## 3 Control Design  
### Sliding Mode Design and Its Input-Output Dynamics  
The super-twisting law is a powerful second-order sliding-mode technique [28]. It can effectively deal with the controlled plant with a relative degree equal to one with respect to the control input. Theoretically, the technique can make the sliding mode variable and its time derivative convergent to the origin in spite of the uncertainties and disturbances. Thus, this technique can be considered as a solution for formation maneuvers of the multi-robot system. In order to implement the control design of the leader-follower pair, the sliding mode vector \(\mathbf{s}_{ik}\), that is, the sliding surfaces, has to be defined in advance. \[\mathbf{s}_{ik}=\left[ \begin{array}{c}s_{ik,1}\\ s_{ik,2}\end{array} \right]=\mathbf{C}_{1}\!\left(\!\left[ \begin{array}{c}l_ {ik}\\ \psi_{ik}\end{array} \right]\!-\!\left[ \begin{array}{c}l_{ik}^{d}\\ \psi_{ik}^{d}\end{array} \right]\!\right)+\mathbf{C}_{2}\!\left(\!\left[ \begin{array}{c}\dot{l}_{ik}\\ \dot{\psi}_{ik}\end{array} \right]\!-\!\left[ \begin{array}{c}\dot{l}_{ik}^{d} \\ \dot{\psi}_{ik}^{d}\end{array} \right]\!\right) \tag{10}\]  
Here, \(\mathbf{C}_{1}\) and \(\mathbf{C}_{2}\) are \(2\times 2\) constant matrices and they need to be pre-defined. \(l_{ik}^{d}\) and \(\psi_{ik}^{d}\) are the desired relative distance and the desired relative bearing angle of this leader-follower pair, respectively. Differentiate the sliding mode vector \(\mathbf{s}_{ik}\) in (10) with the respect to time, the input-output dynamics can be derived as follows. \[\dot{\mathbf{s}}_{ik}=\frac{\partial\mathbf{s}_{ik}}{\partial t}+\frac{ \partial\mathbf{s}_{ik}}{\partial\mathbf{x}_{ik}}\mathbf{f}(\mathbf{x}_{ik}, \mathbf{d}_{ik})+\frac{\partial\mathbf{s}_{ik}}{\partial\mathbf{x}_{ik}} \mathbf{g}(\mathbf{x}_{ik},\mathbf{\Delta}_{k})\mathbf{u}_{k} \tag{11}\]  
In (10) and (11), we have  
\[\frac{\partial\mathbf{s}_{ik}}{\partial\mathbf{u}_{k}}=0\text{ and }\frac{ \partial\mathbf{s}_{ik}}{\partial\mathbf{u}_{k}}=\frac{\partial\mathbf{s}_{ ik}}{\partial\mathbf{x}_{ik}}\mathbf{g}(\mathbf{x}_{ik},\mathbf{\Delta}_{k})\
eq 0\]  
Namely, the relative degree of \(\mathbf{s}_{ik}\) with respect to \(\mathbf{u}_{k}\) is equal to 1. Subsequently, the super-twisting-based sliding mode control design can be available for formation maneuvers of such a multi-robot system. Let  
\[ \begin{array}{l}\mathbf{a}(\mathbf{x}_{ik},\mathbf{d}_{ik},t)=\frac{ \partial\mathbf{s}_{ik}}{\partial t}+\frac{\partial\mathbf{s}_{ik}}{\partial \mathbf{x}_{ik}}\mathbf{f}(\mathbf{x}_{ik},\mathbf{d}_{ik})\\ \mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)=\frac{\partial\mathbf{s}_{ ik}}{\partial\mathbf{x}_{ik}}\mathbf{g}(\mathbf{x}_{ik},\mathbf{\Delta}_{k})\end{array}  \tag{12}\]  
**Assumption 1**. : \(\mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)\) _is a \(2\times 2\) matrix and contains both known and unknown parts, written by (13)._  
\[\mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)=\mathbf{b}_{0}(\mathbf{x}_{ ik},t)+\Delta\mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t) \tag{13}\]  
Here \(\mathbf{b}_{0}(\mathbf{x}_{ik},t)\) is a known positively definite matrix, \(\Delta\mathbf{b}_{0}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)\) is bounded but unknown, and the two parts of \(\mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)\) satisfy  
\[\|\Delta\mathbf{b}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)\mathbf{b}_{0}^{-1}( \mathbf{x}_{ik},t)\|_{2}\leq\gamma(\mathbf{x}_{ik},t)<\gamma_{1}<1 \tag{14}\]  
where \(\gamma_{1}\) is a unknown constant. **Assumption 2**. : \(\mathbf{a}(\mathbf{x}_{ik},\mathbf{d}_{ik},t)\) _is a \(2\times 1\) vector and contains both known and unknown parts, depicted by (15)._  
\[\mathbf{a}(\mathbf{x}_{ik},\mathbf{d}_{ik},t)=\mathbf{a}_{1}(\mathbf{x}_{ik},t )+\mathbf{a}_{2}(\mathbf{x}_{ik},\mathbf{d}_{ik},t) \tag{15}\]  
Here, both \(\mathbf{a}_{1}(\mathbf{x}_{ik},t)\) and \(\mathbf{a}_{2}(\mathbf{x}_{ik},\mathbf{d}_{ik},t)\) are bounded, and they satisfy  
\[