## 3 Fuzzy Adaptive Formation Control Design
### ET Formation Controller Design
\(k_{d1}\) and \(k_{\varphi 1}\) are positive control gains, while the logarithmic barrier function ensures the relative azimuth remains within the predefined constraints, thereby avoiding singularity issues.) **Remark 2**. : _If \(\varphi_{i}=\frac{\pi}{2}\), then there will be singularity problem. We predefine constraints on the relative azimuth. The formation tracking controller is designed by using the barrier Lyapunov synthesis method, ensuring that the relative azimuth remains within a predetermined range._  
**Step 2:** From (9), we have  
\[\dot{z}_{i\varphi}=f_{vi}(v_{i})+m_{1i}\tau_{vi}-\dot{h}_{i,1} \tag{18}\]  
\[\dot{z}_{i\omega}=f_{\omega i}(\omega_{i})+m_{2i}\tau_{\omega i}-\dot{h}_{i,2} \tag{19}\]  
From (9), one has  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]0  
where \(N_{i,s}(\cdot)\) is a continuous function. Since \(f_{wi}(\omega_{i})\) and \(f_{vi}(v_{i})\) are uncertain, the FLSs are utilized for approximating them. Assume that  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]1  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]2  
where \(G_{i,1}\in R^{n}\) and \(G_{i,2}\in R^{n}\) are the ideal weights, \(\phi_{i,1}\in R^{n}\) and \(\phi_{i,2}\in R^{n}\) are the fuzzy basic functions. \(\varepsilon_{i,1}\) and \(\varepsilon_{i,2}\) are the approximated errors, and they satisfy \(\left|\varepsilon_{i,1}\right|\leq\varepsilon_{i,1}^{*}\) and \(\left|\varepsilon_{i,2}\right|\leq\varepsilon_{i,2}^{*}\), with \(\varepsilon_{i,1}^{*}\) and \(\varepsilon_{i,2}^{*}\) being unknown constants. Thus, (18) and (19) can be rewritten as  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]3  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]4  
Consider the log-type barrier Lyapunov function  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]5  
In order to decrease the unnecessary waste of resources resulting from the update of the controller signal, we design the following event-triggered mechanism:  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]6  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]7  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]8  
\[\delta_{i,2}\hat{h}_{i,2}+\hat{h}_{i,2}=\alpha_{i,2},\ \hat{h}_{i,2}(0)=\alpha_{i,2} \tag{11}\]9  
with \(z_{vi}(t)=\tau_{vi}(t)-\theta_{xi}(t_{vik})\) representing the event-triggered error during time interval \([t_{vik},t_{vi(k+1)})\), where \(\kappa=v,\omega\). \(0\leq\lambda_{1\kappa}\leq 1\) and \(\lambda_{2\kappa}>0\) are constants. \(\theta_{xi}(t_{vik})\) are trigger control signals. The virtual controllers, trigger controllers, adaptive formation controllers and adaptive laws are given as  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]0  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]1  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]2  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]3  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]4  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]5  
\[\dot{z}_{id}=-v_{i}\cos\varphi_{i}+v_{i-1}\cos\gamma_{i} \tag{12}\]6  
\[\dot{\hat{G}}  
where \(c_{i,v}>0\), \(c_{i,\omega}>0\), \(\sigma_{i,1}>0\), and \(\sigma_{i,2}>0\) are design parameters. \(\overline{\lambda}_{iv}\geq\frac{\lambda_{2v}}{1-\lambda_{1v}}\) and \(\overline{\lambda}_{i\omega}\geq\frac{\lambda_{2v}}{1-\lambda_{1\omega}}\) are design parameters. \(\varepsilon>0\) are design constants.