Inertial Navigation Systems Analysis (GNSS Technology and Applic....pdf

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Contents

1 Introduction

1.1 THE CONCEPT OF INERTIAL NAVIGATION

1.2 TYPES OF INERTIAL NAVIGATION SYSTEMS

1.3 A CRITIQUE O F PREVIOUS ANALYSIS TECHNIQUES

1.4 A UNIFIED APPROACH TO THE ERROR ANALYSIS

2 Mathematical Notation and

2.1 NOTATIONAL CONVENTIONS

2.2 THE TIME DERIVATIVE OF THE DIRECTION

2.3 COLUMN MATRIX TlME DERIVATIVES

2.4 ANALOGIES TO VECTOR ANALYSIS

2.5 PERTURBATION TECHNIQUES

2.6 SYMBOLOGY

3 Reference Frames

3.1 INERTIAL FRAME (i frame; x, y, z axes)

3.2 GEOGRAPHIC FRAME (n frame; N, E, D axes)

3.3 EARTH FRAME (e frame; x,, ye, ze axes)

3.4 GEOCENTRIC FRAME

3.5 BODY FRAME

3.6 TANGENT FRAME

3.7 REFERENCE FRAME RELATIONSHIPS

3.8 PLATFORM, ACCELEROMETER, AND GYRO FRAMES

4 Geometry of the Earth

4.1 T H E GEOCENTRIC POSITION VECTOR

4.2 THE DEVIATION OF THE NORMAL

4.3 THE EARTH RADIUS MAGNITUDE

4.4 THE EARTH'S GRAVITATIONAL FIELD

4.5 THE EARTH'S GRAVITY FlELD

4.6 ANALYTIC EXPRESSIONS FOR THE SPECIFIC FORCE

5 Single-Degree-of-Freedom Gyroscope Performance

5.1 PRINCIPLE OF OPERATION

5.2 DYNAMIC MODEL FOR THE SDF GYRO

5.3 UNCERTAINTY TORQUE COMPENSATION

5.4 INSTRUMENT A N D SYSTEM REDUNDANCY AND

6 The Space-Stabilized Terrestrial

6.1 DESCRIPTION O F SYSTEM

6.2 MECHANIZATION EQUATIONS

6.3 ERROR ANALYSIS

7 The Local- Level Terrestrial

7.1 DESCRIPTION OF SYSTEM

7.2 MECHANIZATION EQUATIONS

7.3 ERROR ANALYSIS

7.4 T H E TWO-ACCELEROMETER LOCAL-LEVEL .SYSTEM

8 Development of a Unified Error

8.1 A GENERAL TERRESTRIAL NAVIGATOR MODEL

8.2 GENERALIZED MECHANIZATION AND ERROR

8.3 CANONICAL FORM OF THE ERROR EQUATIONS

8.4 SPECIALIZATION OF THE GENERALIZED THEORY

8.5 EFFECT OF ALTIMETER UNCERTAINTY

9 Self-Alignment Techniques

9.1 ANALYTIC COARSE ALIGNMENT METHOD

9.2 PHYSICAL GYROCOMPASS ALIGNMENT7

9.3 ALIGNMENT OF STRAPDOWN SYSTEMS

Appendix A Development of a System Error

A.1 SYSTEM DESCRIPTION

A.2 DERIVATION O F SYSTEM DIFFERENTIAL EQUATIONS

A.3 SO-LUTION OF SYSTEM DIFFERENTIAL EQUATIONS

A.4 APPROXIMATIONS TO THE SOLUTIONS

A.5 DEVELOPMENT OF AN ERROR MODEL

Appendix B State Transition Matrix for Inertial

B.1 FORMULATION IN STATE SPACE NOTATION

B.2 STATE TRANSITION MATRIX

8.3 STATE TRANSITION MATRIX FOR SHORT

8.4 EXAMPLES

Appendix C Statistical Error Analysis Methods

C.1 RESPONSE O F A LINEAR SYSTEM TO RANDOM INRUTS

C.2 RESPONSE TO THE ENSEMBLE OF

C.3 RESPONSE TO W H I T E NOISE

References

Index

Inertial Navigation Systems Analysis

INERTIAL NAVIGATION SYSTEMS ANALYSIS K E N N E T H R. BRITTING, Sc. D. Lecturer in Aeronautics and Astronautics Measurement S y s t e m Laboratory Massachusetts Institute of techno loo?^ WILEY-INTERSCIENCE, a Division of John Wiley & Sons, Inc. N e w York London Sydney Toronto

Copyright 0 1971, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Catalog Card Number: 70- 168635 ISBN 0-471-10485-X Printed in the United States of America

To the memory of KATHERINE ANNE

Foreword Although the technique of inertial guidance can be said to have originated more than sixty years ago with the appearance of the gyrocompass, i t did not attain full navigational status until the impetus of technology after World War I1 made it practical as well as feasible. During this period the basic inertial components-gyroscopes essentially the same but continually improved in performance. Extensive studies led to a fairly complete understanding of the theory of inertial systems. Improve- ment both in inertial components and in the associated signal-processing equipment led from systems with a volume of over a cubic yard t o those with less than a cubic foot. Accuracy and reliability produced systems that not only met the military needs in air and underwater but also allowed the successful rtccomplishment of the Apollo space missions and the installation of inertial systems in commercial aircraft. and accelerometers-remained During this time different general configurations produced systems with very different types of performance, although with the same basic com- ponents. Accordingly, a common basis for meaningful comparison of the performance of these systems was lacking, and discussions by proponents and opponents of a given configuration generated more heat than light. This book, based largely on the author's several years of study leading to his doctorate, is the first definitive attempt that successfully provides a basis for a realistic comparison of performance of various inertial system configu- rations-geometric, semi-analytic, or analytic. The solution is not a simple, rule-of-thumb technique, but i t is of sufficient simplicity and directness for a skilled person to formulate his own comparisons reasonably quickly, effec- tively, and accurately. I n producing this book Dr. Britting presents a "Rosetta Stone" to the inertial guidance profession. As one of his faculty advisors during his doctoral

viii studies, I am very pleased and proud to have the privilege of introducing a former student's noteworthy accomplishment. F O R E W O R D WALTER WRIGLEY, SC. D. Professor of Instrumentation and Astronautics Educational Director, Charles Stark Draper Laboratory Massachusetts Institute of Technology

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