# Dynamic Modeling and Optimization of Tension Distribution for a Cable-Driven Parallel Robot
## 3 Dynamics Model of the Robot
### Force Analysis of the End-Effector  
In fact, since the movement of the end-effector is implemented by releasing and pulling cables, there is a tension that acts on the end-effector for each cable. Additionally, the end-effector is also affected by gravity. The end-effector's force analysis is shown in the Figure 4. Points \(B_{1}\)\(-\)\(B_{8}\) represent the vertexes of the frame, and Points \(P_{1}\)\(-\)\(P_{8}\) are the vertexes of the end-effector. \(T_{i}\) (\(i\) = 1, 2,...,8) is the tension of the \(i\)th cable between Point \(B_{i}\) and \(P_{i}\). The coordinate system \(o_{e}-x_{e}y_{e}z_{e}\) is fixed on the center of the end-effector. The coordinate system \(O_{I}-X_{I}Y_{I}Z_{I}\) is an inertial system denoted by \(\sum_{I}\). For the convenience of dynamics modeling, two assumptions are as follows:  
Figure 3: Schematic diagram of the cable guide path. Figure 2: Machine design schematic. 1. The weight of the cable in the coordinate system \(o_{e}-x_{e}y_{e}z_{e}\) is fixed on the center of the end-effector. The coordinate system \(O_{I}-X_{I}Y_{I}Z_{I}\) is an inertial system denoted by \(\sum_{\mathbf{I}}\). For the convenience of dynamics modeling, two assumptions are as follows: is not considered in dynamics modeling due to the tiny mass of the cable; 2. The cable is an ideal, flexible body that does bending or shearing force. Based on the above assumptions and the Newton-Euler Equation, the dynamics model of the end-effector is given as follows: \[\left\{ \begin{array}{c}m\dot{\mathbf{v}}+\mathbf{G}_{e}=\sum\limits_{i=1}^{8}\mathbf{T}_ {i}+\mathbf{F}_{r}\\ \mathbf{I}\dot{\mathbf{\omega}}+\mathbf{\omega}\times\mathbf{I}\mathbf{\omega}=\sum\limits_{i=1}^{8 }\mathbf{r}_{i}\times\mathbf{T}_{i}+\mathbf{M}_{r}\end{array} \right.\] (1) where \(m\) is the mass of the end effector. \(\mathbf{v}=[v_{x},v_{y},v_{z}]^{\mathrm{T}}\) denotes the linear velocity of the end-effector. \(\mathbf{G}_{e}\) is the gravity of the end-effector. \(\mathbf{I}\) is the inertial matrix, and \(\mathbf{\omega}\) represents the angular velocity. \(\mathbf{r}_{i}\) is the position vector from \(o_{i}\) to \(P_{i}\). \(\mathbf{F}_{r}\) and \(\mathbf{M}_{r}\) represent external force and torque acting as the end-effector, respectively. For ease of analysis and processing, the Equation (1) could be rewritten as follows. : \[\mathbf{M}\dot{\mathbf{x}}+\mathbf{V}\mathbf{x}+\mathbf{G}=\mathbf{J}\mathbf{T}+\mathbf{E}_{r}\] (2) where \(\mathbf{M}\), \(\mathbf{V}\), and \(\mathbf{G}\) are the inertia term, Coriolis term, and gravity term of the end-effector. \(\mathbf{J}\), \(\mathbf{T}\), and \(\mathbf{E}_{r}\) represent the generalized moment arm matrix, tension matrix, and generalized external force/torque matrix. \(\mathbf{x}\) is the generalized velocity of the end-effector. The specific expression of them is as follows: \[\mathbf{M}= \begin{bmatrix}m\mathbf{E}_{3\times 3}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&\mathbf{I}\end{bmatrix} \in\mathbb{R}^{6\times 6},\mathbf{V}= \begin{bmatrix}\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&-(\mathbf{I}\mathbf{\omega})^{\times}\end{bmatrix} \in\mathbb{R}^{6 \times 6},\mathbf{G}= \begin{bmatrix}\mathbf{G}_{e}\\ \mathbf{0}_{3\times 3}\end{bmatrix} \in\mathbb{R}^{6\times 1}\] (3) \[\mathbf{J}= \begin{bmatrix}\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3 \times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3 \times 3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\ \mathbf{r}_{1}^{\times}&\mathbf{r}_{2}^{\times}&\mathbf{r}_{3}^{\times}&\mathbf{r}_{4}^{ \times}&\mathbf{r}_{5}^{\times}&\mathbf{r}_{6}^{\times}&\mathbf{r}_{7}^{\times}&\mathbf{r}_{8}^ {\times}\end{bmatrix} \in\mathbb{R}^{6\times 24}\] (4) \[\mathbf{T}= \begin{bmatrix}T_{1}^{T}&\mathbf{T}_{2}^{T}&\mathbf{T}_{3}^{T}&\mathbf{T}_{4}^{T}& \mathbf{T}_{5}^{T}&\mathbf{T}_{6}^{T}&\mathbf{T}_{7}^{T}&\mathbf{T}_{8}^{T}\end{bmatrix} ^{T} \in\mathbb{R}^{24\times 1}\] (5) \[\mathbf{E}_{r}= \begin{bmatrix}\mathbf{F}_{r}\\ \mathbf{M}_{r}\end{bmatrix}