# 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
From the kinetic energy time history, depicted in Figure 5e, it can be seen that the longitudinal movement of the vehicle causes the greatest consumption of kinetic energy. In this case, the mean kinetic energy values are: \(K_{m}=6.365\) J (and \(K_{m1}=5.133\) J, \(K_{m2}=1.225\) J, \(K_{m3}=0.006\) J). Figure 3: Numerical test 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; **(f)** errors \(\Delta\zeta_{1},\Delta\zeta_{2}\). A slightly greater deformation of the velocity (errors \(\Delta\zeta_{1},\Delta\zeta_{2}\)) occurs than for the linear trajectory, as shown in Figure 5f. The couplings in both directions of horizontal motion are very close to each other. Figure 4: Numerical test for Kambara, QV controller and sine-cosine trajectory (**a–f**), and cycloid-like trajectory (**g–l**): (**a,g**) desired and realized trajectory; (**b,h**) position error states; (**c,i**) velocity error states; (**d,j**) applied force and torque; (**e,k**) kinetic energy time history; (**f,l**) errors \(\Delta\zeta_{1}^{\prime},\Delta\zeta_{2}\). **A cycloid-like trajectory** described by \(p_{d}=\left[0.7\,t-4\cos 0.05\,t,\,20-4\sin 0.05\,t\right]^{T}\), with the start point \(p_{0}=[-3\,\,\,18]^{T}\) for Kambara and XX AUV, was assumed. Figure 5: Numerical test for XX AUV, QV controller and sine-cosine trajectory (**a–f**), and cycloid-like trajectory (**g–l**): (**a,g**) desired and realized trajectory; (**b,h**) position error states; (**c,i**) velocity error states; (**d,j**) applied force and torque; (**e,k**) kinetic energy time history; (**f,l**) errors \(\Delta\zeta_{1},\Delta\zeta_{2}\). The gains set was slightly changed in order to obtain better performance of the controller, namely:  
\[\text{Kambara }\ k_{p_{x}}=k_{p_{y}}=12,\ k_{p_{x}}=k_{p_{y}}=4,\ \ k_{I_{x}}=k_{I_{y}}=0.4, \tag{44}\] \[\text{XX\,AUV }\ k_{p_{x}}=k_{p_{y}}=5,\ \ k_{p_{x}}=k_{p_{y}}=1,\ \ k_{I_{x}}=k_{I_{y}}=0.3. \tag{45}\]  
The used test trajectory is different from the previous two because of its much smaller curvature. In Figure 4g-l, the results for Kambara are depicted. From Figure 4g, it is observed that the task is performed quickly and correctly. However, when tracking the curvature, the error values are larger. The position and velocity error states tend to zero after about 40 s, as shown in Figure 4h,i. The time history of the applied force and torque given in Figure 4j is similar to the history of the sine-cosine trajectory. Figure 4k shows that most of the kinetic energy is absorbed by the forward motion of the vehicle, similarly to the sine-cosine trajectory. However, the shape of the kinetic energy history is different because the change occurs as the shape of the trajectory tracked changes. The mean values of the kinetic energy are: \(K_{m}=44.763\) J (and \(K_{m1}=42.524\) J, \(K_{m2}=2.238\) J, \(K_{m3}=0.001\) J). The maximum values of the velocity deformation (\(\Delta\zeta_{1},\Delta\zeta_{2}\) errors) caused by the couplings are smaller than for the previous two trajectories, but the deformation fluctuates with the evolution of the shape of the trajectory, as shown in Figure 4l. The corresponding results obtained for the vehicle XX AUV are depicted in Figure 5g-l. From Figure 5g, it can be seen that the trajectory is tracked correctly. Similar conclusions can be drawn based on the graph presented in Figure 5h,i, where the positions and velocity error states are given. The applied forces and torque shown in Figure 5j have small values, with the exception of the first phase of the vehicle's motion, and reflect changes in the trajectory shape. In Figure 5k, it can be found that most of the kinetic energy is reduced by longitudinal motion. Due to the shape of the change in this energy, the same effect is found for the Kambara vehicle for the same trajectory.