## 4 Application to Multiple Quadcopter Systems
### Numerical Simulation Results
The figures for the triggering times of these two laws are reported in Table 3. There was a significant reduction in the input update times of the dynamics compared with the statics. It can be seen that by using the proposed method, the triggering times were almost half or one-third that of the conventional scheme. There were dissimilar decreases in the triggering times of all quadcopters because of their different initial tracking errors. In other words, since the reference trajectory of the formation \(r(t)\) was only available to the first quadcopter, the value of its triggering times was slightly greater than that for the others. To be specific, in comparison with periodical sampling where the sampling rate was around 20 to 100 Hz for the outer control loop (10-50 ms for the sampling time) due to GPS sensor, our algorithm provided less triggered times (number of sampling times). As can be shown in Table 3, it took 124 to 115 sampling times for 10 s in the outer control loop (the minimum interval time was approximately 50 ms). It should be noted that fewer sampling times possibly brought twofold beneficial effects. The first one was the decrease of computational burden in the inner loop controller. The other one was that the transmission times between each pair of connected quadcopters and the required sampling rate of its GPS sensors were reduced. By marking \"*\" when events occurred in each quadcopter in Figure 6, not only was it proven in Theorem 1, but also the total corresponding triggering times of each quadcopter were shown to guarantee once again that the Zeno behavior was excluded. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Value** & **Unit** \\ \hline Mass of quadcopter & \(m_{i}=1.568\) & kg \\ Gravitational accelerator & \(g=9.8\) & m\(\cdot\)s\({}^{-2}\) \\ Arm length & \(l_{i}=0.25\) & m \\ Thrust coefficient & \(b_{i}=7.73\times 10^{-6}\) & N\(\cdot\)s\({}^{2}\cdot\)rad\({}^{-2}\) \\ Drag coefficient & \(d_{i}=1.28\times 10^{-7}\) & N\(\cdot\)rad\({}^{-2}\) \\ Moment of inertia along \(x\)-axis & \(\mathcal{I}_{x,i}=0.0119\) & kg\(\cdot\)m\({}^{2}\) \\ Moment of inertia along \(y\)-axis & \(\mathcal{I}_{y,i}=0.0119\) & kg\(\cdot\)m\({}^{2}\) \\ Moment of inertia along \(z\)-axis & \(\mathcal{I}_{z,i}=0.0223\) & kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The quadcopter parameter values in the simulation. Figure 4: Formation flight tracking error of the six quadcopters along the \(x\)-, \(y\)-, and \(z\)-axis. Figure 5: The illustration of formation control. (**a**) Formation flight of the six quadcopters in three-dimensional space. (**b**) Position snapshots of the six quadcopters and \(r(t)\) projected into the \(xy\)-plane. As mentioned in Remark 1, we verified our proposed control algorithm for more general reference trajectories, including multiple straight lines. The following simulation result for the case reference trajectory \(r(t)\) with five straight lines was established to demonstrate the effectiveness of our method. In more detail, the reference velocity could be chosen as: \(v_{0}(t)=[0.8,\ 0.5,\ 1.5]^{T}\ t\in[0,\ 10)\), \(v_{0}(t)=[0.6,0.3,1.5]^{T}\ t\in[10,\ 30)\), \(v_{0}(t)=[1,-0.2,1.5]^{T}\ t\in[30,\ 50)\), \(v_{0}(t)=[0.3,-0.2,1.5]^{T}\ t\in[50,\ 70)\), and \(v_{0}(t)=[-0.4,-0.4,1.5]^{T}\ t\in[70,\ 100]\ (\mathrm{s})\). The formation flight of the probed system tracking the multiple straight lines trajectory is shown in three-dimensional space in Figure 7a. To achieve the asymptotic stability of the closed loop system clearly, Figure 8a-c illustrates the tracking errors of the corresponding quadcopters with multiple line trajectories. Although there were sudden changes of the reference velocities at times \(T=10\) s, \(T=30\) s, \(T=50\) s, and \(T=70\) s, the tracking errors of the considered system along the \(x\)-, \(y\)-, and \(z\)-axes asymptotically converged to the origin.