Mitral regurgitation (MR) is one of the most common valvular abnormalities, and the gold-standard for treatment is surgical mitral valve repair/replacement. Most patients with severe MR are over the age of 75, which makes open-heart surgery challenging. Thus, minimally invasive surgeries using transcatheter approaches are gaining popularity. This paper proposes the next generation of a robotic transcatheter delivery system for the mitral valve implant that focuses on the design of the actuation system, modeling, and task space control. The proposed actuation system is compact while still enabling bidirectional torsion, bending, and prismatic joint motion. A pulley structure is employed to actuate the torsion and bending joints using only one motor per joint in conjunction with an antagonistic passive spring to reduce tendon slack. The robotic transcatheter is also optimized to increase its stability and reduce bending deflection. An inverse kinematics model (with an optimization algorithm), singularity analysis method, and joint hysteresis and compensation model are developed and verified. Finally, a task space controller is also proposed. Experiments, including trajectory tracking and demonstrations of the robot motion in an ex vivo porcine heart and a phantom heart through a tortuous path are presented.