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This paper describes the design of the control loops in a z-axis, MEMS vibrational gyroscope operating in a vacuum enclosure. In this device, a silicon mass is driven through electrostatic actuator so that it has a sinusoidal linear motion, with a controlled speed. The design of a suitable controller, capable of maintaining the required speed and with prescribed restoring capabilities after shocks is briefly described in the paper. Attached to the driving mass, a second mass, free to move in the direction orthogonal to the motion of the first mass, is subjected to a Coriolis force, proportional to the product of the first mass speed by z-axis rotational speed. The sensing of the Coriolis force and, in turn, of the z-axis rotational speed, is performed in closed loop fashion, with a 1-bit quantized actuation. The restoring force that brings the motion of the second mass to zero is equivalent to the output bit stream of a band-pass sigma-delta converter and contains the information of the Coriolis force. The design of this second control loop and a detailed analysis on the signal-to-noise ratio achievable with the proposed design is reported.