# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 4 Numerical Simulations
### Vehicles and Test Conditions
#### 4.1.1 Results for QV Control Algorithm  
**Linear trajectory** described by \(p_{d}=[0.5\,t,\ 0.3\,t]^{T}\), and with the start point \(p_{0}=[-2\ \ 5]^{T}\) for Kambara and XX AUV, was tested. The gains for the controller, to ensure acceptable errors convergence, were selected as follows:  
\[\text{Kambara}\ \ k_{v_{x}}=k_{v_{y}}=12,\ \ k_{p_{x}}=k_{p_{y}}=3,\ \ k_{ Ix}=k_{ly}=0.3, \tag{42}\] \[\text{XX AUV}\ \ k_{v_{x}}=k_{v_{y}}=5,\ \ k_{p_{x}}=k_{p_{y}}=1,\ \ k_{ Ix}=k_{ly}=0.2. \tag{43}\]  
As can be seen in Figure 2a, the desired trajectory is correctly tracked. Observing Figure 2b,c, it may be noted that, after about 50 s, the position and velocity error states are close to zero. The applied force \(\tau_{u}\) and torque \(\tau_{r}\) given in Figure 2d are highest at the beginning of vehicle movement. However, the dominant value is \(\tau_{u}\). Figure 2e shows that the largest part of the kinetic energy is consumed by a variable \(\zeta_{1}\) that refers to the lateral movement of the vehicle. The mean values of kinetic energy are: \(K_{m}=29.767\) J (where \(K_{m1}=18.938\) J, \(K_{m2}=10.825\) J, \(K_{m3}=0.004\) J). Moreover, from Figure 2f, it can be observed that the velocity disturbance values are larger for \(\Delta\zeta_{2}\) (about 0.06 m/s), whereas for \(\Delta\zeta_{2}\), they are much smaller. Thus, the couplings are mainly concerned with lateral velocity. For the same linear trajectory and the XX AUV vehicle, the results of the test are shown in Figure 3. From Figure 3a, it is observed that the vehicle converges to the reference  
Figure 2: Numerical test for Kambara, 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; (**f**) errors \(\Delta\zeta_{1},\Delta\zeta_{2}\). trajectory. The position and velocity error states tend to zero, as seen in Figure 3b,c after about 40 s. The applied force and torque history is presented in Figure 3d. However, in the beginning, both the force and moment contribute to the movement of the vehicle and reach large values. Figure 3e shows that the kinetic energy is mainly dissipated due to the lateral movement of the vehicle but the longitudinal movement is also significant. The mean values of kinetic energy are: \(K_{m}=13.195\) J (and \(K_{m1}=5.165\) J, \(K_{m2}=7.952\) J, \(K_{m3}=0.078\) J). From Figure 3f, it is noted that the dynamical couplings represented by the velocity errors \(\Delta\zeta_{1},\Delta\zeta_{2}\) are comparable, which indicates that the velocity couplings are comparable in both directions (\(x\) and \(y\)). Their values are even higher than those of Kambara. **A sine-cosine trajectory** described by \(p_{d}=[0.5\,t,\ 10\sin 0.02\,t+5\cos 0.01\,t]^{T}\), with the starting point \(p_{0}=[-2\ \ 5]^{T}\) for Kambara and XX AUV, was selected for the second test. The same gain sets as previously, i.e., (42), (43), respectively, were assumed. From Figure 4a, it is found that, for Kambara, the algorithm works correctly. The position and velocity error states are close to zero after about 40 s (Figure 4b,c). The applied force and torque values are similar to the linear trajectory, as depicted in Figure 4d. Figure 4e shows that the kinetic energy is reduced first by longitudinal motion of the vehicle. The mean kinetic energy values obtained are: \(K_{m}=20.693\) J (and \(K_{m1}=18.902\) J, \(K_{m2}=1.790\) J, \(K_{m3}=0.001\) J). However, the speed deformation, represented by variables \(\Delta\zeta_{1},\Delta\zeta_{2}\), is smaller than for the linear trajectory and varies during the motion, as shown in Figure 4f. In Figure 5, the results for the XX AUV model are given. From Figure 5a, it can be noted that the control algorithm works correctly. This fact is confirmed in Figure 5b,c, where the position and velocity error states are shown. Although the applied force and torque are small (Figure 5d), they change as the vehicle moves along the trajectory.