## 3 Main Results
### Distributed Formation Protocol
Suppose that the graph \(\mathcal{G}\) is undirected and strongly connected; there exist positive scalar coefficients \(\gamma\), \(\gamma_{p}\), \(\gamma_{v}\), \(\theta_{p}\), and \(\theta_{v}\) such that:_  
\[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]2 \[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]3 (16) \[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]4  
_Then, under the control law (12) and the dynamic event triggering mechanism (6), the closed loop system (14) is exponentially asymptotically stable, and there is no Zeno behavior._  
Proof of Theorem 1.: From (6), it is obvious that \(\sigma_{i}\chi_{i}(t)+\hat{\chi}_{i}(t)>0\) for all \(t\in\left[t_{k_{i}}^{i},t_{k_{i}+1}^{i}\right)\). Then, (11) offers \(\dot{\chi}_{i}(t)>-\beta_{i}\chi_{i}(t)-\sigma_{i}\chi_{i}(t)\). By using the comparison lemma in, e.g., [29], pp. 102-103, one has:  
\[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]5  
Because of \(\beta_{i},\sigma_{i}\in\mathbb{R}_{+}\), and \(\chi_{i}(0)>0\), it can be derived that \(\chi_{i}(t)>0\). By utilizing [PERSON]'s complement, the condition (17) and Lemma 2 ensure that:  
\[\mathbb{P}=\left[\cdot \begin{pmatrix}\gamma_{p}+\gamma_{v}\gamma\mathcal{M}^{ 2}&\gamma\mathcal{M}\\ \gamma\mathcal{M}&\gamma\mathcal{M}\end{pmatrix} \right]>0. \tag{19}\]Next, let us take a Lyapunov function candidate as follows:  
\[V(t)=\ \frac{1}{2}\bar{\mathbf{e}}^{T}(t)\mathbb{P}\bar{\mathbf{e}}(t)+\sum_{i=1}^{N} \chi_{i}(t), \tag{20}\]  
where \(\bar{\mathbf{e}}(t)=\big{[}\mathbf{e}_{p}^{T}(t),\ \mathbf{e}_{v}^{T}(t)\big{]}^{T}\). The time-derivative of \(V(t)\) along with the solution of (14) is given by:  
\[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\] \[=-\gamma_{p}\gamma\mathbf{e}_{p}^{T}\mathcal{M}^{2}\mathbf{e}_{p}(t)-\bm {e}_{v}^{T}(t)(\gamma_{v}\mathcal{M}^{2}-\gamma\mathcal{M})\mathbf{e}_{v}(t)- \big{(}\gamma\mathbf{e}_{p}(t)+\mathbf{e}_{v}(t)\big{)}^{T}\mathcal{M}\big{(}\gamma_{ p}(\mathcal{L}\otimes I_{n})\widetilde{\mathbf{q}}_{p}(t) \tag{21}\] \[\quad+\gamma_{v}\mathcal{M}\widetilde{\mathbf{q}}_{v}(t)+\gamma_{p}( \Delta\otimes I_{n})\widetilde{\mathbf{e}}_{p}(t)-\widetilde{\mathbf{f}}_{a}(t)\big{)} +\sum_{i=1}^{N}\hat{\chi}_{i}(t),\]  
where \(\widetilde{\mathbf{e}}_{p}(t)=\widetilde{\mathbf{q}}_{p}(t)-\widetilde{\mathbf{r}}(t)\). We remark that \(e_{p,i}(t)\) and \(e_{p,i}(t)\) are only available in the \(i\)th agent if \(\delta_{i}=1\). Thus, let \(\mathbf{z}_{p}(t)=\big{[}z_{p,1}^{T}(t),z_{p,2}^{T}(t),\ldots,z_{p,N}^{T}(t)\big{]} ^{T}\), \(\mathbf{z}_{v}(t)=\big{[}z_{v,1}^{T}(t),z_{v,2}^{T}(t),\ldots,z_{v,N}^{T}(t)\big{]} ^{T}\).