## 2 Modeling of the CDPR System
### Dynamic Modeling for the CDPR System  
To derive the dynamics equation for the CDPR system, first, a static equilibrium model for the system should be established by applying the forces and moments to the end-effector, which consists of internal forces and external forces. These forces and moments are from different parts, with one part from the eight cables and the other from the environment. Considering the static equilibrium, the total forces and moments exerted on the end effector should be zero, and the equilibrium equations can be obtained as follows:  
\[\sum_{i=1}^{8}\overrightarrow{T_{i}}+m\overrightarrow{g}=\overrightarrow{F_{ r}} \tag{6}\]  
\[\sum_{i=1}^{8}\left(\overrightarrow{e_{i}}\times\overrightarrow{T_{i}} \right)=\overrightarrow{M_{r}} \tag{7}\]  
where \(T_{i}\) denotes the tension vector for the \(i\)th cable, \(m\) is the mass of the end-effector, \(g\) is the gravitational acceleration, and \(F_{r}\) and \(M_{r}\) indicate the resultant forces and moment vector, respectively. For convenience, we assumed that the origin geometrical center was located at the center of gravity. Therefore, the moment caused by gravity could be ignored. Combining \(F_{r}\) and \(M_{r}\) together, we obtained the wrench vector:  
\[W_{R}=\left\{\begin{array}{cc}\overrightarrow{F_{r}}&\overrightarrow{M_{ r}}\\ \end{array}\right\}^{T} \tag{8}\]  
From Equations (6)-(8), the static equilibrium equation can be described as follows:  
\[[J]^{T}T+\left\{\begin{array}{c}m\overrightarrow{g}\\ 0\\ \end{array}\right\}=W_{R} \tag{9}\]  
\[J^{T}=\left[\begin{array}{cccc}\hat{L_{1}}&\hat{L_{2}}&\ldots&\hat{L_{8}} \\ \hat{e_{1}}\times\hat{L_{1}}&\hat{e_{2}}\times\hat{L_{2}}&\ldots&\hat{e_{8}} \times\hat{L_{8}}\\ \end{array}\right] \tag{10}\]  
where \(J\) represents the cable tension Jacobian matrix, \([J]^{T}\) represents the transposed matrix of \(J\), and \(\hat{e_{i}}\) and \(\hat{L_{i}}\) denote the unit cable length vector. \(T\) is the vector of cable tensions:  
\[T=\left\{\begin{array}{cccc}t_{1}&t_{2}&\ldots&t_{8}\\ \end{array}\right\}^{T} \tag{11}\]To consider the effects of external force caused by the acceleration of the end-effector, the dynamics equation was derived as shown in Equation (12).  
\[[J]^{T}T+\left\{\begin{array}{c}F_{e}\\ M_{e}\end{array}\right\}=\left[\begin{array}{cc}m&0\\ 0&I\end{array}\right]\ddot{X}+\left[\begin{array}{c}0\\ \omega+I\times\omega\end{array}\right] \tag{12}\]  
where \(F_{e}\) and \(M_{e}\) denote the external force and moment, respectively; matrices m and \(I\) are a [\(3\times 3\)] diagonal matrix of the end-effector mass and tensor of inertia, respectively; \(\ddot{X}\) is the acceleration of the end-effector; and \(\omega\) denotes the angular velocity vector. These are defined as follows:  
\[\ddot{X}=\left\{\begin{array}{cccccc}\ddot{x}&\ddot{y}&\ddot{z}&\dot{\omega}_ {x}&\dot{\omega}_{y}&\dot{\omega}_{z}\end{array}\right\}^{T} \tag{13}\]  
\[\omega=\left\{\begin{array}{cccccc}\omega_{x}&\omega_{y}&\omega_{z}\end{array}\right\} \tag{14}\]  
In this work, the CDPR system was based on translation-only motion with a nearly constant orientation. Thus, the angular velocity \(\omega\) was small enough and the last term, including the angular velocity on the right side of Equation (13), was negligible in this research.