# Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model
## 4 Numerical Simulations
### Algorithm Test with Weak Couplings  
#### 4.2.1 Results for Quasi-Velocity (QV) Algorithm  
The following control parameters were selected for the linear trajectory:  
\[k_{x}=0.2,\ \ k_{\theta}=1.0,\ \ k_{\psi}=0.8,\ \ \lambda_{1}= \lambda_{2}=\lambda_{3}=0.1,\ \ \Gamma_{1}=\Gamma_{2}=\Gamma_{3}=0.1,\] \[\beta_{1}=8,\ \ \beta_{2}=0.2,\ \ \beta_{3}=8. \tag{26}\]  
_Linear trajectory_. The linear trajectory is tracked correctly, as can be observed in Figure 1(a). Also, linear and angular errors, as shown in Figure 1(b),c, decrease to their final values after about 80 s.  
It is evident from Figure 1(d),e that the speed \(u\) has a value that is much lower than that allowed for the ROPOS vehicle. The movement mainly takes place because of this velocity, since others have much smaller values and angular velocities have negligible values. The vehicle mainly performs forward motion, as is clear from Figure 1(f), since the values of the surge force \(\tau_{u}\) are the largest. Kinetic energy (including coupling) is primarily consumed by forward motion, as can be observed from Figure 1(g). Figure 1(h) confirms that the dynamic couplings are very small because the linear velocity deformation values are not significant (the angular velocities are not deformed because the corresponding quasi-velocities are equal to them). For linear trajectories and weak couplings, the settling time for linear position errors is about 60 s, and for angular position errors, it is about 80 s.  
_Complex trajectory_. The control parameters were changed due to the value of couplings, i.e.,  
\[k_{x}=0.2,\ \ k_{\theta}=k_{\psi}=1.0,\ \ \lambda_{1}= \lambda_{2}=\lambda_{3}=0.1,\ \ \Gamma_{1}=\Gamma_{2}=\Gamma_{3}=1.0,\] \[\beta_{1}=\beta_{2}=\beta_{3}=0.6. \tag{27}\]  
Figure 2(a) clearly shows that the trajectory is tracking correctly, which is also reflected by the results in Figure 2(b) (linear position errors). However, it is apparent from Figure 2(c) that the angular position errors change slightly (it is worth noting that the angular velocities are variable in character, as is evident from their calculation).  
Figure 3d,e show that the linear and angular velocities are also allowed, although with slightly higher values. The movement is primarily at forward velocity \(u\). The drive of the vehicle is mainly by means of pitch torque, and is therefore different from the linear trajectory realization. The surge force is also important, as indicated by its values in Figure 3f. However, the kinetic energy consumption is more than double that of the previous case, as illustrated in Figure 3g. It is worth noting (Figure 3h) that the deformation of linear velocities is also small, although their changes are initially oscillatory. This proves that the coupling effect value is detectable regardless of the realized trajectory. For complex trajectories and weak couplings (until a change in the direction of motion), the settling time for linear position errors is about 50 s, and for angular position errors, it is about 80 s. Thus, these times are comparable for linear and complex trajectories.  
Figure 2: QV controller results, linear trajectory, and (26): (**a**) trajectories: \(p_{d}\)—desired, \(p\)—actual; (**b**) linear errors of position; (**c**) angular errors of position; (**d**) linear velocities; (**e**) angular velocities; (**f**) force and torque applied; (**g**) kinetic energy values; (**h**) errors of \(\Delta\zeta_{1},\Delta\zeta_{2},\Delta\zeta_{3}\).