## 3 Distributed Hybrid Formation Control
### Stability Analysis
According to  
\[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]  
We get  
\[\bar{r}_{i}(t)=-k_{r}(\bar{r}_{i}(t)-R_{i})-2\sqrt{k_{r}}\hat{r}_{i}(t)\]  
As a critically damped motion differential equation, its solution is written as  
\[\bar{r}_{i}(t)=R_{i}+((\bar{r}_{i}(0)-R_{i})+(\bar{r}_{i}(0)+(\bar{r}_{i}(0)-R_ {i})\sqrt{k_{r}}))e^{-\sqrt{k_{r}}t} \tag{17}\]  
Seeking the limit of Equation (17), we get  
\[\lim_{t\to+\infty}\bar{r}_{i}(t)=R_{i}\]  
meets the convergence condition (11). The first derivative of \(\bar{r}_{i}(t)\) is  
\[\hat{r}_{i}(t)=-\sqrt{k_{r}}(\hat{r}_{i}(0)+(\bar{r}_{i}(0)-R_{i})\sqrt{k_{r}} )t+\hat{r}_{i}(0)\]  
When  
\[t_{1}=\frac{\hat{r}_{i}(0)}{(\bar{r}_{i}(0)+(\bar{r}_{i}(0)-R_{i})\sqrt{k_{r}} ))\sqrt{k_{r}}}>0 \tag{18}\]  
At \(t_{1}\), \(\hat{r}_{i}(t_{1})=0\), \(\bar{r}_{i}(t)\) reaches its extreme value. To sum up, if the initial conditions satisfy  
\[\forall i\in\mathcal{V},\bar{r}_{i}(0)>0,\hat{r}_{i}(0)>-\sqrt{k_{r}}\bar{r}_{ i}(0) \tag{19}\]  
Then,  
\[\forall i\in\mathcal{V},\forall t>0,\bar{r}_{i}(t)>0\]  
A Lyapunov function candidate is considered as  
\[V(\bar{r}_{i}(t))=(R_{i}-\tilde{r}_{i}(t))^{2}. \tag{20}\]  
Then, the derivative of \(V(\tilde{r}_{i}(t))\) along the trajectories leads to  
\[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]0 \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]1 \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]2 \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]3 \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]4  
where \(\sqrt{k_{r}}>0\). From \(\omega<(\sqrt{k_{r}}+1)(\bar{r}_{i}(t)-R_{i})\), (21) can be rewritten as  
\[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]5  
It can be further proven that \(\tilde{r}_{i}(t)=R_{i}\) is the maximum invariant set in \(\{\tilde{r}_{i}(t)\in M_{1}|\dot{V}(\bar{r}_{i}(t))=0\}\). Additionally, each solution starting in \(\dot{r}_{i}(t)\in M_{1}\) approaches to \(\tilde{r}_{i}(t)=R_{i}\) as \(t\to\infty\) obtained from [PERSON]'s theorem in [40]. Thus, we conclude that if the initial condition meets the condition show as Equation (19), \(\bar{r}_{i}(t)\) satisfies the anti-collision (2) and convergence condition (11) under the designed \(\alpha_{r}^{*}(X_{i})\). 2. Example of \(\omega_{i}^{*}(X_{i})\) and \(\alpha_{\theta}^{*}(X_{i})\) Define \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]6 (23) Then, \[\bar{r}_{i}(t)=\alpha_{r}^{*}(X_{i})\]7 \(\omega_{i}^{*}(X_{i})\) and \(\alpha_{\theta}^{*}(X_{i})\) are rearranged as \[\left\{ \begin{array}{l}\omega_{i}^{*}(X_{i})=\frac{\alpha_{i^{-}}^{*}\dot{ \alpha}_{i^{-}}^{+}-\alpha_{i^{-}}^{*}\dot{\alpha}_{i^{-}}^{-}}{\alpha_{i^{-}} ^{*}+\alpha_{i^{-}}^{*}}=\Delta\dot{\alpha}_{i}\\ \alpha_{\theta}^{*}(X_{i})=k_{\theta}\Delta\alpha_{i}+2\sqrt{k_{\theta}} \Delta\dot{\alpha}_{i}\end{array} \right.\] (24) where \(k_{\theta}>0\). We only consider the initial condition satisfies  
\[\forall i\
eq j,\omega_{i}=\omega_{j} \tag{25}\]  
According to  
\[\dot{\omega}_{i}=\dot{\omega}_{i}^{*}+\alpha_{\theta}^{*}\]  
We get  
\[\omega_{i}=\omega_{i}^{*}+\int\limits_{\tau=0}^{t}\alpha_{\theta}^{*}d\tau+C_ {\theta} \tag{26}\]  
Based on the initial condition, we can obtain \(\forall i\in\mathcal{V},\dot{\alpha}_{i}^{+}=0\), further \(\Delta\dot{\alpha}_{i}=0\). Thus, at \(t=0\), there exists \(\dot{\omega}_{i}=\dot{\omega}_{i}^{*}\) such that \(C_{\theta}=0\) gives. Therefore,  
\[\dot{\omega}_{i}=\dot{\omega}_{i}^{*}+\int\limits_{\tau=0}^{t}\alpha_{\theta} ^{*}d\tau\]  
Take the derivation of (26), we have a critically damped vibration equation for \(\Delta\alpha_{i}\)  
\[\Delta\ddot{\alpha}_{i}=\dot{\omega}_{i}^{*}=-\alpha_{\theta}^{*}=-k_{\theta} \Delta\alpha_{i}-2\sqrt{k_{\theta}}\Delta\dot{\alpha}_{i} \tag{27}\]  
Taking the initial condition \(\Delta\dot{\alpha}_{i}=0\) into consideration, the solution to (27) is  
\[\Delta\alpha_{i}(t)=\Delta\alpha_{i}^{0}(1+\sqrt{k_{\theta}}t)e^{-\sqrt{k_{ \theta}}t} \tag{28}\]  
where \(\Delta\alpha_{i}^{0}=\Delta\alpha_{i}(0)\). For convenience, define  
\[f(t)=(1+\sqrt{k_{\theta}}t)e^{-\sqrt{k_{\theta}}t}\]  
Further, we get  
\[\left\{