# Inverse Optimal Design for Position Control of a Quadrotor
## 4 Simulation Results and Analysis
### Simulation Results  
Figure 3 shows the simulation result obtained by applying the inverse optimal position control of a quadrotor, when the initial and target positions are (0, 0, 0), (1, 1, 1), respectively. More details, the time histories of the \(x\), \(y\), \(z\) for each motion of a quadrotor are shown in Figure 4. From the simulation results, we verify that the proposed control method successfully makes a quadrotor move to the target position. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{Feedback gains of the position controller} \\ \hline \(\gamma_{z}\) & \multicolumn{2}{c}{0.03} \\ \(\gamma_{p}\) & \multicolumn{2}{c}{0.001} \\ \hline \multicolumn{4}{c}{PID parameters of rotational controller} \\ \hline \hline  & \(q=\phi\) & \(q=\theta\) & \(q=\psi\) \\ \(K_{p,d}\) & 3 & 3 & 3 \\ \(K_{i,q}\) & 0.3 & 0.3 & 0.3 \\ \(K_{d,d}\) & 0.2 & 0.2 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Feedback gains of the position and the rotational controllers. Figure 4: The position control results for each of the three axes. The dashed line indicates the goal position of each axis, and the solid line is the result of the simulation about each axis: (**a**) \(x\); (**b**) \(y\); (**c**) \(z\). Figure 3: The inverse optimal position control when the initial and final states are (0, 0, 0) and (\(1,1,1\)), respectively. Figure 5 shows simulation results of the attitude control. We can be found that when the initial state of a quadrotor is \((roll,pitch,yaw)=(0,0,0)\) [rad], the current attitude angles follow the reference attitude angles that enable a quadrotor to converge in the desired position. After a quadrotor reach to the target position, the roll, pitch, and yaw converge to 0, respectively, and then, it maintains a stable posture. Figures 6 and 7 show the altitude and attitude control inputs of a quadrotor. As shown in the simulation result, the altitude control input \(u_{1}\) is positive value and from 5 s when the quadrotor reaches the target position, \(u_{1}\) is maintained at about 9.8 [N], which is magnitude of gravity. Also we can check that the size changes of input \(u_{2}\), \(u_{3}\) and \(u_{4}\) are within a reasonable range. From the simulation results, we verify that the proposed control method successfully makes the position of a quadrotor converge to the references. We conducted additional simulations to verify that the proposed controller has good performance even though the input coupling matrix \(B_{z}(t)\) changes over time. In the additional simulation, the ascending circular trajectory is set as the reference trajectory, we tested whether the quadrotor track the trajectory well in nonlinear motion. Figure 5: The attitude response for the position control. The dashed line indicates the reference attitude angles that enable the quadrotor to converge in the desired position, and the solid line is the result of the simulation about attitude responses of a quadrotor: (a) \(\theta\); (b) \(\psi\); (c) \(\psi\). Figures 8 and 9 show that the quadrotor successfully tracking the given trajectory. Figure 10 shows the reference attitude angle for trajectory tracking control. It can be found that the proposed  
Figure 6: The control input for the altitude of a quadrotor. Figure 7: The control inputs for the attitude of a quadrotor: (**a**) \(u_{2}\); (**b**) \(u_{3}\); (**c**) \(u_{4}\). controller maintains good trajectory tracking performance even when the input matrix is changes with time by nonlinear motion such as Figure 10.  
Figure 8: The trajectory tracking control of a quadrotor. The dashed blue line indicates the reference trajectory and the solid red line is the result of the simulation. Figure 9: The trajectory tracking control results for each of the three axes. The dashed line indicates the reference trajectory of each axis, and the solid line is the result of the simulation about each axis: (**a**) \(x\); (**b**) \(y\); (**c**) \(z\).