# Investigation on the Tip Positioning Accuracy of Cable-Driven Serpentine Manipulators
## 3 Simulation Analysis and Experimental Validation
### Experimental Validation  
According to the experimental prototype shown in Figure 8, we set the experimental conditions as follows:  
* Joint angles: \(-35^{\circ}\leq\theta\leq 35^{\circ}\);
* External forces: 0 N; 0 N; 0 Nm;
* Lower cable tensions(weights): 50; 10; 5 g;
* Cable length error: \(-1\) mm. In order to avoid interference between cables and links, we set variation range of joint angles. It should be noted that the lower cables were in the force control mode, and when the lower cable tensions changed, the upper cable tensions changed; thus, we had to guarantee that cable tensions were all positive. In this simulation example, we added \(-1\) mm error to the first cable, because the first cable was shorter than the third and fifth cables. The same cable length error would cause larger deviation on short cable, which was convenient to measure. Too small deviation resulted in inaccurate results due to the measurement and systematic errors. Compared to the cable length, 1 mm was a reasonable error value. Adding negative error meant that the cable length was shortened, which was convenient in order to adjust the cable length. Figure 7: The influence of the random composite error on the tip positioning accuracy. **(a)** The influence of the cable length error on the tip in the X direction; **(b)** the influence of the cable length error on the tip in the Y direction; **(c)** the influence of the cable length error on the tip in the phi direction. \begin{table}
\begin{tabular}{c c c} \hline \hline
**End Pose** & **Conditions** & **Comparison of Schematic Diagram of Manipulator** \\ \hline At jump point: & At jump point: & At jump point: \\ Add 1 mm error to the 1 st cable; & Add 1 mm error to the 1 st cable; & At jump point: \\ Add -1 mm error to the 3 rd cable; & At jump point: & At jump point point: \\ Add 0.35 mm error to the 2 nd disk’s holes. & At jump point point:  
As shown in Figure 8, joints were connected by bearing and needle roller. It is noted that the friction in this system was minimized by using bearing, pulley, and lubricating grease. The position control device was composed of a retaining bracket with a screwed hole and three screws; we could adjust the cable length by rotating the screws. The force control device was composed of a pulley block and three standard weights (the standard weights hung on the second, fourth, and sixth cables were 5, 10, and 50 g, respectively); we could change the cable tension by changing the number of weights. For the cables in the position control mode, one end of the cable was tied to the disk, and the other end of the cable was driven by the screw, because the cable length was adjusted by rotating the screw. For the cables in the force control mode, one end of the cable was tied to the disk, and the other end of the cable was driven by standard weights, because the cable tension was applied by the weight bypassing the pulley. As shown in Figure 8, if we put the force control cables together (the lower cables), we could make a pulley block to control it, which simplified the prototype design and was convenient to add weights. If we put the position control cables together (the upper cables), we could concentrate the screwed holes on a device to control cable length. The location of the tip was set at the center of the target ball. A laser tracker (API Radian Plus), tracking the target ball on the tip of manipulator, was used to measure the tip position. Three groups of steel cables (0.18 mm in diameter) with tensile structure were used as driving cables. The Young's modulus of steel cables (0.18 mm in diameter) used in the experiments was approximately \(1.8\times 10^{11}\) Pa. Under the maximum external force (50 g), the cable elongation of with length of about 230 mm used in the experiments was approximately 0.0234 mm. Hence, the cable elongation could be very small and negligible.