# Trajectory Tracking Outer Loop Regressor-Based Adaptive Controller for a Quadrotor
## 4 Experimental Results
### Implemented Controllers
#### 4.2.4 Annc  
This controller was introduced in [25], and it can be considered a PID-type controller with an adaptive neural compensation. The scheme is based on adaptive neural networks using the hyperbolic tangent function as an activation function. Similarly to [24], it computes the control inputs for an inner loop controller to achieve the trajectory tracking task. This scheme will be denoted as ANNC (adaptive neural networks controller) from now on in the document. The ANNC was implemented with the following gains  
\[\begin{array}{rl}K_{p}&=\text{diag}\{0.22\;0.165\;0.3\;0.5\},\\ K_{\xi}&=\text{diag}\{0.0041\;0.0041\;0.1655\;0.1655\},\\ K_{d}&=\text{diag}\{0.165\;0.1375\;0.32\;0.1\},\\ \Delta&=\text{diag}_{\mathbf{4}}\{0.001\},\;\kappa=2.5,\;L_{N}=10,\\ B&=\text{diag}_{10}\{0.189\},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;Figure 4: Paths drawn by the quadrotor obtained with the tested controllers.  
Figure 5: Paths drawn by the quadrotor on the \((x,y)\) plane obtained with the tested controllers.  
The tracking of each of the desired signals is shown in Figure 6, where it seems that the performance of all the tested schemes is similar, except for the PDC in the \(z\) coordinate and the yaw angle \(\psi\).  
In Figure 7, the tracking errors obtained with the tested schemes are presented. It can be observed from it that the GRNN, ANNC, and ARBC have better performances than the PDC and ANNISMC in the \(x\) and \(y\) coordinates.  
Figure 6: Closed -loop system time response. Signals \(x(t)\), \(y(t)\), \(z(t)\), and \(\psi(t)\) obtained with the tested controllers.  
Figure 7: Time response of the tracking error obtained with the tested controllers.  
The obtained velocity errors are depicted in Figure 8. Notice that the closest to zero for the \(x,y\) coordinates and the yaw angle \(\psi\) are the ones obtained with the ARBC.  
The provided control inputs by each controller are shown in Figure 9, these signals are the total thrust \(F(t)\) and the torques \(\mathbf{\tau}(t)=[\tau_{\phi}(t)\ \tau_{\phi}(t)\ \tau_{\phi}(t)]^{\top}\).  
Figure 8: Time response of the velocity error obtained with the tested controllers.  
Figure 9: Control input signals computed by the tested controllers.  
The time evolution of the estimated parameters for the ARBC scheme during the trajectory tracking task are shown in Figure 10.  
The root mean square (RMS) value of \(\boldsymbol{e}(t)\) and \(\dot{\boldsymbol{e}}(t)\) was computed in the time interval \(10\leq t\leq 35\) [s] to quantitatively assess the performance of all the tested controllers. Additionally, for every scheme, the relative percentage of improvement (PI%) was computed considering the performance of the PDC scheme as the comparison base point. Furthermore, the average of the PI% values regarding position and velocity errors was calculated. Table 1 presents the quantitative results, where the highest PI% values are in blue, and the lowest are in red. The values in blue mean that the performance on that specific feature was the best, while those in red mean that the performance was the worst. Although the ARBC scheme provided the best results for most features, it did not provided the best average results for position and velocity errors. It is worth mentioning that the ANNISMC, GRNNC, and ANNC schemes are all neural network-based schemes, while the ARBC is a regressor-based adaptive scheme with a much simpler structure and fewer parameters to be set up. Despite its simplicity, it significantly outperformed the other schemes in some features. These results indicate that with the ARBC scheme, a similar performance compared to other complex schemes can be achieved.  
Figure 10: Parameter estimation over time obtained with the ARBC algorithm.