# Fuzzy Risk Evaluation and Collision Avoidance Control of Unmanned Surface Vessels
## 3 Design of Smart Collision Avoidance Control System for USVs
### Scenario 2: Multiple Fixed Obstacles  
In this scenario, multiple fixed obstacles that appear on the sailing course of the controlled USV is adopted. The testing conditions for scenario 2 is listed in Tables 5 and 6.  
Figure 24: Comparisons of the level of collision risk \(\mu_{CRI}\) with/without using the proposed collision avoidance system.  
Figure 25: The comparison of the relative distance with/without using the proposed collision avoidance system.  
For solving this collision problem, the proposed smart collision avoidance control system is used, and the simulation results with the aid of this proposed method are shown in Figure 26. In Figure 26, the black line is the optimal collision avoidance trajectory, which was generated by using (45)-(49) and (51), of the proposed smart collision avoidance system. The blue line represents the tracking trajectory of the controlled USV, which is guided by using the optimal nonlinear control law \(\tau_{\eta 2}=\mathbf{T}_{2}\left(\mathbf{e},t\right)+{\rho_{2}}^{-1}\mathbf{u}_{2} ^{*}\left(\mathbf{e},t\right)\) in (54). Figure 26 reveals the fact that there is no collision that takes place in this scenario after an effective evasion was taken. This achievement strongly relies on the precise tracking ability of the optimal nonlinear control law, because it guides the controlled USV to sail along the no-collision trajectory with pinpoint accuracy. This promising property can be easily found from the tracking errors of the \(x\)-axis, \(y\)-axis, and heading angles \(e_{x}\), \(e_{y}\) and \(e_{\psi}\), shown in Figures 27-29, respectively. In these figures, \(e_{x}\), \(e_{y}\), and \(e_{\psi}\) converge to nearly zero due to the effective trajectory tracking performance delivered by the optimal nonlinear control law.  
\begin{table}
\begin{tabular}{c c c} \hline  & **Center** & **Radius** \\ \hline \multirow{5}{*}{Circle} & (\(-4.1,7\)) & 4.1 \\  & (\(2.5,22.2\)) & 6.4 \\  & (\(11.68,50.43\)) & 7.35 \\  & (\(-4.37,58.36\)) & 7.35 \\ \hline \multirow{5}{*}{Hexagon} & **Vertex** & \\ \cline{2-3}  & & (\(-12.74,37.25\)) \\  & & (\(-9,45.80\)) \\ \cline{1-1}  & & (\(-1.35,47.14\)) \\ \cline{1-1}  & & (\(5.1,40.4\)) \\ \cline{1-1}  & & (\(0.9,34.40\)) \\ \cline{1-1}  & & (\(-6.75,33.05\)) \\ \hline \end{tabular}
\end{table}
Table 6: Testing conditions of fixed obstacles in Scenario 2.  
\begin{table}
\begin{tabular}{c c} \hline  & **USV** \\ \hline Start point (m) & (\(3,-6\)) \\ End point (m) & (\(0,74.5\)) \\ Heading angle (deg) & 49 \\ Initial velocity (m/s) & 0 \\ Final velocity (m/s) & 2.565 \\ \hline \end{tabular}
\end{table}
Table 5: Testing conditions of the controlled USV in Scenario 2.  
Figure 26: History for the testing condition in Scenario 2 with the collision avoidance treatment.  
Figure 30 shows the history of the velocity of the controlled USV. The final sailing velocity was generated as approximately 2.5 m/s, which is close to the specified final  
Figure 28: Tracking error \(e_{y}\) in \(y\)-axis.  
Figure 27: History of the tracking error \(e_{x}\) in \(x\)-axis.  
Figure 29: Tracking error \(e_{\phi}\) in heading angle.  
velocity in the testing conditions. This implied that the controlled USV is able to reach the desired velocity without scarifying velocity during the collision avoidance stage.