# Investigation on the Tip Positioning Accuracy of Cable-Driven Serpentine Manipulators
## 2 Kinematic Model and Position Accuracy Analysis
### Influence Factors of Tip Positioning Accuracy  
The control methods of the CDSM included all cables in the position control mode, all cables in the force control mode, and cables in a hybrid control mode (i.e., some cables in the position control mode and others in the force control mode.). When all cables were in the position control mode, the driving cables could be overly tensioned or relaxed. Additionally, with the number of cables increases, it was more difficult to control. When all cables were in the force control mode, the control was simple, but it was hard to ensure the tip positioning accuracy. Therefore, we chose a hybrid control mode, which could ensure the tip positioning accuracy and reduce the control difficulty as well. In this paper, for the experiment's convenience, we chose for the upper cables (\(L_{ci\_1}\)) to adopt the position control mode, and the force control mode was applied to the lower cables (\(L_{ci\_2}\)). In fact, each cable could use the position control mode or force control mode.  
However, in the practical control process, it was unavoidable to produce control error. For example, between the actual cable length and theoretical cable length existed small difference. In addition, the driving cable passing through the guiding hole on disk may not have been along the center axis of the hole. All these factors affected the tip positioning accuracy of the manipulator.  
This paper studied the influence factors of the tip positioning accuracy in CDSMs' workspace; that is, how the cable length error and cable hole location error to affect the end pose. The processes that study the influence of cable length error, the cable hole location error, and the combined error of two on the tip positioning accuracy are illustrated in Figure 3.  
Figure 3a shows the processes that study the influence of the cable length error. Given the initial pose S\({}_{0}\), we could calculate the initial joint angle \(\theta_{0}\) and initial cable length L\({}_{0}\). When the cable lengths stayed the same (L = L\({}_{0}\)), we added the cable hole location error \(\Delta\)H (i.e., changing the location of the holes A\({}_{i\_1}\), A\({}_{i\_2}\), B\({}_{i\_1}\), B\({}_{i\_2}\)), which influenced the B\({}_{i-1\_1}\)A\({}_{i\_1}\) and B\({}_{i-1\_2}\)A\({}_{i\_2}\) shown in Equations (4) and (5). Then we could obtain a new joint angle \(\theta\) and new pose S by L and \(\Delta\)H. \(\Delta\)S was the influence of the cable hole location error on the tip positioning accuracy.  
Figure 3c shows the processes that study the influence of the combined error of cable length and hole location. Given the initial pose S\({}_{0}\), we could calculate the initial joint angle \(\theta_{0}\) and initial cable length L\({}_{0}\). Then, we added the cable length error \(\Delta\)L and got a new cable length L (L = L\({}_{0}+\Delta\)L). At the same time, we added the cable hole location error \(\Delta\)H to change the B\({}_{i-1\_1}\)A\({}_{i\_1}\) and B\({}_{i-1\_2}\)A\({}_{i\_2}\). Then we  
Figure 3: Flow charts to study the errors. (**a**) The method to study the cable length error; (**b**) the method to study the cable hole location error; (**c**) the method to study the combined cable length and cable hole location errors.  
could obtain the new joint angle \(\theta\) and new pose S by L and \(\Delta\)H. \(\Delta\)S was the influence of the combined error of two on the tip positioning accuracy.  
It should be noted that L in Figure 3b is L = L\({}_{0}\), because when we only studied the cable hole location error, the cable length could not change. However, L in Figure 3c is L = L\({}_{0}\) + \(\Delta\)L, because when we considered the combined error, we needed to change the cable length as well as the cable hole location.