# Robust Backstepping Control Applied to UAVs for Pest Recognition in Maize Crops
## 5 Experimental Results
### Experimental Results Applying Synthesized Controllers  
In this subsection, for comparison purposes, experimental tests in outdoor environment of UAV are presented for the proposed backstepping controller and a PD controller. For the trajectory tracking task on the spaces \((x;y;z)\) and \((\phi,\theta,\psi)\), the UAV follows three paths: 1. The manual takeoff of the UAV, until the desired altitude is reached, and this altitude is defined by the pilot. 2. The quadrotor tracks parametric circle equations given by  
\[x_{ref} = \left(-5\cos\Bigl{(}\frac{\pi}{180}t\Bigr{)}+5\right)m,\] \[y_{ref} = -5\sin\Bigl{(}\frac{\pi}{180}t\Bigr{)}m,\] \[z_{ref} = 2.3m.\]  
3. At the end, when the trajectory is finished, the pilot regains control of the quadrotor and lands it. The programmed sampling time (\(T\)) is \(T=0.01\) s, it was used in all experimental tests. The quadrotor parameters are as follows: mass \(m=1.3\) kg, distance from the motors to the centre of gravity \(l=0.3\) m, constant of gravity \(g=9.81\) m/s\({}^{2}\) and inertia moments \(I_{x}=I_{y}=0.01567\) kgm\({}^{2}\), \(I_{z}=0.028346\) kgm\({}^{2}\). The gains of controllers (13), (16), (26) and (34) are heuristically proposed in Table 1. The delta values (\(\Delta_{i},i=1,2,3,4\)) are the maximum values for the matched disturbances that could be adjusted heuristically such that the closed-loop performance is satisfactory. To compare the behavior of the proposed controller, a PD controller, which is tuned as is proposed in [35], is employed, obtaining the gains by proposing the following temporal parameters: the maximum overshoot and the settling time. Translational and attitude initial values of the drone are near to the origin. Figure 3 displays the translational positions recorded during the experiment following the desired trajectory. The figure is divided in three figures: 1. The first shows the trajectory of the vehicle on the X axis, 2. The second displays the trajectory along the Y axis, 3. The last one illustrates the trajectory tracking task on the Z axis. As shown in Figure 3, the trajectory tracking task in \((X,Y,Z)\) space, in the blue line, displays a reliable performance when it is compared with the reference values in the red line. The position performance of the drone in the \((X,Y)\) space is displayed in Figures 4 and 5. It is noteworthy that these experimental results were obtained in an outdoor environment. Figure 6 shows the relative small errors during the trajectory tracking. During the trajectory tracking task, the references are subject to changes in the \(X\), \(Y\) axes, then the errors may increase or decrease. Nevertheless, for the altitude, the Set Point is constant, then the error is relatively small. Furthermore, the measurements of the position, provided by the GPS device, present a relatively larger deviations (around 1.4 m to 1.6 m) than other devices employed for indoor environment. This explains the errors in the trajectory-tracking displacement task. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Subsystems**} \\ \hline \(\psi\) & \(z\) & \(x-\theta\) & \(y-\phi\) \\ \hline \(k_{\psi}=24.2\) & \(k_{z}=7.39\) & \(k_{5}=0.1\) & \(k_{1}=0.4\) \\ \(\alpha_{7}=5.7\) & \(\alpha_{6}=12.3\) & \(k_{6}=0.13\) & \(k_{2}=.27\) \\ \(\Delta_{1}=3.2\) & \(\Delta_{2}=2.5\) & \(k_{7}=7.3\) & \(k_{3}=5.3\) \\  & & \(\Delta_{3}=5\) & \(\Delta_{4}=2\) \\  & & \(\alpha_{5}=12\) & \(\alpha_{1}=0.009\) \\  & & \(\beta_{1}=0.1\) & \(\beta_{3}=5.2\) \\  & & \(\beta_{2}=0.25\) & \(\beta_{4}=3.64\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Controller gains. Figure 8: Translational positions in the \((X,Y,Z)\) space (PD controllers). Figure 6: Error signals; (Backstepping controllers). Figure 7: Control signals; (Backstepping controllers). Figure 9: 3D view trajectory tracking (PD controllers).