## 4 Dynamic Analysis of CDFM  
A dynamic analysis was conducted to obtain the relation between the motor torque and the speed of the manipulator end. Without considering the energy of friction loss, the energy generated by the motor was equal to the sum of the kinetic energy and potential energy of the system. In the operating space, the dynamic equation established according to energy conservation law is shown as follows:  
\[\textbf{W}=\textbf{E}_{k}+\textbf{E}_{p} \tag{11}\]  
where \(\textbf{E}_{k}\) is the total kinetic energy of the CDFM system, \(\textbf{E}_{p}\) is the total potential energy of the CDFM system, and \(\textbf{W}\) is the work done by the motor. The CDFM had six rods with \(\dot{\theta}=[\dot{\theta}_{1},\dot{\theta}_{2},\dot{\theta}_{3},...,\dot{ \theta}_{6}]\) representing the angular velocity of the rod centroid and \(\textbf{v}_{ci}=[v_{c1},v_{c2},v_{c3},...,v_{c6}]\) denoting the linear velocity of the rod centroid. The kinetic energy of the first \(n\) rods of the system could be derived according to formula (12):  
\[\textbf{E}_{kn}=\frac{1}{2}\sum_{i=1}^{n}m_{i}\varphi_{ci}^{2}+\frac{1}{2} \sum_{i=1}^{n}\textbf{I}_{zzi}\overset{.}{\theta_{i}} \tag{12}\]  
where \(\textbf{I}_{zzi}\) is the rotational inertia of rod (i) around the \(Z\)-axis and \(m_{i}\) denotes the mass of rod (i). The mass and centroid coordinates of each rod of the CDFM are shown below in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**Linkage (i)** & \(\textbf{a}_{i-1}\) **(mm)** & \(\textbf{a}_{i-1}\) **(°)** & \(\textbf{d}_{i}\) **(mm)** & \(\textbf{\theta}_{i}\) **(°)** \\ \hline
1 & 108 & 0 & 0 & \(\theta_{1}\) \\
2 & 108 & 0 & 0 & \(\theta_{2}\) \\
3 & 108 & 0 & 0 & \(\theta_{3}\) \\
4 & 108 & 90 & 0 & \(\theta_{4}\) \\
5 & 108 & 0 & 0 & \(\theta_{5}\) \\
6 & 73 & 0 & 0 & \(\theta_{6}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The D-H parameters of CDFM. \begin{table}
\begin{tabular}{c c c c} \hline
**Rod (i)** & **Mass of Rods (kg)** & **Centroid Coordinates** & **Centroid Coordinates** & **Centroid Coordinates** \\  & **(mm)** & **r\({}_{y}\) (mm)** & **r\({}_{z}\) (mm)** \\ \hline
1 & 0.191 & 0 & 61.056 & 0 \\
2 & 0.191 & 0 & 169.056 & 0 \\
3 & 0.187 & 0 & 276.826 & 0 \\
4 & 0.191 & 0 & 385.056 & 0 \\
5 & 0.191 & 0 & 493.056 & 0 \\
6 & 0.092 & 0 & 575.818 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: The mass and centroid coordinates of each rod of the CDFM. The inertia tensor of each rod of the CDFM was as follows (the unit is kg \(\times\) mm\({}^{2}\)). \[{}^{c}\mathbf{I}_{1}={}^{c}\mathbf{I}_{2}={}^{c}\mathbf{I}_{4}={}^{c}\mathbf{I}_{5}=\left[ \begin{array} []{ccc}205.093&0.071&0\\ 0.071&111.425&0.190\\ 0&0.190&194.877\end{array} \right]\]  
\[{}^{c}\mathbf{I}_{3}=\left[ \begin{array}{ccc}199.522&0.010&0\\ 0.010&108.268&0.094\\ 0&0.094&190.308\end{array} \right]\]  
\[{}^{c}\mathbf{I}_{6}=\left[ \begin{array}{ccc}44.973&-0.033&0\\ -0.033&54.498&0\\ 0&0&43.812\end{array} \right]\]  
The potential energy of the system was composed of gravity potential energy and elastic potential energy. The potential energy of the first \(n\) (\(n\leq 6\)) connecting rod of the system could be determined as follows:  
\[\mathbf{E}_{pn}=\frac{1}{2}\underset{i=1}{\overset{n}{\sum}}K_{2i-1}\Delta\mathbf{x}_{ 2i-1}^{2}+\underset{i=3}{\overset{n}{\sum}}m_{i}H_{ci}g\sin\theta_{i} \tag{13}\]  
where \(\Delta\mathbf{x}_{2i-1}^{2}\) represents the deformation quantity of the spring \(2i-1\); \(H_{ci}\) represents the distance between the centroid of connecting rod (i) and the shaft; and \(K_{2i-1}\) (\(i=1,\ldots,6\)) represents the stiffness coefficient of the spring \(2i-1\), which could be calculated using the following equation:  
\[K_{2i-1}=\frac{Gd^{4}}{8D^{3}n} \tag{14}\]  
where \(G\) is the stiffness coefficient; \(d\) and \(D\) represent the cable diameter and mean diameter of the spring, respectively; and \(n\) is the number of effective cycles.