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Mechanical Engineering Series Frederick F. Ling Series Editor Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

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Mechanical Engineering Series

Frederick F. Ling

Series Editor

SpringerNew York


Hong KongLondonMilanParis


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Mechanical Engineering Series

J. Angeles, Fundamentals of Robotic Mechanical Systems:

Theory, Methods, and Algorithms

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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)

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Anthony Lawrence

Modern Inertial Technology

Navigation, Guidance, and Control

Second Edition

With 201 Figures

, Springer

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Anthony Lawrence

32 Sunny Hill RoadLunenburg, MA 01462


Series Editor

Frederick F. LingErnest F. GJoyna Regents Chair in EngineeringDepartment of Mechanical EngineeringThe University of Texas at AustinAustin, TX 78712-1063USA

andWilliam Howard Hart Professor Emeritus

Department of Mechanical Engineering,Aeronautical Engineering and MechanicsRensselaer Polytechnic InstituteTroy, NY 12180-3590USA

Library ofCongress Cataloging-in-Publication DataLawrence, Anthony, 1935-

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

p. cm. - (Mechanical engineering series)Includes bibliographical references and index.ISBN 0-387-98507-7 (hardcover: alk. paper)1. Inertial navigation (Aeronautics) 1. Title. II. Series:

Mechanical engineering series (Berlin, Germany)TL588.5.L38 1998629.132'51---dc21 98-13047

Printed on acid-free paper.

© 1998, 1993 Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, 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 theformer are not especially identified, is not to be taken as a sign that such names, as understood bythe 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-ready copy prepared from the author's WordPerfect files.

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

ISBN 0-387-98507-7 SPIN 10843117

Springer-Verlag New York Berlin HeidelbergA member ofBertelsmannSpringer Science+Business Media GmbH

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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 pro

ductivity and competitiveness that require engineering solutions, among others.

The Mechanical Engineering Series features graduate texts and research mono

graphs intended to address the need for information in contemporary areas of me

chanical engineering.The series is conceived as a comprehensive one that covers a broad range of

concentrations important to mechanical engineering graduate education and re

search. We are fortunate to have a distinguished roster of consulting editors on theadvisory 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 ofconcentration are applied mechanics, biomechanics, computational mechanics,

dynamic systems and control, energetics, mechanics of materials, processing, ther

mal 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 Engi

neering series to provide our readers with topical monographs as well as graduatetexts in a wide variety of fields.

Austin, Texas Frederick F. Ling

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Mechanical Engineering Series

Frederick F. Ling

Series Editor

Advisory Board

Applied Mechanics


Computational Mechanics

Dynamical Systems and Control


Mechanics of Materials


Production Systems

Thermal Science


F.A. Leckie

University of California,

Santa Barbara

V.C.MowColumbia University

H.T. Yang

University of California,

Santa Barbara

K.M. Marshek

University of Texas, Austin

J.R. WeltyUniversity 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

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Since 1993, when the first edition of this book was published, inertial technologyhas 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


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

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Series Preface



1. An Outline of Inertial NavigationNavigation's Beginnings

Inertial NavigationMaps and Reference FramesThe Inertial Navigation ProcessInertial Platforms

Heading and Attitude Reference SystemsSchuler Tuning

Gimbal LockStrapdown SystemsSystem Alignment

GyrocompassingTransfer Alignment

Advantages and Disadvantages of Platform SystemsAdvantagesDisadvantages

Advantages and Disadvantages of Strapdown SystemsAdvantagesDisadvantages

Aiding Inertial NavigatorsThe Global Positioning System

Applications of Inertial NavigationConclusionsReferences






















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x Contents

2. Gyro and Accelerometer Errors and Their Consequences 25Effect of System Heading Error 25

Scale Factor 26

Nonlinearity and Composite Error 27System Error from Gyro Scale Factor 27Asymmetry 28

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

Random Drift 31

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

g-Sensitivity 34Anisoelasticity 35

Rotation-Induced Errors 36Angular Acceleration Sensitivity 37Anisoinertia

37Angular Accelerometers 38

Angular Accelerometer Threshold Error 39

The Statistics of Instrument Performance 39Typical Instrument Specifications 40

References 42

3. The Principles of Accelerometers 43The Parts of an Accelerometer 43

The Spring-Mass System 44QFactor 47Bandwidth 48

Open-Loop Pendulous Sensors 48

Cross-Coupling and Vibropendulous Errors 48

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

Binary Feedback 51Ternary Feedback 53Pulse Feedback and Sensors 53

The Voltage Reference Problem 54Novel Accelerometer Principles 54

Surface Acoustic Wave Accelerometer 55

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Contents xi

Fiber-Optic Accelerometers 55References 56

4. The Pendulous Accelerometer 57A Generic Pendulous Accelerometer 57

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

The IEEE Model Equations 60

The "Q-Flex" Accelerometer 61The Capacitive Pickoff 62The Forcer 63

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

The Silicon Accelerometer 67References 70

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

Accelerex Signal Processing 78The Kearfott Design 79Silicon Micromachined VBAs 81

Comparison of Free and Constrained Accelerometers 82General Comparison of the SPA and VBA 82Comparison of Performance Ranges 83

Conclusion 83References 84

6. The Principles of Mechanical Gyroscopes 85Angular Momentum 85

The Law of Gyroscopics 86Parasitic Torque Level 87The Advantage of Angular Momentum 87

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

Coriolis Acceleration 90

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xii Contents

The Ice Skater 92Gyroscopes with One and Two Degrees of Freedom 92

Conclusion 93

References 94

7. Single-Degree-of-Freedom Gyroscopes 95

The Rate Gyro 95The Scale Factor 96The Spin Motor 97The Ball Bearings 98Damping 98

The Pickoff 99The Torsion Bar 100

Flexleads 100Rate Gyro Dynamics 100

The Rate-Integrating Gyro 102

The Torquer 102

The Output Axis Bearing 104The Principle of Flotation 105Damping 106Flotation Fluids 107Structural Materials 109The Externally Pressurized Gas Bearing Suspension 110

A Magnetic Suspension 110Self-Acting Gas Bearings 111

Anisoelasticity in the SDFG 113

Anisoinertia in the SDFG 114Vibration Rectification 115

The SDFG Model Equation 117A Digression into Accelerometers 118

The Pendulous-Integrating Gyro Accelerometer 118Conclusion 119References 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 125The Electrically Suspended Gyro 126The Gas Bearing Free Rotor Gyro 128

References 130

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Contents xiii

9. The Dynamically Tuned Gyroscope 131

The DTG Tuning Effect 131

The Tuning Equations 132

DTG Nutation 136Figure 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 143Anisoinertia 144

Pseudoconing 145

The Pickoff and Torquer for a DTG 146

The DTG Model Equation 149

Conclusion 150

References 151

10. Vibrating Gyroscopes 152The 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 173The Optical Resonator 175

The Fabry-Perot Resonator 176

Resonator Finesse 179

The Sagnac Effect in a Resonator 179

Active and Passive Resonators 180

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xiv Contents

Resonator Figure of Merit 181

Optical Fibers 181

Refraction and Critical Angle 182

Multimode and Single-Mode Fibers 183Polarization 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 188The 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 194Acousto-Optic Frequency Shifters 195

Integrated Optics195

Serrodyne Frequency Shifting 197

Fiber-to-Chip Attachment-The JPL IFOG 198

Drift Due to Coil Temperature Gradients 199The Effect of Polarization on Gyro Drift 200The Kerr Electro-Optic Effect 201

The Fundamental Limit of IFOG Performance 202IFOG Shot Noise 202Relative Intensity Noise (RIN) 203

Conclusions 204References 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 213The Magnetic Mirror 215

The Multioscillator 217

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

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14. Passive Resonant GyrosThe 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


15. Testing Inertial SensorsInertial Sensor Test Labs

Performance Test Gear

Environmental Test Gear

Qualification, Acceptance, and Reliability Tests

Accelerometer TestingThe 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 TestVibration Tests

Testing the Dynamically Tuned Gyro

The Eight-Position Test

DTG Rate Testing

Testing Optical Gyros

The Sigma Plot



16. Design Choices for Inertial InstrumentsA Platform or a Strapdown System?

Aiding the IMU

Choice of Sensor Type

Differential Design

Contents xv





















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xvi Contents

Using Resonance

Mechanical or Optical Gyros?

Inertial Memory



Sensor Design Check Lists