## 4 Numerical Simulations
### Simulation Results  
In the first simulation case, the results in Figure 2 show the output performances at three joints of the SMC (blue line), the ASMC (black line), and the proposed control (red line), tracked to the reference signals. Figure 3 presents the error responses of the SMC with a blue line, the ASMC with a black line, and the proposed control with a red line. Figure 4 presents the control signal performance of SMC in Figure 4a, ASMC in Figure 4b, and proposed control in Figure 4c. These results show the effectiveness of the adaptive laws for robust gain in chattering suppression in the ASMC and the proposed controller. Figure 2: Output responses of the SMC, ASMC, and proposed control in (**a**) link 1, (**b**) link 2, (**c**) link 3. In the second case, three wrong connection situations are implemented to prove the superiority of the proposed control. These situations are the wrong connection in an actuator, \(\textbf{B}=diag\Big{(}\left[ \begin{array}{cccc}-1&1&1\end{array} \right]^{T} \Big{)}\), two actuators \(\textbf{B}=diag\Big{(}\left[ \begin{array}{cccc}-1&-1&1\end{array} \right]^{T} \Big{)}\), and all three actuators \(\textbf{B}=diag\Big{(}\left[ \begin{array}{cccc}-1&-1&-1\end{array} \right]^{T} \Big{)}\). Figure 5 shows the output responses of the 3-DOF manipulator with the proposed control with the wrong connection in actuator 1. These results show good tracking performances. Figure 6 presents the control signals of the proposed control at each joint of the manipulator. Comparing to Figure 4c, the control signal of joint 1 in Figure 6 has a different direction, which exhibits the wrong connection in actuator 1. Figure 7 plots the Nussbaum function at each joint to help the controlled system overcome the unknown control direction. The response of the Nussbaum function in Figure 7a changed from \(+1\) to \(-1\) to compensate for the connection issue in actuator 1. Figure 3: Error efforts with the SMC, ASMC, and proposed control in (**a**) link 1, (**b**) link 2, (**c**) link 3. Figure 4: Control performance of (**a**) SMC, (**b**) ASMC, and (**c**) proposed controller in the 3-DOF manipulator. Figure 5: The output responses of the proposed control under the presence of the wrong connection in actuator 1, \(\textbf{B}=diag\bigg{(}\left[ \begin{array}{ccc}-1&1&1\end{array} \right]^{T} \bigg{)}\). Next, the advantages of the proposed control are taken into account with the wrong connections in actuators 1 and 2, \(\mathbf{B}=diag\Big{(}\big{[} \begin{array}{rrrr}-1&-1&1\end{array} \big{]}^{T} \Big{)}\). Figure 8 is the control responses of the proposed controller. The results show that the control signals in actuator 1 and 2 are different directions when compared to the control signals in actuator 1 and 2 with the normal case in Figure 4c. The effectiveness is shown in Figure 9, which presents Nussbaum function responses at each joint with the wrong connection in actuator 1 and 2. The results in Figure 9a,b turn from \(+1\) into \(-1\) to overcome the wrong connections. In the final subcase, the wrong connection at all three actuators, \(\mathbf{B}=diag\left(\big{[} \begin{array}{rrrr}-1&-1&-1\end{array} \big{]}^{T} \right)\), is considered. Figure 10 presents the control signals of the proposed controllers. The sign of the control responses of the proposed control in three actuators is different compared to the control responses in the normal case, as shown in Figure 4c. Additionally, Figure 11 shows the Nussbaum function responses which turn into \(-1\) to help the proposed control to deal with the unknown control direction. Figure 6: Control responses of the proposed controller with the wrong connection in actuator 1, \(\mathbf{B}=diag\Big{(}\big{[} \begin{array}{rrrr}-1&1&1\end{array} \big{]}^{T} \Big{)}\). Figure 7: Nussbaum function performance of the proposed control in (**a**) joint 1, (**b**) joint 2, and (**c**) joint 3. Figure 8: Control responses of the proposed controller with the wrong connection in actuator 1 and actuator 2, \(\textbf{B}=diag\Big{(}\Big{[}