# RBF Neural Network Based Backstepping Control for an Electrohydraulic Elastic Manipulator
## 4 Experimental Results
### The Experimental Procedures  
The equilibrium position of the EEM is considered. Two experiments are carried out to verify the effectiveness of the proposed control, the adaptive backstepping sliding mode control (ABSMC). The system carried a low load of 50N, and the stiffness was changed from low stiffness to high stiffness at the 10 th second in each experiment. Firstly, the desired equilibrium position is defined as \(x_{1d}=12+10\cos(\pi t)\). The initial state of the electrohydraulic elastic manipulator is \(x(0)=\left[ \begin{array}{cc}0&0\end{array} \right]^{T}\). To demonstrate the effectiveness of the proposed controller, it is compared to the PID control and the CBSMC. The parameters of controllers are selected as the PID control: \(K_{p}=10\), \(K_{i}=5\), \(K_{d}=0.1\); CBSMC: \(k_{1}=300\), \(k_{2}=7.2\times 10^{10}\), \(\lambda_{1}=40\), \(\lambda_{2}=4.4\); RBFNN approximations: \(E=\left[e_{1},\frac{e_{2}}{10},\frac{e_{3}}{2}\right]^{T}\)\(v_{11}=\left[1,0,0\right]^{T}\), \(v_{12}=\left[0,1,0\right]^{T}\), \(v_{21}=\left[1,0,0\right]^{T}\),\(v_{22}=\left[0,0,1\right]^{T}\), \(m_{ei}=\left[ \begin{array}{cc}0.1(j-11)&0.1(j-11)\end{array} \right]^{T}\)\((i=1,2;j=1,\ldots,20)\), \(\sigma_{ij}=0.3162\)\((i=1,2;j=1,\ldots,20)\), \(W_{i}=\left[0,\ldots,0\right]^{T}\in R^{20},i=1,2\). **Remark 2**. : _The control parameters are chosen based on a trial and error method. Because the ABSMC were developed based on the CBSMC, so all control gains of the CBSMC are kept in the ABSMC._  
In this research, the experimental results, including the position, position error, torque, and the control signal, are analyzed. They are presented in Figures 7-10, respectively. In each figure, the responses of the PID control, the CBSMC, and the proposed ABSMC with the approximations in mechanical dynamics and hydraulic dynamics are shown. Figure 6: Structure of the EEM test bench. Figure 8: The angle error responses of the PID control, the CBSMC, and the proposed ABSMC. Figure 7: The angular displacement responses of the manipulator with the PID control, the CBSMC, and the proposed ABSMC. Figure 9: The torque responses of **(a)** CBSMC; **(b)** the proposed ABSMC. The error responses of PID, CBSMC, and ABSMC are plotted in Figure 8 with the blue dash line, the black dashed dot line, and the red line, respectively. The results exhibited that the ABSMC with approximators improved the precision of the system more effectively than the PID and CBSMC. Figure 9 shows the torque responses of the CBSMC and the ABSMC, with a virtual torque shown as the black line and estimated torque shown as the dashed red line. The results also show that the proposed control improved the tracking torque error by using the approximation to compensate for the uncertainties in hydraulic dynamics. Figure 10 plots the control signals of the PID control, BSMC, and ABSMC. Furthermore, the responses of weights in the approximators are presented in Figure 11. The results verified the convergence of weights of the approximators. **Remark 3. **_Because the friction was not considered in the CBSMC, the tracking error of the CBSMC did not converge to zero as shown in Figure 8. It was stable at a nonzero value. The proposed control with the approximation in mechanical dynamics compensated for the friction to improve the accuracy of the controlled system._  
Another experiment is executed to discuss the multi-step tracking ability of the proposed controller. The three controllers are then run for the different positions shown in Figure 12, which has a large distance and a maximum angular displacement; both are near their physical limits. The tracking errors are shown in Figure 13. The results show that the PID control cannot handle such a large movement  
Figure 11: The convergence of weights of the approximations (**a**) in mechanical dynamics, (**b**) in hydraulic dynamics. Figure 10: The control signal of (**a**) the PID control, (**b**) CBSMC, and (**c**) the proposed ABSMC. well, with the tracking error around 10.