## 2 Problem Formulations  
### System Description  
The dynamic equations of each 3-DOF manipulator in joint space can be expressed as follows:  
\[\mathbf{M}_{i}(\mathbf{q}_{i})\ddot{\mathbf{q}}_{i}+\mathbf{C}_{i}\big{(} \mathbf{q}_{i}\cdot\dot{\mathbf{q}}_{i}\big{)}\dot{\mathbf{q}}_{i}+\mathbf{G }_{i}(\mathbf{q}_{i})+\mathbf{d}_{i}=\mathbf{\Pi}_{i}\boldsymbol{\tau}_{i} \tag{1}\]  
where \(i(1\leq i\leq 2)\) presents the \(i\)th robot index in the network and \(p\) is the total number of the individual elements. Furthermore, \(\mathbf{q}_{i}=[q_{i1},q_{i2},q_{i3}]^{T}\in R^{3\times 1}\), \(\dot{\mathbf{q}}_{i}=\big{[}\dot{q}_{i1},\dot{q}_{i2},\dot{q}_{i3}\big{]}^{T} \in R^{3\times 1}\), \(\ddot{\mathbf{q}}_{i}=\big{[}\ddot{q}_{i1},\ddot{q}_{i2},\ddot{q}_{i3}\big{]}^ {T}\in R^{3\times 1}\) derive the vector of position, velocity, and acceleration vector in the joint space of the \(i\)th robot, respectively; \(\mathbf{M}_{i}(\mathbf{q}_{i})\in R^{3\times 3}\), \(\mathbf{C}_{i}(\mathbf{q}_{i},\dot{\mathbf{q}}_{i})\dot{\mathbf{q}}_{i}\in R^{ 3\times 1}\) and \(\mathbf{G}_{i}(\mathbf{q}_{i})\in R^{3\times 1}\) are the known symmetric and uniformly positive definite inertia matrix, the known Coriolis and centrifugal matrix, and the known gravitational force vector,respectively; \(\mathbf{d}_{i}=j_{i}^{T}F_{exit}\in R^{3\times 1}\) presents the lumped uncertainties including parametric variation and external disturbances in the \(i\)th robot coordinates; \(j_{i}\in R^{3\times 3}\) is a Jacobian matrix of the ith robot; \(\mathbf{F}_{exit}(t)\in R^{3\times 1}\) is an external force vector at the end-effector of the ith robot; \(\mathbf{\tau}_{i}\in R^{3\times 1}\)\(\mathbf{\tau}_{i}\in R^{3\times 1}\) presents the input torque vectors in the \(i\)th robot coordinates; and \(\mathbf{\Pi}_{i}\in R^{3\times 3}\) is the control direction vector of the ith robot. The properties of the manipulator dynamics are presented as follows [28]:  
**Property 1:** Matrix \(2\mathbf{C}_{i}\big{(}\mathbf{q}_{i},\dot{\mathbf{q}}_{i}\big{)}-\dot{ \mathbf{M}}_{i}(\mathbf{q}_{i})\) is a skew-symmetric matrix, defined as \(\mathbf{x}^{T}\Big{(}2\mathbf{C}_{i}\big{(}\mathbf{q}_{i},\dot{\mathbf{q}}_{i }\big{)}-\dot{\mathbf{M}}_{i}(\mathbf{q}_{i})\Big{)}\mathbf{x}=0\). **Property 2:** The inequalities \(\big{\|}\mathbf{C}_{i}\big{(}\mathbf{q}_{i},\dot{\mathbf{q}}_{i}\big{)}\big{\|} \leq\lambda_{ci}\), \((i=1,2)\) and \(\|\mathbf{G}_{i}(\mathbf{q}_{i})\|\leq\lambda_{gi}\), \((i=1,2)\) are satisfied, where \(\lambda_{c}\) and \(\lambda_{g}\) are known positive constants. **Property 3:** The inequality \(\lambda_{M_{ij}}\leq\|\mathbf{M}_{i}(\mathbf{q}_{i})\|\leq\lambda_{M_{2i}}\) is satisfied where \(\lambda_{Mij}(i,j=1,2)\) are known positive constants. **Assumption 1:** The lumped uncertainties, including parametric variation and external disturbance, \(d_{i}\), are differentiable and bounded. Figure 1 illustrates the system of a dual-arm model using \(\{O_{W}\}\) as the world coordinate system; \(\{O_{bi}\}\)\((i=1,2)\) is the base frame of the \(i\)th robot; \(q_{ij}\) is the rotation angle of the \(j\)th joint of the \(i\)th robot; and \(c_{1}\) and \(c_{2}\) are vectors of each robot's end-effector reference to \(\{O_{W}\}\) with \(n(c_{1})\) and \(n(c_{2})\) as the rotation matrix, respectively. \({}_{b1}^{W}\mathbf{T}(. )\) and \({}_{b2}^{W}\mathbf{T}(. )\) are correspondingly the transformation matrix from the base frame of the ith robot, \(\{O_{bi}\}\)\((i=1,2)\), to the world coordination system, \(\{O_{W}\}\). Let define \(\mathbf{x}_{i1}=\mathbf{q}_{i}\in R^{3\times 1}\), and \(\mathbf{x}_{i2}=\dot{\mathbf{q}}_{i}\in R^{3\times 1}\) with \(i=1,2\).