## 4 Simulations  
In this part, some numerical simulations and comparison are implemented to evaluate the effectiveness of the proposed algorithm by using the MATLAB/Simulink program. The mechanical parameters and hydraulic parameters used for simulating are shown in Tables 3 and 4, respectively. In our simulations, discrete time with the fixed step is \(10^{-4}\) s and the Runge-Kutta method is used for the approximate calculation of some differential problems. We investigate performance of the manipulator in three case studies with the same condition: the initial value of the three joints is \(\left[ \begin{array}{ccc}0.1&0.6&0.8\end{array} \right]^{T}\) (rad), and the initial velocity of the three joints are all zero; the external payload varies in the interval [5,20] kg and is shown in Figure 9; and the time for simulating is 20 s. The comparison between sliding mode control (SMC), fuzzy sliding mode control (FSMC), and adaptive fuzzy sliding mode control (AFSMC) are utilized. The white noise is added to illustrate uncertainty parameters and verify the proposed method. The Controller gains used for simulation are \(k_{qij(i,j=1,2)}=2.1610^{-8}m^{3}/\left(\sqrt{Pa.s.V}\right)\), \(\lambda=\) diag (60, 60, 50), \(k_{qij(i=1,2)}=3  
are described in Figure 10. The role of the proposed controller is validated at the time when the external payload is applied. The error of the AFSMC is the smallest, in which the smallest overshoot is at the time the payload changed, compared with that of the FSMC and SMC. The load torques estimated by using disturbance observer are depicted in Figure 11. And the control input signals which are constrained in the interval [-15, 15] (V) are performed in Figure 12.  
Figure 11: _Cont._  
Figure 10: The tracking performance of the three joints: (**a**) The first joint and (**b**) its error; (**c**) the second joint and (**d**) its error; (**e**) the third joint and (**f**) its error when applying sliding mode control (SMC), fuzzy sliding mode control (FSMC), and adaptive fuzzy sliding mode control (AFSMC). Figure 11: The load torque estimated of the: (**a**) first joint, (**b**) second joint, and (**c**) third joint. Figure 12: The control signal acting on the: (**a**) first joint, (**b**) second joint, and (**c**) third joint. Case study 2: In this case, sine function signals are used as the desired performance. The three joints trajectory are designed as:  
\[\left\{ \begin{array}{ll}q_{1d}=60\sin(\pi t/5-\pi/2)&(\text{degree})\\ q_{2d}=40\sin(\pi t/5-\pi/2)+60&(\text{degree})\\ q_{3d}=60\sin(\pi t/5-\pi/2)+90&(\text{degree})\end{array} \right. \tag{45}\]  
In this case study, the sinusoidal signals are used as a smooth function to investigate the feedback states. The tracking performance of the three joints are described in Figure 13, the load torques estimated are depicted in Figure 14 and the control input signals are depicted in Figure 15. As can be seen in Figure 14, no matter how changed the load torques is, the DO can estimate its change for the compensation. Figure 14: The load torque estimated of the: (**a**) first joint, (**b**) second joint, and (**c**) third joint. Figure 15: _Cont._  
Case study 3: This last simulation is implemented in task space (Cartesian co-ordinate). The end-effector is controlled to track the ellipse curve. The desired trajectory based on the working space of the robot in 3D space is supposed to be:  
\[P=\left( \begin{array}{c}P_{x}\\ P_{y}\\ P_{z}\end{array} \right)=\left( \begin{array}{c}0.3+0.1\sin(\pi t/5)\\ 0.2\sin(\pi t/5)\\ -0.55+0.1\Big{(}P_{x}+P_{y}\Big{)}\end{array} \right)(m) \tag{46}\]  
The Jacobian matrix is expressed in Appendix B. The responses of the end-effector in 3D space and the responses of the three joints are described in Figure 16, the movement of the end-effector in x, y, and z directions are depicted in Figure 17 and the behavior of the end-effector in Cartesian space in shown in Figure 18. The load torques estimated are depicted in Figures 19 and 20 shows the control input signals of the three joints.