## 5 Conclusions  
This paper proposed an adaptive synchronous sliding mode control for a dual-arm robot under the presence of unknown control direction, parametric uncertainties, and external disturbance. The proposed control was constructed by using cross-coupling error, sliding mode control, extended state observer, and Nussbaum function. By defining a cross-coupling mistake to integrate both tracking and synchronous objectives, when the proposed control drives the cross-coupling error forward to zero, both tracking and synchronous errors converge to zero simultaneously. The Nussbaum function managed the unknown control direction. The external state observer was also used to estimate and compensate for the lumped uncertainties, including parametric uncertainties and external disturbance. The stability and robustness of the controlled system were analyzed by the Lyapunov approach in theory. Finally, some simulations on MATLAB Simulink were conducted with a dual 3-DOF manipulator under the above challenges. The results demonstrated that the tracking  
Figure 12: The responses of the lumped uncertainties and the estimated results in (a) joint 1, (b) joint 2, and (c) joint 3 of the second robot.  
Figure 13: The control inputs of the proposed control in (a) the first robot and (b) the second robot.  
and synchronous objectives of the dual arm were achieved with the proposed control under the presence of unknown friction, external disturbance, and unknown control direction.  
Although the Nussbaum function can instantly detect the wrong control direction when they suddenly happen in the robot system, because the wrong control direction is a serious problem, the lumped uncertainties still increase dramatically. As a result, they can affect the stability of the whole system. Although, the ESO was designed in this study to estimate the lumped uncertainties whose time derivatives are bounded. The ESO effectively worked in this case because of the uncertainty caused by the unknown control direction. Then, the control signals of the proposed control dramatically increase before they are stable again. In future work, advanced approaches will be proposed to manage this issue, and the proposed control will be verified on a practical test bench.  
Conceptualization, D.T.T. and H.V.D.; methodology, D.T.T.; software, D.T.T.; validation, D.T.T. and H.V.D.; formal analysis, D.T.T.; investigation, D.T.T.; resources, D.T.T.; data curation D.T.T.; writing--original draft preparation, D.T.T.; writing--review and editing, D.T.T., H.V.D. and K.K.A.; visualization, D.T.T.; supervision, K.K.A.; project administration, D.T.T. and K.K.A.; funding acquisition, D.T.T. and K.K.A. All authors have read and agreed to the published version of the manuscript.  
This work belongs to project grant No.: T2022-59. funded by Ho Chi Minh City University of Technology and Education, Vietnam, and the work was supported by the \"Regional Innovation Strategy (RIS)\" through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (MOE)(2021 RIS-003).  
Not applicable.  
The authors declare no conflict of interest.  
List of abbreviations used in this paper along with their explanation.  
Explanations  
DGF Degree of freedom  
OPCL Open-Plus-Close-Loop  
MRAC-PD Model reference adaptive control--proportional derivative  
DAM Dual-arm manipulator  
NN Neural network  
FLS Fuzzy logic system  
ESO Extended state observer  
MIMO Multiple inputs, multiple outputs  
SSMC Synchronous sliding mode control