## 5 Experimental Results
### Case Study Results  
From the system description above, it can be seen that the delay time on the computer is much more than the interval time in the simulation. The long time delay combined with the influence of the low-resolution encoders discretizes the control signal values. Thanks to the novel combination in PDAFS, the control signals are generated effectively for improving the Delta robot control performance. In this section, the two case experiments are conducted based on the desired trajectories in (62)-(64). These conditions can reduce unwanted vibration for the Delta robot in the practical environment. In the first case without external load, the performance of the proposed controller is validated based on the features of a robust adaptive controller to an unknown disturbance in the real robot. In the remaining case, an external load is attached to the traveling plate. The purpose of the experiment is to validate the adaptability of the PDAFS with an external disturbance. In our experiments, the PDAFS and the compared controller parameters are turned and presented in Table 8.  
Case 1: Helix and parabolic trajectory without external load  
The experiment results of the position and errors in the helix and parabolic path are illustrated in Figures 12 and 13. Figure 14 shows tracking performances in the workspace. The angular errors of PD are stable in the boundedness but non-zero convergent. While the PDFF errors are found with a minor improvement of values, the PDSM errors have some fluctuations in Figures12b,e and 13, but the SM is more stable than PDFF in RMSE comparison. Significantly, the tracking errors of the proposed controllers are under two ticks of the encoder resolution (0.367 degrees/tick) of the RMSE in the helix path and about 1.2 degrees in the second path, the smallest one among those of the other approaches. Moreover, the finite-time convergence property of both synergetic-based controllers is clearly present at the initial times in Figures  
It is noteworthy that the behavior of the proposed controller in Figure 15 is very similar to the control torque signals produced by the remaining schemes. The transitions of velocity efforts to the tracking performance and control signals thus consolidating the analyses in the simulation section. Although the PDSM can guarantee stability, some mechanical stresses caused by the on-off signals are witnessed at nearly 1.5 degrees and over 5 degrees, significantly higher than the values of the PDAFS in Figures 12b and 13, respectively. The vibrations are obtained clearly at the SM control signals in Figure 15. Meanwhile, the smoothing of control output from the synergetic algorithm demonstrates the advantages of chattering noise reduction in maintaining the attractiveness of the finite-time manifold constraint. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Joint 1 (Degrees)** & **Joint 2 (Degrees)** & **Joint 3 (Degrees)** \\ \hline PD & 3.7855 & 3.5320 & 3.8794 \\ PDFF & 2.8982 & 3.2263 & 2.8451 \\ PDSM & 2.2124 & 2.0874 & 2.7929 \\ PDFS & 2.5561 & 2.4066 & 2.6639 \\ PDAFS & 1.1978 & 1.1713 & 1.2120 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: RMSE Comparison among the controllers and PDAFS in Parabolic Case 1. Figure 12: Helix trajectory performance in joint space: (**a**) joint 1; (**b**) joint 2; (**c**) joint 3 and tracking error of joints: (**d**) joint 1; (**e**) joint 2; (**f**) joint 3 in experiment case 1. Figure 13: Parabolic trajectory performance in joint space: (**a**) joint 1; (**b**) joint 2; (**c**) joint 3 and tracking error: (**d**) joint 1; (**e**) joint 2; (**f**) joint 3 in the experiment case 1. Although the controller-based dynamic model can provide performance improvement in tracking control, PDFF improves the least values because of the lack of convergent functions in contrast to the robust PDSM or PDFS. The adaptive unknown torque approximation in the proposed approach reduces the effects of the disturbance, resulting in better dynamic performances than the other four methods.