Differentiable periodic maps

MEMS Vibratory Gyroscopes provides a solid foundation in the theory and fundamental operational principles of micromachined vibratory rate gyroscopes, and introduces structural designs that provide inherent robustness against structural and environmental variations. In the first part, the dynamics of the vibratory gyroscope sensing element is developed, common micro-fabrication processes and methods commonly used in inertial sensor production are summarized, design of mechanical structures for both linear and torsional gyroscopes are presented, and electrical actuation and detection methods are discussed along with details on experimental characterization of MEMS gyroscopes. In the second part, design concepts that improve robustness of the micromachined sensing element are introduced, supported by constructive computational examples and experimental results illustrating the material.

Part I Fundamentals of Micromachined Vibratory Gyroscopes
1 Introduction	p. 3
1.1 The Coriolis Effect	p. 3
1.2 Gyroscopes	p. 4
1.3 The MEMS Technology	p. 5
1.4 Micromachined Vibratory Rate Gyroscopes	p. 6
1.5 Applications of MEMS Gyroscopes	p. 8
1.6 Gyroscope Performance Specifications	p. 8
1.7 A Survey of Prior Work on MEMS Gyroscopes	p. 10
1.8 The Robustness Challenge	p. 14
1.9 Inherently Robust Systems	p. 15
1.10 Overview	p. 16
2 Fundamentals of Micromachined Gyroscopes	p. 17
2.1 Dynamics of Vibratory Rate Gyroscopes	p. 17
2.1.1 Linear Gyroscope Dynamics	p. 17
2.1.2 Torsional Gyroscope Dynamics	p. 22
2.2 Resonance Characteristics	p. 25
2.3 Drive-Mode Operation	p. 28
2.4 The Coriolis Response	p. 29
2.4.1 Mode-Matching and [delta]f	p. 32
2.4.2 Phase Relations and Proof-Mass Trajectory	p. 36
2.5 Summary	p. 42
3 Fabrication Technologies	p. 43
3.1 Microfabrication Techniques	p. 43
3.1.1 Photolithography	p. 44
3.1.2 Deposition	p. 46
3.1.3 Etching	p. 48
3.1.4 Wafer Bonding	p. 51
3.2 Bulk Micromachining Processes	p. 52
3.2.1 SOI-Based Bulk Micromachining	p. 53
3.2.2 Silicon-on-Glass Bulk Micromachining	p. 56
3.3 Surface-Micromachining Processes	p. 59
3.4 Combined Surface-Bulk Micromachining	p. 63
3.5 CMOS Integration	p. 64
3.5.1 Hybrid Integration	p. 64
3.5.2 Monolithic Integration	p. 65
3.6 Packaging	p. 67
3.6.1 Wafer-Level Packaging	p. 68
3.6.2 Vacuum Packaging	p. 69
3.7 Summary	p. 71
4 Mechanical Design of MEMS Gyroscopes	p. 73
4.1 Mechanical Structure Designs	p. 73
4.2 Linear Vibratory Systems	p. 74
4.2.1 Linear Suspension Systems	p. 75
4.2.2 Linear Flexure Elements	p. 83
4.3 Torsional Vibratory Systems	p. 87
4.3.1 Torsional Suspension Systems	p. 88
4.3.2 Torsional Flexure Elements	p. 90
4.4 Anisoelasticity and Quadrature Error	p. 93
4.4.1 Quadrature Compensation	p. 100
4.5 Damping	p. 102
4.5.1 Viscous Damping	p. 102
4.5.2 Viscous Anisodamping	p. 104
4.5.3 Intrinsic Structural Damping	p. 105
4.6 Material Properties of Silicon	p. 107
4.7 Design for Robustness	p. 108
4.7.1 Yield	p. 108
4.7.2 Vibration Immunity	p. 109
4.7.3 Shock Resistance	p. 109
4.7.4 Temperature Effects	p. 109
4.8 Summary	p. 110
5 Electrical Design of MEMS Gyroscopes	p. 111
5.1 Introduction	p. 111
5.2 Basics of Capacitive Electrodes	p. 111
5.3 Electrostatic Actuation	p. 113
5.3.1 Variable-Gap Actuators	p. 113
5.3.2 Variable-Area Actuators	p. 114
5.3.3 Balanced Actuation	p. 116
5.4 Capacitive Detection	p. 117
5.4.1 Variable-Gap Capacitors	p. 117
5.4.2 Variable-Area Capacitors	p. 118
5.4.3 Differential Sensing	p. 119
5.5 Capacitance Enhancement	p. 120
5.5.1 Gap Reduction by Fabrication	p. 121
5.5.2 Post-Fabrication Capacitance Enhancement	p. 122
5.6 MEMS Gyroscope Testing and Characterization	p. 124
5.6.1 Frequency Response Extraction	p. 125
5.6.2 Capacitive Sense-Mode Detection Circuits	p. 133
5.6.3 Rate-Table Characterization	p. 138
5.7 Summary	p. 139
Part II Structural Approaches to Improve Robustness
6 Linear Multi-DOF Architecture	p. 143
6.1 Introduction	p. 143
6.2 Fundamentals of 2-DOF Oscillators	p. 144
6.3 The 2-DOF Sense-Mode Architecture	p. 149
6.3.1 Gyroscope Dynamics	p. 150
6.3.2 Coriolis Response	p. 151
6.3.3 Illustrative Example	p. 155
6.3.4 Conclusions on the 2-DOF Sense-Mode Architecture	p. 157
6.4 The 2-DOF Drive-Mode Architecture	p. 158
6.4.1 Gyroscope Dynamics	p. 159
6.4.2 Dynamical Amplification in the Drive-Mode	p. 162
6.4.3 Illustrative Example	p. 163
6.4.4 Conclusions on the 2-DOF Drive-Mode Architecture	p. 165
6.5 The 4-DOF System Architecture	p. 166
6.5.1 The Coriolis Response	p. 169
6.5.2 Dynamics of the 4-DOF Gyroscope	p. 170
6.5.3 Parameter Optimization	p. 172
6.5.4 Illustrative Example	p. 177
6.5.5 Conclusions on the 4-DOF System Architecture	p. 179
6.6 Demonstration of 2-DOF Oscillator Robustness	p. 180
6.7 Summary	p. 185
7 Torsional Multi-DOF Architecture	p. 187
7.1 Introduction	p. 187
7.2 Torsional 3-DOF Gyroscope Structure and Theory of Operation	p. 189
7.2.1 The Coriolis Response	p. 191
7.2.2 Gyroscope Dynamics	p. 192
7.2.3 Cross-Axis Sensitivity	p. 194
7.3 Illustration of a MEMS Implementation	p. 195
7.3.1 Suspension Design	p. 195
7.3.2 Finite Element Analysis	p. 197
7.3.3 Electrostatic Actuation	p. 198
7.3.4 Optimization of System Parameters	p. 199
7.3.5 Sensitivity and Robustness Analyses	p. 200
7.4 Experimental Characterization	p. 201
7.5 Summary	p. 206
8 Distributed-Mass Architecture	p. 207
8.1 Introduction	p. 207
8.2 The Approach	p. 207
8.2.1 The Coriolis Response	p. 210
8.2.2 Wide-Bandwidth Operation for Improving Robustness	p. 211
8.3 Theoretical Analysis of the Trade-offs	p. 213
8.4 Illustrative Example	p. 215
8.4.1 Prototype Design	p. 215
8.4.2 Experimental Characterization Results	p. 217
8.5 Summary	p. 224
9 Conclusions and Future Trends	p. 225
9.1 Introduction	p. 225
9.2 Comparative Analysis of the Presented Concepts	p. 226
9.2.1 2-DOF Oscillator in the Sense-Mode	p. 226
9.2.2 2-DOF Oscillator in the Drive-Mode	p. 226
9.2.3 Multiple Drive-Mode Oscillators	p. 227
9.3 Demonstration of Improved Robustness	p. 227
9.3.1 Temperature Dependence of Drive and Sense-Modes	p. 228
9.3.2 Rate-Table Characterization Results	p. 229
9.3.3 Comparison of Response with a Conventional Gyroscope	p. 231
9.4 Scale Factor Trade-off Analysis	p. 232
9.5 Future Trends	p. 236
9.5.1 Anti-Phase 2-DOF Sense Mode Gyroscope	p. 237
9.5.2 2-DOF Sense Mode Gyroscope with Scalable Peak Spacing	p. 242
9.6 Conclusion	p. 245
References	p. 247
Index	p. 255

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