## 5 Simulation and Experimental Results
### Real-Time Experiments on Simulator-Experiment Platform  
In these experiments are the formation task of four quadrotors, where quadrotor 1 is a leader. The objective is to track the trajectory \(r(t)\) with constant biases \(d_{10}=[0,\sqrt{2}]^{T}\), \(d_{20}=[-\sqrt{2},0]^{T}\), \(d_{30}=[0,-\sqrt{2}]^{T}\) and \(d_{40}=[\sqrt{2},0]^{T}\).  
Two tests are carried out on the simulator for the sake of comparison. In both tests, the objective is to track a circular trajectory (radius: 2 m; maximum linear velocity: 1.3 m/s; linear acceleration 0.1 m/s2).  
In the formation control without high-order feedforwards, the gains are chosen by \(k_{p}=1.5\) and \(k_{d}=0.1\). We note that these gains are not arbitrarily chosen. For the fairness of comparison, we tune the parameters on the simulator to obtain the gains with best performance.  
We proposed a nonlinear-flatness-based formation controller in (12), where the high-order derivatives are estimated by using (20). In the simulator and the experiment, the control frequency is 50 Hz. Then, the states of the observer (20) are updated as follows  
\[\hat{X}_{i}^{d}(k+1)=(I_{5}+T\cdot A_{t})\hat{X}_{i}^{d}(k)-T\cdot L_{t}\cdot C _{t}(\hat{X}_{i}^{d}(k)-X_{i}^{d}(k))\]  
where \(T=\frac{1}{50}=0.02\) s. The observer gains are given by (38).  
In Figure 5, we observe from Figure 5 that even at the beginning (when \(t<20\) s), the quadrotors are able to follow the given trajectory, after 20 s, the performance becomes not satisfactory, although the formation errors keep stable.  
On the contrary, by using the proposed formation control with the same controller gains, the performance is greatly improved and the quadrotors can always follow the given trajectory (see Figure 6).  
The desired altitude is 1 m for all the quadrotors. The curves of the altitudes of the quadrotors are given in Figure 7. We can observe from Figure 7 (left) that the altitude of the quadrotor 2 oscillates after \(t=25\) s. It is important to note that this phenomena is caused by the actuator saturation of the quadrotors. Considering this problem, we have proposed several bounded formation controllers such as in [13], which will not be detailed in the current paper.  
Figure 5: Formation control without high-order feedforwards.  
Figure 8 shows the animation in the simulator. On the left of Figure 8, the formation control without high-order feedforwards is used, where the quadrotors fail to preserve the rectangle formation pattern after several second of flight. On the contrary, the nonlinear-flatness-based formation control can preserve the rectangle formation and track the RFT that is only given to the leader.  
A real-time experiment is also given in Figure 9, where four quadrotors are used in the tests. The reference is a circular trajectory. Two tests are given for comparison. We can observe that by using our proposed controller, the formation trajectory is more precisely followed. Figure 10 shows a photo of flight in a real-time experiment.  
Figure 6: Our proposed formation control.  
Figure 7: Altitudes of the quadrotors. **Left**: Formation control without high-order feedforwards. **Right**: Our proposed formation control.  
Figure 8: Four quadrotors with rectangle formation pattern, tracking a circular trajectory, using formation control without high-order feedforwards (**left**), and the proposed formation control (**right**).  
Figure 10: A snap of the real flight of four quadrotors’ formation, tracking a circular trajectory while maintaining a rectangle pattern.  
Figure 9: The real-time four quadrotors formation. **Left**: Formation control without high-order feedforwards; **Right**: our proposed method.