A robotic system and methods for performing spine surgery are disclosed. The system comprises a robotic manipulator with a tool to hold a screw and to rotate the screw about a rotational axis. The screw is self-tapping and has a known thread geometry that is stored by a controller. A navigation system tracks a position of a target site. Movement of the robotic manipulator is controlled to maintain the rotational axis of the surgical tool along a planned trajectory with respect to the target site based on the tracked position of the target site. In autonomous or manual modes of operation, the rotational rate of the screw about the rotational axis and/or an advancement rate of the screw linearly along the planned trajectory is controlled to be proportional to the known thread geometry stored in the memory.