Introduction Magnetic Resonance Imaging has better resolution for soft tissue; at the same time, the robot can work in a stable manner for a long time. MRI image-guided breast interventional robots have attracted much attention due to their minimally invasive nature and accuracy. In this paper, a hydraulic-driven MRI-compatible breast interventional robot is proposed to perform breast interventional procedure. Methods First is the analysis of the design requirements of the hydraulic-driven MRI-compatible breast interventional robot, and then the design scheme is determined. Second, the three-dimensional model and the link frames are established. The workspace of the robot end point is solved by MATLAB/Simulink software. Then, the 3D printing technology is used to make a physical model of the MRI-compatible breast interventional robot. After assembly and debugging, the physical model is used for workspace verification, and the simulation result of the workspace shows that it is correct. Finally, the experimental research on the positioning error of the hydraulic drive is carried out, which established the theoretical foundation for the follow-up control research of the robot. Results The positioning error has nothing to do with the motion distance, speed, and length of the selected tubing. The errors are 0.564 mm, 0.534 mm, and 0.533 mm at different distances of 40 mm, 80 mm, and 120 mm, respectively. The errors are 0.552 mm, 0.564 mm, and 0.559 mm at different speeds of 3 mm/s, 5 mm/s, and 8 mm/s, respectively. The errors are 0.564 mm, 0.568 mm, and 0.548 mm for different lengths of 0.5 m, 1 m, and 1.6 m, respectively. Then, the robot's working space on the XOZ plane and the XOY plane meets the conditions. Conclusion The structure of a differential rotary breast interventional robot is determined, with the link frames assigned to the mechanism and the Denavit-Hartenberg parameters given. Workspace simulation of MRI-compatible breast interventional robot is done in MATLAB. The 3D printed MRI-compatible breast interventional robot is assembled and debugged to verify that its working space and positioning error meet the requirements.