## 4 Simulation Studies
### Simulation Verification  
The leader's signal is \(x_{0}=y_{0}=\sin t\). The robots' initial positions are given as \(p_{0}=[0,0,0]^{T}\ p_{1}=[0.1,0,\pi/9]^{T}\ p_{2}=[-0.1,0,\pi/9]^{T}\ p_{3}=[0.5,1, \pi/4]^{T}\ p_{4}=[0.1,0,\pi/4]^{T}\). The initial function is as follows:  
\[ \begin{array}{l}f_{vi}=\frac{2b_{i}c\omega_{i}^{2}-p_{3}(d_{11}+d_{22})-b \omega_{i}(d_{11}-d_{22})}{2(m_{11}-m_{12})}\\ f_{\omega i}=-\frac{2c_{i}(v_{i}\omega_{i}+v_{i}(d_{11}-d_{22})+b\omega_{i}(d_{ 11}+d_{22}))}{2c_{i}(m_{11}-m_{12})}\end{array} \]  
To estimate the unknown nonlinear functions \(f_{vi}\) and \(f_{\omega i}\), assume that the universes of discourse for variables \(\kappa_{i}\) are \([-4,4]\). Define If-Then fuzzy rules as  
\(L^{i}\): if \(\kappa_{i}\) is \(f_{xi}\), then \(y_{i}\) is \(P\ (\kappa=\sigma,\omega;i=1,2,3,4;j=1,2,3,4,5)\). The fuzzy membership functions are given by \(\mu_{f_{1}}=1/(1+e^{[5(3+\kappa_{1})/2]})\), \(\mu_{f_{\xi 2}}=e^{[-(2+\kappa_{2})^{2}/2]}\), \(\mu_{f_{\xi 3}}=e^{[-(3)^{2}/2]}\), \(\mu_{f_{\xi 4}}=e^{[-(2+\kappa_{4})^{2}/2]}\), \(\mu_{f_{\xi 5}}=1/(1+e^{[-(3-\kappa_{5})/2]})\), \(\kappa=v,\omega\). Consequently, the fuzzy membership functions are presented in Figure 2. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(a_{i}\) & 0.75 & m & \(I_{ib}\) & 15.625 & kg \(\cdot\) m\({}^{2}\) \\ \(c_{i}\) & 0.3 & m & \(I_{iu}\) & 0.005 & kg \(\cdot\) m\({}^{2}\) \\ \(r_{i}\) & 0.15 & m & \(I_{im}\) & 0.0025 & kg \(\cdot\) m\({}^{2}\) \\ \(m_{ib}\) & 30 & kg & \(d_{i11}\) & 5 & kg \(\cdot\) m\({}^{2}\)/s \\ \(m_{io}\) & 1 & kg & \(d_{i12}\) & 5 & kg \(\cdot\) m\({}^{2}\)/s \\ \hline \end{tabular}
\end{table} 
Table 2: Robot parameters. The FLS \(\hat{f}_{vi}(v_{i})=G_{i,1}^{T}\textbf{V}_{i,1}(v_{i})\) and \(\hat{f}_{\omega i}(\omega_{i})=G_{i,2}^{T}\textbf{V}_{i,2}(\omega_{i})\) are employed to estimate \(f_{vi}(v_{i})\) and \(f_{\omega i}(\omega_{i})\). **Remark 3**. : _It is also worth pointing out that according to Lemma 1, an FLS can approximate an unknown nonlinear function when the number of If-Then rules is larger, and for every input variable in the FLS, its fuzzy membership functions are chosen as Gaussian types \(\mu f_{i}(\kappa_{i})=\exp[\frac{-(\kappa-\mu_{i})^{T}(\kappa-\mu_{i})}{\gamma _{i}^{2}}],i=1,\ldots,n\), where the center points \(\mu_{i}\) belong to the universe of discourse of the variable, and the width \(\gamma_{i}>0\). Moreover, the defined fuzzy membership functions should satisfy the following conditions._  
1. _Each pair of_ \((\mu f_{k}(\kappa_{i}),\mu f_{j}(\kappa_{i}))\) _is required to be overlapped._
2. _The whole sequence of fuzzy membership functions_ \(\mu f_{1}(\kappa_{i}),\ldots,\mu f_{n}(\kappa_{i})\) _is required to cover the universe of discourse of the variable_ \(x\) _(the universe of discourse usually chosen as a symmetrical interval about zero)._  
The distance and angle of ensuring connectivity, collision avoidance, and the idealize are chosen as \(d_{con,i}=5\) m, \(d_{col,i}=3\) m, \(\varphi_{con,i}=1\) rad, \(d_{des,i}=4\) m, and \(\varphi_{des,i}=0\) rad, respectively. The parameters of the virtual controllers in expressions (16) and (17), along with the adaptive laws stated in (49) and (50), are given as \(k_{A1}=k_{\varphi 1}=9\) and \(\sigma_{i,1}=\sigma_{i,2}=1\) for robots 1-4, \(i=1,2,3,4\). The parameters of trigger controllers (32) and (33) are given as \(\lambda_{iv}=0.01\), \(\overline{\lambda}_{iv}=0.05\), and \(\varepsilon=0.9\). The simulation results are exhibited in Figures 3-8. Figure 3 depicts the curves of NMRs. From Figure 3, it can be seen that Robot 1, Robot 2, Robot 3, and Robot 4 are stable and able to track Robot 0 to maintain the desired formation pattern. In the whole formation control process, there will be no collision between the robots, and the formation can be maintained within the scope of constraints. Figures 4 and 5 show the tracking error of the four robots in terms of distance and angle, respectively.