## 3 Proposed Control Design  
### Control Design  
As presented in Figure 2, the proposed control is constructed by synchronous sliding mode control, Nussbaum gain function, and extended state observer to deal with the challenges in the uncertain dual-arm robot. The synchronous sliding mode control can maintain both movement synchronization and tracking objective in the dual arm because it is developed by cross-coupling errors and the sliding mode control technique. The extended state observer is used to estimate the lumped uncertainties consisting of parameter variation and external disturbance in each robotic arm. Additionally, the Nussbaum gain function helps to avoid the misconnection control signal. As a result, the proposed method will inherit all superiorities of these techniques. Figure 2: Structure of the proposed control. The sliding variable is defined as follows:  
\[\mathbf{s}=\dot{\mathbf{e}}_{c}+\lambda\mathbf{e}_{c} \tag{13}\]  
where \(\lambda\in R^{6\times 6}\) is a positive definite diagonal matrix. The time derivative of the sliding variable (13) is expressed as follows:  
\[\dot{\mathbf{s}}=\ddot{\mathbf{e}}_{c}+\lambda\dot{\mathbf{e}}_{c}=(\mathbf{I}+ \alpha\mathbf{T})\big{(}\ddot{\mathbf{e}}+\lambda\dot{\mathbf{e}}\big{)}=( \mathbf{I}+\alpha\mathbf{T})\big{(}\ddot{\mathbf{x}}-\ddot{\mathbf{x}}_{d}+ \lambda\dot{\mathbf{e}}\big{)} \tag{14}\]  
The second differential of the state variable, \(\mathbf{x}\), is calculated as follows:  
\[\ddot{\mathbf{x}}=\mathbf{M}^{-1}(\mathbf{x})\big{(}\Pi\mathbf{\tau}-\mathbf{C }\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}\dot{\mathbf{x}}-\mathbf{G}( \mathbf{x})-\mathbf{d}\big{)} \tag{15}\]  
where \(\mathbf{x}=\big{[}\mathbf{x}_{11}^{T},\mathbf{x}_{21}^{T}\big{]}^{T}\in R^{6 \times 1}\), \(\dot{\mathbf{x}}=\big{[}\mathbf{x}_{12}^{T},\mathbf{x}_{22}^{T}\big{]}^{T}, \ddot{\mathbf{x}}=\big{[}\dot{\mathbf{x}}_{12}^{T}\quad\dot{\mathbf{x}}_{22}^ {T}\big{]}^{T}\), \(\mathbf{M}(\mathbf{x})= \begin{bmatrix}\mathbf{M}_{1}(\mathbf{x}_{11})&0\\ 0&\mathbf{M}_{2}(\mathbf{x}_{21})\end{bmatrix} \in R^{6\times 6}\), \(\mathbf{C}(\mathbf{x},\dot{\mathbf{x}})= \begin{bmatrix}\mathbf{C}_{1}(\mathbf{x }_{11},\mathbf{x}_{12})&0\\ 0&\mathbf{C}_{2}(\mathbf{x}_{12},\mathbf{x}_{22})\end{bmatrix} \), \(\mathbf{G}(\mathbf{x})= \begin{bmatrix}\mathbf{G}_{1}(\mathbf{x}_{11})\\ \mathbf{G}_{2}(\mathbf{x}_{21})\end{bmatrix} \in R^{6\times 1}\), \(\mathbf{d}= \begin{bmatrix}\mathbf{d}_{1}^{T}&\mathbf{d}_{2}^{T}\end{bmatrix} ^{T }\in R^{6\times 1}\). By substituting (15) into (14), its result is presented as follows:  
\[\dot{\mathbf{s}}=(\mathbf{I}+\alpha\mathbf{T})\Big{(}\mathbf{M}^{-1}(\mathbf{ x})\big{(}\Pi\mathbf{\tau}-\mathbf{G}(\mathbf{x})-\mathbf{C}\big{(}\mathbf{x}, \dot{\mathbf{x}}\big{)}\dot{\mathbf{x}}-\mathbf{d}\big{)}-\ddot{\mathbf{x}}_{ d}+\lambda\dot{\mathbf{e}}\big{)} \tag{16}\]  
The proposed control law is selected as follows:  
\[\mathbf{\tau}=\mathbf{N}(\zeta)\mathbf{\tau}_{0} \tag{17}\]  
where \(\mathbf{\tau}_{0}=\mathbf{M}(\mathbf{x})\big{(}\ddot{\mathbf{x}}_{d}-\lambda \dot{\mathbf{e}}\big{)}+\mathbf{C}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)} \dot{\mathbf{x}}+\mathbf{G}(\mathbf{x})-\mathbf{K}\mathbf{s}-\mathbf{\eta} sign(\mathbf{s})+\mathbf{M}(\mathbf{x})\dot{\mathbf{x}}_{3}\in R^{6\times 1}\), \(\mathbf{N}(\zeta)=diag\big{(}[N(\zeta_{1}),\dots,N(\zeta_{6})]\big{)}\), \(N(\zeta_{i})=e^{\zeta_{i}^{2}}\cos\big{(}\frac{\pi}{2}\zeta_{i}\big{)}\), \(\dot{\zeta}_{i}=\sigma_{i}s_{i}\tau_{0i}\), \((i=1,\dots,6)\). where \(\sigma_{i}(i=1,\dots,6)\) are positive constants. When we apply a controller defined by (17) and an extended state observer (9) for the dual-arm system (2), then the sliding variable, \(\mathbf{s}\), the cross-coupling error, \(\mathbf{e}_{c}\), the synchronous error, \(\mathbf{e}_{s}\), and the tracking error, \(\mathbf{e}\), are bounded. Consequently, the controlled system is demonstrated to be semi-globally uniformly and ultimately bounded (SGUIUB).