In abdominal surgery, intraoperative bleeding is one of the major complications that affect the outcome of minimally invasive surgical procedures. One of the causes is attributed to accidental damages to arteries or veins, and one of the possible risk factors falls on the surgeon's skills. This paper presents the development and application of an Enhanced Vision System for Robotic Surgery (EnViSoRS), based on a user-defined Safety Volume (SV) tracking to minimize the risk of intraoperative bleeding. It aims at enhancing the surgeon's capabilities by providing Augmented Reality (AR) assistance toward the protection of vessels from injury during the execution of surgical procedures with a robot. The core of the framework consists in (i) a hybrid tracking algorithm (LT-SAT tracker) that robustly follows a user-defined Safety Area (SA) in long term; (ii) a dense soft tissue 3D reconstruction algorithm, necessary for the computation of the SV; (iii) AR features for visualization of the SV to be protected and of a graphical gage indicating the current distance between the instruments and the reconstructed surface. EnViSoRS was integrated with a commercial robotic surgical system (the dVRK system) for testing and validation. The experiments aimed at demonstrating the accuracy, robustness, performance, and usability of EnViSoRS during the execution of a simulated surgical task on a liver phantom. Results show an overall accuracy in accordance with surgical requirements (<5 mm), and high robustness in the computation of the SV in terms of precision and recall of its identification. The optimization strategy implemented to speed up the computational time is also described and evaluated, providing AR features update rate up to 4 fps, without impacting the real-time visualization of the stereo endoscopic video. Finally, qualitative results regarding the system usability indicate that the proposed system integrates well with the commercial surgical robot and has indeed potential to offer useful assistance during real surgeries.