## 3 Dynamics Responses of the Workpiece under Different Clamping Robot Configurations  
### Coordinate Systems of the ARGW System  
The diagrams of the body element and the hinge element are shown in Figure 4a and **b**, respectively, which are located in the same world coordinate system \(o_{0}x_{0}y_{0}z_{0}\). In particular, the mass of the body element is \(C\), its dynamics parameters are measured by the model, and the coordinate system of the input end \(lx^{\prime}y^{\prime}z^{\prime}\) and the coordinate system of the output end \(Oxyz\) are always in same direction. Moreover, the translational stiffness and torsional stiffness of the hinge element are identified using the mode test, and the coordinate system of the input end and output end coincide. Furthermore, defining the coordinate system \(Oxyz\) of the body element and the coordinate system \(lx^{\prime}y^{\prime}z^{\prime}\) of the next hinge element always coincide; thus, the coordinate system of the ARGW system can be represented by the \(Oxyz\) of the body element.  
As shown in Figure 5, \(o_{L}x_{L}y_{L}z_{L}(i=1,\,2,\,3,\,4)\) are the coordinate systems of four supporting legs, \(o_{L}x_{L}y_{L}z_{L}i\)\((i=1,\,2,\,3,\,4)\) is the coordinate system of the AGV, \(o_{i}x_{i}y_{i}z_{i}\)\((i=1,\,2,\,\cdots,\,6,\,\mathrm{f})\) are the six-link and flange coordinate systems of the robot,  
Figure 4: The diagrams of the body element and hinge element.  
Figure 3: Topology figure of the dynamics model.  
\(\omega_{\text{end}}x_{\text{end}}y_{\text{end}}z_{\text{end}}\) and \(\omega_{\text{w}}x_{\text{w}}y_{\text{w}}z_{\text{w}}\) are the end gripper and workpiece coordinate systems, respectively.  
When the clamping robot is in the configuration shown in Figure 5, all coordinate systems have the same direction, and the angles of the six joints of the robot are defined as \(0^{\circ}\). After the robot's configuration changes, the transfer direction of the state vector of each joint will change. Depending on the different rotation directions of six joints, the transformation matrices of \(x\), \(y\), and \(z\) directions are defined as follows:  
\[\mathbf{H}_{x}(\theta_{k})=\begin{bmatrix}h_{x}(\theta_{k})&\mathbf{O}_{3 \times 3}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}\\ \mathbf{O}_{3\times 3}&h_{x}(\theta_{k})&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{x}(\theta_{k})&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{x}(\theta_{k})\end{bmatrix},\;\;k=1, 2, 3,4,5,6 \tag{1}\]  
\[\mathbf{H}_{y}(\theta_{k})=\begin{bmatrix}h_{y}(\theta_{k})&\mathbf{O}_{3 \times 3}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}\\ \mathbf{O}_{3\times 3}&h_{y}(\theta_{k})&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{y}(\theta_{k})&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{y}(\theta_{k}) \end{bmatrix},\;\;k=1, 2, 3,4,5,6 \tag{2}\]  
\[\mathbf{H}_{z}(\theta_{k})=\begin{bmatrix}h_{z}(\theta_{k})&\mathbf{O}_{3 \times 3}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{h}_{z}(\theta_{k})&\mathbf{O}_{3\times 3}&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{z}(\theta_{k})&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3}&\mathbf{h}_{z}(\theta_{k}) \end{bmatrix},\;\;k=1, 2, 3,4,5,6 \tag{3}\]  
where \(\theta_{k}\) is the angle of the joint \(k\).  
\[\mathbf{h}_{x}(\theta)=\begin{bmatrix}1&0&0\\ 0&\cos\theta&\sin\theta\\ 0&-\sin\theta&\cos\theta\end{bmatrix},\mathbf{h}_{y}(\theta)=\begin{bmatrix} \cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix},\mathbf{h}_{z}(\theta)=\begin{bmatrix}\cos \theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix} \tag{4}\]