## 3 Characterization of Position-Tracking Error in Robotic Catheter Systems
### Experimental Observation of Master-Slave Position and Velocity under Uniform Translational Motion  
The motion lag study was also performed for position and velocity under uniform translational motion. For this, uniform translational motion in steps of 1-10 mm was issued to the slave robot to quantify the tracking error. In addition, each trial was performed five times for each displacement step value to trigger navigation, while the robotic devices' mean position and velocity output were recorded. In Figure 7A-C, we present the distance traveled by both master and slave platforms when displacement values of 3, 5, and 10 mm were, respectively, set. In Figure 7A, the slave device closely matched the displacement of the master platform for a translation step of 3 mm. In contrast, the slave device response was below the input command per step when actuated with a uniform displacement of 5 mm, as shown in Figure 7B. However, above 5 mm, a more visible gap exists such that it extends to the proportion shown in Figure 7C when operated under a displacement step value of 10 mm. Overall, the accruing error influences the resultant slave positions, as shown in Figure 7B,C.  
In addition, the velocity trajectories of the master and slave platforms for 5 and 10 mm are presented in Figure 8A,B, respectively. In Figure 8A, the variation of the slave robot's velocity was within the threshold of the motion gap observed in Figure 7B. This could infer that the slave's position tracking is smooth for small displacement step values (\(\leq\) 3 mm). However, limiting an operator to this axial displacement range during catheterization could increase the procedural duration and induce boredom in interventionalists, especially when navigating through vessels with a larger diameter. This includes endovascular tool movement from within the radial vessels to the aorta. However, smaller displacement step values are essential for safe tool movement beyond the aorta and within the coronary arteries. Furthermore, this could allow the operator to choose between different displace  
Figure 7: Master–slave position trajectories based on uniform displacement values of (**A**) 3 mm, (**B**) 5 mm, and (**C**) 10 mm.  
ment step values depending on the vessel diameter and the catheterization path while a steady translational motion could be achieved.  
Figure 8B shows a noticeable difference when the slave robot actuates under a higher velocity, such as applying \(>\)30 mm/s. Similarly, this correlates with the evident position-tracking error in Figure 7C. Thus, it is remarked that a velocity of 16 mm/s (or below) at the master side could suitably maintain a minimal position error during master-slave robotic catheterization. Comparing Figures 5-8, our study suggests that both position and velocity contribute to the hysteresis observed at the slave-side robotic device during catheterization. In conclusion, Figures 5-8 indicate that the nonlinearities and uncertainties could be higher and unbounded when applying variable displacement values. This typical scenario usually happens during robot-assisted tool catheterization. We, therefore, proposed a neural-based system for learning the dynamics of the master-slave setup to aptly determine and compensate for position error occurrences in the RCS.