# Investigation on the Tip Positioning Accuracy of Cable-Driven Serpentine Manipulators
## 3 Simulation Analysis and Experimental Validation
### Simulation Analysis  
Taking the proposed manipulator model in Section 3.1 as the research object, the factors affecting the tip positioning accuracy of the manipulator were analyzed by adopting the methods proposed in Section 2.3. In order to simplify the analysis, the end-effector attitude of the manipulator was set as zero; that is, the end-effector attitude always kept parallel to the ground. We analyzed the influence of the cable hole location error and cable length error on the tip positioning accuracy in the manipulator's workspace. Due to this, simulations were performed by taking a 3-DOF planar CDSM as an example; that is, there were 3 joints in total. The minimum joint angle was \(-30^{\circ}\) and the maximum joint angle was \(30^{\circ}\). Every joint moved from \(-30^{\circ}\) to \(30^{\circ}\) with the step size of \(5^{\circ}\). Multiple sets of joint angles were obtained through three nested loops, the corresponding tip positions of those joint angles form the workspace. Firstly, we analyzed the influence of the cable length error:  
* Add \(\pm 1\) mm error to the first cable;
* Add \(\pm 1\) mm error to the third cable;
* Add \(\pm 1\) mm error to the fifth cable,  
where adding \(+1\) mm error meant that the cable length was lengthened by \(1\) mm, and adding \(-1\) mm error meant that the cable length was shortened by \(1\) mm. Next, we analyzed the influence of the cable hole location error:  
* Add \(\pm 0.35\) mm error to the first disk's holes;
* Add \(\pm 0.35\) mm error to the second disk's holes;
* Add \(\pm 0.35\) mm error to the third disk's holes,  
where adding \(+0.35\) mm error meant that the cable hole height was increased by \(0.35\) mm, and adding \(-0.35\) mm error meant that the cable hole height was reduced by \(0.35\) mm. Figure 5 shows the results of adding the cable length error. Figure 6 shows the results of adding the cable hole location error. It should be noted that the points in the pictures were obtained by traversal of joint angles; that is, a combination of the joint angles varying from the minimum to the maximum. These points almost covered the entire workspace of the manipulator's tip. Figure 5: _Cont._Figure 5: The influence of the cable length error on the tip positioning accuracy: **(a)** Adding \(-1\) mm error to the first cable; **(b)** adding \(+1\) mm error to the first cable; **(c)** adding \(-1\) mm error to the third cable; **(d)** adding \(+1\) mm error to the third cable; **(e)** adding \(-1\) mm error to the fifth cable; **(f)** adding \(+1\) mm error to the fifth cable. Figure 6: _Cont._  
Figure 6: The influence of the cable hole location error on the tip positioning accuracy: (**a**) Adding \(-0.35\) mm error to the first disk’s holes; (**b**) adding \(+0.35\) mm error to the first disk’s holes; (**c**) adding \(-0.35\) mm error to the second disk’s holes; (**d**) adding \(+0.35\) mm error to the second disk’s holes; (**e**) adding \(-0.35\) mm error to the third disk’s holes; (**f**) adding \(+0.35\) mm error to the third disk’s holes. The red color represents the influence of the cable length error on the tip in the X direction; blue color represents the influence on the tip in the Y direction; green color represents the influence on the tip attitude. From the above 36 figures, the following conclusions can be obtained:  
1. When we added \(-\)1 mm error to the first cable/+1 mm error to the third cable/\(-\)0.35 mm error to the first disk's holes/\(-\)0.35 mm error to the second disk's holes, it generated jump points (i.e., the variation tendency about deviation of the end pose changed at these points);
2. When we added \(+\)1 mm error to the first cable/\(-\)1 mm error to the third cable/+0.35 mm error to the first disk's holes/+0.35 mm error to the second disk's holes, from both ends of the \(Y\)-axis to \(Y=0\), the deviation of the end pose gradually increased and reached the maximum value nearby \(Y=0\);
3.