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Mechanical Engineering Series
Frederick F. Ling
Series Editor
SpringerNew York
BerlinHeidelbergBarcelona
Hong KongLondonMilanParis
SingaporeTokyo
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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)
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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
USA
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
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.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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Preface
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
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
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Contents
Series Preface
Preface
Introduction
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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Conclusion
References
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
ConclusionReferences
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
Conclusion
References
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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