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ADVISORY COMMITTEE ON ARTIFICIAL LIMBS NATIONAL RESEARCH COUNCIL DETERMINATION OF ACCELERATION BY USE OF ACCELEROMETERS BY N.J RYKER AND S.H. BARTHOLOMEW PROSTHETIC DEVICES RESEARCH PROJECT INSTITUTE OF ENGINEERING RESEARCH UNIVERSITY OF CALIFORNIA SERIES I! ISSUE I7 BERKELEY ECEIVE JUL 16 2008 THE GEORGE E. BRO LIBRARY WN, JR SEPTEMBER 195!
QP303 .R95 1951 ¢.1 Determination of acceleration by use of accelerometers /
CONTENTS Introduction Fage Basic Requirements Ll Description of Equipment | | 3 Use of Equipment to Measure Shank Accelerations ly Comparison of Accelerometer and Grapho-Numerical Differentiation Methods 18 Figures l. 26 36 he Se 6. Te 8. 9. Tables le Ze Operation of an Unbonded Resistance Wire Linear Accelerometer Wiring Diagrams for Combining Outputs of Two Accelerometers Wiring Diagrams Using Strain Analyzer Angular Accelerometer Dynamic Calibration Apparatus System Output vs Square of Input Frequency Dynamic Calibration of Angular Accelerometer Method of Mounting Accelerometers Typical Accelerometer Test RunâNormal Subjectâ Level Walking Typical Accelerometer Record of Prosthetic Shank During Swing Phase of Level Walking Comparison of Static and Dynamic Calibration Comparison of Accelerometer and Grapho-âNumerical Methods
Intraduction In studies of Human Locomotion, linear and angular accelerations have for some time been determined by differentiation of timeâdisplace=â ment data. The problems encountered arise in obtaining satisfactory data and in graphically or numerically operating on this data to obtain second derivitives. A comprehensive study of the various differentiation techniques has been made and results published in a separate project report. The present report presents a method whereby linear and angular accelerasions are measured directly by the use of accelerometers and compares this method with one of the more reliable differentiation procedures. The procedure reported below is essentially the one followed by the project after considerable experimentation. Although deviation from this procedure usually caused difficulty, it is not suggested that the course followed was the only satisfactory one. It is hoped, however, that it may at least serve as a guide for others interested in obtaining reliable dynamic measurements.
Basic Requirements The dynamic quantities investigated were the angular acceleration of the shank of a normal or prosthetic leg and one component of the linear acceleration of a point on the shank. In order to determine definite requirements to be met by the measuring and recording system it was necessary to consider factors common to dynamic measurements in general. These factors and the bearing of each on this particular investigation are as follows: le Range and Sensitivity Requirements Prior studies of the lower leg motion pattern indicated that angular accelerations as high asf100 radians per secondâ could be anticipated. Similarly, linear accelerations were not expected to exceedt75 ft. per second. It was assumed that a variation of not more than 5 percent of the maximim was the smallest value that would be considered significant in each casee Hence it was required that the measuring and recording system: ae Have a range in the measured angular acceleration of +100 radians per second d be sensitive to a change of +5.0 radians per second . be Have a range in the measured linear acceleration of +75 ft. pey secondâ and be sensitive to a change of + ft. per second . 2 Frequency Response If the angular and linear acceleration pattern were represented as a Fourier Series, the dominant components in such a series probably would not occur at frequencies exceeding 30 cps. Hence the measuring and recording system as a whole should respond accurately to an input whose frequency may vary up to 30 cps. 3e Extent of Information and Accuracy Required A contimoous record was desired. A value considered logical for the maximum permissible error was 1 1/2 times the least count or 7 1/2 percent of the required ranges Distortions in the general shape of the record to be as small as possible.
Ze 4. Size of Pickup and Method of Mounting Relatively speaking, the shank of a normal or prosthetic leg is quite a small "structure," in regard to both weight and size. Only a small, light pickup could be mounted on the shank without materially affecting the normal acceleration pattern. The amplifying and recording instruments must be placed elsewhere. The attachment must be rigid and yet not depend on nailing, screw-â ing, or bolting the pickup to the shank. 5. Ease and Precision of Calibration The calibration procedure must be simple and reliable. 6. Special Conditions Determinations to be made indoors or outdoors during moderate weather. There were no extreme climatic or other adverse physical conditions anticipated. 7e Initial Equipment Cost The initial cost of required apparatus was an important factor but was considered in the light of the value of the information gained and the possible advantage of great economy in obtaining and reducing test records. 8. Time and Expense of Record Reduction The expense and time involved in reducing the test records was considered very important. Hence a record that could be quickly interpreted was desirable. Attention will now be directed to a detailed description of the measuring and recording system used in shank angular and linear accel= eration determinations and to the tests that were made at the University of California to verify that this system was satisfactory in view of the criteria established above.
36 Description of Equipment The equipment selected consisted of three Statham linear accelerometers, two type AP and one type C, two Brush Development Company Model BL-310 Strain Analyzers, and one Brush Development Company Type BL-202 two-channel Brush Recorder. At the inception of the testing program, the type AP accelerometers were on hand and had adequate frequency response and sufficient sensitivity. The type © accelerometer which was ordered later in the program had a lower natural frequency but greater sensitivity than the type AP accelerometers and was better adapted to this use. The Strain Analyzers and Brush Recorder were also initially on hand. The manufacturer's descriptive literature indicated that their performance with the Statham accelerometers would be satisfactory. Accelerometer Operation In order to present an intelligent detailed description of the entire measuring and recording system, it will first be necessary to consider the basic features of individual accelerometer operation. linear Accelerometer: = The Statham Model AP linear accelerometer is a force sensitive pickup of the unbonded resistance wire strain gage type. The sensitive element of this instrument is a small weight suspended by four wires, each of which is an unbonded resistance wire strain gage. (Each of these four wires is in reality a loop of several turns but for simplicity may be thought of as a single wire). The operation of the accelerometer may be schematically represented by Fige le A free body diagram of the instrument at rest with the sensitive axis in a horizontal position is shown in Fig. l=a, and the corresponding internal strain gage bridge is shown in Fig. l=b. Also, the free body diagram and strain gage bridge are shown in Fig. l-c and l-d fora case in which the accelerometer is moving with a linear acceleration a in the direction of the sensitive axis which is inclined to the horizontal by an angle 6.
he The instrument is constructed so that when at rest in a horizontal position, the tension in each of the suspending wires is the same and each wire is of equal lengthe Hence each arm of the internal bridge has the same resistance R; the bridge is then in balance and no current flows through the recorder. When subjected to the motion depicted in Fige l-c, the operation of the accelerometer is as follows: 1. The weight Wis subjected to a force F acting as shown with a magnitude given by: Wa F = W 6+â sin 6 + g Eqe (1) Rearranging terms, a+gsin® =F (§) Eq. (2) 2. Each of the four suspending wires contributes equally in resisting the force F. The tension and wire length is diminished in wires (1) and (3), and increased in wires (2) and (). The corresponding bridge resistances are decreased and increased, respectively, by an amount MR which throws the bridge out of balance and causes a current to flow through the recorder, This current, or bridge output, is proportional to the force F and therefore proportional to the quantity F(t). 3. By examination of Eq. (2), it is seen that the bridge output is proportional, not to the true acceleration a, but to an apparent acceleration a, equal to a + g sin @. Therefore, the output of a single linear accelerometer bridge, which is the quantity recorded, is not in general a measure of the true acceleration but of a quantity that must be corrected by the component of the earth's gravitational field parallel to the sensitive axis of the accelerometer. Vhen the instrument is in the position shown in Fige l-c, g sin 6 mst be subtracted to yield the true accelerat- ion, and when oriented with the opposite inclination, the g sin 9 correction must be addede In the special case when the sensitive axis is horizontal no correction is necessary since the g sin 0 term then vanishes.
De In view of the above explanation, it is apparent that the bridge output must be calibrated against some known linear acceler= atione The necessary calibration procedure will be discussed later. A further point to be kept in mind, is that if a single accelerometer is subjected to some general acceleration, only that component of the acceleration parallel to the sensitive axis may be determined. The preceding explanation and description applies to the Statham type AP accelerometer... The same treatment will not be given for the Statham type C accelerometer, because its operation is similar although the mechanical details are quite different. The intelligent use of linear accelerometers depends to a large extent on a clear understanding of the physical quantity that is measured. This is true because, in addition to the determination of true linear acceleration, these instruments may be used in applications where the desired quantity may be obtained without the intermediate step of reducing the output of each individual accelero- meter to the true linear acceleration. An example of this type of application is the determination of angular acceleration by the use of two linear accelerometers. Angular Accelerometer: ~ The angular acceleration o¢ of a rigid body moving in space may be determined from the difference in linear accelerations of two points on the body, A and B, according to the expression: Coc = Ay = 7h Eqe (3) where, oc = angular acceleration in radians per secondâ. âAy - acceleration of point A in a,direction perpendicular to line AB in ft. per second ,. âBy ~ acceleration of point B in a direction perpendicular to line AB in ft. per second$. L «= distance between A and B in feet. If linear accelerometers are placed at points A and B with the sensitive axes normal to the line AB their apparent accelerations an are 3
6. an, = âAy * g sin @ =C (output of bridge A) Eq. (4) âny = âBy + gsin®=C (output of bridge B) Eq. (5) Where C is the linear acceleration calibration constant. By subtraction, aaa a mo- m= Ayo By . output of _ output of C lbridge A bridge B | Eqe (6) and by substitution in Eq. (12), .f output of â output of I bridge A bridge B Eqe (7) ct output of output S| bridge A = bridge B Where C' is the angular acceleration calibration constant. The subtraction indicated in Eq. (7) may be done electrically by wiring the bridges so that the individual bridge outputs are combined into a single output equal to their difference. The bridge wiring diagram with the two accelerometers so connected is shown in Fige 2âb. The combined output, unlike the output of a single linear accelerometer, is a direct measure of the angular acceleration e¢ and no correction due to inclination is necessary. Measuring and Recording System Angular Acceleration: â An angular accelerometer consisting of two Statham type AP linear accelerometers of matched sensitivity and frequency response characteristics was used as the angular acceleration pickup. The outputs of the two AP accelerometers were wired to subtract and fed by a four conductor shielded cable into a Brush Development Company model BL-310 Strain Analyzer. A schematic wiring diagram of this hookup is show in Fig. 3-c. It should be noted that in this arrangement, the a=-c power supply to the accelerometer bridges is supplied by the strain analyzer unit and that the actual measuring bridge
Te contains four accelerometer strain gage resistances and two resistances provided internally in the Strain Analyzer. Two arms of the strain gage bridge in each accelerometer were removed from the circuit by not connecting one of the output leads on each accelerometer. This was necessary because the two internal resistances of the model BL-310 Strain Analyzer could not be removed from the bridge circuit. The output of the angular acceleration dynamic pickup was amplified by the Strain Analyzer and fed to a Brush Development Company Model BL-202 two channel Brush Recorder. This recorder operated at a paper speed of 25 mme per second and produced on one of its channels a continuous record of angular acceleration. Linear Acceleration:-A single model C accelerometer was used as the linear acceleration dynamic pickupe The accelerometer output was fed by four conductor shielded cables to another model BL-310 Strain Analyzer. The bridge power supply was again provided by the Strain Analyzer, and as the actual measuring bridge contained two resistances provided by the Strain Analyzer, one of the leads on the accelerometer was left unconnected. This arrangement with the original measuring bridge (Fige 1=-b) coupled with the Strain Analyzer is shown in Fig. 3-a. Range and Sensitivity The measuring and recording system described above was found to have an adequate combination of range and sensitivity. The gain on the Strain Analyzers was easily adjusted so that values of t75 ft. per secondâ linear acceleration and+100 radians per second angular acceleration were indicated on the Brush Recorder record. Corresponding to these values, the least readings on the record weret2 ft. per secondâ andtS radians per secondâ, respectively. Frequency Response In order to insure that the response of the measuring and record ing system was "flat," or linear, within the prescribed frequency range (up to 30 cps), supplementary tests were made. The accelerometers were rigidly fastened to a vibrating table which was driven by a synchronous
8. motor so that the table top moved vertically with simple harmonic motion. The amplitude and frequency of the displacement were adjustable so that a great range in magnitude of vertical linear acceleration could be obtained. The amplitude of the resulting simusodial acceleration pattern could be calculated for each particular displacement and frequency adjustment. During the test, the measuring and recording system was the same as that used during shank acceleration test runs except that the type AP accelerometers were wired to add outputs. The wiring diagram for this case is shom in Fige 3âbe A table displacement of 0.048 in. was used and the frequency varied from about 10 cps to 0 cps. At each frequency an acceleration run was made and a record obtained on the Brush Recorder. Since the maximim acceleration in simple harmonic motion is a linear function of the frequency squared, the peak deflections on the Brush record were plotted against the square of the frequency. The results are shown in Fige 5<a where it may be seen that the combined output of the type AP accelerometers varied linearly with the square of the excitation frequency up to a value of 1300 cps*. Therefore the response of the measuring and recording system was linear or "flat" for frequencies as high as 1300 = 36 cps. Above this frequency the output of the system departed from a straight line variation with the square of the frequency. This test established the upper limit of the flat frequency. response range as 36 cps, a satisfactory value in view of the required upper limit of 30 cps. Calibration Procedures Static Calibration - Linear Accelerometer: = The Model C accelero- meter was placed at rest in a position such that the sensitive axis was vertical. The Strain Analyzer was balanced so that the bridge output was zeroe Then the accelerometer was inverted still keeping the sensitive axis vertical. The bridge was then out of balance by an amount corresponding to 2 x g or 6h; feet per secondâ.
