# Dynamic Modeling and Optimization of Tension Distribution for a Cable-Driven Parallel Robot
## 5 Simulation Study
### The Simulation for the End-Effector's Motion  
Here, a spatial arc trajectory, as shown in Figure 9, is designed for the end-effector of the cable-driven parallel robot to trace. The arc is determined by \(\mathbf{O}_{c}\), \(\mathbf{R}_{c}\), and \(\mathbf{n}_{c}\) three parameters. \(\mathbf{O}_{c}\)is the center of the arc. \(\mathbf{R}_{c}\) is the vector from the center to the initial point of the arc. \(\mathbf{n}_{c}\) is the normal vector of the plane where the arc is. As a consequence, for any point on the arc, its position vector is obtained as follows:  
\[\mathbf{P}_{c}=\mathbf{O}_{c}+e^{\eta\cdot\mathbf{n}_{c}^{\times}}\cdot\mathbf{R}_{0} \tag{34}\]  
\[e^{\eta\cdot\mathbf{n}_{c}^{\times}}=\mathbf{I}+\sin(\eta)\cdot\mathbf{n}_{c}^{\times}+(1- \cos(\eta))\cdot\mathbf{n}_{c}^{\times}\cdot\mathbf{n}_{c}^{\times} \tag{35}\]  
where \(\eta\) is the corresponding central angle of \(\mathbf{P}_{c}\). Figure 8: The simulation model of the cableriven parallel robot. Figure 9: Simulation trajectory of the end-effector. In the simulation, the parameters of the spatial arc are set as follows:  
\[\left\{\begin{array}{l} \begin{array}{l}\mathbf{O}_{c}=\left[\begin{array}{cc}0.0 \text{m}&0.2\text{m}&0.5\text{m}\\ \end{array}\right]^{T}\\ \mathbf{R}_{0}=\left[\begin{array}{cc}0.0\text{m}&0.0\text{m}&0.2\text{m}\\ \end{array}\right]^{T}\\ \mathbf{n}_{c}=\left[\begin{array}{cc}0&0&1\\ \end{array}\right]^{T},\eta\in\left[0^{\circ},\ 360^{\circ}\right]\end{array} \right. \tag{36}\]  
Combining Equation (34) and Equation (36), the values of \(\mathbf{P}_{c}\) in \(x,y,\) and \(z\) are obtained as follows:  
\[\left\{ \begin{array}{l}\begin{array}{l}P_{cx}=0.2\cos(\omega t)+0\\ P_{cy}=0.2\sin(\omega t)+0.2\\ P_{cz}=0.5\end{array}\right.\end{array} \right. \tag{37}\]  
Here, the simulation step size is set as 0.1 s, and the simulation time is 12 s. According to the simulation results, some states of the end-effector at different times during the simulation are shown in Figure 10. As illustrated in Figure 11, it represents the included angle \(\theta_{i}\) and \(\phi_{i}\) for each cable. Figure 10: Simulated motion trajectory of the end-effector. (Red represents cable; Green represents the simulation trajectory of the end-effector; Blue represents the end effector; Black represents the robot frame). Figure 11: _Cont._  
Figure 10 shows the movement trajectories of the end-effector at different time points. These six subgraphs show changes in the position of the end effector from 0 s to 12 s. Through the observation of these trajectories, it can be seen that the end effector maintains a smooth and continuous trajectory during the whole motion, which indicates the effectiveness of the optimization algorithm. Figure 11 shows the changing process of the angle \(\theta_{i}\) and angle \(\phi_{i}\) under the circular trajectory movement. Each subgraph (a-h) corresponds to the change in angle between cables 1 and 8. As shown in Table 4, it is the variation range of the sum of the eight ropes under the circular trajectory. In Figure 11a,b,g,h, that is, in the time range of 0 s to 12 s for cable 1, cable 2, cable 7, and cable 8, the variation range of angle \(\theta_{i}\) and angle \(\phi_{i}\) is large, indicating that the cable needs to make a large attitude adjustment to adapt to the trajectory motion during the entire motion process. Among them, the extreme value of angle \(\theta_{i}\) of cable 1 appears near 3 s and 9 s, the extreme value of angle \(\phi_{i}\) appears near 2 s and 8 s, the extreme value of angle \(\theta_{i}\) of cable 2 appears near 3 s and 9 s, the extreme value of angle \(\phi_{i}\) of cable 7 appears near 3 s and 9 s, and the extreme value of angle \(\phi_{i}\) appears near 1 s and 7 s. The extreme value of the angle \(\theta_{i}\) of cable 8 appears near 3 s and 9 s, and the extreme value of the angle \(\phi_{i}\) appears near 2 s and 8 s. In Figure 11e,f, the change range of angle \(\theta_{i}\) and angle \(\phi_{i}\) of cable 5 and cable 6 from 0 s to 12 s is relatively gentle.