# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 4 Numerical Simulations
### Vehicles and Test Conditions
#### 4.1.3 Comparison with Algorithm Based on Velocity Transformations  
To compare the results obtained with the proposed control algorithm, simulation tests were performed for analogous operating conditions using the control scheme developed in [48]. The algorithm chosen is a combination of backstepping, control of the integral sliding mode, and velocity transformation. Thus, its working idea is different from that of the proposed controller. In addition, it is also suitable for a fully asymmetric model like the modified controller considered in this paper. This controller is denoted here as VT  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Kambara**} & \multicolumn{3}{c}{**XX AUV**} \\ \hline
**Trajectory** & **L** & **SC** & **C** & **L** & **SC** & **C** \\ \hline mean(\(\sum\Delta\zeta_{12}\)) & 0.2343 & 0.0992 & 0.1056 & 0.1652 & 0.0794 & 0.0704 \\ std(\(\sum\Delta\zeta_{12}\)) & 0.6691 & 0.2615 & 0.3077 & 0.6071 & 0.2359 & 0.2489 \\ mean(\(\sum\Delta\zeta_{34}\)) & 0.0472 & 0.0188 & 0.0198 & 0.0525 & 0.0210 & 0.0233 \\ std(\(\sum\Delta\zeta_{34}  
(based on velocity transformation). Due to a different control idea, limitations on force and torque were taken in \(|\tau_{u}|\leq 120\) N and \(|\tau_{r}|\leq 37\) Nm, respectively. These values can be obtained considering the engines installed in the Kambara vehicle. Constant disturbances were adopted to keep the direction and nature of the operation consistent with previously assumed disturbances, that is, 5 N along the x-axis and \(-10\) Nm along the y-axis. Tests were performed for a Kambara vehicle that realized a linear and sine-cosine trajectory. Figure 6: Numerical test for Kambara, CL controller and linear trajectory (**a–d**), and sine-cosine trajectory (**e–h**): (**a,e**) desired and realized trajectory; (**b,f**) position error states; (**c,g**) velocity error states; (**d,h**) applied force and torque. Figure 7: Numerical test for XX AUV, CL controller and linear trajectory (**a–d**), and cycloid-like trajectory (**e–h**): (**a,e**) desired and realized trajectory; (**b,f**) position error states; (**c,g**) velocity error states; (**d,h**) applied force and torque. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Kambara**} & \multicolumn{3}{c}{**XX AUV**} \\ \hline  & **L** & **SC** & **C** & **L** & **SC** & **C** \\ \hline mean(\(\sum\Delta\xi_{12}\)) & 0.2342 & 0.1131 & 0.1227 & 0.2808 & 0.1266 & 0.1345 \\ std(\(\sum\Delta\xi_{12}\)) & 0.6366 & 0.2799 & 0.2993 & 0.7391 & 0.2945 & 0.3528 \\ mean(\(\sum\Delta\xi_{34}\)) & 0.0495 & 0.0294 & 0.0261 & 0.0514 & 0.0204 & 0.0197 \\ std(\(\sum\Delta\xi_{34}\)) & 0.1955 & 0.1084 & 0.0625 & 0.1752 & 0.0657 & 0.0575 \\ mean(\(\sum\Delta x_{p3}\)) & 0.5491 & 0.3600 & 0.2910 & 0.4617 & 0.2613 & 0.2251 \\ std(\(\sum\Delta x_{p3v}\)) & 0.7363 & 0.2612 & 0.3527 & 0.8204 & 0.3006 & 0.3820 \\ RMS(\(||e||\)) & 0.7120 & 0.3551 & 0.3588 & 0.7225 & 0.3183 & 0.3360 \\ MIAC(\(\tau_{\rm{r}}\)) & 0.4236 & 0.3365 & 0.4858 & 0.1784 & 0.1414 & 0.2083 \\ MIAC(\(\tau_{\rm{r}}\)) & 0.0679 & 0.0439 & 0.0396 & 0.0548 & 0.0347 & 0.0600 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance indexes for CL controller (L-linear, SC-sine-cosine, C-cycloid-like). **Linear trajectory**. The set of control parameters in the algorithm proposed in [48]  
\[\text{Kambara }\ k_{1}=k_{2}=0.20,\ \ k_{3}=10;\ \ k_{4}=1,\ \ k_{5}=5, \tag{48}\]  
was selected in order to guarantee fast response and acceptable tracking errors. The mean values of the calculated kinetic energy were: \(K_{m}=61.319\) J (and \(K_{m1}=58.586\) J, \(K_{m2}=2.510\) J, \(K_{m3}=0.223\) J). The energy values are higher than those with the algorithm proposed in this paper. Thus, the controller described in [48] consumes energy. The results of the simulation are shown in Figure 8.  
From Figure 8a,b, it can be seen that the tracking of the desired trajectory is implemented correctly; however, there is deformation when the vehicle begins to move.