Whole-Angle MEMS Gyroscopes. Doruk Senkal

separation frequency leads to higher transducer sensitivity, while the mechanical bandwidth of the sensor is typically limited by drive‐sense separation (

).

      Nondegenerate mode gyroscopes are typically operated using open‐loop mechanization. In open‐loop mechanization, “drive” mode oscillation is sustained via a positive feedback loop. The amplitidue of “drive” mode oscillations are controlled via the so‐called Amplitude Gain Control (AGC) loop. No feedback loop is employed on the “sense” mode, which leaves “sense” mode proof mass free to oscillate in response to the angular rate input.

      1.1.2 Degenerate Mode Gyroscopes

      Degenerate mode gyroscopes utilize two symmetric modes for detecting angular rotation. For an ideal degenerate mode gyroscope, these two modes have identical stiffness and damping; for this reason typically an axisymmetric or

–

symmetric structure is used, such as a ring, disk, wineglass, etc. Degenerate mode gyroscopes are commonly employed in two primary modes of instrumentation: (i) force‐to‐rebalance (FTR) (rate) mechanization and (ii) whole‐angle mechanization.

In FTR mechanization, an external force is applied to the vibratory element that is equal and opposite to the Coriolis force being generated. This is a rate measuring gyroscope implementation, where the magnitude of externally applied force can be used to detect angular velocity. The main benefit of this mode of operation is to boost the mechanical bandwidth of the resonator, which would otherwise be limited by the close to zero drive‐sense separation (

) of the degenerate mode gyroscope.

      In the whole‐angle mechanization, the two modes of the gyroscope are allowed to freely oscillate and external forcing is only applied to null the effects of imperfections such as damping and asymmetry. In this mode of operation the mechanical element acts as a “mechanical integrator” of angular velocity, resulting in an angle measuring gyroscope, also known as a Rate Integrating Gyroscope (RIG).

Whole‐angle gyroscope architectures can be divided into three main categories based on the geometry of the resonator element: (i) lumped mass systems, (ii) ring/disk systems, and (iii) micro‐wineglasses. Ring/disk systems are further divided into three categories: (i) rings, (ii) concentric ring systems, and (iii) disks. Whereas, micro‐wineglasses are divided into two categories according to fabrication technology: surface micro‐machined and bulk micro‐machined wineglass gyroscope architectures, Figure 1.2 [2].

      String and bar resonators can also be instrumented to be used as whole‐angle gyroscopes, even though these types of mechanical elements are typically not used at micro‐scale due to limited transduction capacity. In principle, any axisymmetric elastic member can be instrumented to function as a whole‐angle gyroscope.

1.2 Generalized CVG Errors

      Gyroscopes are susceptible to a variety of error sources caused by a combination of inherent physical processes as well as external disturbances induced by the environment.

      Error sources in a single axis rate gyroscope can be generalized according to the following formula:

      (1.1)

where

is the measured gyroscope output,

is scale factor error,

is bias error, and

is noise. Without loss of generality, for a whole‐angle gyroscope the error sources can be written as:

      (1.2)

Figure 1.2 Micro‐rate integrating gyroscope (MRIG) architectures.

where

is the measured gyroscope output, corresponding to total angular read‐out, including the actual angle of rotation, errors in scale factor, bias, and noise.

      1.2.1 Scale Factor Errors

      Scale factor (or sensitivity) errors represent a deviation in gyroscope sensitivity from expected values, which results in a nonunity gain between “true” angular rate and “perceived” angular rate. Scale factor errors can be caused by either an error in initial scale factor calibration or a drift in scale factor postcalibration due to a change in environmental conditions, such as a change in temperature or supply voltages, application of external mechanical stresses to the sensing element, or aging effects internal to the sensor, such as a change in cavity pressure of the vacuum packaged sensing element.

      1.2.2 Bias Errors

      Bias (or offset) errors can be summarized as the deviation of time averaged gyroscope output from zero when there is no angular rate input to the sensor. Aside from initial calibration errors, bias errors can be caused by a change in environment conditions. Examples include a change in temperature, supply voltages or cavity pressure, aging of materials, and application of external mechanical stresses to the sensing element. An additional source of bias errors is external body loads, such as quasi‐static acceleration, as well as vibration.

      1.2.3 Noise Processes

      Noise in gyroscopes can be grouped under white noise, flicker (

) noise, and quantization noise. The most common numerical tool for representing gyroscope noise processes is Allan Variance.

      1.2.3.1 Allan Variance

      Originally created to analyze frequency stability of clocks and oscillators, Allan Variance analysis is also widely used to represent various noise processes present in inertial sensors, such as gyroscopes [3]. Allan Variance analysis consists of data acquisition of gyroscope output over a period of time at zero rate input and constant temperature. This is followed by binning the data into groups of different integration times:

      (1.3)

where

is the sampling time,

is the sample number, and

is the bin size. The uncertainty between bins of same integration times is calculated using ensemble average:

      (1.4)

Finally, the calculated uncertainty

with respect to integration time (