University of California, Irvine. Investigation of Factors Affecting Bias Stability and Scale Factor Drifts in Coriolis Vibratory MEMS Gyroscopes

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1 University of California, Irvine Investigation of Factors Affecting Bias Stability and Scale Factor Drifts in Coriolis Vibratory MEMS Gyroscopes Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical and Aerospace Engineering by Alexander A. Trusov Dissertation Committee: Professor Andrei M. Shkel, Chair Professor Faryar Jabbari Professor William C. Tang 2009

2 2009 Alexander A. Trusov

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5 Dedication To my dear friends and family across the world. You are always on my mind. iii

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7 Table of Contents List of Figures List of Tables Acknowledgments Curriculum Vitae Abstract of the Dissertation xiii xix xxi xxiii xxvii 1 Introduction Trends in Silicon Micromachined Gyroscopes Literature Review Research Objective Dissertation Outline Coriolis Vibratory Gyroscopes Newton s Equations of Motion in a Non-Inertial Frame Mechanical Sensor Element Dynamics Forced Vibrations for Constant Rate Measurements Rate Measurement Bandwidth Conclusions Nonlinearity of Capacitive Detection in Resonant MEMS Introduction Capacitive Detection of Periodic Motion Electromechanical Model Motion Detection with Lateral Combs v

8 3.2.3 Motion Detection with Parallel Plates Nonlinearity of Main Sidebands Demodulation Closed Form Solution Linearization Error Nonlinear Measurement Sinusoidal and Periodic Motion Experimental Demonstration Fabrication of Test Structures Structural Characterization Characterization of Detection Nonlinearities Discussions Nonlinearity of Sensing with Pure DC Voltage Differential EAM Using Parallel Plates Choice of Drive DC-Bias Application Terminal Conclusions Capacitive Detection Scheme with Inherent Self-Calibration Introduction Electromechanical Model Motion Detection with Parallel Plates Demodulation of Parallel Plate EAM Conventional Linear Approach SBR Detection Approach vi

9 4.5 Experimental Demonstration Real-Time Algorithm Implementation Discussions Differential EAM Detection with Pure DC Voltage Demodulation Noise Analysis Conclusions Parametric Excitation of Motion with Parallel Plates Introduction Parametric Excitation Experimental Study of Sub-Harmonic Excitation Characterization Test-Bed Structural Characterization High Amplitude Parametric Resonance Frequency Response Norms Numerical Modeling of Non-Linearities Effect of Approximation Order Effect of AC Drive Voltage Amplitude Vd Effect of DC Bias Voltage Vdc Effect of Quality Factor Q Conclusions Robust Gyroscope with Improved Gain-Bandwidth 107 vii

10 6.1 Introduction Structural Design and Dynamics Structural Design Drive-Mode Dynamics Sense-Mode Dynamics Algorithmic Design Procedure Physical Layout Implementations Finite Element Modeling Comparative Performance Analysis Design Parameters and Gain Gain and Bandwidth Analysis Design Conclusions Structural Characterization Fabrication Structural characterization Thermal robustness Rate and noise performance Angular rate Random Noise High Frequency and Tuning Fork Designs Conclusions Design and Packaging Tradeoffs in High-Q Gyroscopes Introduction viii

11 7.2 Experimental Study: Methods Test-bed Gyroscopes Experimental Setup Structural Characterization Measurement and Analysis of Q-Factors at Different Pressures Experimental Study: Results Effects of Actuation Type and Die Attachment Fabrication Imperfections in Anti-Phase Actuated Gyroscopes Modeling Finite Element Modeling of Thermoelastic Dissipation Dissipation thought Substrate in In-Phase Actuated Devices Dissipation thought Substrate in Anti-Phase Actuated Devices Fabrication Imperfections in Anti-Phase Actuated Devices Discussions and Conclusions High-Q Tuning Fork Designs Introduction Design Concept Drive-Mode Design Sense-Mode Design Actuation and Detection Structural and Rate Characterization Prototype Fabrication Structural Characterization ix

12 8.3.3 Rate Characterization Ultra-High Quality Factor Operation Experimental Characterization of Quality Factors Comparison to Single Mass Gyroscope Effect on Gyroscope Scale Factor Effect on Gyroscope Mechanical-Thermal Noise Quadruple Tuning Fork Design Limitations of the Dual Mass Architecture Quadruple Tuning Fork Architecture Advantages of the Quadruple Mass Tuning Fork Conclusions Conclusions Contributions of This Dissertation Future Research Directions Bibliography 225 A Fabrication and Packaging 227 B Stand-Alone DSP for Versatile MEMS Characterization 237 B.1 Introduction B.2 Electronics Design B.2.1 Processor B.2.2 Digital-to-Analog Interface Circuit B.2.3 Analog-to-Digital Interface Circuit B.2.4 Power Handling x

13 B.3 Experimental Characterization B.3.1 Hardware and Software Configuration B.3.2 Test Device B.3.3 Drive-Mode Detection B.3.4 Sense-Mode Detection C Velocity-Feedback Self-Resonance with Carrier 247 C.1 Phase Control and Velocity Feedback C.2 Velocity Feedback Self-Resonance C.3 Analog Circuit Design C.4 Experimental Demonstration C.5 Single-Sided and Differential Operation D Allan Variance Analysis 255 D.1 Allan Variance Analysis Procedure D.2 Random Noise Modes Classification D.3 White Noise D.4 Pink Noise D.5 Red Noise D.6 Random Noise Modes Identification D.7 Application to Rate Gyroscopes D.8 Mechanical-Thermal Noise E Micro Stage for On-Chip Electro-Mechanical Calibration 267 E.1 Conceptual Design of the System E.2 Scale Factor Drifts in Rate Gyroscopes xi

14 E.3 Proposed Calibration Algorithm xii

15 List of Figures 1.1 Calibration of ADXRS150: ideal vs. year 2003 vs. year Theoretical model of a single axis Coriolis vibratory gyroscope Block diagram of rate sensor dynamics Response to a sinusoidal, fixed-frequency input angular rate (a) Sense-mode at 10 khz with Q=100, drive-mode at 10.1 khz (b) Sense-mode at 10 khz with Q=1000, drive-mode at 10.1 khz Frequency response a vibratory gyroscope (a) Sense-mode at 10 khz with Q=100, drive-mode at 10.1 khz (b) Sense-mode at 10 khz with Q=1000, drive-mode at 10.1 khz Schematic of a capacitive MEMS resonator Typical spectra of EAM pick-up signals (a) Using lateral comb capacitors (linear case) (b) Using parallel plate capacitors (nonlinear case) Numerical verification of the derived Fourier series (a) Parallel plate capacitance (b) Parallel plate current Nonlinearity of the main (first order) sidebands in parallel plate EAM. 43 (a) p1(x0) and its linearization (b) Linearization errors Normalized amplitudes of the first three sidebands Fabricated and packaged test resonator (a) SEM micrograph of a quarter of the fabricated device (b) Photograph of the packaged and wirebonded device Characterization of the test structure in air Experimental measurement of linear and parallel plate EAM spectra. 50 (a) Comparison of linear and parallel plate EAM xiii

16 (b) Parallel plate EAM signal at different amplitudes of motion Signal processing for nonlinear sideband measurements Measured nonlinearity of the first two sidebands PSD of differential EAM (simulation) MEMS resonator with parallel plate detection of motion PSD of nonlinear EAM signal (simulation) SideBand Ration (SBR) detection of motion SEM micrograph of a quarter of a fabricated device Measured PSD of nonlinear EAM pick-up signal Simulink model of a resonator with parallel plate SBR detection (a) Top level schematic (b) Robust Detection block (c) Signals block (d) Resonator block Simulink model of a resonator with parallel plate SBR detection (a) Dual-phase mixing block (b) Amplitude from sideband ratio block SBR detection of motion (simulation) SNR of the conventional and SBR motional detection methods Noise figure for SBR measurement compared to conventional method Capacitive MEMS resonator with parallel plate actuators Nonlinear gradient of parallel plate capacitance Distributed mass gyroscope fabricated in EFAB process Characterization of parallel plate frequency tuning in the device Structural characterization of the test device (a) In air xiv

17 (b) In 200 mtorr vacuum Photographs of at rest and at resonant parametrically excited motion Experimental pick-up signal PSD for parametrically excited resonator Frequency responses of parametrically excited vibrations (a) Maximum amplitude of motion norm (b) Comparison of frequency responses with different norms Effect of the AC driving voltage amplitude on the response Effect of the DC bias on the response Effect of the quality factor Q on the response Comparison of the proposed gyroscope and the DVA-based design Frequency scaling of the sense-mode frequency response Diagram of the proposed sensor element with a 2-DOF sense-mode (a) Schematic of the complete structure of the gyroscope (b) Drive- and sense-mode lumped models Layouts of the proposed gyroscope, implementation #1: cross-shaped. 121 (a) With lateral-comb drive electrodes (b) With parallel-plate drive electrodes Layouts of the proposed gyroscope, implementations #2 and # (a) Implementation #2: crab-shaped (b) Implementation #3: frame-shaped FEM of the micromachined gyroscope implementation #1: cross-shaped.122 (a) Sense-mode, in-phase, 3.7 khz (b) Sense-mode, out-of-phase, 4.2 khz (c) Drive-mode resonance, 3.93 khz Frequency responses of the algorithmically designed gyroscopes (a) Effect of the operational frequency scaling (b) Effect of the sense-mode frequency spacing scaling SEM images of a fabricated MEMS gyroscope with a 2-DOF sense-mode Measured frequency responses of a lateral-comb device xv

18 (a) Drive- and sense-modes in air (b) Effect of pressure on sense-mode response Test-bed for measurement of temperature effects on the response Effects of temperature on the drive- and sense-modes in air (a) 1-DOF drive-mode, temperature sensitivity 3000 ppm/deg C (b) 2-DOF sense-mode, temperature sensitivity 351 ppm/deg C Actuation and detection scheme for rate characterization Photograph of the rate table experimental test-bed Calibration rate plot Identification of ZRO random noise modes using Root Allan variance. 138 (a) Time history of the raw and the filtered data (b) Root Allan variance of the raw output Measured ARRW and ARW distributions showing Gaussian PDF (a) ARW (HPF of ZRO) (b) ARRW (LPF of ZRO) Comparative experimental study of noise at ZRO and 50 deg/s (a) Time history (b) Root Allan variance SEM of a fabricated device Experimental characterization Robust tuning fork gyroscope Quality factor of a resonator defined using its frequency response (a) Zoomed out view of the frequency response (b) Zoomed in view of the frequency response SEM images of test-bed MEMS gyroscopes (a) Gyroscope with a 1-DOF drive-mode (b) Gyroscope with a 2-DOF drive-mode Packaged gyroscope used for Q-factor measurements Experimental setup for Q-factors characterization xvi

19 (a) Vacuum chamber (b) Electromechanical schematic of structural characterization Identification of Q-factor as function of pressure (a) Frequency responses and identification of structural parameters. 157 (b) Measured Q(P) and identification of non-viscous asymptote Qlim Experimental study of Q-factor in in-phase and anti-phase gyroscopes. 160 (a) Gyroscope with a 2-DOF drive-mode (b) In-phase actuated gyroscope with a single-dof drive-mode Experimental study of effect of imperfections on Q-factor in gyroscopes.161 (a) Gyroscopes designed in 7 by 7 mm die size (b) Gyroscopes designed in 3.5 by 3.5 mm die size FEM of thermoelastic damping in a gyroscope (a) In-phase mode Qted=1.7e6, resonant frequency is 1.46 khz (b) Anti-phase mode, Qted=1.3e6, resonant frequency is 2.18 khz Lumped element modeling of energy loss through the substrate (a) Lumped model augmented with the die substrate DOF (b) Frequency responses for different die attachment damping values Modeling of energy loss through the substrate in 2-DOF devices (a) Lumped model with two tines and the mobile substrate (b) Ideal high-q frequency response and effect of substrate damping Modeling of dissipation through the substrate in a tuning fork Structural schematic of the proposed gyroscope architecture FEM of the levered tuning fork gyroscope (a) Levered anti-phase drive-mode (b) In-phase drive-mode resonance (c) Linearly coupled anti-phase sense-mode Optical photograph of a packaged tuning fork gyroscope Measured frequency responses of the drive- and the sense-modes in air Optical characterization of the levered anti-phase drive-mode Measured rate response at atmospheric pressure xvii

20 8.7 Measured frequency response of the levered anti-phase drive-mode Measured frequency response of the dynamically balanced sense-mode Measured quality factor versus pressure for an individual tine Scale factor as a function of the sense-mode quality factor Effects of quality factor on mechanical-thermal noise (a) Noise spectral density scaling (b) RMS noise power and ARW scaling Structural schematic of the quadruple tuning fork architecture Quadruple tuning fork principle of operation, FEM (a) Drive-Mode (b) Sense-Mode SEM micrograph of an x-y coupled quadruple tuning fork gyroscope Optical photograph of a packaged quadruple tuning fork gyroscope xviii

21 List of Tables 1.1 Typical performance parameters of state-of-the-art MEMS gyroscopes Performance requirements for three grades of gyroscopes Parameters of the test device with large amplitude of motion SBR detection of the normalized amplitude of motion Parameters of the test-bed devices (in air) (a) 1-DOF drive-mode of gyroscope (b) 2-DOF drive-mode of gyroscope xix

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23 Acknowledgements First of all I would like to thank Professor Andrei M. Shkel. His positive influences range from introducing me to the captivating research area of MEMS to being my advisor and dissertation committee chair. His insightful guidance helped me to successfully reach this exciting destination and learn many lessons along the path. I am also grateful to Professors Faryar Jabbari and William C. Tang for their time and advice. I would like to acknowledge Vu Phan, Mo Kebaili, David Croasley, and Jake Hes of the Integrated Nano-Research Facility (INRF) at University of California, Irvine for assistance with microfabrication process development. I must thank fellow students of the UCI Microsystems lab and Mechanical and Aerospace Engineering department. Cenk Acar, Chris Painter, Max Perez, Monty Rivers and others have helped to create a friendly and intellectually nurturing environment. I am particularly grateful to Adam Schofield as a friend and collaborator throughout these years. This work was partially supported by the National Science Foundation Grant CMS , Systron Donner Automotive contract BEI-36974, University of California Discovery program ELE , Office of Naval Research and Naval Surface Warfare Center, Dahlgren contract N C1014, and Defence Advanced Research Projects Agency Grant Number W31P4Q xxi

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25 Curriculum Vitae Alexander A. Trusov Ph.D. in Mechanical and Aerospace Engineering University of California, Irvine M.S. in Mechanical and Aerospace Engineering University of California, Irvine B.S. in Applied Mathematics and Mechanics Moscow State University, Russia Journal Articles Field of Study Inertial Microsensors Publications C. Acar, A. R. Schofield, A. A. Trusov, L. E. Costlow, and A. M. Shkel, Environmentally Robust MEMS Vibratory Gyroscopes for Automotive Applications, accepted to IEEE Sensors Journal, A. A. Trusov, A. R. Schofield, and A. M. Shkel, Performance Characterization of a New Temperature-Robust Gain-Bandwidth Improved MEMS Gyroscope Operated in Air, in press, Sensors and Actuators A: Physical Journal, A. A. Trusov, A. R. Schofield, and A. M. Shkel, A Substrate Energy Dissipation Mechanism in In-Phase and Anti-Phase Micromachined Z-Axis Vibratory Gyroscopes, IOP Journal of Micromechanics and Microengineering 18 (2008) A. R. Schofield, A. A. Trusov, and A. M. Shkel, Effects of Operational Frequency Scaling in Multi-Degree of Freedom MEMS Gyroscopes, IEEE Sensors Journal, Vol. 8, No. 10, pp , xxiii

26 A. A. Trusov and A. M. Shkel, A Novel Capacitive Detection Scheme with Inherent Self-Calibration, IEEE Journal of Microelectromechanical Systems, Vol. 16, No. 6, pp , December A. A. Trusov and A. M. Shkel, Capacitive Detection in Resonant MEMS with Arbitrary Amplitude of Motion, IOP Journal of Micromechanics and Microengineering, Vol. 17, No. 8, pp , July Conference Papers A. A. Trusov, A. R. Schofield, and A. M. Shkel, Gyroscope Architecture with Structurally Forced Anti-Phase Drive-Mode and Linearly Coupled Anti-Phase Sense- Mode, accepted to International Solid-State Sensors, Actuators and Microsystems Conference TRANSDUCERS 2009, Denver, Colorado, USA, June A. R. Schofield, A. A. Trusov, and A. M. Shkel, Design Trade-offs of Micromachined Gyroscope Concept Allowing Interchangeable Operation in Both Robust and Precision Modes, accepted to International Solid-State Sensors, Actuators and Microsystems Conference TRANSDUCERS 2009, Denver, Colorado, USA, June A. A. Trusov, A. R. Schofield, and A. M. Shkel, Study of Substrate Energy Dissipation Mechanism in in-phase and Anti-Phase Micromachined Vibratory Gyroscopes, IEEE Sensors 2008 Conference, Lecce, Italy, October 26-29, A. R. Schofield, A. A. Trusov, and A. M. Shkel, Micromachined Gyroscope Design Allowing for Both Robust Wide-Bandwidth and Precision Mode-Matched Operation, IEEE Sensors 2008 Conference, Lecce, Italy, October 26-29, A. A. Trusov, A. R. Schofield, and A. M. Shkel, On Mechanisms of Energy Dissipation and Transfer in MEMS Vibratory Gyroscopes Operated in Vacuum, WCCM8 / ECCOMAS 2008, Venice, Italy, June 30 - July 4, A. A. Trusov, A. R. Schofield, and A. M. Shkel, New Architectural Design of a Temperature Robust MEMS Gyroscope with Improved Gain-Bandwidth Characteristics, Solid-State Sensors, Actuators, and Microsystems Workshop 2008, Hilton Head Island, SC, June 1-5, A. A. Trusov, I. Chepurko, A. R. Schofield, and A. M. Shkel, A Standalone Programmable Signal Processing Unit for Versatile Characterization of MEMS Gyroscopes, IEEE Sensors 2007 Conference, Atlanta, GA, USA, October 28-31, xxiv

27 A. R. Schofield, A. A. Trusov, and A. M. Shkel, Multi-Degree of Freedom Tuning Fork Gyroscope Demonstrating Shock Rejection, IEEE Sensors 2007 Conference, Atlanta, GA, USA, October 28-31, A. A. Trusov and A. M. Shkel, The Effect of High Order Non-Linearities on Sub- Harmonic Excitation with Parallel Plate Capacitive Actuators, ASME IDETC/CIE 2007, Las Vegas, NV, USA, September 4-7, A. A. Trusov and A. M. Shkel, A Novel Capacitive Detection Scheme with Inherent Self-Calibration, ASME IDETC/CIE 2007, Las Vegas, NV, USA, September 4-7, A. R. Schofield, A. A. Trusov, and A. M. Shkel, Structural Design Trade-offs for MEMS Vibratory Rate Gyroscopes with 2-DOF Sense Modes, ASME IDETC/CIE 2007, Las Vegas, NV, USA, September 4-7, A. A. Trusov and A. M. Shkel, Parallel Plate Capacitive Detection of Large Amplitude Motion in MEMS, International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS 07), pp , Lyon, France, June 10-14, A. R. Schofield, A. A. Trusov, C. Acar, and A. M. Shkel, Anti-Phase Driven Rate Gyroscope with Multi-Degree of Freedom Sense Mode, International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS 07), pp , Lyon, France, June 10-14, A. A. Trusov, C. Acar, and A. M. Shkel, Comparative Analysis of Distributed Mass Micromachined Gyroscopes Fabricated in SCS-SOI and EFAB, Proceedings of SPIE - The International Society for Optical Engineering, v 6174 II, Smart Structures and Materials Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, p 61742A, Patent Disclosures A. R. Schofield, A. A. Trusov, and A. M. Shkel, Micromachined Gyroscope with 2-DOF Sense Mode Allowing Interchangeable Robust and Precision Operation, Patent Pending, UC Case No A. A. Trusov, A. R. Schofield, and A. M. Shkel, Temperature-Robust MEMS Gyroscope with 2-DOF Sense-Mode Addressing the Tradeoff between Bandwidth and xxv

28 Gain, US Patent pending, UC Case No A. A. Trusov and A. M. Shkel, A Capacitive Detection Scheme with Inherent Self-Calibration for Resonant MEMS, UC Case No ott, US Patent , A1 pending. xxvi

29 Abstract of the Dissertation Investigation of Factors Affecting Bias Stability and Scale Factor Drifts in Coriolis Vibratory MEMS Gyroscopes By Alexander A. Trusov Doctor of Philosophy in Mechanical and Aerospace Engineering University of California, Irvine, 2009 Professor Andrei M. Shkel, Chair This Ph.D. dissertation studies factors affecting bias stability and scale factor drifts in micromachined Coriolis vibratory gyroscopes with electrostatic/capacitive transduction. The work provides improved understanding of the factors affecting accuracy of measurements, leading to novel design approaches in both mechanical and signal processing domains. This thesis developed a capacitive detection scheme with inherent self-calibration. The method, called Side-Band Ratio (SBR) detection, is robust to variations of such critical parameters as the nominal capacitance, frequency and amplitude of the probing voltage, and electronics gain. The SBR technique allows to drastically improve gyroscope scale factor accuracy and repeatability while simultaneously increasing signal-to-noise ratio and bias stability by 20 to 30 db. This thesis proposed a temperature robust gyroscope with improved gain-bandwidth characteristics. The introduced concept was analyzed theoretically and characterized xxvii

30 experimentally. Devices with 2.5 khz operational frequency were designed, fabricated, and tested in air, demonstrating sense-mode 3 db bandwidth of 250 Hz, achieved for the first time without sacrifice of the resonant gain. The new gyroscope s uncompensated temperature coefficients of bias and scale factor were 313 ( /h)/ C and 351 ppm/ C, respectively. The gyroscope provides low temperature sensitivity on par with quartz tuning fork gyroscopes and bandwidth increased beyond the capabilities of conventional gyroscopes. At the same time, rate sensitivity, quadrature, resolution, and angle random walk of the new gyroscope are as good as for the state-of-the-art conventional gyroscopes. This thesis also investigated the effects of design and die attachment on the gyroscope quality factor. The presented family of ultra-high quality factor gyroscopes with reduced sensitivity to packaging stresses utilizes two and four coupled masses actuated in anti-phase by means of a mechanical synchronization system. Prototypes characterized in vacuum demonstrated drive-mode quality factor of 67,000 and ultrahigh sense-mode quality factor of 125,000. The ultra-high mechanical scale factor of 0.4 nm/( /h) translates into drastically improving sensor noise performance and bias stability. xxviii

31 Chapter 1 Introduction This chapter explains the motivation of the research. The discussion begins with a brief review of the trends in silicon micromachined gyroscopes followed by a detailed literature review. Then, the objectives of this research are stated. The chapter is concluded with the outline of the dissertation and brief discussions of the developed results. 1.1 Trends in Silicon Micromachined Gyroscopes Since their initial introduction almost two decades ago in 1991 [1], the resolution of silicon micromachined gyroscopes has been improving by approximately a factor of ten every two years [2]. Most recent literature on state-of-the-art vibratory MEMS gyroscopes with eletrostatic/capacitive transduction reports precision on the order of 10 /h [3] with a trend of achieving better than 1 /h precision in controlled laboratory conditions [4]. Despite the remarkable continuing improvement in the angular rate measurement resolution and precision achieved by micromachined silicon gyroscopes, measurement repeatability, accuracy, and drift of the state-of-the-art MEMS sensors have consistently remained 2 to 3 orders of magnitude worse, with typical values on the order 1

32 5 Year 2003: y=0.0125x Sensor Output, V Ideal Calibration 1 Year 2006: y=0.0121x Input Angular Rate, /s Figure 1.1: Calibration of an Analog Devices ADXRS150 MEMS gyroscope: ideal vs. year 2003 vs. year The gyroscope s measured responses are different from both the nominal data-sheet response and from each other, resulting in up to 10 /s measurement error. of 500 /h [3]. The relatively poor stability and drift of state-of-the-art MEMS gyroscopes, summarized in Table 1.1 using the data presented in [3], limits their usability in a variety of field applications where autonomous and repeatable operation of the sensors is required over extended periods of time in harsh environmental conditions (with ambient vibrations and shocks and fluctuations of such parameters as temperature, pressure, humidity, etc.). In addition to in-operation drifts of the sensor bias and scale factor, silicon micromachined gyroscopes are also prone to long term degradation of performance characteristics. Figure 1.1 shows calibration data of an ADXRS150 one of the most commercially successful MEMS gyroscopes on the present market. This example illustrates a fundamental performance limitation of the current inertial MEMS sensor technology. 2

33 In Figure 1.1, the gyroscope s response was experimentally measured in laboratory conditions (constant room temperature, negligible variations of humidity, etc.) in 2003 and again in 2006 to study the long-term measurement repeatability and drift of the sensor bias and scale factor. The experiment reveals the major issues of the scale factor and bias drift, effectively limiting their current use to mostly Rate and rarely Tactical grade applications, as presented in Table 1.2 based on the data from [4]. The gyroscope s measured responses are different from both the nominal data-sheet response and from each other, which results in up to 10 /s measurement error and would render a sensor useless to the application of its deployment. Conventionally, gyroscopes are calibrated to identify the actual bias and scale factor (and to compensate for their deviations from the respective nominal values) using laboratory equipment before field deployment. Unfortunately, such identification and calibration procedures are costly, time consuming, and require strictly controlled laboratory conditions. Also, factory identification and calibration of a sensor does not prevent subsequent gradual loss of its calibration, which can ultimately render the sensor useless. Currently, the only feasible solution is off-line re-calibration in inertial metrology laboratories through the use of high stability and low-drift reference rotational stages Table 1.1: Typical performance parameters of state-of-the-art MEMS gyroscopes [3]. Parameter Value Rate resolution 5-10 /h/ Hz Bias thermal sensitivity 10 /h/ C Bias repeatability (over temperature and turn off) 30 /h Scale factor thermal sensitivity 250 ppm/ C Scale factor repeatability (over temperature and turn off) 400 ppm 3

34 Table 1.2: Performance requirements for three grades of gyroscopes [2]. Parameter, unit Rate grade Tactical grade Inertial grade Angle random walk (ARW), / h > <0.001 ARW-equivalent rate resolution, /h/ Hz > <0.06 Bias drift, /h <0.01 Scale factor accuracy, ppm <1 Full scale range, /s >500 >400 Maximum shock in 1 ms, g Bandwidth, Hz > (rate tables) which provide an updated measure of the sensor bias and scale factor. To perform the re-calibration, a sensor has to be extracted from its application, brought to a calibration laboratory, re-calibrated, and finally installed back into its application. 1.2 Literature Review In this section we discuss recently published milestones in silicon micromachined gyroscope development in industry and academia. Comprehensive reviews of MEMS gyroscope development prior to are presented in [5, 6] with an extensive review of the industrial trends presented in [7]. Here we discuss the major published results on silicon micromachined gyroscopes, while reviews of literature on the more narrow topics, such as actuation and detection of motion, packaging, and mechanical design, are presented in the chapters dedicated to these specific aspects and subsystems of a silicon micromachined rate sensor. We begin the review with publications 4

35 on the commercially available silicon micromachined Coriolis vibratory gyroscopes. In 2002, Analog Devices reported a surface micromachined polysilicon gyroscope with a 50 /h random bias instability achieved at the averaging time below 1 minute [8]. The device was not vacuum packaged, which in combination with a 4 µm thin structural layer translates into a relatively low sense-mode quality factor on the order of 20, and was single-chip integrated with the electronics. This family of gyroscopes have since gained a widespread acceptance and popularity in consumer and lower grade automotive markets. According to Table 1.1, this device falls into the middle of the Rate grade category, as defined in [2]. Publication [8] also projected that the random bias can be improved by approximately 80 % by introducing new electronics design (this suggests mechanical-thermal noise on the level of 10 /h). Progress on this family of gyroscopes was reported in [9]. The new generation design of the gyroscope has four electrically coupled, low quality factor (on the order of 20) surface micromachined polysilicon Coriolis vibratory gyroscopes integrated on the same die (together with the electronics) in order to suppress the effect of external vibrations; improvement of the bias instability was also reported. A higher performance polysilicon micromachined gyroscope from Bosch was reported in 2005 [10] and 2007 [11]. The device has 1 /h random bias instability, showing its potential for the Tactical grade performance category, Table 1.1. However, this high stability occurs only at extremely long, impractical averaging times on the order 160 minutes, and instability at practical bandwidths was much worse (30 /h for a 10 Hz bandwidth, and more than 100 /h for a 100 Hz bandwidth). This device was not integrated with electronics on the die level (hybrid integration of the MEMS and the electronics dies was 5

36 done on a package level), and its relatively low noise levels were associated with costly vacuum packaging needed to achieve high sense-mode quality factor and mechanical sensitivity. A 2006 publication [3] from Charles Stark Draper Laboratory provides an insightful review of performance characteristics of a few additional commercial MEMS gyroscopes. According to this paper, the uncompensated thermal sensitivities of higher performance quartz tuning fork gyroscopes from Custom Sensors and Technologies (formerly Systron Donner/BEI) was reported at 300 ppm/ C achieved primarily due to extremely low sensitivity of quartz to temperature. Analog Devices automotive grade angular rate sensor ADXRS150 scale factor uncompensated thermal sensitivity was quoted as 1700 ppm/ C. Honeywell s dissolved wafer, single crystal silicon on glass, commercial device based on the Draper laboratory design achieves a 250 ppm/ C scale factor temperature coefficient. The literature survey confirms that in recent years performance of the commercial MEMS gyroscopes continued to improve, reaching the criteria of Tactical grade category; however, performance of these gyroscopes is still two to three orders of magnitude away from the Inertial grade, [2]. The reported progress mostly comes with costly refinements of the fabrication and vacuum packaging processes. Most reported commercial gyroscopes have intricate electronic compensation for such parameters as temperature drift. From the mechanical structure point of view, most commercial gyroscopes consist of several coupled, anti-phase driven (i.e., tuning-fork design) vibratory gyroscope elements (tines) to suppress ambient vibrations in such harsh environment applications as automotive and projectile/missle guidance. 6

37 Next, we review reported progress on gyroscope development from smaller research and development entities and academic research groups. A single crystal silicon bulk micromachined gyroscope with sub- /h bias instability was reported in [12]. The gyroscope operates in a mode-matched regime, which is known to provide increased scale factor, but at the cost of increased sensitivity to environmental changes. The single crystal silicon device was fabricated using a custom process which, unlike most common bulk micromachining techniques, results in a smooth sidewall and bottom surface profiles of the gyroscope structure. The good performance numbers of the device reported for the 10 mtorr vacuum level are attributed to the reduction of quadrature signal due to the improved fabrication. In [13], a field programmable gate array (FPGA) implementation of a digital control scheme for a silicon micromachined gyroscope was reported. Bias instability of 3.2 /hr was achieved using a mechanical sensor element of a conventional MEMS gyroscope aimed for the automotive industry. The good stability of the output signal is attributed to elimination of low-frequency noise of analog components. An interesting implementation of a vacuum packaged single crystal silicon bulk micromachined gyroscope based on a metal interlaid silicon on glass (SOG) fabrication process process was reported in [14]. The device has a 17 /h bias instability, attributed to the 0.1 mtorr vacuum packaging and good geometric profile of the 80 µm thick structures. A single crystal silicon on insulator (SOI) gyroscope with 1 /h bias instability was reported in [4]. The vacuum operated device has a very high quality factor 7

38 on the order of 10 5 and is mode-matched by means of electrostatic tuning to achieve very high mechanical sensitivity to the input angular rate at the cost of bandwidth and robustness. In [15] a gyroscope operated in air is demonstrated to have 1.5 /s bias instability. The device had symmetrically decoupled design architecture and was fabricated using commercial SOIMUMPS with a 2.6 µm minimal feature at 25 µm structural thickness. Later, the same group demonstrated a gyroscope fabricated in the same commercial process achieving 200 /h instability in [16]. This device utilized a tuning fork architecture with a relatively high drive-mode amplitude of up to 45 µm. To summarize, several main directions of research are found in the recent literature on micromachined vibratory gyroscopes. Reported progress in noise performance is mostly achieved by means of improving and fine-tuning the fabrication processes [12, 14, 17, 18] and mechanical trimming [19] to improve geometric profiles of the microstructures leading to lower quadrature signals, and employing costly vacuum packaging [10, 11, 14] to increase the quality factors leading to higher signal-to-noise ratio. Several groups investigated advanced circuit designs and electronic closed loop schemes for low noise detection and control of conventional micromachined gyroscopes [13, 20 24]. Few papers, for instance, [25 27], focus on mechanical design techniques for inherent structural robustness. There is a gap in the literature on factors affecting bias instability of gyroscopes, tradeoffs between random and deterministic drifts, and original structural design approaches for the mechanical sensor element. Novel signal processing schemes utilizing inherent nonlinearities of capacitive transduction that can yield more efficient and robust excitation and detection with increased signal-to-noise characteristics also 8

39 present an opportunity for increasing sensitivity and robustness of MEMS vibratory gyroscopes. 1.3 Research Objective The goal of this dissertation is to study factors affecting bias instability and scale factor drift in micromachined Coriolis vibratory gyroscopes with electrostatic/capacitive transduction, and to propose new transduction techniques, actuation, signal processing, and self-calibration schemes, as well as innovative designs for the mechanical sensor element to improve sensor robustness, accuracy and overall performance. Two different types of bias and scale factor drifts can be considered: deterministic and random (i.e., stochastic). Deterministic drifts include temperature- and pressure-dependent changes of the sensor response characteristics, along with the effects of electromechanical nonlinearities, package stresses and shocks, materials aging, etc. Deterministic drifts define accuracy and repeatability of the sensor measurements and can be minimized by robust mechanical sensor element architecture and signal processing scheme design, automatic re-calibration techniques, and closed-loop operation. Random drifts occur due to the random processes in the electrotonic circuitry (such as shot noise, Johnson-Nyquist noise, flicker noise, etc.), the Brownian motion in the mechanical sensor element structure and the surrounding gas, and in the transduction interfaces between mechanical and electrical subsystems. Random bias and scale factor drifts define precision of the sensor measurement and also affect the long term accuracy and repeatability. Different noise modes i.e., random processes with different autocorrelation laws dominate the output at different time constants. Short term effect of random drifts in the circuitry can be mitigated by increasing the 9

40 mechanical scale factor and sensitivity of the sensor in order to reduce rate-equivalent power of the noise. At high levels of mechanical sensitivity to the input angular rate, mechanical-thermal noise in the sensor element dominates the output. The effect of this noise mechanism can be reduced by proper scaling of the mechanical design. 1.4 Dissertation Outline In Chapter 2, an introduction to Coriolis vibratory gyroscopes, their dynamics, and principles of operation are given. Forced vibrations dynamics for measurement of quasi-static input angular rates are considered together with inherent gain linearity and bandwidth limitations. In Chapter 3, capacitive detection of vibratory motion using Electromechanical Amplitude Modulation (EAM) is investigated. Nonlinearity of detection using parallel plate capacitors is studied in comparison to the linear detection using lateral comb electrodes. An algorithm for measurement nonlinearity compensation is proposed and demonstrated, enabling large amplitude operation with improved signal-to-noise ratio without sacrifice in accuracy. Results of the chapter allow improving accuracy and linearity of gyroscope scale factor. In Chapter 4, a new method of capacitive detection using parallel plate electrodes is proposed. The algorithm is based on sideband ratio (SBR) detection and constructively takes advantage of the inherent nonlinearity of the detection signal. The proposed detection scheme is shown to provide robust, self-calibrated measurement of vibratory motion and allows for improving accuracy and signal-to-noise ratio at the same time. When applied to Coriolis vibratory gyroscopes, the technique drastically reduces scale factor drifts. In Chapter 5, the topic of capacitive actuation of periodic and quasi-periodic mo- 10

41 tion is explored. Nonlinearity of parallel plate capacitance is shown to enable novel actuation schemes. Particular focus of the discussion is on parametric excitation using a higher frequency driving voltage. This approach to actuation provides surprising robustness to pressure variations and enables improved performance due to easy suppression of drive voltage feed-through. Application of the nonlinear actuation method to Coriolis vibratory gyroscopes can significantly increase sensor resolution and thus improve bias stability. In Chapter 6, a new design architecture for a robust micromachined gyroscope is proposed. The design utilizes a single-dof drive-mode with a symmetricallydecoupled suspension and a fully-coupled, 2-DOF sense-mode. The new gyroscope architecture provides scale factor robustness to fabrication imperfections and operational conditions without the sacrifice in bias stability inherent to the previous robust rate sensors. In Chapter 7, design and packaging tradeoffs in micromachined gyroscopes are considered with the focus on exchange of mechanical energy between the MEMS device and its package. The type of actuation is identified as the key factor to dissipation of vibrational energy through the substrate in vacuum-operated micromachined gyroscopes. Anti-phase gyroscope design can eliminate the effect of package stresses and external accelerations to improve scale factor accuracy and bias stability. In Chapter 8, a family of new design architectures for ultra-high, i.e., above 100 thousand, quality factor gyroscopes is proposed. The new architectures utilize levering mechanism to synchronize drive- and sense-mode motion of multiple proof masses and provide complete momentum and torque balance to suppress dissipation of energy through the substrate. The new designs also provide mechanical cancelation of external vibrations and shocks as well as robustness to package stresses. Ultra-high sensitivity to the input angular rate translates into drastic improvements of sensor 11

42 resolution and bias stability. Finally, Chapter 9 concludes the dissertation with a summary of contributions and discussion of ideas for future research directions. 12

43 Chapter 2 Coriolis Vibratory Gyroscopes In this chapter, we review dynamics and principles of operation of Coriolis vibratory gyroscopes with a focus on single z-axis sensors. Equations of motion of the proof mass are derived based on Newton s equations for a non-inertial frame of reference. For the forced vibrations case, a formula for measurement of quasi-constant input angular rates is derived. Gain linearity and dynamic bandwidth limitations are also discussed. 2.1 Newton s Equations of Motion in a Non-Inertial Frame In this section, we review some of the facts of classical mechanics [28] necessary for understanding the principles behind the operation of vibratory gyroscopes. Let Sξηζ denote an inertial reference frame, in which a non-inertial frame Oxyz is moving with a linear acceleration A 0 = (A 0x, A 0y, A 0z ) and an angular velocity Ω = (Ω x, Ω y, Ω z ), where the components of the vectors are given with respect to the moving frame. Coordinates of a point mass in the inertial and non-inertial reference frames are given by vectors ρ = (ξ, η, ζ) and r = (x, y, z), respectively. Newton s equations of motion of the point mass m under the action of force vector 13

44 F with respect to the inertial reference frame Sξηζ are given by ρ = A abs = F /m. According to the rules of differentiation in a moving reference frame, r = A rel = F [ ]) (A m 2 [Ω ṙ] 0 + [Ω [Ω r]] + Ω r. (2.1) For convenience of further analysis, we distinguish between internal forces F int and external forces F ext. The vector based Equation 2.1 can be expressed in scalar form as A ox (t) F ( ) int,x(t) m + x(t) Ω 2 y(t) Ω 2 z(t) 2Ω z (t)ẏ(t) + 2Ω y (t)ż(t)+ (2.2) ( + z(t) Ω x (t)ω z (t) + Ω ) ( y (t) + y(t) Ω x (t)ω y (t) Ω ) z (t) + ẍ(t) = F ext,x(t) m, A oy (t) F ( ) int,y(t) m + y(t) Ω 2 x(t) Ω 2 z(t) + 2Ω z (t)ẋ(t) 2Ω x (t)ż(t)+ ( + z(t) Ω y (t)ω z (t) Ω ) ( x (t) + x(t) Ω x (t)ω y (t) Ω ) z (t) + ÿ(t) = F ext,y(t) m, A oz (t) F ( ) int,z(t) m + z(t) Ω 2 x(t) Ω 2 y(t) 2Ω z (t)ẋ(t) + 2Ω x (t)ẏ(t)+ ( + y(t) Ω y (t)ω z (t) + Ω ) ( x (t) + x(t) Ω x (t)ω z (t) Ω ) y (t) + z(t) = F ext,z(t) m. Equation 2.2 describes the motion of a point mass m in the non-inertial reference frame Oxyz and are used in the following sections to derive the governing equations of a vibratory Coriolis gyroscope in the forced vibrations mode of operation. 2.2 Mechanical Sensor Element Dynamics In this section, we consider a theoretical model [29] of a single axis Coriolis vibratory gyroscope mechanical sensor element depicted in Figure 2.1. In this model, the noninertial reference frame Oxyz is associated with the sensor s package. A point mass 14