## 3 Dynamic Modeling of CDPDR
### Workspace Analysis  
The workspace of CDPDR is closely related to the configuration, size and pose. The analysis of the workspace of CDPDR can provide a theoretical basis for the working path planning. If the motion platform is required to have a certain anti-disturbance ability and move according to the preset trajectory, the cable tension vector should meet the force closure requirements. In other words, exerting tension on the cable at any time can produce force and torque on the motion platform. [PERSON]. [36] proposed the judgment condition of the force closure. Based on the matrix knowledge, it can be concluded that when the rank of Jacobian matrix \(\widetilde{\mathbf{J}}^{T}\) of CDPDR is \(n\), any n of the n + 1 cable tension vectors of CDPDR are linearly independent; satisfying the condition of the cable tension vector closure can be equivalent to  
1. \(\text{rank}\left(\widetilde{\mathbf{J}}^{T}\right)=\text{n}\);
2. The null space vector of the Jacobian \(\text{N}\left(\widetilde{\mathbf{J}}^{T}\right)>0\). A flow chart of the workspace discrimination is shown in Figure 4. First, input the mechanism parameters and take the points evenly within the working range. On this basis,the Jacobian matrix and null space vector are calculated and discriminated. Finally, we confirm that all points are taken, and output the points that meet the requirements to form a working space distribution figure. Set \(\mathrm{l}_{k4k1}=\mathrm{H}\) and \(\mathrm{l}_{k4k3}=\mathrm{D}\), which are determined by the area to be worked. It is assumed that the working surface formed by the linear motion guide is the vertical plane, then \(\mathbf{k}_{1}^{C}= \begin{bmatrix}0&0&\mathrm{H}\end{bmatrix} ^{T}\), \(\mathbf{k}_{2}^{C}= \begin{bmatrix}\mathrm{D}&0&\mathrm{H}\end{bmatrix} ^{T}\), \(\mathbf{k}_{3}^{C}= \begin{bmatrix}\mathrm{D}&0&0\end{bmatrix} ^{T}\) and \(\mathbf{k}_{4}^{C}= \begin{bmatrix}0&0&0\end{bmatrix} ^{T}\). Set the CDPDR motion platform size \(\mathrm{l}_{p4P1}=\mathrm{h}\), \(\mathrm{l}_{p4P3}=\mathrm{d}\). Further \(\mathbf{x}_{\mathrm{P}1}^{\mathbf{p}}= \begin{bmatrix}-\mathrm{d}/2&0&\mathrm{h}/2 \end{bmatrix} ^{T}\), \(\mathbf{x}_{\mathrm{P}2}^{\mathbf{p}}= \begin{bmatrix}\mathrm{d}/2&0&\mathrm{h}/2 \end{bmatrix} ^{T}\), \(\mathbf{x}_{\mathrm{P}3}^{\mathbf{p}}= \begin{bmatrix}\mathrm{d}/2&0&-\mathrm{h}/2 \end{bmatrix} ^{T}\) and \(\mathbf{x}_{\mathrm{P}4}^{\mathbf{p}}= \begin{bmatrix}-\mathrm{d}/2&0&-\mathrm{h}/2 \end{bmatrix} ^{T}\). The distance between pulleys is set as \(\mathrm{H}=1.4\) m and \(\mathrm{D}=1.9\) m. The pulley radius \(r_{pully}=12\) mm. Different sizes and corresponding rotation angles of the motion platform are studied. The specific parameters are shown in the following Table 1.  
Simulation analysis of controllable workspace is shown in Figure 5. When the rotation angle is zero, the larger the size of the motion platform and the smaller the corresponding controllable workspace area. When the rotation angle is \(3^{\circ}\), the simulation analysis of the controllable workspace is as shown in Figure 6. As the rotation angle changes from \(0^{\circ}\) to \(3^{\circ}\), the controllable workspace is smaller, the closer the motion platform width \(\mathrm{D}\) is to height \(\mathrm{H}\), and the smaller the controllable workspace. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **1\% Motion** & **2\% Motion** & **3\% Motion** & **4\% Motion** & **Platform** \\ \cline{2-5}  & **Platform** & **Platform** & **Platform** & **Platform** \\ \hline Width (m) & 0.1 & 0.3 & 0.5 & 0.1 \\ High (m) & 0.25 & 0.25 & 0.25 & 0.5 \\ Rotation angle (deg) & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulation parameters. Figure 4: Flow chart of workspace discrimination. Figure 5: Simulation results when the rotation angle is 0\({}^{\circ}\): (**a**) 1\(\#\) simulation results; (**b**) 2\(\#\) simulation results; (**c**) 3\(\#\) simulation results; (**d**) 4\(\#\) simulation results.