## 5 Simulation Results  
In this section, simulation results are presented to prove the proposed event-triggered control method, and the ship parameters are chosen from [37]. Four USVs were driven to track a leader at a desired distance and angle with the proposed control law in (69)-(72). The total duration of the simulation runs was 120 s, and the sampling time \(t_{s}\) for each time point was 0.01 s. The initial positions of followers were chosen as \(\eta_{1}(0)=[-2\) m, \(-5\) m, \(0\) rad\(]^{T}\), \(\eta_{2}(0)=[-2\) m, \(5\) m, \(0\) rad\(]^{T}\), \(\eta_{3}(0)=[-4\) m, \(-8\) m, \(0\) rad\(]^{T}\) and \(\eta_{4}(0)=[-4\) m, \(8\) m, \(0\) rad\(]^{T}\). The trajectory of leader was generated by \(u_{j}=0.2\) m/s, \(v_{j}=0\) m/s and \(r_{j}=0\) rad/s for \(0\leq t<40\) s; \(u_{j}=0.2\) m/s, \(v_{j}=0\) m/s and \(r_{j}=0.1\) rad/s for \(40\) s\(\leq t<60\) s; \(u_{j}=0.2\) m/s, \(v_{j}=0\) m/s and \(r_{j}=0\) rad/s for \(80\) s\(\leq t<120\) s; and \(\eta_{j}(0)=[0\) m, \(0\) m, \(0\) rad\(]^{T}\). The desired distance and relative angle are expressed by \(\rho_{1j,d}=3\) m, \(\lambda_{1j,d}=1\) rad, \(\rho_{2j,d}=3\) m, \(\lambda_{2j,d}=-1\) rad, \(\rho_{3j,d}=6\) m, \(\lambda_{3j,d}=1\) rad, \(\rho_{4j,d}=6\) m and \(\lambda_{4j,d}=-1\) rad. The parameters of the ETESO were chosen as \(\epsilon_{i}=0.4\), \(c_{q}=0.1\). The parameters of the proposed control law were chosen as \(\epsilon_{f}=0.1\), \(K_{\Theta}=\text{diag}\{5,5\}\), \(\gamma_{ij}=0.001\), \(K_{i,1}=\text{diag}\{5,5\}\), \(k_{i,2}=5\), \(k_{i,3}=10\), \(k_{i,4}=10\), \(k_{i,5}=10\), \(\gamma_{1}=5\), \(T_{u}=10\), \(T_{r}=16\), \(l_{j,h}=0.1\) and \(h=1 selection is similar to parameter selection in \(a_{1}\) and \(a_{2}\). The ETESO and feed-forward compensation are relied upon, the influence of uncertainties is compensated immediately and the closed-loop system possesses strong robustness to different sets of parameters._  
For surface vessels, the effect of gravity can be neglected because the gravity acting on USVs is equal to the buoyancy acting on USVs. USVs are subject to model uncertainties and environmental disturbances. Model uncertainties are induced by model errors and unknown system parameters. Therefore, only centrifugal terms, damping parameters and environmental disturbances are considered. In this paper, the uncertainties \(\sigma_{i,1}\), \(\sigma_{i,2}\) and \(\sigma_{i,3}\) are assumed to be unknown. Environmental disturbances contain wind, wave and current information. The impacts on USVs of wind can be ignored due to a small windward area. Therefore, environmental disturbances are designed as the sum of some sinusoidal signals, which are chosen as \([0.18-0.18\cos(0.01t)\cos(0.015t),0.6+0.18\sin(0.21t)\cos(0.2t),\)\(0.6-0.18\sin(0.2t)\cos(0.23t)]^{T}\). Simulation results are shown in Figures 4-9. Figure 4 shows the formation pattern shaped by the five vessels; all followers can successfully track the leader. Figure 5 shows that the formation tracking errors approach zero, regardless of model uncertainties and environmental disturbances. Figure 6 shows the input signals of four USVs. In the first 24 s, since initial formation tracking errors are large, saturated control inputs suffer from sudden jumps. From 25 s to 40 s, formation tracking errors approach zero, and input signals convergence gradually. At 40s, 60s and 80s, sudden changes in the control forces are caused by a sudden change of the leader's velocity. Figure 4: Formation pattern with four followers and a leader. Comparisons of simulation results are shown in Figures 7-9. Simulations were conducted using the TTETC proposed in [23] and the ETC proposed in this paper. The two controllers have same control parameters but different trigger condition parameters. The TTETC was designed as follows:  
\[\left\{ \begin{array}{ll}\tau_{i,\mu c}(t)=\omega_{\tau u}(t_{k}^{\mu}),&\forall t \in[t_{k}^{\mu},t_{k+1}^{\mu})\\ \tau_{i,rc}(t)=\omega_{\tau r}(t_{k}^{r}),&\forall t\in[t_{k}^{r},t_{k+1}^{r}) \end{array} \right.