Table Of Content'
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Published By:
Strapdown Associates, Inc.
Maple Plain, Minnesota
The Material Presented In This Book Was Prepared Exclusively And Independently
By Paul G. Savage. Reproduction Of Any Part Of This Material Without Permission
From The Author Or Strapdown Associates, Inc., Is Prohibited.
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© Strapdown Associates, Inc. 1997, 1998, 1999, 2000
All Rights Reserved
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o r e w o r
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This two volume text provides a detailed comprehensive discourse on the analytics of
strapdown inertial navigation systems (INS's), the basic technology used on modem day
commercial and military aircraft, guided surface ships and underwater vehicles.
missile~,
Based on his first-hand experience in this field, the author has provided a unique service to the
aerospace industry in preparing this technical dissertation on the algorithms implemented in
the strapdown system· computer, and the analytics (and software) associated with system
software validation, system test, simulation, performance analysis, and the analytical design
methodology used in deriving the strapdown equations. Included is an in-depth chapter
dealing with Kalman filter theory and its application to the aiding of a strapdown INS.
Strapdown Analytics has been prepared for the reader who may not have had experience in
navigation or Kalman filtering. The analytical material presented is derived from scratch,
showing the developmental steps in rigorous detail, without relying on reference material for
supporting analytics. The book is complicated, yet complete and understandable by
analytically inclined graduate students and practicing engineers. The book can be viewed as
the text for an advanced course one might take following the introductory course taught by the
author, Introduction To Strapdown Inertial Navigation Systems.
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b o u t T h e A u t h o r
Paul G. Savage is an internationally recognized expert in the design and test of ring laser
gyro strapdown inertial navigation systems. He is the president of Strapdown Associates, Inc.
(SAl), a Minnesota engineering company he founded in 1980 to further the advancement of
strapdown inertial technology. Since the down-sizing of SAl in 1995, Mr. Savage has
continued to provide inertial navigation consultation services and teach his Introduction To
Strapdown Inertial Navigation Systems course to the aerospace industry. He also serves as a
reviewer for the AIAA Journal of Guidance, Control, and Dynamics.
From 1980 to 1995, SAl, under Mr. Savage's leadership, provided technical engineering
development and educational support for system configuration definition, flight software
development, system simulation, and testing for strapdown inertial system development
programs for military and commercial aircraft, land vehicles, cruise missiles, and ground and
air-launched tactical missiles. Strapdown Associates has held contracts in these areas with
Naval Weapons Center, Naval Air Development Center, McDonnell Douglas, Honeywell,
Sundstrand, Systron Donner, Texas Instruments, Rockwell International, Aerojet, Contraves,
Bendix, Northrop, General Dynamics, Boeing, Oklahoma City Air Logistics Command I
Tinker Air Force Base, Environmental Research Institute of Michigan, ARINC Research,
Baker Hughes, Alliant Techsytems, Lockheed Martin and United Defense.
From 1963 to 1980, ·Mr. Savage was employed at Honeywell Avionics Division as Senior
Principal Engineering Fellow where he led engineering design teams and provided technical
consultation to Honeywell engineering managers for system design, analysis, software
development, simulation, and integration/test in the evolutionary development of laser gyro
strapdown inertial navigation systems for military and commercial aircraft. From 1971
through 1975, Mr. Savage was the engineering manager and system design engineer for the
Honeywell LINS-0 strapdown inertial system, the first to prove the readiness of laser gyro
strapdown inertial navigation technology for 1 nautical-mile-per-hour accuracy aircraft
applications as demonstrated during a landmark flight test series at Holloman Air Force Base
in 1975.
Mr. Savage is a graduate from the Massachusetts Institute of Technology where he received
hjs MS and BS degrees in Aeronautical Engineering in 1960.
Strapdown inertial navigation is a technology for autonomously determining position
location using a portable system containing strapdown inertial angular rate sensors,
acceleration sensors, and a data processing computer. ''Strapdown" refers to the method of
mounting the inertial instruments directly to the user vehicle, rather than on a gimbaled
platform used to isolate the sensors from vehicle rotation in earlier inertial systems. Inertial
navigation systems (often with additional supporting navigation aids - e.g., positioning data
from Global Positioning System (GPS) receivers) are utilized extensively as the basic
navigational element on military and commercial aircraft, guided missiles, surface ships and
underwater vehicles. Advances in inertial sensor and computer technology in the 1970's made
it possible to achieve accuracies with strapdown systems that equaled earlier gimbaled
technology perforntance for lower cost and improved reliability. By year 2000, virtually all
inertial navigation systems had become (or were being retrofitted to) the strapdown type (an
exception being ballistic missile guidance systems which continued to use gimbaled
technology).
The title of this book, Strapdown Analytics, has a double meaning. It deals with the
analytics associated with strapdown inertial navigation systems; it is also a detailed tutorial
treatise describing the analytics used at my company, Strapdown Associates, Inc. (SAl or
"Strapdown" for short), over the 20 year period following its inception in 1980. During this
time, SAl developed strapdown inertial navigation application software in both free-inertial
and Kalman-filter-aided configurations for various user groups including validation software,
system test software, simulators, and associated performance analyses. In addition to being
SAl's president, one of my principal responsibilities at SAl was to prepare the technical
documentation for SAl programs. Since 1981, I have also provided an introductory course to
the general public, Introduction To Strapdown Inertial Navigation Systems, covering the ·
various strapdown system hardware, software, inertial sensor, and system technological
elements. '··'
While teaching the Strapdown Inertial course, I have been frequently asked for references
covering the course topics in more detail (in addition to the handout material provided to
course attendees). While I have been able to recommend several books covering inertial
sensors and Kalman filtering, I have been at a loss to recommend comprehensive documents
providing a detailed account of the analytical aspects of strapdown inertial navigation systems.
At best, I have referred to some papers that have been published over the years by several
authors in the trade journals and at technical symposiums. For SAl customers, I have
referenced the documentation provided with SAl's software package deliverables, which
though fairly comprehensive for their particular functions, are not in the forn1 of tutorial text
material, and presume a certain level of basic strapdown analytical background by the reader.
This book represents a detailed comprehensive tutorial account of my knowledge in the
iv PREFACE
analytical aspects of strapdown inertial navigation. This includes not only the algorithms
implemented in the strapdown system computer, but the analytics (and software) associated
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with system software validation, system test, simulation, and system performance analysis. It
also implicitly illustrates the analytical design methodology I have used in deriving strapdown
equations. The book was prepared using SAl independently developed technical
documentation and my personal notes as a starting base, and then rewritten for tutorial clarity,
expanded to fill in technical voids in equation derivations, modified for technical parameter
compatibility and cross-referencing between sections, enhanced for analytical improvements
uncovered during book preparation, and expanded to cover important material that I had not
previously documented. Beginning with 1500 pages from assorted SAl documents, I had
hoped to develop a text book with a reasonable page count (e.g., 800 pages). Due to the added
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material and ground-rules I set during book preparation (discussed below), the fmal version
was more than 1500 pages, requiring two volumes.
In preparing this book, I set several goals to achieve what I believe is required of a textbook
for easy reader comprehension. Many of the goals were selected to avoid problems I have
experienced in the past when using analytical text documents. The unfortunate penalty was
increased page count which I decided at the onset to be worth the cost. · The goals were as
follows:
• Clearly delineated parameter definitions separated from the main text, including a
parameter index for referencing back to the book location where the parameter was
defined.
• Repeating a parameter definition where needed for clarity in sections that are far
removed from the section in which it was originally defined.
• Deriving equations beginning from basics; avoiding the practice of referencing
supporting equations to other source material. The entire book contains only 40
references, some of which are the basic textbooks I used at college. One of the reasons
for additional supporting documentation has been to substantiate analytical results
obtained. In this book, equations are derived from scratch including supporting
analytics. As such, results are self-verifying without supporting documentation (with
some minor exceptions, in which case supporting documents are referenced; e.g.,
standard gravity models in References 3 and 4). My analytical experience in the
aerospace industry is the principal reference for the analyticaJ. derivations provided,
supplemented with theoretical background material provided by textbooks I have
· referenced in the book (notably, Reference 6 for Kalman filter theory and Reference 3
for vector/coordinate-frame nomenclature).
• A voiding reference material that was not used in the book preparation. Exceptions are
the selected background material on attitude integration algorithms referenced in
Chapter 7, an important part of strapdown analytical development history, and
References 16 and 21, fairly recent textbooks on inertial sensors (as of year 2000). I
PREFACE v
decided at the onset that I would not perform a literature search to find additional
background material that readers might find useful. This book is based on my
knowledge base acquired during 39 years of experience in the aerospace industry, i.e.,
the material I have used on a daily basis in performing my engineering assignments.
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• Providing equation derivations that show the intermediate steps to avoid having the
reader accept results on faith, or spend valuable time filling in the voids for
verification.
• Developing equations in full closed-form wherever possible without reverting to the
use of linearization approximations (an obvious exception is Kalman filter error state
modeling which is inherently based on linearization techniques). Much of the
analytical results developed in this book are designed for translation into software on a
host computer. Advancements in computer technology make it possible to implement
the added closed-form equation computational burden for virtually no cost penalty.
The important benefit is the elimination of linearization process error and associated
documentation/validation requirements for assuring sufficient accuracy.
• Deriving equations in vector format when possible for more generality, and in this
case, to reduce page count.
This book should be viewed as the text one might use for an advanced course on strapdown
inertial navigation, e.g., the course that might follow my introductory strapdown course. It is
not a simple book, covering some very complicated analytical subjects. Nevertheless, it has
been designed for comprehension by engineering analysts or graduate students having only a
basic working knowledge of vector calculus and matrix operations. Previous background in
inertial navigation or navigation in general is not necessarily required. However, it is helpful.
One final word of caution. Although every effort has been made to eliminate errors in the
book (including careful proofing of five technical draft copies and the final print copy), it is
that some errors have passed undetected (e.g., typographical errors in the equations
~evitable
themselves or in the equation number c.ross-references used liberally throughout cthe book).
As such, if readers plan to use the book material for important engineering projects, they are
encouraged to re-verify the material by analytical means and simulation.
Good luck.
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Paul G. Savage
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An important part of the writing of any book is the proof reading. In the case of an
analytical text book this is particularly important because of the many possible areas where
errors can go undetected (e.g., incorrect symbol, misplaced or missing bracket, wrong Font,
missing vector notation, flawed equation derivation, assumed analytic characteristic that was
never proven, incoherent explanation, incorrect equation or section numbers in the text and in
cross-referencing). It is virtually impossible for the writer of such a book to find and correct
all of the possible errors. Writers are so familiar with their own work that they are blind to
many of their errors during the proof-reading process. To prevent this problem, another reader .
(or several) typically assist in the proofing process.
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The proof reader of an analytical text book should have the basic technical ability to
understand the equations presented and their analytical derivations. Ideally, the proof reader
should have had no previous background in the text book material to avoid introducing
inadvertent technical prejudices in the reviewing process. For the proofing process itself, the
ideal proof reader would independently rederive all equations in the book to verify their
accuracy and analytical cross-references, and ·provide comments to the author regarding
understandability of the written text and equation derivation methods. An important implicit
requirement in this regard is confidence on the part of the author that the proof reader will
perform the task thoroughly and dependably with the same concern that the reader has in the
accuracy of the fmal result.
For the Strapdown Analytics book, I have been fortunate to have found someone with
superb technical credentials, and who was willing to provide such a dedicated proof-reading
service; my daughter Kelly M. Roscoe. Kelly received Bachelor and Master of Science
Electrical Engineering degrees with honors from the Massachusetts Institute of Technology
(MIT), and has had technical experience in military system design/management at Rockwell
Seal Beach Facility and advanced optical materials process control at Minnesota Mining & .
Manufacturing. I have Kelly to thank for the accuracy and understandability of the final
published version of Strapdown Analytics. Without her help, the book in its present form
would not have been possible.
I would also like to express my gratitude to my wife Paula for her encouragement during
the four year period that this book was written, and for her direct assistance in the final
formatting and proofing stages of book preparation. Her understanding and patience when my
thoughts were preoccupied with writing was most appreciated.
Lastly, I would like to express my acknowledgment to the Massachusetts Institute of
Technology for my basic engineering education, Lockheed Missiles & Space Company and
Honeywell, Inc. for exposure as an employee to the general business of inertial navigation, to
employees. and customers of my company Strapdown Associates, Inc. (SAl) who made it
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ACKNOWLEDGMENTS VII
possible for SAl to exist, to past attendees of SAl's strapdown inertial navigation course, and
to the technical colleagues I have infortnally known over the years. Involvement with these
people and associated programs during my career provided invaluable experience and the
environment for developing the technical.e xpertise needed to prepare this book.
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Paul G. Savage
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viii ACKNOWLEDGMENTS
