CA1061132A - Apparatus for performing inertial measurements using translational acceleration transducers and for calibrating translational acceleration transducers - Google Patents

Apparatus for performing inertial measurements using translational acceleration transducers and for calibrating translational acceleration transducers

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Publication number
CA1061132A
CA1061132A CA233,895A CA233895A CA1061132A CA 1061132 A CA1061132 A CA 1061132A CA 233895 A CA233895 A CA 233895A CA 1061132 A CA1061132 A CA 1061132A
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Prior art keywords
axis
support
transducer
translational acceleration
transducers
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CA233,895A
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French (fr)
Inventor
Theodore Mairson
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Lockheed Corp
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Sanders Associates Inc
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Priority to CA297,262A priority Critical patent/CA1061133A/en
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01PMEASURING LINEAR OR ANGULAR SPEED, ACCELERATION, DECELERATION, OR SHOCK; INDICATING PRESENCE, ABSENCE, OR DIRECTION, OF MOVEMENT
    • G01P21/00Testing or calibrating of apparatus or devices covered by the preceding groups
    • FMECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F41WEAPONS
    • F41GWEAPON SIGHTS; AIMING
    • F41G7/00Direction control systems for self-propelled missiles
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01PMEASURING LINEAR OR ANGULAR SPEED, ACCELERATION, DECELERATION, OR SHOCK; INDICATING PRESENCE, ABSENCE, OR DIRECTION, OF MOVEMENT
    • G01P15/00Measuring acceleration; Measuring deceleration; Measuring shock, i.e. sudden change of acceleration
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01PMEASURING LINEAR OR ANGULAR SPEED, ACCELERATION, DECELERATION, OR SHOCK; INDICATING PRESENCE, ABSENCE, OR DIRECTION, OF MOVEMENT
    • G01P3/00Measuring linear or angular speed; Measuring differences of linear or angular speeds

Abstract

Abstract of the Disclosure Inertial measurements, including the measurement of angular velocity as well as angular and translational accelera-tion, are made by spinning one or more translational accelera-tion transducers disposed at a point on a spinning member with the sensitive axis of at least one of the transducers sub-stantially parallel but not coincident with the spin axis of the spinning member and combining their output signals. The principles of the invention are also applied to generate components of translational acceleration for the purpose of calibrating translational acceleration transducers.

Description

sackground of the Invention Presently gyroscopes are used in manv arrangements, such as inertial platforms and the like, for the measurement of angular velocity. In such arrar.gements separate gyroscopic instruments are usually used to measure each com?onent of angular velocityO These gyroscopic arrangements are inherently costly and relatively large. Gyroscopes operate by storing a large angular momentum using a flywheel, thus further contributing to their relatively large size. Since the heavy flywheel must be critically balanced, the cost thereof is further increased.
Gyroscopes are further limited in that they cannot withstand a rugged environment because of their relatively delicate bearing and pickoff alignment. For exam~le, I will describe hereinafter apparatus for measuring angular velocity in'pitch and yaw of a gun~firec spinning projectile on which such apparatus is mounted. The apparatus undergoes translational acceleration of 20,000 or more y's while beinc fired from a gun and experiences a large centrifugal acceleration during the ~ measurement due to the spinning of the projectile. This cen~rifugal acceleration component .
, .,.~, RIS :InS
~ L 31L32 imposed on such apparatlls increases in proportion to the distance away from the spin-axis of the projectile on which the apparatus is mounted and would be very large since space limitations would preclude placing the apparatus near the spin-axis. This csntrifugal acceleration can reach 7000 g's on parts of the apparatus wh;ch are one inch away from the spin- 5 axis. Furthermore, the gyroscope's angular mounting tolerances would have to be impractically precise.
As in the spinning projectile application two gyroscopes would usually be required, their combines si~e further precludes their use.
Accordingly, another type of rate sensor is required. 10 Another problem related to kinematics is the calibrating of trans-lational acceleration transducers. Currently, there are four principal methods of calibrating such transducers, These are: (1) static calibration in a centrifuge, (2) calibration versus frequency by rotation in the earth's gravitational field, (3) calibration versus amplitude and frequency on a 15 "shake table" with a precalibrated transducer and (4) calibration versus amplitude and frequency on a "shake table" employing an optical interfero-meter .
Method (1) is unsatisfactory since it does not provide a dynamic calibration. Method (2), of course, is severely limited because calibration 20 can only be made up to the - 1 g of the earth's gravitational forces and admixes cross~axis sensitivity in varying proportion. Method (3), which is a comparison test performed on a shake table, is accurate to only about 1%, due to practical limitations of current art and, furthermore~ this method does not permit the user to locate the physical center of action of 1 25 the instrument under test. Method (4) is quite cumbersome and is not suitable for routine application.

~L~6~L32 Accordingly, it is an object of this invention to provide an improved method and apparatus for making inertial measurements.
According to this invention there is provided a method of inertial measurement which comprises the steps of:
providing a single transducer, for measuring accelerations at a point, rigidly mounted at said point and oriented so that a sensitive axis of the transducer is substantially parallel to and spaced from a first axis; and rotating said transducer about said first axis.
The invention also provides apparatus for use in inertial measurement, comprising: a single support; means for rotating said support about a first axis; and a transducer rigidly mounted at a predetermined point on said support for measuring accelerations at said point, the transducer being oriented so that a sensitive axis thereof is substantially parallel to and spaced from said first axis.
The invention facilitates measuring the motions of a rigid body by using translational acceleration transducers.
The apparatus is relatively low in cost, small in size and rugged, and can be used for measuring angular velocity; pitch and yaw rates; for measuring translational acceleration, angular acceleration and angular rate separately; and for measuring angular velocities of a spinning body about axes perpendicular to the spin axis. It can also be used for dynamically calibrat-ing translational acceleration transducers.
Briefly, this invention teaches how the problems dis-cussed can be solved by measuring the accelerations at a point and combining the measured , .. ..

D -32g5 RIS:ms ~L~16~32 outputs. This technique can provide rreasures of angular velocity as well as angular and translational acceleration The measurements depend on the kinematics of a point in a moving rigid body and do not depend on stored angular momentuIn, as does a gyroscope. The measurements do not require new motion transducers, but can employ various kinds of translational 5 acceleration transducers, including that class of devices which are known as "linear accelerometers".
These have not been used heretofore to measure angular velocity.
Also, the kinematic principles employed in the measurement can be used in a converse manner to generate precise linear accelerations for the purpose 10 of calibrating translational acceleration transducers, thus enhancing the level output of these devices.
In one embodiment, measurement of angular velocity is achieved by spinning a translational acceleration transducer about an axis, the trans-lS ducer being disposed ao that its sensitive axis is parallel to and offset from 15 the spin-axis. The output from the transducer is a sinusoid having a frequency equal to the spin frequency and having an amplitude proportional to the angular velocity about an axis normal to the spin-axis and a phase proportional to the angle that the axis reaches with respect to a reference direction in inertial space, 20 In another embodiment separate measurements of translational acceleration, angular accelerational and angular velocity are achieved by spinning an array of such transducers about an axis.
In a further embodiment of the invention a translational acceleration transducer is calibrated by spinning such transducer at a predetermined 25 rate about one axis and rotating it at a predetermined rate about a second axis which is perpendicular to the first, whereby the spin and rotation rates generate a predetermined linear acceleration which is measured by tlle ¦ tran6ducer under test, l ~.

RIS .ms ~6~

This invention is based on the measurement and/or generation of components of translational accelerations at one or more points in a rigid body. The devices which perform these measurements can be any of a large class of devices which are capable of measuring translational acceleration. The output signals of these devic0s may be any of a large 5 class of signals, including electrical voltages and/or currents, mechanical ¦ displacements, hydraulic pressure or flow, etc. The invention consists of ¦ the positioning and orientation of one or rnore translational acceleration ¦ transducers in a rigid body so that the transducers generate signals which ¦ can be processed to yield inertial measurements, which have heretofore 10 ¦ required equipment which is larger, more expensive and less rugged. The ¦ invention also consists of the positioning and orientation of one or more I translational acceleration transducers in a calibration device which is ¦ capable of generating components of translational acceleration which are known accurately and which can be used for calibrating the transducers 15 ¦ which have b~3en positioned and oriented on the device.
¦ This invention is valid and independent of the following:
(1) the kind of translational acceleration transducer employed, (2~ the nature of the output signals generated by the transducers, (3) the manner of combining the output signals of the transducers, 20 when more than one is employed, and a~) the manner of processing the output signals of the transducers or the combined signals.
Brief Description of the Drawings The above-mentioned and other features and objects of this invention 25 will become more apparent by reference to the following description taken in conjunctil~n with the ac c ompanyiDg dr awinga, in w hich:

5, RIS:ms 1C9~ 32 FIG. 1 is a diagram illustrating the coordinate conventions employed in the discussion of the invention;
FIG. 2 is a diagram illustrating description of motion in an auxiliary coordinate frame;
FIGS. 3A-3C are diagrams for illustrating apparatus for the 5 calibration of translational acceleration transducers;
FIGS. 4~-4C are plan, top and side views, respectively, of apparatus for measuring angular velocity about two axes perpendicular to each other;
FIGS. 5A-5C are plan, top and side views, respectively, of apparatu 10 for separately measuring translational and angular acceleration;
FIG. 6 is a cross-sectional view of apparatus for separately measuring tra~slation acceleration, angular acceleration and angular velocity; and FIG. 7 is an illustration of an embodiment of the invention wherein 15 translational acceleration transducers are used in a spinning projectile to provide angular rate measurements in pitch and yaw Description of Pre erred Embodiments The translational acceleration at a point in a moving rigid body depends on the motion of the body and of the coordinates of the point in the 20body, These relationships can be used in measuring the motions of a rigid body by means of translational acceleration transducers or in calibrating such transducers by means of the motions oi rigid bodies.
In accordance with the principles of this invention translational acceleration transducers are mounted and oriented in a rigid body so that 25they generate significant output signals. These signals are operated on so that undesired effects are suppressed and desired effects are accentuated.
Two general approaches to these -rnatters are identified which are quite independent of the kind of translationa] acceleration transducer employecl.

RIS:ms ¦

I l~ Z
These approaches have been called the Algebraic Method and the Trigono-¦ metric Method.
¦ The Algebraic Method and the Trigonometric Method provide a ¦ theoretical basis for embodiments of the invention. Such embodiments ¦ include a device for calibrating translational acceleration transducers, 5 ¦ a class of devices for generalized inertial measurements on vehicles ~ such as airplanes, ships and the like and a class of devices for generalized ¦ inertial measurements on spinning bodies, such as artillery projectiles.
¦ The generalized inertial measurements include the measurement of ¦ angular velocity as well as angular and translational acceleration. Hereto- lO
¦ fore, the measurement of angular velocity has required stored angular ¦ momentum, as in gyroscopes. The measurement of angular velocity ¦ described here depends only on kinematic effects and the sensing of ¦ translational acceleration at one or more points in a rigid body and does not require stored angular momentum. 15 The motion of a rigid body can be described by three vectors resolved onto a coordinate frame fixed in that body. In Fig. 1, i, j and k are unit vectors in the X, Y and Z directions of a right-handed cartesian coordinate frame. The three vectors are:
(1) a translational acceleration vector, A, associated 20 with the origin, whose components are a, b, c (1) Thus, A = i aO ~ j bo ~ k c O
(2) an angular velocity vector, Jf~,, whose components are p, q, r. Thus, Z5 51, = i p -~ j q ~ k r (2) 2r n s ~S~L32
(3) an an~ular acccleration vector,~, whose components are p, q, r. Thus (3 Ji = i p ~ j q + k r Generalized inertia'l measurement implies the measurement of the nine - components: aO, bo, cO, p, q, r~ p, q, 5 Note that the translational acceleration vector, A, is associated with a particular point in the rigid body, whereas the angular velocity ¦ vector, 3~, and the angular accelerat~on vector, ~, are associated with ¦ the entire rigid body.
0 The concept which is' developed here is that the rneasurement of the 1( ¦ nine components enumerated above can be accomplished by measuring ¦ translational acceleration components, ai, bi, ci at coordinates xi, Yi- Zi' ¦ However, a translational acceleration transducer will measure only one of ' the three components ti. e., ai, bi or ci) or a single 'resultant of the three.
In particular, the rneasurement of translational acceleration somponents ]
¦ suffiçes to measure angular velocity.
¦ The acceleration at a point in a moving coordinate frame has been ¦ given by L. Page, "Introduction to Theoretical Physics" D. Van Nostrand, Inc., New York, 1961. ' ` Ai = A * ~Q X (~x R;) ~ ~; x Ri (4) where A and 3~ are given by equations (1) and (2) and where Ri = i Xi + j Yi *
Ri being the position vector o~ a point in the coordinate frame.
The vector cross products in equation (4) are ZS ¦¦ ¦ i Yi i ¦

'' = i (Zi q - Yi r) + j (xi r ~ Zi P) + k (Yi P iq) (6 .

~ f)~/ ~ 8.

IS ms 1~ 3;~
Q X (`~ X Ri) = ~ i j k Ziq~ r Xir -~iP YiP iq = i (yipq - xiq - Xir ~ Zi pr~

* j (~irq ~ yir - YiP + i qP) ; + Ic (xi rp ~ ZiP ~ Ziq + Yi q 5 ~S~, x Ri = i ~ k ; xPi Yqi rZi ( i q Yi r) + j (~i r - ziP) ~ k (Yip - xi q ) t8) O Collecting terms in equations (1~, (6), (7) and (8) and expressing the results 10 in matrix orm gives ài = a + ~ ( 2 +r2) pcl - r pr ~ q ~xi-bi b - pq * r ( p2+r2;~ Yi (9) Ci c pr - q rq + P ~(P ~qJ i wher~ ai, bi and ci are components of the translational acceleration of a 15 point and xi, Yi and Zi are the coordinates of the point. Equation (9) agrees with a similar exp~ession given by A.G. Webster, "The Dynamics of Particles and of Rigid, Elastical Fluid Bodies", Hafner Publishing Company, lnc., New Yoxk, 1949.
However, not all the terms in this matrix equation are of the same 2( importancç in current art and practice. The terms which are Inost familiar are those due to translational acceleration o the origin and those which result from treating p, q and r as small quantities and ignoring their products and powers. The matrix equation which results from this usual ¦ engi ring treatment is .' 9.
.,.,.~,, b RLS:ms 9~ 3'A~

¦ b ~ Yi This equation is applied frequently in measuring the translational acceleration at a point which is not conveniently accessible (for example, 5 the center-of-gravity of an airplane engine).
The terms which are next in familiarity are the centrifugal terms, which involve the squares of p, q, and r. The matrix equation containing only these terms is:
, ~ ai ~ = (r ~ q ) o o i , 10 ;~ ¦ c ~ -(E +q ~1'5 il (Il) However, this equation affords some practical difficulties because the coefficients are sums of squares.
The least familiar terms are those which involve the products of the 15 angular velocities, i, e,, pq, qr, and rp.The matrix equation containing only these terms is:
ai = Pq pr xi -bi Pq qr Yi (12) - Ci pr qr o i 20 It is these terms which provide a basis for measuring angular velocity by means of measuring translational acceleration, This basis along with the more famili~r terms can provide a means for making generalized inertial measurements by measuring translational accelerations.
The generalized measurements require means for generating desired 1 25 effects and suppressing undesired ones, Two general methods for realizing these capabilities have been identified, They have been called the Algebraic _3285 S:~qls , ' .

~ 32 Mcthod and ~ne Trigonometric Method an~ they are described below. Tilcy can be employed singly or in combination.
l~he most direct method of generating the desired effects is to lnount translational acceleration transducers in a rigid body and to describe the motion of the rigid body, the coordinates of the points at which the transducer 5 are mounted and the translational accelerations at these points. These des-criptions are all resolved onto a single coordinate frame which is fixed in the body. The signals from the transducers, so mounted, can then be combined to isolate desired signals and suppress undesired effects.
Tbe Algebraic Method lC
The algebraic method can be described in terms of pairs of accelera-tion transducers, but its application is not restricted to pairs, and a generalization will be described later.
A pair of transducers can be mounted in a rigid body. The orientation of each member of the pair are identical as are two out of three posltion lc coordinates. The third position coordinates are equal in absolute magnitude, but opposite in sign~ The output signals from the two transducers are sub-tracted. For example, let transducers Number One and Number Two be oriented to measure the translational acceleration component, a. Let their position coordinates be 2 Xl = X X2 = X
- . Yl = Y Y2 = Y (13) Zl = +~ 12 Z2 = ~ 12 then al = a -(q *r ) x + (pq -r)y ~ (pr + q )~ 12 (14) a2 = a -(q +r ) x + (pq -r)y - (pr ~ q) ~ 12 (15) _3Z85 1 . :
:[S ,'nl s ¦
~t6~2 Consider that ihe desired effect is the product, pr, and that the other effects are urlwanted and are to be suppressed. Subtracting~

al -a2 = 2 ~ 12 (pr + q); pr ~ q 2~12 (16) In like manner, a pair of transducers can be oriented to measure translationa acceleration con ponent c and located at the following position coordinates 5 X3 = f~34 X4 = -~34 Y3 = Y Y4 = Y (17 . Z3 = Z Z4 = Z
then .0 c3 = cO + (pr ~q~ ~34 ~ (rq + p~ y -(p t q ~z (18) 1 C4 = c~ - (pr -q) ~ 34 + (rq ~ p~ y -(p * q )Z (19) and c -c4 =2~ 34 (pr q~; P 34 (20) Also, pr = 4a +-4 ~ (21) 15 a~Z c 3 c4 (22) This scheme can be extended to provide measures of the terms qr, pq, r and P
The translational accelerations at the origin can be measured by placing transducers at the origin oriented to measure the translational accele 2 retion components, a, b, c. This approach is mathematically correct, but physically impractical since physical transducers cannot occupy the same position. A practical approach is to employ pairs of transducers, properly oriented and positioned so that two out of three position coordinates are zero, ,5 the third coordinates are nonzero. Thus, if 2 x ' o X2 =
Yl = Y2 = o (23) 21 = ~ 1 Z~ =~2 ~2 L lZ.

. RIS:Ins ~
~ 32 l al - aO + (pr + q) ~1 (24) ¦ a2 = aO + (pr + q)~2 (25) ~Solving Lor a d pr ~ 1~ (Z6) 1 ~ (27) 5 In liks manner, if x3 = ~l3 x4 = ~4 O (28) Z3 =~ 74 =o Then 10 c3 = cO + (pr - q~3 (29) c4 = cO + (pr -q)a4 (30) Solving for cO and P qc ~ ~ ca~ ~3 c = ~ _ _ (31) 3 4 (3 2) 15 Finally , 1 1 a 2 c - c q 1 1 - 2 _ c3 _c4 (34) The scheme can be extended to measure b as well as qr, pq, p and r.
Note that the theory for the Algebraic Method irnplies transducers whos 20 output signals are proportional to a component of translational acceleration along a line through a point. Existing transducers approximate these charac-teristics, but it is not clear at this writing how precise these approximations can be The matter of calibration and errors is discussed later.

- RlS:,ns ~ 06~32 To surnmarize (1) The 19lgebraic Method describes the motion of a rigid body and l the acceleration at a point in that body as well as the coordinates ¦ of each point in terms of a single coordinate frame, fixed in the ¦ body, 5 1 (2) The Algebraic Method isolates signals proportional to pq, qr, rp, , ¦ q, r, a, b, c by placing translational acceleration transducers in an array in the rigid body and :combining their output signals.
l (3) Placing transducers in an array in a rigid body implies:
l (a) mounting each transducer at predetermined position coordinat 10 l (b) orienting each transducer so that its sensitive axis is aligned ¦ along a preferred direction.
The placement and combination techniques which have been described in detail I involve only weighted sums and differences of a rather simple kind. However, ¦ it is obvlous that more complex summations can be mathematically equivalent. 15 The Tri~onometric Method.
I .
¦ A less direct method is to mount translational acceleration transducer in a rigid body and to describe the coordinates of the points at which the l transducers are mounted and the translational accelerations measured at ¦ these points with ~espect to a coordinate frame fixed in the body. However, 20 ¦ the motions of the body are described with respect to a second coordinate ¦ frame, which will be described presently.
¦ Let the unprimed coordinate frarne already introduced be fixed in the ¦ body. This frame will be used to describe the coordinates of points in the ¦ body and the translational accelerations at these points. 25 ¦ An auxiliary coordinate frame (the primed coordinate frame) is intro-¦ duced and the motions of the body are described wi-th respect to this primed -3~35 lS:ms ~6~3Z

flan~e. These motions are a ', b ~, c ', p', q', r~, p', q', r'.
The auxiliary frame is selected so that its origin corresponds to that of the unprimed Irame and one of its axes coincides with an axis of the unprimed frame. However, the angular velocities of the two frames about the coincident axes are independent of one ano-her, It will be convenient to 5 select the coincident axes - so that their symbols are equivalent (i. e., X and X1, yand y1 or Z and Z~). The discussion presented here will be 1n terms of coincident X and X' axes, but the argument is general and applies to the other possible choices as well.
O The arrangement of the two coordinate frames is shown in Fig. 2.
The origins of the two frames are coincident. So are the X and X~ axes.
However, the angular velocities of the two frames are not necessarily equal (i. e., p and p' are not necessarily equal) and a relative angular displacement,0, res~lts. The magnitude of this angular displacement, 0, is 0 = 0 (o) +; (p - p~) dt ~i = p - p1 (35) 15 The motion components, resolved onto the primed coordinate frame, can then -be resolved onto the unprirned coordinate frame by means of the following transformations a = a 1 (1 b = b~ cos 0 + c10 sin 0 (36~ 2 c =~b1o sin 0~c10 cos 0 q = ql cos 0 ~ rt sin 0 r =-q1 sin 0 ~ r~ c06 0 q = q~ cos 0 * r' sin 0 ~ (p -p') ( ql sin 0 + rt cos 0) .r =lq~ sin 0 + r' cos 0 * (p - p~) (_ql cos 0 r~ sin 0~ (38) These expressions can then be substituted into the equation or the transla-tional acceleration derived previously. The result of this substitution is ~u~r~narized in the table below:

15. _ _1~ _____ s ms ~06113Z
...
Acceleratior~ Accel~ration Coel~icient of y lCo~f~ici~nt of y. Coe~ficien1: of z. C omp o ne rl t A t O r ig i n 2 2 _ _ :: ai al _(ql + r1 ) [ (2p-p')q~ -r] cos 0 {(2p-p')q' -rl]sin0 . . +[(2p-pl)rl+ql]sin 0 +[(2p p',~+q~] cos 0 :'': . .
bi O 0 (p'q~*r~)cos 0 2 _p 5 .. . + ~'r ~ in 0 ~12 ~ os 2 0 ~in 2 ::
~ ~ + qlr1 sin 0 + q1r1 cos 2 0 :~ .
,:, Ci -b'o s~n0 ~p'q~+r~)sin 0 P -~ +q1 * rl3 ~' ~ l l ~ p'r'-q~ c s 0 1 -~in Z 0 IL~c05 Z
o L +q~r1cos 2 0 -q~r~ sin 2 0 lO
It is convenient to separate the terms in this table into three subsets::
. (1) terms which are independent of 0 (2) terms which contain sin 0 and cos 0 ~3) terms which contain sin 2 0 and cos 2 0 This has been done and the results are presented in equations (39), (40) and 15 (41)-T~:rms independent of 0 ai- I = a~ + _(ql2+rlz~-. O Xi .
bi . _( 2+ , Z+ ,2) ~P Yi (39) Ci ' l 1 p _~Z+ j . Zi . 20 . ' : Terms containin~ sin 0 and cos 0 rai ~ . ' * [ (2~p' ) q' -r ' ] cos0 - [~Zp Oq' -r ~] sin 0 x.
b' ~[(2p-p')r'+q']sin p ~(2~P')r'~q']cc~;0 1 biocos0 (plq' + r ~ )c os 0 0 0 Y- 4 l ~c'Osin 0 +~ -q~)sin 0 1 ( o ¦ C -b~os1110 -~ +r~sin ~5 0 0 z.
l 1 . +cOcos 0 *~pr~-q~)cos0 . .

! _ _ ___I_ _ ' 32~5 S~ s ~36~.32 , Terms contailling sin 2 0 and cos 2 0 = I 1 ( ,2 r,2)coS 2 0 2 2 ~ _xi -b. O ~ ~(q' -r' )cos 2 0 y- ~41) ~q'r' sin 2 0 Tq'r' cos 2 01 - I L ~(q'2~r'2)sin 2 0 1(ql2-r~2)COS 2 0 Zi -q'r~ sin 2 0 ¦ - -~q'r' cos 2 0 These equations are quite general and do not involve any simplifying approxi- 5 mations. They separate the dynamic effects in a manner quite different from that described in the Algebraic Method.
The separation of terms afforded by this Trigonometric Method ( so called because of the role played by the angle 0 and the sines and cosines of 0 and 2 0) can be accentuated in the following manner 1 (1) The angular velocity; p, is made large, much larger than q and r.
(2) The angular velocity, p', is rnade small so that p'~ o .
Under these conditlons the terms independent o 0 become ~ero-frequency tern ls,those dependent on sin ,~ and cos 0 become single frequency terms (of frequenc Yf _ p/2~), and those dependent on sin 20 and cos 2 0 become double frequency terms .
The effect of a large angular velocity, p, is to translate certain effects to the frequency f = p/2~ and to arnplify the effects of the angular velocity components, q' and r', by a factor, 2p. The effects of the zero- 2 frequency terms and the single frequency terms can be separated readily by well-known signal processing techniques. The effects of the double-frequency terms can be separated by signal processing techniques and, in many cases can be ignored as negligible second order effects.
Z5 The separation of the single-frequency terms from one another can be 2 ~ accomp ~hed by the .Algebraic Method already de~cribed.
. ,, ~ 17. _ . RIS:Ins ~ 6~

To summarize:
I (1) The Trigonometric Method describes the coordinates of points ¦ and the translational acceleration at these points in terms of a¦ coordinate frame fixed in a moving rigid body (the unprimed coordinate frame?- 5 (2) The Trigonometric Method describes the motion of the moving body in terms of an auxiliary coordinate frame (the primed coordinate frame).
(3~ The origins of the primed and unprimed coordinate frames coincide. So do a pair of coordinate axes (one from each frame) lO
However, the angular velocities about the coincident a~es are not necessarily equal and a relative angular displacement, 0, results.
t4) The translational accelerations at points in the body ~unprimed coordinate frame) can be described by three sets of terms:
(i) terms independent of 0 ~zero-frequency terms) 15 (ii) terms containing sin ~ and cos 0(single-frequency terms) (iii) terms containing sin 20 and cos 20 (double-frequency terms) (5~ The zero-frequency, single-frequency and double-frequency terms can be separated from one another by well-known signal processing techniques. The single frequency terms can be 20 separated from one another by the Algebraic Method.
An ideal translational acceleration transducer should have the followin~
characteristics:
(1) It should have an origin which possesses the properties of a mathematical point. 25 (2) It should have a sensitive axis, which is a straight line, passing through its origin.

J~3285 ~ ~,'j;11~S
~6~1L132 (3) l;S sensitivity in physical unit~ of output signal per physical ~Init of translational acceleration should be finite and constant in the direction of its sensitive axis and zero ~or all directions normal to the sensitive axis, An ideal transducer would be subject to the following errors in being applied S
in either the Algebraic Method or the Trigonometric Method.
(l) a position error, whose components are ~xi, ~yi.dzi (2) an orientation error, due to angular rotations of its sensitive axes whose components are d0i, dei, d~)i O (3) an uncertainty in its sensitivity of,~ ki. lO
The effect of these errors on the generalizecl inertial measurements can be analyzed in a straightforward manner.
Howe~er, physical transducers do not have these ideal characteristics.
Moreover, the state of the transducer art does not provide definitive data on !5 ' the characteristics of physical transducers. This is due to limitations in the 15 current art of calibrating transducers.
The equations developed above for the Trigonometric Method are applied to the design of a calibration device.
Referring now to FIGS. 3A-3C, there is illustrated thereby a device which can be used for generating oscillating linear acceleration components 20 by means of angular velocities, and thus, employed in the calibration of translational acceleration transducers. The basic elements include a face-plate 10 which is mounted on a spin-motor 12 for rotation thereby. The plane of face-plate 10 is normal to the axis of rotation 14 of spin motor 12. The transducer 16 is mounted on face-plate 10 so that its sensitive axis is normal 25 to face-plate lO and parallel (but not coincident) to the axis of rotation 14 thereof. Commercial transducers such as translational accelerometers often have a threaded fitting which can be mounted through a hole in the face-plate :1~ ~9.

RIS :m s ~ 32 and fastened with a nut, however, any fastening means will suffice The assembly, consisting of face-plate 10, spin-motor 12 and transducer 16, is ~ :mounted on a turntable 18, using bolts or other fastenerL s, which is rotated bya yaw-motor 20 which is supported by a frame (not shown). The rotation axis 14 or spin-motor 12 and the rotation axis of the turntable 20 intersect at an 5 angle of 90 .
The primed coordinate frame of the previous discussion is associated with turntable 18. The Zl-axis coincides with the yaw-axis and the X~-axis coincides with the spin-axis, The unprimed coordinate frame is associated with face-plate 10 and 10 orthogonal Y and Z axes may be inscribed on the face-plate. The X axis coincides with the spin-axis.
The yaw axis lies in the piane of the face-plate and the origins of the two coordinate frames are coincident.
Transducer 16 is mounted on face-plate 10 at the following coordinates 15 ., Xi = ~

Yi= ~ (42) Zi ~ J
The following magnitudes are assigned to the dynamic variables p' = q~ = r' = p = 0~ 20 p ~ 0 ~ a ' = b' = 0 ql .= O ~ Cl ~
r~ ~ 0 In other words, spin-motor 12 spins at constant angular velocity, p, and yaw-motor 20 rotates at constant angular velocity, r'. Equations (39), (40), and 25 l (41) become:

D - 3 2 ~ 5 :
' ~ IS ~ S
~ 32 Terms inde~ ndent ,.f 0 bi = 2 (44) Ci = Zi (P + Z
Terms ~vhich contain sin 0 and cos 0 ~ -ai = 2 pr~ :~i cos 0 bi = c ' sin 0 i (45) Ci =co COs ~
Terms which contain sin 20 and cos 2 ~
_ .
ai = ' ' 1( bi = 2 Zi r sin 2 0 (46) Ci = ~ Zi r' cos 2 lZ ~
If transducer 16 is oriented with its sensitive axis normal to the face-place, it will Ineasure ai, where ai = 2 pr' Zi cos 0 (47) ]
and this will be a sinusoida1 signal with Irequency equal to p/2~ Thus, the frequency can be controlled by setting the spin rate after which the amplitude of excitation can be controlled by setting r'. A typical magnitude of excitatio which might be generated is p = 1570 sec (15, 000 RPM~
r' _ 1. 75 sec (100 deg/sec) Zi' = 1.,0 2 pr' Zi = 2 x 1570 x 1. 75 x 1/12 = 457. 9 i~t z = 14. 22 g~s sec The entire assembly may be tilted to provide a different orientation with respect to gravity.
The principle employed in the calibration device shown in FIGS. 3A-3 can be applied in a device which gives an approximate measure of angular velocity. Such a device i8 illustrated in FIG. 4. It consists of a case 22 - 3 ;~ 8 5 IS m s ~196~3;;~:

containing a spin-motor 2~, a ace-p]ate 26 an d a translational acceleration transducer 28, The transdllcer is lnounted on face-plate 26 with its sensitive axis normal to the face-plate, If case 22 is turning at a constant angular velocity the measurement thereof will be exact. Only when the case is turning at an accelerated or 5 decelerated rate will the measurement be approximate, The prilned coordinate frame is associated with the case and lines ar inscribed on the case to indicate the location of the coordinate axes, The unpri[ned coordinate frame is associated with the face-plate, The Y and Z axes may be inscribed on the face-plate and the X-axis coincides wit 1( the spin-axis.
The origins of the two coordinate frames are coincident, The transducer is mounted on the face-plate at the following coordinates Xi = ' lc ~i = (48) . ' ' ' Zi ~ ' - ' and the spin-motor is rotated at constant speed so that , p - p' = constant (4g) The motions of the case are p', q', r', p', q', r', a, b, c .
The signal generated by the transducer is ai = aO'~ Zi[( 2p -p')ql-r'] sin 0 +Zi [~P -p')rl+q'~ cos 0 (50) when 2 p)~ p' and r',,~" 0 and q'~ 0, this becomes ai = aO + 2PZi (-q~ sin 0 + rl cos 0) (51) This signal contains a term which is independent of 0 and terms which contain sin 0 and cos 0, . .

_~ ~

IS:~ns ,., ~)6~.3;~
Thus, a sil~gle instruInent consisting of a case, a spin-rnotor, a face-plate and a transducer can provide approxirr-ate measures of q' and r~., There are many alternative lnethods of processing the output sigllals to separate the term independent of ~ from the terms which contain sin ~5 and cos 0. mese will becc~ obvious to those s~illed in the art, 5 The approxiInations made in this instrument can be corrected in a manner which is described hereinafter.
A common application of acceleronleters is the measurement of trans-lational and angular acceleration, These measurements can be accomplished by spinning translational acceleration transducers embodied in a device which 10 is illustrated in FIGS. 5A-5C.
The devlce consists of a case 30 containing a spin-motor 32 and an extended axle 34 for mounting two transducers 36, 38 on the spin-axis, The transducers are mounted so that their sensitive axes are parallel to one another and normal to the spin-axis, 15 The primed coordinate frame is associated with the case and lines are inscribed on the case to indicate the location of the coordinate axes.
The unpri~rled coordinate frame is associated with the spin axis. The X-axis coincides with the spin axis and the X, Y and Z axes are ~mtually ortho-~0 gonal. 20 The origins of the two coordinate frames are coincident.
Transducers 36 and 38 are assigned the following coordinates:
38 36 ' x2= ~x X3= -x ;5 Y2= Y3 = ~52~ 2 22 = Z53 =
The transducers are oriented æo that their sensitive axes are parallel to the Z axis, thu6, they meaæure cz and C3.

,' - 23.
_~ _____ ~85 ;:ms jj 3~:
The transducer output signals are cz = -b' sin 0 ~ c' cos 0 + x [-~p'q'+rl) sin 0 + (p'rl-q)cos ~] (53) - c3 = -b' sin 0~c'O sin ~-x[-(p'ql+r')sin 0 ~- ~p'rl-q') cos 0]
Rearranging equations (30) b: sin 0 + c~O cos 0 = 2 (54) L

-(p'q~+r') sin 0 + (p'rl-q') cos 0 = 2 3 (55) Usually p', q', r' are small quantities whose products can be neglected.
Neglecting p'q' and ~'r' in (55) gives .
- r' sin 0 -q' cos 0 = 2x (56) `` 10 The magnitudes o bl, cl, q', r' can be sampled at appropriate rnagnitudes 1 of sin 0 and cos 0, or by other signal processing techniques. Thus, a single instrument consisting o a case, a spin-motor and two translational accelera-tion transducers mounted on the spin-axis can provide measurements of the coInponents of translational acceleration, b' and c', and of the components of angular acceleration, q' and r'. 1 The two devices, illustrated in FIGS. 4 and 5, can be combined in a single device, which is illustrated in FIG. 6. This combined device provides p', q', p', r', b', and c' . It also eliminates the approximation errors incor _ porated in the device illustrated in FIG. 4.
The combined device consists of a case 40, a spin-motor 42 and an extended axle 44 for mounting three transducers. One transducer 46 is mounted so that its sensitive axis is parallel to the spin axis. The other two transducers48 and 50 are mounted so that their sensitive axes are parallel to one another and normal to the spin-axis. The sensitive axes of the three transducers are coplanar.
The primed coordinate frame is associated with the case and lines are inscribed on the case to indicate the location of the coordinate axes.
The unprimed coordinate frame is associated with the spin axis. The . 24.

__ _ ~

I.S~ s ., X-axis coincides with th~ spin a~is and the X, Y and Z axes are mutually o rthog ona 1.
Transducers 46, 48 and 50 are ~ssigned the Iollo~ing coordinates:
46 50, 48 Xl = X2 =x x3 = -x Yl= o Y2 ~ Y3 = ~ , Zl = z Z2 = Z3 =

Transducer 46 is oriented so that it measures al. Transducers50 and 48 are .
oriented so that they measure c2 and C3, respectively.
10The transducer output signals are . 10 al = a -z [(2p-p')qt-rl] sin 0 ~z ~(2p -p~ r~+q'] cos 0 (573 cz = b~ sin 0 + c~ c06 0 + x[-(p~q~+r') sin 0+ (p'r~ cos 0~ (58 C3 = -bl sin 0 ~c~ cos 0 -x~-(p'q~+r') sin 0+(p'r'-g')cos 0] (59) . All of the terms involve sin 0 and cos 0 except a, which can be suppressed by a high pass filter. The terms can be combi:ned in the following manner 15 -b~ sin 0 * c~ cos 0 = 2 * 3 -(p'ql+r~ sin 0 + (p~rl -q') cos 0 = z 3 a -[(2p -p') q'-r'] sin 0 ~ [(2p-p )r +q ] cos 0 = z -qlsin0* r' cos 0=Zp LZ * zx J

20and neglecting p~ q~ and ~ r~ 2 -r' sin 0 - q' cos 0 = 2 - 3 6ummarizing 3 -b~ sin 0 * c~0 cos 0 = 2 (60) -r~ sin 0 - q cos 0 = z C2 C3 (61) 2~i -q~ 6in 0 + 1~ c08 0 = 2 ( I + ~ ) (62) The magnitudes o b~, c', q', 'r', q' and r' can be sa~npled at appropriate . ~ , ,1 .~, ;. Z5.
__ ~

RIS:ms 3~96~ 3~
magnitudes of sin 0 and cus 0, or by other signal processing techniql~es.
Thus, a single instrument consisting of a case, a spin motor and three translational acceleration transducers can provide measurements of bl and c~ components of translational acceleration ~1 and r~ components of angular acceleration 5 q' and r1 components of angular velocity The three devices illustrated by FIGS. 4, 5 and ~> are motion sensors of a rather general type. They incorporate measurements of angular velocity which heretofore have been accomplished by means of gyroscopes. The principles of the gyroscope and that of measuring angular velocity by ~neans 10 of transducers are both based on the same laws of acceleration.
The gyroscope and the transducer arrays described here are mathe-matically equivalent but they are physically different, The gyroscope depends on angular momentum, which is made large so that certain physical effects (such as those due to friction~ are suppressed. The transducer scheme depends 1-onl~ on kinematic effects and the measurement of accelerations at one or more points.
The several schemes described above can be applied to spinning projectiles. The unprimed coordinate frame is associated with the body of the projectile and the transducers are mounted in the body. The body 20 becomes the colmterpart of the spinning coordinate frame in such application.
It is convenient to conceptualize a nonspinning (primed) coordinate frame, whose origin and X' axis coincide with the origin and X-axis of the unprimed frame. This primed coordinate frame has no physical counterpart, but it is a useful theoretical construct. It has the further advantage of being Z5 consistent with the theory already presented.
Two modifications of the schemes which have been described are required for app]ications in spinning projectiles. These are (1~ The spin-rate o~ the projectile must be measured in order to evaluate q ' and r' from 2pq' and 2 pr'.
(2) Certain precau~ions may be required in order to compensate fox uncertainties in the location of the spin axisn These modifications can be incorpoxatea in a scheme which employs five accelerometers. This scheme is illustrated in FIG. 7.
Five accelero~eters 52, 54, 56, 58 and 60 are mounted on a spinning projectile which requires a rate sensor to measure pitch and yaw. Conventional gyroscopes are unsatisfactory in that in addition to being large, they are difficult to gun harden. Since the projectile itself is spinning, it acts as the spin motor of the earlier descxibed embodiments. The transducers are assigned the following coordinates and orienta-tions.
Accelero~eter x y z Orientation Number ,, ~
52 1 0 0 Zl a 54 1 0 0 z a ~0 56 x 0 0 I c 58 -x 0 0 1 c 0 0 z I c ~ __ ~ ~ .
The transducer output signals are al = zl[-(2pq'-r~) sin 0 + (2 pr'+q'~ cos 0]
a2=z2[-(2pq'-r') sin 0 + (2pr'~q') cos 0]
c3=-b'o sin 0 + c'O cos 0 + x(-r'sin 0 ~q' cos 0) c4=-b'o sin 0 + c'O cos 0 -x(-r' sin 0 -q' cos 0) "~

3285 l l :rn~

~ c5 = _~j (p2 ~ q~ + r~ ~ ~
These signals can be combined yielding the following results : -b~ sin 0 ~ cl cos 0 = 2 ~63) -ql cos 0 -r~ sin 0 = 3~- 4 (64, __ _ -q sin ~1 + r~ cos 0 = 1 ( I ~ 2 ~ c3 _ c4 ) (65) P=~ i (66j In such an application the sign of p is known and only the magnitude needs be dete rmined.
It should be noted that in equation (65) the difference, Zl ~ Z2 must be known, rather than either coordinate separately. Thus, the measurement of angular velocity does not depend on an accuLate knowledge of the location of the~ spin axis, Transducer 60 can be deleted if the spin rate is known or can be measured by other means.
While I have described above the principles of my invention in connec-tic~n with specific apparatus, it is to be clearly understood that this descriptio~
is made only by way of example and not as a limitation of the scope of my invention as set forth in the accompanying claims.
'.' .'' '`

~ ~ 28.

__ ~

Claims (21)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
1. A method of inertial measurement which comprises the steps of:-providing a single transducer, for measuring accel-erations at a point, rigidly mounted at said point and oriented so that a sensitive axis of the transducer is substantially parallel to and spaced from a first axis; and rotating said transducer about said first axis.
2. A method as claimed in claim 1 which further comprises rotating said transducer about a second axis orthogonal to said first axis simultaneously with the rotation about said first axis.
3. A method as claimed in claim 2 in which said trans-ducer comprises a translational acceleration transducer.
4. A method of generating a translational acceleration at a point, comprising a method of inertial measurement as claimed in claim 3, in which:-said transducer is rigidly mounted at said point on a support so that said sensitive axis is spaced a distance Z
from said first axis; and said transducer and support are rotated at a constant angular velocity p about said first axis and at a constant angular velocity r' about said second axis, whereby the translational acceleration generated at said point is a sinusoid the amplitude of which is 2pr'z and the frequency of which is p/2.pi..
5. A method of calibrating a translational accelera-tion transducer, comprising a method of inertial measurement as claimed in claim 2, in which a translational acceleration trans-ducer to be calibrated constitutes said transducer for measuring accelerations at a point and is rigidly mounted at said point on a support for rotation with said support about said first and second axes.
6. A method of measuring angular velocity about two axes, comprising a method of inertial measurement as claimed in claim 1, in which said transducer comprises a translational acceleration transducer rigidly mounted at said point on a support, said transducer is rotated with said support about said first axis, and said first axis is orthogonal to said two axes, whereby the transducer produces an output which is a measure of the angular velocity about said two axes.
7. A method of measuring angular velocity, angular acceleration, and translational acceleration with respect to two axes, comprising a method of inertial measurement as claimed in claim 1, in which said first axis is orthogonal to said two axes and said transducer comprises a translational acceleration transducer rigidly mounted at said point on a support, further comprising providing second and third translational accelera-tion transducers rigidly mounted on said support at respective points on said first axis with sensitive axes of the second and third transducers orthogonal to said first axis, said support and all of said transducers being rotated about said first axis whereby the transducers produce outputs which are a measure of said angular velocity and angular and translational accelera-tions.
8. A method of measuring as claimed in claim 6 or 7 in which said two axes are orthogonal to one another.
9. Apparatus for use in inertial measurement, com-prising:-a single support;
means for rotating said support about a first axis; and a transducer rigidly mounted at a predetermined point on said support for measuring accelerations at said point, the transducer being oriented so that a sensitive axis thereof is substantially parallel to and spaced from said first axis.
10. Apparatus as claimed in claim 9 wherein said transducer comprises a translational acceleration transducer.
11. Apparatus as claimed in claim 10 and further comprising a second support, on which said means for rotating the first-mentioned support is mounted, and means for rotating said second support about a second axis which is orthogonal to said first axis.
12. Apparatus as claimed in claim 11 wherein the means for rotating the first-mentioned support and the second support are first and second motors, respectively.
13. Apparatus as claimed in claim 12 wherein said first motor has a shaft to which the first-mentioned support is fixed for rotation therewith.
14. Apparatus as claimed in claim 11, 12 or 13 and further comprising means for tilting said first-mentioned and second supports, and said means for rotating said supports, about a third axis which is orthogonal to said second axis.
15. Apparatus as claimed in claim 11, 12 or 13 for calibrating at least one translational acceleration transducer, in which said transducer is a translational acceleration trans-ducer to be calibrated.
16. Apparatus as claimed in claim 10 for measuring angular velocity, angular acceleration, and translational acceleration with respect to two axes with respect to which said first axis is orthogonal, the apparatus further comprising second and third translational acceleration transducers rigidly mounted on said support at respective points on said first axis with sensitive axes of the second and third transducers orthogonal to said first axis.
17. Apparatus as claimed in claim 16 and further comprising a fourth translational acceleration transducer rigidly mounted on said support with a sensitive axis thereof parallel to and spaced from that of the first-mentioned transducer and said first axis.
18. Apparatus as claimed in claim 10, 16, or 17 wherein said means for rotating said support comprises a motor having a shaft to which said support is fixed for rotation therewith.
19. Apparatus as claimed in claim 10, 16, or 17 and further comprising a case containing said support, said means for rotating said support, and said transducer, said means for rotating said support being fixed to said case.
20. Apparatus as claimed in claim 10, 16, or 17 wherein said means for rotating said support comprises a spinning body to which said support is secured and which spins about said first axis.
21. Apparatus as claimed in claim 10, 16, or 17 wherein said means for rotating said support comprises a spinning body to which said support is secured and which spins about said first axis, the apparatus further comprising means for measuring the spin rate of the spinning body.
CA233,895A 1974-11-29 1975-08-21 Apparatus for performing inertial measurements using translational acceleration transducers and for calibrating translational acceleration transducers Expired CA1061132A (en)

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DE3828968A1 (en) * 1988-08-26 1990-03-01 Bayerische Motoren Werke Ag Device for testing acceleration sensors
JPH0833408B2 (en) * 1990-03-29 1996-03-29 株式会社日立製作所 Angle detection device, translational acceleration detection device, and vehicle control device
US6904377B2 (en) * 2003-03-17 2005-06-07 Northrop Grumman Corporation Method for measuring force-dependent gyroscope sensitivity
DE102005025478B4 (en) * 2005-06-03 2007-04-19 Albert-Ludwigs-Universität Freiburg Method and device for determining the relative position, speed and / or acceleration of a body
JP5346910B2 (en) * 2010-11-24 2013-11-20 株式会社ソニー・コンピュータエンタテインメント CALIBRATION DEVICE, CALIBRATION METHOD, AND ELECTRONIC DEVICE MANUFACTURING METHOD
KR20120063217A (en) * 2010-12-07 2012-06-15 삼성전기주식회사 Apparatus for apply multiaxial-inertial force
JP5816900B2 (en) * 2011-10-04 2015-11-18 多摩川精機株式会社 Gyro characteristic measurement method using 2-axis orthogonal double turntable
JP6548218B2 (en) * 2015-06-23 2019-07-24 国立大学法人東京工業大学 Multi-axis gyro sensor characteristic evaluation apparatus and method
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US3015946A (en) 1957-01-28 1962-01-09 Boeing Co Device and method for producing low rate angular acceleration
US3071975A (en) * 1961-02-27 1963-01-08 Percy F Hurt Accelerometer
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GB1530993A (en) 1978-11-01

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