A method for inverse dynamic analysis using accelerometry

A method for inverse dynamic analysis using accelerometry

Pergamon J. Biomechanics,Vol.29, No. 7, pp. 949-954,1996 Copyright© 1996ElsevierScienceLtd. All rightsreserved Printedin Great Britain 0021-9290/96 $...

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J. Biomechanics,Vol.29, No. 7, pp. 949-954,1996 Copyright© 1996ElsevierScienceLtd. All rightsreserved Printedin Great Britain 0021-9290/96 $15.00+ .00










A n t o n J. v a n den Bogert, L y n d a Read a n d B e n n o M. Nigg Human Performance Laboratory, University of Calgary, Canada Abstract--A method was developed to calculate total resultant force and moment on a body segment, in three dimensions, from accelerometer data. The method was applied for an analysis of intersegmental loading at the hip joint during the single support phase of working and running, using four triaxial accelerometers mounted on the upper body. Results were compared to a conventional analysis using simultaneously recorded kinematics and ground reaction forces. The loading patterns obtained by both methods were similar, but the accelerometry method systematically underestimated the intersegmental force and moment at the hip by about 20%. This could be explained by the inertial and gravitational forces originating from the swing leg which were neglected in the analysis. In addition, the accelerometry analysis was not not reliable during the impact phase of running, when the upper body and accelerometers did not behave as a rigid body. For applications where these limitations are acceptable, the accelerometry method has the advantage that it does not require a gait laboratory environment and can be used for field studies with a completely body-mounted recording system. The method does not require differentiation or integration, and therefore provides the possibility of real-time inverse dynamics analysis. Copyright © 1996 Elsevier Science Ltd. Keywords: Accelerometry; Inverse dynamics; Hip joint.

INTRODUCTION Inverse dynamics analysis is a standard tool for biomechanical studies of locomotion, both in two (Winter, 1989) and three dimensions (Bogert, 1994). The traditional approach is to use kinematics of the body segments, combined with measurements of external (ground reaction) forces to obtain a resultant (intersegmental) force and moment at the joint of interest. The intersegmental moment provides information about muscle function and the intersegmental force, combined with muscle forces, can be used to estimate the joint contact force (Crowninshield et al., 1978; Paul, 1976). Typically, kinematics are measured with automated three-dimensional video systems, or with film, and ground reaction forces with a force platform installed in a laboratory. Several methodological difficulties are associated with this approach to inverse dynamics analysis. A typical kinematic data acquisition system has an accuracy of 0.1% of the field of view. For a gait analysis laboratory, resolution is typically 1 mm in a field of view of 1 m. Kinematic data are differentiated twice in order to obtain the inertial forces on the body segments, requiring severe low-pass filtering to prevent amplification of random measurement errors (Woltring, 1985). High-frequency peaks in the acceleration may be removed by this filtering process. For example, a 1 g sinusoidal acceleration at 15 Hz corresponds to a displacement of only 1.1 mm. This acceleration will therefore be barely detectable when the resolution of kinematic measurements is 1 mm. The severity of this problem depends on the mass of the body segment, because acceleration is multiplied by segment mass in the inverse dynamics analysis. Obviously, the problem is most severe for an analysis involving movements of the trunk. Another practical problem is the small field of view of camera-based kinematic measuring system. When considering the feasibility of inverse dynamics analysis of skiing, it was recognized that speeds could reach values of 15 ms -1. This

implies a data collection period of only 67 ms at a 1 m field of view. Enlarging the field of view would cause a proportional increase in measuring errors, with the consequences described earlier. In order to overcome these limitations in frequency response and duration of measurement, an inverse dynamics analysis was developed based on three-dimensional accelerometry. Accelerometer data can be conveniently collected using telemetry or a portable data-logger system. A similar method was presented by Kane et al. (1974) for analysis of forces and moments during impact on a tennis racket. In this paper, we present a generalization of this method, which allows any configuration of at least nine transducers and uses redundancy of the signals to a greater extent. The method is applicable to analyse loading of a single body segment. As an example, the intersegmental force and moment at the hip were determined in walking and running, by considering the upper body as one rigid segment. Results were validated by simultaneous analysis using conventional video and force platform data.

METHODOF ANALYSIS Estimation of kinematic variables

When a uniaxial accelerometer is attached to a rigid body segment at a known position r and orientation u in a segmentfixed coordinate system, it will produce the following signal: s = u.(R-t(a-

g) + t b x r + tox(toxr))


where R is the attitude matrix of the body segment with respect to the ground, a is the acceleration of the origin of the segment coordinate system (which will be chosen as the centre of mass CM of the body) with respect to ground, g = (0, 0, - g)X is the gravitational field, and to is the angular velocity of the body, expressed in the body-fixed coordinate system, u is a unit vector pointing along the sensitive axis of the transducer. Acknowledging the fact that gravitational force and inertial forces due to linear acceleration cannot be separated, this equation was

Received in final form 2 October 1995. Address correspondence to: Dr A. J. van den Bogert, Faculty of Kinesiology, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada.



Technical Note

Fig. 1. Front and back view of subject wearing the accelerometer system. Three triaxial accelerometers are mounted on the back, one on the front. Signals are recorded by a miniature data logger carried in a belt pack.

rewritten as s = u. (So + tb x r + to x(to x r)).


When signals from n accelerometers, attached to the same rigid body, are recorded, n of these (algebraic) equations in the nine unknown kinematic variables (So, to and th) are obtained. The Levenberg-Marquardt method for non-linear least squares problems* was used to solve the kinematic variables from the system of n (~> 9) equations (Levenberg, 1944; Marquardt, 1963). When the system is overdetermined (n > 9), the residuals of the least-squares solution provide information about the consistency of the equations, i.e. the rigidity of the accelerometer configuration. This was quantified by calculating a root-meansquare (RMS) value of the residuals from each sample of accelerometer signals and subsequent averaging over all samples in a recording. A pilot study involving computer generated rigid-body motions was done to verify the reliability of this method for measuring kinematics. It was found that four triaxial accelerometers (n = 12) in a non-coplanar configuration provided sufficient redundancy to eliminate singularities in the system of equations. With the theoretical minimum of three triaxial accelerometers, singularities would cause high sensitivity to measurement errors when the instantaneous axis of rotation is nearly parallel to a line connecting two accelerometers. With n/> 12, the linear system of equations given by Hayes et al. (1983) was used to obtain a good initial guess for the iterative solution of the non-linear equation (2).

the sum of all moments on the segment, expressed in the bodyfixed coordinate system, m is the segment mass, and I is the inertia matrix. Transforming equation (3) into the segment coordinate system, we obtain F' = R - 1 F = m . R - l ( a - g )

= m'so.


The equations (4) and (5) produce the six load components {F', M' }, acting at the centre of mass of the rigid body. If all force is instead assumed to be transmitted through a joint at location rj in the segment coordinate system, the equivalent moment with respect to the joint centre can be obtained: Mj = M' - (rj x F') = I. th + to × (I. to) - m(rj x So).


The force F' is not affected by moving the point of application to the joint centre. When a suitable segment coordinate system is chosen, the three components of the moment vector can be interpreted as moments about the flexion-extension, ab-adduction, and internal-external rotation axes, respectively. Equations (5) and (6) will be used to calculate intersegmental joint force and moment from the kinematic data (So, to and th) obtained from the accelerometer signals. Note that the attitude information R does not occur at the right-hand side of equations (5) and (6) and is therefore not needed for computation of resultant force and moment, when these vectors are quantified with respect to a segment-fixed coordinate system. Also note that the analysis assumes that force and moment exist at one joint only, which limits the applicability.

Equations of motion The equations of motion of a rigid body in three dimensions are, for translation and rotation respectively (Haug, 1989): F + m'g = re'a,


M ' = l.tb + to x (I.to),


where F is the sum of all forces acting on the segment, except for gravity, expressed in the ground-based coordinate system. M ' is

* A public-domain implementation of this algorithm in Fortran can be obtained from Netlib (Dongarra and Grosse, 1987) by sending the command 'SEND LMDIF1 F R O M MINP A C K ' by electronic mail to [email protected] .COM.

APPLICATION TO THE HIP JOINT The method was applied for analysis of forces and moments at the hip joint during single limb support in one male subject with a total body mass of 72kg. Four triaxial accelerometers (EGAXT-*-10, ENTRAN, Fairfield NJ) were attached to a semi-rigid frame which was tightly strapped to the upper body (Fig. 1). The segment coordinate system of the trunk was defined to be coinciding with a previously defined right-handed laboratory system in a well-defined standing position. The positive coordinate axes were chosen: X = left/lateral, Y = posterior, Z = vertical. Three or four reflective markers were attached to each triaxial accelerometer at known positions, measured with respect to the transducer's own coordinate system. The threedimensional coordinates of these markers were measured during standing using an automated three-dimensional video system

Technical Note (Motion Analysis Corp., Santa Rosa, CA). The least-squares method of Veldpaus et al. (1988) was applied to these threedimensional coordinates to determine the position r of the seismic mass and orientation u of each individual transducer with respect to the segment coordinate system. In the same standing position, coordinates of reflective markers attached to anatomical landmarks were used to calculate coordinates of the CM of the trunk (Clauser et al., 1969) and of the left hip joint centre (as recommended by Bell et al., 1990) with respect to the segment-fixed coordinate system. The origin of the segment coordinate system was then shifted to the centre of mass by subtracting the coordinates of the centre of mass from all measurements. In the same standing position, the accelerometer signals were recorded to provide a known reference level s = - (u.g) for the accelerometer signals. After completing these 'subject calibration' measurements, the reflective markers were removed and accelerometer data collected during walking (1.5 m s - 1) and running (3.5 m s- 1). The twelve accelerometer signals were recorded by a portable data acquisition system (Tattletale 6F, Onset Computer Corporation, North Falmouth, MA) at a sampling rate of 300 Hz and transferred to a SUN SparcStation IPC computer for further processing. Intersegmental force and moment at the hip were calculated according to equations (5) and (6), with the following body segment parameters: m = 50.30 kg and I = diag(1.2611,1.3560, 0.3119) kgm 2. The mass represents the total mass of head, arms and trunk for this subject (Clauser et al., 1969), combined with the data acquisition system (3.8 kg). The inertia matrix was obtained using data from Whitsett (1963h and is expressed with respect to a coordinate system where X is medio-lateral, Y is anterior-posterior, and Z is the inferior superior axis. Simultaneous with the accelerometer data collection, ground reaction forces were measured by a Kistler 9287 force platform, and three-dimensional kinematics of left foot, shank and femur were recorded by the video system. These data were used as input for a standard three-dimensional dynamics analysis of the left hip joint, using the KinTrak software package (Motion Analysis Corp., Santa Rosa, CA}. Smoothing of kinematic data in KinTrak was done at a cut-off frequency of 10 Hz for walking and 20 Hz for running. No smoothing was applied to the accelerometry results at any stage. Using accelerometry, the intersegmental force and moment at the hip are calculated in a coordinate system fixed to the upper body, while the force-plate-based analysis is carried out in a coordinate system attached to the femur. For this reason, only the magnitude of the force and moment vectors will be reported. Note that the accelerometer analysis provided the total force and moment on the upper body, in contrast to the force platform and kinematics analysis which provides information about the left hip. Comparable results for both methods are therefore only expected during the single support phase of the left leg. The accelerometry analysis neglects loads transmitted from the right (swing) leg to the upper body completely. The influence of this error was assessed from video data. Marker coordinates were low pass filtered at 7 Hz for walking and 15 Hz for running, which was found to be appropriate due to the different frequency content of the swing motion. Accelerations of the CM of the segments were calculated, multiplied by segment masses, and summed to obtain an estimate of the total inertial and gravitational force generated by the swing leg at the contralateral hip joint.

RESULTS Figure 2 shows results from the inverse dynamics analysis of walking, using both the accelerometry and the video-force plate data. The difference between intersegmental forces computed using the two methods is typically 20 to 25%, differences in intersegmental moment are somewhat less. During walking, the accelerometer system proved to be fairly rigid, with an average RMS residual of 0.32 m s-z obtained from the system of equations (2) for this particular recording. Results from other trials BH 29-7-G


from the same subject were similar: accelerometry results were consistently lower than video-force plate analysis and RMS residuals were between 0.27 and 0.34 m s-2. Results for a typical running trial are presented in Fig. 3. Again, the accelerometry system produces lower intersegmental forces than the video-force plate system, with the exception of the impact phase where the accelerometer system shows a high peak. Differences of similar magnitude occur in the moment; the accelerometry system produces larger moments during the first half of the support phase, and lower values during the second half. This typical trial had an average RMS residual of 1.07 m s-2, while other trials had RMS residuals between 0.86 and 1.22 ms-2. In running, the rigidity of the accelerometer system was relatively poor during the impact phase, where RMS residuals consistently reached peak values of 5 to 5.5 m s-2. The magnitude of intersegmental force generated by the contralateral (swinging) leg at the hip joint are shown in Fig. 4, calculated from video data of typical walking and running trials. Peak forces are 180 N for walking and 300 N for running.

DISCUSSION Although accelerometry has long been recognized for its potential use in human movement analysis (Morris, 1973), it has not found widespread use. Major difficulties associated with accelerometry are integration drift, which makes it impossible to obtain position and attitude information over a long time interval, and the influence of transducer rotation with respect to the gravitational field. In this paper, we have shown theoretically that these problems do not affect the use of accelerometers for inverse dynamics analysis when the intersegmental loads are reported in a segment-fixed coordinate system. In essence, the resultant loading of a body segment are deduced from loads acting on a number of test masses (inside the accelerometers) distributed over the segment of interest. This procedure, similar to other methods for inverse dynamics analysis, is based on the assumption that the body segment is rigid. The use of four triaxial accelerometers allowed us to solve for the nine kinematic variables (linear acceleration and gravity, angular velocity and angular acceleration) without performing integration or differentiation. A precursor of this method was used by Kane et al. (1974} to analyse resultant loads on a tennis racket. Four triaxial transducers were used, but the equations (2) were not solved for the three components of to, but for the six products tnlco/. This reduced the problem to a linear system of twelve equations with twelve unknowns, but did not allow redundancy to be used for reduction of errors or evaluation of the rigidity assumption. Kane's method has been proposed for application to human movement (Hayes et al., 1983), but no actual application was developed. Cross-sensitivity of the accelerometers (quoted by the manufacturer as < 3%) may influence the accelerometry results, because some activities are characterized by very small acceleration components perpendicular to the sagittal plane. It is therefore important that results are quantified as magnitude and direction of force and moment, rather than as cartesian components. Cross-sensitivity as well as noise may cause large relative errors in to if the final (centripetal) term in equation (1) is relatively small, as is typically the case. In fact, the components of to were found to be very noisy during most activities. In such cases, however, to is small and has only a minor influence on the inverse dynamics analysis, as shown by equation (41. Similar arguments can be made for the angular acceleration th. It can be shown that, as long as the accelerometers are distributed over a volume with approximately the same size as the body segment under consideration, error propagation is not a problem for determination of force and moment, although some of the individual kinematic variables So, t o and ¢b may be inaccurate, depending on the type of movement. The accuracy of the accelerometry method will now be evaluated by comparison to the video and force plate analysis, but the reader should be aware that the true value of intersegmental


Technical Note

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200 |













time [s]


120 ....

Video/force plate

100 i






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Fig. 2. Magnitude of intersegmental force and moment at the hip during walking, computed from accelerometer signals (continuous line) and video-force plate recordings (dashed line). The solid bar indicates the duration of the single support phase, which typically starts at 18% and ends at 82% of the support phase.

loads is unknown and both methods suffer from inaccuracies. When calculating resultant hip forces and moments using accelerometry, an estimate is made of the loads generated by the weight and movements of the upper body. The video and force plate analysis incorporates the loads transmitted from the ground as well as weight and movement of the limb segments. According to Newton's third law, these methods should give identical results. However, both methods are affected by errors caused by inaccuracy of the measuring system and the assumptions of that body segments are rigid. Measuring errors have a negligible influence on the accelerometry results, but the video data are differentiated twice before inertial forces can be obtained. Based on a video digitizing accuracy of 1 mm and the filter cut-off frequencies, this could have led to errors of 3.9 and 15.8 m s- 2 in the worst case, for walking and running respectively. This corresponds to inertial forces of 30 and 120 N for the thigh segment, and moment errors can be estimated through multiplication by the distance between segment CM and hip joint. The accelerometry analysis assumed that head, arms, trunk and accelerometer frame move together as one rigid unit. The validity of this assumption depends on the activity. During gait,

the movement of the head is closely coupled to the trunk but the swinging motion of the arms will introduce considerable inertial forces which are not detected by the accelerometers. Due to symmetry of the movement, however, the horizontal components of inertial forces in the two arms will cancel out and produce no net effect at the hip joint. The vertical components will be of considerably lesser magnitude. The arm movement will, however produce a torque about a vertical axis. Assuming a 1 Hz, 0.2 m amplitude, sinusoidal motion the CM of each arm (4 kg; Winter, 1979), and 0.45 m shoulder width, this undetected torque can be estimated as 14 N m, which might help explain the underestimation of moment during walking and running (after the impact phase). The influence of the contralateral (swing) leg is potentially large, as suggested by Fig. 4, which shows that the major part of the difference between video-force plate analysis and accelerometry analysis (shown in Fig. 3) can be explained by the magnitude of forces generated by the swing leg at the contralateral hip. The residuals of the least-squares solution indicated the effect of linear and angular deformation in the frame. This was only a problem during the impact phase of running where the rigid-body assumption was violated by as much as 0.5 g, producing an error in the inverse dynamics

Technical Note




, - - - -



Video/force plate



E (D

f~ J2

1400 1200 1000 8OO 600 4OO 200 0 0






t i m e [s]


250 - - - -

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E o E



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, I i;" t I | I i






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Fig. 3. Magnitude of intersegmental force and moment at the left hip during running, computed from accelerometer signals (continuous line) and video-force plate recordings (dashed line).

analysis of 50% of upper body weight in the force. The corresponding error in intersegmental moment can be estimated by multiplying this value by the distance (10 cm) between the hip joint and force vector at that time, drawn from the segment CM. Although the accelerometers could be mounted on a more rigid frame, we felt that such a frame would no longer reliably represent the motion of the upper body, which is not rigid. The semi-rigid construction used in our experiments allowed every accelerometer to remain in contact with the trunk and, if done properly, can provide a good representation of the effective acceleration of the total trunk mass, including rigid and nonrigid components. Further improvement can be achieved by using more than four triaxial accelerometers, distributed over the trunk. This possibility is included in the least-squares method presented in this paper. No conclusions should be drawn from the comparison between the two methods during the impact phase of running, since both methods are potentially very inaccurate at that time due to large accelerations of the non-rigid masses and/or low-pass filtering of kinematic data with a high frequency content. The magnitude of the effects discussed in this paragraph may be different for activities other than walking and running. After completing initial data collection on skiing (alpine and cross-country), RMS residuals of the kinematic analysis, equation (2), were between 0.3 and 0.5 m s- 2 and no impact-related problems were found. The effect of arm

and leg swing on analysis of alpine skiing is expected to be very small. For cross-country skiing, the effect will be similar to running. An additional difference between the two methods may have been caused by the location of the hip joint. Although both methods used the same hip joint centre, located during standing, this point may not coincide with the true centre of rotation and differences may occur during movement because one joint centre was related to an upper body coordinate system and the other to a femur coordinate system. This would only affect the moments, however. Accelerometers on the lower extremity have been combined with goniometer and force plate recordings for a two-dimensional inverse dynamics analysis of the knee and hip joint (Gilbert et al., 1984), resulting in good estimates for the inertial forces in the limb segments. The method could only be applied to the support phase of gait, because it required a fixed centre of rotation between the lower leg and the ground. This assumption was not required in a similar method described recently (Ladin and Wu, 1991), where additional kinematic data were collected using an optical recording system. These methods, however, still require a gait laboratory environment and can hardly be applied in field studies. For applications where a completely body-fixed recording system is required and the limitations of the method are acceptable, the technique described in this paper presents


Technical Note











o 100 ,2o 80

~ 150

,2 lOO

6O 40


20 0













time [s]








time [s]

Fig. 4. Intersegmental force at the right hip during the single support phase of the left leg, calculated from kinematic data. Results are shown for one typical walking trial and one typical running trial.

a very attractive opportunity to perform inverse dynamics analysis. The system allows continuous recording, for as long as the memory in the data logger allows. The method does not require integration, differentiation or filtering of the data. Each sample is therefore analysed by itself, with no dependence on previous or future samples. This leads to the possibility of performing inverse dynamics analysis in real time, which could be used to provide immediate feedback to a subject or patient when movements or postures occur that produce undesirable loads. The technique would also be applicable to analyse loading at the lower back, as well as the hip joint, provided that no significant external loads are present on the hands or other parts of the upper body. It is concluded that four triaxial accelerometers can be used for inverse dynamics analysis of the hip joint. It should be emphasized that the method can only be applied during the single support phase of gait or other movements. For walking and running, this technique systematically underestimates the forces and moments by about 20%, because forces generated by the swing leg are neglected. Acknowledgements--This study was financially supported by the Swiss Society for Orthopaedics (SGO) and the National Scientific and Engineering Research Council (NSERC) of Canada. This work would not have been possible without technical support of Andrzej Stano. REFERENCES

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Gilbert, J. A., Maxwell, G. M., McElhaney, J. H. and Clippinger, F. W. (1984) A system to measure the forces ad moments at the knee and hip during level walking. J. orthop. Res. 2, 281-288. Haug, E. J. (1989) Computer-Aided Kinematics and Dynamics of Mechanical Systems, Vol. 1: Basic Methods, p. 422. Allyn and Bacon, Boston. Hayes, W. C., Gran, J. D., Nagurka, M. L., Feldman, J. M. and Oatis, C. (1983) Leg motion analysis during gait by multiaxial accelerometry: theoretical foundations and preliminary validations. J. biomech. Engng 105, 283-289. Kane, T. R., Hayes, W. C. and Priest, J. D. (1974) Experimental determination of forces exerted in tennis play. In: Biomechanics I V (Edited by R. C. Nelson and C. A. Morehouse), pp. 284-290. University Park Press, Baltimore. Ladin, Z. and Wu, G. (1991) Combining position and acceleration measurements for joint force estimation. J. Biomechanics 24, 1173-1187. Levenberg, K. (1944) A method for the solution of certain non-linear problems in least squares. Quart. Appl. Math. 2, 164-168. Marquardt, D. W. (1963) An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math. 11, 431-441. Maxwell, S. M. and Hull, M. L. (1989) Measurement of strength and loading variables on the knee during alpine skiing. J. Biomechanics 22, 609-624. Morris, J. R. W. (1973) Accelerometry--A technique for the measurement of human body movements. J. Biomechanics 6, 729-736. Paul, J. P. (1976) Approaches to design. Force actions transmitted by joints in the human body. Proc. R. Soc. Lond. B 192, 163-172. Veldpaus, F. E., Woltring, H. J. and Dortmans, L. J. M. G. (1988) A least-squares algorithm for the equiform transformation from spatial marker co-ordinate. J. Biomechanics 21, 45-54. Whitsett, C. E. (1963) Some dynamic response characteristics of weightless man. AMRL Technical Documentary Report, pp. 63-68. Wright-Patterson Air Force Base, OH. Winter, D. A. (1979) Biomechanics of Human Movement. Wiley, New York. Woltring, H. J. (1985) On optimal smoothing and derivative estimation from noisy displacement data in biomechanics. Hum. Mvmt Sci. 4, 229-245.