THEORY OF INERTIAL NAVIGATION-Aided Systems

426 Pages·2016·16.85 MB·English

by  V. D. AndrEEV

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THEORY OF INERTIAL NAVIGATION
Aided Systems
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Translated from Russian
Published for the National Aeronautics and Space Administration
and the National Science Foundation, Washington, D.C.
by the Israel Program for Scientific Translations
TECH LlSRARY UAFB, NM
00b8995
V. D. Andreev
THEORY OF
INERTIAL NAVIGATION
Aided Systems
( Teo riy a In-rtsial'no i navigatsii : korre kt ir uemye s istr my)
Izdatel'stva "Nauka"
hloskva 1967
Translated frotn Russian
Israel Program for Scientific Translations
J e ros a le in 1' )3 <.
NASA TT F-564
TT 69-55027
Published Pursuant to an Agreement with
THE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
and
THE NATIONAL SCIENCE FOUNDATION, WASHINGTON, D. C.
Copyright 0 lHtiY
Israel Program tor Scientific Translations Ltd.
IPST Cat. No. 5444
Translated by Z. Lerman
Printed in Jerusalem by IPST Press
Binding: Wiener Bindery Ltd., IeNSalem
Available from the
U. S. DEPARTMENT OF COMMERCE
Clearinghouse for Federal Scientific and Technical Information
Springfield, Va. 22151
PREFACE
hlodern aircraft, rockets, and sea vessels require high-precision self-
contained navigation and guidance systems, which are further expected to be
noise-proof, can be set up at a short notice, and will function unattended.
The last years have been marked by considerable advances in the field
of navigation and guidar.ce systems for moving objects. Alongside with the
development of new principles, considerable efforts have been devoted to
inertial navigation systems which determine the current position of the
object by straightforward integration of the accelerations measured on
board the moving object.
Inertial systems are currently the most promising and universal means
of self-contained, autoromous navigation. However, the dynamic properties
of completely autonomous inertial systems are such that the errors of their
elements lead to an inevitable buildup of position errors with time.
"hen a self-contained navigation system is allowed to function continuously
for some time, the measurement errors in the navigation parameters may
eventually exceed the permissible limit value. The inertial system is therefore
never made completely autonomous: it is generally aided by external sources
of guidance information, such as astronomical (stellar) correction, altimeter
correction, Doppler correction, and radio-navigation aiding.
Introduction of external aiding information into the inertial system alters
the unperturbed operation equations of the system (its algorithm). The
dynamics of its perturbed operation, i.e., the equations of errors, are also
altered. Different exte,rnal sources of guidance information may differently
affect the error equations, and in the final account we thus end up with
different dependences of the navigation errors on instrumental errors and
on the errors in the external guidance information. The number of possible
alternatives here is fairly large.
Some particular protllems related to the theory of aided inertial systems
are treated in previously published books.;' These, however, mainly cover
isolated, individual topics, and the analysis is generally far from being
rigorous or amenable to far-reaching generalizations. The aim of the
present book is conversely to present a systematic and rigorous treatment
of the fundamental problems in the theory of aided inertial system, proceed-
ing from a single point of view. The main emphasis is on the analysis of
the various means of introducing the correction signals into the system and
the investigation of the error equations. The ultimate goal of this study is
a comparative analysis of the dependence of navigation errors on instrumen-
tal errors for various e.sterna1 sources of guidance information.
The book is largely based on my original researches, carried out and
partly published in recent years. It is a direct continuation of my "Theory
See, ?.e.. \IC Clure, C.L. Theory of Inertla1 Guidance. - Prenrlce Hall. 1960.
iii
of Inertial Navigation: Autonomous Systems," published in Moscow in 1966
by Nauka Publishing House, and naturally draws upon the results and
techniques of the previous volume.
The book comprises seven chapters. There are also three Appendices,
covering a number of topics which are not an organic part of the theory.
Chapter 1 is introductory in nature: it reiterates the fundamental results
of the theory of autonomous inertial systems.
Chapter 2 is concerned with the general aspects of the application of
external guidance information and with the theory of inertial systems where
an altimeter provides additional information on the distance of the moving
object from the Earth's surface. Systems with three arbitrarily oriented
accelerometers and systems with two horizontal accelerometers are
considered. The ideal (unperturbed) operation equations are derived, as
well as the corresponding error equations. The stability of the system is
investigated and the solution of the error equations is derived.
Chapter 3 discusses Schuler-tuned gyropenduhm systems, particuIar
examples of which are the common Schuler pendulum and various
classical gyroscopic instruments, such as the two-gyro vertical and
the Geckeler- Anschutz three-degrees-of-freedom gyrocompass horizon.
Particular stress is laid on the dynamic analogy between the Schuler-tuned
gyropendulum systems and the two-accelerometer (or two-axis) inertial
guidance systems. Ishlinskii" was the first to call attention to the possible
analogy between these two categories of guidance systems, and I proved the
existence of complete dynamic analogy between them. Consequently, a
number of important results from the theory of Schuler-tuned systems can
be readily generalized to two-accelerometer inertial systems.
Chapter 4 deals with various simplified forms of the ideal operation
I
equations of inertial systems, including the simplifications associated with
the nearly spherical figure of the Earth and the nearly central field of its
gravitation. Simplifications resulting from particular constraints imposed
on the object trajectory are also considered. These include close trajecto-
ries of objects moving near the Earth's surface, trajectories which are
close to some orthodromy on the Earth's surface, and cases of constrained
velocity. By examining the simplifications in the ideal operation equations
(i.e., in the algorithm of the navigation system), we can learn to simplify
the design of the inertial system by omitting or replacing some of its
elements.
Chapter 5 presents the theory of inertial systems where altimeter
correction is reinforced by additional information from a Doppler velocity
I
meter. Introduction of damping in the perturbed operation equations of
inertial systems inevitably leads to velocity errors. Suitable guidance
information supplied by a Doppler velocity meter will eliminate these errors.
The residual errors will then depend only on the instrumental inaccuracy of
the system elements and the inevitable errors in the Doppler information.
The corresponding relations are expressed by the error equations, which
are derived and investigated for different Doppler correction techniques.
Chapter 6 is concerned with astronomical (stellar) correction procedures,
whereby telescopes are used to fix the directions to certain stars. Combined
astro-Doppler correction is also considered in this chapter. It is shown
that, unlike Doppler and altimeter correction routines, astrocorrection
* I s h 1 ins k i I, A.Yu. The equations of the problem of determination of the position of a moving object using
gyroscopes and accelerometers. - Prikladnaya Matematika I Mekhanika, Vol. 20, No. 6. 1957.
affects only the second group of error equations. The resulting changes in
the error equations are studied for linear and relav-type correction routines.
.A phase astrocorrection system using star trackers is dealt with in a
separate section.
Chapter 7 deals with the dynamics of autonomous and aided inertial
systems with random errors. The analysis is carried out within the frame-
work of the correlation theory of random processes. The main object of
this chapter is to study the dispersion of the navigation errors as a function
of the statistical characteristics of the instrumental errors for various
correction techniques.
Appendix I at the end of the book describes the effect of the scattered
soIar radiation in the armosphere on the normal operation of a phase astro-
correction system. Appendix I1 is concerned with close navigation using a
directional gyro and a Doppler velocity meter. Appendix I11 investigates
the problem of direct accelerometer guidance, with or without external
guidance information. The material presented in these Appendices has been
previously published in different periodical journals (the third appendix, in
cooperation svith I. C'. Novozhilov).
One more point should be borne in mind. The present volume is a direct
continuation of my "Theory of Inertial Guidance: Autonomous Systems" and
largely draws upon the results of the previous book. However, Chapter 1
summarizes in a compact form all the fundamental results of the theory of
autonotnous inertial systems and the book can therefore be read as an
independent, self-contained monograph.
This volume appears to be the first attempt at a systematic discussion
of the theory of aided irertial systems. As such, it is not free from errors,
and all remarks and criticism will be appreciated.
In conclusion I would like to acknowledge the great help of A.Yu. Ishlinskii,
whose assistance in formulating a number of problems analyzed in this book uas
invaluable. I am also grateful to N.P. Bukanov, G.I. Vasil'ev-Lyulin,
E.A. Devyanin, -4.P.D ern'yanovskii, I.M. Lisovich, I.V. Novozhilov,
N.A.P arusnikov, P.V. Tarasov, and V.1'. Shed'ko for their part in the
discussion of various sections of the book.
I-. Andveec

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.
Chapter 1. ELEXIENTS OF THE THEORY OF AUTOKOR-IOUS
I NERT IAL SAL.1 GATI ON SY ST E 11s . . . . . . . . . . . . . . . . . . . . . . 1
1.1. The ideal operation equations of inertial navigation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
systems. 1
T 1.1 The fundamental equation cf inertial navigation and its integration in fixed-orientation
r;.f?rcnce a.xes (1). 1.1.2. Fundamental coordinate systems fi7ed to the Earth (3). 1.1.3. The
xabitational field of the Earth (7). 1.1.4. Ideal operation equdtions of inrrtidl systems
intasurin? the Cartesian coordinates of the object (10 ). 1.1.5. Determination of curvilinear
cc'~rdrndtes! 13). 1.1.6. The ideal operation equations in orthodromic, geocentric, and geo-
sra?hical coordinates I 17 I
S 1.2. The equations of errors for inertial navigation systems. . . . 22
1.2.1. Perturbed inercial systems. ?.lain instrumental errors (22!. 1.2.2. Po5itic.n error
equations I 231. 1 2.3. Alternative forms of the pasition error equations ('26). 1.2.4. Orien-
tation error equations for the sensiiive axe and fixed directions in space (29)
$1.3. Some results of the analysis of error equations of
autonomous inertial systems . . . . . . . . . . . . . . . . . . . . . . . . . . 31
L3.1, The second <roup of error 2quations (31I . 1.3.2. Stability of the first group of equations
for latitudinal motion (33j . 1.3.:, Solution of the first group of error equations for a stationary
<?hjecta nd for an object moving along the arc of a fixed-attitude great circle (35). 1.3.4. Inte-
gration of the first zroup ofequatims for Keplerian motion (391. 1.3.5. Position and orien-
tation errors as a function of instrunental and initial setting errors 144).
Chapter 2. AIDED INERTLAL SYSTERIS: ALTIhIETER
CORRECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
I2.1. General considerations pertaining to aided inertial
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B 2.2. Ideal operation equations and error equations for inertial
......................
systems with altimeter correction 52
2.2.1. Three-accelerometer system (521. 2.2.2. Two-accelerometer system (.54).
'$2.3. Stability analysis and integration of the first group of
. . . . . . . . . . . . .
error equations for two-accelerometer systems 60
2.3.1. Stability analysis in cases leading ro equations with constant coefficients (60).
2.3.2. Solution of rhe fiat group of error equations in cases which lead to equations with constant
coefficients (65). 2.3.3. Integrction of the first group of error equations for Keplerian motion (70)
2.3.4. Investigation of the first group of equations for arbitrary motion near the Earth (74),
vii
2.4. Stability analysis and integration of the first group of
error equations for a three-accelerometer system with
altimeter correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4.1. Stability analysis (81). 2.4.2. Internal dissipation forces (83). 2.4.3. Stability of the
first group of equations for a close object (85). 2.4.4. Solution of the first group of equations
for arbitrary motion at a constant distance from the Earth's center (86). 2.4.5. Integration of
the first group of error equations for Keplenan motion (89).
§2.5. Dependence of navigation errors on instrumental and
setting inaccuracies. Comparison with autonomous inertial
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.5.1. Dependence of position and orientation errors on setting errors (94). 2.5.2. Navigation
errors due to accelerometer and gyroscope errors (98). 2.5.3. Altimeter errors (100).
Chapter 3. SCHULER-TUNED GYROPENDULUM SYSTEMS.
ANALOGY WITH TWO-ACCELEROMETER INERTIAL SYSTEMS. 103
S3.1. Schuler pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1.1. The conditions of a relative equilibrium with the pendulum axis caged to the geocentric
vertical (103). 3.1.2. Equations of small oscillations of the Schuler pendulum about the relative
equilibrium (110). 3.1.3. Equations of free oscillations of the Schuler pendulum about a position
of relative equilibrium: the case of finite angles (115 ).
s3.2. The two-gyro vertical and the Geckeler-Anschutz
gyrocompass horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.1. The two-gyro vertical (117). 3.2.2. The Geckeler-Anschutz gyrocompass horizon (122).
3.3. A general Schuler -tuned system. . . . . . . . . . . . . . . . . . . . 126
3.3.1. Conditions of existence of relative equilibrium (126). 3.3.2. Perturbed motion about the
relative equilibrium (128).
P 3.4. Analogy between Schuler-tuned systems and two-accelero-
.............................
meter inertial systems 131
3.4.1. Comparison of the first group of error equations with the equations of oscillations of a
gyropendulum system about relative equilibrium (131 1. 3.4.2. Sufficient stability conditions
for two-accelerometer inertial systems (135). 3.4.3. Damping of oscillations in a Schuler-tuned
system (137).
Chapter 4. SIMPLIFIED OPERATION EQUATIONS OF IDEAL AND
PERTURBED TWO-ACCELEROMETER INERTIAL SYSTEMS . . . 140
.........................
$4.1. General considerations. 140
S 4.2. Simplifications in an inertial system measuring the
orthodromic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2.1. Exact ideal operation equations. Error equations (141). 4.2.2. Some particular cases of
azimuthal orientation of accelerometers (151). 4.2.3. Simplifications which follow from the
nearly spherical figure of the Earth (153). 4.2.4. Simplifications associated with the closeness
of the trajectory to the Earth's surface and the orthodromy plane (158). 4.2.5. Different methods
of accounting for the horizontal component of the Earth's gravitation (165). 4.2.6. Other
simplifications (171).
14.3. Simplifications in an inertial system which measures the
............................
geographical coordinates. 174
4.3.1. Exact ideal operation equations. Error equations (174). 4.3.2. Simplifications of the
equations for geographical coordinates (178).
3 4.4. Determination or orthodromic coordinates with accelero-
meters oriented in the plane of the geographical horizon . . . . . . . 180
Cliaptet, 5. IAERTIAL. SYSTEZTS it.ITH IIOPPLER CORRECTION, 188
i5.1. -4syniptotic stability of inertial systenis. l-elocity errors . 188
5 5.7. &,sic informaticn on Doppler velocity meters . . . . . . . . . 193
E,3.:3. Uoppler damping of an inertial system . . . , . . . . . . . . . . 196
,.
I Lfnpcrtiirbed (ideal 1 operation equdtions. Error equations 196 I. 5.3.2. Ided operation
I. .q, .., ._,~t.i -, %vd~,in!jci r ire'rnr, .,orf tehqeu aetrimonr :i LioIiri aitnio onrst.h oEdricihiiti n'ssttermui n(?enUtUdIl. c r5ro.3o. 3( .2 154 t!s.h iIity analysii(2(23,.
S; 5,4, Changing the nat8iral frequencies of a Doppler-aided
sy sten1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 19
,;.! li?al .gperatim equations. Error equdtiOriS ( 21'3 ). 5.4.:. Aninalysir of error equations f C22!.
4 5.5, Daniping with simultaneous frequency change . . . . . . . . . . 225
Ideal <,peration equations. Error equdtionz (2251 . 5.5.2. Orthodromic coordinates 123?!.
. >tabilit) dnalysir i 2335 1. j..5.4. The solution of error equations. The effect of imtm-
'iierital errors:. Choice of correcti>n coefficients (245).
$5.6. Sonie additional remarks on Doppler correction . . . . , . . . 252
...-:,,! c'orrection techniques uninr a mscopically stabilized platform (232). 5.62. Doppler
.
,>rrectim increasinv tile order of the ror equations. Linear correction w;th variable coefficients.
!:onIiniar correction (259I .
Chapter 6. ASTROKORIICAL CORRECTION OF INERTL4L
NAk'IGATION SYSTE %IS. CO3IEiIKE D ASTRO- DOPPLER
CORRECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
S 6.1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5 6.2. Earth's motion. Determination Of directions to celestial
bodies in axes fixed to the Earth . , . . . . . . . . . . . . . . . . . . . . . 265
6.2.1. Simplified description of the Earth's motion (266 ). 6.2.2. Detsmiination of directions co
celestial hodies '26:!. 6.2.3. Earth's motion and measurement of time (252). 6.2.4. Fiying the
position on the Earth from astronoriical observations (275 ).
9 6.3. Astrocorrection using two stars. . . . . . . . . . . , . . . , , . , 276
6.3.i , Information obtained from telescopic observations. htrocorrection techniques (276 I.
6.9.2. iimultaneous linear correction using two stai-s f 2301. 6.3.3. qimultaneous "re13y-type"
t:orrection using two stars (286!. G.3.4. Alternate two-star correction (288). 6.3.5. The
correctins toques (290).
§ 6.4. Astrocorrection with phase indication of stars . . . . . . . . . 291
6.4.1 The principle of phase indication (2911. 6.4.2. The dymmics of a closed phase strocor-
rection system near the state of equilibrium. Stability dnalpi (2931.
§ 6.5. Combined astro-Doppler correction . . . . . . . . . . . . . . . . 300
6.5.1. System with simultaneous clelivery of Doppler and astmcorrection signak to gyro
rorquen(300). 6.5.2. One-star correction ($\IT).
IT