## 4 Simulation Results  
Several computer simulations are performed to demonstrate the effectiveness of the control scheme proposed for non-strict trajectory tracking of the fully actuated USV. Here, the USV is ordered to track a moving region. The trajectory of the moving region is composed of a straight line and a sinusoidal curve. The traditional RC method is used for the comparative study to illustrate further the performance of the proposed NTTC method. The CyberShip II (CS2) model from the Marine Cybernetics Laboratory (MCLab, Trondheim, Norway) of NTNU (Norwegian University of Science and Technology, Trondheim, Norway) is used for the simulations. CS2 is a 1:70 scale replica of a supply ship. Its mass is \(m=23.8\) kg, its length is \(L_{\rm CS2}=1.255\) m, and its breadth is \(B_{CS2}=0.29\) m. It is fully actuated with two main propellers, two aft rudders, and one bow thruster. The parameters of the vehicle, which are obtained from [28], are shown in Table 1.  
The input forces of the USV are bounded as \(\overline{\tau_{u}}_{max}=20\) N, \(\overline{\tau_{v}}_{max}=15\) N and \(\overline{\tau}_{max}=15\) N\(\cdot\)m. The initial position of the USV is \(x_{0}=-10\) m, \(y_{0}=-10\) m, and the initial yaw is \(\psi_{0}=0^{\circ}\). The initial velocities of the USV are \(u_{0}=v_{0}=r_{0}=0\). The trajectory of the desired moving region \(\eta_{0}(t)=[x_{0}(t),\ y_{0}(t),\ \psi_{0}(t)]^{T}\) is specified as a straight line and a sinusoidal curve given by the following equations:  
\[[x_{0}(t),\ y_{0}(t)]^{T}=\left\{ \begin{array}{c}[0.4t,\ 0]^{T},\ 0\leq t \leq 200\\ [0.4t,K_{r}\sin(\omega(t-80))]^{T},\ t>200\end{array} \right.,\]  
where \(K_{r}=100\), and \(\omega=0.01\) rad/s. The desired yaw \(\psi_{0}(t)\) is set according the Line-of-Sight (LOS) guidance law [26] during the desired straight-line trajectory, and is set to be the tangential angle of the desired trajectory during the desired curved trajectory. The desired yaw \(\psi_{0}(t)\) is specified as follows:  
\[\psi_{0}(t)=\left\{ \begin{array}{c}atan2(y_{0}(t)-y,\ x_{0}(t)-x),\ 0\leq t \leq 200\\ \omega K_{r}\cos(\omega(t-80)),\ t>200\end{array} \right..\]  
The desired regional bounds \(\mathbf{R}=\big{[}R_{p},R_{p},R_{yaw}\big{]}^{T}\) of the desired moving region are expressed as \([2,\ 2,\ pi/36]^{T}\), and the region scaling factor is \(\mathcal{L}(t)=\mathbf{I}_{3\times 3}\). \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m/\)kg & \(23.8000\) & \(x_{\rm g}/\)m & \(0.0460\) \\ \(l_{x}/(\)kg\(\cdot\)m\({}^{2}\)) & \(1.7600\) & \(X_{u}\) & \(-0.7225\) \\ \(X_{\rm in|u|}\) & \(-1.3274\) & \(X_{muu}\) & \(-5.8664\) \\ \(Y_{v}\) & \(-0.8612\) & \(Y_{|v|v}\) & \(-36.2823\) \\ \(Y_{r}\) & \(0.1079\) & \(N_{v}\) & \(0.1052\) \\ \(N_{|v|v}\) & \(5.0437\) & \(X_{u}\) & \(-2.0\) \\ \(Y_{v}\) & \(-10.0\) & \(Y_{r}\) & \(-0.0\) \\ \(N_{v}\) & \(-0.0\) & \(N_{r}\) & \(-1.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the USV. The parameters of potential energy functions are \(\left[k_{p1},\ k_{p2},\ k_{p3}\right]^{T}=\left[2,\ 2,\ 400\right]^{T}\). The positive constant parameter of the reference vector \(\
u_{re}\) is expressed as \(=5\). The parameters of the update law are selected as \(\boldsymbol{\Gamma}=diag\big{(}\big{[}7.5\times 10^{-9},\ 2\times 10^{-8},4\times 10^{-9 },1.5\times 10^{-7},1\times 10^{-8},1.5\times 10^{-8},1\times 10^{-8},1.5\times 10^{-7} \big{]}\big{)}\). The positive constant matrix \(\boldsymbol{\beta}_{v}\) of the control law is \(\boldsymbol{\beta}_{v}=diag\big{(}\big{[}10,\ 10,\ 1000\big{]}\big{)}\). The parameters of the switching gain control term are specified as \(\boldsymbol{\omega}=\left[0.5,\ 0.5,\ 0.5\right]^{T}\) and \(\boldsymbol{\sigma}=\left[0.1,\ 0.1,\ 0.1\right]^{T}\). The total simulation time is 2000 s. Figures 4-12 show the simulation results. Figure 4: Results of the traditional RC method.