## 2 Cable-Driven Parallel-Series Mechanism
### Model Details of the Mechanism  
To model the motion of the proposed CDPS mechanism, a simplified structure schematic diagram is established according to the mechanism model, as shown in Figure 2. This structure is divided into five parts to illustrate the modeling details. 1. Four cables are modeled with negligible mass and diameter in the inverse kinematics model. The fixed points of the cables on the lower and upper platforms are defined as \(P_{1}\), \(P_{2}\) and \(Q_{1}\), \(Q_{2}\). The corresponding relationship between the four cable numbers and the fixed points on the platform is shown in Figure 2. In the initial state, the fixed-point connection line of the moving platform is perpendicular to the connection line of the base. The distances of \(P\) and \(Q\) to the concentric points of the upper and lower platforms are \(p\) and \(q\), respectively. The length of the cable between \(Q\) and \(P\) is defined as \(L\). \(T\) is the force of the cable. The unit vector along the direction of the cable is defined as \(u_{i}\). 2. The lower platform is considered the fixed reference, and its thickness is neglected. \(O_{1}X_{1}Y_{1}Z_{1}\) is defined as the local coordinate system, and origin \(O_{1}\) is located at the center of the platform. Figure 1: Application and schematic of the cable-driven parallel-series (CDPS) mechanism: (**a**) CDPS mechanism as the wrist joint in the robotic arm, and (**b**) structure of the CDPS mechanism. The \(Y_{1}\) axis is along the direction of \(O_{1}P_{2}\). The \(X_{1}\) axis is perpendicular to the \(Y_{1}\) axis. The \(Z_{1}\) axis is determined by the right-hand principle based on the \(X_{1}\) and \(Y_{1}\) axes. An insulation plate is attached to the upper surface of the lower platform to provide electrical insulation for the live-line operation. 3. The upper platform is the movable part of the mechanism, and its thickness is neglected. \(oxyz\) is defined as the local coordinate system; origin \(o\) is located at the geometric center of the moving platform. The \(x\)-axis is along the \(oQ_{2}\) direction, and the \(y\)-axis is perpendicular to the \(x\)-axis. The \(z\)-axis is determined by the right-hand principle based on the \(x\) and \(y\) axes. An insulation plate is attached to the bottom surface of the upper platform to provide electrical insulation for the live-line operation. 4. The cylindrical compression spring connects the concentric lower and upper platforms. The force and torque generated by the cylindrical compression spring support the load and participate in the movement of the upper platform. The combined bearing between the spring and the moving platform rotates around the \(z\)-axis and moves along the \(z\)-axis. Therefore, the spring does not participate in the \(z\)-axis rotation and only rotates in the \(oxy\) plane providing the roll and pitch angles. 5. Two shafts connected by a universal joint in the spring restricts the lateral bending motion of the structure to simplify the inverse kinematics model of the spring lateral bending motion. The upper shaft passes the center of the moving platform through the combined bearing, and the lower shaft is fixed at the center of the lower platform. The upper and lower shafts are perpendicular to the two platforms. A global coordinate system \(OXYZ\) is defined at the center of the universal joint with identical axis directions to the local coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\). The distances from the global origin \(O\) to the local origins \(O_{1}\) and \(o\) are defined as \(d_{1}\) and \(d_{2}\). \(d_{1}\) is equal to the length of the lower shaft, and \(d_{2}\) is variable because the upper shaft is connected with the upper platform by the linear bearing, which slides along the upper shaft during the motion. The upper shaft is made of an insulating material to provide the electrical insulation between the upper and lower platforms.