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A spot check for estimating stereophotogrammetric errors U. Della Croce
A. Cappozzo
Dipartimento di Scienze Biomediche, Universit~ degli Studi di Sassari, Italy
Abstract--Good practice rules in the management of a m o v e m e n t analysis laboratory recommend that photogrammetric measurement errors are assessed, prior to every experimental session, using an ad hoc experiment referred to as a spot check. The paper proposes an inexpensive and easy to make spot check. The test uses a rigid rod carrying two markers and a target point taken on the line joining them and coinciding with the rod tip. The latter point is placed in a fixed and measured position in the laboratory frame and the markers are tracked while the rod is kept stationary and while it is manually made to rotate about the target point. Several target points are used within the measurement volume. The instantaneous errors with which the laboratory co-ordinates of the latter points are reconstructed are determined and submitted to statistical analysis. A normalisation procedure is illustrated that aims at making the test results independent from the geometry of the test object. The experimental and analytical methods underlying the proposed spot check were validated experimentally in two m o v e m e n t analysis laboratories using repeated tests. A rod, 1.5m long, carrying four markers was used. In this way, several test-object geometries were tested. Results confirmed that the photogrammetric error could be divided into a zero-mean random and a systematic component. It was shown that the normalisation procedure was effective for the standard deviation of both error components when the two markers were located at a distance between them 1.5 times larger than the distance of their centroid from the tip of the rod. The systematic component bias could not be normalised, however a conservative value of it could be estimated. The two above-mentioned normalised standard deviations and the bias value can be taken as descriptors of the photogrammetric error of the specific measuring system tested. These parameters m a y also be used to assess the precision and the accuracy with which the laboratory position of a target point, defined relative to any specified marker cluster, m a y be reconstructed during m o v e m e n t analysis. K e y w o r d s - - M o v e m e n t analysis, Stereophotogrammetry, Experimental errors, Spot check Med. Biol. Eng. Comput., 2000, 38, 260-266 J
1 Introduction HUMAN OR animal movement analysis requires reconstruction of the 3D trajectory of target (T) points, mostly anatomical landmarks, in an inertial system of reference. This objective is pursued using a stereophotogrammetric system and point markers attached to the surface of the body segments of interest. These point markers rarely coincide with the T points. The T points are either internal to the body or, if superficial, they may not be visible to at least two cameras simtfltaneously. Thus, a cluster of surface markers is associated with a body segment that complies with the technical requirements of the measuring system and that define a local, segment-embedded, flame. Thereafter, the positions of the T points relative to the latter flame are determined using appropriate measurements (anatomical landmark calibration). Knowledge of the local flame position and orientation in the Correspondence should be addressed to Dr U. Della Croce; emaih
[email protected] First received 2 June 1999 and in final form 4 February 2000 © IFMBE:2000
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stereophotogrammetric calibration-object flame and of the T points' local positions allows for the reconstruction of the global trajectories of the latter points in each sampled instant of time (CAPPOZZO et at., 1995). in most cases a more convenient inertial flame than the often arbitrary stereophotogrammetric calibration object flame is used that is consistent with the analysed motor act principal displacements or that coincides with the set of axes in which external forces are measured (dynamometer flame). This frame, referred to as the laboratory flame, may be identified using a cluster of markers suitably located in the measurement field. T points' trajectories are therefore conveniently represented in the latter flame through obvious co-ordinate transformation. Reconstructed T point trajectories are affected by errors intrinsic in the measuring system and by errors associated with the specific experimental protocol used. The latter errors are mainly associated with the movement of the surface markers relative to the skeleton (LAFORTUNE and LAKE, 1991; KARLSSON and LUNDBERG, 1994; CAPPOZZO et at., 1996; HOLDEN et at., 1997; REINSCHMIDT et at., 1997), and with anatomical landmark identification and calibration uncertainties (DELLA CROCE et at., 1999). These were not relevant to Medical & Biological Engineering & Computing 2000, Vol. 38
the objectives of the present study. The former errors are referred to as photogrammetric errors and are considered here to include the error with which instantaneous marker positions are reconstructed in the stereophotogrammetric calibration object frame, the propagation of this error to the estimated position and orientation of the body segment local frames and laboratory frame, and, finally, to the estimated T point coordinates in the latter frame. Errors affecting marker reconstructed position may have a random origin associated with electronic noise, with the imprecision with which marker images are reduced to image points (marker flickering), and with the quantisation inherent in the digitising process which transforms the marker image co-ordinates into their numerical values (FURN1~E, 1997). These errors, as observed over time, may be hypothesised to have a zero mean value. They will be referred to as instrumental random errors (IRE). Other errors are of a systematic nature such as those caused by photogrammetric calibration inaccuracies or nonlinearities that this calibration could not take care of (for instance, lens distortion). These errors vary depending on the instantaneous location of the marker within the measurement field (GAZZANI, 1993). They will be referred to as instrumental systematic errors (ISE). Good practice rules in the management of a movement analysis laboratory, both in a research and clinical context, recommend that the above-mentioned photogrammetric errors are assessed prior to every experimental session, using an ad hoc experiment referred to as a spot check, in addition, in a data sharing or storage context, the results of this spot check should be attached to the relevant biomechanical data set (CAPPOZZO and DELLA CROCE, 1994). This allows the remote user to apply these data properly and assess their propagation through the specific procedure used for the estimation of functional variables such as those that describe joint kinematics and dynamics (CAPPOZZOet al., 1997). A classic method to estimate the instrument error of a stereophotogrammetric system on a routine basis, consists in moving a rigid rod c a w i n g two markers at a known distance throughout the measurement volume. The inter-marker distance is then estimated in each sampled instant of time and its sample mean and variance calculated (MORRIS and MACLEOD, 1990; BORGHESE and FERRIGNO, 1990; EHARA et al., 1995). These results are taken as a compact description of the overall performance of the system. However, this test provides no information about accuracy, and, above all, it is not consistent with the actual objective of the measurement, as described above, in that it does not assess the error with which a T point is reconstructed in the laboratory frame. Other authors have assessed the effect of stereophotogrammetric errors on the estimation of the relative orientation of two rigid bodies. This was done using mechanical devices that simulated two adjacent body segments and the intervening joint instrumented with rotational transducers. Measured angles were compared with their values obtained using reconstructed marker positions and, in this way, the relevant accuracy was assessed (SAMUELSON, 1987; DELUZlO et al., 1993). The parameters thus obtained were, of course, dependent on the geometry of the marker clusters and on the data processing algorithm used to calculate angles. in this study a spot check ofa stereophotogrammetric set-up is proposed. This check entails the execution of a simple, inexpensive, test (movement analysis laboratory, MAL-test) and straightforward calculations, it provides quantitative information of both the accuracy and precision with which the instantaneous position o f a T point may be reconstructed. In addition, it allows for the identification of different error components and of the relevant properties. The possibility of effectively comparing the results of spot checks obtained in different laboratories through normalisation is also tackled. Medical & Biological Engineering & Computing 2000, Vol. 38
2 M A L test The MAL test is based on recording the movement of a rigid object carrying a marker cluster (MC) and on which a T point is defined. This MC-T ensemble simulates a body segment. The test is to be carried out in the same experimental conditions that apply to the actual biomovement experiments with regard to frame rate and average marker speed. The MC-T system proposed for this test is the simplest ensemble possible (Fig. 1): a rigid rod carrying a cluster of two markers (A and B) and the T point taken on the line joining the two markers and coinciding with the rod tip. To assess both accuracy and precision of the stereophotogrammetric set-up, the instantaneous position of the T point in the laboratory frame must lend itself to be reconstructed using photogrammetric information and to be determined through an accurate direct measurement. To satisfy these requirements the T point is placed in a fixed point (F). The rod is made to rotate manually around its tip (movement test) at a speed which would approximate marker average speed during the biomovement experiment. This rotation may either be a pseudo circle (Fig. 1) or composed of two orthogonal arches, both pivoting around point F. The orthogonal arches allow for a considerable variation of all three marker laboratory co-ordinates as opposed to the circular trajectory which is characterised by a quasi-constant vertical co-ordinate. However, the circular trajectory is easier to perform and only a single data capture session is required. A record while the rod is kept stationary is also required (static test). For each experiment the following quantities are determined: • the geometry of the test object, that is the relative distances between points A, B and T (ab, bt, respectively), by direct measurement; • the laboratory co-ordinates of point F (i.e. the position vector YF), by direct measurement; • the 3D co-ordinates of markers A and B (i.e. the position vectors YA and YB) reconstructed using stereophotogrammetric data, during both the movement and the static test.
3 Analysis A general expression relating the T point laboratory position vector (YT) and local position vector (XT), expressed in a local frame fixed with respect to the MC-T system, is:
YT=RxT +P
( 1)
•
J
AO G
BO ,y
J Z
Fig. 1 The rod used as test object and the proposed circular movemerit pelformed during the experiment
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where p is the laboratory position vector of the origin of the local frame and R is the orientation matrix of the local frame with respect to the laboratory frame. Defining the direction of the line joining A and B as the M C X axis, and the midpoint C between them as its origin, and by calling ct the distance between C and T the following expressions for the right terms of eqn 1 are obtained: YA + YB P-2
[ci]
XT=
R=[r
1 r 2 r3]
(2)
-
-
(3)
(4)
(5)
the following equation is obtained: YT = ( 1 + k)yB + ( 1 - - k)yA
(6)
from which:
dyT ~ (l+k)dyB+ (1-k)dyA
(7)
where dy A and dy B are the errors affecting the reconstructed marker positions and dy r is the error that affects the reconstructed point T position vector, in each sampled instant of time. The errors dy a and dy m and dy r have the following properties: (a) the components of dy A and dy B may be considered
uncorrelated (CAPPOZZOet al., 1983); (b) given the property (a), it is reasonable to assume that the generic component j of dy B and the generic component i of dy B with i ~ j , are also uncorrelated. Properties (a) and (b) entail that the errors affecting point T co-ordinates may be dealt with separately, Thus, the estimates of the bias error (b) and the standard deviation (s) of each coordinate component of the random variable dy r can be calculated as follows: b; =
1 ~ dyTj ,
Sj z (xl'~ 1~ tNyTj-- tl~NyTjtt2t 1/2
bj = 1[(1_ k) S dyAj+ (1_~_k) S dyBj]
(1
Sj = ~S
(8)
(9)
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(11)
Hypothesising that k >> 1/2, and dividing by k the following normalised bias and standard deviation of the components of the error vector dy r are obtained:
bij = ~1 ~ (dyBj- dyAj )
(12)
S(j =1 -'~---~~ ((dyBj-- dyAjt d3'N -
( 13 )
Thus, from these equations it can be noted that under the hypothesis that the stereophotogrammetric errors comply with the above mentioned properties and the test object geometry is such that k > 1/2, that is ct > ab/2, the normalised bias and the standard deviation that describe point T co-ordinates error are independent from the geometry of the test object. As mentioned previously, the photogrammetric error may be looked upon as having both a random (IRE) and a systematic (ISE) component. Given their origins, the ISE and IRE contributions to the total error can be considered additive and uncorrelate& then the following holds: s2 =
2 + Ssj 2 sRj
j = X, Y, Z
(14)
where the subscripts R and S denote the random and the systematic component of the error, respectively. Eqn 14 is valid for both static and movement tests, however, for the former test the ISE contribution to the total standard deviation can be assumed to be equal to zero. Thus, the two error components may be described separately using the following parameters: • for the IRE: the standard error sRj = sj estimated during the static test using eqn 9; • for the ISE: the bias error bj and the standard error 9 1/9 ssj = ( s 9j - s~j) -, where both bj (eqn 8) and sj (eqn 9) are estimated during the movement test. The expected error in the instantaneous T point co-ordinates can also be effectively described by using the concepts of precision and accuracy which summarise the statistical parameters referred to above. Precision refers to the degree of dispersion of estimated values about the relevant mean value. in the present case, the IRE standard error can be used to describe the relevant mean precision (rap) over the observed interval of time. Thus, for each laboratory co-ordinate:
nlpj = sRj where j = X, Y, Z and N is the number of samples. The error components dyTj can be calculated by subtracting the reconstructed point T position vector YT from the measured nominal position vector YF in each sampled instant of time. The parameters bj and sj allow for the assessment of the stereophotogrammetric error.
(10)
((~-~-k)dyBj-~-(~-k)dyAj
--(1S(~-~-k)dyBj-~- (~-k)dyAj
where r 1, r 2 and r 3 are the unit vectors of the local frame axes. Among these unit vectors only r 1 is determined (r 1 ---(YA--YB)/ab), while r 2 and r 3 are irrelevant. Substituting eqns 2, 3 and 4 into eqn 1, and defining the test-object geometry index
ct k= ab
Substituting eqn 7 into eqns 8 and 9, point T co-ordinates bias error and standard deviation can be expressed as a function of the marker co-ordinates errors in the following form:
(15)
Accuracy refers to the degree of closeness of an estimated value to the relevant true value. In the present case, a mean value of the accuracy (ma) of co-ordinate j, throughout the tested volume, can be calculated as:
maj=(b2 + S2j) 1/2
(16)
Medical & Biological Engineering & Computing 2000, Vol. 38
Of course, in principle, the normalisation with respect the test object geometry referred to above is applicable also to SRj and Ssj, and to mpj and maj.
The rod was inclined by approximately 25 degrees with respect to the vertical and made to rotate around its tip by manually moving the other end along a full circle or along two orthogonal arches in the X Y and Z Y planes, respectively. The above-mentioned movement was done manually at three different speeds within the range 0.2-0.5 cycles s -1 in order to assess the sensitivity of the results to this parameter. An additional data acquisition trial while the rod was kept stationary in the vertical position was performed. Ten spot checks were carried out in each laboratory in five different days.
4 Experiments
Experiments were carried out in two different laboratories equipped with optoelectronic photogrammetric systems that are commercially available. Both used retro-reflective markers. The measurement volumes were similar (approximately 2 . 5 x 2 x l m 3) and suitable for human locomotion studies. Two cameras were used with their optical axes at approximately 30 degrees. The photogrammetric systems were calibrated using the procedures recommended by the respective manufacturers. The laboratory frame was defined making reference to a force plate embedded in the floor in the centre of the photogrammetric measurement volume and point F was defined on it with an accuracy better than 1 mm. This frame is indicated in Fig. 1 and had the Z axis towards the stereo pair. The transformation matrix between the photogrammetric frame and the laboratory frame was determined using the reconstructed co-ordinates of four markers located at the four comers of the force plate and a least squares approach (CAPPOZZOet al., 1997). All acquisitions were carried out at 100 frames s -1. A paradigmatic case was chosen whereby the size of the test obj ect and the amplitude of its movement were made to resemble a human lower limb during the stance phase of a gait cycle. A rigid rod 1.5 m long with a tip at one end was used. Five markers were located at 0.4, 0.6, 0.8, 1 and 1.2 m respectively from the tip. In this way, several combinations for distances ab and ct were obtained and, using the same trial, it was possible to test the sensitivity of the results to these parameters and the practical feasibility of the normalisation procedure illustrated above. The geometry parameters of the test object were assessed with an accuracy better than 0.5 mm.
5 Results
and discussion
The dependency of the estimated errors on the type of rod trajectory (pseudo circle or orthogonal arches) and on movement speed was found to be negligible, therefore, in the following, only the circular movements of the rod at approximately 0.3 circles s -1 will be analysed. The results obtained during the repeated tests in the two laboratories, although numerically slightly different, led to the same general conclusions. Thus, only the data obtained in one laboratory and relative to a randomly selected trial will be used to exemplify the present discussion. Figs 2 and 3 show examples of time histories of the reconstructed T point laboratory co-ordinates during a static and a movement test (pseudo circle), respectively. The results obtained using two different MCs are shown. The magnitude of the errors that affected the estimated point T co-ordinates depend on the MC-T system geometry, as expected. These magnitudes are different depending on the co-ordinate. This is consistent with the different sensitivity to experimental errors that the stereophotogrammetric model exhibits while reconstructing each marker co-ordinate (WOLTRING, 1990).
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in the traces of Fig. 2 the random component of the error is overwhelming. This is mostly represented by high-frequency noise superimposed on the so-called flickering effect which causes sudden shifts of the co-ordinate value. For obvious geometrical reasons these errors increase as the value of k increases. The traces in Fig. 3, in addition to the highfrequency noise, exhibit a low-frequency error associated with the fact that the MC during the movement assumes different positions in the measurement volume. The latter error component is, therefore, to be associated with the residual measurement volume deformation referred to previously as systematic error. These results confirm the appropriateness of dividing the photogrammetric error into the two additive and uncorrelated components referred to above as iRE and iSE. Fig. 4 shows typical plots of the iSE bias and standard deviation and the iRE standard deviation against the parameter k. These plots indicate that both iSE and iRE standard deviations may be considered to be linearly correlated with k and that for these parameters the normalisation with respect to the test object geometry is feasible. This, which is theoretically true for k >> 1/2, appears to hold in practice for k equal to or greater than 1.5. Evidence for this circumstance is provided by the figures in Table 1. The iSE bias exhibited no correlation Table 1 ISE and IRE normalised standard deviations (S~sj and s~j, respectively) for each co-ordinate, as obtained through a linear regression Ovith intersect imposed equal to zero) o f the relevant values associated with different MC- T system geometries (1.5 < k < 5.5). The coejficients o f determination 0") are also given
ISE !
X Y Z 264
5 Conclusions The results of this study led to the following operational suggestions:
IRE !
with k. This indicates that this parameter cannot be normalised. it is evident that the conditions that must apply for the normalisation to be feasible are not satisfied by the errors that affect this parameter, in addition, by comparing the results of tests carried out with different locations of point F, i.e. exploring different portions of the measurement volume, it was seen that the iSE bias assumed values that could not be correlated with any of the geometrical or positional variables involved. This finding suggests that the spot check should include the performance of several movement tests in different regions of the measurement volume and the maximum iSE bias taken as a relevant conservative estimate. Given the results discussed above, precision may be normalised with respect to the test object geometry, but accuracy cannot (see eqns 15 and 16). However, given the above-mentioned iSE and iRE normalised standard deviations and a conservative estimate of the iSE bias, it is possible to assess the precision and accuracy with which the laboratory position ofa T point, e.g. an anatomical landmark, defined on a subject's body within a specified MC-T system may be estimated during the analysis of a biomovement. The smallest linear dimension of the MC will be divided by the distance between the MC centroid and the T point thus obtaining a k value. The normalised iSE and iRE standard deviations will be multiplied by this index and then, using eqns 15 and 16, an estimate of the relevant precision and accuracy obtained.
SS, rain
r
SR, rain
r
2.6 1.2 3.3
0.984 0.978 0.979
1.2 0.1 1.7
0.947 0.949 0.901
• a test object such as the one depicted in Fig. 1 should be used with the markers positioned so that the index k is greater than 1.5; • the MAL test should consist of a static test and several movement tests carried out in different locations of the measurement volume; Medical & Biological Engineering & Computing 2000, Vol. 38
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References
BORGHESE,N. A., and FERRIGNO,G. (1990): 'An algorithm for 3-D
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Plots" o f the ISE bias error (by), the ISE standard deviation (ss/), and the IRE standard deviation (sR/) o f the A; Yand Z co-ordinates o f point T against the MC-T system geometo; index (k), obtained during a circular movement • X: • Y: OZ
• the IRE and the ISE normalised standard deviations and a conservative value of the ISE bias are estimated as illustrated previously; • these three parameters are taken as descriptors of the photogrammetric error of the specific measuring system tested and attached to the relevant biomechanical data files; • these parameters provide an overall performance index of the measuring set-up and, in addition, may be used to assess the precision and accuracy with which the laboratory position of a target point defined within any specified MC-T system may be reconstructed. Ac~Tlowledgments~his work was carried out within the CEC programme A I M ~ r o j e c t A-2002 CAMARC-II. The constructive discussions which the authors have had with the project partners regarding the problem addressed in this paper are gratefully acknowledged. Copies of the CAMARC II Internal Reports and Deliverables quoted in this paper may be requested from the Project Co-ordinator Professor Tommaso Leo, Universit/t degli Studi di Ancona, Dipartimento di Elettronica ed Automatica, Via Brecce Bianche, 1-60131 Ancona, Italy.
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automatic movement detection by means of stmadard TV cameras', IEEE Trans. Biomed. Eng., 37, pp. 1221-1225 CAPPOZZO, A., LEO, T., and MACELLARI, V. (1983): 'The CoSTEL kinematic monitoring system: performance and use in human movement measurements' in MATSUI, H. and KOBAYASHI, K. (Eds): 'Biomechaaaics' (Human Kinetics Publishers Inc., Champaign, Illinois), pp. 1067-1074 CAPPOZZO, A., and DELLACROCE, U. (1994): 'PGD lexicon, Version 1.1 ~ in Measurement and data processing methodology in clinical movement analysis. CEC project CAMARC II (A-2002)/AIM, Internal Report CAPPOZZO, A., CATANI, E, DELLA CROCE, U., mad LEARDINI, A. (1995): 'Position and orientation of bones during movement: anatomical frame definition and determination', Clin. Biomech., 10, pp. 171-178 CAPPOZZO, A., CATANI, E, LEARDINI, A., BENEDETTI, M. G., mad DELLA CROCE, U. (1996): 'Position and orientation of bones during movement: experimental artefacts' Clin. Biomech., 11, pp. 90-100 CAPPOZZO, A., CAPPELLO,A., DELLACROCE, U., mad PENSALFINI,E (1997 ): ' Surface-maxker cluster de sign criteria for 3-D bone movement reconstruction', IEEE Biomed. Eng., 44, pp. 1165-1174 DELLA CROCE, U., CAPPOZZO, A., mad KERRIGAN, D. C. (1999): 'Pelvis and lower limb anatomical landmark calibration precision and its propagation to bone geometry and joint angles', Med. Biol. Eng. Comput., 37, pp. 155-161 DELUZIO, K. J., WYSS, U. R, JIAN LI and COSTIGAN,P. A. (1993): 'A procedure to validate three-dimensional motion assessment systems', J. Biomech., 26, pp. 753-759 EHARA, Y., FUJIMOTO, H., MIYAZAKI,S., TANAKA,S., and YAMAMOTO, S. (1995): 'Comparison of the performance of 3D camera systems', Gait & Posture, 3, pp. 166-169 FURNt~E, H (1997): 'Real-time motion capture systems', in ALLARD, P., CAPPOZZO, A., LUNDBERG, A., and VAUGHAN, C. L. (Eds): 'Three-dimensional analysis of human locomotion' (John Wiley & Sons, London), pp. 85-108 GAZZANI, E (1993): 'Comparative assessment of two algorithms for calibrating stereophotogrammetric systems', J. Biomech., 26, pp. 1449-1454 HOLDEN, J. E, ORSINI,J. A., SIEGEL,K. L., KEPPLE,T. M., GERBER,L. H., and STANHOPE,S. J. (1997): 'Surface movement errors in shaxtk kinematics and knee kinematics during gait', Gait & Posture, 5, pp. 217-227 KARLSSON, D., and LUNDBERG, A. (1994): 'Accuracy estimation of kinematic data derived from bone anchored external markers'. 'Proc. Third Int. Syrup. 3-D Analysis of Human Movement', July 5-8, 1994, Stockholm, Sweden, pp. 27-30 LAFORTUNE,M. A., and LAKE, M. J. (1991): 'Errors in 3d analysis of human movement'. Proc. Int. Syrup. 3-D Analysis of Human Movement, July 28-31, 1991, Montreal, Quebec, Canada, pp. 5556 MORRIS, J. R. W., and MACLEOD,A. (1990): 'An investigation of the sources and characteristics of noise in a video-based kinematic measurement system' in 'Models, connections with experimental appaxatus and relevant DSP techniques for functional movement analysis'. CEC project CAMARC II (A-2002)/AIM, Internal Report REINSCHMIDT, C., BOGERT,A. J. VAN DEN, MURPHY,N., LUNDBERG, A. N., and NIGG, B. M. (1997): 'Tibiocalcaxleal motion during rtmning-measured with external and bone markers', Clin. Biomech., 12, pp. 8-16 SAMUELSON,B. (1987): 'A device for the three dimensional registration of human movement', J. Ergonomics, 30, pp. 1655-1670 WOLTRING,H. (1990): 'Model and measurement error influnce in data processing' in BERME,N. and CAPPOZZO,A. (Eds): 'Biomechaxfics of human movement-applications in rehabilitation, sports and ergonomics' (Bertec Corporation, Worthington, Ohio, USA), pp. 203-237
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Authors" biographies UGO DELLA CROCE received his PhD degree in biomedical engineering from the University of Bologna, Italy. In 1994-95 he held an appointment at the Neuromuscular Research Center, Boston University. Since 1996 he has been a research associate at the Department of Physical Medicine and Rehabilitation, Harvard Medical School and since 1997 assistant professor at the University of Sassaxi, Italy. His interest is in biomechanics, including posmrography, electromyography and image analysis applications for motion measurements.
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AURELIO CAPPOZZO received his degree in electronic engineering in 1970 from the Polytechnic of Milan and a PhD degree in bioengineering in 1982 from the University of Strathclyde, Glasgow. He was a research professor and then associate professor of medical physics at the University of Rome, 'La Sapienza', from 1971 to 1994. He is now a full professor of bioengineering at the Medical School of the University of Sassari, Italy. His major research interest is in human movement biomechaxfics.
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