## 3 Main Results
### Distributed Formation Protocol
Because (32) is a sufficient condition of (6), one has:  
\[=-\gamma_{p}\gamma\mathbf{e}_{p}^{T}\mathcal{M}^{2}\mathbf{e}_{p}(t)-\bm {e}_{v}^{T}(t)(\gamma_{v}\mathcal{M}^{2}-\gamma\mathcal{M})\mathbf{e}_{v}(t)- \big{(}\gamma\mathbf{e}_{p}(t)+\mathbf{e}_{v}(t)\big{)}^{T}\mathcal{M}\big{(}\gamma_{ p}(\mathcal{L}\otimes I_{n})\widetilde{\mathbf{q}}_{p}(t) \tag{21}\]0 \[=-\gamma_{p}\gamma\mathbf{e}_{p}^{T}\mathcal{M}^{2}\mathbf{e}_{p}(t)-\bm {e}_{v}^{T}(t)(\gamma_{v}\mathcal{M}^{2}-\gamma\mathcal{M})\mathbf{e}_{v}(t)- \big{(}\gamma\mathbf{e}_{p}(t)+\mathbf{e}_{v}(t)\big{)}^{T}\mathcal{M}\big{(}\gamma_{ p}(\mathcal{L}\otimes I_{n})\widetilde{\mathbf{q}}_{p}(t) \tag{21}\]1 \[=-\gamma_{p}\gamma\mathbf{e}_{p}^{T}\mathcal{M}^{2}\mathbf{e}_{p}(t)-\bm {e}_{v}^{T}(t)(\gamma_{v}\mathcal{M}^{2}-\gamma\mathcal{M})\mathbf{e}_{v}(t)- \big{(}\gamma\mathbf{e}_{p}(t)+\mathbf{e}_{v}(t)\big{)}^{T}\mathcal{M}\big{(}\gamma_{ p}(\mathcal{L}\otimes I_{n})\widetilde{\mathbf{q}}_{p}(t) \tag{21}\]2  
By taking \(\varepsilon=\frac{\sigma_{i}}{\alpha_{i}}\chi_{i}(0)e^{-(\beta_{i}+\sigma_{i}) \mathcal{T}}\), this implies that (33) contradicts (29). Therefore, the aforementioned Zeno behavior is eliminated in the proposed approach. Consequently, it can be seen from (27) that the formation tracking error \(\mathbf{\varepsilon}(t)\) exponentially asymptotically converges to the origin. In general, the proposed distributed control law navigates the group of agents to track the predefined formation. Further, the internal dynamics (11) is also asymptotically stable. **Remark 3**. : _Intuitively, from (30), we can conclude that the larger initial values \(\chi_{i}(0)\) result in a larger inter-event time. Further, one can obtain from proof of Theorem 1 that there always exists the static event trigger \(\hat{\chi}_{i}(t)\leq 0\) that has inter-event time larger than the dynamic event triggering mechanism._  
**Remark 4**. : _In the following, let us provide a discussion on the choice of parameters in order to tune the behaviors of the considered system. While \(\gamma\) solely guarantees the existence of \(\gamma_{p},\gamma_{v},\theta_{v}\) and does not impact the performance of the system, the others influence the inter-event time and a decay rate \(\kappa\) of the Lyapunov function (26). Actually, because \(\mathcal{M}\) in (16) and (17) absolutely depends on the network topology of the considered multi-agent system, we should firstly choose \(\gamma\), \(\gamma_{p},\gamma_{v}\), and \(\theta_{v}\) to satisfy (16) and (17). After selecting \(\sigma_{i}\) in the proposed triggering law (6), \(\beta_{i}\) and \(\theta_{p}\) are possibly adjusted to compromise the decay rate of the Lyapunov function (26) and the value of the minimum inter-event time._  
**Remark 5**. : _Obviously, because the system requires extensive communication among agents, communication delays and package losses clearly affect the performance of the system. In this note, we studied the application of multi-agent systems to a multiple quadcopter system. Due to the much larger dynamics of mechanical systems, compared with that of communication, it can be assumed that there is ideal information transmission among agents. On the other hand, there have been several studies [20; 30; 31; 32] addressing these aforementioned problems in consensus multi-agent systems. Thus, we leave these as future research into the formation control of multi-quadcopter systems._