US-12618866-B2 - Time-switched frequency modulated accelerometer long-term stabilization

US12618866B2US 12618866 B2US12618866 B2US 12618866B2US-12618866-B2

Abstract

A method for improving long-term stability in a time-switched frequency modulated accelerometer includes demodulating a frequency of the time-switched frequency modulated accelerometer in an anti-phase vibrational mode one or more instances. Assuming an initial phase offset value, demodulating occurs in-phase and out-of-phase with a modulation reference to extract the in-phase and out-of-phase responses. The method further includes calculating a slope between the in-phase and the out-of-phase responses where if the slope is not equal to 1 or −1, the phase offset value is adjusted, and the demodulating is repeated until the slope is equal to 1 or −1. If the slope is equal to 1 or −1, the selected phase offset value is kept as a selected phase offset value. The method further includes using a linear model to detect bias drift with the out-of-phase response, thereby compensating for bias drift in the in-phase response.

Inventors

Andrew B Sabater

Assignees

Naval Information Warfare Center Pacific

Dates

Publication Date: 20260505
Application Date: 20230628

Claims (13)

1 . A method for improving long-term stability in a time-switched frequency modulated accelerometer, comprising: demodulating a frequency of the time-switched frequency modulated accelerometer in an anti-phase vibrational mode one or more instances, wherein assuming an initial phase offset value, demodulating occurs in-phase and out-of-phase with a modulation reference to extract in-phase responses and out-of-phase responses; calculating a slope between the in-phase responses and the out-of-phase responses, wherein one of the following occurs: i) if the slope is not equal to 1 or −1, the phase offset value is adjusted, and demodulating the in-phase and out-of-phase responses from the frequency of the time-switched frequency modulated accelerometer in an anti-phase vibrational mode is repeated and the slope is calculated until the slope is equal to 1 or −1; or ii) if the slope is equal to 1 or −1, keep the initial phase offset value as a selected phase offset value; and using a linear model to detect bias drift with the out-of-phase response, thereby compensating for the bias drift in the in-phase response of the time-switched frequency modulated accelerometer and a stability of the time-switched frequency modulated accelerometer is maintained.
2 . The method of claim 1 , wherein a computer processor, an application-specific integrated circuit, a field programmable gate array, or a microcontroller is used to demodulate the frequency of the time-switched frequency modulated accelerometer, calculate the slope, and use the linear model with the out-of-phase response to compensate for bias drift of the in-phase response.
3 . The method of claim 1 , wherein the slope is calculated using an equation (I): m = ∑ i = 1 n ( IP i - IP _ ) ⁢ ( OOP i - OOP _ ) ∑ i = 1 n ( OOP i - OOP _ ) 2 ( I ) where m is the slope, n is a total number of measurements of the in-phase response and the out-of-phase response, IP i is an i-th measurement of the in-phase response, OOP i is an i-th measurement of the out-of-phase response, IP is a mean value of the in-phase response, and OOP is a mean value of the out-of-phase response.
4 . The method of claim 1 , further including calculating a correlation coefficient to validate the slope between an in-phase channel and an out-of-phase channel.
5 . The method of claim 4 , wherein the correlation coefficient is calculated using equation (II): m = r ⁢ s IP s OOP ( II ) where m is the slope, r is the correlation coefficient, s IP is an uncorrected sample standard deviations of the in-phase response, and s OOP is an uncorrected sample standard deviations of the out-of-phase response.
6 . The method of claim 4 , wherein a computer processor, an application-specific integrated circuit, a field programmable gate array, or a microcontroller is used to calculate the correlation coefficient.
7 . The method of claim 1 , wherein the linear model is defined by equation (III): IP c = IP - m × OOP ( III ) where IP c is a compensated in-phase response, IP is the in-phase response, m is the slope, and OOP is the out-of-phase response.
8 . The method of claim 1 , wherein the time-switched frequency modulated accelerometer is a single-body multi-mode mechanical resonator including: two or more spring mass devices, wherein the two or more spring mass devices include two or more lumped masses and one or more anchor springs where the one or more anchor springs attach the two or more lumped masses to a fixed substrate and the two or more spring mass devices have an anti-phase mode and an in-phase mode; one or more coupling springs, wherein the one or more coupling springs are attached to each lumped mass and couple the motion of the two or more lumped masses; one or more phase 1 tuning electrodes and one or more phase 2 tuning electrodes, wherein the one or more phase 1 tuning electrodes and the one or more phase 2 tuning electrodes are attached to each lumped mass and apply forces to the in-phase mode that yield equal in magnitude, but opposite in sign frequency shifts of the anti-phase mode; a plurality of interdigitated shaped combs, wherein the plurality of interdigitated shaped combs is attached to each lumped mass and compensates for a frequency drift of the anti-phase mode and cancels first-order displacement effects of the plurality of interdigitated shaped combs; a plurality of interdigitated drive combs, wherein the plurality of interdigitated drive combs are attached to each lumped mass and are aligned to the anti-phase mode, with a frequency selective excitation, can excite the anti-phase mode with a low voltage when the multi-mode mechanical resonator is operated within a low-pressure environment; a plurality of interdigitated sense combs, wherein the plurality of interdigitated sense combs are attached to each lumped mass and are aligned to the anti-phase mode, are grounded when the multi-mode mechanical resonator is polarized, and are capable of cancelling common-mode noise effects.
9 . The method of claim 8 , wherein the two or more spring-mass devices have a motion described with equations (IV) and (V): M 1 ( x 1 ‶ - x 1 ⁢ Accel ‶ ) = - K 1 ( x 1 - 0 ) - K c ( x 1 - x 2 ) - F 1 ⁢ A + F 1 ⁢ B ( IV ) M 2 ( x 2 ‶ - x 2 ⁢ Accel ‶ ) = - K 2 ( x 2 - 0 ) - K c ( x 2 - x 1 ) - F 2 ⁢ A + F 2 ⁢ B ( V ) where M 1 is a lumped mass of a first mass, M 2 is a lumped mass of a second mass, K 1 is a first anchor spring stiffness, K 2 is a second anchor spring stiffness, K c is a coupling spring stiffness, x 1 is a displacement of the first mass, x 2 is a displacement of the second mass, x″1 is an acceleration of the first mass, x″2 is an acceleration of the second mass, x″ 1Accel is a proper acceleration of the first mass, x″ 2Accel is a proper acceleration of the second mass, F 1A is a phase 1 electrostatic force applied to the first mass, F 1B a phase 2 electrostatic force applied to the second mass, F 2A is a phase 1 electrostatic force applied to the second mass, and F 2B is a phase 2 electrostatic force applied to the first mass.
10 . The method of claim 8 , wherein a moving finger and a stationary finger have a gap between the stationary finger and the moving finger represented by an equation (VI): h ⁡ ( x ) = g 0 / ( 1 + x / x CL ) ( VI ) where g 0 is the gap between a straight comb and the maximum distance between the straight comb and a curved comb and x CL is a dimensional constant wherein the straight comb is the stationary finger or moving finger and the curved comb is different than the straight comb and either the stationary finger or the moving finger.
11 . A system for improving long-term stability in a time-switched frequency modulated accelerometer, comprising: the time-switched frequency modulated accelerometer; and an electronic device, wherein the electronic device demodulates a frequency of the time-switched frequency modulated accelerometer, calculates a slope, and uses a linear model with an out-of-phase response to compensate for bias drift of an in-phase response of the time-switched frequency modulated accelerometer; wherein the time-switched frequency modulated accelerometer is a single-body multi-mode mechanical resonator including: two or more spring mass devices, wherein the two or more spring mass devices include two or more lumped masses and one or more anchor springs where the one or more anchor springs attach the two or more lumped masses to a fixed substrate and the two or more spring mass devices have an anti-phase mode and an in-phase mode; one or more coupling springs, wherein the one or more coupling springs are attached to each lumped mass and couple the motion of the two or more lumped masses; one or more phase 1 tuning electrodes and one or more phase 2 tuning electrodes, wherein the one or more phase 1 tuning electrodes and the one or more phase 2 tuning electrodes are attached to each lumped mass and apply forces to the in-phase mode that yield equal in magnitude, but opposite in sign frequency shifts of the anti-phase mode; a plurality of interdigitated shaped combs, wherein the plurality of interdigitated shaped combs is attached to each lumped mass and compensates for a frequency drift of the anti-phase mode and cancels first-order displacement effects of the plurality of interdigitated shaped combs; a plurality of interdigitated drive combs, wherein the plurality of interdigitated drive combs are attached to each lumped mass and are aligned to the anti-phase mode, with a frequency selective excitation, can excite the anti-phase mode with a low voltage when the multi-mode mechanical resonator is operated within a low-pressure environment; a plurality of interdigitated sense combs, wherein the plurality of interdigitated sense combs are attached to each lumped mass and are aligned to the anti-phase mode, are grounded when the multi-mode mechanical resonator is polarized, and are capable of cancelling common-mode noise effects.
12 . The system of claim 11 , wherein the two or more spring-mass devices have a motion described with equations (III) and (IV): M 1 ( x 1 ‶ - x 1 ⁢ Accel ‶ ) = - K 1 ( x 1 - 0 ) - K c ( x 1 - x 2 ) - F 1 ⁢ A + F 1 ⁢ B ( IV ) M 2 ( x 2 ‶ - x 2 ⁢ Accel ‶ ) = - K 2 ( x 2 - 0 ) - K c ( x 2 - x 1 ) - F 2 ⁢ A + F 2 ⁢ B ( V ) where M 1 is a lumped mass of a first mass, M 2 is a lumped mass of a second mass, K 1 is a first anchor spring stiffness, K 2 is a second anchor spring stiffness, K c is a coupling spring stiffness, x 1 is a displacement of the first mass, x 2 is a displacement of the second mass, x″ 1 is an acceleration of the first mass, x″ 2 is an acceleration of the second mass, x″ 1Accel is a proper acceleration of the first mass, x″ 2Accel is a proper acceleration of the second mass, F 1A is a phase 1 electrostatic force applied to the first mass, F 1B a phase 2 electrostatic force applied to the second mass, F 2A is a phase 1 electrostatic force applied to the second mass, and F 2B is a phase 2 electrostatic force applied to the first mass.
13 . The system of claim 11 , wherein a moving finger and a stationary finger have a gap between the stationary finger and the moving finger represented by an equation (V): h ⁡ ( x ) = g 0 / ( 1 + x / x CL ) ( VI ) where g 0 is the gap between a straight comb and the maximum distance between the straight comb and a curved comb and x CL is a dimensional constant wherein the straight comb is the stationary finger or moving finger and the curved comb is different than the straight comb and either the stationary finger or the moving finger.

Description

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT The invention described herein may be manufactured and used by or for the government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. Licensing and technical inquiries may be directed to the Office of Research and Technical Applications, Naval Information Warfare Center Pacific, Code 72120, San Diego, CA, 92152; (619) 553-5118; NIWC_Pacific_T2@us.navy.mil. Reference Navy Case Number 211116. BACKGROUND Inertial navigation provides a means to navigate independent of global positioning satellites (GPS) and is immune to certain issues related to the use of GPS. Inertial navigation is based on fusing measurements from accelerometers and gyroscopes to estimate position, velocity, and attitude. In order to extract the actual acceleration of a vehicle while moving, a model for local gravity is needed to compensate the output of the accelerometers. The position estimate provided by inertial navigation can be degraded for a variety of factors internal and external to these sensors. Internal factors include Gaussian noise processes associated with electronics of the sensors and a slow drift due to temperature variations or packaging material aging. External factors include initialization errors (an error in the initial estimate for position, velocity or attitude) or deflection of vertical (DOV) errors associated with modeling errors for local gravity. However, position drift due to bias changes can be mitigated by creating stable accelerometers. DESCRIPTION OF THE DRAWINGS Features and advantages of examples of the present disclosure will be apparent by reference to the following detailed description and drawings, in which like reference numerals correspond to similar, but in some instances, not identical, components. Reference numerals or features having a previously described function may or may not be described in connection with other drawings in which they appear. FIG. 1 is an example of a flow diagram of a method for improving long-term stability in a frequency modulated accelerometer; FIG. 2 is an example of a graph showing the uncalibrated Output (Hz) (Y-axis) vs. Phase (deg) (X-axis) of the mean value of the in-phase and out-of-phase channels when setup to measure −1 g for different phase offset values; FIG. 3 is an example of a graph showing the R Value (Y-axis) vs. Phase (deg) (X-axis) of the correlation coefficient (R value) and slope between the in-phase and out-of-phase channels for different phase offset values; FIG. 4 is an example of a graph showing the uncalibrated Output (Hz) (Y-axis) vs. Time (sec) (X-axis) of the in-phase channel response to a −1 g input with the phase reference selected where the slope between the in-phase and out-of-phase channels is −1; FIG. 5 is an example of a graph showing the Allen Deviation (g) (Y-axis) vs. Tau (sec) (X-axis) of the in-phase channel and in-phase channel compensated with the out-of-phase channel when set to measure −1 g; FIG. 6 is 2D schematic an example of a time-switched frequency modulated accelerometer in a static mode; and FIG. 7 is 2D schematic an example of a time-switched frequency modulated accelerometer in an anti-phase vibrational mode. DETAILED DESCRIPTION The current approach for selecting the phase offset when using frequency-modulated accelerometers is to maximize the in-phase scale factor, which, in turn, improves sensitivity of the time-switched frequency modulated accelerometer. This approach is justified based on the assumption that the out-of-phase response provides no useful information after the phase offset is tuned to maximize in-phase scale factor as these responses should be orthogonal. However, as discussed in the method herein, this is not correct. The in-phase channel and out-of-phase channel can be used to observe bias drift over time in a time-switched frequency modulated accelerometer. In the method herein, the magnitude of the correlation coefficient is nearly one for most values of the phase offset. A rapid phase value transition to zero occurs where the slope is about zero. This demonstrates that, in general, the in-phase and out-of-phase channels are correlated. As such, the correlated signal can be used to observe bias drift. A linear model can then be implemented to improve long-term stability of time-switched frequency-modulated accelerometers. The advantages of this model are it is simple enough for real-time operation and reduces turn-on and off bias drift. To initialize a time-switched frequency modulated accelerometer, a positive-feedback loop to sustain the oscillations of the anti-phase vibrational mode of the time-switched frequency modulated accelerometer at a constant amplitude is implemented. Then, with a pair of tuning electrodes that are aligned to the in-phase mode, voltages equal in magnitude but opposite in phase are applied. These tuning electrodes are aligned to the in-phas