## 2 Problem Formulation
### Control Problem Formulation  
Considering the dynamic mode (6) of the leader-follower formation system, a relative-degree is calculated by  
\[L_{f}h_{1}(x_{ik})=\frac{\partial h_{1}(x_{ik})}{\partial x}f(x_{ik},d_{ik}) \tag{8}\]  
\[L_{f}h_{2}(x_{ik})=\frac{\partial h_{2}(x_{ik})}{\partial x}f(x_{ik},d_{ik}) \tag{9}\]  
\[L_{B_{k,1}k}L_{f}h_{1}(x_{ik},d_{ik})=L_{B_{k,1}k}\left(L_{f}h_{1}(x_{ik},d_{ik })\right)=\frac{\partial\left(L_{f}h_{1}(x_{ik},d_{ik})\right)}{\partial x_{ik }}\cdot B_{ik,1K} \tag{10}\]  
\[L_{B_{k,1}k}L_{f}h_{2}(x_{ik},d_{ik})=L_{B_{k,1}k}\left(L_{f}h_{2}(x_{ik},d_{ik })\right)=\frac{\partial\left(L_{f}h_{2}(x_{ik},d_{ik})\right)}{\partial x_{ik }}\cdot B_{ik,1K}. \tag{11}\]  
Here, \(r_{K}\) (\(K=1\), 2) is the smallest integer so that the least one of the control inputs appears in \(y_{K}^{r_{K}}\) (\(K=1\), 2), then  
\[\left[ \begin{array}{c}y_{1}^{r_{1}}\\ y_{2}^{r_{2}}\end{array} \right]=\left[ \begin{array}{c}L_{f}^{r_{1}}h_{1}(x_{ ik})+\sum\limits_{K=1}^{m}L_{B_{k,1}k}L_{f}h_{1}(x_{ik},d_{ik})u_{k}\\ L_{f}^{r_{2}}h_{2}(x_{ik})+\sum\limits_{K=1}^{m}L_{B_{k,1}k}L_{f}h_{2}(x_{ik},d_{ ik})u_{k}\end{array} \right]. \tag{12}\]  
Here, \(r_{1}=r_{2}=2\), \(m=2\). \(L_{B_{k,1}k^{\prime}}\), \(L_{f}\) are lie derivatives. Further, the input-output dynamic equation is depicted by  
\[\ddot{y}_{ik}=d_{ik}+H(x_{ik},d_{ik})u_{k}. \tag{13}\]  
Here, \(d_{ik}=\widetilde{L}_{ik}(I_{3}+\Delta_{i})u_{i}+\widetilde{P}_{ik}+ \widetilde{F}_{ik}\), \(H(x_{ik},d_{ik})=G_{ik}(I_{4}+\Delta_{k})\). Here, \(I_{3}\), \(I_{4}\) are 2 \(\times\) 2 matrices, \(u_{i}\) is the control input of \(i\)th robot, and other matrices are described by  
\[G_{ik}=\left[ \begin{array}{cc}\cos\varphi_{ik}&r\sin\varphi_{ik}\\ -\frac{\sin\varphi_{ik}}{I_{ik}}&\frac{r\cos\varphi_{ik}}{I_{ik}}\end{array}  \right],\widetilde{L}_{ik}=\left[ \begin{array}{c}-\cos\psi_{ik}&0\\ \frac{\sin\varphi_{ik}}{I_{ik}}&-1\end{array} \right],\,\widetilde{F}_{ik}=\left[ \begin{array}{c}F_{1}\\ F_{2}\end{array} \right],\,\widetilde{P}_{ik}=\left[ \begin{array}{c}P_{1}\\ P_{2}\end{array} \right].\]  
**Hypothesis 1. **\(H(x_{ik},d_{ik})\) _has the normal part \(G_{ik}\) and nonlinear part \(G_{ik}\Delta_{k}\), and meets the following in equation_  
\[\delta_{ik}I\leq H(x_{ik})H^{-1}_{ik}\leq\delta^{-1}_{ik}I. \tag{14}\]  
_Here, \(\delta_{ik}>1\), \(H_{dik}=H\Big{(}x^{d}_{ik}\Big{)}\), \(I\) is the 2 \(\times\) 2 square matrix,\(x^{d}_{ik}\) denotes the desired state vector in (6)._  
**Remark 1. **\(G_{ik}\Delta_{k}u_{k}\) _is the matched uncertainties in (13), meaning the parameter fluctuations of the follower robot \(k\). The term \(d_{ik}\) depicts the unmatched uncertainties in the leader-follower formation system, and consists of three parts. Due to the formation, framework (13) is applied to the follower robot, and the information of leader robot \(u_{i}\) can't be matched._  
The terms \(\widetilde{F}_{ik}\), \(\widetilde{P}_{ik}\) denote the model uncertainties and external disturbances caused by slipping, friction, and obstacles etc., which are also hard to be matched. **3. Control Design**  
Due to the inherent characteristics of centralization, the scheme mainly depends on the leader robots and exists as the \"single point of failure\" problem. In order to develop derivative and integral terminal sliding mode approach to coordinate the leader robot \(i\) and follower robot \(k\), a recursive structure of the terminal sliding function for high relative-degree MIMO systems (with \(r_{1}\), \(r_{2}>1\)) is designed as  
\[e_{ik,DO1}=y_{ik,1}-y^{d}_{ik,1},\ \ e_{ik,DO2}=y_{ik,2}-y^{d}_{ik,2}, \tag{15}\]  
\[e_{ik,D11}=\dot{e}^{P_{11}/q_{11}}_{a,D11}+\gamma_{ik,1}e_{ik,D01},\ \ e_{ik,D12}=\dot{e}^{P_{12}/q_{12}}_{a,D02}+\gamma_{ik,2}e_{ik,D02}, \tag{16}\]  
\[s_{ik}=\left[ \begin{array}{c}s_{ik,1}\\ s_{ik,2}\end{array} \right]=\left[ \begin{array}{c}e_{ik,D_{11}}+\lambda_{ik, 1}e_{ik,I1}\\ e_{ik,D_{12}}+\lambda_{ik,2}e_{ik,I2}\end{array}