## 3 Main Results
### Nonlinear Controller Design and Stability Analysis
\begin{array}{l}\dot{V}^{i}(t)=\bigg{(}\mathbf{\mathcal{T}}_{0i}^{ \mathcal{L}_{t}}\widetilde{X}_{i}^{\mathcal{L}_{t}}\bigg{)}^{\mathrm{T}}\big{(} \mathbf{\Lambda}_{L}\big{(}\eta_{i_{k}}a_{i_{k}}\big{)}\otimes\mathbf{I}_{2}\big{)} \widetilde{\mathbf{q}}_{i}^{\mathcal{L}_{t}}\\ \quad\quad+\gamma_{k}(\mathbf{I}_{L}\otimes\mathbf{P}_{e})\bigg{(}\widetilde{T}_{ij}^ {\mathcal{L}}\otimes\mathbf{P}_{2}\bigg{)}\mathbf{u}_{*}^{i_{\mathcal{L}_{t}}}\\ \quad\quad-\gamma_{k}(\mathbf{I}_{L}\otimes\mathbf{P}_{e})\bigg{(}\widetilde{T}_{ij}^ {\mathcal{L}}\otimes(\mathbf{P}_{e}\mathbf{P}_{1}\mathbf{P}_{e})\bigg{)}D_{1}\Big{(}\mathbf{q }_{i}^{\mathcal{L}_{t}}\Big{)}\\ \quad\quad+\gamma_{k}\bigg{(}\widetilde{\mathbf{\varphi}}_{i}^{\mathcal{L}_{t}} \widetilde{\mathbf{q}}_{i}^{\mathcal{L}_{t}}+\mathbf{I}_{L}\otimes\mathbf{P}_{1}D_{1} \Big{(}\mathbf{q}_{i}^{\mathcal{L}_{t}}\Big{)}\bigg{)}\\ \gamma_{k}=\eta_{0}\bigg{(}\widetilde{\mathbf{q}}_{i}^{\mathcal{L}_{t}}\bigg{)}^{ \mathrm{T}}\big{(}\mathbf{\Lambda}_{L}\big{(}\eta_{i_{k}}a_{i_{k}}\big{)}\otimes \mathbf{I}_{2}\big{)}\end{array} \right. \tag{20}\]  
where \(\mathbf{u}_{i}^{\mathcal{L}_{t}}\) can be rewritten as  
\[\mathbf{u}_{i}^{\mathcal{L}_{t}}=\widetilde{\mathbf{K}}_{i}^{2}\mathbf{1}_{L \times 1}\otimes\mathbf{P}_{1}k_{i}^{2}\mathbf{q}_{i}^{\mathcal{L}_{t}}+\widetilde{\mathbf{K}}_ {i}^{4}D_{1}\Big{(}\mathbf{q}_{i}^{\mathcal{L}_{t}}\Big{)} \tag{21}\]  
With \(\tilde{\mathbf{T}}_{0i}^{\mathcal{L}}=l_{ii}\mathbf{\Lambda}_{L}\Big{(}k_{i_{k}}^{2} \lambda_{i_{k}}\Big{)}\), \(\widetilde{\mathbf{K}}_{i}^{1}=\mathbf{\Lambda}_{L}\Big{(}k_{i_{k}}^{1}\Big{)}\), \(\widetilde{\mathbf{K}}_{i}^{2}=\left[k_{i_{k}}^{1}\right]_{L}^{1}\), \(\widetilde{\mathbf{K}}_{i}^{3}=\mathbf{\Lambda}_{L}\Big{(}k_{i_{k}}^{2}\Big{)}\), \(\widetilde{\mathbf{K}}_{i}^{4}=\mathbf{\Lambda}_{L}\Big{(}k_{i_{k}}^{3}\Big{)}\). Redefining \(\mathbf{k}_{ii}\) and \(\mathbf{k}_{i_{k}}\) as \(\mathbf{k}_{ii}=\left[k_{ii}^{1},\mathbf{k}_{ii}^{2},0_{3}\right]\) and \(\mathbf{k}_{i_{k}}=\left[k_{ij_{k}}^{1},\mathbf{k}_{ii_{k}}^{2},\mathbf{k}_{ij_{k}}^{3}\right]\) with  
\[\left\{ \begin{array}{l}\mathbf{k}_{ii}^{1}=\sum_{j\
eq i}k_{ij}^{1}\mathbf{\lambda}_{ i}^{1}\mathbf{\mathcal{T}}_{0i}-\sum_{j\
eq i}k_{ij}^{2}\lambda_{ij}P_{e}\mathbf{ \mathcal{T}}_{0i}\\ \mathbf{k}_{ij_{k}}^{1}=k_{ij}^{1}\mathbf{\lambda}_{i}^{1}\mathbf{\mathcal{T}}_{0i}-k_{ij}^ {2}\mathbf{\lambda}_{ij}P_{e}\mathbf{\mathcal{T}}_{0i}\\ \mathbf{k}_{ii}^{2}=\left[\begin{array}{c}k_{ij}^{2}\\ 0\end{array}\right],\mathbf{k}_{ii}^{2}=\frac{1}{v_{i}^{2}}\left[\begin{array}{ ccc}cos\Big{(}\psi_{r}^{i_{k}}\Big{)}v_{t}^{i}&0\\ -\sin\Big{(}\psi_{r}^{i_{k}}\Big{)}&0\end{array}\right]\\ \mathbf{k}_{ij_{k}}^{3}=\frac{1}{v_{i}^{2}}\left[\begin{array}{c}cos\Big{(}\psi_{ r}^{i_{k}}\Big{)}v_{t}^{i}&0\\ -\sin\Big{(}\psi_{r}^{i_{k}}\Big{)}&v_{t}^{i}\end{array}\right]\\ \end{array} \right. \tag{22}\]  
\[\text{where }\mathbf{\lambda}_{i}^{1}=\frac{1}{l_{ii}v_{t}^{i}} \begin{bmatrix}v_{t}^{i}&0 \\ 0&1\end{bmatrix} ,\mathbf{\lambda}_{ij}=\frac{-1}{l_{ii}v_{t}^{i}} \begin{bmatrix}\psi_{t}^ {j}v_{t}^{i}&0\\ 0&\psi_{t}^{j}\end{bmatrix} .\]Next, we have  
\[ \begin{split}\dot{V}^{i}(t)&=\eta_{0}\sum_{k=1}^{L} \gamma_{i_{k}}\mathbf{P}\mathbf{r}\mathbf{k}_{i_{k}}^{2}\mathbf{q}_{i}^{k}\\ &+\eta_{0}\left(\widehat{\mathbf{q}}_{i}^{\angle_{t}}\right)^{\mathrm{ T}}\mathbf{\Lambda}_{L}\left(\eta_{j_{k}}a_{i_{k}}\right)\widehat{\mathbf{\varphi}}_{i}^{ \angle_{t}\sim_{t}}\mathbf{q}_{i}^{\angle_{t}}\\ &-\eta_{0}\sum_{k=1}^{L}\gamma_{i_{k}}\mathbf{P}\mathbf{r}i_{k}\mathbf{\lambda }_{i_{k}}(\mathbf{P}_{e1})^{\mathrm{T}}\mathbf{\mathcal{T}}_{0i}\widehat{\mathbf{X}}_{i_{ k}}^{t}\\ &\eta_{0}\sum_{k=1}^{L}\left(\gamma_{i_{k}}\xi_{i_{k}}D_{1}\left( \mathbf{q}_{i}^{k}\right)+\gamma_{i_{k}}\mathbf{P}\mathbf{r}\mathbf{P}_{1}\mathbf{q}_{i}^{t}\right) \\ &+\sum_{k=1}^{L}\left(\eta_{j_{k}}-\eta_{0}k_{j_{k}}^{1}\right) \left(\mathbf{\mathcal{T}}_{0i_{k}}\widehat{\mathbf{X}}_{i_{k}}^{t}\right)^{\mathrm{ T}}\left(\gamma_{i_{k}}\right)^{\mathrm{T}}\end{split}  \tag{23}\]  
where  
\[\left\{