## 4 Simulation Results
### XX AUV Model Vehicle  
In order to test performance of the control algorithms the model of XX AUV underwater vehicle, described in [68], was chosen. It is a torpedo-shape vehicle. Its parameters are collected in Table 2. The parameters set means, that \(N_{1}=47.5\) kg, and \(N_{2}=94.1\) kg, \(N_{3}=13.3\) kg\(\cdot\)m\({}^{2}\). For the point \(h\), value \(l=1.5\) m was assumed. For this vehicle \(\left\|\mathrm{Y}^{-1}\right\|=1.0280\), which means about 5% couplings (1 is equivalent to 0% whereas 1.0732 to 10%). This value means that the couplings are very weak. The set of gains applied for the QV controller was (50) whereas for the CL one (51) both for tracking of the linear and the sine trajectory. However, due to the dynamics of the XX AUV for the QV algorithm, it was assumed that \(s_{f}=10^{-1}\), \(s_{f}N_{1}=4.75\), and \(s_{f}N_{3}=1.33\). **Linear trajectory**. The results for linear trajectory tracking are presented in Figure 6 (QV controller). From Figure 6a, it can be concluded about the correct work of the control algorithm because the desired trajectory is tracked. Figure 6b,c show that after about 40 s, the position and velocity error states are close to zero. The applied force and torque (Figure 6d) have great values only if the vehicle starts, next their values are small. It results from the assumed initial point. It is clear from Figure 6e that the largest part of kinetic energy is dissipated in lateral motion \(v_{r}\) and half less in longitudinal motion \(u_{r}\). The error \(\Delta\xi_{2}\) shown in Figure 6f has small values, which means that the impact of dynamic parameters on lateral movement is also very small. The kinetic energy average values are \(K_{m}=39.45\) J (and \(K_{m1}=11.80\), \(K_{m2}=27.63\), \(K_{m3}=0.02\)). Figure 7 shows results obtained using the original controller [56]. As it is observed from Figure 7a, the desired trajectory is tracked but the position and velocity error states (Figure 7b,c) go to zero slower than for the QV control algorithm, i.e., after about 90 s. The applied force and torque (Figure 7d) have comparable values as for the previous controller. Other information is not available. **Sine trajectory**. For the QV control algorithm, the results are shown in Figure 8, whereas for CL algorithm in Figure 9.  
It can be clearly seen from Figure 8a that the QV controller works correctly. This observation is confirmed in Figure 8a,b, where the position and velocity error states are presented (the steady state is achieved after about 50 s). Moreover, the deviations of these errors are small taking into account the realized trajectory. The applied force and torque have great values only in the first phase of the vehicle's motion (Figure 8d). The kinetic energy consumption is presented in Figure 8e. As can be seen, the dominant consumption  
Figure 5: Simulation results for ROPOS original controller (CL) and sine trajectory: **(a)** desired and realized trajectory; **(b)** position error states; **(c)** velocity error states; and **(d)** applied force and torque. of this energy is caused by transverse movement of the vehicle, i.e., \(v_{r}\). Only in the first phase the forward motion (\(u_{r}\)) causes the highest energy consumption. From Figure 8f, it can be concluded about the weak influence of dynamic parameters on the change in lateral velocity of the vehicle. The kinetic energy average values are \(K_{m}=19.04\) J (and \(K_{m1}=5.34\), \(K_{m2}=19.69\), \(K_{m3}=0.01\)). Figure 6: Simulation results for XX AUV QV controller and linear trajectory: (**a**) desired and realized trajectory; (**b**) position error states; (**c**) velocity error states; (**d**) applied force and torque; (**e**) kinetic energy time history; and (**f**) error \(\Delta\zeta_{2}\). Figure 7: Simulation results for XX AUV original controller (CL) and linear trajectory: (**a**) desired and realized trajectory; (**b**) position error states; (**c**) velocity error states; and (**d**) applied force and torque.