## 1 Introduction
In 2020, an adaptive neural-network-based dynamic surface control [14] was developed for a dual-arm robot against parameter variation, external disturbance, and actuator nonlinearities. The neural network was used to approximate the uncertainties in the robotic system. In 2021, a consensus sliding mode control [15] was applied for multiple manipulators against external disturbances. The proposed approach was developed by combining graph theory and sliding mode control. In 2022, a synchronous control [16] was proposed for a 2-DOF master-slave manipulator against the asymmetric nonlinear dead-zone characteristics. The dead-zone issues were managed by the recursive least squares method. In 2023, a multi-objective synchronization control [17] was proposed for a dual-robot to obtain high-accuracy motion and force compliance behavior simultaneously. The proposed method was developed based on an integral of the past synchronization force error and the integral of the future predicted synchronization motions error for designing the impedance-based force feedback law and motion feedback law, respectively. In 2023, an adaptive synchronous sliding mode control [18] was designed for a dual-arm robot against parameter variations. The approach used an adaptive law to adapt the robust gains in the sliding mode control to manage uncertainties effectively. Additionally, a cross-coupling error in the control design guarantees synchronization between two robots. Based on the above analysis, synchronous control in multiple manipulators takes into account external disturbance, modeling error, and actuator issues such as dead zone, saturation, and friction. They were usually considered as lumped uncertainties which are estimated by universal approximations such as neural networks, fuzzy logic systems, and adaptive laws. In order to apply neural networks (NNs) and fuzzy logic systems (FLSs) as universal approximators in practice, it requires the practitioner's considerable knowledge to choose the suitable parameters relating to the structure of the NNs and FLSs, fuzzy rules, membership functions, learning rates, initial weighting vector, etc. In addition, some intensive computations are also required when these approximators are used. Notably, the extended state observer (ESO), introduced by [PERSON] [19] in 2009, has been conducted to estimate both the unmeasured state and lumped uncertainties in nonlinear systems [20; 21; 22] in an infinite time. In [20], an ESO was designed to approximate external disturbances such as nonlinear loads and parameter perturbations in voltage source converter stations. In [21], an ESOwas developed to reject the lumped disturbance, such as external disturbances and model parameter perturbation, and estimate the unmeasured velocity states in underactuated underwater vehicles. In [22], [PERSON] et al. used the ESO to estimate not only the unmeasured states but also the lumped uncertainties, including the input dead zone, the unknown frictions, and the external disturbances. In practice, the sign of the control gain is often unknown, also named the control direction. When this issue occurs, it can lead to system instability. The Nussbaum gain technique, which was first introduced in [23] has been widely investigated to manage unknown control directions with nonlinear systems [24], nonlinear switched stochastic systems [25], nonlinear MIMO systems [26], and robotic manipulators [27]. To the authors' knowledge, synchronization control for multiple robots with unknown control directions and uncertainties has not been fully conducted in the existing literature. Based on the above discussions, a novel adaptive synchronous sliding mode control is proposed for a dual-arm system against modeling error and unknown control directions. In order to guarantee both tracking and synchronous objectives in the robotic manipulators, the proposed control is designed by using sliding mode control and cross-coupling error. By forcing the cross-coupling error toward zero, tracking and synchronous errors converge to zero.