# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 4 Numerical Simulations
### Vehicles and Test Conditions  
In order to show the performance of the controller, vehicle models with different dynamics were selected.  
**The vehicle Kambara** was described in [53; 54]. The vehicle has length \(L_{v}=1.2\) m, width \(W_{v}=1.5\) m, and height \(H_{v}=0.9\) m. For Kambara, the matrix \(M\) is taken to be a diagonal one. In order to consider the matrix \(M\) with off-diagonal elements, it is assumed that \(m_{13}=my_{g}=10\) kg m, which corresponds to a center of gravity shift of 0.09 m, \(m_{23}=mx_{g}=-35\) kg m, which corresponds to a center of gravity shift of 0.3 m. The set of parameters is shown in Table 2. This set of parameters enables calculation of the elements of the diagonal matrix \(N\), i.e., \(N_{1}=175.4\) kg, and \(N_{2}=140.8\) kg, \(N_{3}=6.80\) kg m\({}^{2}\).  
The **XX AUV** model of the second vehicle was taken from [34; 55]. It has a torpedo-like shape (length approximately 1.2 m). The parameters used for the simulations are presented in Table 2. Due to the matrix symmetry \(M\)\(m_{13}=m_{31}\), \(m_{23}=m_{32}\), it was assumed also that \(m_{23}=mx_{g}=3.0\) kg m, which corresponds to a center of gravity shift of about 0.07 m (in the cited references, this quantity is absent, but it is needed for this test).  
From this set of parameters, the calculated elements of the diagonal matrix \(N\) are \(N_{1}=47.5\) kg, \(N_{2}=94.1\) kg, and \(N_{3}=13.1\) kg m\({}^{2}\). Simulations using Matlab/Simulink (time step \(\Delta t=0.05\), and using the ODE3 Bogacki-Shampine method) were carried out under additional assumptions:  
1. The starting points were selected taking into account the dynamics of each vehicle model.
2. The simulation time \(t=200\) s was chosen to show the effects of the use of the controller.
3. The disturbances were assumed to be close to [38], i.e., \(V_{x}=0.05\) m/s, \(V_{y}=-0.10\) m/s. The forces and torques were limited to reduce their initial values, that is, \(|\tau_{u}|\leq 15\) N and \(|\tau_{r}|\leq 15\) Nm.
4. The test was performed based on the software referred to in [56], modified to obtain results using the equations expressed in QV.  
For the point \(h\) value \(l=1.2\), m was chosen for Kambara and \(l=0.7\) m for XX AUV. Due to the dynamics of each vehicle \(s_{f}=10^{-2}\), \(s_{f}N_{1}=1.754\), and \(s_{f}N_{3}=0.068\) were assumed for Kambara, whereas \(s_{f}=10^{-1}\), \(s_{f}N_{1}=4.75\), and \(s_{f}N_{3}=1.33\) were assumed for XX AUV.  
The norm of the matrix \(\left\|\mathrm{Y}^{-1}\right\|\) depends on the vehicle parameter set and is determined for each vehicle. Some information about the proposed couplings which can be deduced from this norm is given in Table 3. For Kambara, one gets \(\left\|\mathrm{Y}^{-1}\right\|=1.136\), which means more than 14% couplings (1.000 is equivalent to 0%, whereas 1.932 to 100%). This value means that the couplings are weak. For XX AUV, one obtains \(\left\|\mathrm{Y}^{-1}\right\|=1.043\), which means almost 5% couplings. This value means that the couplings are very weak, but almost weak. Of course, it is possible to take the values of the parameters so that the couplings in the system are larger, but, for practical reasons, this seems unrealistic.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Kambara** & **XX AUV** & \\ \hline
**Parameter** & **Value** & **Value** & **Unit** \\ \hline \(m_{11}\) & 175.4 & 47.5 & kg \\ \(m_{13}\) & 10 & 3.0 & kg m \\ \(m_{22}\) & 140.8 & 94.1 & kg \\ \(m_{23}\) & \(-35\) & 5.2 & kg m \\ \(m_{33}\) & 16.07 & 13.6 & kg m\({}^{2}\) \\ \(d_{11}\) & 120 & 13.5 & kg/s \\ \(d_{13}\) & 10 & 10.0 & kg/s \\ \(d_{22}\) & 90 & 50.2 & kg/s \\ \(d_{23}\) & 10 & 41.4 & kg m/s \\ \(d_{31}\) & 10 & 10.0 & kg m/s \\ \(d_{32}\) & 10 & 17.3 & kg m/s \\ \(d_{33}\) & 18 & 27.2 & kg m\({}^{2}\)/s \\ \hline \end{tabular}
\end{table}
Table 2: Parameters for Kambara and XX AUV model.