## 4 Numerical Simulation and Experimental Studies
### Numerical Simulations
\begin{table}
\begin{tabular}{c c} \hline \hline
**Controller** & **Parameters** \\ \hline \multirow{2}{*}{PID} & \(K_{P}=[60,24,15]^{T},K_{l}=[32,10,8]^{T}\), \\  & \(K_{D}=[0.4,0.2,0.1]^{T}\) \\ \hline \multirow{4}{*}{BSMC} & \(c_{1}=diag([60,60,40]),\)\(c_{2}=diag([250,1000,50])\), \\  & \(\eta_{1}=50 eye(3),\)\(\eta_{2}=10^{2}eye(3)\), \\  & \(K_{1}=diag([30,25,30]),\)\(K_{2}=diag([10^{3},10^{3},5.10^{4}])\); \\ \hline \multirow{4}{*}{ABSMC} & \(\Gamma_{11}=2 eye(30),\)\(\Lambda_{11}=2,210^{-3}eye(30),\) \\  & \(\Gamma_{21}=10.eye(90),\)\(\Lambda_{21}=10^{-6}eye(90),\) \\  & \(\Gamma_{31}=10^{-6}eye(90),\)\(\Lambda_{31}=10^{-5}eye(90),\)\(\epsilon_{1}=10^{-4}[200,1,1]^{T},\) \\  & \(\epsilon_{10}=10^{-2}\epsilon_{1}\)\(\Gamma_{41}=6.10^{3}eye(3),\) \\ \cline{1-1}  & \(\kappa_{1}=diag([40,40,28]),\) \\ \cline{1-1}  & \(\Gamma_{12}=20e^{-4}eye(30),\)\(\Lambda_{12}=4.4e^{-3}eye(30),\) \\ \cline{1-1}  & \(\Gamma_{22}=10 oye(60),\)\(\Lambda_{22}=10^{-6}eye(60),\) \\ \cline{1-1}  & \(\Gamma_{32}=10^{-6}eye(60),\)\(\epsilon_{2}=[1,1,1]^{T},\epsilon_{20}=10^{-2}\epsilon_{2},\) \\ \cline{1-1}  & \(\Gamma_{42}=6.10^{5}eye(3),\)\(\kappa_{2}=diag([0.2,0.2,0.14]).\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Control Parameters. Figure 3: Payload performance. **Remark 3**. : _The proposed control is developed from the BSMC, so some parameters of SMCs in the proposed control are inherited from the BSMC._  
Figures 4 and 5 plots the position responses and the position errors of the joints with three controllers, PID, BSMC, and ABSMC. The results showed that the nonlinearities, uncertainties and variant payloads impacted the accuracy of the controlled system with PID control. The nonlinearities which are caused by variant payloads are dealt by the BSMC. However, the remained errors are still significant. The proposed control with adaptive laws compensated the uncertainties and improved the accuracy of the controlled system, essentially. Figures 6 and 7 respectively show the torque performances and torque errors at each joint of the two controllers which are the BSMC and the proposed control. The virtual torques were calculated by (14) and (33) with the BSMC and the proposed control, respectively. The estimated torques were computed based on pressures from two chambers. The results proved that the proposed control with the adaptive mechanisms regulated the torque responses better than the BSMC. Figure 8 shows the control signals of the BSMC and the proposed controller. Figure 5: Joint errors of PID, BSMC, and ABSMC in (**a**) joint 1, (**b**) joint 2, (**c**) joint 3. Figure 6: _Cont._  
Figure 8: _Cont_. Figure 6: Torque responses with **(a)** BSMC, and **(b)** ABSMC. Figure 7: Torque errors with **(a)** BSMC, and **(b)** ABSMC. **Remark 4**. : _The simulation results proved that the proposed control compensated all uncertainties more effectiveness than the PID control and BSMC. The RBFNNs exhibited the approximately ability with the smooth uncertainties and the adaptive switching gains also demonstrate their ability for compensating the unsmooth uncertainties. However, the learning rates of adaptive laws have not mentioned in this paper. They will be intensively study in future work._