## 5 Simulation and Experimental Results
### Simulation Results  
We reconsider the formation of four quadrotors in Figure 1 (left). The reference formation trajectory RFT is given by \(r(t)=[3\sin(0.1t),3\cos(0.1t),1,0]^{T}\). The four quadrotors should keep a desired biases w.r.t. RFT defined by \(d_{i0}\), \(i\in\{1,2,3,4\}\). As shown in Figure 1 (right), quadrotor 3 has two neighbors, i.e., 2 and 4. The quadrotor 3 is expected to keep inter-distances \(d_{32}=[\sqrt{2},-\sqrt{2}]^{T}\) and \(d_{34}=[-\sqrt{2},-\sqrt{2}]^{T}\) with respect to 2 and 4. It is easy to verify that \(d_{32}=d_{30}-d_{20}\) and \(d_{34}=d_{30}-d_{40}\).  
The tuning process of the controller gains in Table 2 is given as follows.
* **Controller gains tuning on simulator:** Let \(\bar{X}_{i}^{d}\), \(\bar{X}_{i}^{d}\), \(\bar{X}_{i}^{d}\), \(\bar{X}_{i}^{d}\), and \(\hat{Y}_{i}^{d}\), \(\bar{Y}_{i}^{d}\), \(\bar{Y}_{i}^{d}\), \(\bar{Y}_{i}^{d}\) be null. Give some small \(X_{i}^{d}\) and \(Y_{i}^{d}\), for example a small step signal, then, tune the gains \(k_{jX}\) and \(k_{jY}\), \(j=\{1,2,3,4\}\). Observe the oscillation of the rotation angles during translational motion, adjust the gains \(k_{3X}\), \(k_{4X}\) and \(k_{3Y}\), \(k_{4Y}\) to reduce the oscillation.
* **Implement the controller gains on real quadrotor:** Test on single quadrotor, slightly tune the parameters \(k_{1X}\), \(k_{2X}\) if necessary. Then, if the performance is satisfied, implement them on the formation tracking control.  
The observer gain matrix is selected according to the pole assignment as follows  
* **Pole assignment:** The gains of the observer are dominated by the finite time \(T_{converge}\) that we want to have. Once it is fixed, the gains of the observer can be calculated by the technique of pole assignment, such that the nearest pole of \(A_{o}\) to imaginary axis is placed to \(-\gamma\).  
In the following simulation and experiments, we assume that the position and velocity in the guidance vector are measurable, i.e.,  
\[C_{t}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\end{bmatrix}\]  
the observe gains are selected as follows,  
\[L_{t}=\begin{bmatrix}6&1&1&1.02&0.198\\ 1.804&9.02&14.432&6.675&0.934\end{bmatrix}^{T} \tag{38}\]  
Some constant bounds of the saturation functions are selected as \(b_{4}=0.000035\), \(b_{3}=0.00035\), \(b_{2}=0.0035\), and \(b_{1}=\infty\). The saturation functions are introduced to regulate the large initial deviation of the observer. There bounds are tuned progressively until a satisfactory performance is achieved.  
We respectively show in Figures 2 and 3 the simulation results (i) without estimation of high-order derivatives and (ii) with estimation of high-order derivatives. We observe from Figure 2 that the quadrotors track the RFT with delays and the inter-distances are not well maintained. On the contrary, in Figure 3, the delays are small and the quadrotors keep around the desired inter-distances.  
In the following subsection, we will implement the formation control protocol on the real-time simulator-experiment platform.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(k_{4X}\) & \(k_{3X}\) & \(k_{3X}\) & \(k_{4X}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) & \(k_{4Y}\) \\ \hline
1.0 & 2.5 & 1.8 & 1.3 & 1.0 & 2.5 & 1.8 & 1.3 & 17.3 & 10.2 & 3.1 & 2.2 \\ \hline \end{tabular}
\end{table}
Table 2: The controllers parameters.  
Figure 3: Rigid formation of 4 quadrotors with high-order derivatives estimation, the objective is to track the FFT \(r(t)\) and meanwhile keep inter-distance between quadrotors.  
Figure 2: Rigid formation of 4 quadrotors without high-order derivatives estimation, the objective is to track the RFT \(r(t)\) and meanwhile keep inter-distance between quadrotors.