## 3 Dynamics Responses of the Workpiece under Different Clamping Robot Configurations
### Dynamics Response of the Workpiece
Therefore, the geometric equation of the workpiece can be obtained as follows:  
\[\mathbf{G}_{31-32}\mathbf{Z}_{32,0}+\mathbf{G}_{31-1}\mathbf{Z}_{1,0}+\mathbf{G}_{31-4}\mathbf{Z}_{4,0 }+\mathbf{G}_{31-7}\mathbf{Z}_{7,0}+\mathbf{G}_{31-10}\mathbf{Z}_{10,0}=0_{6\times 1} \tag{8}\]  
where  
\[\mathbf{G}_{31-1}=\mathbf{H}_{31,\,30}\mathbf{U}_{31,30}^{-1}\mathbf{T}_{31-1}\]  
\[\mathbf{G}_{31-4}=\mathbf{H}_{31,\,30}\mathbf{U}_{31,30}^{-1}\mathbf{T}_{31-4}\]  
\[\mathbf{G}_{31-7}=\mathbf{H}_{31,\,30}\mathbf{U}_{31,30}^{-1}\mathbf{T}_{31-7}\]  
\[\mathbf{G}_{31-10}=\mathbf{H}_{31,\,30}\mathbf{U}_{31,30}^{-1}\mathbf{Equations (8) and (10) are the geometric equations of the ARGW system. By combining the geometric equations with Equation (6), the overall transfer matrix can be written as follows:  
\[\left[ \begin{array}{cccccc}-\mathbf{I}_{12}&\mathbf{T}_{31-32}&\mathbf{T}_{31-1}&\mathbf{T}_{3 1-4}&\mathbf{T}_{31-7}&\mathbf{T}_{31-10}\\ \mathbf{O}_{6\times 12}&\mathbf{G}_{31-32}&\mathbf{G}_{31-1}&\mathbf{G}_{31-4}&\mathbf{G}_{31-7}&\mathbf{G} _{31-10}\\ \mathbf{O}_{6\times 12}&\mathbf{O}_{6\times 12}&\mathbf{G}_{13-1}&\mathbf{G}_{13-4}&\mathbf{O}_{6 \times 12}&\mathbf{O}_{6\times 12}\\ \mathbf{O}_{6\times 12}&\mathbf{O}_{6\times 12}&\mathbf{G}_{13-1}&\mathbf{O}_{6\times 12}&\mathbf{G} _{13-7}&\mathbf{O}_{6\times 12}\\ \mathbf{O}_{6\times 12}&\mathbf{O}_{6\times 12}&\mathbf{G}_{13-1}&\mathbf{O}_{6\times 12}&\mathbf{O} _{6\times 12}&\mathbf{G}_{13-10}\end{array} \right]\left[ \begin{array}{c}\mathbf{Z}_{31,0} \\ \mathbf{Z}_{32,0}\\ \mathbf{Z}_{1,0}\\ \mathbf{Z}_{4,0}\\ \mathbf{Z}_{7,0}\\ \mathbf{Z}_{10,0}\end{array} \right]=0 \tag{13}\]  
It can be simplified as follows:  
\[\mathbf{U}_{\text{all}}\mathbf{Z}_{\text{all}}=0_{72\times 1} \tag{14}\]  
By removing the 0 element and its corresponding array in the boundary conditions, Equation (14) is further simplified as follows:  
\[\overline{\mathbf{U}}_{\text{all}}\overline{\mathbf{Z}}_{\text{all}}=0 \tag{15}\]  
Then, the eigenfrequency equation of the ARGW system can be written as follows:  
\[\text{det}\overline{\mathbf{U}}_{\text{all}}=0 \tag{16}\]  
Solving Equation (16), the natural frequency \(\omega^{k}(k=1,2,3\cdots 6)\) of the ARGW system can be obtained. Then, the state vector of the workpiece can be solved by taking \(\omega^{k}\) into Equation (13). However, to analyze the vibration suppression effect of the ARGW system, it is necessary to convert the state vector to the physical coordinate system based on the dynamics model. The dynamics equation of rigid body element \(i\) is as follows:  
\[\mathbf{M}_{i}\mathbf{v}_{i,tt}+\mathbf{C}_{i}\mathbf{v}_{i,t}+\mathbf{K}_{i}\mathbf{v}_{i}=\mathbf{f}_{i} \tag{17}\]  
where \(\mathbf{M}_{i}\), \(\mathbf{C}_{i}\) and \(\mathbf{K}_{i}\) are the mass matrix, damping matrix, and stiffness matrix of the rigid body \(i\). Additionally, \(\mathbf{v}_{i}\) is the displacement generated by the external forces \(\mathbf{f}_{i}\), and \(\mathbf{v}_{i,t}\), \(\mathbf{v}_{i,tt}\) are the first and second derivatives of \(\mathbf{v}_{i}\) with respect to time \(t\). By superimposing the dynamics equation of each rigid body element according to the topology structure of the ARGW system, the overall dynamics equation can be written as follows:  
\[\mathbf{M}\mathbf{v}_{tt}+\mathbf{C}\mathbf{v}_{t}+\mathbf{K}\mathbf{v}=\mathbf{f} \tag{18}\]  
According to the MSTMM, Equation (18) can be expressed as follows:  
\[\bar{q}^{k}(t)+2\zeta^{k}\omega^{k}\bar{q}^{k}(t)+\left(\omega^{k}\right)^{2} q^{k}(t)=\sum_{i}\left(\left\langle\mathbf{f}(t),\mathbf{V}_{i}^{k}\right\rangle/M_{i}^{k}\right) \tag{19}\]  
where \(\bar{q}^{k},\bar{q}^{k}\), and \(q^{k}\) are the generalized acceleration, velocity, and displacement, respectively. The eigenvector \(\mathbf{V}_{i}^{k}\) contains only the liner displacement and angular displacement of the state vector \(\mathbf{Z}_{i}^{k}\).