## 3 Main Results
### Distributed Formation Protocol
From (9) and (10), it can be established that:  
\[\mathbf{z}_{p}(t)=\mathcal{M}\mathbf{e}_{p}(t),\ \mathbf{z}_{v}(t)=\mathcal{M}\mathbf{e}_{v}(t). \tag{22}\]  
Additionally, by letting \(\mathbf{q}(t)=\big{[}q_{1}^{T}(t),q_{2}^{T}(t),\ldots,q_{N}^{T}(t)\big{]}\), it can be derived from (8) that:  
\[\mathbf{q}(t)=-\gamma_{p}(\mathcal{L}\otimes I_{n})\widetilde{\mathbf{q}}_{p}(t)- \gamma_{v}\mathcal{M}\widetilde{\mathbf{q}}_{v}(t)-\gamma_{p}(\Delta\otimes I_{n} )\widetilde{\mathbf{e}}_{p}(t)+\widetilde{\mathbf{f}}_{a}(t). \tag{23}\]  
From (9) and (10), (21) can be rewritten as:  
\[\dot{V}(t) =-\gamma(\gamma_{p}-\theta_{p})\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2} \mathbf{e}_{p}(t)-\mathbf{e}_{v}^{T}(t)\big{(}(\gamma_{v}-\theta_{v})\mathcal{M}^{2} -\gamma\mathcal{M}\big{)}\mathbf{e}_{v}(t)-\gamma\mathbf{z}_{p}^{T}(t)\big{(}\theta_{ p}\mathbf{z}_{p}(t)-\mathbf{q}(t)\big{)} \tag{24}\] \[\quad-\mathbf{z}_{v}^{T}(t)\big{(}\theta_{v}\mathbf{z}_{v}(t)-\mathbf{q}(t) \big{)}+\sum_{i=1}^{N}\hat{\chi}_{i}(t).\]  
Further, by substituting the dynamic event-triggered law (6), (7), and (23) into (24), it can be established:  
\[\dot{V}(t)= -\gamma(\gamma_{p}-\theta_{p})\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2} \mathbf{e}_{p}(t)-\mathbf{e}_{v}^{T}(t)\big{(}(\gamma_{v}-\theta_{v})\mathcal{M}^{2} -\gamma\mathcal{M}\big{)}\mathbf{e}_{v}(t)\] \[-\sum_{i=1}^{N}\hat{\chi}_{i}+\sum_{i=1}^{N}\left(-\beta_{i}\chi_ {i}(t)+\hat{\chi}_{i}(t)\right)\] \[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\]0  
where:  
\[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\]1  
By invoking the conditions (15) and (16), the matrix \(\mathbb{M}\) is positive definite. From the fact that \(\mathbb{M}\geq\frac{\hat{\lambda}_{\text{min}}(\mathbb{M})}{\lambda_{\text{ max}}(\mathbb{P})}\mathbb{P}\) where \(\mathbb{P}\) is also a positive definite matrix, let us select:  
\[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\]2Then, (25) offers:  
\[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\]3  
for \(\forall t\geq 0\). Therefore, by applying the comparison lemma again to Inequality (26), one has:  
\[\dot{V}(t) =(\gamma_{p}+\gamma_{v}\gamma)\mathbf{e}_{p}^{T}(t)\mathcal{M}^{2}\bm {e}_{v}(t)+\gamma\mathbf{e}_{v}^{T}(t)\mathcal{M}\mathbf{e}_{v}(t)+\gamma\mathbf{e}_{p}^{T} (t)\mathcal{M}\hat{\mathbf{e}}_{v}(t)+\mathbf{e}_{v}^{T}(t)\mathcal{M}\hat{\mathbf{e}}_{v} (t)+\sum_{i=1}^{N}\hat{\chi}_{i}(t)\]4  
Next, we are going to prove that there is no Zeno behavior in the proposed dynamic event triggering mechanism.