I. Biomechtmics, 1973, Vol. 6, pp. 729-736. Pergamon Press. Printed in Great Britain
`
`TECHNIQUE FOR THE
`ACCELEROMETRY-A
`MEASUREMENT OF HUMAN BODY MOVEMENTS*
`
`J. R. W. MORRIS Nutheld Department of Orthopaedic Surgery, University of Oxford, and Department of Engineering Science, University of Oxford Abstract-A summary of the indications for new systems of measurement
`
`is given, with par-
`A study
`
`al.,
`
`of the movement of the shank, or lower leg, using accelerometers is reported. The paper con- cludes that improved transducers will allow this method to be extended to the study of the movement of other parts of the body. An Appendix shows how the signals from six accelero- meters may be used to define completely the movement of a body in space. INTRODUCTION MANY bioengineers involved with the study of human movement have at some time attemp- ted to use an accelerometer for that quantita- tive measure of that movement. Some of the attempts have been reported (Saunders et al., 1953; Gage, 1964) but, certainly, by far a larger number are remembered only as failures. The probable reasons for these failures are worth examining, because, in many areas, the poten- tial advantages of accelerometry over kinephotography (Sutherland et
`1972. Inertial guidance systems use transducers of the ‘force-feedback’ type with a steady- state response and high sensitivity but again the bulk of the associated electronics and the enormously high price of such accelerometers makes them unsuitable. Strain-gauge ac- celerometers which deform elastically due to inertial force are certainly the most suitable type of transducer, and these are available at low cost in a variety of configurations. A can- tilever type with semiconductor strain ele- ments was used in the experiments reported here. Another reason why accelerometry has not been more widely used in biomechanics is the widespread misconception that, by analogy with inertial guidance, gyroscopes are needed for the measurement of angular movement. The true situation is quite different. Gyro- scopes are used in aerospace inertial guidance precisely because the movements are largely translational, and rotations are small and slow, and therefore difficult to measure. In gait, however, the acceleration of a point of the leg is normally due largely to rotational move- ments with the translational components be- coming large only when the system is chang- ing the number of its degrees of freedom by contact with the external environment. It is
`1972), electrogoniometry (Kettelkamp et aL, 1970) and other current methods appear numerous. Probably the commonest cause of failure with the method is the use of an unsuit- able transducer. Most applied mechanics laboratories possess a piezoelectric ac- celerometer designed for the study of vibra- tion and it is usually this device which is first used by the experimenter. Since these devices might more properly be called ‘jerkometers’, needing as they do a charge-integrating am- plifier in order to measure acceleration, the benefit of their small size is often lost. Fur- thermore, the absence of a true steady-state response and the low sensitivity of such de- vices make them wholly unsuitable for the ex- amination of muscular-controlled movements.
`
`*Received 7 December
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`ticular reference to the advantages and potential hazards in the use of accelerometers.
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`730
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`J. R. W.
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`MORRIS
`
`sufficient that six simultaneous independent measurements are made of the translational components of the movement of a rigid body, moving unconstrained in three-dimensional space, for the movement of that body to be determined absolutely with respect to a refer- ence coordinate system. These six measure- ments can all be made with accelerometers. (See Appendix). mental and clinical use which could be oper- ated simply and with minimal disturbance of gait in situations outside the biomechanics laboratory. The study of all animal movement is com- plex both analytically and numerically. The analysis may be simplified by approximations such as rigid-body assumptions, but the num- erical effort involved is always considerable. The use of digital computation is therefore clearly indicated. Furthermore, biological data is often ill-ordered and unpredictable, and the ability of the experimenter to interact with the automatic computational process in order to make algorithmically complex decisions con- tributes a great saving of time and effort.
`I The particular study reported here involves the examination of the movement of the lower leg, or shank. The aim was to develop a sys- tem of measurement suitable for both experi- A well recognised difficulty associated with the kinephotographic measurement of gait has been that encountered in the single and double differentiation of position data with respect to time. There are numerous sources of noise at the upper end of the frequency spectrum of the data (Gutewort, 1971). Since these noise components are preferentially increased by time-differentiation, the signal-to-noise ratio of the data is decreased. Whenever accelera- tions have been obtained from photographic position data, relatively severe mathematical filtering has been employed (Paul, 1965) so that the transfer function of the differentiation process has a frequency spectrum as shown in Fig. 1. The break-point, 00, is chosen as the highest frequency compatible with subjec- tively noise-free velocity and acceleration. The value of wo, which would allow true dif- ferentiation of the whole signal band is almost certainly higher than that normally used.
`
`Upper
`
`frequency
`
`break-
`
`poini
`
`db
`
`Max. signal
`
`frequency
`
`Fig. 1. Band-limited Time-differentiation, showing a possible discrepancy between the
`differentiation break-point and the maximum sign&ant signal frequency.
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`t
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`ACCELEROMETRY 731 An analogous noise problem occurs when acceleration is measured and then integrated to determine velocity and position. The noise frequency which is then the most significant is the lowest. However, in gait analysis, the low- est signal frequency of interest is known to be the double-step frequency. Cutting-off fre- quency components below this value prevents the determination of the moving body’s abso- lute position in space. But more important is the knowledge of movements within a double-step and this can be determined with considerable accuracy.
`A detailed algebraic description of the analysis procedure is given in the Appendix. Only such details as are necessary to give a description of the technique are included in this section. Accelerometers of the type shown in Fig. 2 are used to obtain data on the accelerations of the leg between knee and ankle. The net acceleration field, due to move- ment and gravitation, causes the cantilever to bend in the plane of the sensitive axis. Mean signal-to-noise ratio within the signal fre- quency band is better than 40db. Five ac- celerometers are mounted on the perspex plat- form shown in Fig. 3. No attempt is made to measure transverse rotations of the shank. Such measurements would require a larger di- mension of the platform in the plane normal
`
`METHOD
`
`to
`
`the long axis of the accelerometer platform. Since such rotations are relatively small (Le- vens et al., 1948), they may reasonably be as- sumed to be zero. However, results so far ob- tained with five transducers are good enough to suggest that the inclusion of a sixth to allow measurement of transverse rotations would be justified and this improvement is planned. The platform is mounted over the flat, antero-medial surface of the tibia. Silicone rubber caulking is coated onto the contact side of the platform to provide a high friction inter- face and the platform is held in place by a moulded “Plastazote” cast also coated with silicone rubber. The effect of the mounting technique is to provide heavy mechanical damping between the accelerometers and the shank. Signals from the accelerometers can be re- corded either on a portable subject-carried tape recorder, or passed by a lightweight cable to a fixed recorder. The entire analysis of the signals is done on a small interactive digital computer with analogue input facilities and a visual display. The data is first searched visually for an event of particular interest, and a period of 2.56 set real-time data is selected and sampled at 10 msec. intervals, and digitised. One cycle of any periodic function can be clearly recog- nised on the computer v.d.u. and cursors set to mark the beginning and end of a cycle. Such \
`
`‘\ ’
`
`Cantilever
`
`lnertiol
`
`Fig. 2. A schematic representation of the type of accelerometer used. (The length of the
`cylindrical casing is 14 mm).
`
`Strain elements
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`J. R. W. MORRIS
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`functions (Fig. 4a) are then filtered mathemat- ically (Fig. 4b) to make their values equal at the beginning and end of the cycle. This pro- cess removes drift and sets a lower frequency limit on the signal pass-band corresponding to the double-step period. The stance phase of the walking cycle can generally be divided into three distinct periods. The first, immediately following “heel-strike”, is short and lasts until the foot is flat on the ground. The second, or “foot-flat” period lasts until “heel-off” and is of approxi- .- 6
`
`rad/tec
`
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`
`a
`
`mately the same duration as the third period, which lasts from “heel-off” until “toe-off”. During the second period the shank of the leg moves with pure rotation about a point whose position is known. This point lies within the talus between the axes of the ankle and subta- lar joints (Wright et
`
`al.,
`
`1964). Since the posi- tion of this origin of rotation is known, the knowledge of the motion of the shank be- comes mathematically redundant. The six parameters of motion which are measured or assumed to be zero are not then independent. Their interdependence allows the angular pos- ition of the leg with respect to the fixed axes to be calculated. If the start of the cycle of shank movement is chosen to occur during the ‘foot-flat’ period of the stance phase, the ini- tial conditions needed to solve the simultaneous-differential equations for the in- stantaneous direction-cosine matrix are known. Figure 5 shows a display of the direc- tion cosine matrix for a typical walking cycle. With the solution for the direction cosine matrix available for each sample time, it is possible to solve for the translational compo- nents of the limb’s movement. The angle which the axis of each transducer makes with the vertical is known, and the component of
`
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`Fig. 4a, b. A period of 2.56 set of angular velocity data
`before and after filtering. The upper and lower traces in Fig. 5. The direction cosine matrix of the shank for one
`step, corresponding to time between the cursors in Figs.
`each picture show coronal and sagittal plane rotations, re-
`4a and b.
`spectively.
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`“% :
`

`

`Fig. 3. The accelerometer mounting platform, showing five accelerometers and other as-
`sociated electrical components.
`
`(Facing p. 732)
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`

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`ACCELEROMETRY
`
`733
`
`the gravity vector measured by each trans- ducer can be calculated and subtracted from the net measured acceleration field. An origin is arbitrarily chosen at a fixed position in the moving body and the translational movement of this origin is calculated by double integra- tion of the acceleration of the body referred to this point. The accelerations, velocities and translations of the mid-shank origin are shown in Fig. 6. Besides the graphical and numerical rep- resentation of the measured movement, a true-speed moving picture of the limb element can be simulated on the computer v.d.u. A set of 26 points is used to represent a shank and foot and can be seen moving as though viewed from any desired direction. The effect is similar to viewing a photographic record of the movement of a leg mounted with marker dots. Ankle articulation is simulated simply by preventing the toe from going below the “ground”, a line drawn at the lowest excursion of the heel. Otherwise the foot is kept perpen- dicular to the shank. (This approximation is used only to make the simulation more visu- ally acceptable, since the actual amount of ankle dorsi and plantar flexion is unknown.)
`
`- 17 m/sec2
`
`:
`
`,A? m/set
`
`,- 613;lm
`
`Simulated cinephotography is particularly useful in recognising unusual features of gait before returning to study the time-derivative functions in more detail. Figure 7 shows a time-exposure of the movement simulation over one cycle. Since the space coordinates of all 26 points are known, the movement may be displayed in any two-dimensional plane with- out difficulty, simply by effecting the approp- riate coordinate transformation.
`
`CONCLUSION
`
`The purpose of gait studies is twofold. The results obtained from normal subjects have been used to make estimates of the forces in joints, muscles and ligaments (Paul, 1%5; Morrison, 1970). This is a highly complex pro- cedure both experimentally and analytically. Very little is known of the kinematics of walk- ing in circumstances other than the laboratory. While kinematic measurements alone do not allow joint forces to be estimated, there are many areas of knowledge, for instance con- cerning the aetiology of osteo-arthrosis, (Radin et al., 1972), which may only be clarified by gait studies made on subjects in normal environmental conditions. Portable measurement systems, such as that described here are the only way such measurements can be made. Apart from measurements on normal sub- jects, gait studies infrequently find a place in clinical orthopaedics. The reason for the lack of acceptance of current methods is that they are either too complex in application, or else are simple but produce very little information. This problem is widely recognised (Suther- land, 1972), and it is hoped that this technique ,. .* ,‘. *.: . I . . - . . : . -.*.* ,. ,: ,’ , . . . , .* . . _ .; 1: *I . . I P)._ - - . ’ . ., . . . . . ,I :. :- .‘.G). .‘,/, 1 , ;: ‘, .-: : . .;. .*. *.,;,: ..: . _ . . . . . . . A. . - - - - * * - . - _ . _ rr.*; . . .
`
`Fig. 7. Multiple exposure of movement simulation. Frame
`rate: rate: 12.5/set.
`
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`
`-17
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`
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`-62
`Fig. 6. The translational components of the move-
`ment of (a mid-shank origin. The rows, from the top, show
`forwards-backwards, sideways and vertical movkments.
`Scales indicate units.
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`734
`
`J. R. W. MORRIS
`
`SUMMARY
`
`A,
`
`will help overcome some of the current diffi- culties. Figures 4b, 5 and 6 show, for the same step, the angular velocities, direction cosines, trans- lational acceleration, velocities and positions of the shank. Comparison studies of a wide variety of types of step indicate that the angu- lar and translational velocities are the func- tions which reveal most about the gait of the subject. These are shown in Fig. 4b and the centre column of three graphs in Fig. 6. The cycle lasts from foot-flat to foot-flat, passing through a smooth toe-off at
`a muscle driven initial swing-phase to point B and a sinusoidal, free swing-phase to point C. A small positive angular velocity in the saggittal plane, in con- junction with a low vertical velocity, prepares the leg for a low energy heel-strike at point D. These are clear characteristics of normal gait and each of them changes characteristically with specific abnormalities. used to study aperiodic movements, for in- stance of the upper limbs.
`A technique for using accelerometers to study the total movement of the shank of the leg has been developed. Signals from the transducers are analysed on an interactive dig- ital computer. Graphs of angular velocity, di- rection cosine, translational acceleration, vel- ocity and position are produced as well as a visual representation of the movement. It is hoped to use the technique extensively in a clinical orthopaedic environment and to de- velop further the method for use in the study of the movement of other parts of the body.
`
`assistance of Drs. J. J. O’-
`Acknowledgements-The
`Connor and C. Ruiz in the Department of Engineering
`Science, and of Professor R. B. Duthie in the Depart-
`ment of Orthopaedic Surgery is gratefully acknowledged.
`
`The purpose of the study was to show that accelerometers could be used to provide suffi-
`
`REPERENCES
`Bortz, J., Snr. (1970) A new concept in Strapdown inertial
`navigation. NASA TR R-329.
`Gage, H. (1964) Accelerographic analysis of human gait.
`ASME Paper 64, WA/HUF-8.
`Gutewort, W. (1971) The numerical presentation of the
`kinematics of human body motions. In Biomechunics
`Vol. II, pp. 290-298. University Park Press.
`Jordan, 3. (1%9) Direction cosine computation error.
`NASA TR R-304.
`Kettelkamp, D., Johnson, R., Smidt, G., Chao, E. and
`Walker, M. (1970) An electrogoniometric study of knee
`motion in normal gait. J. Bone Jnt Surg. 52A, 775-790.
`Levens, A., Inman, V. and Blosser, J. (1948) Transverse
`rotation of the lower extremity in locomotion. J. Bone
`Jnt. Surg. 3OA, 859-872.
`Morrison, J. (1970) The mechanics of the knee joint in
`relation to normal walking. J. Biomechanics 3,51-61.
`Paul, J. (1965) Bioengineering studies of the forces trans-
`mitted by joints. In Biomechanics and Related Bioen-
`Topics, pp. 369-380. Pergamon Press, Oxford.
`Radin, E., Paul, I. and Rose, R. (1972) The role of mechan-
`ical factors in the pathogenesis of primary osteoar-
`thritis. Lancet 7749, 519-522.
`Saunders, J., Inman, V. and Eberhart, H. (1953) The major
`determinants in normal and pathological gait. J. Bone
`Jnt Surg. 3SA, 543-558.
`Sutherland. D. and Hagy, J. (1972) Measurement of gait
`movements from motion picture film. J. Bone Jnt Surg.
`54A, 787-797.
`Wright, D., De&, S. and Henderson, W. (1964) Action of
`the subtalar and ankle joint complex during the stance
`phase of walking. J. Bone Jnt Surg. 46A, 361-382.
`
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`cient information to define the movement of a
`segment of the body. It has been shown that
`they can be used in this way. A common criti-
`cism of the technique has been that it would
`be impossible to predict the movement of a
`limb element without some form of surgical
`bony attachment for the transducers. Clearly,
`the mounting site used in this case is chosen
`specifically to minimise the effects of soft tis-
`sue movements. By covering the whole area
`adjacent to the mounting platform with the
`high-friction cast, the skin movements are
`spatially integrated and heavily damped. It is
`thought that, with careful site-selection, and
`the appropriate form of fixation, non-invasive
`gineering
`measurements could be made on the move-
`ment of other parts of the body. New trans-
`ducers, with strain-elements diffused into a
`semiconductor cantilever substrate, have far
`better drift characteristics than those at pres-
`ent in use. Using these transducers, it should
`be possible to reduce low-frequency cut-off
`point in the signal spectrum by several orders
`of magnitude. Accelerometers could then be
`

`

`ACCELEROMETRY 735 0123 0’ 1’2’3’
`
`f
`8
`
`am
`
`pm are
`
`not necessarily coincident. (Body-fixed frame: 0123. Fixed reference frame 0’1’2’3% and a component, f, due to the acceleration of the body- fixed origin (See Fig. 8). .
`
`am=bXr+wX(wxr)+g+f
`(3) and
`flm=chxs+wx(uxs)+g+f.
`
`(4) Clearly:
`
`ffm, t3m NOMENCLATURE a Cartesian coordinate system fixed to a body moving in space a Cartesian coordinate system used as a fixed reference frame an acceleration vector relating point 0 to point 0’ an acceleration vector defining the gravitational field position vectors defining the position of points in the 0123 reference frame. inertial measurements of acceleration taken at points in the moving body accelerations of points in the moving body the angular velocity vector of 0123 w.r.t. 0’1’2’3’ expressed in 0123 coordinates time-derivative of o Kronecker delta Levi-Civita density direction cosine transformation for vectors from 0123 to 0’1’2’3’ coordinates . . time-derlvatrve of [A], A,, transform of matrix [A] identity matrix. APPEND= Euler’s Theorem states that the general displacement of a rigid body with one point fixed can be achieved by a rotation about a suitable fixed axis through the fixed point. Since the direction of the axis can be defined by two ang- les and the rotation by a third, the movement should be defined by three independent measurements, for instance by orthogonal measurements of acceleration. Three further independent measurements are needed to define the movement of the ‘body-fixed’ point with respect to some other frame of reference. These three independent variables may also be measurements of acceleration. We define ‘fixed’ reference frames as having an acceler- ation field, known as the gravitational field, which is a function of position. In order to determine movements within such a reference frame from unreferenced meas- urements of the local acceleration field, the local gravita- tional field must be specified. One indirect specification of the gravitational field is the assumption that it does not vary within the dimensions of the body whose movement is to be determined. This specification may be used to I’ Fig. 8. Points of measurement defined w.r.t. two reference frames. The points of measurement of the components of
`
`where C is a constant vector, and the integral is over time. Equation (6) may be rewritten in tensor notation: where: i,j,k.l.mcC (1,2,3) and the integral is over time, or matrix form: calculate the rotation of the body from the signals from three orthogonal parallel pairs of accelerometers. The acceleration of a point with respect to an origin to which it is fixed is defined, in vector notation, as:
`Since the angular velocity and acceleration do not change with position, B=~Xs+wX(wxs) (2) Unreferenced, or inertial, measurements of acceleration, as indicated above, contain a gravitational component, g, where: .t, is the qth component of t for measurements
`pth direction and integrals are over time. The only limitation on the points of measurement is that the measurement axes are orthogonal. Various simplifica- tions of equations (6,7 and 8) are possible by making cer- tain components of t equal to zero. For instance, col- linearity makes pf4=pf4&Q where S,, is the Kronecker delta. This simplifies the equations to a non-differential form, but this form, while simpler to solve, may give a Pess accurate solution if 0 X (w X t) Q g. Equation (7) can be solved numerically and a descrip- tion of the possible methods is not appropriate here ex-
`
`am-@m=hXt+wx(wxr)
`
`I=r-s
`
`[(am-/3m)-wx(wxt)]+c
`
`(5)
`
`(6)
`
`in
`
`cr=hxr+wx(uXr)
`
`(1)
`
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`and
`where
`.
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`736 J. R. W. cept to say that in this particular case 1t,, &, &,, &and *f, were made equal to zero and oj was also assumed to be zero. A simple, first-order, trapezoidal integration proce- dure was found to be adequate. Once the angular velocity vector is known, the solution of the movement of the rigid body is that conventionally used in ‘strapped-down’ inertial navigation systems (Bortz, 1970). The direction cosine transformation, A, is found from the matrix equation: [Al = rolrhl which, again, may be rewritten: (9) i, = O&qEprr (10) The method of solution of this set of nine equations for the components of A is, as before, based on simple trapezoidal integration. Knowledge of the direction cosines relating the body- fixed reference frame to the fixed frame allows the meas- ured acceleration field from a single orthogonal triad of transducers to be transformed into the fixed coordinate system. Since the gravitational field is specified in terms of this coordinate system, this component may be sub- tracted from the transformed, measured field in order to determine the translational acceleration of the body-fixed origin with respect to the fixed reference frame. Simple in- tegrations then yield the velocity and position of this origin. Thus the movement of the body with 6 degrees-of- freedom relative to a fixed reference frame can be calcu- lated from 6 acceleration field measurements made on the body. Two further points of importance should be mentioned here. The first concerns the estimation of initial conditions in the integral equations. Separately determined con- straints are always required for such estimations. Conven- tionally in inertial navigation systems, these initial condi- tions are found from exterior measurements of initial ac- celeration, velocity and position. In this case, however, such information is not complete, but, as is mentioned in the main text, the cyclic nature of the movement allows MORRIS the initial angular velocity and rotation, translational ac- celeration and velocity to be made equal to the final values of these functions. The initial position of the leg is of importance only in a rotational sense (in order that the gravitational component may be removed from the accel- eration field). As is indicated in the main text, the redun- dancy of the information on movement during the foot- flat phase of the walking cycle allows the initial direction cosine to be found as follows. The origin of rotation of the shank during this phase is fixed in the stationary reference frame. The term f in equation (3) is therefore zero, so this equation becomes: am=~xx++x((wxr)+g. (11) Since dr, w and r are known, g may be determined in terms of am. The relative magnitudes of the components of g are related to the fixed-frame gravity vector by three terms in the direction cosine matrix. Referring to Fig. 8; A,, = g,; A32 = g2 and Aa3 = g3. The remaining six terms re- late the directions of the three moving axes to the direc- tions of the two fixed axes perpendicular to the gravity vector. This relationship may be defined quite arbitrarily, most conveniently by making Al2 = 0. This has the effect of making axis 2’ perpendicular to axis 1.
`
`(13) A,, = AXA,,- AzzA~, (14) AD = AXA,, (15) A,, = A,A,,-ADAz. (16) Thus the initial direction cosine can be found. The second point of importance is a check on the or- thogonality of the A matrix. If the matrix is orthogonal: [AllAl’= VI. (17) If not the matrix may be corrected to orthogonality by the iterative procedure (Jordan, 1%9) [Al.+, = [Al.([Jl-;([Al:[Al. -VI)). (18)
`
`AZ=+-
`Am = - A,,A,z/Am
`
`(12)
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