## 4 Simulation Examples  
In this section, a string of four USVs are used for the numerical simulations to demonstrate the effectiveness of the proposed method. The communication relationship of the four USVs is shown in Figure 4.  
For the platoon formation control, every USV is a leader for its nearest successive USV except for the last one. So when an arbitrary USV is absent in practice, its follower should follow the virtual leader, which is at the forefront, and change the desired distance to one large enough. With this strategy, even if one of USVs is absent, the formation pattern and safety can be guaranteed. The ship model parameters used for simulation in this work are those of Cybership-II, which is a 1:70 scale supply vessel replica built in a marine control laboratory in the Norwegian University of Science and Technology [38]. For the sake of convenience of comparison, the desired reference trajectory is chosen to be similar to [6],  
\[\eta_{0}=\left\{ \begin{array}{l}[1.2t,0,0]^{T},\text{if }t\leq t_{c},\\ [1.2t_{c}+60\sin t^{\prime},60\left(1-\cos t^{\prime}\right),t^{\prime}]^{T}, \text{if }t>t_{c},\end{array} \right.\]  
where \(t_{c}\geq 0\) is a time constant and \(t^{\prime}=0.02(t-t_{c})\). The desired distance between the follower and the leader is considered to be 5 m. The initial states of the USVs are given as \(\eta_{0}=[0,0,0]^{T}\), \(\eta_{1}=[0,5.9,0]^{T},\eta_{2}=[0,11.8,0]^{T},\eta_{3}=[0,17.7,0]^{T}\)and \(\upsilon_{1}=\upsilon_{2}=\upsilon_{3}=0\). The time constant \(t_{c}=200\) s. Figure 4 illustrates the validity of the position and velocity information among USVs. The maximum thrust force is 2 N. In this paper, we consider the environment disturbances as \(w_{i}=[0.1\sin t,0.1\sin t,0]^{T}\). The minimum collision distance is 4 m and the maximum connectivity distance is 6 m. So the distance error should be satisfied by the following constraints:  
\[d_{i,col\,\min}<d_{i}<d_{i,con\,\max}.\]  
If the inequality is satisfied during the entire process of moving, it means that the collision distance and effective connectivity distance will not be violated, so the output constraint performance can be guaranteed. In order to verify the effectiveness of the proposed controller, two cases are evaluated for Matlab simulation studies. Case I (model-based control): For the model-based control, the control parameters are chosen as follows: \(k_{d1}=12,k_{d2}=k_{d3}=1,k_{y1}=1,k_{y2}=k_{y3}=0.5,k_{a1}=k_{a2}=k_{a3}=1\) and \(k_{b1}=k_{b2}=k_{b3}=0.5\). The control gains are set to \(K_{2i}=diag\)[6; 6; 4]. The parameters \(k_{ai}\) and \(k_{bi}\) are the distance constraint and angle constraint selected based on the communication equipment or other plant requirements. The value of \(k_{di}\) and \(k_{yi}\) affect the virtual control of \(\alpha_{1i},\ \alpha_{2i}\) and \(\alpha_{3i}\), if \(k_{di}\) and \(k_{yi}\) are too large, the virtual control will become large which may lead to the control input violating the physical limit value and affecting the stability of the system. If \(k_{di}\) and \(k_{yi}\) are too small, the control input will not drive the USV to the desired velocity. Figure 5 shows that the unmeasurable surge, sway and yaw velocity of the leader USV can be measured by the designed finite time observer with zero estimate error. From Figure 6, it is obvious that the USVs can move in a desired formation pattern under the model-based method. We can observe that from Figure 7 the distance between the successive vehicles is limited to 4 (collision avoidance distance) to 6 (the maximum connection distance) during the whole moving process. So the distance error constraints \(e_{di}<k_{ai}\) and \(e_{di}>-k_{ai}\), \(\forall t>0\) and relative angle error constraints \(-k_{bi}<e_{yi}<k_{bi}\), \(\forall t>0\) are realized. Figure 8 shows yaw angle tracking errors along with proposed boundary. Case II (adaptive NN control): Secondly, to demonstrate the control integrated with NNs, we choose the formation pattern to be the same as in case I.