## 3 Distributed Hybrid Formation Control
### Stability Analysis  
**Theorem 1**. : _Assuming the control for robot \(p_{i}\) presented in (12) has appropriate tuning functions \(\mathbf{v}_{i}^{\theta}(t)\) and \(\mathbf{v}_{i}^{*}(t)\), the control objective can be reached._  
Proof. : For any robot \(p_{i}(i\in\mathcal{V})\), the relative position \(\tilde{r}_{i}(t)\) satisfies  
\[\dot{\tilde{\mathbf{r}}}_{i}(t)=\tilde{\mathbf{v}}_{i}(t)=\mathbf{v}_{i}(t)- \mathbf{v}_{c}(t)\]  
Then,  
\[\ddot{\tilde{\mathbf{r}}}_{i}(t) =\dot{\mathbf{v}}_{i}(t)-\dot{\mathbf{v}}_{c}(t)=\dot{\mathbf{v}}_ {i}^{\theta}(t)+\dot{\mathbf{v}}_{i}^{*}(t)\] \[=(\omega_{i}^{*}(X_{i})\tilde{r}_{i}(t)\vec{e}_{\theta})+\!\! \alpha_{i}^{*}(X_{i})\]  
Together with \(\frac{d}{dt}\vec{e}_{\theta}=-\omega_{i}\vec{e}_{r}\), it yields  
\[ \begin{array}{l}\ddot{\tilde{\mathbf{r}}}_{i}(t)=\dot{\omega}_{i}^{*}(X_{i} )\tilde{r}_{i}(t)\vec{e}_{\theta}+\omega_{i}^{*}(X_{i})\dot{\tilde{r}}_{i}(t) \vec{e}_{\theta}-\omega_{i}\omega_{i}^{*}(X_{i})\tilde{r}_{i}(t)\vec{e}_{r}\\ \qquad+(\dot{\alpha}_{r}^{*}(X_{i})-\dot{r}_{i}(t)\omega_{i}+\dot{r}_{i}(t) \omega_{i}\omega_{i}^{*}(X_{i}))\vec{e}_{r}\\ \qquad+(\dot{\alpha}_{\theta}^{*}(X_{i})\tilde{r}_{i}(t)+\omega_{i}^{*}(X_{i} )\dot{\tilde{r}}_{i}(t)\vec{e}_{\theta}\\ \qquad=(\dot{\alpha}_{r}^{*}(X_{i})-\dot{\tilde{r}}_{i}(t)\omega_{i})\vec{e}_{ r}+(\dot{\omega}_{i}^{*}(X_{i})\tilde{r}_{i}(t)\ +\dot{\alpha}_{\theta}^{*}(X_{i})\tilde{r}_{i}(t)\\ \qquad+2\omega_{i}\dot{\tilde{r}}_{i}(t))\vec{e}_{\theta}\end{array}  \tag{13}\]  
Besides, \(\dot{\tilde{\mathbf{r}}}_{i}(t)\) can also be rewritten as  
\[\dot{\tilde{\mathbf{r}}}_{i}(t)=\dot{\tilde{r}}_{i}(t)\vec{e}_{r}+\tilde{r}_{ i}(t)\omega_{i}\vec{e}_{\theta}\]  
Consequently,  
\[\ddot{\tilde{r}}_{i}(t)=(\ddot{\tilde{r}}_{i}(t)-\tilde{r}_{i}(t)\omega_{i}^{ 2})\vec{e}_{r}+(\dot{\omega}_{i}\ddot{r}_{i}(t)+2\omega_{i}\dot{\tilde{r}}_{i }(t)\vec{e}_{\theta} \tag{14}\]  
By combining (13) and (14), we get  
\[\left\{ \begin{array}{l}\alpha_{r}^{*}(X_{i})-\tilde{r}_{i}(t)\omega_{i}= \ddot{\tilde{r}}_{i}(t)-\tilde{r}_{i}(t)\omega_{i}^{2}\\ (\dot{\omega}_{i}^{*}(X_{i})\dot{r}_{i}(t)+\dot{\alpha}_{\theta}^{*}(X_{i}) \ddot{r}_{i}(t)+2\omega_{i}\dot{\tilde{r}}_{i}(t)=\dot{\omega}_{i}\dot{r}_{i} (t)+2\omega_{i}\dot{\tilde{r}}_{i}(t)\end{array} \right.\]  
which is simplified as  
\[\left\{ \begin{array}{l}\ddot{\tilde{r}}_{i}(t)=\alpha_{r}^{*}(X_{i})+-\dot{ \tilde{r}}_{i}(t)(\omega_{i}^{2}-\omega_{i})\\ \omega_{i}=\dot{\omega}_{i}^{*}(X_{i})+\dot{\alpha}_{\theta}^{*}(X_{i})\end{array} \right. \tag{15}\]  
That is to say, under the designed control law, we can separate the control of \(\omega_{i}\) from \(\tilde{r}_{i}(t)\). Equation (15) indicates that the control of \(\omega_{i}\) only relates to \(\omega_{i}^{*}(X_{i})\) and \(\alpha_{\theta}^{*}(X_{i})\), meanwhile the control of \(\tilde{r}_{i}(t)\) only relates to \(\alpha_{r}^{*}(X_{i})\). Namely, the formation control problem is decomposed into the radial control and the circumferential control. However, Equation (15) does not satisfy with the anti-collision condition and the convergence condition. Thus, specific forms of \(\omega_{i}^{*}(X_{i})\), \(\alpha_{r}^{*}(X_{i})\), and \(\alpha_{\theta}^{*}(X_{i})\) are given to meet both convergence condition the anti-collision condition. (1) Example of \(\alpha_{r}^{*}(X_{i})\)  
\(\alpha_{r}^{*}(X_{i})\) has the form as  
\[\alpha_{r}^{*}(X_{i})=-k_{r}(\tilde{r}_{i}(t)-R_{i})-2\sqrt{k_{r}}\dot{r}_{i}(t) \tag{16}\]  
where \(k_{r}>0\). When the initial condition fulfills \(\forall i\in\mathcal{V}\) and \(\hat{r}_{i}(0)>-\sqrt{k_{r}}\hat{r}_{i}(0)\), the movement of all robots satisfies the convergence condition and the anti-collision condition.