## 4 The Simulation Results
 ^{T}= \begin{bmatrix}-2&2&5\end{bmatrix} ^{T}\) in degrees. The desired trajectories for simulation use the information of helix and parabolic trajectory, as shown in Figures 3 and 4, which is generalized by Table 4. The objective of these cases is to validate the effect of the initial condition position and different types of movements on the controllers' ability. The values in Tables 1 and 2 are used as the nominal parameters of the Delta robot. The PDAFS parameters (42) in detail as (36)-(38) are selected as \(\mathbf{K}_{P}=\text{diag}\{18,18,18\}\) and \(\mathbf{K}_{D}=\text{diag}\{1.8,1.8,18\}\); the constraint in (25) takes the parameters as \(\beta^{-1}=2000\), \(r=5,\;\;p=7\). Otherwise, the manifold in (23) takes \(\mathbf{\Lambda}=\text{diag}\{10,10,10\}\). The RBF network has 7 nodes with the center vectors chosen as \(\mathbf{c}_{q}= \begin{bmatrix}-0.03,\,-0.02,\,-0.01,\,\,0,\,0.01,\,\,0.02,\,\,0.03 \end{bmatrix} ^{\text{T}}\), \(\mathbf{c}_{q}= \begin{bmatrix}-0.3,\,-0.2,\,-0.1,\,\,0,\,0.1,\,\,0.2,\,0.3\end{bmatrix}  ^{\text{T}}\). The initial neural network weights \(\hat{\mathbf{W}}_{i}\) are selected as zeros. The updated weight law is given by (39) and (41) with the adaptation rate \(\gamma=0.0005\). These simulations are implemented with the interval time \(\Delta t_{s}=10^{-4}\) s. The parameters of the proposed controller and the compared controllers are generalized in Table 5.  
For validating the effectiveness of the PDAFS, the tracking errors in the joint space of the five controllers are illustrated and compared in Figure 7a-c for the helix case and Figure 7d-f for the pick and place object assumption. From the tracking error figures, all controllers provided stable trajectory tracking control. Whereas the PD has a large offset error stability, the dynamic model included in PDFF and PDSM provides a minor improvement in the tracking errors. Thanks to the switching function hidden in SM control, the tracking errors of PDSM are better than PDFF. However, PDFS and PDAFS give more \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Helix Trajectory** & **Parabolic Trajectory** \\ \hline Initial point \((x_{0},y_{0},z_{0})\) & (0.05, 0, \(-0.35\)) m & (\(-0.05,\,0.05,\,-0.3\)) m \\ Final point \(\left(x_{f},y_{f},z_{f}\right)\) & (0.05, 0, \(-0.25\)) m & (0.05, \(-0.05,\,-0.3\)) m \\ Number of circles \(k\) & 4 & None \\ Duration \(t_{f}\) & 10 s & 1 s \\ Maximum speed \(v_{max}\) & 0.24 m/s & 0.4 m/s \\ Maximum acceleration \(a_{max}\) & 1.2 m/s\({}^{2}\) & 2.8 m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameters of helix and parabolic trajectory for Delta robot. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **Torque Calculation** & **Parameters** \\ \hline PD & \(\mathbf{\Gamma}_{\text{PD}}=\mathbf{K}_{P}+\mathbf{K}_{D}\mathbf{e}\) & \(\mathbf{K}_{P}=\text{diag}\{18,18,18\}\), \(\mathbf{K}_{D}=\text{diag}\{18,18,18\}\) \\ \hline PDFF & \(\mathbf{\Gamma}_{\text{PDFF}}=\mathbf{\Gamma}_{\text{PD}}+\mathbf{\Gamma}_{\text{A}}\) & \(\mathbf{K}_{P};\mathbf{K}_{D}\) and Table 2 \\ \hline PDSM & \(\mathbf{\Gamma}_{\text{PDSM}}=\mathbf{\Gamma}_{\text{PD}}+\mathbf{\Gamma}_{\text{A}}+\mathbf{ k}\text{sign}(\mathbf{\
abla})\) & \(\mathbf{\Lambda}=\text{diag}\{10,10,10\}\); \(\mathbf{k}=\text{diag}\{1.3,1.3,1.3\}\) \\ \hline PDFS & \(\mathbf{\Gamma}_{\text{PDFS}}=\mathbf{\Gamma}_{\text{PD}}+\mathbf{\Gamma}_{\text{FS}}\) & \(\mathbf{\Lambda}=\text{diag}\{10,10,10\}\); \(\beta^{-1}=2000\), \(r=5,\;p=7\) \\ \hline PDAFS & \(\mathbf{\Gamma}_{\text{PDAFS}}=\mathbf{\Gamma}_{\text{PD}}+\mathbf{\Gamma}_{\text{FS}}+ \mathbf{\Gamma}_{D}\) & \(\mathbf{\epsilon}_{q}=\begin{bmatrix}-0.03,\,-0.02,\,-0.01,\,\,0,\,0.01,\,\,0.02,\,\,0.03\end{bmatrix}^{\text{T}}\), \(\mathbf{c}_{q}=\begin{bmatrix}-0.3,\,-0.2,\,-0.1,\,\,0,\,0.1,\,\,0.2,\,0.3\end{bmatrix} ^{\text{T}}\), \(\mathbf{b}_{i}=0.02\), \(\gamma=0.0005\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The parameters of the proposed controller and compared controllers in simulation.