Continuous online self-calibrating resonant FM microelectromechanical systems (MEMS) accelerometer

A self-calibration method for an accelerometer having a proof mass separated by a gap from a drive electrode and a sense electrode includes initializing the accelerometer to resonate, applying a first bias voltage to the sense electrode and a second bias voltage to the drive electrode to obtain a first scale factor, measuring a first acceleration over a first time interval, swapping the first bias voltage on the sense electrode with the second bias voltage previously on the drive electrode and the second bias voltage on the drive electrode with the first bias voltage previously on the sense electrode so that a bias voltage on the sense electrode is set to the second bias voltage and a bias voltage on the drive electrode is set to the second bias voltage to obtain a second scale factor, measuring a second acceleration over a second time interval, and calculating a true acceleration.

STATEMENT REGARDING FEDERAL FUNDING

TECHNICAL FIELD

This disclosure relates to accelerometers and microelectromechanical systems (MEMS).

BACKGROUND

Accelerometers in the prior art have performance limitations due to bias errors. The bias errors may be time varying bias errors also known as bias drifts, which particularly limit performance, because with time varying bias errors the accelerometer cannot be entirely calibrated before use.

Reference [1], below, which is incorporated herein by reference, describes a prior art MEMS accelerometer fabricated on silicon-on-insulator (SOI) wafers or other suitable material using micromachining techniques in combination with an electronic circuit capable of driving the primary in-plane resonance mode of the accelerometer structure into sustained oscillations and frequency modulation (FM) readout of the up-converted inertial acceleration signal, which is mapped onto FM sidebands of the primary resonance oscillation frequency. It should be noted that some of the inventors of the present application are also inventors of the patent application referenced in Reference [1] below. The accelerometer structure is designed such that it has a high quality factor (Q>10,000) primary in-plane resonance mode with a natural frequency greater than 10 kHz. The resonant MEMS accelerometer is sealed in a vacuum package at less than 1 milliTorr to preserve the high Q of the silicon structure. In contrast to static MEMS accelerometers, which are limited to dynamic ranges <106by their amplitude modulation (AM) readout mechanism, the FM mechanism employed in this prior art accelerometer allows dynamic ranges greater than 109. Such a large dynamic range is necessary to enable large input ranges (>±100 g to ±1000 g) while simultaneously preserving the ability to accurately resolve small inertial signals below 1 μg, where 1 g=9.81 m/s2. However, this prior art accelerometer has performance limitations due to time varying bias errors.

Reference [2], below, which is incorporated herein by reference, describes a prior art self-calibration method for an inertial instrument which has two inertial sensor devices. The reference describes an example gyroscope, but also mentions applying the technique to accelerometers. A disadvantage of the described method is that two inertial sensor devices are required.

References [3] and [4], below, which are incorporated herein by reference, describe a resonant MEMS accelerometer consisting of two independent resonators each formed from two coupled masses. This prior art describes a resonant MEMS accelerometer which uses the nonlinearity of a biased capacitive transduction gap to create a frequency shift of a resonance mode of the structure. However, references [3] and [4] do not disclose any method to perform self-calibration.

References [5] and [6], below, which are incorporated herein by reference, describe a typical approach to resonant MEMS accelerometers in which a mechanically-induced frequency shift is the main sensing mechanism.

References [7] and [8], below, which are incorporated herein by reference, describe “best-in-breed” prior art static MEMS accelerometers. As discussed above, a disadvantage of static MEMS accelerometers is their relatively low dynamic range, which may be defined as the ratio of the total input range divided by the smallest measurable signal.

REFERENCES

What is needed is an improved accelerometer that mitigates bias errors including time varying bias errors also known as bias drifts. The embodiments of the present disclosure answer these and other needs.

SUMMARY

In a first embodiment disclosed herein, a method for self-calibration of an accelerometer having a proof mass separated by a gap from a drive electrode and separated by a gap from a sense electrode comprises initializing the accelerometer to resonate, applying a first bias voltage to the sense electrode and applying a second bias voltage to the drive electrode to obtain a first scale factor Γ=+|Γ|>0, wherein a scale factor is a sensitivity of the accelerometer, a partial differential

dfhdain,
wherein fhis a harmonic frequency of the accelerometer, and wherein ainis an acceleration along an input axis (IA) of the accelerometer, measuring a first acceleration â1over a first time interval, swapping the first bias voltage on the sense electrode with the second bias voltage previously on the drive electrode and the second bias voltage on the drive electrode with the first bias voltage previously on the sense electrode so that a bias voltage on the sense electrode is set to the second bias voltage and a bias voltage on the drive electrode is set to the first bias voltage to obtain a second scale factor Γ=−|Γ|>0, measuring a second acceleration â2over a second time interval, and calculating a true acceleration

In another embodiment disclosed herein, a method for self-calibration of an accelerometer having a proof mass separated by a gap from a drive electrode and separated by a gap from a sense electrode comprises initializing the accelerometer to resonate, applying a first bias voltage VBto the sense electrode and applying a second bias voltage VCto the drive electrode to obtain a first scale factor Γ=+|Γ|>0, wherein a scale factor is a sensitivity of the accelerometer, a partial differential

dfhdain,
wherein fhis a harmonic frequency of the accelerometer, and wherein ainis an acceleration along an input axis (IA) of the accelerometer, measuring a first acceleration â1over a first time interval, adjusting the first bias voltage VBto the sense electrode and the second voltage VCto the drive electrode to obtain a second scale factor Γ=A|Γ|, where

A=Γ2Γ1,
measuring a second acceleration â2over a second time interval, and calculating a true acceleration

In yet another embodiment disclosed herein, an accelerometer for providing continuous self-calibration comprises a proof mass suspended from a frame, a drive electrode separated from the proof mass by a first gap, a sense electrode separated from the proof mass by a second gap, a phase lock loop circuit coupled to the sense electrode and having a phase lock loop output, a calibration dither generator for generating a dither output for modulating a scale factor, wherein the scale factor is a sensitivity of the accelerometer, a partial differential

dfhdain,
wherein fhis a harmonic frequency of the accelerometer, and wherein ainis an acceleration along an input axis (IA) of the accelerometer, a drive bias offset voltage VC, a summer coupled to the phase lock loop output, the dither output, and the drive bias offset voltage VCfor providing a summer output having the sum of the phase lock loop output, the dither output, and the drive bias offset voltage VC, wherein the summer output is coupled to the drive electrode, and wherein the phase lock loop circuit outputs the harmonic frequency of the accelerometer.

These and other features and advantages will become further apparent from the detailed description and accompanying figures that follow. In the figures and description, numerals indicate the various features, like numerals referring to like features throughout both the drawings and the description.

DETAILED DESCRIPTION

The present disclosure improves upon prior art accelerometers, including the one described in Reference [1], above, by including in the accelerometer continuous online self-calibration to estimate and mitigate bias errors, including time varying bias errors commonly known as bias drifts, which can be key performance limiters for MEMS accelerometers. Continuous and online self-calibration is preferred because the operation of the accelerometer is not interrupted for calibration, enabling the measurements to be made continuously and accurately in real time.

Static accelerometers have a low Q factor to avoid ringing in the accelerometer response. In the MEMS accelerometer described in Reference [1], the MEMS design and vacuum packaging produce silicon accelerometer structures with Q factors greater than 10,000 with natural frequencies greater than 10 kHz. Static MEMS accelerometers are limited to dynamic ranges <106by their amplitude modulation (AM) readout mechanism. The FM mechanism employed in the accelerometer of Reference [1] provides dynamic ranges greater than 109. Such a large dynamic range is necessary to enable large input ranges (>±100 g to ±1000 g) while simultaneously preserving the ability to accurately resolve small inertial signals below 1 μg. In addition, since the device described in Reference [1] operates at a higher natural frequency than a static accelerometer, the device is 25 to 2500× less sensitive to spurious vibration signals which can corrupt a true inertial acceleration signal.

FIGS. 1A and 1Bshow a concept of operation for the accelerometer described in Reference [1]. The resonant MEMS accelerometer described in Reference [1] and shown inFIG. 1Acan achieve navigational grade performance (sub-micro-g) and at the same time accept high input ranges (up to ±1000 g and beyond). The present disclosure improves the accelerometer of Reference [1] by providing continuous online self-calibration of the accelerometer device to remove bias errors.

As shown inFIG. 1A, the accelerometer has a proof mass10suspended by springs11from a frame12. The drive electrode14and the sense electrode16are separated by a gap from the proof mass10. Voltages VC18and VB20are applied to the drive electrode14and sense electrode16, respectively. An amplifier22and frequency detector24are connected to the sense electrode16, and an output23of the amplifier22may be fed back to the drive electrode14.

Preferably the accelerometer structure ofFIG. 1Ais completely symmetric by design, except when appropriate combinations of VBand VCare applied to the electrodes to purposefully unbalance the structure in one direction or the other. However, although the description describes a symmetric resonator as a preferred embodiment, the techniques disclosed herein may also be used with an asymmetric resonator. Asymmetry in the structure may arise, for example, if the accelerometer structure is imperfectly fabricated, even though the design of the accelerometer structure is for a symmetric structure. Any asymmetry in the resonator structure may be accounted for by first adjusting the combination of VBand VCas appropriate, and then applying the methods of the present disclosure. Therefore the methods of the present disclosure are equally applicable for an asymmetric resonator structure.

FIG. 1Bshows the concept of operation for the resonant MEMS accelerometer ofFIG. 1A. The acceleration signal is mapped from near direct current (DC) to a signal contained in the sidebands around the MEMS device's resonance frequency. Acceleration is detected through a change of the device's natural frequency and can be read out in the time domain using a frequency detector circuit24, which may be digital or analog circuitry.

The scale factor Γ or the sensitivity

dfhdain
is a critical parameter for operation of the resonant MEMS accelerometer, and has particular characteristics in the accelerometer that can be exploited to enable self-calibration, as explained further below. The scale factor Γ in units of Hz/(m/s{circumflex over ( )}2) for the accelerometer ofFIG. 1Ais determined by the following equation:

The leading terms outside the bracket on the left are constants for a particular accelerometer device. The terms inside the bracket are shown below.

In the above expression, xsis the static displacement of the accelerometer proof mass10evaluated at zero input acceleration and can be found from the equation:

If one swaps the voltages VB20and VC18in the above, and then solves for the resulting displacement, one finds:

Comparing the two previous equations, one can draw the conclusion that
xs′−xs

when the drive electrode14voltage VC18and sense electrode16voltage VB20voltages are swapped, meaning one takes the voltage value that was applied to drive electrode14and applies it to sense electrode16and vice versa.

Armed with this result, performing the voltage swap yields:

Comparing this expression with the previous expression for Γ leads us to the conclusion that
Γ′=−Γ

when the drive electrode14and sense electrode16voltages18and20are swapped. This is an extremely important result and enables self-calibration of the accelerometer shown inFIG. 1Athrough several different approaches, as described further below.

The meaning of the above result in the physical context of the operation of the accelerometer is described below. All accelerometers have what is known as an input axis (IA). For example, inFIG. 1A, the arrow26designated ain26denotes acceleration along the IA of the accelerometer. In the convention used inFIG. 1A, the IA arrow26is pointed to the right, meaning that the accelerometer is intended to assign a corresponding positive sign to positive accelerations in that direction. Therefore, drive electrode14voltage18and sense electrode16voltage20values are set to VB20=V1and VC18=V2, respectively, to produce a positive Γ. Since voltage sources VB20and VC18are adjustable in the voltage value that they produce, their values can be swapped such that VB20=V2and VC18=V1, producing a new scale factor Γ′=−Γ, in accordance with the above theory. Since these two accelerometer biasing cases have equal scale factors in magnitude, but with opposite sign, the result is equivalent to flipping the direction of the accelerometer input axis (IA). In other words, the second voltage biasing condition is a way to virtually rotate the IA so that it is 180° opposite to the IA of the first voltage biasing condition. This virtual method of rotation is equivalent to physically rotating the accelerometer in the first biasing condition by 180°.

In the preceding discussion, a method has been described for using our symmetric resonant accelerometer to reverse its input axis (IA)26through swapping of the bias voltage levels on the drive14and sense 16 electrodes. The input axis (IA) reversal is a consequence of the flipping of the sign of the scale factor upon swapping the bias voltages, leading to Γ′=−Γ. Next it is described how the reversal of the sign of the scale factor enables self-calibration of the accelerometer by removal of an unknown but fixed bias offset over two sequential acceleration measurements.

The following procedure is called a full IA reversal. First, two back to back readings are taken of the acceleration indicated by our accelerometer, â1and â2, where the hat symbol ({circumflex over ( )}) indicates a measurement or reading from the accelerometer. These measurements are each made over a finite time interval, which may be the same duration, and the time interval is also chosen such that the sequential measurements have approximately the same input acceleration (ain) and bias offset (Bf). This means the measurements are made fast enough that the acceleration measured is the same and substantial bias drift has not occurred between the measurements. However, the scale factor Γ used to make each measurement is allowed to change between sequential readings. The measurements so obtained can be expressed as:
â1=Γ1−1(Γ1ain+Bf)
â2=Γ2−1(Γ2ain+Bf)

Note that the bias offset represents an additive process in the above after the input acceleration is converted to a frequency by the scale factor Γ. Therefore, to obtain the acceleration measurement, the frequency reading is multiplied by the inverse scale factor.

During measurement interval 1, the scale factor Γ1=Γ, and during measurement interval 2, the scale factor Γ2=Γ′=−Γ through appropriate combinations of VB20and VC18as explained previously.

Substituting into the Above Yields:
â1=Γ−1(Γain+Bf)
â2=−Γ−1(−Γain+Bf)

This can be rearranged to get:

This is an unbiased, meaning calibrated, estimate of the true acceleration along the input axis (IA) from measurements â1and â2.

It is important in high performance applications to monitor the bias offset to observe if it is drifting. This can be achieved by determining

Rearranging gives the bias estimate as:

The above expresses the bias offset in terms of a frequency. Alternatively, the bias offset can be expressed in native acceleration units by multiplying through by the inverse scale factor:

A flow chart describing the self-calibration algorithm using full IA reversal as described above is shown inFIG. 2. In step30, the resonant FM accelerometer is initialized to be operational or in a resonant mode and therefore resonating. Then in step32bias voltages VBand VCare adjusted to produce Γ1=+|Γ|>0. Next in step34an acceleration measurement â1is made over a first interval. Then in step36VBand VCare adjusted by setting bias voltage VBto be equal to the previous Vcand by setting bias voltage Vcto be equal to the previous VBto produce Γ2=−|Γ|>0. Next in step38, an acceleration measurement â2is made over a second interval. The first and second interval may have equal time spans. Then in step40, the true acceleration

ain=a^2+a^22
is calculated. Next in step42the bias offset

Ba=a^2-a^22
may be optionally calculated. Finally, in step44the true acceleration ainand bias offset Bamay be outputted from the accelerometer. This self-calibration may be continually repeated as shown by the loop46from step42to step32.

The accelerometer self-calibration method disclosed in the previous section is made possible by flipping the sign of the scale factor on an otherwise symmetric accelerometer device. This method is useful if the transitions between positive and negative scale factors during each measurement interval can be made quickly with respect to the application needs. However, during the transition from positive to negative scale factor and vice versa, the adjustment of the bias voltages VB20and VC18might be made instantaneously or smoothly over time. In the first case, the sudden switching of the bias voltages VB20and VC18can perturb the control loops of the accelerometer, which would then require subsequent time to re-stabilize to their nominal operating points. Thus, in the simplest case, the transition periods can be thought of as dead measurement time, since it would be difficult to extract meaningful acceleration measurements before the accelerometer is in stable operation. In the second case, the bias voltages are transitioned gradually or smoothly over time between measurements. This enables the control loops to track the accelerometer in a smooth manner and avoid the need to re-stabilize the control loops. However, if the time needed to make a smooth transition is too great, it slows the ability to obtain low noise stable acceleration measurements at constant scale factors.

If the restrictions caused by the adjustment of the bias voltages VBand VCmade instantaneously or smoothly over time are too strict for a particular application, an alternative self-calibration method may be used, which is called partial input axis reversal. The advantage of partial input axis reversal that will become apparent is that the bias voltage changes do not have to be very abrupt and preserve the loop stability when transitioning scale factors. Smoothly transitioning the scale factor from fully positive to fully negative as described in the second example above can be thought of as an extreme case of partial input axis reversal.

To mathematically describe partial input axis (IA) reversal, one starts by making two acceleration measurements in sequential intervals, as before:
â1=Γ1−1(Γ1ain+Bf)
â2=Γ2−1(Γ2ain+Bf).

One also sets Γ1=Γ as before. However, instead of setting Γ2=−Γ, one sets Γ2==AΓ such that

A=Γ2Γ1.
A is known because of the choice of VBand VC. For example, for the embodiment shown inFIG. 4, the magnitude of Γ can be ensured to not equal 0 by setting VB=0 according to the equation above in paragraph [0030]. This is a preferred embodiment since it maximizes the sense current obtained from the device. However, the methods can work for any VBnot equal to VC; however, if VBis not equal to 0, there is less signal to work with and it may require the ability to set a negative bias potential which makes the implementation more complicated.

Making these substitutions gives:

One can recognize that the first term in each measurement is the same. Therefore:

The above can be rearranged to estimate the bias offset:

The next step is to add the two acceleration measurements:

Since Bfis known, this becomes

This can be rearranged to find ain:

Comparison with the full IA reversal equation confirms that the above matches for the case when A=−1. Once again, one can also write the bias offset in terms of natural acceleration units:

A flow chart describing the self-calibration algorithm using partial IA reversal is shown inFIG. 3. In step50, the resonant FM accelerometer is initialized to be operational or in a resonant mode and therefore resonating. Then in step52bias voltages VBand VCare adjusted to produce Γ1=+|Γ|>0. Next in step54an acceleration measurement â1is made over a first interval. Then in step56VBand VCare adjusted by setting bias voltage VBand by setting bias voltage Vcto produce Γ2=A|Γ|, where A=Γ2/Γ1. Next in step58, an acceleration measurement â2is made over a second interval. The first and second interval may have equal time spans. Then in step60, the true acceleration

ain=(a^1+a^2)2-(1+1A1-1A)⁢(a^1-a^2)2
is calculated. Next in step62the bias offset

Ba=(a^1-a^21-1A)
may be optionally calculated. Finally, in step64the true acceleration ainand bias offset Bamay be outputted from the accelerometer. This self-calibration may be continually repeated as shown by the loop66from step62to step52.

Having disclosed the mathematical principals underlying accelerometer self-calibration based on full and partial input axis reversal, it is useful to describe a practical implementation of an accelerometer device based on these principals. The example disclosed below is only one example of how one might choose to implement the invention.

FIG. 4shows an embodiment of a continuously self-calibrating accelerometer employing partial IA reversal.FIG. 5shows the same embodiment but identifies which parts or domains are implemented in digital electronics, analog electronics, or the electromechanical resonator acting as the acceleration transducer.

As inFIG. 1A, the accelerometer electromechanical resonator10is implemented as a proof mass suspended by springs11which are fixed on one side to the accelerated frame12as indicated by the ainarrow26. The proof mass10displaces relatively in the opposite direction in response to an acceleration of its support frame12. Drive electrode14and sense electrode16are configured around the proof mass10to actuate and sense motion of the proof mass10. The proof mass10is driven into small harmonic oscillations using an amplitude feedback loop. Analog electronics70, as shown inFIG. 5, which may include amplifiers, including buffer amplifier71, as shown inFIG. 4, filters, and so on, are used on the drive side to buffer and condition the voltages applied to the drive electrode14. Analog electronics72, as shown inFIG. 5, which may include amplifiers, including pick-off amplifier73, as shown inFIG. 4, filters, and so on, are used on the sense side to amplify the motional current that is picked off the device and convert the current to a voltage that can be read by the analog to digital converter (A/D)74. As shown inFIG. 5, other parts may be implemented digitally, as shown by digital domain120.

A phase locked loop (PLL), shown inFIG. 4, may be used to track the phase and frequency of the harmonic motion xh75of the proof mass10. The phase locked loop (PLL) includes phase controller78, and numerically controlled oscillator (NCO)80. The harmonic frequency fh82can be read directly from the numerically controlled oscillator (NCO)80. The NCO80is also used to generate the in phase (I)84and quadrature (Q)86reference signals for the demodulator76and the modulator88. The demodulator76accepts the I84and Q86reference signals and compares them with its input from the analog to digital converter (A/D)74. The result is an in-phase (I) component90of the signal amplitude as well as the quadrature (Q) component92.

The Q component92is fed to the phase controller78, which may be implemented, for example, as a proportional-integral (PI) controller, which generates an error signal94based on the difference between the phase of the measured motion and the phase of the reference signal. The error signal94from the phase controller is used to adjust the NCO80until the Q component of the A/D read by the demodulator76is nulled to zero, indicating phase lock.

The I component90is fed to the amplitude controller91, which compares it to the desired amplitude set point and generates an error signal50used to control the wave amplitude produced by the modulator88. The desired output100of the modulator88after being converted to a voltage through the digital to analog (D/A) converter102can be expressed as vdrive=Vdrivecos(ωht).

A summer104is used to add the modulator output signal100to the drive bias offset106, which ultimately creates the direct current (DC) bias VC108on the drive electrode14. The calibration dither generator110produces a signal112which is used to modulate the scale factor to implement smooth, continuous partial input axis reversal. The summer104adds the output112of the calibration dither generator110to the sum of the modulator output signal100and the drive bias offset106, and the resulting sum of all these signals is fed to the D/A converter102, where it is buffered and conditioned by buffer amplifier71and applied to the drive electrode14to control the static gap dimension between the proof mass10and the drive electrode14and induce a small harmonic oscillation so that the harmonic frequency82of the accelerometer system can be read out.

As shown inFIG. 4, the (DC) bias VB107on the sense electrode16is 0 volts or ground in this embodiment. This ensures maximum transduction efficiency since the transduction gap voltage is equal to Vgap=VP−VB=VP−0=VP, where Vpis the potential of the proof mass10, as shown inFIG. 4.

In one embodiment, the calibration dither generator can be implemented to have an output112vcal=Vcalcos(ωcalt), where wcal=2πfcal<<ωhand fcalis less than the bandwidth of the PLL loop so that the PLL can track the calibration dither. The total voltage applied to the drive electrode14can now be expressed as vdrive_total=VC+Vdrivecos(ωht)+Vcalcos(ωcalt). When cos(ωcalt) is at its peak, the total effective bias voltage will be VC+Vcal, and when cos(ωcalt) is at its trough, the total effective bias voltage will be VC−Vcal. The effect on the scale factor is such that Γ1=(1+C)Γ0=Γ at the peak and Γ2=(1−C)Γ0=AΓ at the trough, where Γ0is the nominal scale factor at a bias of VCand

A=Γ2Γ1=(1-C)⁢Γ0(1+C)⁢Γ0=(1-C1+C).
Substituting into the equations above for partial IA reversal, by measuring fh82at the peak (â1) and at the trough (â2) of cos(ωcalt), the unbiased accelerometer measurement is:

and the bias can be estimated as:

The peak and trough of cos(ωcalt) can be obtained in several ways. First, since the calibration dither signal is generated in a known manner, one knows exactly when cos(ωcalt) is at its peak or trough and can sample fh82at those instants. A second method that would have less measurement noise would be to demodulate the output of the first demodulator a second time with respect to cos). The twice demodulated amplitude in this way would be proportional to â1−â2and therefore enable determining Ba, which could then be subtracted from the acceleration determined by fh82. A third method would be to employ signal processing to fit a sinusoid with frequency to the fcalto the fhdata stream and determine the local max (â1) and min (â2) from the sine wave fit. Additional terms like a linear ramp could be added to improve the fit if necessary.

The above methods are conceptually illustrated inFIGS. 6 and 7.

FIG. 6shows waveforms illustrating partial input axis (IA) reversal self-calibration of a symmetric resonant FM accelerometer by application of a dither voltage112proportional to cos(ωcalt) to the drive electrode14in the absence of applied external acceleration26(ain=0) but in the presence of an unknown fixed bias offset Bn130. The vertical distance between the measurements â1′132and â2′134where the (′) indicates the frequency domain equivalent, hat ({circumflex over ( )}) indicates an estimated quantity from the measurement indicated by gray points on the fhwaveform82is equal to Bf, which is the equivalent bias offset in the frequency domain. The green points indicate moments in time when the fh82waveform is equal to an unbiased estimate of ainin the frequency domain.

FIG. 7shows waveforms illustrating partial input axis (IA) reversal self-calibration of a symmetric resonant FM accelerometer by application of a dither voltage proportional112cos(ωcalt) to the drive electrode14in the presence of an unknown and time varying bias offset Ba140and a time varying applied external acceleration26. The vertical distance between the measurements â1′142and â2′144, where (′) prime indicates the frequency domain equivalent, hat ({circumflex over ( )}) indicates an estimated quantity from the measurement indicated by gray points on the fhwaveform is equal to Bf, which is the equivalent bias offset in the frequency domain. The green points146indicate moments in time when the fhwaveform is equal to an unbiased estimate of ainin the frequency domain. In the first 5 cycles of cos(ωcalt), Ba140is shown to increase from a negative offset to a positive offset, and the effect on the fh82waveform is to change the peak to trough distance which indicates Bf. It can be seen that even though the peak to trough distance is changing, the unbiased acceleration measurement points (green points)146indicate ain=0. In the last 3 cycles of cos(ωcalt), Ba140is held fixed while the applied acceleration is linearly ramped, first increasing for one cycle, and then decreasing for two cycles. The fh82response shows that the unbiased acceleration measurement points track the input acceleration signal, while the peak to trough distance stays the same, in agreement with the fixed Ba140signal. This example shows how unbiased acceleration measurements can be made using partial IA reversal, even in the presence of an unknown and time varying bias offset.

The accelerometer in accordance with the present disclosure may be a resonant accelerometer including a proof mass, one or more springs connecting the proof mass to an anchor, and one or more capacitive transduction gaps between the movable proof mass and fixed electrodes, wherein the static displacement of the proof mass in response to acceleration applied to the anchor modifies the electrostatic stiffness imparted by one or more of the capacitive transduction gaps on the proof mass, resulting in a corresponding change in the resonance frequency of the combined electromechanical system.

The accelerometer can be configured to produce a first scale factor through application of a first set of voltages to the capacitive transduction gaps and a second scale factor through application of a second set of voltages to the capacitive transduction gaps.

The accelerometer is first configured to produce the first scale factor and obtain a first measurement of acceleration and then configured to produce the second scale factor and obtain a second measurement of acceleration. The second scale factor may be equal in magnitude to the first scale factor but with opposite sign, which is full input axis reversal, or the second scale factor may be equal to a predetermined fraction of the first scale factor, which is partial input axis reversal.

The first and second measurements of acceleration may be combined to create an unbiased estimate of the true acceleration, and may be combined to create an estimate of the bias offset over the interval of measurement. The scale factor configuration and measurement procedure may be repeated indefinitely or until a break condition is triggered.