## 4 Simulation Results
### Simulation Tests Conducted
#### 4.3.2 Comparison with Terminal Sliding Mode Controller  
In order to compare the performance and capabilities of the proposed control scheme, the simulation tests were conducted for the same hovercraft model and the same desired \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Sine** & **Complex** \\
**Index** & & **Trajectory** & **Trajectory** \\ \hline MIA & \(\Delta p_{x}\) & 0.0045 & 0.0072 \\  & \(\Delta p_{y}\) & 0.0398 & 0.0371 \\  & \(\Delta\zeta_{1}\) & 0.0244 & 0.0327 \\  & \(\Delta\zeta_{2}\) & 0.0244 & 0.0327 \\ MIAC & \(T\cos\psi\) & 5.1426 & 5.9542 \\  & \(T\sin\psi\) & 2.4981 & 2.9009 \\ RMS & \(||e||\) & 0.1365 & 0.1307 \\ KE & \(K_{m}\) & 0.7055 & 1.1793 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance for IQV controller (with friction). Figure 4: Changes in kinetic energy (\(K_{1},K_{2},K_{3},K_{T}\)) during motion—results with 50% parameter increase in parameter values: (**a**) sine trajectory; (**b**) complex trajectory. trajectories but using a different control method. An algorithm based on the terminal sliding mode control (TSMC) method proposed in [44] was chosen. The following considerations were followed:  
1. According to the cited reference, the TSMC method provides robustness against unmodeled dynamics, model uncertainty and external interference from ocean currents and waves. It is also known to be superior to the conventional sliding mode control technique in terms of finite-time convergence and high-precision steady-state tracking. It has been successfully applied in many works. 2. Despite the fact that the approach based on TSMC is designed for underactuated underwater vehicles, it has been successfully implemented for the hovercraft in [27]. 3. In the cited article, several algorithms based on different methods were investigated through simulation, and the mentioned TSMC method proved successful in tracking different trajectories while two others proved ineffective. The comparative of the TSMC algorithm tests was performed taking into account three issues: 1. The model with diagonal inertia matrix \(M_{diag}\) and without friction. 2. The model with diagonal inertia matrix \(M_{diag}\) with friction. 3. The model with non-diagonal inertia matrix \(M_{sym}\). During the testing, it was noted that the TSMC algorithm is very sensitive to changes in control parameters, and numerical problems also arise (e.g., the integration step of the integration method). For this reason, the values of a set of these parameters were assumed similar to those given in [27] (the model of the tested vehicle has dynamics very similar to the model used in that paper). On the other hand, changing the gain of the controller led to a loss of trajectory tracking efficiency. The control scheme of the TSMC algorithm was introduced and studied in [44]. The simulation investigation used software from [45] (Matlab/Simulink source codes). The following set of control parameters was assumed for the trials, which provided satisfactory trajectory tracking results: \[l_{x}=l_{y}=1.1,\ \ k_{x}=k_{y}=1.0,\ \ W_{1}=W_{2}=1.0,\] (43) \[\beta_{1}=\beta_{2}=0.6,\ \ q_{1}/p_{1}=1/2,\ \ q_{2}/p_{2}=3/5,\] (44) and \(\tilde{e}_{u}(0)=0.05\), \(\tilde{e}_{v}(0)=0\). The saturation of the control signals was limited to \(|\tau_{u}|\leq 30\) N and \(|\tau_{r}|\leq 30\) Nm to obtain comparable constraints with the previous simulations. **Sine trajectory without friction** and using \(M_{diag}\). Due to numerical problems, acceptable results could not be obtained as in the case of the algorithm using IQV for the same step time. Therefore, this time was reduced to \(\Delta t=0.003\) s (ODE 3 integration method). As can be seen in Figure 5a, the trajectory tracking is correct and the position errors are close to zero after about 10 s, as shown in Figure 5b. However, the time to obtain the error convergence is longer than with an IQV-type algorithm.