## 2 Modelling
### Model of a Leader-Follower Formation Pair  
Consider a multi-robot system composed of N robots. Each robot is the same as the robot in Figure 1, where the robot \(i\) is assigned as the leader and takes charge of other robots, that is, there exist N\(-\)1 leader-follower pairs in this multi-robot system. Figure 2 illustrates a leader-follower pair made of the leader \(i\) and its follower \(k\)[27]. Some symbols in Figure 2 are represented as follows. For each individual robot, the subscripts \(i\) and \(k\) are adopted to describe the individual variables of the leader and the follower, respectively. The subscript \(ik\) is employed to depict the relative variables of the pair. Here, the relative distance \(l_{ik}\) means the distance between the leader's center and the follower's front castor, formulated by  
\[l_{ik}=\sqrt{(x_{i}-\overline{x}_{k})^{2}+\left(y_{i}-\overline{y}_{k}\right) ^{2}} \tag{6}\]  
Here,  
\[\overline{x}_{k}=x_{k}+r\cos\theta_{k}\]  
\[\overline{y}_{k}=y_{k}+r\sin\theta_{k}\]  
The relative bearing angle \(\psi_{ik}\) of the leader-follower pair is determined by  
\[\psi_{ik}=\pi+\zeta_{ik}-\theta_{i} \tag{7}\]  
Here,  
\[\zeta_{ik}=\arctan\frac{y_{i}-y_{k}-r\sin\theta_{k}}{x_{i}-x_{k}-r\cos\theta_ {k}}\]  
The purpose of the paper was to investigate an adaptive-gain super-twisting sliding mode control design for formation maneuvers of this multi-robot system. Motivated by this purpose, the formation  
Figure 2: Sketches of a leader-follower pair. objective of the leader-follower scheme was that each leader-follower pair of the multi-robot system has to keep the desired relative distance and the desired relative bearing angle in spite of uncertainties and disturbances. In order to focus on the objective, some ideal conditions were taken in the multi-robot system: (1) There are neither collisions nor communication delay; (2) the follower is well-known, that is, it knows its position and velocity, meanwhile, it can obtain the position and velocity of the leader as well. Define a vector \(\mathbf{x}_{ik}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]^{T}\). Let \(x_{1}=l_{ik}\), \(x_{2}=\dot{l}_{ik}\), \(x_{3}=\psi_{ik}\) and \(x_{4}=\dot{\psi}_{ik}\). According to the formation objective, the relative distance \(l_{ik}\) and the relative bearing angle \(\psi_{ik}\) are determined as the formation control output. Then, the dynamics of formation maneuvers of the multi-robot system can have the form of (8) in light of the leader-follower scheme [14]. \[ \begin{array}{l}\dot{\mathbf{x}}_{ik}=\mathbf{f}(\mathbf{x}_{ik},\mathbf{d}_ {ik})+\mathbf{g}(\mathbf{x}_{ik},\mathbf{\Delta}_{k})\mathbf{u}_{k}\\ \mathbf{y}_{ik}=\mathbf{h}(\mathbf{x}_{ik})\end{array}  \tag{8}\]  
Here, \(\mathbf{x}_{ik}\) is the system state vector and \(\mathbf{y}_{ik}\) is the system output vector. Further,  
\[ \begin{array}{l}\mathbf{f}(\mathbf{x}_{ik},\mathbf{d}_{ik})=\mathbf{A}_{ik} \mathbf{x}_{ik}+\mathbf{B}_{ik,2}\mathbf{d}_{ik}\\ \mathbf{g}(\mathbf{x}_{ik},\mathbf{\Delta}_{k})=\mathbf{B}_{ik,1}+\mathbf{B}_ {ik,1}\mathbf{\Delta}_{k}\end{array} \]  
\(\mathbf{A}_{ik}\), \(\mathbf{B}_{ik,1}\), \(\mathbf{B}_{ik,2}\), and \(\mathbf{h}(\mathbf{x}_{ik})\) are depicted by  
\[\mathbf{A}_{ik}=\left[ \begin{array}{cccc}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{array} \right]\mathbf{,\ B}_{ik,2}=\left[ \begin{array}{cccc}0&0\\ 1&0\\ 0&0\\ 0&1\end{array} \right]\mathbf{,\ B}_{ik,1}=\left[ \begin{array}{cccc}0&0\\ \cos\varphi_{ik}&r\sin\varphi_{ik}\\ 0&0\\ \frac{-\sin\varphi_{ik}}{l_{ik}}&\frac{r\cos\varphi_{ik}}{l_{ik}}\end{array}  \right]\mathbf{,\ h}(\mathbf{x}_{ik})=\left[ \begin{array}{cccc}x_{1}\\ x_{3}\end{array} \right]\]  
where \(\varphi_{ik}=\psi_{ik}+\theta_{ik}\). \(\mathbf{d}_{ik}\) is the lumped term of all the uncertainties and disturbances in the leader-follower pair. \[\mathbf{d}_{ik}=\mathbf{L}_{ik}(\mathbf{I}_{2}+\mathbf{\Delta}_{i})\mathbf{u }_{i}+\mathbf{F}_{ik}+\mathbf{P}_{ik} \tag{9}\]  
In (9),  
\[\mathbf{L}_{ik}=\left[ \begin{array}{cccc}0&0\\ -\cos\psi_{ik}&0\\ 0&0\\ \frac{\sin\psi_{ik}}{l_{ik}}&-1\end{array}