Brain tumors are a major problem affecting the health status of the nation. To determine the extent of brain tumor in order to determine the next step in treatment, a puncture biopsy procedure of brain tumor tissue is often required. Magnetic resonance imaging (MRI) is commonly used to detect brain tumors due to its better soft-tissue resolution. Therefore, research related to MRI-compatible robots is necessary. Based on an MRI-compatible hydraulically driven puncture surgery robot, the kinematic model of the robot is derived based on the principle of hydraulic linker, and the dynamic model of the robot is obtained based on the relevant theory of fluid dynamics. In order to realize the accurate control of the designed robot system, the state feedback $H_{\mathrm{\infty }}$ control rate of this hydraulically driven system is designed according to the $H_{\mathrm{\infty }}$ control theory, which enables the robot to track the target signal quickly and stably. Finally, the average positioning accuracy of the designed robot system in x-axis, y-axis, z-axis, pitch-axis and roll-axis are obtained through experimental studies, and they are, respectively, 0.41 mm, 0.6 mm, 0.67 mm, 0.886° and 1.17°. Experimental results verify the performance of robot-assisted positioning puncture needles, and the dynamic model and control method of the robot have been given, which provide certain reference value for the research of the control algorithm of puncture robots.