# Backstepping Based Formation Control of Quadrotors with the State Transformation Technique
## 4 Simulation Results and Analysis
### Simulation Results and Analysis
In this case, both the PID control method and the formation control method proposed in this paper show good performance of the formation control. Figures 12 and 13 show the simulation results for the quadrotor tracking the circular trajectory four times in 30 s. In this case, unlike the previous case, larger reference angles are required to track the circle trajectory faster. When the change of the attitude angle is large, in the case of deriving the reference angle using the PID control method applied to simplified dynamics with SAA, the formation control performance is degraded as shown in Figure 13a. As shown in Figure 13a, the formation errors for the \(x\)-, \(y\)- and \(z\)-axes do not converge completely to zero. On the other hand, when the reference angles for formation control are derived using the proposed state transformation technique, the formation errors for the \(x\)-, \(y\)- and \(z\)-axes converge completely to zero as shown in Figure 13b. The simulation results are shown in Figures 10-13. Furthermore, if we set up more rotating trajectories  
Figure 11: The formation errors for each of the three axis about the 4\(\pi\) circular trajectory: (**a**) PID control method; (**b**) formation control method proposed in this paper. Figure 10: The results of the comparison simulation for the formation control of quadrotors about the 4\(\pi\) circular trajectory: The red dashed line is the result of the formation control using the PID control method, and the green dashed line is the result of the formation control using the proposed formation control method with the state transformation technique (STT). within the same time, the formation cannot be maintained, and the states of follower quadrotors diverge when using the PID control method applied to the simplified dynamics with SAA. Figures 14-18 show performance comparison simulations for tracking a circular trajectory at low speed (4\(\pi\)/10 s) and tracking a circular trajectory at high speed (6\(\pi\)/10 s) using the proposed formation control method with the state transformation technique and backstepping-based formation control with SAA. The formation control problem covered in this simulation is the same as before, with the leader tracking the ascending circular trajectory followed by two follower quadrotors. For fair performance comparison, the conditions of each follower were the same. We set the initial position of each follower to \((x_{i},y_{i},z_{i})\) = (0,1,0) (m) and the desired relative distances between the follower quadrotor and the leader to \((d_{i,x},d_{i,y},d_{i,z})\) = (1,1,0.4) (m). Furthermore, the same backstepping control method was used to control the altitude, \(x,y\) position and attitude of each follower, and the same feedback gain values were also used. Figure 4:0 The results of the comparison simulation for the formation control of quadrotors about the 8\(\pi\) circular trajectory: the red dashed line is the result of the formation control using the PID control method, and the green dashed line is the result of the formation control using the proposed formation control method with the state transformation technique (STT). Figure 4:1 The formation errors for each of the three axis about the 8\(\pi\) circular trajectory: (**a**) PID control method; (**b**) proposed formation control method in this paper. Figure 14 shows the result of the formation control using the state transformation technique proposed in this paper and the formation control using SAA. Simulation results for the \(x\)-, \(y\)- and \(z\)-axis errors in each case are shown in Figure 15a,b, respectively. As shown in Figures 14-16, when the leader flies a \(4\pi\) circular trajectory in 10 s, the large reference attitude angles are not required, and the follower quadrotors trail the leader in a circular motion pattern with small attitude angles (as shown in Figure 16). In this case, the proposed formation control using the state transformation technique and the conventional formation control using SAA show similar formation control performance.