Mechanical Engineering Series

Frederick F. Ling Series Editor

Springer Science+Business Media, LLC

Mechanical Engineering Series

J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms

P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory

J.M. Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis

I.J. Busch-Vishniac, Electromechanical Sensors and Actuators

J. Chakrabarty, Applied Plasticity

G. Chryssolouris, Laser Machining: Theory and Practice

V.N. Constantinescu, Laminar Viscous Flow

G.A. Costello, Theory of Wire Rope, 2nd ed.

K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems

M.S. Darlow, Balancing of High-Speed Machinery

J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability

IF. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed.

P.A. Engel, Structural Analysis of Printed Circuit Board Systems

A.C. Fischer-Cripps, Introduction to Contact Mechanics

J. Garcia de Jal6n and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge

W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach

K.C. Gupta, Mechanics and Control of Robots

J. Ida and J.P .A. Bastos, Electromagnetics and Calculations of Fields

M. Kaviany, Principles of Convective Heat Transfer, 2nd ed.

M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed.

E.N. Kuznetsov, Underconstrained Structural Systems

P. Ladeveze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation

(continued after index)

Anthony Lawrence

Modern Inertial Technology N avigation, Guidance, and Control

Second Edition

With 201 Figures

Springer

Anthony Lawrence 32 Sunny Hill Road Lunenburg, MA 01462 USA

Series Editor Frederick F. Ling Emest F. Gloyna Regents Chair in Engineering Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712-1063 USA

and William Howard Hart Professor Emeritus Department of Mechanical Engineering,

Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY 12180-3590 USA

Library of Congress Cataloging-in-Publication Data Lawrence, Anthony, 1935-

Modem inertial technology : navigation, guidance, and control / Anthony Lawrence - 2nd ed.

p. cm. - (Mechanical engineering series) Includes bibliographical references and index. ISBN 978-1-4612-7258-8 ISBN 978-1-4612-1734-3 (eBook) DOI 10.1007/978-1-4612-1734-3 l. Inertial navigation (Aeronautics) 1. Title. II. Series:

Mechanical engineering series (Berlin, Germany) TL588.5.L38 1998 629. 132'5 l-dc2 1 98-13047

Printed on acid-free paper.

© 1998, 1993 Springer Science+Business Media New York Origina1ly published by Springer-Verlag New York, Inc. in 1998 Softcover reprint ofthe hardcover 2nd edition 1998 AII rights reserved. This work may not be translated or copied in whole or in part without the writ­ten permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especialIy identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Anthony K. Guardiola; manufacturing supervised by Jeffrey Taub. Camera-readv copv prepared from the author's WordPerfect files.

9 8 7 6 5 4 3 (Corrected third printing, 2001)

ISBN 978-1-4612-7258-8 SPIN 10843117

Series Preface

Mechanical Engineering, an engineering discipline borne of the needs of the in­dustrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of pro­ductivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research mono­graphs intended to address the need for information in contemporary areas of me­chanical engineering.

The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and re­search. We are fortunate to have a distinguished roster of consulting editors on the advisory board, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the next page of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, ther­mal science, and tribology.

I am pleased to present this volume in the Series: Modern Inertial Technology: Navigation, Guidance, and Control, Second Edition, by Anthony Lawrence. The selection of this volume underscores again the interest of the Mechanical Engi­neering series to provide our readers with topical monographs as well as graduate texts in a wide variety of fields.

Austin, Texas Frederick F. Ling

Mechanical Engineering Series

Frederick F. Ling Series Editor

Advisory Board

Applied Mechanics

Biomechanics

Computational Mechanics

Dynamical Systems and Control

Energetics

Mechanics of Materials

Processing

Production Systems

Thermal Science

Trihology

F.A. Leckie University of California, Santa Barbara

V.C.Mow Columbia University

H.T. Yang University of California, Santa Barbara

K.M. Marshek University of Texas, Austin

J.R. Welty University of Oregon, Eugene

I. Finnie University of California, Berkeley

K.K. Wang Cornell University

G.A. Klutke Texas A&M University

A.E. Bergles Rensselaer Polytechnic Institute

W.O. Winer Georgia Institute of Technology

Preface

Since 1993, when the first edition of this book was published, inertial technology has changed in two ways. First, the maturing of the Global Positioning System (GPS) has encouraged electronics manufacturers to produce simple, inexpensive ($100) position indicators for the general public. Also, silicon micromachined gyroscopes and accelerometers have come of age and are now mass-produced. Together, these developments have impacted the low-cost, low-accuracy inertial system market.

Secondly, the Interferometric Fiber Optic Gyroscope (IFOG) has become a reliable and accurate sensor and has found a market in heading and attitude reference systems. Different IFOG technologies have converged to a fairly standard instrument.

In this second edition, we have generally updated each chapter and expanded the text and references relating to the micromachined sensors and the IFOG. While we cannot describe some proprietary design features, there is enough public literature available so that the reader can understand recent technological advances.

We decided not to remove descriptions of some of the older technology (floated gyros, for example), as these may well be in the inventory for years to come. Also, the Pendulous Integrating Gyroscope Accelerometer (PIGA), based on this technology, has not yet been bettered as a precise accelerometer, although engineers are still attempting to make a "modem," solid-state, less expensive, and more reliable replacement.

There were a few errors in the first edition that have been corrected. Our thanks to those who took the time to point them out.

Whitman, MA Anthony Lawrence

Contents

Series Preface

Preface

Introduction

1. An Outline of Inertial Navigation Navigation's Beginnings Inertial Navigation Maps and Reference Frames The Inertial Navigation Process Inertial Platforms

Heading and Attitude Reference Systems Schuler Tuning Gimbal Lock

Strapdown Systems System Alignment

Gyrocompassing Transfer Alignment

Advantages and Disadvantages of Platform Systems Advantages Disadvantages

Advantages and Disadvantages of Strapdown Systems Advantages Disadvantages

Aiding Inertial Navigators The Global Positioning System Applications of Inertial Navigation Conclusions References

v

vii

1

4 4 6 6 8 9

11 12 12 13 15 16 17 17 17 17 18 18 18 19 20 22 22 23

x Contents

2. Gyro and Accelerometer Errors and Their Consequences 25 Effect of System Heading Error 25 Scale Factor 26 Nonlinearity and Composite Error 27

System Error from Gyro Scale Factor 27 Asymmetry 28

Bias 28 System Error from Accelerometer Bias 28 Tilt Misalignment 30 System Error from Accelerometer Scale Factor Error 30 System Error from Gyro Bias 30

Random Drift 31 Random Walk 32 Dead Band, Threshold, and Resolution 32 Hysteresis 33 Day-to-Day Uncertainty 33 Gyro Acceleration Sensitivities 34

g-Sensitivity 34 Anisoelasticity 35

Rotation-Induced Errors 36 Angular Acceleration Sensitivity 37 Anisoinertia 37

Angular Accelerometers 38 Angular Accelerometer Threshold Error 39

The Statistics of Instrument Performance 39 Typical Instrument Specifications 40 References 42

3. The Principles of Accelerometers 43 The Parts of an Accelerometer 43 The Spring-Mass System 44

QFactor 47 Bandwidth 48

Open-Loop Pendulous Sensors 48 Cross-Coupling and Vibropendulous Errors 48 Pickoff Linearity 50

Closed-Loop Accelerometers 50 Open-Loop Versus Closed-Loop Sensors 50 Sensor Rebalance Servos 51

Binary Feedback 51 Ternary Feedback 53 Pulse Feedback and Sensors 53

The Voltage Reference Problem 54 Novel Accelerometer Principles 54

Surface Acoustic Wave Accelerometer 55

Contents xi

Fiber -Optic Accelerometers 55 References 56

4. The Pendulous Accelerometer 57 A Generic Pendulous Accelerometer 57

Mass and Pendulum Length 57 Scale Factor 58 The Hinge 59 The Pickoff 59 The Forcer and Servo 60

The IEEE Model Equations 60 The "Q-Flex" Accelerometer 61

The Capacitive Pickoff 62 The Forcer 63

Other Electromagnetic Pendulous Accelerometers 66 Moving Magnet Forcers 66 Electrostatic Forcers 66

The Silicon Accelerometer 67 References 70

5. Vibrating Beam Accelerometers 72 The Vibration Equation 72 The Resolution of a Vibrating Element Accelerometer 74 The Quartz Resonator 75 VBAs in General 76 The Accelerex Design 77

Accelerex Signal Processing 78 The Kearfott Design 79 Silicon Micromachined VBAs 81 Comparison of Free and Constrained Accelerometers 82

General Comparison of the SPA and VBA 82 Comparison of Performance Ranges 83

Conclusion 83 References 84

6. The Principles of Mechanical Gyroscopes 85 Angular Momentum 85 The Law of Gyroscopics 86

Parasitic Torque Level 87 The Advantage of Angular Momentum 87

The Spinning Top-Nutation 88 Equations of Spinning Body Motion 89

Coriolis Acceleration 90

xii Contents

The Ice Skater 92 Gyroscopes with One and Two Degrees of Freedom 92 Conclusion 93 References 94

7. Single-Degree-of-Freedom Gyroscopes 95 The Rate Gyro 95

The Scale Factor 96 The Spin Motor 97 The Ball Bearings 98 Damping 98 The Pickoff 99 The Torsion Bar 100 Flexleads 100 Rate Gyro Dynamics 100

The Rate-Integrating Gyro 102 The Torquer 102 The Output Axis Bearing 104 The Principle of Flotation 105 Damping 106 Flotation Fluids 107 Structural Materials 109 The Externally Pressurized Gas Bearing Suspension 110 A Magnetic Suspension 110 Self-Acting Gas Bearings 111 Anisoelasticity in the SDFG 113 Anisoinertia in the SDFG 114 Vibration Rectification 115 The SDFG Model Equation 117

A Digression into Accelerometers 118 The Pendulous-Integrating Gyro Accelerometer 118

Conclusion 119 References 120

8. Two-Degree-of-Freedom Gyroscopes 122 The Two-Degree-of-Freedom (Free) Gyro 122 The External Gimbal Type 123 Two-Axis Floated Gyros 124 Spherical Free Rotor Gyros 125 The Electrically Suspended Gyro 126 The Gas Bearing Free Rotor Gyro 128 References 130

Contents xiii

9. The Dynamically Tuned Gyroscope 131 The DTG Tuning Effect 131 The Tuning Equations 132

DTG Nutation 136 Figure of Merit 136

Damping and Time Constant 137 Biases Due to Damping and Mistuning 137 Quadrature Mass Unbalance 139 Synchronous Vibration Rectification Errors 140

Axial Vibration at IN 140 Angular Vibration at 2N 141

Wide Band Vibration Rectification Errors 142 Anisoelasticity 143 Anisoinertia 144 Pseudoconing 145

The Pickoff and Torquer for a DTG 146 The DTG Model Equation 149 Conclusion 150 References 151

10. Vibrating Gyroscopes 152 The Vibrating String Gyro 153 The Tuning Fork Gyro 154

The Micromachined Silicon Tuning Fork Gyro 156 Vibrating Shell Gyros 158 The Hemispherical Resonator Gyro 159

Scale Factor 160 Asymmetric Damping Error 160

The Vibrating Cylinder (START) Gyro 162 The Advantages of Vibrating Shell Gyros 163 The Mu1tisensor Principle and Its Error Sources 164 Conclusion 167 References 167

11. The Principles of Optical Rotation Sensing 169 The Inertial Property of Light 169 The Sagnac Effect 170

Sagnac Sensitivity-The Need for Bias 172 The Shot Noise Fundamental Limit 173 The Optical Resonator 175

The Fabry-Perot Resonator 176 Resonator Finesse 179 The Sagnac Effect in a Resonator 179 Active and Passive Resonators 180

xiv Contents

Resonator Figure of Merit 181 Optical Fibers 181

Refraction and Critical Angle 182 Multimode and Single-Mode Fibers 183 Polarization 183 Birefringent Fiber for a Sagnac Gyro 185

The Coherence of an Oscillator 185 Types of Optical Gyro 185 Conclusion 186 References 186

12. The Interferometric Fiber-Optic Gyro 188 The History of the Fiber-Optic Gyro 188 The Basic Open-Loop IFOG 189 Biasing the IFOG 190

Nonreciprocal Phase Shifting 190 The Light Source 192 Reciprocity and the "Minimum Configuration" 193 Closing the Loop-Phase-Nulling 194

Acousto-Optic Frequency Shifters 195 Integrated Optics 195 Serrodyne Frequency Shifting 197

Fiber-to-Chip Attachment-The JPL IFOG 198 Drift Due to Coil Temperature Gradients 199 The Effect of Polarization on Gyro Drift 200 The Kerr Electro-Optic Effect 201 The Fundamental Limit of IFOG Performance 202

IFOG Shot Noise 202 Relative Intensity Noise (RIN) 203

Conclusions 204 References 205

13. The Ring Laser Gyro 208 The Laser 208

Stimulated Emission 208 The Semiconductor Laser 211

The Ring Laser 212 Lock-In 213

Mechanical Dither 213 The Magnetic Mirror 215

The Multioscillator 217 Shared-Mirror RLG Assemblies 219 The Quantum Fundamental Limit 220 Quantization Noise 222

Conclusion References

14. Passive Resonant Gyros The Discrete Component Passive Ring Resonator

The PARR Fundamental Limit The Resonant Fiber-Optic Gyro The Micro-Optic Gyro

The MOG Fundamental Limit IFOG, RFOG, and MOG Size Limits Fundamental Limits for RFOG, IFOG, and RLG Conclusion References

15. Testing Inertial Sensors Inertial Sensor Test Labs

Performance Test Gear Environmental Test Gear Qualification, Acceptance, and Reliability Tests

Accelerometer Testing The Accelerometer Acceptance Test Procedure

Centrifuge Tests Gyroscope Testing

Testing the SDF Rate Gyro Testing SDF Rate-Integrating Gyros

Tombstone Tests The Six-Position Test The Polar-Axis (Equatorial Tumble) Test The Servo Table Scale Factor Test Vibration Tests

Testing the Dynamically Tuned Gyro The Eight-Position Test DTG Rate Testing

Testing Optical Gyros The Sigma Plot

Conclusion References

16. Design Choices for Inertial Instruments A Platform or a Strapdown System? Aiding the IMU Choice of Sensor Type

Differential Design

Contents xv

222 223

225 225 227 227 231 233 235 235 237 237

239 239 240 241 242 243 243 246 247 247 248 249 249 251 252 252 253 254 256 256 256 258 258

260 261 261 262 262

xvi Contents

Using Resonance Mechanical or Optical Gyros?

Inertial Memory Lifetime

Reliability Redundancy

Sensor Design Check Lists Conclusions Reference

Index

262 263 263 263 264 264 265 266 267

268