Capsulotomy in cataract surgery is delicate and susceptible to surgeon fatigue, limited perception accuracy, and physiological hand tremors. To aid cataract surgery, we propose a novel vision-guided hybrid robot system. The robot, consisting of a PPRR/PPRRP parallel unit and an RP serial unit (R: rotation pair, P: prismatic pair), adopts a parallel configuration as the base. Capsulotomy has high requirements for position accuracy, thus, a robot control algorithm with the RCM constraint is proposed. This algorithm reduces the position error of the robot by limiting the position error within tolerance, thereby increasing the motion accuracy and smoothing the trajectory. Instead of obtaining position information based on the visual perception of the surgeons, we propose an edge detection method based on the U-Net to provide visual feedback to the robot.Experimental validations on a standard circle and ex-vivo pig eyeballs are performed to verify the effectiveness of the proposed robot control algorithm with the RCM constraint and the accuracy of the proposed robot system. When a circular trajectory of 5 mm is tracked, the maximum position error of the actuator end point is 0.084 mm and that of the RCM point is 0.292 mm. When the edge of the anterior capsule is tracked, the mean tracking error of the robot is 0.129 mm and that of the doctor is 0.764 mm.