## 5 Experiments
### Experiment Results  
In the experiments, we consider three case studies. The parameters of the system are same as defined in Tables 3 and 4. The initial of the three joints is set when the actuators reaches the limited position (maximum extend for cylinder and maximum angular position for rotary actuators). The parameter of the sliding surface mentioned in Equation (17) is \(\lambda=\text{diag}\) (35, 30, 25), and the parameters of the robust terms in Equation (21) are \(K_{1}=diag(40,40,40)\) and \(\eta=\text{diag}\) (1.8, 1.8, 1). The values of \(K_{1}\) and \(\eta\) are different from the simulation because the real system has many uncertainties because of its structure and the different drivers used to run the manipulator, the gain is scheduled such that the system can operate smoothly. All the controller gains are kept same in the three case studies.  
Case study 1: Sinusoidal signal and no load verification.  
The desired trajectory is set as:  
\[\left\{\begin{array}{ll}q_{1d}=30\sin(\,2\pi t/5\,)&(\text{degree})\\ q_{2d}=30+20\sin(\,2\pi t/5\,+\,\pi/2\,)&(\text{degree})\\ q_{3d}=40+20\sin(\,2\pi t/5\,)&(\text{degree})\end{array}\right. \tag{47}\]  
In this case study, no load is attached into the end-effector. Therefore, the comparison is implemented between the conventional back-stepping slide ding mode control (BSMC) and fuzzy back-stepping sliding mode control (FBSMC). The response of the three joints is described in Figure 22.  
Case study 2: The desired trajectory is set as same sinusoidal signal as the first case study, but 5 kg external payload is attached. The response of the three joints is depicted in Figure 23.  
Figure 22: Cont.  
Figure 23: The tracking performance of the three joints in case of sinusoidal signal with load: (**a**) The first joint and (**b**) its error; (**c**) the second joint and (**d**) its error; (**e**) the third joint and (**f**) its error in comparison between BSMC and FBSMC and AFBSMC.  
Figure 22: The tracking performance of the three joints in case of sinusoidal signal without load: (**a**) The first joint and (**b**) its error; (**c**) the second joint and (**d**) its error; (**e**) the third joint and (**f**) its error in comparison between SMC and FSMC.  
Case study 3: Step signal is applied instead of using smooth signal like the two above case studies or tooth saw profile like in the simulation. The controller parameters are still kept same as the original value setup above. The response of the system is shown in Figure 24.  
As can be seen in the three case studies, depicted from Figure 22 to Figure 24, especially in the second and third case studies (Figures 23 and 24), the response of the three joints is quite similar with the simulations, in which the errors when employing adaptive fuzzy back-stepping sliding mode control (AFBSMC) is the smallest compared with that of the fuzzy back-stepping sliding mode control (FBSMC) and back-stepping sliding mode control (BSMC). The error of the third joint in case of load and without load is quite similar. This can be explained as the short length of the third joint gets less affected from the load. But because of the length in the second link and dynamics coupling torque between the third and the second link, the error of the second joint is a little greater than that of when no load affected. When utilizing AFBSMC, the error decreased significantly. Same profile in the first joint in comparison between with and without load.  
Figure 24: The tracking performance of the three joints in case of pulse signal with load: **(a)** The first joint and **(b)** its error; **(c)** the second joint and **(d)** its error; **(e)** the third joint and **(f)** its error in comparison between BSMC and FBSMC and AFBSMC.