## 3 Controller Design
**Step 1:** Differentiating \(q_{i,e}\) along (17) and (19) yields  
\[\psi_{i,e} =\sum_{j=0,j\
eq i}^{n}c_{ij}(\psi_{i}-\psi_{ij,a}),\]6 \[\psi_{i,e} =\sum_{j=0,j\
eq i}^{n}c_{ij}(\psi_{i}-\psi_{ij,a}),\]7 \[\psi_{i,e} =\sum_{j=0,j\
eq i}^{n}c_{ij}(\psi_{i}-\psi_{ij,a}),\]8 \[\psi_{i,e} =\sum_{j=0,j\
eq i}^{n}c_{ij}(\psi_{i}-\psi_{ij,a}),\]9 \[u_{i,e} =u_{i}-\tilde{\beta}_{i,1}-\gamma_{1}\tanh(\alpha_{i,1}),\]0 \[u_{i,e} =u_{i}-\tilde{\beta}_{i,1}-\gamma_{1}\tanh(\alpha_{i,1}),\]1 \[u_{i,e} =u_{i}-\tilde{\beta}_{i,1}-\gamma_{1}\tanh(\alpha_{i,1}),\]2  
where  
\[A_{ij}=\left[ \begin{array}{cc}\frac{\rho_{ij,e}}{a_{ij}^{2}- \rho_{ij,e}^{2}}&\frac{\lambda_{ij,e}}{b_{ij}^{2}-\lambda_{ij,e}^{2}}\\ \end{array} \right], \tag{37}\]  
\[B_{ij}=\text{diag}\{1,\frac{1}{\rho_{ij}}\}\,\]  
\[\Xi_{ij,1}=\left[ \begin{array}{cc}\cos(\psi_{i}-\lambda_{ij})&- \sin(\psi_{i}-\lambda_{ij})\\ \sin(\psi_{i}-\lambda_{ij})&\cos(\psi_{i}-\lambda_{ij})\\ \end{array} \right], \tag{38}\]\[\Xi_{ij,2}=\left[ \begin{array}{cc}\cos(\psi_{j}-\lambda_{ij})&-\sin(\psi_{j}- \lambda_{ij})\\ \sin(\psi_{j}-\lambda_{ij})&\cos(\psi_{j}-\lambda_{ij})\end{array} \right], \tag{39}\]  
\(z_{i}=[u_{i},v_{i}]^{T}\), \(z_{j}=[u_{j},v_{j}]^{T}\) and \(D_{ij,d}=[\rho_{ij,d},\lambda_{ij,d}]^{T}\). Then, \(\dot{q}_{i,e}\) is rewritten as follows:  
\[\dot{q}_{i,e}=A_{i}B_{i}(-\Xi_{i,1}z_{i}+\Xi_{i,2}-D_{i,d}), \tag{40}\]  
where  
\[A_{i}=\left[ \begin{array}{cccc}c_{i,h_{i,1}}A_{i,h_{i,1}},&\cdots&c_{i,h_{i, B}}A_{i,h_{i,B}}\end{array} \right]\in\mathbb{R}^{1\times 2n}, \tag{41}\]  
\[B_{i}=\left[ \begin{array}{cccc}B_{i,h_{i,1}}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&B_{i,h_{i,B}}\end{array} \right]\in\mathbb{R}^{2n\times 2n}, \tag{42}\]  
\[\Xi_{i,1}=\left[ \begin{array}{c}\Xi_{ih_{i,1},1}\\ \vdots\\ \Xi_{ih_{i,B},1}\end{array} \right]\in\mathbb{R}^{2n\times 2}, \tag{43}\]  
\[\Xi_{i,2}=\left[ \begin{array}{c}\Xi_{ih_{i,1},2}z_{h_{i,1}}\\ \vdots\\ \Xi_{ih_{i,B},2}z_{h_{i,B}}\end{array} \right]\in\mathbb{R}^{2n\times 1}, \tag{44}\]  
and  
\[D_{i,d}=\left[ \begin{array}{c}D_{ih_{i,1},d}\\ \vdots\\ D_{ih_{i,B},d}\end{array} \right]\in\mathbb{R}^{2n\times 1}. \tag{45}\]  
In these expressions, \(h_{i,1},\ldots,h_{i,B}\) are the elements of the set \(h_{i}=\{h_{i,1},\ldots,h_{i,B}\}=\{j|c_{ij}\
eq 0\}\). Define \(f_{i}=[f_{i,1}\;\;f_{i,2}]^{T}=A_{i}B_{i}\Xi_{i,1}\), and \(f_{i}\) is bounded and \(\|f_{i}\|<f_{i}^{*}\), \(f_{i}^{*}\) is an unknown positive constant. **Remark 4**. : _From the definition of \(\Xi_{i,1}\), \(A_{i}\) and \(B_{i}\) and Equations (41)-(43),one notes that \(\|\Xi_{i,1}\|\leq 1\), \(A_{i}\) and \(B_{i}\) are bounded, so \(\|f_{i}\|<f_{i}^{*}\) is reasonable._  
In order to stabilize \(\dot{q}_{i,e}\), the desired virtual control \(\beta_{i}\) is given as:  
\[\beta_{i}=\Xi_{i,1}^{-1}(K_{i}B_{i}^{-1}P_{i}+\Xi_{i,2}-D_{i,d})+\tilde{f}_{i,2}, \tag{46}\]  
where \(\beta_{i}=[\beta_{i,1},\beta_{i,2}]^{T}\), \(\tilde{\Xi}_{i,1}^{-1}=\Xi_{i,1}^{T}(\Xi_{i,1}\Xi_{i,1}^{T})^{-1}\), \(K_{i}\in\mathbb{R}^{2n\times 2n}\) is a positive definite matrix,  
\[P_{i}=\left[ \begin{array}{cccc}P_{i,h_{i,1}}&\cdots&P_{i,h_{i,B}}\end{array}  \right]^{T}\in\mathbb{R}^{2n\times 1}, \tag{47}\]  
\(h_{i,1},\ldots,h_{i,B}\) are the elements of the set \(h_{i}=\{j|c_{ij}\
eq 0\}\), \(P_{i,h_{i}}=[\rho_{ii,e},\lambda_{ii,e}]\), \(\tilde{f}_{i,2}=[0,\;\tanh(f_{i,2}/e_{i,1})]^{T}\), and \(\epsilon_{i,1}>0\) is a constant. Notice that the matrix \(B_{i}\) is invertible owing to \(\rho_{ij}\
eq 0\). Notice that based on Lemma 1, we have \(\ln\frac{k^{2}}{k^{2}-z^{2}}<\frac{z^{2}}{k^{2}-z^{2}}\) and define \(k_{i,1}=\lambda_{\min}(K_{i,1})\), then \(-A_{i}K_{i,1}P_{i}<-2k_{i,1}q_{i,e}\). **Step 2:** Differentiating \(\psi_{i,e}\) along (21) yields  
\[\dot{\psi}_{i,e}= \dot{\phi}_{i,1}r_{i}-\dot{\phi}_{i,2}, \tag{48}\]  
where \(\dot{\phi}_{i,1}=\sum_{j=0,j\
eq i}^{n}c_{ij}\) and \(\dot{\phi}_{i,2}=\sum_{j=0,j\
eq i}^{n}c_{ij}\psi_{ij,d}\).