Self-calibrating three axis angular rate sensor

A self-calibrating three axis angular rate sensor in which a single proof mass is supported from an outer case by a set of spring elements having substantially equal mechanical translational spring rates in all directions. Each axis is electromechanically coupled such that each of three sets of electrical terminals acts as a combined electrical driving and electrical detection means with respect to motion of the proof mass. The proof mass motion in combination with a general three axis angular rate input results in coriolis forces acting upon the proof mass. When the electrical terminals are driven with a multiplicity of additive sinusoidal time function electrical voltages, measurements of these voltages and resulting electrical currents comprise observable data for a computed extraction of three axis angular rate inputs. In addition, the computed extraction provides an accurate measure of variations of electrical parameters of the electromechanical coupling, of variations of mechanical translational spring rates and damping parameters, and of variations of electromechanical coupling coefficients of the set of spring elements. The computed extraction results in outputs of three axis angular rate which are substantially independent of these electrical and mechanical parameter variations. The measurements of electrical and mechanical parameter variations are compared with their respective nominal values in order to establish an overall data validity check.

BACKGROUND--FIELD OF INVENTION 
This invention relates to a vibratory, coriolis force type angular rate 
sensor, specifically to a self-calibrating, three axis implementation 
thereof. 
BACKGROUND--DESCRIPTION OF PRIOR ART 
Vibrating mass coriolis force type angular rate sensors or gyros have 
inherent advantages in terms of simplicity of design, low cost, and long 
life. The use of coriolis force in vibratory gyroscopes is well known. For 
example, Soderkvist, U.S. Pat. No. 5,251,483 relates to a 
piezoelectrically driven tuning fork type coriolis force type gyro. The 
accuracy of this type of sensor, however, has been limited by variations 
in its physical and electrical parameters with time, temperature, and 
usage. These variations result in changes in the sensor scale factor, the 
sensor angular rate bias offset, and the misalignment angles of the axis 
about which the angular rate is to be measured. Moreover, because these 
sensors measure only a single axis of input angular rate, three sensors 
are generally required for any systems that must measure angular rate 
along all three physical axes. Means for angularly orienting the input 
angular rate sensing axes of three sensors with respect to one another 
with sufficient accuracy must also be provided. 
Coriolis force type angular rate sensors typically require one transducer 
or one set of transducers to perform the drive motion function and a 
separate sensor or set of sensors to detect the coriolis force generated 
motion resulting from an input angular rate. Even the application of 
feedback control methods, as disclosed in Fersht, et al., U.S. Pat. No. 
5,056,366, to null the coriolis force induced motions of the driven 
element, do not eliminate scale factor errors due to variations in 
piezoelectric coupling. These methods also do not eliminate angular rate 
bias errors due to variations in the combination of mechanical coupling 
and mechanical damping. In general, any mechanical coupling between driven 
elements and detection elements other than desired coriolis force coupling 
results in sensor errors. Since this undesired coupling may change with 
time and temperature, these errors have limited the accuracy attainable 
from coriolis force vibratory angular rate sensors. 
Most coriolis force type angular rate sensors are mechanized in a tuning 
fork, or similar, arrangement, in which pairs of inertial elements are 
driven in opposite time phase with respect to one another. This is to 
establish a high mechanical "Q," i.e., to minimize mechanical energy 
losses of the vibrating inertial elements through the supporting base 
structure. This is critically important for tuning fork type angular rate 
sensors because any combination of these energy losses and unwanted 
mechanical coupling between the driven inertial elements and the detection 
elements results in the detection of a nonexistent angular rate. 
One form of tuning fork type angular rate sensor, for example, Weinberg, et 
al, U.S. Pat. No. 5,388,458, employs a quartz resonant oscillator. This 
methodology forces the excitation frequency to constantly be near the 
natural frequency of the driven tines of the tuning fork sensor. While 
this objective is achieved, there result significant limitations. These 
include a need for amplitude control of the feedback oscillator as well as 
the utilization of only a single frequency per axis. This significantly 
limits the ability to separate angular rate from variations in sensor 
electrical and mechanical parameters. 
Stewart, et al, U.S. Pat. No. 5,065,627, discloses an inertial measurement 
unit with three axes of angular rate as well as three axes of acceleration 
outputs. The methodology described requires a multiplicity of mechanical 
parts, including two sets of three separate pendulous accelerometers. Each 
said set of accelerometers must be mounted on a separate driven 
oscillatory rotary element. These driven mounting structures must include 
highly accurate angular orientation provisions, and must provide for 
balancing with extreme precision centrifugal forces acting upon the 
accelerometer pendulosities with respect to the two sets of 
accelerometers. There are no provisions here for self-calibration. 
Dunn, U.S. Pat. No. 5,359,893, employs a micromachined structure to 
implement a two axis vibratory rotation gyro. By adding an additional 
identical gyro mounted orthogonal to the first, three axes of output 
angular rate are obtained. No provisions are made, however, for 
self-calibration. There is, therefore, a need for a coriolis force 
vibratory gyro which maintains the inherent advantages of mechanical 
simplicity and low cost, but which overcomes the errors due to variations 
of the critical mechanical and electrical parameters with time and 
temperature. There is also a need for an intrinsically three axis single 
proof mass or inertia sensor which continuously maintains mutual axis 
orthogonality self-calibration. 
OBJECTS AND ADVANTAGES 
Accordingly, several objects and advantages of this invention include 
significant improvement in accuracy over previous vibratory type angular 
rate sensors. This is accomplished by incorporation of means to 
continuously estimate substantially all critical mechanical and electrical 
parameters of a coriolis force type angular rate sensor. Three mutually 
orthogonal axes of angular rate information from a single mechanical 
sensor unit are provided, and said measures of angular rate remain 
substantially independent of variations in critical mechanical and 
electrical parameters. 
The mechanical sensor unit of this invention requires only a single proof 
mass. The high mechanical "Q" requirement of conventional vibrating 
angular rate sensors with its implied requirement for a tuning fork type 
mechanical configuration is minimized here. This is done by the continuous 
estimation process of said critical mechanical parameters. In this way, 
although significant input-to-output coupling may be present, the desired 
coriolis force effects are separable from the mechanical damping and 
angular cross coupling force effects. 
This invention utilizes one set of electromechanical transducers per axis 
as both a means of excitation and a means of detection of vibration of a 
proof mass this obviates the need for separate excitation and detection 
transducers. 
This invention requires only conventional electromechanical technology, 
such as the use of piezoelectric quartz or piezoceramics transducer 
elements. These transducers are mechanically connected to a single common 
proof mass element as well as to a common outer case along all three axes. 
The digitally computed outputs include three mutually orthogonal angular 
rates as well as mechanical parameter outputs of all direct axis and cross 
axis mechanical translational spring rates, all direct axis and cross axis 
mechanical damping parameters, and all direct axis and cross axis 
electromechanical coupling coefficients, such as piezoelectric coupling 
coefficients. The electrical parameters in this context are the electrical 
capacity and resistance values of piezoelectric beams. 
Another advantages of this invention is the integrated and complementary 
use of mechanical sensor and microprocessor technologies. This enables an 
accurate and continuous modeling of a relatively simple electromechanical 
three axis sensor, thus providing the needed calibration for greatly 
improved accuracy over techniques used heretofore. This calibration also 
incorporates the automatic continuous maintenance of sensor axis mutual 
orthogonality. 
In addition, this invention provides, as a direct output from this sensor 
parameter and coefficient evaluation process, a continuous measure of the 
validity of the output data, as any out of tolerance measurements indicate 
a risk of an impending sensor malfunction. 
Still further objects and advantages will become apparent from a 
consideration of the ensuing description and accompanying drawings.

REFERENCE NUMERALS 
10 three axis mechanical sensor unit 
12 six-channel analog-to-digital voltage conversion subsystem 
14 angular rate microprocessor 
16 outer case 
18 proof mass 
20 beam 
20A beam 
20B beam 
20C beam 
20D beam 
20E beam 
22 clamping block 
22A clamping block 
22B clamping block 
22C clamping block 
22D clamping block 
22E clamping block 
22F clamping block 
22G clamping block 
22H clamping block 
221 clamping block 
22J clamping block 
22K clamping block 
22 multi-frequency sinusoidal voltage source subsystem 
26 electrode 
26A electrode 
26B electrode 
26C electrode 
26D electrode 
26E electrode 
26F electrode 
26G electrode 
26H electrode 
26I electrode 
26J electrode 
26K electrode 
28 wire lead 
28A wire lead 
28B wire lead 
28C wire lead 
28D wire lead 
28E wire lead 
28F wire lead 
28G wire lead 
28H wire lead 
28I wire lead 
28J wire lead 
28K wire lead 
30 electrical feed-through element 
30A electrical feed-through element 
30B electrical feed-through element 
30C electrical feed-through element 
30D electrical feed-through element 
30E electrical feed-through element 
30F electrical feed-through element 
30G electrical feed-through element 
30H electrical feed-through element 
30I electrical feed-through element 
30J electrical feed-through element 
30K electrical feed-through element 
SUMMARY 
A self-calibrating three axis angular rate sensor in which a single proof 
mass is supported from an outer case by a set of spring elements. These 
spring elements provide mechanical support of the proof mass with 
substantially equal mechanical translational spring rates in all 
directions. The set of spring elements is electromechanically driven at or 
near the mechanical suspension system natural translational frequency 
along all three axes by electrical excitation. This excitation is made up 
of a multiplicity of additive separate frequency sinusoidal signals with 
respect to time at substantially constant driving frequencies. The 
resulting three axis proof mass motion results in coriolis forces upon the 
proof mass in resonse to three axis angular rate inputs. The electrical 
voltages and currents are measured and computed at each frequency, and the 
resulting values used to compute the three input angular rates as well as 
the critical electrical parameters, mechanical parameters, and 
electromechanical coupling coefficients. Said computed parameters and 
coefficients are compared with their nominal values in order to establish 
an overall data validity check. 
Description 
A self-calibrating three axis angular rate sensor is made up of a three 
axis mechanical sensor unit 10, a six-channel analog-to-digital voltage 
conversion subsystem 12, a multi-frequency sinusoidal voltage source 
subsystem 24, and an angular rate microprocessor 14, as shown in FIG. 1. 
FIGS. 2 and 3 show three axis mechanical sensor unit 10 made up of an outer 
case 16, a proof mass 18, and six compliant length extensional mode 
piezoelectric beams: beam 20, beam 20A, beam 20B, beam 20C, beam 20D, and 
beam 20E. Because of design symmetry of three axis mechanical sensor unit 
10, FIG. 3, which is a three axis diametrically opposing view with respect 
to FIG. 2, is identical to FIG. 2 except for identification of specific 
parts. 
Said beams are arranged in three pairs of two substantially collinear 
beams, with said resulting collinear directions substantially mutually 
orthogonal. Said beams comprise a set of mechanical spring elements having 
substantially equal mechanical translational spring rates along all 
directions connecting proof mass 18 to outer case 16. 
Clamping blocks 22, 22A, 22B, 22C, 22D, 22E, 22F, 22G, 22H, 22I, 22J, and 
22K together with their mechanical interfaces with outer case 16 and proof 
mass 18, comprise controlled compression clamping attachments for beam 20, 
20A, 20B, 20C, 20D, and 20E mechanical terminations, as shown in FIGS. 
2,3, and 4. The preferred embodiment of this invention employs length 
extensional mode piezoelectric beams to make up said set of spring 
elements with electromechanical coupling means. 
Piezoelectric beams 20, 20A, 20B, 20C, 20D, and 20E are each a 
piezoelectric material rectangular solid made up of either crystalline 
alphaquartz or a piezoceramic material, such as lead zirconate titanate. 
Length extensional mode piezoelectric beams are fabricated by emplacement 
of electrode pairs across the narrowest dimension of said piezoelectric 
material rectangular solid, and by selection of piezoelectric axes 
orientation such that application of electrical voltage to said electrodes 
results in a mechanical stress substantially along the largest dimension, 
the length axis, of said rectangular solid. As shown in FIGS. 2 and 3, 
said length axis of each beam is substantially oriented along a separate 
axis of a mutually orthogonal three axis coordinate system XYZ fixed with 
respect to outer case 16. 
Piezoelectric beams 20, 20A, 20B, 20C, 20D, and 20E have substantially 
identical physical dimensions, crystal axis orientations, as well as 
symmetrical installations. This results in substantially equal mechanical 
translational spring rates in all directions with respect to mounting of 
proof mass 18, and it also results in consequent substantially equal 
translational mode natural mechanical resonant frequencies along X, Y, and 
Z axes. Translational mode natural mechanical resonant frequency for any 
axis is calculated as 
##EQU1## 
where Y.sub.11 is direct axis mechanical translational spring rate, and m 
is value of proof mass 18. Translational mode natural mechanical resonant 
frequency is set to a value one or more orders of magnitude greater than 
any external vibration isolator roll-off frequency for anticipated input 
translational vibration motion frequency spectra to three axis mechanical 
sensor unit 10. Roll-off frequency in this context is defined as that 
frequency above which the magnitude of vibration experienced by three axis 
mechanical sensor unit 10 is decreasing with respect to frequency at a 
rate of at least 12 decibels per octave. For expected values of said 
roll-off frequency of 100 Hz or less, a translational mode natural 
mechanical resonant frequency is generally set a value between 1000 Hz and 
10,000 Hz. 
Because of expected variations of physical dimensions and mounting of beams 
20, 20A, 20B, 20C, 20D, and 20E due to reasonably set tolerances, actual 
translational mode natural mechanical resonant frequencies along 
directions X, Y, and Z may differ by small amounts, depending upon said 
tolerances. 
Length axes of beam pair 20-20A, beam pair 20B-200 and beam pair 20D-20D 
are oriented substantially along X, Y, and Z axis directions of three axis 
mechanical sensor unit 10, respectively, as shown in FIGS. 2 and 3. 
Because of anticipated small errors in manufacture and assembly, length 
axes of said beam pairs are generally not precisely aligned with 
directions X, Y, and Z. Length axes of said beam pairs also substantially 
intersect proof mass 18 center of mass. 
Piezoelectric beams 20, 20A, 20B, 20C, 20D, and 20E are plated with two 
sided, symmetrical, metallic electrode pairs 26-26A, electrode pair 
26B-26C, electrode pair 26D-26E, electrode pair 26F-26G, electrode pair 
26H-261, and electrode pair 26J-26K, respectively, as shown in FIGS. 2 and 
3. This establishes a means for externally connecting electrical voltages 
and currents to said electrode pairs.. Each of said electrodes is made up, 
for example, of a thin layer of fired or vacuum deposited silver, gold, 
aluminum or electroless nickel. Connected to electrodes 26, 26A, 26B, 26C, 
26D, 26E, 26F, 26G, 26H, 26I, 26J, and 26K are wire leads 28, 28A, 28B, 
28C, 28D, 28E, 28F, 28G, 28H, 28I, 28J, and 28K, respectively. 
Wire leads 28, 28A, 28B, 28C, 28D, 28E, 28F, 28G, 28H, 28I, 28J, and 28K 
connect to electrical feed-through elements 30, 30A, 30B, 30C, 30D, 30E, 
30F, 30G, 30H, 301, 30J and 30K, respectively, as shown in FIGS. 2 and 3. 
All of said electrical feed-through elements are mounted in outer case 16. 
External terminals of electrical feed-through elements are connected in 
pairs to multi-frequency sinusoidal voltage source subsystem 24 and six 
channel analog-to-digital voltage conversion subsystem 12 as shown in FIG. 
7. Said pairs are electrical feed-through elements 30A-30B, 30-30C, 
30E-30F, 30D-30G, 30I-30J and 30H-30K, as shown. 
FIG. 5 shows orientation of crystallographic axes for X-axis beam pair 
20-20A and electrical parallel connection of electrodes 26, 26A, 26B, and 
26C. Y and Z axis corresponding elements are similarly interrelated. While 
other combinations of crystallographic axes orientations and electrical 
connections are possible, in the preferred embodiment of this invention a 
configuration shown in FIG. 5 has been selected. Any other similar 
combination may also be employed provided that crystallographic axes 
orientation of beam 20A is substantially identical to that of beam 20 and 
upper electrode of beam 20 is connected to lower electrode of beam 20A and 
vice versa. Piezoelectrically induced length and shear motions of beams 20 
and 20A, in this arrangement, result in only a net translational force 
upon proof mass 18 and substantially no torques upon said proof mass. 
One lead from each of said pairs 30A-30B, 30E-30F, and 30I-30J is connected 
to one of a set of known value scaling resistors RS1, RS2, and RS3, 
respectively, as shown in FIG. 7. Said resistors are connected to separate 
multi-frequency sinusoidal voltage sources 1, 2, and 3, respectively, of 
multifrequency sinusoidal voltage source subsystem 24. Remaining leads 
from each said pair of external terminals of electrical feed-through 
elements 30-30C, 30D-30G, and 30H-30K are connected to a common electrical 
ground as shown. 
Each side of scaling resistors RS1, RS2, and RS3 is connected to separate 
voltage analog-to-digital converter channels. This results in six channels 
of analog-to-digital voltage conversion. Use of these electrical 
connections results in a direct measure of electrical voltage across each 
pair of piezoelectric terminals. Also obtained is an indirect measure of 
electrical current flowing through said pairs of piezoelectric terminals 
by measurement of voltage across each said series connected, known value, 
scaling resistor. 
Electrical drive voltages are each made up of a multiplicity of additive 
separate frequency sinusoidal signals with respect to time. These 
electrical drive voltages are applied to external terminals of electrical 
feed-through element pairs 30A-30B, 30E-30F, and 30I-30J. While other 
choices of driving frequency sets can be used, in the preferred embodiment 
of this invention each multi-frequency sinusoidal voltage source outputs 
three separate and unique, substantially constant frequency, substantially 
constant amplitude, and substantially sinusoidal waveform voltage signals. 
Two of said separate frequencies for each axis are set to be near said 
translational mode natural mechanical resonant frequency nominal value. A 
third frequency for each axis is set at a factor of approximately 1.9 
times said translational mode natural mechanical resonant frequency 
nominal value. This factor value of 1.9 is not critical. 
Mechanical motion responses of proof mass 18 with respect to driving 
voltage frequency near said resonant frequency are, as desired, 
particularly sensitive to electrical and mechanical parameters to be 
measured. Setting said third frequency at said factor of 1.9 is to obtain 
an observable and accurate measure of electrical capacity parameters of 
said beams 20, 20A, 20B, 20C, 20D, and 20E. 
Six total frequencies which are near said translational mode natural 
mechanical resonant frequency are separated from each other by 
approximately one percent of said translational mode mechanical sensor 
natural mechanical resonant frequency nominal value. The optimum selected 
value for said separation is proportional to ratio of nominal direct axis 
mechanical damping parameter of beams 20, 20A, 20B, 20C, 20D, and 20E to 
nominal value of proof mass 18. Said six frequencies mean value is set at 
said translational mode natural mechanical resonant frequency nominal 
value. Accordingly, nine separate driving frequencies are synthesized and 
set from angular rate microprocessor 14 by a crystal oscillator timing 
reference shown in FIG. 10. Said crystal oscillator timing reference 
provides time synchronization for said nine separate driving frequencies. 
In addition, said crystal oscillator timing reference provides time 
synchronization for sampling rate of six channel analog-to-digital voltage 
conversion subsystem 12. 
Six channel analog-to-digital voltage conversion subsystem 12 provides, as 
shown in FIG. 10, separate time periodic data sample measurements of 
voltage outputs of multi-frequency sinusoidal voltage source subsystem 24 
and of beam piezoelectric voltages at electrical feed-through element 
pairs 30A-30B, 30E-30F, and 30I-30J, shown as E1, E2, and E3, 
respectively, in FIG. 8. All six input voltage signals are converted 
simultaneously and periodically at a sampling rate which is preferably at 
least 400 times greater than angular rate microprocessor 14 overall 
computation output rate. Said sampling rate is also preferably at least 
five times greater than translational mode natural mechanical resonant 
frequency. 
Operation--General 
Operation of a self-calibrating three axis angular rate sensor is carried 
out by: 
(1) Piezoelectrically driving proof mass 18 of mechanical sensor unit 10 
with known frequency sinusoidal voltages, as shown in FIG. 7. 
(2) Utilizing a set of equations which define relationship of proof mass 
motion to measured piezolectric beam voltages E1, E2, E3 and currents Im1, 
Im2, Im3 shown in FIG. 8. 
(3) Fitting said piezoelectric beam voltage measurements to a set of 
sinusoidal time functions using angular rate microprocessor 14 as shown in 
FIGS. 10 and 11, and, by use of said fitted sinusoidal functions, 
(4) Computing input angular rate about three axes of coordinate system XYZ 
shown in FIGS. 2 and 3, as shown in FIGS. 10 and 11 by use of said 
measurement set of sinusoidal functions. 
(5) Computing specific electrical and mechanical parameters and 
coefficients of three axis mechanical sensor unit 10, as shown in FIGS. 10 
and 11 by use of said measurement set of sinusoidal functions. 
This operation is described in detail in the following paragraphs: 
Operation--Piezoelectric Drive of Proof Mass 
Rectangular solid piezoelectric materials with specifically selected 
crystal axis orientations experience a length expansion along an axis 
orthogonal to direction of electric field. For example, for an x-cut 
crystalline alpha-quartz bar, an applied positive voltage field along 
crystal xc-axis direction results in a contraction along crystal yc-axis 
shown in FIG. 5. A negative applied voltage along this same crystal x-axis 
results in an expansion along crystal yc-axis. In this way, as also 
illustrated by FIG. 5, a positive voltage at electrical feed-through 
elements 30A and 30B with respect to electrical feed-through elements 30 
and 30C results in a length contraction of beam 20A and a length expansion 
of beam 20. These changes in length of beams 20A and 20 result in a 
"push-pull" force upon proof mass 18 in the negative X direction of 
coordinate system XYZ shown in FIGS. 2, 3, and 5. Similar relationships 
between applied voltages and resulting forces upon proof mass 18 exist for 
Y-axis and Z-axis of said XYZ coordinate system. It must be noted here 
that crystal xc-axis and crystal yc-axis are not necessarily related to 
X-axis and Y-axis, respectively, of said XYZ coordinate system. 
Accordingly, as shown in FIG. 7, an application of electrical voltages with 
respect to ground at electrical feed-through element pairs 30A-30B, 
30E-30F, and 30I-30J results in forces upon proof mass 18 in directions X, 
Y, and Z, respectively of said XYZ coordinate system, the direction of 
said forces depending upon polarity of said applied electrical voltages. 
Therefore, application of sinusoidal voltages from multi-frequency 
sinusoidal voltage source subsystem 24 to electrical feed-through element 
pairs 30A-30B, 30E-30F, and 30I-30J shown in FIGS. 7 and 8 results in 
piezoelectrically generated sinusoidal forces along all three axes of 
proof mass 18. These forces result in sinusoidal translational motions and 
velocities of proof mass 18. Said translational velocities of proof mass 
18, in combination with input angular rates about XYZ coordinate system 
shown in FIG. 6, result in generation of coriolis forces acting upon proof 
mass 18. FIG. 6 illustrates said coriolis force generation for angular 
rates .omega..sub.x, .omega..sub.Y, and .omega..sub.z about X, Y and Z of 
said XYZ coordinate system in combination with velocities VX, VY, and VZ 
of proof mass 18 with respect to outer case 16 along said X, Y, and Z axis 
directions. Generated coriolis forces are a negative vector cross product 
of a vector input angular rate to three axis mechanical sensor unit 10 and 
vector velocity of proof mass 18 with respect to outer case 16 multiplied 
by twice the effective mass of proof mass 18. For said assumed angular 
rates and said velocities the coriolis forces generated upon proof mass 18 
are, as shown in FIG. 6: 
X-axis component of coriolis force, 
EQU CF.sub.x =-2m (.omega..sub.y 
V.sub.z -.omega..sub.z V.sub.y), 
Y-axis component of coriolis force, 
EQU CF.sub.y =-2m (.omega..sub.z 
V.sub.x -.omega..sub.x V.sub.z ), 
and 
Z-axis component of coriolis force, 
EQU CF.sub.z =-2m (.omega..sub.x 
V.sub.y -.omega..sub.y V.sub.x). 
It is seen that each said coriolis force component is a function of two 
separate products of said angular rate and said velocity components. 
Multi-frequency sinusoidal voltage source subsystem 24 of FIG. 1 supplies 
driving voltages for each axis, as shown in FIG. 7, applying corresponding 
driving forces along length axes of each beam of three axis mechanical 
sensor unit 10. FIG. 8 shows electrical voltage and current interface to 
all twelve electrical feed-through element pairs 30A-30B, 30-30C, 30E-30F, 
30D-30G, 30I-30J, and 30H-30K. 
Also with reference to FIG. 8, piezoelectric coupling of externally 
connected electrical voltages and currents to three axis mechanical sensor 
unit 10 mechanical parameters takes place analytically by means of a three 
axis electromechanical piezoelectric transformer matrix. Said matrix takes 
into account angular misalignments between coordinate system XYZ and 
length axis orientation of beams 20, 20A, 20B, 20D, 20D, and 20E, as shown 
in FIGS. 2 and 3, as well as inherent direct axis and cross axis 
coefficients of piezoelectric coupling. Because said angular misalignments 
and said piezoelectric coupling coefficients are not immediately known and 
vary with time, they are, as an important part of the operation of this 
invention, automatically and continuously computed, as shown below by 
Equations (1) through (28) and associated discussion of said equations. 
Said piezoelectric coupling coefficients relate input vector voltage to 
output vector force and identical said piezoelectric coupling coefficients 
relate output mass vector velocity to input vector electrical current. For 
this reason each collinear beam pair 20-20A, 20B-20C, and 20D-20E can be 
considered as an equivalent single piezoelectric beam because of a 
parallel electrical connection of each said pair of beam electrodes as 
exemplified by FIG. 5. Said parallel electrical connection sums electrical 
current for each said beam pair for a common vector velocity of proof mass 
18, and sums forces upon said proof mass 18 for a common driving voltage. 
Operation--Equations Description 
FIG. 8A shows V1, V2, and V3 translational velocity vector components of 
proof mass 18 with respect to outer case 16 corresponding to 
piezoelectrical currents I1, 12, and 13, respectively, shown in FIG. 8; 
FIG. 8A also shows piezoelectrically generated F1, F2, and F3 force vector 
components corresponding to voltages E1, E2, and E3 shown in FIG. 8. 
Because these are piezoelectrically related electrical voltages, 
electrical currents, forces and velocities, the vector combinations V1-F 
1, V2-F2, and V3-F3 are collinear, as shown in FIG. 8A. These 
relationships, expressed mathematically, are as follows; 
##EQU2## 
Where N1, N2, and N3 are piezoelectric coupling coefficients for collinear 
beam pairs 20-20A, 20B-20C, and 20D-20E, respectively. 
Velocities V1, V2, and V3 are related to VX, VY, and VZ, corresponding 
velocity components along X, Y, and Z axes, respectively by sets of 
direction cosines a11, a12, a13, etc. as follows: 
EQU V1=a11*VX+a12*VY+a13*VZ (7) 
EQU V2=a21*VX+a22*VY+a23*VZ (8) 
EQU V3=a31*VX+a32*VY+a33*VZ (9) 
Similarly, force vector components FX, FY, and FZ along X, Y, and Z axes, 
respectively are expressed mathematically as: 
EQU FX=a11*F1+a21*F2+a31*F3 (10) 
EQU FY=a12*F1+a22*F2+a32*F3 (11) 
EQU FZ=a13*F1+a23*F2+a33*F3 (12) 
Substituting Equations (1) through (6)into Equations (7) through (10) 
result in matrix equations as follows: 
##EQU3## 
Equations (13) and (14) can be expressed as: 
##EQU4## 
and a.sup.T is the matrix transpose of a. 
Making use of classical dynamics for suspended proof mass 18 and known form 
of piezoelectric coupling coefficients results in two steady-state 
frequency response matrix equations as follows: 
EQU F=a.sup.T P=KU+jhDU-h.sup.2 m(W.cndot.U)-W.sup.2 U)-jhmj(2W.times.U)(20) 
EQU L=jhaU=L.sub.M -EG-jhEC (21) 
Where h is frequency of operation in Hz multiplied by 2.pi., m is known 
value of proof mass 18, j=.sqroot.-1, 
##EQU5## 
represents measured electrical currents Im1, Im2, and Im3 shown in FIG. 8 
divided by their respective piezoelectric coefficients N1, N2, and N3, 
##EQU6## 
represents reciprocals of products of piezoelectric leakage resistances 
R1, R2, and R3 shown in FIG. 8 and their respective piezoelectric 
coefficients N1, N2, and N3, 
##EQU7## 
represents electrical capacities C1, C2, and C3 shown in FIG. 8 divided by 
respective piezoelectric coefficients N1, N2, and N3, 
##EQU8## 
represents displacement of proof mass 18, where X, Y, and Z are 
displacement motions of proof mass 18 with respect to outer case 16 along 
these said respective axes, as shown in FIGS. 2 and 3, symmetrical K 
matrix 
##EQU9## 
represents relationship between mechanical spring forces on proof mass 18 
and a general three axis vector displacement of said proof mass 18 with 
respect to XYZ coordinate system shown in FIG. 2. K11, K22, and K33 are 
direct axis mechanical spring rate parameters along said X, Y, and Z axes, 
respectively; cross axis mechanical spring rate parameters K12, K13, and 
K23 take into account both misalignment angles of beams 20, 20A, 20B, 20C, 
20D and 20E and general anisotropic spring rates of piezoelectric 
materials, 
symmetrical D matrix 
##EQU10## 
relates mechanical damping forces on proof mass 18 to said X, Y, and Z 
axis velocity components. D11, D22, and D33 are direct axis mechanical 
damping parameters; D12, D13, and D23 are cross axis mechanical damping 
parameters, 
a set of three input angular rates to be measured about said X, Y, and Z 
axes is denoted by vector 
##EQU11## 
where .omega..sub.x, .omega..sub.y, and .omega..sub.z are input angular 
rates about said X-axis, said Y-axis and said Z-axis, respectively. 
Equation (20) states that piezoelectrically coupled vector force F is equal 
to summed spring force vector KU, damping force vector jhDU, acceleration 
of proof mass 18 force vector -h.sup.2 mU, proof mass 18 centrifugal force 
vector -m(W(W.cndot.U)-W.sup.2 U), and proof mass 18 coriolis force vector 
-jhm(2W.times.U). Scalar term Win Equation (20) represents magnitude of 
total vector input angular rate. 
Term m(W(W.cndot.U)-W.sup.2 U) in Equation 20 represents centrifugal force 
acting upon proof mass 18, and term jhm(2W.times.U) represents coriolis 
force acting upon proof mass 18. In these terms denotes vector dot product 
and x denotes vector cross product. Because, in the preferred embodiment 
of this invention, coriolis force is many orders of magnitude greater than 
centrifugal force, centrifugal force can generally be safely neglecting in 
modeling this process. FIG. 8 illustrates force summation of Equation (20) 
with centrifugal force neglected. 
Equation (21) states that piezoelectrically coupled electrical current 
related vector I, is equal to summed measured current related vector 
L.sub.M, current related vector of piezoelectric electrical leakage 
resistances -EG, and current related vector of piezoelectric electrical 
capacities -jhEC, as shown in FIG. 8. 
Direction cosine a matrix precisely defines XYZ coordinate system 
orientation shown in FIGS. 2 and 3. Three elements, a21, a31, and a32, of 
matrix a are defined to be zero, specifically requiring: 
(1) Said X-axis orientation defined as direction along which only voltage E 
1 results in a piezoelectric force. This is also direction perpendicular 
to the plane containing E2 and E3 induced piezoelectric force vectors. 
(2) Said Y-axis orientation defined as direction in which only voltages E 1 
and E2 result in a force along said Y-axis. This is also direction in said 
plane containing E2 and E3 induced piezoelectric force vectors, which 
direction is perpendicular to E3 induced piezoelectric force vector. 
(3) Said Z-axis orientation defined as perpendicular to both said X-axis 
and said Y-axis, forming mutually orthogonal axes. 
Because there are only two independent direction cosine elements per row, 
and there are three rows in direction cosine matrix a there are, in 
general, a total of six independent direction cosine elements. Since three 
said elements are set to zero by conditions (1) through (3), above, there 
remain only a total of three unknown direction cosine elements. 
While in the preferred embodiment of this invention piezoelectric coupling 
is used, in general, electromechanical coupling coefficients for any 
selected form of electromechanical coupling have directly related matrix 
structures. 
An example of expansion of one matrix term of Equation (20), KU, from 
matrix to complex form is shown in FIG. 9. K matrix direct axis and cross 
axis spring rate parameters are shown in FIG. 9. Also shown are X, Y, and 
Z axis force components on proof mass 18 resulting from all spring rate 
parameters and displacements X, Y, and Z of proof mass 18. Since said 
proof mass 18 displacements in general have both cosine and sine time 
function components, and because Equations (20) through (28) are defined 
to represent steady-state frequency domain sinusoidal quantities, FIG. 9 
expressions for force components contain both real and imaginary terms, 
representing said cosine and sine terms, respectively. Expansions of 
remaining terms are carried out using identical methodology. 
For each of nine separate driving frequencies generated by multi-frequency 
sinusoidal voltage source 24 shown in FIG. 7, six equations defining input 
voltages E1, E2, and E3 relationship to corresponding six input currents 
Im1, Im2, and Im3, as shown in FIG. 7, are derived as follows: 
(1) Matrix equation (20), above, is expanded into three complex equations, 
as exemplified by the expansion of KU shown in FIG. 9. Each of these 
complex equations is then written as two scalar equations, representing 
real and imaginary parts of said complex equations, resulting in a total 
of six scalar equations for each separate frequency. 
(2) Matrix equation (21), above, is likewise expanded into three complex 
equations, which are then solved for X, Y, and Z. These complex 
expressions for X, Y, and Z are then separated into real and imaginary 
parts of X, Y, and Z. Said real and imaginary parts are expressed as 
functions of angular rate inputs, electrical and mechanical parameters, 
piezoelectric coupling coefficients, and measured voltages as defined by 
Equations (22) through (28). For each separate frequency, cosine 
magnitudes of said measured voltages are directly equivalent to real parts 
of said measured voltages. Negative sine magnitudes of said measured 
voltages are directly equivalent to imaginary parts of said measured 
voltages. 
(3) Said expressions for real and imaginary parts of X, Y, and Z are then 
substituted into said six equations defined in paragraph (1), above. This 
results in six scalar equations that include only variables in said 
angular rate inputs, electrical and mechanical parameters, direction 
cosines, piezoelectric coupling coefficients, and measured voltages. 
Since there are nine separate frequencies and six equations per frequency, 
as described in paragraph (3), above, there results a total of nine times 
six or 54 scalar equations. Said54 equations are used to solve for 3 
angular rates as defined by Equation (28), 6 electrical parameters as 
defined by Equations (23) and (24), 6 mechanical compliance parameters as 
represented by Equation (26), mechanical damping parameters as defined by 
Equation (27), 3 direction cosines as defined by Equation (19) and 3 
piezoelectric coupling coefficients as defined by Equations (1) through 
(6), a total of 27 outputs. 
Operation--Voltage Measurement Fitting and Angular Rate and Parameter 
Computation 
As shown in FIG. 10, angular rate microprocessor 14 accepts all six outputs 
of six-channel analog-to-digital voltage conversion subsystem 12. Angular 
rate microprocessor 14 also provides temporary storage for these outputs 
over said overall computation output rate period as shown in FIG. 11. 
While this output rate period can be set to different constant values, in 
the preferred embodiment of this invention said period is set to a 
constant value of approximately 0.01 seconds. Angular rate microprocessor 
14, as shown in FIG. 11 flowchart, uses each constant computation time 
block of analog-to-digital voltage conversion samples as input data for a 
linear least squares fit to a known set of nine driving frequencies. This 
least squares fit is performed separately and independently for each six 
analog-to-digital voltage conversion system output. This computation 
process results in a total of six channels times nine frequencies times 
two outputs per frequency (sine and cosine components), or 108 computed 
sine and cosine magnitudes. 
These magnitudes are then used as inputs for computing needed said 27 
output estimates using said 54 scalar equations generated. Although a 
number of algorithm choices exist for providing said 27 output estimates, 
including a Kalman filter, in the preferred embodiment of the invention 
this solution is carried out using a linear least squares fit 
computational process, based upon a linear expansion of said 54 scalar 
equations with respect to an assumed reference point. 
As shown in FIG. 11, not only are X, Y, and Z angular rates computed for 
each output rate period, but 24 physical parameters and coefficients are 
compared with their known nominal values. If, for any reason, one or more 
of these parameters or coefficients falls outside of a specified tolerance 
range, then proper operation of three axis mechanical sensor unit 10 is 
suspect, and a "bad data" discrete is output as a warning. 
Because of significant separation of translational mode natural mechanical 
resonant frequency from said roll-off frequency of anticipated input 
vibration motion frequency spectrum, and because of extremely narrow 
frequency bands used by measurement means described above errors in output 
data caused by externally induced vibration forces acting upon proof mass 
18 are minimized. 
Conclusions, Ramifications, and Scope 
It can be seen that, according to the invention, a digital output three 
axis angular rate sensor is provided, the accuracy of which is 
substantially independent of variations of mechanical sensor critical 
electrical and mechanical parameters and electromechanical coefficients. 
These variations have, until now, limited the performance of vibratory 
angular rate sensors. Satisfactory comparisons of computed outputs of said 
electrical and mechanical parameters and electromechanical coefficients 
with their normally expected values enable a verification of the validity 
of the basic angular rate output data. The invention makes use of 
available electronic technology to electrically excite a single proof mass 
into motion along three axes in order to induce coriolis forces in 
response to input angular rates. This electrical excitation is made up a 
multiplicity of sinusoidal signals with respect to time. Provided are the 
necessary means for voltage measurements and real time computation of the 
three axis input angular rates, using a microprocessor. Also provided are 
means for computation of said parameter and coefficient variations. Data 
extraction accuracies are attained which heretofore have not been 
obtainable with conventional methodologies. 
Although the description above contains many specificities, these should 
not be construed as limiting the scope of the invention, but as merely 
providing illustrations of some of the presently preferred embodiments of 
this invention. Various other embodiments and ramifications are possible 
within its scope. For example, the use of electrostatic or electromagnetic 
electromechanical coupling in three axis mechanical sensor unit 10 in 
place of the piezoelectric coupling described as the preferred embodiment 
makes possible the same basic functions and outputs of three axis 
mechanical sensor unit 10, six-channel analog-to-digital voltage 
conversion subsystem 12, and angular rate microprocessor 14, and 
multi-frequency sinusoidal voltage source subsystem 24. 
Thus the scope of the invention should be determined by the appended claims 
and their legal equivalents, rather than by the examples given.