## 2 Modeling of the CDPR System
### Kinematics of the Redundant-Actuated CDPR System  
Briefly reviewing the kinematic model, the geometry of the CDPR robot is described by its proximal anchor points on the robot frame and the distal anchor points on the end-effector, which were defined by the vector. Figure 2 shows that B was the origin of the global coordinate, and the vector \(\overset{\rightarrow}{k_{i}}\) was described from the frame B to anchor point A of the frame. The vector denoted as \(\overset{\rightarrow}{G}\) represented the vector from B to the center of the end-effector fixed frame E, the vector \(\overset{\rightarrow}{e_{i}}\) was connected from the end-effector fixed frame E to the anchor point C of the end-effector, and it described the position vector of the anchor point from E. In addition, the vector of cable length denoted as \(\overset{\rightarrow}{L_{i}}\) was derived from the given vector. The cable length of the cables could be obtained by solving the inverse position kinematics (IPK). By applying a vector loop, the cable vector \(\overset{\rightarrow}{L_{i}}\) was established as follows:  
\[\overset{\rightarrow}{L_{i}}=\overset{\rightarrow}{k_{i}}-(\overset{ \rightarrow}{G}+\overset{\rightarrow}{e_{i}}) \tag{1}\]  
where the index \(i\) denotes the cable number. In a typical CDPR robot system, the pulleys have an important role, and they are used to guide the cable at point A towards the end-effector; the proximal anchor points of the robots are simplified to be ideal points. Thus, the starting point of the cable described by the position vector is not a constant but depends on the current pose of the robot and the wrapping angle of the cable around the pulley is a variable. Moreover, in a CDPR system, the real cables achieve a reasonable lifetime only when a minimum bending radius is exceeded [23]. Thus, determining the starting point of the cable is essential to improve  
Figure 1: Schematic diagram of the CDPR system: (**a**) diagram of 6–DOF CDPR; (**b**) asymmetric pulley locations. the accuracy of its position and the effect of pulley kinematics in this system. A model of the pulley kinematics is presented in Figure 3, and the kinematic equation is given as follows:  
\[L_{comp}=L_{c}+r\theta \tag{2}\]  
where \(L_{comp}\) is a cable length compensation, \(L_{c}\) is the free cable length from point A to point B, \(r\) is the pulley radius and \(\theta\) is the wrapping angle of the cable. For the two right-angled triangles ADO and AOE shown in Figure 3, OA is the common side. Therefore, based on the Pythagorean theorem for the two right-angled triangles, the following can be obtained:  
\[(d_{xy}-r)^{2}+d_{z}^{2}=L_{i}^{2}=L_{c}^{2}+r^{2} \tag{3}\]  
Considering the tetragon EODA, there are two angles that are right-angles. Thus, the enclosed angle \(\theta_{1}+\theta_{2}\) at point A equals the sought complementary angle \(\theta\) at point A.  
Figure 3: Diagrammatic representation of the pulley kinematics. Figure 2: Diagrammatic representation of the kinematic parameter. O [5; 23; 24]. Using the elementary trigonometric function, the cable wrapping angle around the pulley is given by Equation (4). \[\theta=\theta_{1}+\theta_{2}=\arccos\frac{L_{c}}{\sqrt{\left(d_{xy}-r\right)^{2}+d _{z}^{2}}}+\arccos\frac{d_{z}}{\sqrt{\left(d_{xy}-r\right)^{2}+d_{z}^{2}}} \tag{4}\]  
where \(d_{xy}=\sqrt{d_{x}^{2}+d_{y}^{2}}\). Given the coordinates of the point at pulley O whose coordinates are O(\(x_{2}\), \(y_{2}\), \(x_{2}\)), at the anchor point of end-effector A(\(x_{1}\), \(y_{1}\), \(z_{1}\)), and at the center of end-effector G(\(x_{0}\), \(y_{0}\), \(z_{0}\)), then the values \(d_{x}\), \(d_{y}\), and \(d_{z}\) are coordinates in the \(x\), \(y\), \(z\) directions, which are calculated by the equation: \(L(x,y,z)=\) O \(-\) A \(-\) G. \(d_{z}\) represents coordinates of point E with respect to the frame point \(B\) coordinates.