMC)—linear and sine trajectory. (**a**,**b**) Desired and realized trajectory; (**c**,**d**) position errors; (**e**,**f**) applied forces/torques. Figure 11: Simulation results for ROPOS (FTSMC)—linear and sine trajectory. (**a,b**) Desired and realized trajectory; (**c,d**) position errors; (**e,f**) applied forces/torques. Figure 10: Simulation results for Kambara (fast TSMC (FTSMC))—linear and sine trajectory. (**a,b**) Desired and realized trajectory; (**c,d**) position errors; (**e,f**) applied forces/torques. For Lyapunov MBC (linear trajectory), the following values of the parameters were assumed: \(a=0.6\) m (Kambara) and \(a=0.9\) m (ROPOS) \(\delta_{x}=0.005\) m, \(b_{T}=0.5\). Originally, the algorithm was appropriate for a hovercraft, therefore, two underwater vehicles are tested then averaged inertial parameters have been used, i.e., \(m=158.1\) kg, \(J=16.07\) kgm\({}^{2}\) (Kambara) and \(m=9217\) kg, \(J=7457\) kgm\({}^{2}\) (ROPOS). The controller gains were (K-Kambara, R-ROPOS):  
Figure 2:0  
The parameters estimated during simulations were off. From Figure 1(a), one can see that for Kambara the controller works correctly. This is confirmed in Figure 1(c) where both position errors tend to constant values close to zero (note that \(\delta_{x}\) is assumed arbitrarily). From Figure 1(e), it is observed that the forces have maximal or almost maximal values while the vehicle is in motion. Comparing the above results with the results obtained from ROPOS (Figure 3), it can be noticed that the tracking task is realized after a longer time than before (after about 20 s) as it is given in Figure 2(a),\(c\) and the applied forces have maximal values for a very long time (Figure 2(e)). Lyapunov MBC (sine trajectory). The same conditions of simulations, as for the linear trajectory tracking, were assumed.