## 3 Characterization of Position-Tracking Error in Robotic Catheter Systems
### Experimental Observation of the Master-Slave Position and Velocity Based on an Operator Hand Motion  
To quantify the occurrence of position-tracking errors during axial translations with our custom RCS, we segmented the master's stroke into several discrete points to have the RCS navigate a well-controlled path length each time. An operator was made to navigate the master through 30, 45, 70, 100, and 120 mm lengths. Each navigation trial was performed five times, and the average displacement data were recorded for the navigation length. In this procedure, the operator randomly moved the cylindrical knob in the master device from a starting position to each discrete point. This triggers the slave robot to translate along the linear path. Ideally, the master-slave positions should align from the start to the end of a given translational motion with an equal magnitude of distance traveled at both ends. However, we observed that distinct position errors existed in the trials. From the statistical analysis outlined in Table 1, the mean and standard deviation of the slave device's position error for each discrete point indicate that the tracking error increases as the length of the path navigated increases. Similarly, based on the operator's hand motion, the average resultant errors (ARE) for 70 and 120 mm were 10.26 and 18.61 mm, respectively, as shown in Figure 5A,C, while the percentage resultant error (ARE/distance) varied from 14.6 to 19.9%.  
The mean velocity of the master-slave devices obtained when distances of 70 and 120 mm were navigated are shown in Figure 6A,B. The interval between the master-slave velocities was greater in Figure 6A. Relating this to Figure 5A, it indicates that the motion gap increases with the velocity gap in the master-slave platforms for varying translational motion but not necessarily with the catheterization distance. Thus, this elucidates why Figure 5A (70 mm) has a more significant motion gap than Figure 5C (120 mm). In addition,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Distance (mm)** & **Error (mm)** & **Resultant Error (\%)** \\ \hline
30 & \(3.84\pm 2.03\) & 19.90 \\
45 & \(6.38\pm 2.30\) & 14.83 \\
70 & \(7.43\pm 3.72\) & 14.66 \\
100 & \(10.67\pm 4.30\) & 15.11 \\
120 & \(13.14\pm 6.11\) & 15.51 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average position error of the slave robotic device for given distances.  
the magnitude of the interval between consecutive points during a catheterization stroke is proportional to the motion gap experienced by the slave robot. Again, we have observed that for a shorter distance (30-70 mm), a novice operator is more likely to move the master knob faster (and at a higher velocity) than for a longer distance (100-120 mm). This could be why fewer motion gaps occurred between the master and slave trajectories when longer distances were moved. Nonetheless, we found that the magnitude of the resultant position error increases with catheterization length.  
Figure 5: Master–slave position trajectories based on an operator’s hand motion for a distance of (**A**) 70 mm, (**B**) 100 mm, and (**C**) 120 mm.  
Figure 6: Master–slave velocity based on an operator’s hand motion for a distance of (**A**) 70 mm and (**B**) 120 mm.