## 3 Fuzzy Adaptive Formation Control Design
### Stability Analysis
\begin{array}{ll}\overline{m}_{1i}z_{iv}\tau_{vi}-\overline{m}_{1i}z_{iv}\rho_ {vi}&=\overline{m}_{1i}z_{iv}\big{(}\frac{\theta_{ii}}{1+\lambda_{ii}\theta_{vi 1}}-\frac{\overline{\lambda}_{ii}\delta_{ii2}}{1+\lambda_{ii}\delta_{ii1}}- \rho_{vi}\big{)}\\ &\leq\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_{ii1}}\big{(} |z_{iv}\rho_{vi}|-z_{iv}\rho_{vi}\text{tanh}\big{(}\frac{z_{iv}\rho_{vi}}{ \varepsilon}\big{)}+\big{|}z_{iv}\overline{\lambda}_{vi}\big{|}\\ &-z_{iv}\overline{\lambda}_{vi}\text{tanh}\big{(}\frac{z_{iv}\delta_{ii}}{ \varepsilon}\big{)}\big{)}-\frac{\overline{m}_{1i}z_{iv}\overline{\lambda}_{ vi}\delta_{ii2}}{1+\lambda_{ii}\delta_{vi1}}\\ &\leq 0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_{ vi1}}+(\frac{\overline{m}_{1i}z_{iv}\overline{\lambda}_{vi}\delta_{ii2}}{1+ \lambda_{ii}\delta_{vi1}})^{2}+\frac{1}{4}\\ \end{array}  \tag{44}\]  
\[ \begin{array}{ll}\overline{m}_{1i}z_{iv}\tau_{vi}-\overline{m}_{1i}z_{iv} \rho_{vi}&=\overline{m}_{1i}z_{iv}\big{(}\frac{\theta_{ii}}{1+\lambda_{ii} \delta_{vi1}}-\frac{\overline{\lambda}_{ii}\delta_{ii2}}{1+\lambda_{ii}\delta_ {vi1}}-\rho_{vi}\big{)}\\ &\leq\frac{\overline{m}_{ii}(1+\lambda_{i})}{1+\lambda_{ii}\delta_{vi1}}\big{(} |z_{iv}\rho_{vi}|-z_{iv}\rho_{vi}\big{)}\text{tanh}\big{(}\frac{z_{iv}\rho_{ vi}}{\varepsilon}\big{)}+\big{|}z_{iv}\overline{\lambda}_{vi}\big{|}\\ &-z_{iv}\overline{\lambda}_{vi}\text{tanh}\big{(}\frac{z_{iv}\delta_{ii}}{ \varepsilon}\big{)}\big{)}-\frac{\overline{m}_{1i}z_{iv}\overline{\lambda}_{ vi}\delta_{ii2}}{1+\lambda_{ii}\delta_{vi1}}\\ &\leq 0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{vi})}{1+\lambda_{ii}\delta_{ vi1}}+\overline{m}_{1i}\big{(}\frac{z_{iv}\overline{\lambda}_{vi}\delta_{vi2}}{1+ \lambda_{ii}\delta_{vi1}})^{2}+\frac{1}{4}\\ \end{array}  \tag{45}\]  
According to [PERSON]'s inequality \(a^{2}+b^{2}\leq 2 ab\), it is available  
\[ \begin{array}{ll}z_{iv}\varepsilon_{i,1}+z_{iv}\varepsilon_{i,2}+\chi_{i,1} (N_{i,1}(\cdot)-\frac{\chi_{i,1}}{\delta_{i1}})+\chi_{i,2}(N_{i,2}(\cdot)- \frac{\chi_{i,2}}{\delta_{i2}})\\ \leq\frac{1}{2}z_{iv}^{2}+\frac{1}{2}z_{iv}^{2}+\frac{1}{2}\varepsilon_{i,1}^{2 }+\frac{1}{2}\varepsilon_{i,2}^{2}+(\frac{1}{2}-\frac{1}{\delta_{i,1}})\chi_ {i,1}^{2}+\frac{1}{2}N_{i,1}^{2}\\ +(\frac{1}{2}-\frac{1}{\delta_{i,2}})\chi_{i,2}^{2}+\frac{1}{2}N_{i,2}^{2}\\ \end{array}  \tag{46}\]  
\[\widetilde{G}_{i,s}^{\top}\hat{G}_{i,s}\leq\frac{1}{2}\big{(}\big{\|}\hat{G}_ {i,s}^{*}\big{\|}^{2}-\Big{\|}\widetilde{G}_{i,s}\Big{\|}^{2}\big{)} \tag{47}\]  
where \(p_{1}\) and \(p_{2}\) are positive design parameters. From (30)-(37) and (43)-(47), we have  
\[ \begin{array}{ll}\dot{V}_{2}&\leq(\frac{1}{2}+c_{i,d})z_{id}^{2}+(\frac{1} {2}-c_{i,\varphi})z_{iv}^{2}+(1-c_{i,\varphi})z_{iv}^{2}+(1-c_{i,\omega})z_{iv }^{2}\\ &-\frac{\sigma_{1,1}}{2}\Big{\|}\widetilde{G}_{i,1}\Big{\|}^{2}-\frac{\sigma_ {1,2}}{2}\Big{\|}\widetilde{G}_{i,2}\Big{\|}^{2}+(\frac{1}{2}-\frac{1}{ \delta_{i,1}})\chi_{i,1}^{2}+(\frac{1}{2}-\frac{1}{\delta_{i,2}})\chi_{i,2}^{2 }\\ &+\frac{1}{2}N_{i,1}^{2}+\frac{1}{2}N_{i,2}^{2}+\frac{1}{2}\varepsilon_{i,1}^{2 }+\frac{1}{2}\varepsilon_{i,2}^{2}+\frac{\sigma_{1,2}}{2}\Big{\|}\hat{G}_{i,1} ^{*}\Big{\|}^{2}+\frac{\sigma_{2}}{2}\Big{\|}\hat{G}_{i,2}^{*}\Big{\|}^{2}\\ &+0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_{vi1}} +0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_{vi1}} +\frac{1}{2}\end{array}  \tag{48}\]  
Let  
\[ \begin{array}{ll}\overline{\Pi}=\frac{1}{2}N_{i,1}^{2}+\frac{1}{2}N_{i,2}^{2 }+\frac{1}{2}\varepsilon_{i,1}^{2}+\frac{1}{2}\varepsilon_{i,2}^{2}+\frac{ \sigma_{1,2}}{2}\Big{\|}\hat{G}_{i,1}^{*}\Big{\|}^{2}+\frac{\sigma_{2,2}}{2} \Big{\|}\hat{G}_{i,2}^{*}\Big{\|}^{2}\\ &\quad+0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_ {vi1}}+0.557\ell\frac{\overline{m}_{ii}(1+\lambda_{ii})}{1+\lambda_{ii}\delta_ {vi1}}+\frac{1}{2}\end{array}