# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 4 Numerical Simulations
### Discussion of Results  
Models of two vehicles with different dynamics and three types of desired trajectories were investigated. The test results of the QV-based control scheme can be summarized as follows. 1. It was found that the performance of the control algorithm is affected by both vehicle dynamics and the selection of the desired trajectory. 2. The curvature of the trajectory (smaller radius of curvature) causes more interference at the beginning of the vehicle movement, and this, in turn, forces the controller to increase the control signals. The result can be a reduction in the trajectory-tracking accuracy unless the value of the coefficient \(l\) is reduced. 3. Not only are the gains of the controllers \(k_{v_{x},k_{y_{x}}},k_{p_{x}},k_{p_{y}},k_{l_{x}},k_{l_{y}}\) important for tracking the desired trajectory, but the value of the \(l\) parameter is also essential. If \(l\) has a large value (\(l>1\)), then, in the case of trajectories with small curvature (which should be determined after performing a simulation to check the accuracy of trajectory tracking), tracking occurs after a short time, but is inaccurate during movement (tracking errors in the \(x\)-\(y\) plane increase). The value of the coefficient \(s_{f}\) depends on the weight of the vehicle, and, more specifically, on its dynamic parameters. 4. It turned out that the algorithm is somehow robust to changes in the appropriately selected set of control gains. This means that the same set of gains can be applied to different trajectories and only slightly modified to track trajectories with sufficiently small curvature. 5. Investigating the distribution of kinetic energy consumption across variables representing quasi-velocities and velocity deformations caused by dynamic couplings in the vehicle provides additional insight into its dynamics. The time history of this energy indicates for which QV the reduction is greatest (in which direction of motion), and how the changes occur for different shapes of the desired trajectory. 6. The variables \(\Delta\zeta_{i}\) also reflect the shape of the trajectory because they can change as the vehicle moves. Simulation tests performed under other operating conditions and also for another vehicle indicate the suitability of the proposed algorithm, both for control purposes and for study of the dynamics of marine vehicles. **Analysis of the results obtained from the QV-based controller and the classical one. **  
1. Comparative observations of the figures show that both the QV and CL controllers work correctly, although, in some cases, the error convergence time may be shorter for the QV controller than for the CL controller. This may be due to the fact that coupling in the lateral direction is included in the control equation. 2. The proposed performance indexes provide more information. It can be seen that, for both algorithms, their effectiveness depends on the parameters of the vehicle model and the trajectory being tracked. 3. Based on the values of the indexes in Tables 4-6, it can be observed that: * After applying the CL algorithm, the smaller index values are not determined by the dynamic parameters of the vehicle because some of the smaller index values apply to the Kambara vehicle and some to XX AUV. Therefore, it is difficult to identify the relationship between the performance and the vehicle dynamics. Having applied the algorithm using QV, it is clear that smaller index values were obtained more often for the XX AUV vehicle than for the Kambara. From this it can be concluded that the effectiveness of the proposed controller depends, to some extent, on the dynamic parameters of the vehicle, which is not seen for the CL one. * The calculated index values show that both algorithms track linear trajectories with less accuracy than curvilinear trajectories. * The base indexes (that is, without force and torque) are mostly lower for each vehicle after using the QV controller when we compare them with the indexes obtained from the CL algorithm.