## 5 Simulations and Discussion
### Discussion  
In simulations, the uncertainties are selected as \(\Delta m_{o^{{}^{\prime}}}=0.8\sin(t)\), \(\Delta F_{ni}=0.8\sin(t)\) and \(\Delta F_{ei}=0.8\cos(2t)\), \(i=1,2,3\). The time histories of the states (i.e., \(\|r(t)\|\) and \(\|\hat{r}(t)\|\)) with the optimal adaptive robust control (under \(\gamma_{\rm opt}=6.78\), \((\beta_{1},\beta_{2},\beta_{3})=(1,1,1)\)) are shown in Figure 3. It can be seen that the trajectory tracking error \(\|r\|\) enters a small zone round 0 after 1s and stays there after, the trajectory tracking error \(\|\hat{r}\|\) increases at first and then approximately close to 0 after 1s, hence, they are ultimately bounded for all \(t>0\) and uniformly ultimate bounded for \(t\leq 1\)s. Figure 4 shows the time history of the adaptive parameter \(\hat{\rho}\) in (24) and (25). It increases quickly from the initial value 0.02 to the maximal value 1.17, and decreases after 4s as the existence of the leakage term. Figure 5 shows the corresponding time histories of the control torques \(\tau_{1}\), \(\tau_{2}\) and \(\tau_{3}\). From Figure 5, it can be concluded that the deterministic and fuzzy performance of Delta robot could be guaranteed by a small control effort. Let \(\Delta_{m}=\max_{t}|\Delta m_{o^{{}^{\prime}}}(t)|\), \(\Delta_{fn}=\max_{t}|\Delta F_{n}(t)|\), \(\Delta_{f_{e}}=\max_{t}|\Delta F_{e}(t)|\) stand for the upper bounds of the uncertain factors \(m_{o^{{}^{\prime}}}\), \(F_{n}\) and \(F_{e}\). The effects of uncertainty bounds on \(\beta_{\max}\) are demonstrated in Figures 6-8. These figures show the phenomenon that \(\hat{\rho}_{\max}\) increase slightly as the uncertainty bounds (\(\Delta_{m}\), \(\Delta_{fn}\) and \(\Delta_{f_{e}}\)) increase form 0 to 0.8. Figure 9 compares the end-effector trajectories under two control strategies: one is \(C_{1}+C_{2}\), the other is \(C_{1}+C_{2}+C_{3}\). The trajectory under \(C_{1}+C_{2}\) control (without the optimal robust control portion) could not follow the desired trajectory and go far away from it. The trajectory under \(C_{1}+C_{2}+C_{3}\) control (including the optimal robust control portion) in (20) can approximately track the desired trajectory, which verifies the effectiveness of the proposed optimal control scheme. The time histories of the trajectory tracking error \(\|r\|\) and \(\|\hat{r}\|\) with different weighting factors combinations \((\beta_{1},\beta_{2},\beta_{3})\) are shown in Figures 10 and 11. It can be seen that the lager \(\gamma_{\rm opt}\) can guaratee a better system performance of Delta robot. The corresponding histories of control torques \(\|\tau\|\) in different simulations are show in Figure 12. Obviously, the lager \(\gamma_{\rm opt}\) requires the higher control cost. Hence, the aforementioned results demonstrate that the proposed control scheme can regulate the Delta robot with a high precision regardless of the uncertainty. The membership functions in this section are chosen as a triangular type to verify the proposed control scheme. In practice, this choice is not obligatory. The designers may select the suitable membership function based on the study of the uncertainty in the system. Furthermore, the different \(\gamma_{\rm opt}\) with different weighting factors could significantly influence the control cost and the system performance (which are considered as a trade-off). The choices of weighting factors depend on the demands of the specific application scenarios. \begin{table}
\begin{tabular}{c c c} \hline \((\beta_{1},\beta_{2},\beta_{3})\) & \(\gamma_{\rm opt}\) & \(J_{\rm min}\) \\ \hline \((1,1,100)\) & 1.87 & 832.17 \\ \((1,1,10)\) & 3.51 & 336.26 \\ \((1,1,1)\) & 6.78 & 157.15 \\ \((1,100,1)\) & 12.54 & 772.78 \\ \((100,1,1)\) & 27.14 & \(5.63\times 10^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Weighting factors/optimal gain/minimum cost. Figure 4: The adaptive parameter \(\beta\). Figure 3: The trajectory tracking errors. Figure 5: The control torques \(\tau\). Figure 8: Maximum of \(\hat{\rho}_{\max}\) with respect to \(\Delta_{m}\) and \(\Delta_{fn}\).