## 4 Application to Multiple Quadcopter Systems
### Numerical Simulation Results  
The parameter values of each quadcopter based on the F450 frame are given in Table 2[33], and each quadcopter's initial states were chosen such that they lied outside their predefined spots in the formation to validate the proposed algorithm strongly: \(p_{1}(0)=[0,\,2,\,0]^{T}\), \(p_{2}(0)=[1.5,\,1,\,0]^{T}\), \(p_{3}(0)=[1.5,\,0,\,0]^{T}\), \(p_{4}(0)=[0,\,-1,\,0]^{T}\), \(p_{5}(0)=[-1,\,0,\,0]^{T}\), and \(p_{6}(0)=[-1,\,1.5,\,0]^{T}\) (m). The formation shape is defined by a regular hexagon with the length of the sides \(d_{0}=3\)m. It is easily seen that \(f_{p,1d}=\left[0,\,3d_{0}/2,\,0\right]^{T}\), \(f_{p,2d}=\left[d_{0}\sqrt{3}/2,\,d_{0},\,0\right]^{T}\), \(f_{p,3d}=\left[d_{0}\sqrt{3}/2,\,0,\,0\right]^{T}\), \(f_{p,4d}=\left[0,\,-d_{0}/2,\,0\right]^{T}\), \(f_{p,5d}=\left[-d_{0}\sqrt{3}/2,\,0,\,0\right]^{T}\), and \(f_{p,6d}=\left[\,-d_{0}\sqrt{3}/2,\,d_{0},\,0\right]^{T}\) (m). Each \(i\)th quadcopter time-varying spot can be set as \(f_{p,i}(t)=f_{p,iid}\left(1-\exp\left(-10\,t\right)\right)\), \((i\in\mathbb{S})\). Additionally, based on Theorem 1, all coefficients are selected as: \(\gamma=0.2\), \(\gamma_{v}=3\), \(\gamma_{p}=2\), \(\theta_{v}=1.3\), \(\theta_{p}=1.3\), \(\beta_{i}=5\), \(\sigma=0.2\), \(\chi_{i}(0)=10\quad(i\in\mathbb{S})\), \(\delta_{1}=1\), and \(\delta_{j}=0\)\((j\in\mathbb{S}\), \(j\
eq 1)\). The reference velocity is set as \(v_{0}=[1,\,1,\,2]^{T}\). Furthermore, a connected network of six quadcopters is considered with the Laplacian matrix:  
\[\mathcal{L}=\left[ \begin{array}{cccccc}2&-1&0&0&0&-1\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&0\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ -1&0&0&0&-1&2\end{array} \right], \tag{41}\]  
where each quadcopter communicates with its two neighbors, e.g., the first quadcopter exchanges its information with the second and sixth quadcopters. Figure 3: The control diagram of the multiple quadcopter system based on Robot Operating System (ROS). Figure 4a-c shows six different dotted color lines describing the tracking errors of the quadcopters projected on the \(x\)-, \(y\)-, and \(z\)- axes, respectively. The initial state of each quadcopter was arranged outside its desired position in the formation to demonstrate our method strongly. By using the cascade control structure, the actual attitude of each quadcopter automatically tracked its desired value generated by the outer loop. Hence, although the reference attitude of each quadcopter remained unchanged between two consecutive triggering times, the formation tracking errors of each quadcopter were almost smooth. Based on Theorem 1, it was proven that the investigated system asymptotically achieved the predefined time-varying formation flight as time went to infinity. It can be seen that these figures confirmed the asymptotic convergence of the closed loop system at the origin from the eighth second, leading to the achievement of the time-varying formation flight for the six quadcopter system. Figure 5a illustrates the overall formation flight for the probed system within ten seconds of simulation. By using dotted blue lines to mark the shape of the six quadcopters at times T = 0, T = 2, T = 5, and T = 10 (s), the transition of the time-varying formation is easily observed in three-dimensional space. Additionally, by projecting into the \(xy\)-plane, the position snapshots of the time-varying formation flight are shown in Figure 5b. Beginning from an arbitrary hexagon at T = 0 s, they almost reached the predefined regular hexagon at T = 5 s after magnifying their shape at T = 2 s. Finally, the predefined regular hexagon was totally achieved at T = 10 s.  
As mentioned in [24], the minimum execution time of the proposed method was greater than that of the static law. However, we could not say anything about the total triggering times for these two algorithms. These times could be computed in the simulations of two cases, i.e., using the proposed and static triggering laws for the same scenario of the formation flight within 10 s of simulation.