## 3 Control Design
### Adaptive-Gain Super-Twisting Sliding Mode Design  
According to the super-twisting law, the following sliding mode formation control approach was addressed. Considering the input-output dynamics (17), here the variable \(\omega_{k}\) is related to the control input \(\mathbf{u}_{k}\) so that their control design is equivalent to each other. Therefore, the following sliding mode control is formulated.  
\[\omega_{k}=\omega_{k1}+\omega_{k2} \tag{18}\]  
Here,  
\[\omega_{k1}=-\alpha_{k}\sqrt{\|\mathbf{s}_{ik}\|_{2}}\mathrm{sgn}(\mathbf{s} _{ik})\]  
\[\dot{\omega}_{k2}=-\frac{\beta_{k}}{2}\mathrm{sgn}(\mathbf{s}_{ik})\]  
where \(\alpha_{k}\) and \(\beta_{k}\) are the adaptive gains to be deduced from the closed-loop system stability. The signum function \(\mathrm{sgn}(\mathbf{s}_{ik})\) in (18) is determined by \(\mathrm{sgn}(\mathbf{s}_{ik})=\left[\begin{array}{cc}\mathrm{sgn}(s_{ik,1})& \mathrm{sgn}(s_{ik,2})\end{array}\right]^{T}\).  
From (17) and (18), the input-output dynamics can be re-written as  
\[\begin{array}{l}\dot{\mathbf{s}}_{ik}=-\alpha_{k}\sqrt{\|\mathbf{s}_{ik}\|_ {2}}\mathbf{b}_{1}\mathrm{sgn}(\mathbf{s}_{ik})+\mathbf{a}_{1}+\omega_{k},\\ \dot{\omega}_{k*}=-\frac{\beta_{k}}{2}\mathbf{b}_{1}\mathrm{sgn}(\mathbf{s}_{ ik})+\dot{\mathbf{a}}_{2}+\dot{\mathbf{b}}_{1}\omega_{k2}\\ \omega_{k*}(0)=0\end{array} \tag{19}\]  
Here, \(\mathbf{a}_{1}(\mathbf{x}_{ik},t)\), \(\dot{\mathbf{a}}_{2}(\mathbf{x}_{ik},\mathbf{d}_{ik},t)\), and \(\dot{\mathbf{b}}_{1}(\mathbf{x}_{ik},\mathbf{\Delta}_{k},t)\) are abbreviated by \(\mathbf{a}_{1}\), \(\dot{\mathbf{a}}_{2}\), and \(\dot{\mathbf{b}}_{1}\) for brevity, and \(\omega_{k*}=\mathbf{a}_{2}+\mathbf{b}_{1}\omega_{k2}\).  
**Assumption 3**.: \(\dot{\mathbf{b}}_{1}\omega_{k2}\) _in (19) is bounded but its boundary is unknown, that is,_  
\[\|\dot{\mathbf{b}}_{1}\omega_{k2}\|_{\infty}\leq\delta_{3} \tag{20}\]  
_Here, \(\delta_{3}\) is positive but unknown._  
Substituting \(\omega_{k2}\) in (18) into (20) yields  
\[\|\dot{\mathbf{b}}_{1}\omega_{k2}\|_{\infty}\leq\frac{\|\dot{\mathbf{b}}_{1}\| _{\infty}}{2}\int_{0}^{t}\beta_{k}\mathrm{d}t\leq\delta_{3}\]  
This fact indicates the adaptive gain \(\beta_{k}\) is bounded as well, i.e.,  
\[|\beta_{k}|\leq\beta^{*} \tag{21}\]  
Here, \(\beta^{*}\) is positive but unknown.  
**Assumption 4**.: _The adaptive gain \(\alpha_{k}\) is bounded, that is,_  
\[|\alpha_{k}|\leq\alpha^{*} \tag{22}\]  
_Here, \(\alpha^{*}\) is positive but unknown._  
Note that the uncertainties and disturbances of the input-output dynamics (17) exist in the terms \(\mathbf{a}_{2}\) and \(\mathbf{b}_{1}\). From Assumptions 2 and 3, the lumped boundary of the uncertain terms in (17) can be deduced from (16) and (22).  
\[\|\dot{x}\|_{\infty}=\|\dot{\mathbf{a}}_{2}+\dot{\mathbf{b}}_{1}\omega_{k2}\|_ {\infty}\leq\|\dot{\mathbf{a}}_{2}\|_{\infty}+\|\dot{\mathbf{b}}_{1}\omega_{k2} \|_{\infty}\leq\delta_{2}+\delta_{3}=\delta_{4} \tag{23}\]where \(\dot{\chi}=\dot{\mathbf{a}}_{2}+\dot{\mathbf{b}}_{1}\alpha_{k2}\).  
Until now, the formation control design has been equivalent to the deduction of the adaptive-gain laws in (18) that can make \(\mathbf{s}_{ik}\) and \(\dot{\mathbf{s}}_{ik}\) convergent to zero in finite time despite the uncertainties and disturbances, where the input-output dynamics are decided by (19) and the unknown boundaries \(\delta_{1}\), \(\gamma_{1}\), and \(\delta_{4}\) are given by (14), (16), and (23). Finally, the adaptive gains are designed by  
\[\dot{\alpha}_{k} =\begin{cases}\xi\sqrt{\frac{\gamma_{1}}{2}}\text{sgn}(\|\mathbf{ s}_{ik}\|_{2}-\mu)&\text{if }\alpha_{k}>\alpha_{m}\\ \eta&\text{if }\alpha_{k}<\alpha_{m}\end{cases} \tag{24}\] \[\beta_{k} =2\varepsilon\alpha_{k}\]  
Here, \(\gamma_{1}\) is determined by (14); \(\varepsilon\), \(\xi\) and \(\eta\) are arbitrary positive constants; \(\alpha_{m}>0\) is an arbitrary small constant and the initial condition of \(\alpha_{k}\) at \(t=0\) satisfies \(\alpha_{k}(0)>\alpha_{m}\).