## 4 Simulation Results and Discussion  
In this section, the formation is composed of one leader and four followers. From the defined heterogeneous communication and avoidance ranges, the directed communication graph is given in Figure 2. The model of USV is taken from [28], and the main parameters are \(M_{i}=\text{diag}\{23.8,33.8,2.764\}\),  
\[D_{i}= \begin{bmatrix}2&0&0\\ 0&7&0.1\\ 0&0.1&0.5\end{bmatrix} .\]  
Environmental disturbances are chosen as a superposition of zero mean white noise and constant interference. Standard deviation of white noise is chosen as 0.4, constant interference is chosen as 0.1. The parameters for ESO are set to \(K_{i,\rho 1}=7\cdot I_{3}\), \(K_{i,\rho 2}=45\cdot I_{3}\), and \(K_{i,\rho 3}=80\cdot I_{3}\). The control forces and moment are constrained as \(\tau_{i,\max}=[\tau_{i,u\max},0,\tau_{i,r\max}]^{T}\), \(\tau_{i,\min}=[\tau_{i,u\min},0,\tau_{i,r\min}]^{T}\), \(\tau_{i,u\max}=-\tau_{i,u\min}=2\text{N}\), and \(\tau_{i,r\max}=-\tau_{i,r\min}=1.5\text{Nm}\). The design parameters of control laws are chosen as \(K_{i}=0.3\cdot I_{2}\), \(k_{i,2}=6\), \(k_{i,3}=8\), \(k_{i,4}=5\), \(k_{i,5}=5\), \(l_{i,1}=l_{i,2}=l_{i,3}=l_{i,4}\)= 0.01, and \(\gamma_{ij}=0.01\). The heterogeneous communication ranges of USVs are \(L_{0}=8\) m, \(L_{1}=7\) m, \(L_{2}=7\) m, \(L_{3}=5\) m and \(L_{4}=5\) m. The avoidance ranges are \(R_{i}=1\) m and \(i=0,1,2,3,4\). The path of the leader is generated by \(u_{0}=0.1\) m/s and \(r_{0}=0\) rad/s for \(0\leq t<40\) s, \(u_{0}=0.1\) m/s and \(r_{0}=0.1\text{sin}(\pi t/20)\) rad/s for \(40\leq t<80\) s, \(u_{0}=0.1\) m/s and \(r_{0}=0\) rad/s for \(80\leq t<140\) s. The initial positions of USVs are chosen as \(\eta_{0}(0)=[0\) m, 0 m, 0 rad\(]^{T}\), \(\eta_{1}(0)=[0\) m, \(-5\) m, \(0\) rad\(]^{T}\), \(\eta_{2}(0)=[0\) m, 5 m, 0 rad\(]^{T}\), \(\eta_{3}(0)=[7\) m, 8 m, 0 rad\(]^{T}\), \(\eta_{4}(0)=[7\) m, \(-8\) m, 0 rad\(]^{T}\). Desired distances and angles are chosen as \(\rho_{10,d}=\rho_{20,d}=\rho_{31,d}=\rho_{42,d}=5\) m, \(\lambda_{10,d}=-25\pi/180\) rad, \(\lambda_{20,d}=25\pi/180\) rad, \(\lambda_{31,d}=-25\pi/180\) rad, and \(\lambda_{42,d}=25\pi/180\) rad. The formation tracking result is shown in Figure 3. Figures 4a,c, 5 and 6 show the the distance errors \(\rho_{ij}-\rho_{ij,d}\), and Figures 4b,d, 7 and 8 show the angle errors \(\lambda_{ij}-\lambda_{ij,d}\). As shown in Figure 3,  
Figure 4: Distance and angle between the leader and USV 1, USV 1 and USV 3. Figure 5: Distance between the leader and USV 2. Figure 6: Distance between USV 2 and USV 4. Figure 8: Angle between USV 2 and USV 4. Figure 7: Angle between the leader and USV 2. Figure 9: Control input of USV 1. Figure 11: Control input of USV 3. Figure 12: Control input of USV 4. Figure 10: Control input of USV 2. Comparison with the controller proposed in [14] is given though Matlab simulation in Figures 13-16; the simulation result is provided to validate the effectiveness and feasibility of the proposed controller. Comparison is designed under the same environmental disturbances, and the standard deviation of white noise is chosen as 0.4 while the constant interference is chosen as 0.1, with the same initial position \(\eta_{1}(0)=[0\) m, \(-5\) m, \(0\) rad\(]^{T}\) and the same objective \(\rho_{10,d}=5\) and \(\lambda_{10,d}=-25\pi/180\) rad. Figure 13 shows that USV 1 could track the leader accurately in a different controller. Figure 14 shows that the controller proposed in [14] accomplishes faster convergence and tracing speed. Not taking input saturation into consideration, the result is a more aggressive tracking trajectory, which is difficult to implement. In Figure 15, control inputs under different controllers are given. A local enlarged drawing of the control input is given in Figure 4:0 In the beginning, the input signal in this paper is saturated, and the position of the USV changes relatively slowly. At about 60 s, the follower catches up with the leader. The input signal in [14] is huge, and the position of the USV changes rapidly.