## 3 Fuzzy Adaptive Formation Control Design
### Stability Analysis  
The properties of the above control scheme are provided in the following theorem. **Theorem 1**. : _For the NMRs (1), under above assumption 1, if the virtual controllers are adopted by (16) and (17), the trigger controllers are adopted by (32) and (33), the actual controller are adopted by (34) and (35), and the parameter updating laws are adopted by (36) and (34), and then the whole fuzzy adaptive ET formation control scheme can ensure the following:_  
1. _The multiple underactuated NMRs are stable._
2. _Each follower NMR can track the leader NMR._
3. _The Zeno behavior can be excluded._  
Proof. : Differentiating (14) and (15) yields  
\[\dot{V}_{11i}=\frac{{z_{i}}{\dot{z}_{i}}}{{k_{d1}}^{2}-{z_{i}}^{2}}=\frac{{z_{ i}}{k_{d1}}^{2}-{z_{i}}^{2}}{[-(z_{iv}+\alpha_{vi})\cos\varphi_{i}+v_{i-1}\cos \gamma_{i}]} \tag{38}\]  
\[\dot{V}_{12i}=\frac{{z_{i}}{\dot{\varphi}_{i}}}{{k_{\varphi}}^{2}-{z_{i}}^{2} }=\frac{{z_{i}}{\dot{\varphi}}}{{k_{\varphi}}^{2}-{z_{i}}^{2}}{[-(z_{i\omega}+ \alpha_{vi})+\frac{v_{i}}{d_{i}}\sin\varphi_{i}-\frac{v_{i-1}}{d_{i}}\sin \gamma_{i}]} \tag{39}\]  
In the intervals \(|z_{i\emptyset}|\leq k_{\partial 1},\partial=d\), \(\varphi\), we have  
\[\log\frac{{k_{\partial 1}}^{2}}{{k_{\partial 1}}^{2}-{z_{i\emptyset}}^{2}}\leq \frac{{z_{i\partial}}^{2}}{k_{\partial 1}^{2}-z_{i\partial}^{2}} \tag{40}\]  
Substituting the virtual control laws (16), (17), and (40) into (38) and (39), we obtain  
\[\dot{V}_{11i}=-\frac{{z_{i}}{\dot{\sigma}}{z_{iv}}\cos\varphi_{i}}{k_{d1}^{2} -z_{id}^{2}}-\frac{{k_{d1}}{z_{id}^{2}}}{k_{d1}^{2}-z_{id}^{2}}\leq-{k_{d1}} V_{11i}-\frac{{z_{i\partial}}{z_{iv}}\cos\varphi_{i}}{k_{d1}^{2}-{e_{di}^{2}}} \tag{41}\]  
\[\dot{V}_{12i}=-\frac{{z_{i\varphi}}{\dot{\sigma}}{z_{i\omega}}}{k_{\varphi 1}^{2}-z_{i \psi}^{2}}-\frac{{k_{\varphi 1}}{z_{i\psi}^{2}}}{k_{\varphi 1}^{2}-z_{i\psi}^{2}} \leq-{k_{\varphi 1}}V_{12i}-\frac{{z_{i\varphi}}{\dot{\sigma}}{z_{i\omega}}}{k_{ \varphi 1}^{2}-z_{i\psi}^{2}} \tag{42}\]  
Differentiating (25) yields  
\[ \begin{array}{rl}\dot{V}_{2i}&=\dot{V}_{11i}+\dot{V}_{12i}+z_{iv}{\dot{z}_{ i\upsilon}}+z_{i\omega}{\dot{z}_{i\upsilon}}-\widetilde{G}_{i,1}^{T}\dot{G}_{i,1}- \widetilde{G}_{i,2}^{T}\dot{G}_{i,2}+\chi_{i,1}\dot{\chi}_{i,1}+\chi_{i,2}\dot {\chi}_{i,2}\\ &=\dot{V}_{11i}+\dot{V}_{12i}+z_{iv}(\widetilde{G}_{i,1}^{T}\dot{\psi}_{i,1}(v _{i})+\dot{\varepsilon}_{i,1}+m_{11}\tau_{vi}-\dot{h}_{i,1})\\ &+z_{i\omega}(\hat{G}_{i,2}^{T}\phi_{i,2}(\omega_{i})+\dot{\varepsilon}_{i,2} +m_{2i}\tau_{\omega i}-\dot{h}_{i,2})\\ &+\chi_{i,1}(N_{i,1}(\cdot)-\frac{\dot{\chi}_{i,1}}{\dot{\lambda}_{i,1}})+ \chi_{i,2}(N_{i,2}(\cdot)-\frac{\dot{\chi}_{i,2}}{\dot{\lambda}_{i,2}})\\ &+\widetilde{G}_{i,1}^{T}(z_{iv}\dot{\phi}_{i,1}(v_{i})-\dot{G}_{i,1})+ \widetilde{G}_{i,2}^{T}(z_{i\omega}\dot{\varphi}_{i,2}(\omega_{i})-\dot{G}_{ i,2})\end{array}  \tag{43}\]  
According to [27], define the set \(\Omega_{i,s}=\big{\{}\sum\limits_{i=1}^{N}\big{(}z_{i\kappa}^{2}+\widetilde{G}_{i,s}^{T}\widetilde{G}_{i,s}+\chi_{i,s}^{2} \big{)}\leq 2\xi\big{\}}\), where \(\kappa=v,\omega\), \(s=1,2\), \(\xi>0\) is a constant and \(V(0)<\xi\). Since \(\Omega_{i}\) is compact in \(R^{\dim(\Omega_{i})}\), there exists a constant \(\overline{N}_{i,s}\) with \(|N_{i,s}(\cdot)|\leq\overline{N}_{i,s}\) on \(\Omega_{i,s}\). (\(\dim(\Omega_{i})\) denotes the dimension of \(\Omega_{i}\). )Invoke the fact that \(0\leq|\theta|-\theta\text{tanh}(\theta/\ell)\leq 0.2785\ell\). According to (43), and event-triggered mechanism (26)-(29), we have  
\[