## 4 Numerical Simulation and Experimental Studies
### Experimental Results  
Furthermore, the proposed controller and backstepping controller are practically conducted on the hydraulic manipulator with load and without load of 20 N. The test bench includes a hydraulic power, a computer and a 3-DOF manipulator as shown in Figure 9. The hydraulic flow rates which are supplied to the actuator from the hydraulic power unit are driven by the servo valves. The computer is equipped PCI cards such as PCIE 6363, and Quad04 to provide the control signal to the servo valves and read the pressure sensors and encoder sensors at each joint. The control algorithms are practically carried out in MATLAB Simulink with the Real-time Windows target toolbox at sampling time of 10 ms.  
Figure 8: Control signals of (**a**) BSMC, and (**b**) ABSMC. Figure 9: Structure of the test bench. The control parameters are set to be backstepping sliding mode control: \(c_{1}=diag([60,70,35])\), \(c_{2}=diag([250,850,45])\), \(\eta_{1}=5 eye(3)\), \(\eta_{2}=10^{2}eye(3)\), \(K_{1}=diag([30,25,30])\), \(K_{2}=diag(\left([10^{3},10^{3},5.10^{4}]\right))\); proposed control: \(\Gamma_{11}=2 eye(30)\), \(\Lambda_{11}=2,2.10^{-3}eye(30)\), \(\Gamma_{21}=10.eye(90)\), \(\Lambda_{21}=10^{-6}eye(90)\), \(\Lambda_{31}=10^{-5}eye(90)\), \(\varepsilon_{1}=10^{-4}[200,1,1]^{T}\), \(\varepsilon_{10}=10^{-2}\varepsilon_{1}\)\(\Gamma_{41}=5.10^{3}eye(3)\), \(\kappa_{1}=diag([40,40,28])\), \(\Gamma_{12}=20e^{-4}eye(30)\), \(\Lambda_{12}=4.4e^{-3}eye(30)\), \(\Gamma_{22}=10 eye(60)\), \(\Lambda_{22}=10^{-6}eye(60)\), \(\Gamma_{32}=10^{-6}eye(60)\), \(\Lambda_{32}=10^{-5}eye(60)\), \(\varepsilon_{2}=[1,1,1]^{T}\), \(\varepsilon_{20}=10^{-2}\varepsilon_{2}\), \(\Gamma_{42}=5.10^{3}eye(3)\), \(\kappa_{2}=diag([0.2,0.2,0.14])\). **Remark 5**. : _The initial weighting vectors of the RBFNNs are selected to be zero. Additionally, because the proposed control is developed based on the backstepping sliding mode control, some parameters of the proposed control are inherited from the backstepping sliding mode control._  
**Remark 6**. : _The parameters of the controllers are adjusted when the manipulator operates without load. The parameters are kept in other case._  
Figures 10-12 shows the joint responses, errors responses and control signals of the BSMC and proposed control when the hydraulic manipulator works without payload. In Figure 10, sinusoidal signals, \(x_{1d}=[30\sin[0.628t],20[1+\cos[0.628t]],20[2-\sin[0.628t]]]\)(Deg. ), are set up as trajectories in the joints of the hydraulic manipulator and each subfigure presents for each joint response in the manipulator. The results show that the output responses of the BSMC and the proposed control track the reference signals. In Figure 11, the error performances of joints are provided in subfigures. The results demonstrate that the proposed control approximate the uncertainties to enhance accuracy of the control performance. The Figure 12 presents the control signals of the controllers. Next experiments, the controllers are conducted on the hydraulic manipulator when the manipulator carry a payload of 20 N. The references are still sinusoidal signals as mentioned in previous case. Figures 13 and 14 present the error performances and control signals of the BSMC and the proposed control. The results in Figure 13 again prove that the adaptive approximators in the proposed control compensate not only the uncertainties such as unknown friction, modeling error and leakage but also the variant payload. Figure 11: _Cont._  
Figure 10: Joint responses of BSMC and proposed control in (**a**) joint 1, (**b**) joint 2, (**c**) joint 3 without load. Figure 11: Error performance of BSMC and proposed control in (**a**) joint 1, (**b**) joint2, and (**c**) joint 3 without load. Figure 12: Cont. Figure 12: Control efforts of (**a**) BSMC and (**b**) Proposed control without load. Figure 13: Cont. Figure 14: Control efforts of (**a**) BSMC and (**b**) proposed control with load.