## 4 Simulation Results
### Simulation Tests Conducted
#### 4.3.1 Test of Quasi-Velocity Based Controller  
The set of control parameters to ensure acceptable tracking results was (for both kind of desired trajectories)  
\[k_{1}=5.0,\,\,\,k_{2}=5.0. \tag{42}\]These gains were obtained by tuning the control algorithm, i.e., increasing the values from 1 to such values that guaranteed acceptable trajectory tracking accuracy (taking into account the other operating conditions). **Sine trajectory**. As is evident in Figure 2a, the set trajectory is tracking correctly and the position errors (presented in Figure 2b) in the x and y directions are tending to near zero after about 5 s. As can be observed in Figure 2c, the non-steady state for the control signals disappears after about 15 s of movement. Figure 2d demonstrates that at a similar time, this condition disappears for hovercraft velocities. Additional information about the effect of dynamic parameters on movement is obtained from changes in the value of \(\Delta\zeta_{1},\Delta\zeta_{2}\) as seen in Figure 2e. Because the displacement values of the center of mass position are the same in the x and y directions, \(\Delta\zeta_{1},\Delta\zeta_{2}\) are therefore symmetrical. Figure 2f presents the evolution of kinetic energy when the vehicle is in motion. Using this graph, it is possible to determine which variable causes more or less kinetic energy consumption, assuming dynamic parameters and a desired trajectory. This can be used to design the vehicle and for planning its movement. Figure 2: Simulation results for sine trajectory: (**a**) desired (\(p_{d}\)) and realized (\(p\)) trajectory; (**b**) position errors \(\Delta p_{x},\Delta p_{y}\) (in inertial frame); (**c**) input signals components (applied force) \(T\cos\psi,T\sin\psi\); (**d**) actual velocities \(u,\sigma,r\); (**e**) changes in quasi-velocity (\(\Delta\zeta_{1},\Delta\zeta_{2}\)); (**f**) changes in kinetic energy (\(K_{1},K_{2},K_{3},K_{T}\)) during motion. **Complex trajectory**. Figure 3 shows the results of the analogous tests conducted for the complex trajectory. It can refer, for example, to a situation in which the vehicle should avoid a fixed obstacle. As can be seen in Figure 3, the algorithm works properly and the desired trajectory is tracked efficiently. This observation is also confirmed in Figure 3. However, from Figure 3, it is apparent that the values of the applied force components do not reach a steady state but are constantly changing and, moreover, are larger than for a sine trajectory. A similar phenomenon can be seen in Figure 3, which shows the velocities of the vehicle. The changes in the \(\Delta\zeta_{1},\Delta\zeta_{2}\) values are larger than before but also symmetrical (for the same reason as previously), as observed in Figure 3. The kinetic energy changes, as shown in Figure 3, have larger values than for the sinus trajectory, and their waveforms are considerably different from the previous ones. So, the selection of the trajectory significantly affects the consumption of this energy. In order to obtain more objective information, other indexes were used to evaluate the achieved results. The corresponding values of the controller's performance are summarized in Table 2.  
Figure 3: Simulation results for complex trajectory: (**a**) desired (\(p_{d}\)) and realized (\(p\)) trajectory; (**b**) position errors \(\Delta p_{x},\Delta p_{y}\) (in inertial frame); (**c**) input signals components (applied force) \(T\cos\psi,T\sin\psi\); (**d**) actual velocities \(u,v,r\); (**e**) changes in quasi-velocity (\(\Delta\zeta_{1},\Delta\zeta_{2}\)); (**f**) changes in kinetic energy (\(K_{1},K_{2},K_{3},K_{T}\)) during motion. From the values of the MIA indexes, it can be seen that the tracking accuracy is comparable for both selected trajectories. The \(\Delta\zeta_{1},\Delta\zeta_{2}\) indexes have larger values for the complex trajectory, which means that the velocities are more deformed for such a trajectory.