## 4 Controller Design  
An output feedback controller combined with the ETESO was developed based on an event-triggered strategy. As shown in Figure 3, the characteristic of the proposed controller are as follows. The estimator is used to estimate the velocity of the leader. The ETESO was designed to provide the velocity, yaw rate and uncertainties. The transmitted signal contains an event-triggered strategy that reduces sensor-to-observer communication costs.,The triggering condition of the controller can largely reduce the action times of the actuators. The control law design process is divided into the following steps. Step 1: Define the formation tracking error vector:  
\[P_{ij}=[\rho_{ij,e},\lambda_{ij,e}]^{T}, \tag{49}\]  
where \(\rho_{ij,e}=\rho_{ij}-\rho_{ij,d}\), and \(\lambda_{ij,e}=\lambda_{ij}-\lambda_{ij,d}\). Using (13) and (14), the time derivative of \(P_{ij}\) is given as  
\[\dot{P}_{ij}=B_{ij}(-\Xi_{ij,1}z_{i}+\Xi_{ij,2}z_{j}-D_{ij,d}), \tag{50}\]  
where \(B_{ij}=\text{diag}\{1,\frac{1}{\dot{P}_{ij}}\}\),  
\[\Xi_{ij,1}=\left[ \begin{array}{cc}\cos(\psi_{i}-\lambda_{ij})&-\sin(\psi_{i} -\lambda_{ij})\\ \sin(\psi_{i}-\lambda_{ij})&\cos(\psi_{i}-\lambda_{ij})\end{array} \right], \tag{51}\]  
\[\Xi_{ij,2}=\left[ \begin{array}{cc}\cos(\psi_{j}-\lambda_{ij})&-\sin(\psi_{j }-\lambda_{ij})\\ \sin(\psi_{j}-\lambda_{ij})&\cos(\psi_{j}-\lambda_{ij})\end{array} \right], \tag{52}\]  
\(z_{i}=[u_{i},v_{i}]^{T}\), \(z_{j}=[u_{j},v_{j}]^{T}\) and \(D_{ij,d}=[\dot{\rho}_{ij,d},\dot{\lambda}_{ij,d}]^{T}\). Additionally, \(f_{i}=[f_{i,1}\)\(f_{i,2}]=P_{ij}^{T}\dot{P}_{ij}\dot{\varpi}_{ij,1}\), and \(\|f_{i}\|\leq f_{i}^{*},f_{i}^{*}\) is an unknown positive constant. **Remark 3**. : _The definitions of \(\Xi_{ij,1}\), \(P_{ij}\) and \(B_{ij}\), indicate that \(\|\Xi_{ij,1}\|=1\), \(P_{ij}\) and \(B_{ij}\) are bounded, so \(\|f_{i}\|\leq f_{i}^{*}\) is reasonable._  
_Choosing \(\beta_{i}\) as a virtual controller, \(\beta_{i}\) is designed as:_  
\[\beta_{i}=\Xi_{ij,1}^{-1}(B_{ij}^{-1}K_{i,1}P_{ij}+\hat{\Theta}_{i}-D_{ij,d})+ \tilde{f}_{i,2} \tag{53}\]  
_where \(\Xi_{ij,1}^{-1}=\Xi_{ij,1}^{T}(\Xi_{ij,1}\Xi_{ij,1}^{T})^{-1}\), \(K_{i,1}\in\mathbb{R}^{2\times 2}\) is a positive definite matrix, \(\tilde{f}_{i,2}=[0,\ \tanh(f_{i,2}/\epsilon_{f})]^{T}\), \(\epsilon_{f}>0\) is a constant, \(\hat{\Theta}_{i}\) is the estimator of \(\Theta_{i}\), \(\Theta_{i}=\Xi_{ij,2}z_{j}\) and it updated as_  
\[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]  
_where \(\tilde{P}_{ij}=B_{ij}[\rho_{ij,e}\tanh(\rho_{ij,e}/\epsilon_{\rho}),\lambda_{ ij,e}\tanh(\lambda_{ij,e}/\epsilon_{\lambda})]^{T}\), \(K_{\Theta}\) is a positive definite matrix, and \(\epsilon_{\rho}\) and \(\epsilon_{\lambda}\) are positive constants._  
**Remark 4**. : _The adaptive term \(\hat{\Theta}_{i}\) is employed to reduce the volume of data contained in the formation communication [32] and estimate the information (\(\Theta_{i}\)). \(\Theta_{i}\) can be estimated because \(\Xi_{ij,2}z_{j}\leq|z_{j}|\), and \(z_{j}\) is bounded, ensuring that formation tracking control is achieved without velocity of the leader and that the amount of data exchanged between USVs can be reduced. Unlike  
Figure 3: Structure of the proposed ETC. the controller presented in [32], the proposed controller was designed without the velocities of follower. The ETESO can reduce sensor-to-observer communication costs, and the ETC can largely reduce the action times of actuators._  
Step 2: An error is defined as  
\[\psi_{i,e}=\psi_{i}-\psi_{ij,a}, \tag{55}\]  
where  
\[\psi_{ij,a}=[\arctan 2(g_{ij,1},g_{ij,2})-\psi_{j}]\tanh(g_{ij,3})+\psi_{j}, \tag{56}\]  
where  
\[g_{ij,1} =\rho_{ij}\,\sin(\lambda_{ij})-\rho_{ij,d}\,\sin(\lambda_{ij,d}), \tag{57}\] \[g_{ij,2} =\rho_{ij}\,\cos(\lambda_{ij})-\rho_{ij,d}\,\cos(\lambda_{ij,d}),\] (58) \[g_{ij,3} =\{(\rho_{ij,d}-\rho_{ij})^{2}+(\lambda_{ij,d}-\lambda_{ij})^{2} \}/\gamma_{ij}, \tag{59}\]  
where \(\gamma_{ij}\) is a positive constant.