## 4 The Simulation Results  
In this section, helix and pick-place characteristics of the nonlinear dynamics of the Delta robot are calculated as the desired trajectory to validate the PDAFS control performance. The planning trajectories followed by a quintic function of time as [6] can \begin{table}
\begin{tabular}{c c c c} \hline
**PDAFS** & **Torque Contributions** & **Features** & **Advantages** \\ \hline PD & DC Motors & Offset-error convergence & Guarantee the stability \\ \hline FS & Delta manipulator & Finite-time and asymptotic convergence & Manifold surface Chattering noise elimination \\ \hline RBF networks & Generalized disturbance & Zero-error convergence & Fast function regression Online estimation \\ \hline \end{tabular}
\end{table} 
Table 3: The contribution and advantages of object-oriented model control in PDAFS. generate smoothing velocities and accelerations, which are applied to reduce vibration in the system. The fifth polynomial function describes the quintic function, as below. \[\left\{ \begin{array}{l}x_{H}(u)=R\cos(k2\pi u)\\ y_{H}(u)=R\sin(k2\pi u)\\ z_{H}(u)=hz_{0}u\end{array} \right. \tag{62}\]  
\[\left\{ \begin{array}{l}x_{P}(u)=x_{0}+(x_{F}-x_{0})u\\ y_{P}(u)=y_{0}+(y_{F}-x_{0})u\\ z_{H}(u)=z_{0}-4 hn^{2}+4 hu\end{array} \right. \tag{63}\]  
\[u(t)=10\frac{t^{3}}{t^{5}_{f}}-15\frac{t^{4}}{t^{4}_{f}}+6\frac{t^{5}}{t^{5}_{ f}}\,\left(0\leq t\leq t_{f}\right) \tag{64}\]  
where \(x_{H},y_{H},\)\(z_{H}\) and \(x_{P},y_{P},z_{P}\) are the desired positions in the Cartesian coordinate system of the helix and parabolic curves at the determined time. The coordinates \(x_{0},y_{0},z_{0}\) and \(x_{F},y_{F},z_{F}\) are initial and final positions. \(R\) is the radius, \(k\) is the number circle of the helix, whereas \(h\) is the height of parabolic on the z-axis, \(u(t)\) is the quintic function of time \(t\), and \(t_{f}\) is the duration of the movement. The characteristic curves of the nominal robot model in Figures 3 and 4 provide the positions in the Cartesian and angular coordinate systems, velocities, accelerations, and desired joint torques for the Delta manipulator. The parameters are calculated via inverse kinematics [2,3] and Equations (12)-(15), respectively. For simulating the dynamic Delta robot model, we use the robot model in Figure 5, which includes the three dynamic DC motors (2) and the Delta manipulator (15). We denote \(\tau_{Li}\) with \(i=1,\ 2,\ 3\) as the load torque applied to the motor in (2). The proposed control scheme of the Delta robot is shown in Figure 6. For demonstrating the effectiveness and advantages of the proposed PDAFS scheme, the simulation results are instructive to compare with that of proportional derivative (PD), PD feedforward (PDFF) [11], PD sliding mode (PDSM) [13], and PDFS regulators. In particular, the performance quantity of the robot trajectory calculated by the \(i\)-th joint root mean square error (RMSE) formula, is expressed as:  
\[\text{RMSE}_{i}=\sqrt{\frac{1}{n}\sum_{k=1}^{n}\left(q_{di_{k}}-q_{i_{k}} \right)^{2}} \tag{65}\]  
Figure 3: Helix trajectory characteristics: position transitions in workspace (**a**), versus time (**b**); in joint space (**c**), angular velocities (**d**); angular accelerations (**e**); torques of delta manipulator (**f**). Figure 4: Parabolic characteristics: position transition in workspace (**a**), versus time (**b**); in joint space (**c**), angular velocities (**d**); angular accelerations (**e**); torques of delta manipulator (**f**). Figure 5: The Delta robot model in this paper. Figure 6: The proposed control scheme. The controller parameters are chosen constantly in the two cases. Moreover, the adaptability of RBF neural networks is evaluated by the assumption of external disturbance as (66) in the Delta robot model. \[ \begin{cases}\tau_{D1}=0.4\sin(\dot{q}_{1})\\ \tau_{D2}=0.8\sin(\dot{q}_{2})\\ \tau_{D3}=1.1\sin(\dot{q}_{3})\end{cases}  \tag{66}\]  
With the initial angular error vector such that \(\mathbf{e}(0)= \begin{bmatrix}\mathbf{e}_{1}(0)&\mathbf{e}_{2}(0)&\mathbf{e}_{3}(0)\end{bmatrix}