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Dissecting Tuned MEMS Vibratory Gyros

  • Dennis Kim
  • Robert T. M’Closkey
Chapter

Abstract

Vibratory rate sensors exploit coriolis force coupling terms that exist between two lightly damped modes when the equations of motion are written in a sensor-fixed coordinate system. Feedback plays a central role in the function of these sensors because one mode is driven at its natural frequency to a target amplitude, despite the fact that the natural frequency may drift or disturbances may be present that would otherwise cause the response amplitude to change. This excitation loop is present in all vibratory rate sensors. The highest performance vibratory gyros, however, employ a force-to-rebalance controller that attenuates the motion of the second mode. In this case, two closed-loop signals are demodulated in order to estimate the angular rate of rotation of the sensor. This chapter explains in detail the advantages of operating high quality factor vibratory gyros in a “tuned” configuration in which the modal frequencies of both modes are equal. In high Q resonators, however, the bandwidth of the sensor is too low to be useful for the vast majority of applications and so a force-to-rebalance controller is necessary to increase the sensor bandwidth. We address some common misconceptions concerning the use of feedback and unequivocally show that the force-to-rebalance controller does not degrade the signal-to-noise ratio for frequencies below the mechanical bandwidth of the resonances. As a consequence of this property, the estimated change in angle of the gyro, that is the integrated estimated angular rate, employing force-to-rebalance feedback asymptotically approaches the optimum noise characteristics associated with the open-loop gyro. The analysis is complemented by experiments conducted with the Disk Resonator Gyro (DRG). Tuned sensor dynamics are of paramount importance so we present a systematic method that efficiently determines the biasing voltages that are necessary for tuning the DRG. This approach has been successfully applied to other vibratory gyro prototypes, but its most attractive feature is the fact that it lends itself to automation and, thus, is suited to a production environment. The zero rate bias and quadrature terms are also derived from a generic sensor model, and it is shown that aside from the dynamics of the sensor’s resonator, there are two critical dynamic elements in vibratory gyros that can impact the stability of the zero rate bias. Any change in phase of these dynamic elements will mix the zero rate bias and quadrature signals so as to produce a change in the apparent rate bias and, thus, these elements are a source of low-frequency drift in the rate signal.

Keywords

Angular Rate Signal Conditioning Loop Gain Noise Density Sensor Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

The authors thank Dorian Challoner at the Boeing Satellite Development Center for his enthusiastic support. This work was supported in part by the National Science Foundation (ECS 0601622) and the UC MICRO program in conjunction with The Boeing Company.

References

  1. 1.
    Y.-C. Chen, R.T. M’Closkey, T.A. Tran, and B. Blaes. A control and signal processing integrated circuit for the JPL-Boeing micromachined gyroscopes. IEEE Trans Cntrl Sys Tech, vol. 13, no. 2:286–300 (2005).CrossRefGoogle Scholar
  2. 2.
    K. Fearnside and P.A.N. Briggs. The mathematical theory of vibratory angular tachometers. IEE Monograph, no. 264:155–166 (1957).Google Scholar
  3. 3.
    T.B. Gabrielson. Mechanical-thermal noise in micromachined acoustic and vibration sensors. IEEE Trans Elec Dev, vol. 40, no. 5:903–909 (1993).CrossRefGoogle Scholar
  4. 4.
    J. Hale. Ordinary differential equations. Wiley, New York (1969).MATHGoogle Scholar
  5. 5.
    J.B. Johnson. Thermal agitation of electricity in conductors. Phys Rev, vol 32:97–109 (1928).CrossRefGoogle Scholar
  6. 6.
    D.W. Jordan and P. Smith. Nonlinear ordinary differential equations, 2nd ed. Oxford University Press, Oxford (1987).MATHGoogle Scholar
  7. 7.
    D.-J. Kim and R.T. M’Closkey. A systematic method for tuning the dynamics of electrostatically actuated vibratory gyros. IEEE Trans Cntrl Sys Tech., vol. 14, no. 1:69–81 (2006).CrossRefGoogle Scholar
  8. 8.
    R.T. M’Closkey, A. Vakakis, and R. Gutierrez. Mode localization induced by a nonlinear control loop. Nonlinear Dynamics, vol. 25, no. 1:221–236 (2001).CrossRefMATHGoogle Scholar
  9. 9.
    C.T. Morrow. Zero signals in Sperry tuning fork gyrotron. J. Acoust Soc Am, vol 27:581–585 (1955).CrossRefGoogle Scholar
  10. 10.
    G.C. Newton Jr. Theory and practice in vibratory rate gyros. Control Eng, vol 10:95–99 (1963).Google Scholar
  11. 11.
    D. Scwhartz, D.-J. Kim, and R.T. M’Closkey. Frequency tuning of a disk resonator gyroscope via mass matrix perturbation. ASME J. Dyn Sys Meas Cntrl, vol. 131, no. 6 (2009).Google Scholar
  12. 12.
    A. Sharma, M.F. Zaman, and F. Ayazi. A sub-0. 2o/hr bias drift micromechanical silicon gyroscope with automatic CMOS mode-matching. IEEE Trans Sld St Crct, vol. 44, no. 5:1593–1608 (2009).Google Scholar
  13. 13.
    A.C. To, W.K. Liu, G.B. Olson, et al. Materials integrity in microsystems: a framework for a petascale predictive-science-based multiscale modeling and simulation system. Comput Mech (2007). doi: 10.1007/s00466-008-0267-1Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mechanical and Aerospace Engineering DepartmentUniversity of CaliforniaLos AngelesUSA

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