Note: Descriptions are shown in the official language in which they were submitted.
6~7
1 elated Paten-t Application
Thi.s appl:ica-tion rela-tes tv the app:Lican-ts U.S.
patent ~,445,376 which issued May 1, 1984 directed to an
apparatus and method for measuriny specific force and anyular
5 rate of a moving body, and to a processor therefore partic-
I ularly useful in inertial navigation systems.
Field of the Inventiorl
.
The present invention relates to apparatus for
measuring the inertial specific force and angular rate of a
moving body by means of a plurality of accelerometers mounted
on mutually perpendicular axes. The invention is particularly
useful in the apparatus and method as applied to a high-
precision, nongyroscopic Inertial Measurement Unit tIMU) for
an Inertial Navigation System (INS~, as described in U.S.
paten~ 4,445,376 as well as the article by Shmuel J. Merhav
entitled "A Nongyroscopic Inertial Measurement Unit," published
in the AIAAJ. of Guidance and Control, May-June, 1~82, pp.
.
227-235, and is therefore described below with respect to
such patent.
! 20 The above-cited patent discloses a method and
apparatus for measuring the specific force vectox and angular
rate vector of a moving body by means of a plurality of
cyclically driven accelerometers. The embodiment described
therein uses rotating accelerometers which, broadly, had been
proposed as early as 1965, bu-t which had not yet matured as a
practical technology, as discussed in U.S. patent 4,445,376.
That patent wa5 particularly directed to a novel manner of
processing the acc~lerometer output signals so as to derive
the specif ic force vector F and the angular rate vector
Q COmpQnentS thereof in such a manner that the unwanted
components are suppressed to a sufficiently lo~ level so as
to be compatible with INS precision re~uirements. Briefly,
the,an~ular rate vector components of Q ti.e., Qi' wherein
i - x, y, z) are derived from each of the acceleromet~r output
35, si~nals ('a'), by: tl) multiplying the accelerometer output
signal by the function sgncos~t to produce the prodllct ~i~n~l
a sgncos~t, and (2~ inte~rati.ng the product ~ignal ov~r the
cyclic period~ The specific orce vector components of F
7(~6'7
1 ~i.e., F1 wherein i = x, ~, z) are derived by integratîng
the respective accelerometer output signals ('a') over the
cyclic period. U.S. pa-tent 4,~45,376 includes a discussion
and a mathematical analysis of the dynamics involved, and
shows tha-t a number of important advantages are obtained
which make the described method and appara-tus particularly
suitable for non~yroscopic Inertial Navigation Systems (INS).
Reference may be had to U.S. patent 4,445,376 and the article
by S.J. Merhav, cited above, for Eurther details of the
described technique Eor sig~al separation and of the ad-
vanta~es ob-tainable thereby.
An object of the present invention is to provide new
apparatus for measuring the specific force and angular rate
of a moving body enabling a number of-further important ad-
vantages to be attained as will be described more particu-
larly below. The apparatus of the present application is
particularly useful with the technique of signal separation
described in the above-cited patent and is therefore described
; below with respect to that technique, but it will be appre-
ciated that the invention of the present application, or
features thereof, could also be advantageously used in other
applications.
Brief S~unmary of the Invention
Briefly, the present invention provides apparatus for
measuring the specific force and angular rate o a moving body
by means of a plurality of accelerometers mounted on mutually
perpendicular axes and cyclincally driven by drive means in
mutually perpendicular planes, characteri~ed in that each
accelerometer is mounted for vibratory movement and is driven
by the drive means along an axis of vibration in its respective
plane rather than being rotated in its plane, as in the embodi-
ment of the invention described in U.S. patent 4,445,376~
Several arrangements constituting further features
of the invention are described below fo~ effecting the
vibratory movement of the accelerometers.
Thus, in one described embodiment, the accelerometer
is resiliently mounted by resilient means, such as a diaphragm,
constrainin~ the accelerometer to move only along the axis of
vihration, the drive means being connectable to a source of
sinusoidal current for vibrating the accelerometer
'~''
~2~ 6'~
along the axis of vibrution. E~tch reference QXiS of the movirg body m~y be
provided with two such vibrating assemblies mounted in COaxiQI back-t~back
relationship, the drive means of ~ne vibrating its ~ssemb]y in synchronism with,but in opposite direction to, the drive means of the other, whereby one tssembly
5 serves ns a counterbalancing mass for dynamica11~ balancing the other ~ssembly.
App~LrF,tus constructed in accordance with the foregoing features
enables a nurnber of important advantages to be ~tti~ined, particularly when
6pp'ied to nongyroscopic Iner~ial hlavigation Systems (lNSj. Thus, it enables the
accelerometers to be cyclically driven without rotsting or sliding mechQnical
10 joints, thereby obviating the need for slip rings or other sllding electrical~ con~acts. In addition, the described arrangement provides an acc eler~meter
2~ 5~ assembly which is inherently rigid along the sensitive ~xi~ ~hich permits the
vibratory motion to be imparted to the accelerometers at amplitudes,
frequencies and phase angles that may be Yery precisely controlled, anc1 which
15 ma~e the ~ccelerometers substantially insensitive to extern~l forces, shock and
vibrstion. A still further advantage, particularly in the bac~ ,back arrang~
ment, is that it generates the required vibratory motion in a mPnner sueh th~t
the dyn~tmical forces are precisely balanced. Th-o foregoing advantsges provide A
much higher mefln-time-between-f~ilure (MTBF) than the gyr~type lMUts~ or the
20 nongyro-type 1;~1U's having rotating accelerometers.
A second embodiment of the invention is described wherein each
~5~ accelerometer is mounted on a support;ng member ~n~}e ~bout ~t l'OtQ-
tional axis perpendicular to its axis of vibration, with the drive means oscill~tting
i~s supporting :nemb~r tllrough a sm~ll sngular motion about its rot~tional axis.
25 The supporting member also includes a coun~erbalancing mass on ~he opposite
side o~ its rotational ~xis for counterb~ancing the Qccelerome~er rn~ss. In thisdescribed embodiment, the drive means ~omprises an electric~' torque motor
driYeR by the sinusoidsl current to execute a sm~ ngular oscillatory motion ~
~ew degrees) which is almost linear. This embodiment has tll~ ~urther advan-
ages o- s-lbstantially complete immunity io external linesr vibration ~nd shockJsimplicit~ of construction, snd high precision at low cost.
A third embodiment of the invention is des~ribed below Irlcluding a
tunin~ fork, which embodiment ~lso permits precise belancing o~ the dynamic~l
forces. Thus, the flccelerometer includes a m~s~ mounted for vibratory
35 movement on r~ first prong of the tuning fork, the second prong o~ the tunin~ fork
including Q counterbr~ nring ma~s c~using the two prongs to vibr~te ~t ~
pre~etermined natur~l fre(Juerlcy. ln this described embodim~nt. the ele~tricllldriving member i~ c~rried orl one prong of the tuning fork, ~nd the eleclrieal
pick-off member may be cArried on the other prong. Such An ~rrangement cRn
include nn elec~ric~l feedback loop from the pick-off member to the drive
member to form therewith an electromechanic~l oscill~tor whose oscillations are
sust~ined by the feedback loop.
The tuning-fork embodiment provides, in addition to R bfllnnced
dynamical sy,sten) bec~use of eountermoving m~sses, a number of ~ddition~l
advQntages. Thus, since the power re~uired to drive the driving member is only
thAt needed to replenish the energ~ loss due to friction7 the arrangernent
requires but ~ small amount of po~ er. Fur~her, since the device acts as ~
lû sharply tuned oscillator, it will reject mechanical di~turuances along the
sensitive sxis unless they are exactly at the resonant frequency. Still further,since the ~rrangement operates at iîs natursl frequency, it CQn be used to
synchronize the multivibr~tor which controls the signal processor, thPreby
avoiding phase lags which might affect the ~ccuracy of the angul~r rste e~nd
15 specific force vectors derived from the sccelerometer output sigrlals.
Frorn the Qbove, it will be sppreci~ted that the "vibratory move-
ment" applied to the ~ccelerometers may not only be ~ ~ure rectilinear
movement, sueh as in the first of the abov~mentioned embodiments, ~ut mP~y
Qlso be a substantial]y rectilinear movement (e.g., ~m~ll Angular oscill~tory
20 motions which are almost rectilineflr) such BS in the second and third of theabove-mentioned embodiments of the invention. This will be more apparent
from the detailed description below of e~ch of these three embodiments.
A further improvernent to the rate and force sensor utilizing
vibrating sccelerometels m~y be obtQined by utilizing p~ired accelerometers for
25 each axis for which angular r~te information is desired. There are three
arrQngements ot paired accelerometers described herein which csn provide a
signifieant increase in the a~curac~ of both the r~te ~nd ~orc~ sign~l obtained.The first ~uch arrangernent cal~s for two a~celerometers mounted together with
thPir inrut- or force-sensing ~xes par~llel to the axis al:o~ which they are
30 vibrated. A second arrAngemen~ hss both acce~erometers mounted b~ck to b~ck
~ith their force-sensing axes opposite one another and norrnall to ~n axis aboutwhich they ~re vibrated. The third arrangement calls for the accelerometers to
be mounted bacl; to bacli uith their forc~sensing axes opposite to one another
snd having the acceler~ineters vibrated in a linear dire~tion normal to the force-
35 sensi ng axes.
The paired rlrrangement des~ribed above makes it possible to
further sepnrate the foree si~nAls from the rate signQls by surnming snd
-- 5 --
.
1 differencin~ the output signals of the paired accelero-~
meters prior -to having these signals input to a signal
separation circuit.
Further features and advantages of the :invention
will be apparent from the description below.
Brief Description of the Drawings
The invention is herein described, by way of example
only, with reference to the accompanying drawings, wherein:
Fig.l is a diagram which will be helpful in ex~
plaining the principle of signal separation described in U.S.
patent 4,445,376 and used in the preferred embodiment of the
present invention as described herein;
Fig. 2 is a block diagram illustrating one form o
nongyroscopic Inertial Measuring System based on the principle
of signal separation described in U.S. patent 4,445,376 and
also included in the preferred embodiment of the present
invention;
Fig. 3 is a diagram similar to that of Fig. 1 but
modified so as to include vibrating accelerometers in
accordance with the present invention, rather then rotating
accelerometers as in Fig.l;
E'ig. 4 illustrates one form of vibrating accelero-
meter assQmbly constructed in accordance with the present
invention;
Fig~ 5 is a diagram of a closed-loop acceleromet~r
assembly drive constructed in accordance with the present
invention;
Fig. 6 schematically illustrates the use o two
vibrating accelerometer assemblies, each in accordance with
the construction illustrated in Fig~ 4, for example, mounted
in back-to-back relationship for balancing the dynamical
forces;
Fig. 7 illustrates a second form of vibrating
accelerometer assembly constructed in accordance with the
present invention, based on the use of an electrical torque
motor rotàtionally driving the accelerometer and a counter-
balancing mass through a small an~ular oscillatory motion;
~7(~6~
1 Fig. 8 illustrates a t~,ird form oE vibratiny
accelerometer assembly constructed in accordance with -the
present invention, based on the use of a tuniny fork for
precisely balancing the dynamical forces;
Fig. 9 is a block diayram illustrating an electro-
mechanical oscillator arrangement including a tuning-fork
accelerometer assembly such as illustrated in Fig. 8;
Fig. 10 is a simplified, perspective diagram of
paired accelerometers having their force-sensing axes
parallel to an axis of angular vibration;
Fig. 11 is a simplified, perspective diagram o
paired accelerometers arranged back to back with their force-
sensiny axes normal to an axis of angular vibration;
Fig. 12 is a simplified; perspective diagram of
paired accelerometers arranged back to back with their force-
sensing axes normal to the direction of linear vibration;
Fig. 13 is a diagram similar to that of Fig. 3
used to illustrate the operation of the paired accelero-
meter arrangement of Fig. lQ in a three-axis-rate senSQr;
FigO 14 is a diagram of a type similar -to that of
Fig 3 illustrating the operation of the paired accelero-
meter of Fig. 11 in a three-axis-rate sensor;
Fig. 15 is a diagram of the type similar to that
of Fig. 3 illustrating the operation oE the paired accelero-
meter arrangement of ~ig. 12 in a ~hree-axis-rate and force
sensor;
Fig. 16 is a side view of a mechanism ~or implement-
ing the paired accelerometer arrangement of Fig. 10;
Fig. 17 is a side view of a mechanism for implement-
ing the paired accelerometer arran~ement of E`ig. 11;
~"iCJ. 18 is a side view of a mechanism for implement
ing the paired accelerometer arran~ement oE Fi~. 12 t and~
Fig. 19 is a block diagr~m of a signal separatin~
circuit of the type shown in Fig. 2 including the addition of
a preprocessor circuit for summing and differencing the out-
put si~nals of the accelerometers.
.~
,~
~'7~6'7
1 Descrip-ti.on of PreEerred Embodiments
Overall Sys-tem and Principle of Si~nal S parat:ion
(Figs. 1 and 2)
.
Before describing -the various ernbodiments of the
inven-tion as lllustra-tecl in Figs. 3-8, it is be:Lieved that
a preliminary discussion of the principles of signcll
separation and of the overall system for measurlng specific
force and angular rate of a moving body, both as described
- in V.S. pa-tent 4,445,376 and article, will be helpful in
better understanding the present invention and its advan-
tages, particular:Ly when the present invention is embodi~d
in such a measuring system.
With reference to Fig. 1, the general equation for
total acceleration measurable at a point mass moving
in a rotatinq system is
a = F + n x r + 2~ x dr¦ +Q xtnxr)-t d
_ . .~
dt b dt~ b
- where Q is the angular rate or velocity vector of -the system,
F is the specific force vector and _ is the instantaneous
distance of the point mass from the center oE ro-tation o
-the system. In particular, r = p + Q, wherein p is the
instantaneous vec-tor distance of the poi.nt mass from its
center of revolution, and Q the fixed distance of the
element's center of revolution to the system center of
rotation. The index b indicates differentiation with r~spect
to the rotating body axes. Equation (1~ can now be rewritten
as follows: 2
a = F + Qx(p+ Q)+ 2Q x dP¦ +~ x (nx(p+~])-~ d P¦ (2
- dt b dt2lb
~1~>~7( ~
substitutincJ
P x ¦ pXcos~t pxsin~l)ti I i
Py = pysin~t O pyCos~t ~ ¦
Pz pzcos~t pzsin~t O J kJ
where i, ~ and k are the unit vectors in the -~x, ~y
and +~ directions, respectively, incorporating the noise
components nx, ny, nz, respectively, resolvincJ a into
ax, ay, and a~, let-ting Px = P~ = Pz = P, and rearranging
termsJ we have
x x ~Qz(q+pr)+2~ p cos~t ~ ~ 2~
~ 3)
-~2~p sinwt ~ 2q ) +P P(q cos~ + r sin~t)+nx
ay -Fy +QX(r-tqp)+2~ P cos~t ~ ~ 2- )
: ~ \ (4)
~2~p sin~t p+ ~ ) +q p(r cos~t ~ p sin~t~ny
az =Fz -~Qy(p~rq~=2~ p cos~t ~ ~ 2 )
~2~p sin~t ~+ 2P ) +r ~)(p cos~t + q sin~t)+n~
Each of the noise si~nals n = ~nx~ n~, n~, is assumed
to consist of thxee components as follows~
n=n -~n +n
- -d -v -r
where:
nd-Low frequency (drift) noise
nv-Periodic or random vehicular vi.bration noise
nr-Random zero-mean high-frequency sensor noise
Thus, in the rotatin~ accelerometer system illustrated
in U.S. patent 4,445,376 in addition to the underlined
~5 desirable te~ls in Equations (3) - (5), there are a variety
of additional undesi:rable terms potentially contributing
i 3 ~
. 9 _
1 to errors. These primari.ly result from the dynamical
terms containiny p, q, r and -their deriva-tives and from the
sensor noise componen-ts contained in n.
As further described in U.S. patent ~,~45,376, an
i.mportant feature of that invention is that it provides
means for separating Fx from q, Fy from r, and Fz from p,
in such a manner that the undesirable terms are suppressed
to a sufficiently low level so that the effect of the cross-
product terms qp, qr and rp is substantially eliminated.
An important advantage in the use of vibrating
accelerometers in accordance w.ith the present invention
is that the orthogonal terms, e.g., 2r~psinwt and
p(pq-r)cos~t in Equation (3) above (and the corresponding
terms in Equations ~4) and (5) above) actually drop out,
thereby even further reducing this source of error in the
rotating accelerometer arrangement. That is to say, since
thes~ orthogonal terms result from the ro-tational movement
of the accelerome-ters, they are not present in the invention
of this patent application involving a vibrational movement
of the accelerometers.
Fig. 2 of the present application (which corresponds
to Fig. 4 of U.S. patent 4,445,376) illustrates in block
diagram form one nongyroscopic Inertial Measurin~ Unit (IMU)
implementing the above-described principles of signal
separation with respect to one channel, namcly that of
accelerometer A~, it being appreciated tha-t the cther two
channels, for accelerators A ~ and ~, are similarly
construct~d.
The unit illustrated in Fig. 2 includes three major
subsystems, namely: a control pulse generator, generally
designated ~; an electronechanical drive, generally de-
signated 3, for rotating each of the accelerometers of the
triad illustrated in Fig. l; and an electronic siqnal~
separa-tion processor, generally designated 4.
The control pnlse genera-tor 2 is driv~n ~y a free-
running, multi vibrator 21 having a high precision reference
~.~
- 10 -
1 frequency 4f (~ = l/T). The multivibrator controls a
square wave generator 22 which generates squarewaves at
a frequency f. These square waves are used as synchronizincJ
pulses. They are applied to a reset-and integrade control
pulse generator 23 and to a sampling pulse generator 24,
which yenerators are used to control the operation of
processor 4, as will be described more particularly below.
Synchronizing pulses from the square-wave genera-tor
22 are also applied to a drive signal generator 31 within the
electronechanical drive system 3. The output pulses from
generator 31 drive the accelerometers of the assembly 32,
such that the accelerometers are rotated about their respec-
tive axes at a predetermined frequency (~) equal to 2 ~f-
Thus, when the body to which the accelerometer assembly 4 is
strapped down is subject to a specific force Fz and angular
roll rate p, it produces a resultant output of az.
The accelerometer output az is fed to the processor
4 for separating therefrom the specific force vector Fz and
; angular rate vector p in such a manner so as to substantially
suppress the undesirable components of signal az in accor-
dance with the Equations ~3) - (5) discussed above. In this
case, we are considering the Fx and p components, so that
equation ~5) is the pertinent one.
Thus, processor ~ includes a multiplier or sign-
switching circuit 41 for multiplying the introduced values
az by the zero mean periodic f~nction ~Isgncos~t~ outputting
the product signal az sgncos~t. This latter signal is fed
to an integrating circuit 42 which integrates the product
signal over the cyclic period T. The integrating circuit
42 is reset at the end of period T by the control pulse
generator 23, but before being reset, it outputs its contents
to a sample-and-hold circuit 43, which latter circuit is
controlled by the sampling pulse generator 24. As described
above, this processing of the accelerometer output si~nal a7
causes the contents of the sample-and-hold circuit 43 to
correspond -to the angular rate component "p".
~. .
- ~-o~ -
1 The accelerometer outpu-t sicJnal aæ is also fed
to a second channeL wi-thin processo~ 4 including a second
inte~ratin~ circuit 44 which inte~rates tllat signal over
the period T. In-tegrating circuit 44 is also reset at the
end of period T by the control pulse generato.r 23, but just
beEore being reset, it outputs its contents t~ another
sample-and-hold circuit 45 cont~ollecl by the sampling
pulse generator 24. It will be appreciated from the previous
discussion, that the contents of the sample-and-hold circuit
45 will correspond to the specific Eorce vector Fz.
Reference may be had to ~.S, patent 4,445,376 for a
further description of the overall system and the advantages
provided by the principle of si~nal separation on which that
system is based, it being appreciated that the same ad-
vantages would apply to the present invention when imple-
mented in such a measuring system in addition to the ~urther
advantages attainable by the invention of the present appli-
cation, as described more particularl~ below.
Principle of Using Vibrating Accelerometers (Fig. 3)
.. _ _ . .... ... . ...
Briefly, the invention of the present application
utilizes vibrating accelero~eters for generating the
accelerometer output signals from which are derived the
components of the specific force vector F and the components
of the anJular rate vector Q while subs-tantially s~ppressing
the undesira~le components of such signals. This is illus-
trated in Fiy. 3, which is similar to ~le diayram of Fi~. 1
but includes vibrating accelerometers rather than rotatin~
accelerometers. Fig. 3 thus illustrates a triad of
accelerometers A , Ay, Az arran~ed to vibrate at an amplitude
30 "p" and re~uency "~" perpendicular to the ~x,y), (y,æ) and
~x,z) planes, respectively, with the sensitive .input axes
aligned as shown in the x, y, z directions
Specific vibratin~ accelerometer mechanizations are
illustrated in Figs. ~I through 9, to be described below.
These figures illustrate only one channel, nc~ely th~t o~
accelerometer A~, wherein the accelerometer-sensitîve axis
- lOh -
1 Eor the specific :Eorce vec-tor is the Z-axis, and the axis
of vibration is the Y-axis, the senseitive axis ~or the
angular rate vec-tor belng the X-axis. Thus, accelerometer
A~, vi.brating along the Y--axi.s~ measures the inert:ial
specific force and angular rate oE the moving body with
respect to reference axes z and x. It will be appreciated
that the other two channels, i.e., for accelerometers
Ax and Ay, are similarly constructed and provide corres-
ponding measurements for their respective axes. Preferably,
the freq~ency of vibration (.~) of the accelerometers in
all the below-described embodiments is 30-60 H2, and the
displacement during their vibratory motions is typically in
the range of 0.25-3 n~l.
As indicated earlier, one of the main advantages in
the use of vibrating accelerometers, over rotational
accelerometers, is the orthogonal terms in Equatiorls
(3) - (5) (e.g., 2r~psin~t and p(p~ - r) cos~t in Equation
3) do not even exist, thereby inherently permitting greatly
improved overall performance. Many other advantages are
described more particularly below.
Vibrating-Accelerometer Arrangement of Figs. 4 through 6
The accelerometer assembl~ illustrat~d in Fig. ~ t
therein generally designated 50, comprises an outer cylin-
drical housing 52 enclosing an accelerometer unit 54
containing an a~celerometer proof mass 56. The accelerometer
. .,:
. ~"
unit 54 is ~arried by ~ mount;ng plate 5~ resilientl- mourlted within housing 52 by
means of s resilient diflphr~gm 60, which dinphrflgm constrains the movement Or
the accelerometer unit 54 only to the Y-axis, tllis being the axis of vibration,and perpendicular to the Z-axis, which is the specific-folce vector-sensitive axis
5 for the accelerometer ~ssembly, as noted above.
The driving m~ans for vibrating the accelerometer unit 54 alon~
the Y-axis comprises a permanent magnet 62 of cylindrical conStrUctiGrl fixed
~,~ith3n housing 52 flt one end thereo} arld formed witll a cylindrical air g~p 64
coaxial with th~ Y-nxis oi vibration of the accelerometer assernbly. The drive
10 means further includes a driving coil 66 carried on 6 ~ -lindrical bobbin 6e fixed
to the accelerometer mounting plate ~8 within the cylindrical air gep 64 ~nd
COQX;~I with the Y-axis of vibration. Driving coi] ~fi is sdapted to receive
sinusoidal driving current producing a force which ~uses the accelerometer
unit 54, inc]uding its proof mass 56 and mounting plate 58, to move sinusoidally15 ~long the Y-axis o~ vibration as constrained by the spring force Or diaphrsgm~.
The accelerometer assembly illustrated in ~ig. 4 further includes
~ich-off m~ans disposed within housing 52 and coupled to the accelerometer
unit 54 and its mass 56 for measuring the rate of displseement thereof along theY-AX;S O~ vibration. Such pick-off means may comprise ~nother permanent
20 magnet 70 (or an ironless field coil) and ~ pick-off coil 72 cooperable therewith
st the other end of housing 52. Permanent magnet '~0 is also of cylindrieQI
constructionS but of much smaller d;mensions th~n the drive magnet Ga, and is
also formed with a cylindrical air gap 74, ancl the pick-off coil 72 is also carried
on a cylindrical bobbin 7fi secured to the accelerometer unit 54 so that the pich~-
25 vff coil 7~ i~ disposed ithin air gap 74 and is coaxial to the Y-axis of vibration
of the sccelerometer assembly.
The &ccelerometer assembly illustrated in Fi~;. 4 further includes ~
first group of external terminals 77 ~onnected by eleetrical le~ds ~not shown~ to
the driving coi] 66 for feeding in the driving curre~nt s~nd the pick-off coil 76
30 measuring the motion of the accelerometer 54~ ~nd ~ second gro~p o~ external
terminals 78 connected by electrical leads ~not shoY.n~ to lntern~l terrninals ?9
carried by the accelerometer ~nit 54 for feeding the supply voltages snd ouSput
sign~ls to snd Srom the ~ccelerometer unit 54.
5~ uill thus be seen that the sinusoidhl driv,ng curren~ Imsin~` t)
35 ~ed into the dri~in~ coil 66 exerts a force proportion~l to it and cnuses the~ccelerometer unit 5~, including its mass ~ Qnù mounting plate 58, to Yibrste
sinu~oidall~ ~)cn~ the Y-~is ~s constrained by the resilient diaphra~m 60 in
m~nner silnilar l~ e excitation o~ a loudst eaker. Tl~t- p;ck-off coil 7~, which
'7
, " .
moves with the accelerometer unit 54 inducec ~ voltnge proportional to the
velocity Or the sinusoidal motion of the ~ccelerome~er unit ~ the me~sured
velocity of motion QlOng the Y axis of vibr~tion (namely y) being outputted ViA
terlninAls 77. Thus if the motion due to I = Im sin ~t is y = ~ msin (e)t ~ ~ ) = P sin
((I)t I ~ ) wherein ~ is a phase shift due to the dynamic lag of the movin~
ass~mbly the corresponding velocity is y - pw cos (~t ~ ~ ). The sign~l
out?utted b~ the pick-off coil 72 eonstitlltes this messured Yelocity (y) of themo~d;lg Hssembly includ ng t~e ac~elerometer unit 54.
F:ig. 5 illustr~tes how this measured velocity (Y)J outputted by the
pick-off coil 72 is also used to enforce a c]osed-loop controlled motion of the
accelerometer unit 5~ by controlling the supply ~f ~he driving current to the
driving coil 66.
rhus the control signal gener~tor 80 synchronized by input
pulses 81 gener~tes the signal V = Vm cos~t ~hich is fed vi~ lead û2 to a
differenti~l power amplifier 84. The output of ~mplifier 84 is connected by
lead 86 to the drive coil 66 of the ~cceleromete assembly illustr~ted in Fig. 4
thereby driving the ssserr)bly by means Or Q cllr~ent I = Im sin(~t ~ p ~. The
l~tter eurrent produces the force ~ = Fm sin~wt ~ ~ ~ ~hich causes the motion
~indicated by output arrow 88) y = Pwcos ~wt + ~ ~. The latter motion is meaiured
by the pick-off ~oil 72 in Fig. 4 and is fed, via lesd ~0~ into ~nother input
terminal of the differentiQl power amplifier 84. The small difference between
the signal on le~d 90 and the sign~l V = Vmcos~ t is nmplified to cre~te the drive
sign~l I = Imsin~;lJ t + ~ )
The errangement of Fig. 5 is a negative feedbsck loop having 8
2~ tot~l ioo~ ~in L deterniincd by the gQin of ampliiier 8~ and the p;ck-off 72. To
those iamiliQr ~ ith the Hrt it is clear that by providing L ~> l it is assured that ;~
will closely follc w ~ = ~ m cos~t ~nd that a poss.~ e force disturbence Fd~ shown
schematic~lly v;a the dotted line 92 will excite a disturbance in the velocit~y y
c ~Y ;d ~ FdlL. Since L 1. Yd can be suppl ssed to a negli~ibly low leYel,
thus foreing the aecelerometer unit 54 ~nd rnass 56 in Fig. 4 to perform for 81lpracti~Ql purpo~cs as a rigid body with respect to dissurbing forces ~long the Y-
axis.
Is i.~ thus seen that the closed-loop drive s~s~em of Fig. 5 providss
Q well re~ulateà and con~rolled sinusoid~l line~r ~elocity ~long the Y-axis. The3j a:nF)lifie ~ i; so designed in terms of the ~requency respollse that the closed-
IQOP ~ra~s~er function of Fig. ~;9 denoted by ~ y ~)/V~ is ~lAt UP ~SO ~
~ ~27(~
-13-
bandwidth "b", such thEIt b"" arld In~ phase shift ~ w) at the driving
:~frequency Ll), is pr~cticQII!,~ zero. The high-g~in, feedbnt~k drive system genc rates
the reguired motion with A negligible nonlineflr ~nd pllase distortion.
Two RSpectS Or major importance will now be demonstrated:
(1) large bandwidth and small pl)ase distortion of the closed-
loop system; and
(2) immunity to nonlinear distortion due to the velocity pich-off
coil 72.
(1~ l/\ ilh r espect to aspect (I ) above, the open -loop transfer
10 function of the electrod,vllarnic drive iâ gi~en by:
F = B ~ J ~ B ~ u!Rc
wherein:
B = magnetic induction;
Q = length o~ drivine coil;
R~- resistance of driving coil;
J ~ u/Rc; Qnd
u = input volt~ge at coil terminals.
2û
Now, if:
m = mass of moving assembly;
b = damping coefficient; and
c = spring constant of diaphragm
then:
~-m~by~cy~ m(s2~ms+m)y;
30 B~ U-~s2~2wn~5~ ~)n )Y
K ~-I B ~ b
-
y =
s ~ 21l~ r~ ~5 ~ (') n
~h~
Denoting the input driving volta~e by Vi (output of the drive sign~l
generator 80 in Fig~ 5), we can express the closed-loop transfer function as:
AKs
y 52~n~S+~n2 = A~s
U 1 ~ S2 f (2W n ~ + A~H) s + n
S ~2Wn~S~ ~I)n
where H is the scale actor of the velocity pick-of ~2 in Fig. 5.
It is easily verified that K can be quite large. By its definition, and
using electromagnetic units
K lOmRc WRC Wp
15 where W = weight of moYing mass;
p = resistivity of copper coil; and
S = coil wire cross section;
thus, for B = 104 Gsuss; W - 100 gr; S = 10 4cm2; p (for copper~ = 1.6 x 10 ~:
K = -6 = 116 = 6000
Assuming, e.g., A ~ 100, clearly AKH is in the order of 1Q5 ~o 106.
Thus, AKH 2~n~ ~nd the relation y ~ 1 Yi holds over a Yery ~Yide
bandwidth so that phase distortion ¢, can be made negligibly small for the
25 excitation frequency of 30-60 Hz.
~ 2) With respect to ~spect ~2) above, since y - ~ Yi we can
write Y = h~li where h - 1 . Let a possible nonlinearîty be expressed ~s follows:
Y hoYi hlVi
For Vi = Vimcos ~1) t we have:
y = hoV~mcosl.~ t + hlYim2cos2 ~ t -
hovimcos~t ~im + ~lm cos2~t
1 The operation of sgncos ~tof the signal
separation processor, and the in~egration over the cyclic
period T, clearly cau~es the contributions due to
h V 2 h V 2cos2~t
_ 1 lm and 1 lm - to drop. Thus, -the velocity
pick-up sensor 72 is not critical in requirements of
linearity.
It will be appreciated that when -the invention
is applied to a strapdown-type of inertial guidance
system, an accelerometer assembly 50, as illustrated in
FigO 4, including a closed-loop drive as illustrated in
Fig. 5, would be applied for each o~ the three axes, with
the outer housing 52 of the accelerometer assembly mounted
to the vehicle ti.e., the moving body). In a stable
gimballed platform-type of inertial guidance system, the
outer housing 52 of the accelerometer assembly 50 f~or each
axis would be applied to the inner gimbal of the p:atform~
In either application, the vibratory motion of
the accelerometer assembly may cause reaction forces
acting on the support of the outer housing 52. In order
to a~oid these undesired unbalancing forces, two v.ihrating
assemblies, such as shown at 50a and 50b in Fig~ 6, may be
mounted back to back, with one assembly including an
accelerometer, as described abov~, vibrating in
synchronism with, but in opposite direction to, the other
assembly vibrating a counterbalancing mass ~ynamical~y
balancing the acc21erometer assembly.
Oscillating Accelerometer Arrangement o~ Fig. 7
Fig. 7 illustrates a second type of vibratin~
acceleromèter arrangement, namely, one in which the
accelerometer is rotated by an electrical torque motor
through a s~all angular oscillatory motion ~e.g., a few
degrees~ which makes the vibratory motion almost
rectilinear. The sensitive axis of the accelerometer is
parallel to this axis of rotation. A suitable balancing
mass is provided to dynamically balance the accelerometer
- 15 -
6'~
during its oscilla-tory motion so -tha-t no external ~orces
are imparted to the body to which the assembly is mounted.
Thus, the accelerometer assembly il:l.ustrated in
Fig. 7, therein generally designated 100, comprises and
outer cylindrica]. housing 102 rotatably mounting a shaft
104 via rotary bearings 106 and 108. To shaft 104 is
secured a disc or plate 110, which serves as a supporting
member for supporting an accelerometer unit 112 having a
proof mass 114. Disc 110 also carries a counterbalancing
mass 116 on the opposite side of the disc.
Disc 110 is driven through a small angular
oscillatory motion by means of an electrical torque motor
including a stator 118 fixed to housing 102 and a rotor
120 fixed to shaft 104. A pick~off rotor 122 is secured
to the
- 15a -
~r!~
-16
opposite end of shaft 104 and is dispo~ed Y.lthin a pick-off stQtor 124 seeured to
housing 102.
The electrical corInections may be the same as illustrated in Fig. S,
wherein the differentisl power amplifier ~4 feeds the driving current ~o the
S conductors of the torque motor ststor 118 to drive its rotor 120 and, thereby, the
aceelerometer 112 and courlterbslancing mass 116 secured to the motor
rotor 120, throu"h a small angular oscillatory motion having an amplitude "'~"'.This will CAUSe the displscement y ~ r~, where "r" is the radius from the axis of
rotation 130 of shaft 104 to the center of gravity of the accelerometer proof
10 mass 114. Thus, if ~ m~in~ t, then; ~ mCOS,~, t, pointing into the paper
plane. It u ill be appreciated that the oscilIatory movement of the accel-
erator 112 into and out of the paper planr is substantially.linear along an axisperpendicular to the sensi~ive Z-axis of the accelerometer.
As one example, "r" may be ~bout 3 cm; the angular oscillatory
15 motion may be a few degrees; and the amplitude of displacement of the
accelerometer may be 0.25-3 mm.
The pick-Dff 122 fixed to shaft 104 senses the angular velocity ~ =
w~4 mcosll) t. As in the case of the piek-off 72 in Fig. 5, the output of the pick-
off 122 in Fig. 7 m~y be connected a~s a feedback into differential amplifier 84,
20 to which the driving voitage Vj - Ymcos~t is fed vi~ lead 82. Thus, the angul~r
velocity ~, in the Fig. 7 ~rrangement is made to follow olosely the driving
vo]tage Vj = Vmcosw t.
This arrangement illustrated in Fig. 7 has a number of ~dv~ntages
over that descriL)ed aboYe with respec~ to the Figs. 4-1;7 ineluding greHter
2j simplicitv of meehanical parts, high precision of motion implernentation, And sut)stanti~lly eomplete immunity to linear a~celerationLs in 8~1 sxes.
Ti~n;;~-Foik Arrangement of Fi~s. ~ and 9
Fig. 8 illustr~tes a thir~ type of vibratir)g accelerometer ~ssembly,
n~me]y~ one using a tuning fork, hich may "^ provided for each ser~iitive axis of
30 the mo-in~ t)ody in order to provide dynamic balancing of the ~orces~ ~s weli as
important ~dvant~ges to be des~ribed below. Fig. Q illustr~te~ ~ m~nner Gf
connecting the vibr~ting accelerometer sssembly of ~ig. 8 ~o ~ to form an
electromechanicRI oscill~tor for sust~ining oscillations of the tuning-fork
sccelcrcrr;eter assembly with but A smsll amount of input power, sufficient to
35 rep eni~h ~he cnergy loss due to friction.
~ ith reference ~irst to Pi~ 8, the vibr~ting aceelerorncter
ascc mbl ;. therein genernlly design~ted 200, c.~mprises ~n outer cylinàri~Rl holls-
in~ 'IJ" ir. uhich is mounted ~I tuning fork ~4, including A p~ir of prongs 20~,
~7~
1 204b. The prongs extend parallel to the sensi-tive axis
for the r~spective accelerometer assembly, this being the
Z-axis in Fig. 8, and therehy perpendicu].arly to the axis
of vibration of the accelerome-ter a.ssembly, this being the
Y-axis in Fig. 8. The tuning fork 204 is mounted within
housing 202 by means of a moun-ting post 206 secured to an
intermediate web 204c of the tuning fork~
Housing 202 further includes another post 208
aligned with post 206 but spaced from it and also from web
204c of the tuning fork 204. Post 208 is used for
mounting, on one s:ide, a permanent magne-t 210 cooperable
with a drive coil 212, andl on the other side a permanent
magnet 214 cooperable with a pick-off coil 216. The t~o
permanent magnets 210 and 214 are of cylindrical
lS configuration and include cylindrical air gaps within
which are disposed their respective drive coil 212 and
pick-off coil 216, each of the latter _oils being ~arried
on cylindrical bobbins 218 and 220 secured to the inner
faces of the two prongs 204b and 204a.
To the outer face of prong 204b of the tuning
fork is secured, by maans of a mountin~3 222, an
accelerometer unit 224 having a mass 226. In a similar
manner, there is secured to the outer face of prong 2.04a
of the tuning fork another mass 228 to provide a
counterbalance for accelerometer unit 224 and its mass
226.
The electrical connections to the driving ~oil
212 and pick-off coil 216, as well as to the accelerome~er
unit 224, are provided by terminals 230 and 232 extanding
3Q externally of housing 202 and terminals 234 internally of
the housing and connected to the accelerometer unit 224,
It will be appreciated that the tuning fork 2Q4
in Fig. 8 vibrates at its natural fr~quency, and thereby
causes the accelerometer unit 224 and its mass 226 on
prong 204b to move in synchronism with, but in opposite
directions to, the counterbalancing mass 228 on prong
204a. Thus, no net force is exerted on the housing 2Q2,
- 17 _
,~
~'7(~
1 and therefore, on any support to which the vibrating
accelerometer assembly 200 is secured. As described abo~e
with respect to Figs~ 4-6, this support would be the
moving body i-tself in a strapdown application, and the
inner gimbal of a platform in a stable gimballed platform
applicationO The arrangement illustrated in Fig. 8 thus
provides a high degree of dynamic balancing~
Because of friction and damping, the
oscillations of the tuning fork 204 would decay to ~0'~ in
a relatively short period of time. To sustain the
oscillations indefinitely, the vibrating accelerometer
assembly 200 illustrated in Fig. 8 may be connected to
form an electromechanical oscillatorJ as illustrated in
Fig, 9.
Thus, as shown in Fig. 9, the signal acros~ the
pick-off coil 216 is fed to an amplifier 240, the output
o which is~ connected to the input of the ~riving coil
212. The amplifier 240 is of a polarity of reinforce any
initial displacement of the prongs 204a, 204~ o~ tl~e
tuning fork 204. Thus, the system operates as an
electromechanical oscillator having a frequency de~ermined
by the natural frequency of the tuning fork. This natural
frequency can be used to synchroni~e precisely the
$requency of the free-running multivibrator 21 in '?ig. 2.
Preferably, amplifier 240 is of the nonl:inear
type, such as including a saturation device, so as to
force the complete electro~echanical oscillator to
stabilize at a finite amplitude.
It will thus be seen that the tuning-fork
accelerometer assembly arrangement illustrated in E'igs~ 8
and 9 provides a balancing dynamical system because of the
countermoving masses J and requires but a small amount o~
power for the driving coil 212, merely to repleni~h the
energy 10s5 due -to friction. The assembly, being a
sharply tuned oscillator, rejects mechanical disturbances
in the axis of vibration tY-axis in Fig. 8~ unless they
are exactly at the resonant frequency. Further, the
- 18 -
, .
~2'7( ~6~
1 struc-ture is inherently rigid in the accelerometer-
sensi-tive axis, namely the Z--axis. Moreover, since -the
assembly operates at the natural frequency and
synchroni~es the multivibra-tor 21 of the control pulse
generator 2 in FigO 2, there are no phase lags involved
bet~een cos ~-t and sgncos ~t. Still further, the
accelerometer assembly is extremely simple and can be
constructed at low cost.
The advan-tages, attainable by the use of
vibrating accelerometers, as described above, enable the
construction of IMU's theoretically having a much higher
rnean-time-between-failure (MTBF) than either the
gyroscope-type or of the rotational accelerometer-type
[MU's.
While the invention has been described with
respect to several preferred embodiments, it will be
ilppreciated that these are set forth purely for purposes
of example. Thus, there are many other possible
~- arrangements for producing the vibrational motion, e.g~,
]~y the use of mechanical devices, such as cams or linkage~
for transforming rotary motion, such as from an electrical
~preferably synchronous~ motor, to the vibrational
motion. Many other variations, modifications and
applications of the invention will he apparent.
~5 Paired Accelerometer Arran~ements
Significant improvements in signal stren~th ~or
bo~h the force and the rate channel can be achieYed while
at the same time a reduction in ~ignal noise can be
obtained when pairs of accelerometers, instead of a s~ngle
-19
a~e~-erome~er as shown in Fig. 3, are used for each axis of rotation. Simplifiedillustr~tions of three arrangements of paired accelerometers ~re provided in
Figs. 10-12. A significant advantage of utilizing ~ccelerometers arrQnged in
pairs, as shown in Figs. 10-12, is that the noise present in both the force and the
rate ehannel o~ Fig. 2 is increQsed onl~ by the square root of two while the
erfective force and rate signals are doubled, thus providing an effective signal to
noise increase of a square root of two. In addition, common ~cceler~tion
d.s~urbances in the rate channel due to external forces th~ m~y result frorn
vehicular snd mechanization sources are substantially cancelled in this type o~
arrangement.
The first arrangemPnt of paired acce]erometers is illustrated in
Fig. 10 where a pair of accelerometers 300 and 302 are mounted on an angul~rly
rotating base 304 which vibrHtes about the 2-axis 305 as in~icated by lhe
arrows 308. The force-sensing axes A1 and A2 of the sccelerometers 300 and
302 are aligned so as to be p~rallel to the Z-axis 306 about which the support 304
vibrates. Since the ~rrangement in ~ig. 10 includes a pair of accelerometers
lhith their force-sensing axes Al and A2 parallel to the vibraticn axis 306, this
arrangement will hereinafter be referred to 85 the PAPYA arrangem~nt.
The second arrsngement of paired accelerometers is shown in
Fig. 11 where two accelerometers 310 and 312 are mounted on a support 314
which Yibrates ar)gularly about the Z-axis indicated at ~18 QS suggested by the
arrows 316. In this ~rrangement, the accelerometers 310 cnd 312 are secured to
the support 314 in a back-to-back arrangemer)t such that the ~orce-sensing axes
Al and A~ ~re p~rallel, but opposite in direction, ~nd are n~rmal to the axis 318
of ~ngulhr vibration. This arrangement will hereinafter be termed the PAN~'A
arrangement to denste a p~ir of a~celerome-ers subje~t to ~ngular motion with
Iheir force-sensing ~xis normal to the axis of vibration or angular motion.
The tllird arrangement is illustrated in ~ig. 12 where a pair of
accelerometets 320 ans 322 are arrAnged b~ck to b~ck with their forc~sensing
~,u s~;es Ay and Ay locQted in par~llel but opposite directions. In this arrangement,
the sccelerometers 3~0 and 322 are caused to vibrste in a linear direction alongthe X-~xis, as indieated by arr~ws 324 and 326. For convenien~e, this ~rrange-
ment ~ ill hereinafter be refelred to as PLN~,'A due to the f~ct that i~ is an
arrangement of paired accelerometers caused to vibrate in a linear m~nner along
~ vibrQtioll axis ~hich is norm~l ts the force-sensing axis.
Figs. 13-15 correspond to the P.~P~'A, PAM~'A snd PLN~'A arrange-
ments of ~igs. 10-12, respectively, and pro~ide 8 conce?tual il]llstratiOn ot` hou~
the paired accelerometers can be ~rranged in triadc. ~n ]~igs. 13-15, the
'7(~
accelerometers are denoted by their force-sensing axes Ax,
Ax, Al, A2, Al, and A2 -to provide force sensing and
angular ra-te sensing along and about the orthogonal axes
X, Y and ZO The arrangements shown in Figs. 13-15 are
suitable for use in an inertial reference systern that can
in turn be used in an Inertial Navigation Sys~em.
In the PAPVA mechanism shown in Figs. 10 and 13,
six accelerometers are required, namely, Ax, Ax, Ay A2, Azl
and Az. The accelerometer pairs are vibrated at an
angular constant frequency and constant angul~r
amplitudes6 M. The principle of force and angular rate
signal separa-tion is substantially the same as illustrated
in Fig~ 2. The accelerometer outputs contain the same
basic information for angular rotation Q and force F,
although the actual signal content is somewhat different.
In developing equations describin~ the signal
content of the accelerometers in Fig. 13, the
instantaneous distance of each accelerometer from the
vehicular center of rotation is given by:
-Ax~~ LCos6 lz~LSin~
xl0 -LCos6 1z~LSin6
r 1¦1x-~LSin6 0 LCos6 ~ 1 ~6)
-Ayllx-Lsin6 0 -LCos6 ~lJ
rA ¦LCos6 ly~LSin~ 0
rA-I.Cos6 ly-LSin~ 0
and defining:
6 = 6MS~t (7
~ =
30 while assuming 6 1
Sin 6 ~ 6 = 6M S~t (9~
Co~ 6 ~ 1-1/2 6~12S2~t (10)
and defining:
L ~M ~ P/~
- 20 - -
'7( ~
21
Then substituting Equations (6~ - (10) into Equation (1) and expandin~, the
followin~ Elccelerometer output equations resu]t:
a1. =Fx + lz(pr+~q) + ~ S wt(pr+q) + L(qp-r) + 2~ 2 C~ t(q~ ~Msw t r)
~(/2)L ~ 2S2~ (qp-r~- 2w ~ Gl) t(l/2) q~l2S2~t-q ~]~)
a~2 = Fx ~ 3 (pr~q) - ~ S ~,)t(pr+q) - L(qp-r) -2~1~ 2 C~,~ t(g+ ~ S~t ~r~
+(~ M2S2w~(gp-;)~ 2~ C~t(l/2) ~M2S2~1)tq (12)
ay = Fy + lx(pq+r) + ~Sw t(pq+r) + L(qr-p) + 2w~C ~,~t(r+ ~ S ~"t . p)
~1~L ~M~S2w t(qr-p~ - 2w 2 C~t(1/2) ~M2S2ll)tor (13)
a2 = Fy + Ix(pq+r) - 2 S~t(pq~r) - L(qr-p~ - 2w 2 C~,~t(r+ ~1S:,~ t^p)
t(~ L ~M~S~ t(qr-p) + 2~ C~I~ tt1/2) ~ 2S2~ t-r (14)
al = Fz + Iy~qr+p~ + ~S(IJ t(qr+p) + L(pr-q) ~ 2~1~ 2 C ~t(p~ ~MS~, t- q) -
20L~ M252~t(pr-q) - 2 ~ 2 CL~)t~l/2) ~M S ~ t~p (15)
~2 = ~ ~ I (qr+p) ~ 2 S ~" t~qr+p) - L(pr -q) - 2 ~Q C w ttp~ ~ MS lll t q) +
S (1~ttpr--q) ~ 211) 2 C ll)tll/2~ ~l S ~vt~p ~16)
25Before entering the si~nal processor of Fi~. l9, the acceleron e~er
signals are preprocessed as sums and differences as sho~;n in ~ in
accordance with the following matrix equation:
~1 '1~1 0 0 0 0 ~1 ,
ay D û 1 - I O O a2
azP O O 0 0 1 - 1 ~1
3~ aFx = 1/2 l/2 D 0 0 0 by ~I iJR
. a '~ o O ~I" 1/2 0 0 al ¦
Z 0 ~) 0 l/2 1'2 ~_
7( ~
-22- .
Assuming ~g~in F and S~ are subst~nti~lly cnnst~nt in the Interv~l T, ~ll time
derivatives YAnish in iEgu~tions (11) - ~16). Substituting accordingly Into
Equation (17~ and expanding:
~ = 2wp Cw t~q~ S~ PSw t-pr + ~Lqp ~
- L ~M~S ll) t qp - 2~p Cw t(1/2) ~M~S2~1) t-q (1~)
s~ = 2 ~P C~ t (r~ ~;,qS~ t~p) + p S~ tpq ~ 2Lrg -
- L ~M2S~ t rg - 2 wp Cll) t(l/2) ~M2S2~ t r (l9)
= 2 ~p C~ t(p~ ~MS!L~ t- q~ + p S~l~ tqr ~ 2Lpr --
- L ~M2S2~ t pr - 2 ~ p C~ t(lt2) ~ M2S2~" t~p (20)
F
axX~ Fx ' lzpr
n Y = F * I qp . . (21)
2~ a~Z = Fz ~ lyqr
Thus, two adv~ntsges are obtained through the pRired mechani~ation, ~ll
specific force components ~re removed ~rom the acceleration signals in
3:quations (18~ - (2û) and all angul~r rate comp~nents are remov~d from si~nals
25 in Equation ~1). This sig,nifica..~l~ improves the ~ecoupling of ~ fro:n n .
Common mode noise terms due to Yehicle noise ~e also remove~ frQm ~he ~?
hannel as c~n be seen from Equations ~18) - (20). To obtain estirn~tes of p, q
and r ~s defined b~ Eqll~tion (22) belos,
p = ~ PSgn(Cw t~dt
q = ~p ~ aq Sgn(C~ t)dt (223
T
r - 8 p ~ ay Sgn ~ t)dt
~ ' .
-23 -
p q r
a2 ~ ~x ~ ay ~n Equa~ions (18) - (20) ~re substituted into EquAtion (22), nnd to obtain
~n estimate of ~ y ~nd F~ as de~ine~l by Equation t23) below,
~x T Jo~t
~ T Jo~3t (23)
3~z T ~o~dt
F F
AXx ayYand a2Z in Equa~ion ~21~ are ~ubs~itL~ted into
Equation ~23). T~e coYresponding results are:
p = p(1 - ~6 ~ 2)
9 = 9~1 - 1/6 ~M ) . (24)
1/8 ~ 2)
and:
FX=F ~1 pr
Fy - Fy + ~xPq ~25)
Fz=F~lyqr
Thus, p, q ~nd r ~re determined precisely except for a constant known scale
factor and FX9 Fy and Fz are the same as determined before.
It can be seel~ from the equations sboYe th~t the ou2puts obtained
are in many respects equiv~ent to the meehaniz~tion using a single accelero-
meter. Also, the effect of vehicular noise in the S~ ~hannel is almost carlcelled
by virtue of the common mode rejection obt~ined through the ~ccelerometer
30 pairing RS indicated by Equ~tion (24). However, gradients in velliculRr noisealong 1, st;ll retain some noise in the Q channel. Thus, since L will normall~,~ be
a few cenlirneters, Yehieul~r noise is not entirely c~ncelled. Through possible
residu~l anE~ular vibr~tion of the drive axis, residual, synehronous ~nd uncontrol-
led noise mQy b~ re~ained and appear as An unlinown bias in the S2 ch~nnel.
35 Also, ~he mechanization in Fig. 13 in principle is dynRrnicall~ ~aIsnced.
As illustrated in Fig. 1~, six accelerometer~ Al, A2, Ay~ Ay2, ~1
~nd A~ ~Lre used in the PANVA rnechanizati~n. Again, the nccelerorneter pairs
are ~ibrated ~ angul~r frequency~ and ar;gular ~mplitudep . The ~ccelero~
meter ~utputs contain _ and F infQrmation ns before but with different
1 additional dynamic terrns. As wi-th the PAPVA mechaniza-tion
of Fig. 13, the basic principle o:E signal separation i5
not changed. This mechaniæation also has the advantage of
essentially perfect vehicular noise rejection.
In accordance wi-th the arrangement shown in Fig.
14, the instantaneous distance of each accelerometer from
the vehicular center of rotation is:
-Ax LCos~ 0 1z+Lsin6
rAX -LCos6 o l~-Lsin6
-Ay = lx+LSin~ LCos6 0 ~1~
Ay lx-LSin~ ~LCos6 0 i (26)
-Az 0 ly+LSin~LCos6 i k
f -Az 0 ly-LSin6-LCos6
Since, in this mechanization~ the accelerometer input axe.s
15 change direction with res?ect to the body axes, the sensed
components are modulated. For example, the input axis
varies in accordance with 5 : ~Cos 6.i, O.j, Sin ~.K].
Deno-ting the total acceleration that would be
sensed along the body axes in the case of ideal parallel
20 motion by al, ay,and al~ the actual acceleration sensad by
the angularly vibrating accelerometers is given by:
al Cos6 0 Sin8
` ax .~Cos~ 0 -Sin5
ay Sin6 Cos~ 0 x
ay -Sin~ -Cos6 o ~Y (27)
az Sin6 Cos~
a2 0 -Sin~-Cos6 a~
-2~-
2~
With EquAt~ons (7) - (10) and substituting Equ~itions (26) and (27)
lnto Equation (1) ~nd exp~nding~ the ~ctual nccelerometer palr 4utputs ~or ~1 ~nd
~ csn be represented by:
~1 ~ Fx '~z (pr+q) - L(q2~r2) ~ p S ~ t pr + 2~ ~C O t o tl ~
S~t~ Qz(p~+q2)] + 1/2 ~M2Sa~ tl-Fx-~(pr~q) (28)
~ 2L(r~-p2)J - L 4~22~ ~C~ t
n2 = -Fx Qz(pr~q) - L(q2fr2) + pSwt~pr ~ ~ ~C~t-q ~
Swt[-Fz+lz(p2~q2)] ~1/2 ~ 252wt[Fx~z~pr~q) (29)
2L~r~-p~ L 4~ 2C2 L~ t
15 Similar equations for fly, 8y, ~1l and ~2 result from this substitution.
Since the PA~VA mech~nization of Eigs. 11 ~nd 14 is back to ba~k,
the preprocessing ~peration of Fig. 19 for the P~NVA ~rangernent ;s repr~
sented by the following m~trix equ~tion:
aq . 1 1 O O û O ~1
ay ~0 0 . 1 - 1 0 ~ .~2 .
2 5 Fx = 1/2 - 1/ 2 1l 0 0 2 (3 0)
¦ 8yFY 1~2-112 0 û a~
~Z o ~ o 0 1~2- lJ2
Substituting ~:quations (28) and ~9) ~long with similar equations for ay, a2~ fll
Bnd 8z into Equation (30~ and Qssuming F and Q const~nl: throughout Tt results
in:
sx ~PC~tq+2ps~tDpr- 2L(q2~r~3~J~)~5 2S2"t I~IL( 2 2
- 2L-w 2 ~;~,12c2 ~t (31)
~' ~
--26 -
~27~67
wlth similar equations for ~ Qnd aP ~nd Xqu~tion (~2) below ~or ~Fx snd sirnil~requAtlons for llyY ~nd flx~
~x ~ = ~x ~ Qzpr ~ ~MS ~ t I Fz~Q~z(p2~g2)] ~1/2 ~M~sa ll~ t t- PiQ ~z~ Pr] (32)
Substituting the equations ~or the angul~r r~te ~omponent of the acceler~tions,
such QS Equation (31~, and th~ e~uations for the force component o~ the
~ceeler~tions, such as Equ~tion (32~, into Equ~tions (22) and (23) results in:
p= p
q_ q (33)
r= r
t5 Fx = (Fx Qzpr~ (~ 4
y ~Fy xpq] (~ ~ ) 134)
F~ = (Fz~Qyqr) (~ _ ~411
~D Thus, in the PANV~ mechanization, the sngular rates ~re determined ex~ct]y,
and specific f~rces are determined ~o R known sc~le factor.
From the a~ove equations, it is ~pparent that in ~he back-t~bacl;
mechanizsti~ns, all ~ehicular noise is elirninated in the Q ch~nnel. This is true
for possible gradients or angulQr sccelerations. Possible residual synchronous
'~5 noise of the Yibr~tion axis, including angular noise norm~l to vibr~tion axis, is
also eliminQted in the Q ~hannel. The b~ck-t~b~ck mech~nizat50ns are alsb
sensiSiYe to possible rectification ef~ects in the F channel resultin~ from
peri~dic components normal to the input axes of the ~ccelerometer pairs.
~ioweYer, this effect can be compensated for ele~tronically. Although the
~ccelerometer pairs ~e subjectcd to centrifugal foree due to the ~n~ular
motion, the fset ~ th~' frequency 2L~ malces it posslble to eliminate this effec~
in the processor of Fig. 19. The phase angle of sign~ls in the proce~or ~ill n~thaYe an effect here. Howwer, at, for example, L=lSmm and ~l/lS r~d and ~or
h) ` 200 radls~it ~mounts to 0.3g, Ulus consuming part of tl e e~fec~iYe range ot the
3S ~ccelerometer. IS should be noted that residual harmonics in the S~t mo~ion
retain uneven in-phase components in the eentripetal acceleration~ thusl contri-buting t~ possible null point offset.
The PLNVA mechsnisrns shown In Figs. 12 ~nd 15 ~re simil~r to
PANVA me hanisms except that motion of the ~ccelerometers 320 and 322 Is
lineQr. In ~ccord~nce with ~ig. 15, th2 Inst~ntuneous àistance to the vehicul~r
center of rot~tion is:
~ ~1 L 0 lz~ ~ S :~ t
~4x -L 0 ~z- ~S~ t
r 2 ¦~ lx +~S t L 0 ~ I (35
r 1lx- ~Sw t -L 0
I ~2 0 ly ~ 2 Sw t L
Z l 0 Iy _ P~ Sw t -L
- Substituting Equation (35)ahove into Equation (1) ~nd expandin~
the following equa~tions for al and a2 result:
~91 = F~ (p}~q) ~ P Sw t(pr~q) ~ 2~Cw t-q - L(q2~r2~ 136)
~2 = ~~x ~ lz(pr+q~ ~ ~ S;~Jt(pr+q~ + 2w ~ Cw t-q - L(q2~r~ 137)
25 with similar es~ ations fsr ay, ay, al snd a2 also resulting from the su~stitution.
Substitutin" Equations (36~ ~nd (37) along with the equations for
~y~ ay~ al and a2 into Equation l30), snd assuming F ~nd Q ~re const~nt
throu~hout T, results in:
a~ - 2wp&~ t~q ~ pS~ t~pr - 2L(q2+r2) (38
8y = 2 w pC w t r `t p SW t pg - 2L(p2~r2~ ~39~
3S aP ~ p Cw t~ p + p S~ t ~r - 2L(p2+q2) (~0)
~' ' .
7( ~'7
ax ' F x + lzpr
ayY = Fy -~ lxPq (42)
azZ = Fz + lyqr (43)
Here p, q, r, Fx, Fy and Fz are obtained as before by
substitutin,g Equations (38) - ~43) into Equations ~22) and
(23~, respectively. The results are:
q = q (44)
r = r
Fx = Fz + lzpr
Fy o Fy ~ 1xpq ~45)
Fz = Fæ ~ lyqr
In this mechani~ation, there:Eore, all common
mode vehicular noise components are essentially eliminated
15 , as in the PANVA mechanization. The small w periodic
deviation from exact colinearity of the input axes varies
in accordance with S ~t. Thus, pos~ible noise due to
angular acceleration is ~liminated by the Sgn (cw t)
operation of the processor in Fi5~. 19. Centripetal force
due to the periodic excitation is none~istent as in the
- PA~VA mechanization.
In Figs. 16 through 18 are illustrated apparatus
for implementing the PAPVA mechanization cf Fig. 10, the
PA~VA mechanization of Fig. 11, and the PLNV~
2.5 mechaniæation of Fig. 12, xespectively. The apparatus f~r
mechani2ing ~he PAPVA mechanism is sho~n in Fig. 16 and
includes a housing 330 having a pair of input/output plugs
332 and 334. Secured to the hou.~ing 330 by means o a
pair of bearings or flexible joints ~36 and 33~ is a shaft
340. The paired accelerometers 300 and 302 ar~ mounted on
- 28 -
s,
~" r
1 the accelerome-ter support frame or mernber 304 which in
turn is secured to the shaft 340 for rotation therewith.
rotational vibration of -the shaft 340 is provided by a
motor that includes a rotor 342 connected to the shaft 340
and a s-tator 344 attached to the housing 330. Signals
providing either position or velocity information for a
feedback signal to a drive servo that would control the
amplitude ~m of the shaft 340 vibration can be obtained by
the pick-off arrangement indicated generally at 346.
- 2Ba -
, ~
--2~3--
An apparatus for implementing the PANV~ mecharlizntion is
provided in Fig. 17, wherein the accelcrometers 310 and 312 fLre mounted on the
support 314 which in turn is secured to a shaft 348. The shaft 3~8 is rotfltablysecured within a housing 350 by rmeans of a pair OI bearings or flexible joints 352
s and 354. Angular vibration of the shaft 348 and, hence, the accelerometers 310
and 312 is provided by an electric motor which includes a rotor 356 securcd to
the shaft 348 and a stator 358 secured to the housin~ 35û~ Signals of the motionof the accelerometers 310 and 312 can be obtained by the pick-off arrangement
indicated generally at 360 to provide negative feedback for a drive servo
controlling the amplitude ~ of the shaft 348 vibration.
An ~rrangement for implementing the PL~VA mechanization of
Fig. 12 is illustrated in Fig. 18. In this particul~r implementation, linear
translation of the accelerometers 320 and 322 along the axes 324 and 326, as
shown in Fig. 12, is provided oy a mechanism that includes a support frame 362
that holds the accelerometers 320 and 322 whicll is secured to a housing 364 by
means of ~ pair of flexures 366 and 368. Abutting the lower portion of the
accelerometer support frame 362 is a linkage element 3~0 that in turn is secure~to a sh~ft 372. The shsft 372 is rotatably secured within the housing 364 by
meens of A pair of bearings or flexible joints 374 and 376. An electric motor
~0 inclLldin~ a rotor 378 attached to the shaft 372 and a stator 380 attached to the
housing 364 will cause the shaft 3?2 to rotate or vibrate back and forth throu~rh a
very limited angular rotation. As the shaft 372 rotates back and forth througl~ a
small an,,le, the Iinkage element 370 will cause the accelerometers 320 and 322
to move in directions essentially normal to the force sensin~-axes Al and Aa .
As a resultt substantially linear movement of the accelerometers 320 and 322
can be achieYed in Q direction normal to their îorce~ensing axes by using the
mechanism of Fig. 18. Signals representing angular position or velocity of the
shaft for use by a drive serv~ can be obtained by means of the picl;~off
arrangement indicated generally at 382.
The preferred embodiment of a signal processor for separating the
~orce signals F from the ~ngular rate sign~ls Q for the paired ~ccelerometer
mechanizations of Figs. 10 through 13 is illustrated in Fig. 19. The basic
operation of the processor circuit shown in Fig. 1~ is the same as the signal
separation circuit of Fig. ~. For example, the control pulse generator ~ i~ the
same as shown in Fig. 2 with a line 384 connecting the square wave gener~tor 22,as shown in Fig. 2~ to the drive signal ~enerator 31~ In a similar manner, the
output of the reset ~nd integrate control pulse generator 23 is transrnitted on a
line 386 from the control puLsc generator 2, and the ou~pu~ of the samp1inc3 ~r
.'~ ' .`" - " , ' ' '~
1 p~llse generator 24 ls -transmitted on a line 388. Since
the pai.~ed acce].erome-ter mechaniza-tions make use of two
accelerometers, there are -two accelerometer assemblies 390
and 392 shown in Fig. 19 tha-t correspond to accelerometers
300 and 302 in Fig. 10 and accelerometers 310 and 312 in
Fig. 11 and accelerometers 320 and 322 in Fig~ 12.
Accelerometer output signals al and a2 are output from the
accelerometer assemblies 390 and 392 on a pair of lines
394 and 396, respectively.
Signal separation is performed in the circuit of
Fig~ 19 generally by the sarne means as the circuit in Fig.
2 except that the force channel producing the Fz signal on
a line 398 and the angular rate channel for producing the
p signal on a line 400 are represented in FigO 19 as two
separate circuits~ As shown in Fig. 19, a force channel
ci.rcuit 402 includes the integrating circuit 44 and the
s=ample-and-hold circuit 45 of the electronic signal
separation processor 4 of Fig. 2, ~ith the signals on
li.nes 386 and 388 being applied to the integrating circuit
44l and the sample-and-hold circuit 45 as shown in Fig.
2. In a similar mannex, an angular rate channel circuit
404 includes the integrating circuit 42 and th~ sample~
and-hold circuit 43 oE Fig~ 2, as well as the sign
switching or mult.iplying circuit 41~ The signals on lines
3#6 and 388 are applied to the integrating circuit 42 and
the sample-and-hold circuit 43, as ~ell as the pulse
si.gnal on line 3~4 in the sa~e manner as shown in Fig~ 2.
One of the key advantages of ~he paired
accelerometer machanization is the ability to use sum and
diference techniques to separate those signals which
pertain primarily to translational motion from the signal~
which pertain primarily to angular motions. To be able to
cancel linear specific force signals that are O~ltpUt from
paired accelerometers, it is necessary that the force-
sensing axes be as nearly parallel as possible and that
the effective centers of mass be close together as well.
Whether the force~sensing axes of the accelerometers are
in the same or opposite directions is a matter of
- 30 -
,~
~ 7(~ t
1 convenience i.n designing the mounting of the
accelerometers. In either case, the separation process is
made possible by designl.ny the mechanism that produces the
vibratory motion so that the driven velocity vec-tors are
at all times equal and opposite when measured in the frame
of reference of the housing.
A preprocessor circuit or performing the sum
and differenc~ functions is illustrated in the dashed line
406 of Fig. 19. The preseparation or preprocessor circuit
406 includes two summing junctions 408 and 410. The
particular preprocessing circuit 406 shown in Fig. 9 is
utilized for mechanizations where the force-sensing axes
are in the same direction, such as in the PAPVA
mechanization shown in FigO 10 and , as such, implements
the logic o Equation (17). Here the summing junction 408
acts to provide a si.gnal to the force channel 402 that
represents the sum of the accelerometer signals on lines
394 and 396. In a similar manner, the summirlg junction
410 provides a signal to the angular rate channel 404 that
represents the difference between the accelerometer
signals on lines 394 and 396. It is assumed that the
nonrotational specific force signals will be substantially
equal on lines 394 and 396 so that the summed signal on
line 412 will, in effect, provide twice the sensitivity
25 for the specific force being measured by accelerometers
along the force-sensing axes. Likewise, the diference
signal on line 414 will be su~stantially free of
components representing specific force. Conversely, a
purely rotational motion will produce two sinusoidal
Coriolis accelerations along the force sensitives axes o
the accelerometers with a phase di~ference of 180, This
phase difference occurs because the Coriolis acceleratiolls
are the vector product of angular rate and relative
velocity and, in this case, the an~ular rate i5 common,
while the relative velocities are 180~ out of phase. As a
result, the output oE the summing junction 408 on line 412
will be substantially free of components representing
- 31 -
:~ ~2'7t.~
1 angular ro~a-~ion. For the same reason, the output of the
summing juncti.on 410 on line 4:L4 will provi.de a signal to
the angular rate channel 404 with twice the sensitivity
~or angular rateO
In the mechanization, such as the PANVA of Fig.
11 and the PLNVA of Fig. 12, where the force-sensing axes
have the opposite sense, the same principles apply except,
of course, the signs of the signals are reversed. Thus,
in the preprocessor 406 for the PANVA and the PLNVA
mechanizations, the summing junction 408 would diEference
the acceleration signals on line 394 and 396 and the
summing junction 410 would add the signals on lines 394
and 396 conforming generally to the relations expressed in
Equation (30). As a result, the summed signal from
summing junction 410 will contain angular rate informati.on
only while the difference signal from summing junction 408
contains specific force information only. Therefore, it
may be seen that the preprocessor 406 has the effect of
. separating the specific force signal ~rom the angular rate
signal before the signals are applied to the force channel
402 and the angular rate channel 404.
A further advantage of the circuit arrangement
shown in Fig. 19 is that the sum and difference techniques
provided by th preprocessor 406 can he used to facilitat~
the scaling of the signals applied to the force channel
402 and the angular rate channel 404. Scaling is
illustrated by means of a pair of scaling amplifiers 416
and 418. The scaling amplifiers 416 and 418 can be used
to scale the level of signals being applied to the ~orce
channel 402 and ~he angular rate
- 31a
~ .
1 b~3'~
- 32 -
1 channel 404 without concern for the magnitude of the si~nal
output from the accelerometers. This is particularly impor-
tant when one considers that the amplitudes of -the si~nals
representing specific force Ez may be up to 100 times
greater than the signal amplitudes relating to angular rate
p. Thus, the values of the amplifier gain constants KF and
KQ can be adjusted to the expected signal amplitudes on lines
412 and 414 to permit the maximum resolution of the signals
without overranging the circuits 402 and 404. Similarly,
Inertial Navigation System gains KF and K~ can be switched
to increase sensitivity and , hence, to improve resolution
during the navigation system adjustment process. During a
mission, it may be necessary to temporarily reduce the
sensitivity of either the force-sensin~ or rate-sensing
channels to avoid overranging the circuits 402 and 404
during transient maneuvers of thP vehicle containing the
navigation system. F and p have been chosen by way of
illustrat:ion of one of the component pairs of F and Q.
~dentical considerations pertain to Fx and q and Fy and r.
Since one of the principal uses of the rate
signals ~ produced by the accelerometer systems di cussed
above is in Inertial Navigation Systems, the effect of
noise and error signals on the navi~ation system is a
significant concern. As it turns out, accelerometer noise
present in the ou-tput Qf the an~ular rate channel 404 is a
principal factor in the accuracy of an Inertial Navigation
System using accelerometers to determine an~ular rate. The
effect of accelerometer noise for a given acceleromet~r level
of noise is inversely proportional to the vibration amplitude.
It has been found, for example, utilizing the QA-200Q
accelerometer commercially available from Sunclstrand Data
Control, Inc., the positional error is about nautical miles
per hour for a vibxation amplitude of approximately 1.25 ~n.
