## 5 Simulations and Comparison
### Vehicle Models and Test Conditions  
Simulation studies were performed to include at least partially real conditions. Two underwater vehicle models known from the literature were selected. Moreover, the technical capabilities resulting from the design of each vehicle were taken into account, namely the values of forces and moments which can be obtained from the thrusters and the possible vehicle velocities. The parameters of SIRENE [59] and Kambara [60] are given in Table 1. Since the test of the control algorithm was performed assuming the inertia matrix as symmetric, additional parameters were assumed in the simulations, namely: \(m_{23}=m_{32}=-700\) kgm (for SIRENE), \(m_{23}=m_{32}=-35\) kgm (for Kambara) and values \(Y_{r}=N_{v}=10\), \(Y_{r|v}=Y_{v|r}=Y_{r|r}=N_{r|v|}=N_{v|r}=10\) (the same for both vehicles). It is worth noting that the parameters of the models of the two vehicles are significantly different, which allowsverification of the proposed algorithm for vehicles with different dynamic parameters. For tracking, the desired trajectory position profiles are assumed, described as \(p_{d}=[x_{r},y_{r}]^{T}\):  
\[p_{d1} =[0.4\,t,\ 0.2\,t]^{T}, \tag{62}\] \[p_{d2} =[0.6\,t-3\cos(0.02\,t),\ 10-3\sin(0.02\,t)]^{T}, \tag{63}\]  
linear and cycloid trajectory, respectively. The starting points (the same for both vehicles) were \(p_{01}=[-5,\ 2,\ 0]^{T}\) and \(p_{02}=[-5,\ 12,\ 0]^{T}\). These points were assumed in order to ensure realistic condition of work.  
Moreover, time of motion was \(t=200\) s (for linear trajectory), \(t=500\) s (for cycloid trajectory) with the time step \(\Delta t=0.03\) s, and using the method ode 3 Bogacki-Shampine (in Matlab/Simulink). The simulation test was done based on software given in [61] but adapted for use with IQV.  
Two types of disturbances were considered, i.e., constant and variable of the form (as an example of Assumption 2):  
\[f_{u}=12\ \text{N},\ \ f_{v}=-1.5\ \text{N},\ \ f_{r}=2\ \text{Nm}, \tag{64}\] \[f_{u}(t)=2+1.5\sin(0.3\,t)+0.5\cos(0.2\,t)\ \text{N},\] (65) \[f_{v}(t)=1+0.5\sin(0.1\,t)+0.3\cos(0.3\,t)\ \text{N},\] (66) \[f_{r}(t)=1+\sin(0.2\,t)+0.2\cos(0.4\,t)\ \text{Nm}, \tag{67}\]  
For evaluating the tracking control performance the following indexes were assumed--_MIA_ (mean integrated absolute error), _MIAC_ (mean integrated absolute control), _RMS_ (root mean square of the tracking error), i.e.,  
\[MIA =\frac{1}{t_{f}-t_{0}}\int_{t_{0}}^{t_{f}}|\pi_{e}(t)|dt,\ \ \ \ M IAC=\frac{1}{t_{f}-t_{0}}\int_{t_{0}}^{t_{f}}|\tau(t)|dt,\] \[RMS =\sqrt{\frac{1}{t_{f}-t_{0}}\int_{t_{0}}^{t_{f}}}\|e(t)\|dt,\ \ \ \ K_{m}=\text{mean}(KE), \tag{68}\]  
where \(\pi_{e}=x_{e},y_{e},\psi_{e},\ldots,\ \|e(t)\|=\sqrt{x_{e}^{E}+y_{e}^{E}}\ (x_{e}^{E},y_{e}^{E}\) mean the position error in the reference frame) and \(KE\) is the kinetic energy, respectively.  
A heuristic method was used to select control parameters, as in [51]. This method is similar to the trial-and-error method, but also reflects the dynamics of the vehicle model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**SIRENE**} & \multicolumn{2}{c}{**Kambara**} \\ \hline
**Symbol** & **Value** & **Value** & **Unit** \\ \hline \(L\) & 4.0 & 1.2 & m \\ \(b\) & 1.6 & 1.5 & m \\ \(h\) & 1.96 & 0.9 & m \\ \(m_{11}\) & 2234.5 & 175.4 & kg \\ \(m_{22}\) & 2234.5 & 140.8 & kg \\ \(m_{33}\) & 2000 & 16.07 & kgm\({}^{2}\) \\ \(X_{u}\) & 0 & 120 & kg/s \\ \(Y_{v}\) & 346 & 90 & kg/s \\ \(N_{r}\) & 1427.2 & 18 & kgm\({}^{2}\)/s \\ \(X_{u|u|}\) & 35.4090 & 90 & kg/m \\ \(Y_{v|v|}\) & 667.5552 & 90 & kg/m \\ \(N_{r|r|}\) & 26,036 & 15 & kgm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of SIRENE and Kambara.