## 4 Workspace Analysis
Moreover, when the calculated cable tension values exceeded the limited value, the end-effector at these positions was eliminated from the workspace domain and appeared vacant in the figures. The simulation results of the wrench-feasible workspace considering nonlinear tension are shown in Figures 7 and 8. In the case of the end-effector moves in the X, Y, and Z directions, the figures show the maximum cable tension distribution of the eight cables when applying various accelerations for a given position motion of the end-effector. As shown in these figures, the average cable tension was \(\sim\)100 N at an acceleration of 10 m/s\({}^{2}\) and \(\sim\)200 N at an acceleration of 100 m/s\({}^{2}\). Moreover, the total maximum tensions increased significantly from 600 N to 1000 N when the acceleration varied from 10 m/s\({}^{2}\) to 100 m/s\({}^{2}\). In addition, maximum tension values appeared at the vertices and edges of the workspace. The negative cable tension disappeared in the moving direction due to the constraint of the optimal method. Considering the case of X-direction acceleration (Figure 7a,b, and Figure 8a,b), the same directions between the front view and back view surface were compared, and the tension distribution of the back view surface was slightly higher than that in the opposite surface. The tension region had a C shape, with the edges created by the contiguous faces, and the surrounding area had a higher tension distribution than that of the front view surface. This tension distribution was similar for both cases of X-direction acceleration. In particular, the vertical edge was the edge created between the back view surface of the X-direction and the front view surface of the Y-direction, which had the highest tension distribution in this position because of the optimal method and the variable cable length; thus, the cable was expected to maintain a positive tension for the acceleration in the X-direction--the greater the acceleration, the greater the tension distribution in this region. This finding means that the CDPR system is restricted from working in this area and the tension must be decreased so that the wrench-feasible workspace can be improved. Similarly, the range of acceleration in the Y-direction (Figure 7c,d, and Figure 8c,d) and the surface of the front view of the Y-direction showed a higher tension distribution compared to the opposite surface in the same direction of movement. Although the applied acceleration was different, the tension distribution on the faces in the same direction was similar. In addition, the tension distribution in the vicinities of the edges on these surfaces had high tension values because the cable maintained a positive tension for Y-direction acceleration. In particular, a high-tension distribution phenomenon occurred at the bottom edges on these surfaces. Here, as the acceleration increased, the tension on the line increased. In particular, in the case of 100 m/s\({}^{2}\), the tension distribution reached the maximum value; therefore, the CDPR system should be restricted from working in this region. Overall, neither tension distribution case exceeded the allowable tension of 4000 N. Figure 9a,b show the quarter section of the wrench-feasible workspace in the simulation results, and they provide detailed information on the internal workspace and show that both quarter sections were filled with blue boxes. The cable tension distribution in the interior of the workspace is shown to be relatively low, with an average value of approximately 100 N. Therefore, these results were desirable. Additionally, Figure 9c,d show the workspace results at Z = 0.2 m and Z = 0.4 m, respectively. It is obvious that the blue boxes comprised the bulk of the internal workspace and the tension was similar in these two diagrams. However, the high-tension distribution appeared at the corner of its slice and the vicinities of this position.