## 2 Problem Formulation
### Leader-Follower Formation Framework  
In this section, the kinematics model of the leader-follower formation system is given. The leader-follower formation mechanism is displayed in Figure 2. In the leader-follower formation system, there is a leader robot, and others are selected as follower robots. The \(i\)th robot is set as the leader robot, and the \(k\)th robot is picked up as the representative of all follower robots. The relative distance \(l_{ik}\) and relative bearing angle \(\varphi_{ik}\) between the leader robot and follower robot are defined in Figure 2. The relative distance \(l_{ik}\) means the distance between the center of the leader robot \(i\)th and the front castor of the follower robot \(k\)th, described by  
\[l_{ik}=\sqrt{\left(x_{i}-\overline{x}_{k}\right)^{2}+\left(y_{i}-\overline{y}_ {k}\right)^{2}}. \tag{3}\]  
Figure 1: Sketches of the mobile robot. Here, \((x_{i},y_{i})\) denotes the center of the leader robot \(i\), and \((\overline{x}_{k},\overline{y}_{k})\) represents the caster position of the follower robot \(k\). The calculation of \(\overline{x}_{k},\overline{y}_{k}\) has the following form  
\[ \begin{array}{l}\overline{x}_{k}=x_{k}+r\cos\theta\\ \overline{y}_{k}=y_{k}+r\sin\theta\end{array} . \tag{4}\]  
Here, \(r\) is the radius of the round robot, and \((x_{k},\,y_{k})\) denotes the center of the follower robot \(k\). Simultaneously, \(\psi_{ik}\) is formulated by  
\[\psi_{ik}=\pi+\zeta_{ik}-\theta_{i}. \tag{5}\]  
Here, \(\theta_{i}\) denotes the orientation angle of the leader robot \(i\), \(\zeta_{ik}=\arctan\frac{y_{i}-y_{k}-r\sin\theta_{i}}{x_{i}-x_{k}-r\cos\theta_ {k}}\). In this paper, the derivative and integral terminal sliding mode controller is designed so that the follower robots can follow the leader robot with desired relative distance and angler. Therefore, the following conditions are satisfied: The collisions between the robots are avoided. There is no communication delay between the leader robot and the follower robot. Each follower robot knows its position, velocity, and corresponding information of the leader robot. According to leader-follower formation mechanism, the robots move along a specified trajectory with desired relative distance and bearing angle. It is necessary to shape the dynamics of the leader-follower formation system. Differentiate (3) and (5) twice with the respect to time \(t\), and substitute (2) into the second derivative of (3) and (5). Define the state variable \(x_{ik}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]^{T}\), where \(x_{1}=l_{ik}\), \(x_{2}=\psi_{ik}\), \(x_{3}=\dot{l}_{ik}\), \(x_{4}=\dot{\psi}_{ik}\). The dynamic model of the formation system has the form of  
\[ \begin{array}{l}\dot{x}_{ik}=f(x_{ik},d_{ik}(t))+B_{ik,1}u_{k}\\ y_{ik}=h(x_{ik})\end{array} . \tag{6}\]  
Here, \(B_{ik,1}\) is a \(2\times 2\) matrix whose columns are smooth vector fields \(B_{ik,1K}\). \(h(x_{ik})\) is the output equation of formation system. Here,  
\[f(x_{ik},d_{ik}(t))=A_{ik}x_{ik}+B_{ik,2}\widehat{d_{ik}}(t)+B_{ik,1}\Delta_{k }u_{k}\]  
\(A_{ik}\), \(B_{ik,1}\), \(B_{ik,2}\), \(h(x_{ik})\) are described by  
\[A_{ik}=\left[ \begin{array}{cccc}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{array} \right],\,B_{ik,2}=\left[ \begin{array}{cccc}0&0\\ 1&0\\ 0&0\\ 0&1\end{array} \right],\,B_{ik,1}=\left[ \begin{array}{cccc}0&0\\ \cos\varphi_{ik}&r\sin\varphi_{ik}\\ 0&0\\ \frac{-\sin\varphi_{ik}}{l_{ik}}&\frac{d\cos\varphi_{ik}}{l_{ik}}\end{array}  \right],\,h(x_{ik})=\left[ \begin{array}{cccc}x_{1}\\ x_{3}\end{array} \right].\]  
Figure 2: Sketches of the leader–follower coordinated framework. Here, \(\varphi_{ik}=\psi_{ik}+\theta_{ik}\), \(\widetilde{d}_{ik}(t)\) denotes the uncertainties of the leader-follower formation system (6), written by  
\[\widetilde{d}_{ik}(t)=L_{ik}(I_{2}+\Delta_{i})u_{i}+F_{ik}+P_{ik} \tag{7}\]  
\[L_{ik}=\left[ \begin{array}{cc}0&0\\ -\cos\psi_{ik}&0\\ 0&0\\ \frac{\sin\psi_{ik}}{I_{ik}}&-1\end{array} \right],F_{ik}=\left[ \begin{array}{ c}0\\ F_{1}\\ 0\\ F_{2}\end{array} \right],\,P_{ik}=\left[