## 2 Problem Formulation
### Leader-Follower Formation Model  
In the leader-follower formation model, the relative configuration between the leader and the \(i\)-th ASV follower is defined by \(q_{i}=\left[\begin{array}{cc}\rho_{i}(t)&\psi_{i}(t)\end{array}\right]^{T}\), where \(\rho_{i}(t)\) and \(\psi_{i}(t)\) denote the relative distance and relative bearing respectively. To design this formation, position and relative bearing errors in the Earth-fixed frame \(\left[O_{B},X_{B},Y_{E}\right]\) are transformed to the Body-fixed frame of \(i\)-th ASV, as follows (Figure 4).  
\[e_{i}(t)=\left[\begin{array}{c}e_{i1}(t)\\ e_{i2}(t)\\ e_{i3}(t)\end{array}\right]=\left[\begin{array}{ccc}\cos\varphi_{i}&\sin \varphi_{i}&0\\ -\sin\varphi_{i}&\cos\varphi_{i}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{c}x_{d}-x_{i}\\ y_{d}-y_{i}\\ \varphi_{d}-\varphi_{i}\end{array}\right],i=1,2,\ldots,N \tag{18}\]  
Therefore, the distance and bearing of \(i\)-th ASV can be described as below:  
\[q_{i}(t)=h_{i}(e_{i}(t))=\left(\begin{array}{c}\rho_{i}(t)\\ \psi_{i}(t)\end{array}\right)=\left(\begin{array}{c}\sqrt{e_{i1}^{2}(t)+e_{ i2}^{2}(t)}\\ atan2(e_{2}(t),e_{i1}(t))\end{array}\right)i=1,2,\ldots,N \tag{19}\]  
Time derivative of (19) results in (20) and (21) equations.  
\[\dot{q}_{i}(t)=R_{i}\
u_{i}+\delta_{i} \tag{20}\]  
\[\ddot{q}_{i}(t)=\ \dot{R}_{i}\
u_{i}+R_{i}\dot{\
u}_{i}+\dot{\delta}_{i} \tag{21}\]  
where:  
\[R_{i}=\left[\begin{array}{cc}\frac{-\
u_{i2}(t)}{\rho_{i}(t)}&0\\ \frac{e_{i2}(t)}{\rho_{i}^{2}(t)}&-1\end{array}\right] \tag{22}\]  
\[\delta_{i}=\left[\begin{array}{c}\frac{-\
u_{i2}(t)+u_{d}[e_{1}\cos e_{2}+e_ {2}\sin e_{2}]+v_{d}[e_{1}\sin e_{2}+e_{2}\cos e_{2}]}{\rho_{i}(t)}\\ \frac{-\
u_{i2}(t)+u_{d}[e_{1}\sin e_{2}-e_{2}\cos e_{2}]+v_{d}[e_{1}\cos e_{2} -e_{2}\sin e_{2}]}{\rho_{i}^{2}(t)}\end{array}\right] \tag{23}\]  
Figure 4: Leader-follower formation model.  
From Equation (20) and (21), one can easily find \(v_{i}\) and \(\dot{v}_{i}\) as follows:  
\[v_{i}=R_{i}^{-1}\Big{(}\dot{q}_{i}-\delta_{i}\Big{)} \tag{24}\]  
\[\dot{v}_{i}=R_{i}^{-1}\ddot{q}_{i}-R_{i}^{-1}\dot{R}_{i}R_{i}^{-1}\Big{(}\dot{q }_{i}-\delta_{i}\Big{)}-R_{i}^{-1}\dot{\delta}_{i} \tag{25}\]  
Substituting (24) and (25) into a dynamic model (7) results in the following model as the leader-follower formation model in ASVs.  
\[M_{i}(e_{i})\ddot{q}_{i}+C_{i}\Big{(}e_{i},\dot{q}_{i}\Big{)}\dot{q}_{i}+D_{i} \Big{(}e_{i},\dot{u}_{d},\dot{v}_{d},\dot{v}_{i}\Big{)}+R_{i}^{-T}\tau_{w1i}(t )=\overline{\tau}_{ai}(t) \tag{26}\]  
where:  
\[M_{i}(e_{i})=R_{i}^{-T}M_{1i}R_{i}^{-1} \tag{27}\]  
\[C_{i}\Big{(}e_{i},\dot{q}_{i}\Big{)}=-R_{i}^{-T}M_{1i}R_{i}^{-1}\dot{R}_{i}R_ {i}^{-1} \tag{28}\]  
\[\overline{\tau}_{ai}(t)=R_{i}^{-T}\tau_{ai} \tag{29}\]  
\[D_{i}\Big{(}e_{i},\dot{u}_{d},\dot{v}_{d},v_{i}\Big{)}=-R_{i}^{-T}M_{1i} \frac{\partial\big{(}R_{i}^{-1}\delta_{i}\big{)}}{\partial\dot{v}_{i}}J_{i}R_ {i}^{-1}\dot{q}_{i}+R_{i}^{-T}M_{1i}\frac{\partial\big{(}R_{i}^{-1}\delta_{i} \big{)}}{\partial\dot{v}_{i}}\Big{)}J_{i}R_{i}^{-1}\delta_{i}\]  
\[-R_{i}^{-T}M_{1i}\frac{\partial\big{(}R_{i}^{-1}\delta_{i}\big{)}}{\partial \dot{v}_{i}}\Delta_{i}-R_{i}^{-T}M_{1i}\frac{\partial\big{(}R_{i}^{-1}\delta_{ i}\big{)}}{\partial\dot{u}_{d}}\dot{u}_{d}-R_{i}^{-T}M_{1i}\frac{\partial \big{(}R_{i}^{-1}\delta_{i}\big{)}}{\partial\dot{v}_{d}}\dot{v}_{d}\]  
\[-R_{i}^{-T}M_{1i}\frac{\partial\big{(}R_{i}^{-1}\delta_{i}\big{)}}{\partial \dot{v}_{i}}\dot{v}_{i}+R_{i}^{-T}C_{1i}R_{i}^{-1}\dot{q}_{i}-R_{i}^{-T}C_{1i} R_{i}^{-1}\delta_{i}+R_{i}^{-T}D_{1i}R_{i}^{-1}\dot{q}_{i}-R_{i}^{-T}D_{1i}R_{i}^{-1} \delta_{i}\]