## 3 Mechanism Insulation Analysis  
Since the cable-driven parallel-series mechanism is used as the joint of the robotic arm for live-line operation, the insulation performance of the joint is essential to protect the safety of the mechanism operator and power equipment. The CSM has been used to analyze the potential and electric field distribution of the mechanism. The ring charges are used in the platforms, insulating plates, and conductive springs due to their axis-symmetrical geometries. The line charges are used to fit the shape of the insulating cables. The contour points are set to the interfaces between different materials to satisfy the boundary conditions. The schematic for the insulator analysis is shown in Figure 4. The electric field distribution is calculated and measured along the field line with a 10 mm gap to the insulating plate.  
Figure 4: The contour points and charge distribution in the mechanism.  
The charge value is calculated by the following equation:  
\[[Q]=[P]^{-1}[V], \tag{22}\]  
where \(P\) is the potential or field coefficient matrix, \(Q\) is the simulating charge array, and \(V\) is the values of contour points on the interfaces between different materials.  
The insulating plates and insulating cables are considered the insulating materials in the mechanism to provide the mechanic support and electric insulation. The charges and contour points on the interface between the platform and these solid insulating materials satisfy the following boundary condition:  
\[\sum_{i=1}^{n_{e}+n_{a}}P_{ij}Q_{j}=V, \tag{23}\]  
where \(n_{e}\) and \(n_{a}\) are the numbers of charges in the electrode and air, respectively; \(P_{ij}\) is the potential coefficient; \(Q_{j}\) is the simulating charge; \(V\) is the potential on the interface between the electrode and the insulating materials.  
The charges and contour points on the interface between the platform satisfy the following boundary condition:  
\[\sum_{i=1}^{n_{e}+n_{i}}P_{ij}Q_{j}=V, \tag{24}\]  
where \(n_{i}\) is the number of charges in the insulating plate; \(V\) is the potential on the interface between the electrode and air.  
The charges and contour points on the interface between the solid insulating materials and air satisfy the following condition to satisfy Dirichlet and Neumann boundary conditions:  
\[\sum_{i=1}^{n_{e}+n_{i}}P_{ij}Q_{j}-\sum_{i=1}^{n_{e}+n_{a}}P_{ij}Q_{j}=0, \tag{25}\]  
\[\varepsilon_{a}\sum_{i=1}^{n_{e}+n_{i}}f_{ij}Q_{j}-\varepsilon_{r}\sum_{i=1}^ {n_{e}+n_{a}}f_{ij}Q_{j}=0,, \tag{26}\]  
where \(f_{ij}\) is the normal flux density coefficient; \(\varepsilon_{r}\) and \(\varepsilon_{a}\) are the relative permittivities of the insulating material and air, respectively.  
The charges and contour points on the interface between the insulating plate and the conductive spring satisfy the following boundary condition:  
\[\sum_{i=1}^{n_{e}+n_{a}+n_{a}}P_{ij}Q_{j}=V_{s}, \tag{27}\]  
where \(n_{s}\) is the number of charges in the conductive spring; \(V_{s}\) is the unknown floating potential of the spring.  
The charges and contour points on the interface between the air and the conductive spring satisfy the following boundary condition:  
\[\sum_{i=1  
This model is used to calculate the electric potential and field distributions of the mechanism and analyze the insulation performance based on the maximum electric field strength.