## 2 Problem Formulation  
### System Descriptions  
Consider a group of NMRs with one passive wheel and two actuated wheels in Figure 1, the mathematical model of the \(i\)th \((i=1,\ldots,n)\) robot is given by [21]  
\[\left\{ \begin{array}{l}\dot{\eta}_{i}=J_{i}(\eta_{i})\overline{\omega}_{i}\\ M_{i}\dot{\overline{\omega}}_{i}=-C_{i}(\dot{\eta}_{i})\overline{\omega}_{i}-D _{i}\overline{\omega}_{i}+\tau_{i}\end{array} \right.\]  
where \(\eta_{i}=[x_{i},y_{i},\psi_{i}]^{T}\) is the \(i\)th robot's posture consisting of the position \((x_{i},y_{i})\) and heading angle (orientation) (\(\psi_{i}\)) of the robot coordinated. \(\overline{\omega}_{i}=[\overline{\omega}_{1i},\overline{\omega}_{2i}]^{T}\) are the angular velocities of the two wheels, \(\tau_{i}=[\tau_{i,1},\tau_{i,2}]^{T}\) is the control input. \(M_{i}\), \(C_{i}(\dot{\eta}_{i})\), and \(D_{i}\) are the \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Controlled Nonlinear NMRs** & \
\begin{tabular}{c} **Event-Triggered** \\ **Mechanism** \\ \end{tabular} & **State Constraints** & **First-Order Filter** &
\begin{tabular}{c} **Unknown** \\ **Nonlinear** \\ **Function** \\ \end{tabular} \\ \hline The proposed control method & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ The control method in [17] & \(\surd\) & \(\times\) & \(\surd\) & \(\surd\) \\ The control method in [21] & \(\times\) & \(\times\) & \(\times\) & \(\surd\) \\ The control method in [22] & \(\times\) & \(\times\) & \(\times\) & \(\surd\) \\ \hline \hline \end{tabular} where \(\surd\) indicates that the situation presented was considered in the corresponding article in Table 1.  
\end{table} 
Table 1: The comparison results of control algorithm. inertia matrix, the centripetal and Coriolis matrix, and the damping matrix, respectively. The matrices \(J_{i}(\eta_{i})\), \(M_{i}\), \(C_{i}(\dot{\eta}_{i})\), and \(D_{i}\) in system (1) are given by  
\[ \begin{array}{c}J_{i}(\eta_{i})=\frac{r_{i}}{2}\begin{bmatrix}\cos\psi_{i}& \cos\psi_{i}\\ \sin\psi_{i}&\sin\psi_{i}\\ b_{i}{}^{-1}&-b_{i}{}^{-1}\end{bmatrix}&M_{i}=\begin{bmatrix}m_{11i}&m_{12i} \\ m_{12i}&m_{11i}\end{bmatrix}\\ C_{i}(\dot{\eta}_{i})=\begin{bmatrix}0&c_{i}\dot{\psi}_{i}\\ -c_{i}\dot{\psi}_{i}&0\end{bmatrix}D_{i}=\begin{bmatrix}d_{11i}&0\\ 0&d_{22i}\end{bmatrix}\end{array} \]  
where \(r_{i}\) and \(b_{i}\) shown in Figure 1 are the wheel radius and half-width of the \(i\)th mobile robot, respectively. We consider that the NMR model (1), the robot parameters \(m_{11i}\), \(m_{12i}\), \(b_{i}\), and \(r_{i}\), the centripetal and Coriolis coefficient \(c_{i}\), and the damping coefficients \(d_{11i}\) and \(d_{22i}\) are uncertain constants. To facilitate control development, we can convert the wheel angular velocities \((\overline{\omega}_{1i},\overline{\omega}_{2i})\) to the robot's linear velocity \(v_{i}\) and angular velocity \(\omega_{i}\) by the following equation:  
\[\overline{v}_{i}=B_{i}^{-1}\overline{\omega}_{i}\]  
with \(\overline{v}_{i}=\left[v_{i},\omega_{i}\right]^{T}\) and \(B_{i}=\frac{1}{r_{i}} \begin{bmatrix}1&b_{i}\\ 1&-b_{i}\end{bmatrix} \), where the matrix \(B_{i}\) is invertible, since the determination of matrix \(B_{i}\) is \(\det(\tilde{B}_{i})=-2b_{i}/r_{i}\). After the above description, we can obtain the following dynamic equations:  
\[\left\{ \begin{array}{l}\dot{x}_{i}=v_{i}\cos(\psi_{i})\\ \dot{y}_{i}=v_{i}\sin(\psi_{i})\\ \dot{\psi}_{i}=\omega_{i}\\ M_{i}\overline{\omega}_{i}=-C_{i}(\dot{\eta}_{i})\overline{\omega}_{i}-D_{i} \overline{\omega}_{i}+\tau_{i}\end{array} \right.