The currently available robotic systems rely on rigid heart stabilizers to perform totally endoscopic coronary artery bypass (TECAB) surgery on beating hearts. Although such stabilizers facilitate the anastomosis procedure by immobilizing the heart and holding the surgery site steady, they can cause damage to the heart tissue and rupture of the capillary vessels, due to applying relatively large pressures on the epicardium. In this paper, we propose an advanced robotic approach to perform TECAB on a beating heart with minimal invasiveness. The idea comes from the fact that the main pulsations of the heart occur as excursions in normal direction, i.e., perpendicular to the heart surface. We devise a 1-DOF flexible heart stabilizer which eliminates the lateral movements of the heart, and a 1-DOF compensator mechanism which follows the heart trajectory in the normal direction, thus canceling the relative motion between the surgical tool and the heart surface. In fact, we bring a compromise between two radical approaches of operating on a completely immobilized beating heart with no heart motion compensation, and operating on a freely beating heart with full compensation of heart motion, considering the invasiveness of the first and the technical challenges of the second approach. We propose operating on a partially stabilized beating heart with unidirectional compensation of the heart motion; the flexible stabilizer would exert much less holding force to the heart tissue and the robotic system with unidirectional compensator would be technically feasible. In the proposed approach, a motion sensor mounted on the stabilizer measures the heart excursion data and sends it into a control unit. A predictive controller uses this data to generate an automated trajectory. The slave robots follow this trajectory, which is superimposed on the surgeon's tele-operation commands received from a master console. Finally, the tool-activation units in the slave robots actuate the articulated laparoscopic tools to perform the anastomosis procedure. The evaluation of the hypothesis showed that our solution for the robotic TECAB on beating heart is both practical and cost effective. We showed in an in-vivo study that the flexible stabilizer can effectively restrict the heart lateral movements, while allowing for its normal excursion. We found readily available linear motors which could afford the high forces, speeds and accelerations required for following the heart trajectory. Finally, we showed that the tool-activation unit is capable of providing the maneuverability and workspace required for the most challenging task of CABG procedure, i.e., anastomosis suturing.