## 3 Proposed Control Design
### Simulation Results  
In order to show the superiorities of the proposed control, a synchronous sliding mode control (SSMC) with the Nussbaum function (27) was conducted on the dual robotic arm system on MATLAB Simulink with the same working conditions. The controlled system in MATLAB Simulink is presented in Figure 3. The input torques of two robots are limited by a saturation block from \(-100\) to \(+100\) (Nm). In advance, the dynamics of each robot are  
Figure 3: Structure of the controlled system in MATLAB Simulink. separated into different MATLAB function blocks, and two-step blocks set up the control directions in each robot. \[\tau_{1}=\mathbf{M}(\mathbf{x})\big{(}\ddot{\mathbf{x}}_{d}-\lambda\dot{\mathbf{ e}}\big{)}+\mathbf{C}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}\dot{\mathbf{x}}+ \mathbf{G}(\mathbf{x})-\mathbf{K}\mathbf{s}-\eta sign(\mathbf{s}) \tag{27}\]  
**Remark 2:** In order to guarantee equality in the comparisons, the control parameters of the proposed control are inherited from the SSMC. Because adjusting the parameters of the controllers is a complex task, the trial-error approach is used to select the appropriate parameters of the synchronous sliding mode control. The parameters of these controllers are presented in Table 1.  
Figures 4 and 5 present the output response in the joint space of two robotic arms. The black, blue, and red lines correspondingly describe the references and output responses of the SSMC and the proposed control. In the first period, although the two robots had different modeling errors, the SSMC and proposed control are suitable against these issues. In the second period, an external force was added at the end-effector of the second robot. The proposed control with ESO coped well with this challenge. However, the SSMC could not handle it; the output responses in three joints were degraded significantly. Because the SSMC was developed based on the cross-coupling error between two robots, the output responses of the first robot were also degraded. At the 20 th and 30 th seconds, the wrong control direction arises in the two robots. Although the Nussbaum functions were used to deal with these problems, they cannot manage instantly. As a result, the impulses arose in these instances. Consequently, the results show that these responses track the predefined references under the presence of friction, external force, and wrong input connections in two robotic arms. In order to evaluate the efficiency of the proposed control clearly, the tracking errors in the joint space of two robotic arms are presented in Figures 6 and 7 with the black lines and red lines of the SSMC and the proposed control, respectively. In the first scenario, when the main challenge is the unknown frictions, the tracking errors in both robotic arms of the proposed control are better than the SSMC. In the second scenario, an external force is applied at the end-effector of the second robotic arm along the \(y\)-axis. Consequently, the tracking errors in the joints of the second robot increased significantly with the SSMC. Because the SSMC used the cross-coupling error to form the relationship between two respective joints in the dual arm, the influences on the second robot also affect the first robot. As a result, the tracking errors in the first robot are also degraded, similar to the second one with the SSMC. In the proposed control, because the ESO perfectly estimated the lumped uncertainties caused by the unknown friction and external force, the tracking accuracy in both robots is maintained like the first scenario. In the third and fourth scenarios, the wrong control direction issue arises in each robotic arm's first actuator. The stability in two robotic arms with two controllers is guaranteed because the Nussbaum function was used for designing these controllers. The Nussbaum function calculates the sign of the control input at each robotic arm joint. It is one with the right connection and minus one with the