## 4 Simulation Results
### Simulation Tests Conducted
#### 4.3.4 Discussion of Results  
Tests were conducted on one hovercraft model and two desired trajectories selected for tracking. The goal was to compare the performance in both cases based on runs of different quantities, and to indicate the information possible with the criteria and indicators introduced. Based on the results obtained, the following was determined regarding the control algorithm:  
* It provides information on the asymmetric movement of the hovercraft (with a displacement in the center of mass caused, for example, by an additional load);
* It makes it possible by simulating the effect of a set trajectory on the vehicle motion, velocity deformation and kinetic energy consumption for the vehicle and for the individual quasi-velocity corresponding to motion in each direction;
* A small effect of the inaccuracy of the model parameters (with a 50% increase in their values) on the tracking accuracy was noticed, except for the kinetic energy values;
* If possible for the realization of the set task of trajectory tracking, it is better to choose trajectories described by less complicated mathematical functions.  
The fact that no differences were observed in the accuracy of tracking the set trajectory, but the dynamics indexes changed, can be explained by the reason that the proposed control scheme is based on the control of kinematic quantities, but at the same time it allows the detection of changes caused by modifications in vehicle dynamics.  
_Comment_. Only an approximate friction model is included in the proposed control algorithm, which limits the controller's applicability under conditions where disturbances may be negligible. Such a situation is the case, for example, in laboratory conditions or under favorable atmospheric conditions that will not disturb the work of the controller. In addition, a _tanh_ continuous function approximation was used in the simulations to avoid model discontinuities.  
Based on the comparative tests conducted, some observations can be made about the performance of the proposed IQV algorithm and the TSMC method. The results relating to trajectory tracking errors are given in Table 5. The percentage increase UP was calculated according to the following formula:  
\[UP=\left(1-\frac{W_{IQV}}{W_{TSMC}}\right)\cdot 100\%. \tag{45}\]  
As can be seen from this table, the IQV controller yielded lower values than the TSMC controller for all cases considered.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Sine** & **Complex** \\
**Index** & & **Trajectory** & **Trajectory** \\ \hline MIA & \(\Delta p_{x}\) & 0.0043 & 0.0047 \\  & \(\Delta p_{y}\) & 0.0375 & 0.0391 \\ MIAC & \(T\cos\psi\) & 4.7606 & 5.5258 \\  & \(T\sin\psi\) & 1.8247 & 2.7876 \\ RMS & \(||e||\) & 0.1392 & 0.1316 \\ KE & \(K_{m}\) & 0.6813 & 1.0801 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance for classical controller (with friction).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Sine Trajectory** & **Sine Trajectory** & **Complex Trajectory** \\
**Index** & & **without Friction** & **with Friction** & **without Friction** \\ \hline MIA & \(\Delta p_{x}\) & 93.7\% & 96.3\% & 72.3\% \\  & \(\Delta p_{y}\) & 58.3\% & 62.8\% & 23.8\% \\ RMS & \(||e||\) & 46.8\% & 48.6\% & 18.0\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Percentage increase in IQV tracking performance over TSMC.  
Table 6 shows a comparison of the percentages of the indexes used in the tests. The minus symbol for the IQV controller indicates that the values were worse. Although in 3 cases the IQV controller gave worse index values, these are position errors of 0.2 mm to 2.5 mm. It is also significant that the initial position in the y direction was 1 m.