## 3 Control Design with a Symmetric BLF
### Control Design  
In the case where matrices \(M_{i}\), \(C_{i}(v_{i})\) and \(D_{i}(v_{i})\) are all known, let \(z_{1i}=[z_{11i},z_{12i}]^{T}=[e_{di},e_{\psi i}]^{T}\), and \(z_{2i}=v_{i}-\alpha_{i}=[z_{21i},z_{22i},z_{23i}]^{T}\), where \(\alpha_{i}=[\alpha_{1i},\alpha_{2i},\alpha_{3i}]^{T}\) is a stabilizing function which will be outlined later. Consider the symmetric BLF candidate as  
\[V_{1i}=\frac{1}{2}\log\frac{k_{ai}^{2}}{k_{ai}^{2}-e_{di}^{2}}+\frac{1}{2}\log \frac{k_{bi}^{2}}{k_{bi}^{2}-e_{yi}^{2}}, \tag{12}\]where \(k_{ai}\) and \(k_{bi}\) are positive constants used to constrain \(e_{di}\) and \(e_{\psi i}\), i.e., \(|e_{di}|<k_{ai},|e_{\psi i}|<k_{bi}\), respectively. Differentiating \(V_{1i}\) with respect to time we have  
\[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]  
According to Equation (8), differentiating \(e_{di}\) and \(e_{\psi i}\) with respect to time, we can obtain  
\[\dot{e}_{di} = -\left(z_{21i}+\alpha_{1i}\right)\cos\left(\psi_{i}-\varphi_{i}\right)\] \[+\left(z_{22i}+\alpha_{2i}\right)\sin\left(\psi_{i}-\varphi_{i} \right)+\dot{x}_{i-1}\cos\varphi_{i}+\dot{y}_{i-1}\sin\varphi_{i},\] \[\dot{e}_{\psi i} = \dot{\psi}_{i-1}-\left(z_{23i}+\alpha_{3i}\right). \tag{14}\]  
The virtual control \(\alpha_{i}\) is designed as follows:  
\[\alpha_{1i} = \cos(\psi_{i}-\varphi_{i})[k_{di}e_{di}(k_{ai}^{2}-e_{di}^{2})+ \dot{x}_{i-1}\cos\varphi_{i}+\dot{y}_{i-1}\sin\varphi_{i}],\] \[\alpha_{2i} = -sin(\psi_{i}-\varphi_{i})[k_{di}e_{di}(k_{ai}^{2}-e_{di}^{2})+ \dot{x}_{i-1}\cos\varphi_{i}+\dot{y}_{i-1}\sin\varphi_{i}],\] \[\alpha_{3i} = k_{\psi i}e_{\psi i}(k_{bi}^{2}-e_{\psi i}^{2})+\dot{\psi}_{i-1}. \tag{15}\]  
Substituting Equations (14) and (15) into Equaiton (13) yields  
\[\dot{V}_{1i}=-k_{di}e_{di}^{2}-k_{\psi i}e_{\psi i}^{2}+\Theta_{1i}, \tag{16}\]  
where  
\[\Theta_{1i} = \frac{e_{di}\left(-z_{21i}\cos\left(\psi_{i}-\varphi_{i}\right)+ z_{22i}\sin\left(\psi_{i}-\varphi_{i}\right)\right)}{k_{ai}^{2}-e_{di}^{2}}\] \[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]0  
The speed constraints of \(\
u_{i}\) are not considered in this work, so another [PERSON] candidate function is chosen as follows:  
\[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]1  
Then the time differentiation of Equation (17) can be written as  
\[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]2 \[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]3  
The model-based control law can be designed as:  
\[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]4 \[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]5  
where  
\[\dot{V}_{1i}=\frac{e_{di}\dot{e}_{di}}{k_{ai}^{2}-e_{di}^{2}}+\frac{e_{\psi i} \dot{e}_{\psi i}}{k_{bi}^{2}-e_{\psi i}^{2}}. \tag{13}\]6 \[\Theta_{2i} = \left[ \begin{array}{cc}\frac{e_{di}(-\cos(\varphi_{i}-\varphi_{ i}))}{k_{ai}^{2}-e_{di}^{2}}&\frac{e_{di}\sin(\varphi_{i}-\varphi_{i})}{k_{ai}^{2}-e_{di}^{2}}& \frac{-e_{pi}}{k_{bi}^{2}-e_{pi}^{2}}\end{array} \right]^{T}.\]  
Substituting Equation (19) into Equation (18), Equation (18) can be rewritten as  
\[\dot{V}_{2i}=-k_{di}e_{di}^{2}-k_{\psi i}e_{\psi i}^{2}-k_{2i}z_{2i}^{T}z_{2i}. \tag{20}\]Based on Lemma 1, we can obtain that the error \(e_{di}\) can be constrained in the range of \((-k_{ai},k_{ai})\) and \(e_{yi}\) could be constrained in the range of \((-k_{bi},k_{bi})\). However, the hydrodynamic damping \(D_{i}(v_{i})v_{i}\) is a continuous nonlinear function related to the USV speed.