## 4 Experimental Investigation
The ARC controller, with limited ability to compensate the nonlinearities, showed a better performance than the DRC both in transient response and steady-state. The steady-state error of the ARC was about \(0.8\) mm at the set point \(90\) mm and \(0.25\) mm at the point \(-90\) mm. It is obvious the ARNNC controller performed the best of three controllers. The transient behavior was improved significantly, and the state-state error was extremely small which was within \(0.1\) mm at the two different positions due to the further compensation for unmodeled dynamics, random disturbances, and residual errors of the adaptive part. It is noted that the tracking performance of ARNNC was improved gradually as the estimated weights of NN compensator converge to the optimal values. Case study \(\Pi\)--Sinusoidal trajectory (free load): In this case, comparative experiments were carried out for tracking sinusoidal trajectory with three different frequency \(0.125\) Hz, \(0.25\) Hz, \(0.5\) Hz and amplitude \(90\) mm. The tracking results of three different trajectories are depicted in Figures 5-7, respectively. As shown in the pictures, the DRC controller performed the worst of three in all three experiments and tracking performance of the ARC controller was improved but not enough. The tracking error of the ARNNC controller was slightly less than ARC at the beginning of experiments and reduced as it trained on-line. After a few seconds of the training process, the tracking error of the ARNNC was much smaller than the DRC and ARC controller. It is noted that the tracking performances of DRC and ARC degraded significantly as the reference trajectory speeded up. The \(s_{1r}\) of the DRC was about \(0.609\%\) at \(0.125\) Hz, \(0.949\%\) at \(0.25\) Hz and increased rapidly to \(2.931\%\) at \(0.5\) Hz. The ARC controller experienced a similar situation. Inaccurate estimation of the velocity by the nonlinear differentiator, which led to inaccurate model compensation was the main factor for this degradation. The ARNNC controller with better compensation for unmodeled dynamics, random disturbances, and estimated model errors performed much better than the DRC and ARC controllers, especially in the case with a frequency of \(0.5\) Hz that \(s_{1 rms}\) of the ARNNC was about \(22.5\%\) of the ARC and \(19.6\%\) of the DRC. Figure 4: Tracking performances of smoothed-step (**a**) Smoothed-step tracking responses of different controllers. (**b**) Tracking errors of different controllers. (**c**) Zoomed responses from \(43.5\) to \(46\) s.  
Case study III--Sinusoidal trajectory (load): To further test the performances of the controllers, another series of trajectory tracking experiments of the sinusoidal reference signal with frequency 0.5  Hz and amplitude 90 mm were performed. As shown in Figures 6, 8 and 9, free load, 5 kg load, and 8 kg load conditions were carried out. The \(s_{1 rms}\) of the DRC controller with free load, 5 kg load, and 8  kg load, was 0.8539 mm, 1.2565 mm, and 1.5386 mm, respectively. Compared to the DRC, both the ARC and ARNNC performed much better, and the performance of ARC and ARNNC changed little when the load increased. The \(s_{1 rms}\) of ARC in three different conditions was 0.6291 mm, 0.6429  mm and 0.7242 mm, which were much smaller than the DRC. Nevertheless, the ARNNC achieved the best performance due to its ability to compensate for unmodeled dynamics, the unknown disturbances, and residual errors. \(s_{1 rms}\) of the ARNNC in the three conditions was 0.1901 mm, 0.2325  mm, and 0.2345 mm, which were about 30% of the ARC. And \(s_{1 rms}\) of the ARNNC at 8 kg load condition was only about 15.2% of the DRC. Figure 5: Tracking performances of 90 Sin(0.25\(\pi\)t - 0.5\(\pi\)) (**a**) Tracking errors of different controllers. (**b**) Zoomed tracking errors from 0 to 10 s. (**c**) Zoomed tracking errors from 30 to 40 s.  
Figure 6: Tracking performances of 90 Sin(0.5\(\pi\)t - 0.5\(\pi\)) (**a**) Tracking errors of different controllers.