Minimally invasive surgery (MIS) during cardiovascular interventions reduces trauma and enables the treatment of high-risk patients who were initially denied surgery. However, restricted access, reduced visibility and control of the instrument at the treatment locations limits the performance and capabilities of such interventions during MIS. Therefore, the demand for technology such as steerable sheaths or catheters that assist the clinician during the procedure is increasing. In this study, we present and evaluate a robotically actuated delivery sheath (RADS) capable of autonomously and accurately compensating for beating heart motions by using a model-predictive control (MPC) strategy. We develop kinematic models and present online ultrasound segmentation of the RADS that are integrated with the MPC strategy. As a case study, we use pre-operative ultrasound images from a patient to extract motion profiles of the aortic heart valve (AHV). This allows the MPC strategy to anticipate for AHV motions. Further, mechanical hysteresis in the steering mechanism is compensated for in order to improve tip positioning accuracy. The novel integrated system is capable of controlling the articulating tip of the RADS to assist the clinician during cardiovascular surgery. Experiments demonstrate that the RADS follows the AHV motion with a mean positioning error of 1.68 mm. The presented modelling, imaging and control framework could be adapted and applied to a range of continuum-style robots and catheters for various cardiovascular interventions.