Coriolis Vibratory Gyroscopes: Theory and Design

122 Pages·2016·4.377 MB·English

by  Vladislav Apostolyuk (auth.)

Vladislav Apostolyuk
Coriolis
Vibratory
Gyroscopes
Theory and Design
Coriolis Vibratory Gyroscopes
Vladislav Apostolyuk
Coriolis Vibratory
Gyroscopes
Theory and Design
123
Vladislav Apostolyuk
Department ofAircraft Control Systems
andInstruments
National Technical University of Ukraine
Kiev
Ukraine
ISBN978-3-319-22197-7 ISBN978-3-319-22198-4 (eBook)
DOI 10.1007/978-3-319-22198-4
LibraryofCongressControlNumber:2015946586
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thankstotheadvancesinmicromachiningfabricationtechnologiesandsignificant
cost reduction due to mass production, miniature sensors of angular rate, or
gyroscopes,foundtheirwayintotheeverydaylifeofeveryuserofmoderngadgets,
such as smart phones, tablets or even wristwatches. Often without realising, many
ofusarecarryinginourpocketsfullyequippedwithallnecessarysensorscomplete
inertial navigation systems that not so long ago were available only for advanced
vehicles in sea, land, air or space. Accelerometers and gyroscopes are found in
specifications of any gadget supposed to react to user movements. And one of the
most commonly used type of gyroscopes used to developed these systems is
Coriolis vibratory gyroscope (CVG).
Needless to say that such a progress is a rewarding result of still ongoing
research and development work of many scientists around the world. There are
manybooksondesignandfabricationofmicromachinedinertialsensorspublished
by now, covering many important aspects of creating miniature inertial sensors.
This book is intended to complete them with such aspects of gyroscopes devel-
opmentastheoreticalanalysis,mathematicalmodellingandanalyticaldesign,when
desired performances are achieved by mathematical model analysis, rather than by
trial and error in prototyping. Mathematical models and theoretical analysis are
usedbothforoptimaldesignofsensitiveelements,aswellasdevelopmentofsignal
processing and control.
Chapter 1 gives an overview and classification of the most commonly used
designs of CVG sensitive elements. Chapter 2 deals with deriving sensitive
elements motion equations, and combining them into a single set of equations,
coveringmostofthesensitiveelementstypes.InChap.3thesegeneralisedmotions
equations are solved for terms that can be used for angular rate measurements.
Chapter 4 covers all aspects of CVG mathematical modelling in terms of demod-
ulatedenvelopesignals,thusprovidingthemeansforefficientanalysisofsensitive
elements dynamics as well as control and signal processing systems development.
CalculationofmajorCVGperformancesisdemonstratedinChap.5,alongwithan
analytical approach to sensitive element design. The final Chap. 6 covers signal
processingandcontrolsystemsdevelopment.ThelatterChaps.5and6arebasedon
v
vi Preface
the results from previous chapters, and are the only two chapters that can be read
independently of each other.
This book requires from the reader at least basic knowledge of higher mathe-
matics, mechanics and control systems theory. So unfortunately, not everyone
interested in vibratory gyroscope theory would be an intended reader, but rather
graduate students and higher. At the same time, readers will not be required to
follow the references and obtain additional information elsewhere as there are no
references in this book. Many results presented in the book could be referenced to
the previous publications of the author, but these references were intentionally
omitted in order to relieve the reader from the burden of searching for important
information on the subject elsewhere. Nevertheless, a time ordered list of recom-
mendedimportantpublicationsonCoriolisvibratorygyroscopesisgivenattheend
for those who may need it.
Finally,thereisnofinishlineinresearch,developmentandwritingbooksabout
them. So feel free to express your opinions, give suggestions and ask questions.
Hope you will have plenty of them.
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Principle of Operation and Classification. . . . . . . . . . . . . . . . . . 1
1.2 CVG Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Sensitive Element Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Motion Equations of Coriolis Vibratory Gyroscopes . . . . . . . . . . . 15
2.1 Translational Sensitive Element Motion Equations . . . . . . . . . . . 15
2.2 Rotational Sensitive Element Motion Equations. . . . . . . . . . . . . 19
2.3 Tuning Fork Sensitive Element Motion Equations . . . . . . . . . . . 23
2.4 Ring-Shaped Sensitive Element Motion Equations . . . . . . . . . . . 25
2.5 Generalised Motion Equations. . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Sensitive Element Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Primary Motion of the Sensitive Element . . . . . . . . . . . . . . . . . 29
3.2 Sensitive Element Motion on a Rotating Base . . . . . . . . . . . . . . 31
3.3 Modelling Proof Mass Motion Trajectory . . . . . . . . . . . . . . . . . 38
®
3.4 Numerical Simulation of CVG Dynamics Using Simulink . . . . . 44
4 CVG Dynamics in Demodulated Signals . . . . . . . . . . . . . . . . . . . . 47
4.1 Motion Equations in Demodulated Signals . . . . . . . . . . . . . . . . 47
4.2 CVG Transfer Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Amplitude and Phase Responses . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Stability and Transient Process Optimisation . . . . . . . . . . . . . . . 53
4.5 Simplified CVG Transfer Function and Its Accuracy . . . . . . . . . 57
4.6 Trajectory Rotation Transfer Function. . . . . . . . . . . . . . . . . . . . 58
vii
viii Contents
5 Sensitive Element Design Methodologies . . . . . . . . . . . . . . . . . . . . 63
5.1 Optimal Excitation of the Primary Oscillations. . . . . . . . . . . . . . 63
5.2 Scale Factor and Its Linearity . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Resolution and Dynamic Range. . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Dynamic Error and Bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Signal Processing and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1 Process and Sensor Noises in CVG . . . . . . . . . . . . . . . . . . . . . 87
6.2 Sensor Noise Optimal Filtering . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Process Noise Optimal Filtering. . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Optimal Kalman Filter Synthesis . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 Cross-Coupling Compensation. . . . . . . . . . . . . . . . . . . . . . . . . 98
6.6 Temperature Errors Compensation . . . . . . . . . . . . . . . . . . . . . . 105
6.7 Whole-Angle Force Rebalance Control. . . . . . . . . . . . . . . . . . . 111
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 1
Introduction
It is commonly accepted that the conventional gyroscope, as an instrument for
determining orientation, was inspired by a top, known to practically all ancient
civilisation.Whenatoprotates,itpreservesitsverticalorientationevenifthebase
is tilted. This natural behaviour was used since eighteenth century A.D. to imple-
ment artificial horizon. The technical term gyroscope was introduced by Léon
Foucault,whenin1852heusedafastspinningrotorinsidegimbalstodemonstrate
the Earth’s rotation. The word gyroscope consists of two Greek words:
gyros—“rotation” or “circle” and scopeo—“to see” or “observe”. Although he did
notinventthegyroscopeitself,1yearearlier,in1851,heusedapendulum,known
as Foucault pendulum, to demonstrate the Earth’s rotation in another experiment.
Curiouslyenough,thispendulumiscommonlyconsiderednowadaysasaprototype
to modern Coriolis vibratory gyroscopes.
1.1 Principle of Operation and Classification
Conventionalmechanicalgyroscopeconsistsofarotormountedinsideatwo-frame
gimbal that provides the rotor with two angular degrees offreedom in addition to
one, corresponding to its own rotation. Such design is often referred to as a free
gyroscope. Free gyroscope preserves orientation of axis of its own rotation rela-
tively to the inertial reference frame, usually assumed to be fixed to stars. This
phenomenon is used to determine orientation of the user relatively to the inertial
frame when direct observation of orientation is impossible. When outer frame of a
freegyroscopeisremovedandtheremainingoneprovidesonlyoneangulardegree
offreedom,andwhenthebasestartstorotate,thespinningrotormovestoalignthe
axisofitsownrotationwiththeaxisoftheexternalangularrate.Thisphenomenon
is used to build such gyroscopic instruments as artificial horizons and gyroscopic
compasses,wheretheexternalangularrateisprovidedbytheEarth’srotationitself.
However,ifaspringisattachedtotheremaininggimbalframe,thuspreventingthe
alignment between rotor and angular rate axes, the final deflection of the frame is
linearly related to the magnitude of the external rate and allows its measurement.
Such an instrument is called angular rate sensor. It turned out that using angular
©SpringerInternationalPublishingSwitzerland2016 1
V.Apostolyuk,CoriolisVibratoryGyroscopes,
DOI10.1007/978-3-319-22198-4_1
2 1 Introduction
rate sensors instead offree gyroscopes is a cheaper, although less accurate, way to
buildinertialnavigationinstruments.Manydifferentdesignsofangularratesensors
were developed since, and among them vibratory gyroscopes occupy niche of
modern electro-mechanical sensors.
The main idea behind vibratory gyroscopes is to replace continuously spinning
rotor with vibrating structure, and make use of the Coriolis effect, causing sec-
ondary motion that is related to the external angular rate. This type of angular rate
sensors is commonly referred to as Coriolis vibratory gyroscopes (CVG).
In most of Coriolis vibratory gyroscopes, the sensitive element can be repre-
sentedasaninertiaelementmandelasticsuspensionwithtwoprevalentdegreesof
freedom: vertical and horizontal, as shown in Fig. 1.1.
Massiveinertia elementmisoftencalledproofmass,similarly tothetermused
in accelerometers. The sensitive element is driven to oscillate at one of its modes
with prescribed amplitude. This motion is usually called primary mode or primary
motion. When the sensitive element rotates about a particular fixed-body axis,
which is called sensitive axis (out-of-plane axis in Fig. 1.1), the so-called Coriolis
effect causes the proof mass to move in an orthogonal direction, thus resulting in
secondary motion. This effect is quantified by a fictitious Coriolis force
~F ¼(cid:2)2(cid:3)m(cid:3)~X(cid:4)~v: ð1:1Þ
C
Here,~visthevectoroftheprimarymotionandX~istheexternalangularrate.In
fact,wehavetwomotionsthataredynamicallycoupledviaCorioliseffect.Ifthere
is no external angular rate (X¼0), secondary motion is absent, since there is no
Coriolis coupling.
In vibratory gyroscopes, this primary motion is designed to be oscillatory. As a
result,secondarymotionbecomesoscillatoryaswell.Hereafter,excitedoscillations
are referred to as primary oscillations and oscillations caused by angular rate are
referred to as secondary oscillations or secondary mode.
Fig.1.1 PrincipleofCVG
Primary
operation
motion
Secondary
m motion
Ω