# Backstepping Based Formation Control of Quadrotors with the State Transformation Technique
## 4 Simulation Results and Analysis
### Simulation Results and Analysis  
Figures 4 and 5 show the simulation results obtained by applying the backstepping-based quadrotor formation control using the state transformation technique. As shown in Figures 4 and 5, the initial position of the leader and follower quadrotors is \((x_{i},y_{i},z_{i})\) = (0,1,0) (m), and the reference trajectory is set to the ascending 720-degree circular trajectory. From the simulation results, we can see that the leader tracks the predefined trajectory well, and the follower quadrotors maintain a constant distance from the leader quadrotor, as well as the formation shape. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value (Unit)** \\ \hline \(m\) & 1.0 (kg) \\ \(g\) & 9.806 (m/s\({}^{2}\)) \\ \(l_{xx}\) & \(2.3\times 10^{-3}\) (kg \(\cdot\) m\({}^{2}\)) \\ \(l_{yy}\) & \(2.3\times 10^{-3}\) (kg \(\cdot\) m\({}^{2}\)) \\ \(l_{zz}\) & \(5.09\times 10^{-3}\) (kg \(\cdot\) m\({}^{2}\)) \\ \(J\) & \(6.5\times 10^{-5}\) (kg \(\cdot\) m\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The model parameters for the dynamics of a quadrotor. Figure 4: The simulation result for the formation control of quadrotors in three dimensions. Figure 5: The simulation result for the formation control of quadrotors in each of the three axis: The solid line indicates the trajectory of leader quadrotor, and the dashed lines are the trajectories of each follower quadrotor. Figures 6 and 7 show the simulation results for the attitude control of Followers 1 and 2, respectively. The initial state of the follower quadrotors is \((roll,pitch,yaw)=(0,0,0)\) (rad), the current follower attitude angles track the reference attitude angles enabling the follower quadrotors to follow and maintain the set distance from the leader. Figures 8 and 9 show the altitude and attitude control inputs of the leader and follower quadrotors, respectively. As shown in the simulation result, the altitude control input \(u_{1}\) is a positive value and close to about 9.8 N, which is the magnitude of gravity force. Furthermore, we can see that the size changes of inputs \(u_{2}\), \(u_{3}\) and \(u_{4}\) are within a reasonable range. Figure 6: The reference angles of the first follower quadrotor for the proposed formation control: the dashed line indicates the reference attitude angles that enable the quadrotor to converge to the desired formation, and the solid line is the result of the simulation about attitude responses of the first follower quadrotor. Figure 7: The reference angles of the second follower quadrotor for the proposed formation control: the meanings of the dashed line and the solid line are the same as in Figure 6.  
Figures 10-13 show performance comparison simulations for tracking a circular trajectory at low speed (\(4\pi/30\) s) and tracking a circular trajectory at high speed (\(8\pi/30\) s) using the proposed control method and PID control method with the small angle assumption. In this simulation, we set the same ascending circular trajectory as before. We set the initial position of each follower and the leader to be equal to \((x_{i},y_{i},z_{i})\) = (0,1,0) (m) and the desired relative distances between the follower and the leader to \((d_{i,x},d_{i,y},d_{i,z})\) = (1,1,0.4) (m). In addition, for fair performance comparison, the feedback gain of PID controllers used in this simulation is selected by changing slightly based on the gain of PID controller used in [32]: these gains were tuned to ensure acceptable tracking performance considering reality with minimizing the overshoot and undershoot error signals. Figure 8: The control input for the altitude of the leader and follower quadrotors. Figure 9: The control inputs for the attitude of the leader and follower quadrotors. Figures 10 and 11 show the simulation results for quadrotor to track a circular trajectory two times in 30 s. As shown in Figures 10 and 11, there is no significant difference between two cases when the reference angles for the formation control are small.