The complex, deformable, and dynamic cardiovascular system makes precise control of flexible medical instruments a challenging task. An innovative robot-assisted catheterization system (RACS) based on braided sleeves was thus developed to aid interventionalists during cardiovascular procedures. The RACS allows navigating instruments through the vasculature with continuous uninterrupted motion. The complete design and characterization of the developed RACS are presented and experimentally evaluated. The RACS' capability to track dynamic motion is demonstrated. Hereto, heart motion is estimated and fed into a dedicated motion controller. It is further demonstrated that tissue tracking performance can be improved by compensating for hysteresis present along the instrument's length and incorporating a rate-dependent Bouc-Wen model in the motion control strategy. Here, a practical method based on multi-core FBG-inscribed fiber technology is presented for continuous instrument pose tracking. A velocity controller is cascaded with a force control loop to enable instrument tip contact force tracking. An experimental comparison between three different force control strategies was carried out on a benchtop laboratory setup resembling a dynamically moving heart wall. Results confirmed the superior performance of the cascaded force-velocity controller with Bouc-Wen hysteresis compensation. Finally, the RACS system was validated through in-vivo experiments on a living swine which demonstrated its ability to successfully navigate instruments toward the heart.