# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 4 Numerical Simulations
### Discussion of Results
The advantage is more evident for the actual errors (mean(\(\sum\Delta p_{xy}\)) and RMS(\(||e||\))). It is worth noting that the errors in the position and velocity states indicate whether the algorithm correctly performs the tracking task, but the actual errors are larger, which is, however, in line with the assumptions of both the original (CL controller) and the modified (QV controller) methods. Since the QV-based control scheme takes into account both the longitudinal and lateral asymmetry of the vehicle, while CL only takes into account the longitudinal asymmetry, it would seem that it is the CL algorithm that should be more effective in performing the task of trajectory tracking. The fact that this is very often not the case can be explained by modifications to the controller that use the dynamic parameters of the vehicle. * Both controllers, i.e., QV and CL, primarily use the driving force, but smaller values are provided by the QV algorithm. In contrast, the average torque is much lower (close to zero) for the CL controller. The reason for this is the taking of vehicle dynamics into account, as it is necessary to provide larger values of this torque. * Compared to the other control scheme tested, here denoted VT, it turned out that, for the assumed operating conditions and desired trajectories, the considered QV algorithm gave slightly better performance with respect to the tracking errors with significantly lower values of the control signals. * It is worth noting that a properly selected set of control coefficients allows some flexibility because it can be applied to different trajectories (at most with a slight change in value). * Using the QV controller, more information is obtained on the dynamics of the vehicle than the CL controller. This is one of the essential differences between the two control schemes. **Limitations and disadvantages of the algorithm using QV** can be summarized in several points. * According to the idea presented in [38] and used to design the QV-based controller, only the convergence of the position and velocity state errors to zero is guaranteed. This means that the real position errors (represented by the values of RMS(\(||e||\)) and mean(\(\sum\Delta p_{xy}\))) can tend to certain constant values. This is what would explain their larger values than the values of the state errors. * In addition to the controller parameters \(k_{yz},k_{yz},k_{yz},k_{yz},k_{1x},k_{1y}\) and the \(l\) parameter, the quality of the results is also affected by the value of the additional factor \(s_{f}\), so the total number of parameters is one more than for a classic controller. However, the advantage of \(s_{f}\) is that it is adopted according to the dynamic parameters of the vehicle and can be tuned even if the algorithm does not work properly with another set of gains. * It takes quite a long time to reach the convergence values of the position and velocity states, even with the correct choice of controller gains. This time can be reduced by decreasing the values of \(k_{yz},k_{zy}\), but then oscillations and rapid changes in the error values will occur. To avoid this phenomenon, the values of \(k_{yz},k_{zy}\) can be increased, but this will increase the convergence time of the state errors to zero. Therefore, a compromise between all gain values is important.