Jour nal of Sports Sciences, 1996, 14, 219-231

Quanti® cation of the uncertainties in resultant joint m om ents com puted in a dynamic activity JO H N H . C H A L L IS 1 * an d DAVID G. K E RW IN 2 1

Applied Physiolog y Research U nit, School of Sport and Exercise Sciences, University of B irm ingham, Edgbaston, B ir m ingham B 15 2TT and 2 B iom echanics Research Laborator y, Loughborough U niversity, Loughborough LE 11 3TU, U K

Accepted 26 October 1995

Resultant joint m oments are an important variable with which to examine human movem ent, but the uncertainty with which resultant joint m oments are calculated is often ignored. This paper presents a procedure for exam ining the uncertainty with which resultant joint moments are calculated. The uncertainty was calculated by changing the parameters and variables required to compute the resultant joint m oments, by amounts relating to their estim ated uncertainties, and then quantifying the resulting change in the resultant joint m oments. The procedure was applied to the elbow joint during loaded elbow ¯ exion executed at m aximum volitional speed. For this activity, the estimated m oments were most sensitive to uncertainties in the derivatives of the position data. A number of other sources of error and uncertainty were identi® ed which warrant further investigation. The protocols outlined in this study are applicable to other activities. K eywords : Errors, resultant joint moments, uncertainties.

Introduction Resultant joint m oments have been com puted to provide insight into the functions of m uscle and the coordination of hum an movem ent, for example the clinical assessm ent of hum an gait (e.g. W inter, 1980; Lai et al., 1988). The resultant joint m om ents are the result of m uscle forces, ligam ent forces and forces due to articular surface contact acting about a speci® ed joint. It is com m on to assum e that these mom ents are caused entirely by the muscles crossing the joint. T his is done by assuming that ligam ent forces are insigni® cant and that the articular contact force acts through a single point only, the pre-selected joint centre, m aking the m oment arm zero and therefore its contribution to the resultant inter-segm ental mom ent zero. These assum ptions are com m only m ade, in which case the resultant joint m om ent represents the m uscle m om ent at the joint. Resultant joint m om ents have been used to exam ine a variety of sporting activities, including jogging (W inter, 1983), weightlifting squat (C appozzo et al., 1985) and baseball pitching (Feltner, 1989). The *Address all correspondence to John H. Challis, Biom echanics Laboratory, The Pennsylvania State U niversity, U niversity Park, PA 16802, U SA. 0264-0414 /96

©1996 E. & F.N. Spon

value of such an analysis was stated by W inter (1980), who claim ed: `One of the m ost valuable biomechanical variables to have for the assessm ent of any hum an m ovem ent is the tim e histor y of the mom ents of force at each joint’ . T he analysis of resultant joint m om ents can lead to increased understanding of the roles played by m uscle groups in producing m otion, and thus can be used to increase the understanding of any m ovement. T hese analyses can focus on m any different aspects of hum an activity. For exam ple, the use of heel inserts to raise the heel during heel-toe running has been hypothesized to reduce Achilles tendon forces. Reinschm idt and N igg (1995) used ankle joint resultant m om ents to infer the load on the Achilles tendon during running to exam ine this hypothesis. M any m otor tasks can be executed with a variety of m otor patterns; Young and M arteniuk (1995) exam ined how a kicking skill was learnt by exam ining how the sequencing of resultant joint m om ents changed during the learning of the task. These two exam ples indicate som e of the ways in which resultant joint m om ents can be used to exam ine hum an m ovem ent, but this is only a sm all indication of their potential. Resultant joint m oments m ay be com puted not only for their own sake but also as an interim step in the com putation of other variables, for exam ple the

220 com putation of joint powers (e.g. Van Ingen Schenau et al., 1990: cycling) and the estim ation of individual m uscle forces (e.g. Challis and Kerwin, 1993: dum bbell curl). Given the widespread applicability of resultant joint m om ents in the analysis of human m ovem ent, an exam ination of the uncertainty associated with their calculation seem s justi® ed. A num ber of researchers have examined the error or uncertainty with which resultant joint m om ents are determ ined. N ote that it is com m on to use the term s `error’ and `uncertainty’ inter-changeably (e.g. Taylor, 1982; Barford, 1985), although Holm an (1989) has argued that an error is some incorrect assessm ent of a value, w hich can be quanti® ed, whereas an uncertainty is the possible value an error m ay have. In this paper, the de® nitions of Holm an (1989) w ith regard to error and uncertainty are used. To com pute resultant joint m oments, a num ber of assum ptions are m ade about the hum an m usculoskeletal system , and the m easurem ents m ade of that system . Cappozzo (1991) discussed the in¯ uence on gait analysis of m any of the errors arising from these assum ptions. Others have quanti® ed the effect of one assum ption on the com puted m om ent; for exam ple, Kadaba et al. (1990) assessed the in¯ uence of uncertainty in the reconstruction of em bedded reference fram es on resultant joint m om ents. In a sporting context, resultant joint mom ents are often higher than those found in ever yday activities, and it is therefore possible that these assum ptions m ay be untenable to a greater extent. It is the aim of this paper to present a procedure for assessing the uncertainty with which resultant joint m om ents are determ ined; an exam ple of the application of this procedure is presented for a dynam ic activity.

Sources of error and uncertainty Resultant joint m om ents are norm ally calculated during hum an m ovement by taking m easurem ents of the positions of body segm ent landm arks using an im agebased motion analysis system (e.g. W inter, 1983; Cappozzo et al., 1985). T he processing of these landm ark position data gives the positions of the centres of m ass and the orientations of the segments; these data are num erically differentiated to give the ® rst and second derivatives of the positions of the centres of m ass and the orientations of the segm ents. T he required body segm ent inertial param eters can be calculated from a variety of sources, for exam ple regression equations based on cadaver data (e.g. C lauser et al., 1969; Hinrichs, 1985), or from modelling the body segm ents as geom etric solids (e.g. Jensen, 1978; H atze, 1980). T he sources of error and uncertainty m ay be due to:

Challis and Kerwin 1. 2. 3. 4. 5.

The m otion m easurem ent system . The force measurem ent system . The de® nition and com putation of body axes. The estim ation of joint centroids. The determ ination of body segm ent inertial parameters. 6. The com putation of derivatives. Suitable calibration should perm it quanti® cation of the errors introduced due to the m otion m easurem ent system and force m easurem ent system . To a large extent, any uncertainties in the m otion measurem ent system will be incorporated into any estim ation of the uncertainties in the de® nition of the body axes and joint centres. W ith regard to force m easurem ent in the case exam ined here, the distal rigid body was in free space; that is, there were no external forces which norm ally require the use of a force m easurem ent system for their determ ination. C onsequently, this source of uncertainty was elim inated.

Theoretical analysis To determ ine resultant joint mom ents, the hum an body is m odelled as a system of articulated rigid bodies. To determ ine the m om ents, the kinem atics and inertial properties of the rigid body must be known. The resultant joint forces are therefore calculated from the dynam ic equilibrium condition that the sum of all the forces acting on a segm ent m ust equal the product of the m ass of the segm ent and the linear acceleration vector of the segm ent. T he resultant joint m om ents are calculated from the condition that the sum of all the m om ents acting about the segm ent centre of m ass m ust be equal to the rate of change of the angular mom entum vector. o Fi = m ´ a o M ci =

(1)

dL c dt

(2)

where o F i = sum of all the forces acting on the segm ent; m = m ass of the segm ent; a = acceleration vector of the segm ent; o M c i = sum of all the mom ent vectors acting about the centre of m ass of the segment; and L c is the angular m om entum vector of the segm ent. Expansion of equation (1) and (2) gives the following: m ´ a = F D + FP + F G dL c dt

= M I = M P + M D + (r 1 ´ F D ) + (r 2 ´ F P )

(3) (4)

221

Uncertainties in resultant joint moments where F D = force vector acting at the distal joint of the body segm ent; F P = force vector acting at the proxim al joint of the body segm ent; F G = force vector acting on the segm ent due to gravity; M I = inertia m om ent vector acting about the centre of m ass of the segm ent (from Euler’s equations); M P = mom ent vector acting about the proxim al joint centre; M D = m om ent vector acting about the distal joint centre; r 1 = displacement vector from the distal point of force application to the centre of m ass of the segm ent; r 2 is the displacement vector from the proxim al point of force application to the centre of m ass of the segment; and ´ refers to the vector cross-product. T hese forces and m om ents acting on a rigid body are illustrated in Fig. 1. Substitution of the appropriate values in equations (3) and (4) gives the resultant joint m om ents, but before such a substitution is performed the param eters and variables m ust be com puted. The param eters are inertial parameters of the segm ent, the variables are the location of the proxim al and distal joint centres, location of segm ent centre of mass, linear acceleration of the segm ent centre of m ass, and angular velocity and acceleration of the segm ent. For each of the param eters or variables, there will be an uncertainty associated with their estim ation, and there will be propagation of these uncertainties in any variable determ ined from their com bination. T he angular velocities and linear and angular accelerations are not norm ally m easured directly, but estim ated by differentiating the m easured

position and orientation data, which adds m ore uncertainty into the m om ent estim ation procedure. The uncertainty in any param eter determ ined from the com bination of param eters or variables can be determ ined using standard equations (Barford, 1985). If the required variable is P, which is a function of n variables and or param eters, then P = f(X 1 , X 2 , . . ., X n ) ( d P) 2 =

o

n

i = 1

3 ¶ X ´d X 4 ¶ P

i

2

(5) (6)

i

where X i = variable or param eter; d P = uncertainty in the variable P; ¶ P/¶ X i = partial derivative of function P with respect to X i ; and d X i is the uncertainty in variable or param eter X i . If the variable P is dependent on a large num ber of other variables, then equation (6) becom es quite unw ieldy. Perform ing such an analysis assum es the uncertainties in the variables are random and uncorrelated. Another way to assess the in¯ uence of these uncertainties and a potentially sim pler one, which does not require that the uncertainties are random and uncorrelated, is to perform a sensitivity analysis, where the param eters and variables are changed (perturbated) by am ounts relating to the estim ated uncertainty w ith which they were m easured. The change in the required variable as a consequence of this perturbation is then quanti® ed. It was this type of perturbation analysis which was undertaken in this study.

M ethods

F igure 1 Free body diagram showing forces (single arrowheads) and m oments (double arrowheads) acting on a body segment. Legend for all symbols is incorporated into the text.

This section describes the m ethod used to determ ine the kinematics and kinetics of the activity studied. A fem ale subject (height 1.68 m , body mass 65.1 kg) perform ed an elbow ¯ exion at m axim um volitional speed while grasping a 17.37 kg dum bbell. To obtain the kinematic data describing the action, the subject was ® lm ed at a nom inal fram ing rate of 200 fram es per second using a Locam II 16-mm cine-cam era. A tim e base for the activity was obtained from a set of tim ing lights placed in the ® eld of view of the cam era; this con® rmed the sam ple rate was 200 Hz. Two im ages of the activity were recorded by the use of an appropriately placed m irror. T he cam era position relative to the m irror was selected so that the m arkers on the subject were visible from both views for m ore of the arm orientations than any other cam era position. The positioning gave an angle of convergence between the two views of 130¡. C am era calibration and point reconstruction were perform ed using the direct linear transform ation (DLT; Abdel-Aziz and Karara, 1971).

222 C ontrol points were provided by a calibration structure (1.00 ´ 0.60 ´ 1.00 m ) containing 51 points in know n locations. The structure ® lled the whole of the space in which the activity took place, and the control points were distributed throughout this space. For digitizing, the ® lm was projected on to a digitizer (Term inal D isplay System s HR 48), w hich had an active area of 1.20 ´ 0.90 m and a resolution of 0.025 m m . The projected images were half life-size. For the activity exam ined, a dum bbell was grasped in the hand, and the w rist was rigidly ® xed allow ing the forearm , hand and dum bbell to be considered as one segm ent, and the upper arm the other. T he locations of three signi® cant body landm arks on the hum erus and on the forearm were used to de® ne mem ber reference fram es (the points digitized are listed in Table 2). M atrices were calculated to transform points from the inertial reference fram e to m em ber reference fram es. From these m atrices the position and attitude of the forearm relative to the upper arm were obtained. T he axes used to de® ne the rotations at the elbow joint are shown in Fig. 2. D ata sm oothing and differentiation were performed using a generalized cross-validated quintic spline (Woltring, 1986). To determ ine the inertial properties of the forearm , it was m odelled as a series of 13 truncated cones. As the hand was gripping the dum bbell, it was modelled as a truncated cone to represent the heel/base of the hand and a hollow cylinder representing the ® ngers grasping

Challis and Kerwin the dum bbell. M easurem ents to represent these geom etric shapes were m ade directly on the subject, and density values were obtained from Clauser et al. (1969). A diagram of the model is presented in Fig. 3. The inertial properties of the dum bbell were determ ined by direct m easurem ent. The subject was instructed to allow no wrist m ovem ent during the activity exam ined. Validity of this assumption was con® rm ed by checking kinematic data; therefore, the inertial properties of the forearm , hand and dumbbell were com bined to represent one rigid body. For this distal rigid body, the inertial properties of the dumbbell were dominant over those of the forearm and hand. As it was possible to determ ine with accuracy the inertial properties of the dum bbell, the inertial properties for the distal rigid body (dum bbell, hand, forearm ) were anticipated to be m ore accurate than if the distal rigid body had been composed of hum an body segm ents only. Inverse dynam ics was used to determ ine the resultant joint m om ents at the elbow joint. T herefore, the forces acting through, and the m om ents acting about, the elbow joint centre were calculated using equations (3) and (4).

Estim ation of uncertainties This section describes the procedures used to estim ate the uncertainty in the m easurem ent of the variables and param eters used to com pute the resultant joint m om ents. De® nition of body axes The de® nition of body axes perm its the calculation of joint angles and the location of the centre of m ass of the

F igure 2 Upper lim b with the axes for the elbow joint de® ned.

Figu re 3 Diagram showing the con® guration of the geometric solids used to model the forearm and hand for the determination of inertial properties.

223

Uncertainties in resultant joint moments distal rigid body; the position of the latter is de® ned in the body ® xed reference fram e. Lansham m ar (1980) presented a series of form ulae which give an estim ate of the noise that can be exp ected to rem ain in a noisy signal after sm oothing and or differentiation. The basic form at of the form ulae is:

s

2 k

³

s

2

´t ´w

(2 k + 1 ) b

p (2k + 1)

(7)

where k = the order of the derivative; s 2k = the variance of the noise affecting the k th derivative, after processing; s 2 = the variance of the additive white noise; t = the sam ple interval; w b = the bandw idth of the signal in radians ( w b = 2´p ´f b ); and f b is the sample bandwidth in H ertz. The assumptions implicit in using these form ulae are outlined in the discussion. The frequency com ponents of the signals describing m ovement of the centre of m ass, and the orientation of the forearm in relation to the upper arm , were all determ ined, and the bandwidth of the signals com puted using standard frequency analysis techniques (Rabiner and G old, 1975). If the noise contam inating a signal is white, this m eans the noise is not correlated between sam ples and has a m ean value of zero. On summ ing repeat m easures the true underlying signal is additive, and the noise is statistically independent so that it tends to its m ean value which is zero. Therefore, by taking the average of a number of repeat sam ples, the signal-to-noise ratio is im proved. If n repeat m easures are taken, the true signal size is increased by a factor n, the variance of the noise is increased by the same factor, w hile the standard deviation of the noise is increased by a factor equal to the square root of n. O n taking the average, the signal am plitude to noise standard deviation ratio is increased by the square root of n. To analyse the random noise introduced in the digitizing process, four frames were random ly selected from the ® lm of the activity and digitized 10 tim es. From the digitized coordinates of the points, a three-dim ensional reconstruction was perform ed, and then using the procedures already described the position of the centre of m ass and the joint orientation angles were determ ined. For each of the four fram es, 10 estim ates of these variables were m ade. T he variances of these estim ates were calculated for each fram e, and then the m ean variances for the four fram es determ ined. Estimation of joint centres T he resultant joint m om ent is calculated with reference to som e pre-de® ned joint centre; the com puted resultant joint m om ent values are therefore dependent on the

location of the joint centre, so accurate assessm ent of the joint centre location is essential. T he instantaneous joint centres for two-dim ensional and three-dim ensional analyses (Suh and Radcliffe, 1978) represent points through which the resultant joint forces pass, assum ing the joint is a kinematic connection (i.e. w here any force associated with this connection does not generate or absorb power). T he location of such a point can be calculated using kinematic data. Such a procedure was not taken in the present study, but the m ean centroid of rotation was de® ned as the proxim al point of `force application’ ; the position of this centroid was obtained from the literature and de® ned as the centre of the trochlea of the humerus. This location corresponds with the location of the centre of rotation as determ ined by Fischer (1909) and C hao and M orrey (1978) during ¯ exion. The location of the centroid also corresponds well with the location of the instantaneous helical axes during ¯ exion-extension and supinationpronation calculated by Youm et al. (1979). If the joint centre had been incorrectly located, this could have resulted in an uncertainty in the estim ated resultant joint m om ents. T he location of the joint centre was therefore sequentially varied by 0.01 m from its pre-selected position for all three axes on the distal m ember reference frame. T his variation of 0.01 m was greater than the errors anticipated from the threedim ensional m easurem ent system , and so re¯ ects the com bined effects of errors in the m easurement system and variations in the location of the joint centre. T hese variations in the location of the joint centre could be caused by incorrect initial positioning of the joint centre or by m ovem ent of the joint centre away from its assum ed ® xed location during the course of the m ovem ent. Sm all variations of a few m illim etres in the location of the joint centre and the instantaneous helical axis have been reported in the literature for the elbow joint during ¯ exion and supination (e.g. C hao and M orrey, 1978; Youm et al., 1979). Deter mination of body segment inertial parameters The body segm ent inertial parameters were determ ined by m odelling the segm ents as geom etric solids. The only direct assessm ent of the accuracy of modelling the segm ents as geometric solids was obtained by com paring the volume of the segm ent as determ ined by the m odel w ith the volum e obtained by im mersing the hand and forearm in water, and m easuring the displaced volum e of water. As there were no criterion values for the rem aining inertial properties of the subject’s forearm and hand, it was not possible to assess the m odel’s accuracy directly. H owever, the relative

224 in¯ uence of these param eters on the inertial properties of the w hole system could be exam ined, since it was assum ed that the inertial param eters of the dum bbell were accurately determ ined. To exam ine the effect of the accuracy of density estim ates, different density values were used as inputs to the m odel of the forearm and hand. This inform ation was com bined with the details of the dum bbell to com pute the inertial param eters of the whole system . The percentage differences were evaluated between the whole system inertial param eters as originally calculated using the density values of C lauser et al. (1969) and the whole system inertial param eters com puted using other density values. This procedure exam ines the in¯ uence of constant density values, but it is also possible to use variable density values in geom etric m odels (e.g. Hatze, 1980). If volume is well estim ated and an appropriate m ean density value is used, m ass will be estimated with good accuracy. Variable densities in the truncated cones m aking up the segm ents could be have an effect on the centre of mass location and the m oments of inertia of the segm ents. The forearm was further divided into 16 sections so that the variable density values of Rodrigue and Gagnon (1983) could be used as inputs to the m odel of the forearm . As this paper had no density values for the hand, it was not possible to perform a full analysis of the effect of this variable density data on the w hole system. To assess the in¯ uence of inaccuracy in any one of the param eters, each of the param eters com puted using the density data of C lauser et al. (1969) was changed by 5 and 10% and the in¯ uence of this on the whole system exam ined. T hese changes are arti® cial because it is unlikely that any one param eter would change without another also changing. D ata smoothing and computation of the derivatives U sing the form ulae of Lansham m ar (1980), the variances in the derivatives were estimated. T he m ultiple digitization data were also used to exam ine the effectiveness of the spline smoothing routine. T he root m ean square differences were calculated for locating all digitized points w ith one digitization and the m ean of 10 digitizations. As four fram es were digitized 10 tim es, the calculated differences were then averaged for all four fram es and for each axis. T his gave an indication of the noise that m ay occur across all the fram es. To assess the perform ance of the data-sm oothing routine, the root m ean square difference was also com puted between the points with and without sm oothing for all frames digitized; again these differences were then averaged for the three axes.

Challis and Kerwin Sensitivity analysis of mom ents The resultant joint mom ents were com puted using the original parameters and variables. Resultant joint m om ents were also com puted w ith one or m ore of the param eters or variables changed by a value indicated by the uncertainty analysis; these m om ents were then com pared with those com puted using the original param eter and variable set. D ue to the nature of the activity examined and the way in which the axes were de® ned, the m om ents about the abduction-adduction axis were sm all, generally ± 1 N m, and so the analysis was perform ed for the ¯ exion-extension and supination-pronation axes only.

Results The follow ing two sub-sections give the results for the estim ation of the uncertainties in the kinematic data and body segm ent inertial param eters, and the ® nal sub-section gives the results for the sensitivity analysis of the resultant joint mom ents. Kinematic data This sub-section reports the results for the analysis of the uncertainties in the kinem atic data; com bining those errors and uncertainties for the de® nition of the body axes with those from the determ ination of the derivatives of the data resulting from these data. The results are presented in Table 1. T his analysis was only for four fram es of data, and it is assum ed for the rem ainder of the paper that these ® gures are re¯ ective of all the fram es of data. An indirect indication that this assum ption is valid is provided by an analysis of the sm oothing routines. The noise added to the digitized body landm arks was estim ated by com paring the result of one digitization with the m ean of 10 digitizations; this was then com pared with the difference between the values of the points from one digitization before and after sm oothing. T he results of this analysis are given in Table 2, and indicate that the sm oothing routine rem oved the m ajority of the noise in the data. Deter mination of body segment inertial parameters The inertial properties of the forearm, hand and dum bbell are presented in Table 3. The only direct check on the accuracy of the anthropom etric m easures was direct m easurem ent of the volum e of the forearm and hand. T he volum e of displaced water when the forearm and hand were im m ersed was m easured as 1.299 ´ 10-3 m 3 . T his com pared favourably with the

225

Uncertainties in resultant joint moments

Table 1 The minimum standard deviation of the noise affecting variables, calculated using the equations of Lansham mar (1980)

Standard deviation of noise for derivatives Standard deviation of noise Position

Signal bandwidth

0

1

m s -2

m

Hz

m

0.0016 0.0050 0.0019

6.3 6.3 4.7

0.0004 0.0013 0.0004

0.009 0.029 0.007

0.28 0.88 0.16

Angles a

rad

Hz

rad

rad s-1

rad s -2

F/E AB/AD S/P

0.0075 0.0112 0.0134

3.0 4.0 2.6

0.0013 0.0022 0.0022

0.014 0.033 0.020

0.21 0.63 0.26

x y z

m s

2 -1

a

The angles refer to rotations about the ¯ exion-extension (F/E) axis, abduction-adduction (AB /AD ) axis and supination-pronation (S/P) axis.

value of 1.297 ´ 10-3 m 3 calculated with the geom etric m odel, a difference of 0.15% . Changes in the inertial properties of the segm ent due to varying the density data are presented in Table 4.

Table 2 The root mean square difference for the m ean of 10 digitizations compared with the results for one digitization (averaged for four fram es) and the root m ean square difference for all fram es between the smoothed and nonsmoothed signal for digitized points

Root mean square differences (m) Digitized point Styloid process of radius Styloid process of ulna Olecranon process of ulna M edial epicondyle of humerus Lateral epicondyle of humerus Lesser tubercle of hum erus M edial end of dumbbell Lateral end of dumbbell

M ean data

Smoothed data

0.0016 0.0040 0.0030

0.0019 0.0030 0.0017

0.0033

0.0021

0.0029 0.0042 0.0021 0.0029

0.0010 0.0034 0.0021 0.0021

The effect of different density values on the m ass and location of the centre of m ass of the whole system was less than 1% in all cases exam ined. The largest difference in estimated m om ent of inertia due to varying density was 3.4% about the x-axis. T his occurred when the density values were 5% different from the selected values. W ith the other cadaver density data sets, the m axim um difference in any of the com puted param eters was equal to or less than 2.3%. T he effect of variable density values on the inertial properties of the forearm was sm all, giving percentage Table 3 The inertial properties of the forearm, hand and dumbbell

Inertial property Mass (kg) Centre of m ass (m )a I x (kg m 2 ) I y (kg m 2 ) I z (kg m 2 )

Forearm

Hand

Dumbbell

0.998 0.106 0.0052 0.0052 0.0006

0.411 0.060 0.0005 0.0006 0.0003

17.370 0.000 0.0470 0.1736 0.1736

a

T he centre of m ass location is the distance from the proximal joint centre to the centre of mass along the longitudinal axis only, as the segm ent was assum ed to be sym metrical about the other axes.

Table 4 The effect of different density values for the forearm and hand on the inertial parameters of the whole system, expressed as the percentage difference between the inertial param eters calculated using the density values of Clauser et al. (1969)

Percentage difference Density data

M ass

Centre of mass

Ix

Iy

Clauser et al. (1969), increased by 5% Clauser et al . (1969), decreased by 5% Dempster (1955) Chandler et al. (1975)

0.4 0.4 0.3 -0.5

-0.2 0.2 -0.1 0.2

1.7 3.4 1.6 -2.3

0.8 1.7 0.8 -1.1

Iz

0.0 0.0 0.0 0.0

226

Challis and Kerwin

differences of -2.4% for m ass, -1.3% for centre of m ass location, -0.4% for transverse m om ent of inertia, and -2.2% for longitudinal m om ent of inertia. The results for changing one of the param eters only are shown in Table 5. T he m ass of the w hole system was not ver y sensitive to changes in the m ass of the forearm and hand. The effect of this param eter was greatest on the transverse m oments of inertia (I x and I y ) of the whole system . The accuracy of m ass estim ation is dependent on the accuracy of the volum e and density values. It has been established that volum e has been well estim ated by the m odel, and the in¯ uence of the density estim ates has already been reported. For sub-

sequent analyses, the inertial param eters were changed by amounts greater than their estim ated uncertainties (see Table 5): m ass ± 1%, location of centre of m ass ± 1% , transverse m om ents of inertia ( I x and I y ) ± 5%, and the longitudinal m om ent of inertia (I z ) ± 3% . Sensitivity analysis of mom ents The resultant joint m om ents at the elbow joint for the activity examined are illustrated in Fig. 4. T he in¯ uence of the uncertainties in each of the identi® ed param eters and variables on the com puted resultant joint

Table 5 The effect of a change in any one of the param eter values on the inertial parameters of the whole system, expressed as the percentage difference between the inertial param eters of the whole system calculated using the density values of Clauser et al. (1969), and the perturbated values

Percentage difference Varied param eter M ass (5%) M ass (10% ) Centre of mass (5%) Centre of mass (10%) M oment of inertia I x and I y (5%) M oment of inertia I z (10% )

M ass

Centre of mass

Ix

Iy

Iz

0.4 0.9 0.0 0.0 0.0 0.0

-0.2 -0.4 0.2 0.5 0.0 0.0

1.7 3.4 -3.7 -7.2 1.0 1.9

0.8 1.7 -1.8 -3.5 0.5 1.0

0.0 0.0 0.0 0.0 0.0 0.1

F igure 4 The graphs show the resultant joint moments for the loaded elbow activity exam ined, from the arm being straight at the subject’ s side until it is raised to shoulder height. (a) The ¯ exion-extension moment, where a positive moment is a ¯ exion m oment and a negative one an extension moment. (b) The supination-pronation moment, where a positive moment is a supination m oment and a negative one a pronation moment.

227

Uncertainties in resultant joint moments Table 6 The variation in the resultant joint moment when different inertial values were used for the distal rigid body (forearm, hand and dum bbell)

Flexion moment (N m) Parameter and percentage change M ass (+1% ) M ass (-1%) C entre of mass (+1%) C entre of mass (-1%) M oments of inertia, x and y axes (+5%) M oments of inertia, x and y axes (-5%) M oment of inertia, z axis (+3%) M oment of inertia, z axis (-3%) M ass (+1%); centre of m ass (+1%); moments of inertia, x and y axes (+5%), z axis (+3%) M ass (-1%); centre of mass (-1%); moments of inertia, x and y axes (-5%), z axis (-3%)

Supination m oment (N m)

Mean difference

Standard deviation

Mean difference

Standard deviation

0.3 -0.3 0.3 -0.3 0.0 0.0 0.0 0.0

0.15 0.15 0.15 0.15 0.01 0.01 0.01 0.01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.04 0.04 0.01 0.01 0.01 0.01 0.03 0.03

0.6

0.31

0.0

0.05

0.6

m oments was assessed, as well as their com bined effect. T he results for this analysis are presented in Tables 6, 7 and 8. Table 6 shows that both the resultant joint m om ents about the ¯ exion and supination axes were fairly insensitive to inaccuracies in the inertial param eters of the distal segm ent. The m axim um difference for any of the cases exam ined for the resultant joint mom ents about the ¯ exion-extension axis was less than 1.0 N m , and less than 0.6 N m for the supination-pronation axis. Table 7 show s that the resultant joint m om ents were m ore sensitive to uncertainties in the kinematic variables than to uncertainties in the inertial param eters. T he m ean differences between the resultant joint m oments calculated using the original and perturbated

0.31

0.0

0.05

kinematic variables were sm all for the m om ents about both the ¯ exion-extension and supination-pronation axes. For the resultant joint mom ents about the ¯ exion-extension axis, the standard deviations of the differences were larger than the m ean difference values, indicating the sensitivity of these m easures to uncertainties in the kinematic variables. T he m axim um absolute difference for the one standard deviation case was 4.8 N m , and for the two standard deviation case 9.6 N m . For the supination-pronation axis, the m ean differences between the resultant joint m om ents calculated using the original and perturbated kinematic variables were sm aller than the standard deviations of the differences. The m aximum absolute difference for the standard deviation case was 0.5 N m , and for the

Table 7 The variation in the resultant joint moment when perturbated kinematic data and inertial param eters were used for the distal rigid body (forearm, hand and dumbbell)

Flexion moment (N m) Am ount added to kinematic variables Plus 1 Minus Plus 2 Minus Plus 1 Minus Plus 2 Minus a

S .D .

1 S .D . S .D . 2 S .D . a S . D . and inertia values 1 S . D . and inertia values a a S . D . and inertia values 2 S . D . and inertia values a

Supination m oment (N m)

Mean difference

Standard deviation

Mean difference

Standard deviation

-0.4 0.4 -0.8 0.8 -1.0 -0.2 -0.6 0.2

3.72 3.72 7.44 7.44 3.79 3.52 0.31 7.16

-0.2 0.2 -0.5 0.5 -0.2 0.3 0.0 0.5

0.17 0.17 0.35 0.35 0.19 0.18 0.05 0.34

The inertial param eters of the distal segment were varied as follow s: m ass of the system by ± 1% ; location of the centre of mass by ± 1% ; mom ents of inertia about the x and y axes by ± 5% ; and the m oment of inertia about the z axis by ± 3% . S .D ., standard deviation.

228

Challis and Kerwin

two standard deviation case 1.1 N m . The resultant joint m om ents about the supination-pronation axis were m uch sm aller than those about the ¯ exion-extension axis (see F ig. 4), and relatively the m om ents about the supination-pronation axis were less sensitive to uncertainties in the kinem atic variables than the m oments about the ¯ exion-extension axis. Com bining kinem atic uncertainties with inertial param eter uncertainties did not change signi® cantly the mean difference and standard deviation between the original and perturbated resultant joint m om ents. Table 8 show s that the differences in estim ated m oments due to incorrect location of the joint centre were sm all. T he m aximum absolute differences were 1.4 N m for the ¯ exion-extension axis and 2.5 N m for the supination-pronation axis. W hen com bined with anticipated uncertainties in the inertial param eters, there was a sm all increase in the differences between the m om ents. W hen uncertainties in the location of the joint centre were com bined with uncertainties in the kinem atic variables, there was a large increase in the difference between the original m om ent and perturbated mom ent. T his demonstrates that the m oments estim ated in this study were m ost sensitive to inaccuracies in the kinematic variables. G reater differences m ay

have arisen if the joint centre location had been varied by different am ounts throughout the m ovem ent. To com plete this analysis, the com bined effects of inaccuracies in the inertial param eters and kinem atic variables were assessed. This was done by perturbating all the inertial param eters by the am ounts used previously and all the kinematic variables by one standard deviation. These results, presented in Table 8, con® rm that the resultant joint m oments are m ore sensitive to the anticipated uncertainties in the kinematic variables than to uncertainties in the inertial param eters, the location of the joint centre, or the com bined effect of these param eters or variables.

D iscussion This section will discuss the results, and m ake som e general points about the potential accuracy of variables derived from m easurem ents of hum an body m otion. Kinematic data In the preceding analysis, the assum ption has been m ade that the noise calculated for the four random ly

Table 8 The variation in the resultant joint moment when joint centre location was varied. In addition, perturbated kinematic data and inertial parameters were used for the distal rigid body

Flexion moment (N m) C hange in joint centre location a M oved +0.01 m along x axis M oved -0.01 m along x axis M oved +0.01 m along y axis M oved -0.01 along y axis M oved +0.01 m along z axis M oved -0.01 m along z axis M oved +0.01 m along x, y and z axes M oved -0.01 m along x, y and z axes M oved +0.01 m along x, y and z axes, plus inertial values M oved -0.01 m along x, y and z axes, minus inertial values M oved +0.01 m along x, y and z axes, plus 1 S . D . of kinematic error Moved -0.01 m along x, y and z axes, minus 1 S . D . of kinem atic error Moved +0.01 m along x, y and z axes, plus positive kinematic and inertial values errors Moved -0.01 m along x, y and z axes, plus negative kinematic and inertial values errors a

Supination m oment (N m)

Mean difference

Standard deviation

Mean difference

Standard deviation

0.0 0.0 0.0 0.0 -0.8 0.8 -0.8 0.8

0.00 0.00 0.06 0.04 0.48 0.48 0.44 0.46

-0.4 0.2 0.0 -0.1 0.0 0.0 -0.4 0.1

1.05 0.95 0.89 0.92 0.00 0.00 1.34 1.27

-0.2

0.15

-0.5

1.37

0.8

0.46

-1.1

3.78

0.6

3.79

-0.6

3.70

0.6

3.79

0.3 -0.7 0.1

-0.8

0.3

M ovem ents in joint centre were m ade along the m ember reference fram e for the distal segm ent.

1.37 1.52 1.27

1.55

0.13

229

Uncertainties in resultant joint moments selected fram es gave an indication of the random noise present in the whole of the ® lm sequence. This assum ption was validated to som e extent by the results presented in Table 2. T here are three major assum ptions associated with the form ulae of Lansham m ar (1980): 1. The signal is band-lim ited. 2. The noise contam inating the signal is w hite. 3. The frequency response of the ® lter or differentiator is ideal. Slepian (1976) has shown that, from a m athematical point of view, no signal is band-lim ited. Effectively, frequencies above the N yquist frequency have to be ignored. It is not possible for the noise contam inating a signal to be perfectly w hite; for exam ple, the noise m ay be correlated between sam ples. T he assum ption that the ® lter is ideal m eans that the signal passes unattenuated up to the speci® ed ® lter cut-off, after which all of the rem aining signal is rem oved irrespective of whether it is noise or true signal. For a differentiator to be ideal, the signal m ust be suitably am pli® ed up to the cut-off frequency, after which no signal is allowed to pass. Such ® lters and differentiators are not practically realizable. These assumptions mean that the form ulae give only m inim um estim ates. Estimation of joint centroids Resultant joint m om ent values are dependent on the location of the joint centre, a point through which the line of the resultant joint force vector passes for a kinem atic connection. T he instantaneous joint centre can be calculated using the kinem atic data required for com puting the resultant joint m om ent. U nfortunately, the instantaneous joint centre or centroid can m ove throughout the m ovement, w hich makes reference to anatomical landm arks very dif® cult. In this study, variation in the centroid m ade little difference to the values of the resultant joint m om ents, but for another joint this m ay not have been the case. The assum ption about the line of action of the resultant force passing through the instantaneous joint centre or centroid is dependent on the joint being a kinematic connection. This assum ption applies only if there is no frictional loss of energy or storage of energy in elastic cartilage or joint ligam ents. T he presence of signi® cant am ounts of joint degeneration m ay invalidate the assumption that the resultant joint force vector passes through the instantaneous joint centre or centroid, particularly if the degeneration causes signi® cant amounts of joint friction. T he calculation of resultant joint mom ents in such cases requires investigation to determ ine its in¯ uence on resultant joint m om ents, particularly given the

problem of locating the instantaneous centre of rotation. Deter mination of body segment inertial parameters The analysis showed that the resultant joint m om ents were more sensitive to uncertainties in the location of the centre of m ass of the forearm and hand, than to any other inertial param eters. In this study, the centre of m ass location of the body segm ents was determ ined by m odelling them as geom etric solids; the accuracy of the location of the centre of mass was therefore dependent on how well the shape of the segm ents was m odelled (if density is assum ed to be constant). It would appear that the centre of mass location is well predicted by a num ber of techniques; M ungiole and M artin (1990) showed that a variety of techniques produced estim ates for the location of the centre of m ass within 2.5% of one another. As volum e was accurately m odelled in the present study, if the assum ption of constant density is valid, the centre of m ass location can be regarded as accurate. The change in centre of mass location w hen using variable density for the forearm was shown to be sm all. If the centre of m ass had been poorly located, it would have resulted in large uncertainties in the m om ent of inertia values about the transverse axes (I x and I y ) (see Table 5). T he inertial param eters required for this analysis were determined with less uncertainty than is com m on in hum an m ovem ent analyses due to the dominant effect of the inertial properties of the dum bbell on those of the distal rigid body. This perm itted the focus of the study to be on other relevant factors. G iven another activity, without the dom inating in¯ uence of an object with known inertial param eters, the in¯ uence of inaccuracies in the body segm ent inertial param eters m ay have been m uch higher. Lack of segment rigidity The preceding analysis has not explicitly considered m arker movem ent, which m ay occur due to lack of segm ent rigidity. To exam ine marker movem ent due to skin m ovem ent, two m arkers were selected w hich were on the same bone and suf® ciently far apart to assum e that the uncertainties associated with estim ating their positions were independent. T he m arkers considered were the m arker on the lesser tubercle of the hum erus and the m arker on the lateral epicondyle of the hum erus, the distance between them being 0.3005 m (m easured accurately using a calliper with a resolution of 0.02 m m). T he standard deviation over all the trials in comparing the calculated length w ith actual length was 0.0078 m when the data were not sm oothed and 0.0064 m when the data were sm oothed. For a rigid

230 rod of sim ilar length m oved throughout the calibrated space, the standard deviation in com paring know n length with actual length was 0.0035 m for the sm oothed data, show ing that for the m arkers on the body there was m ovement of the m arkers relative to the body ® xed reference fram e. This analysis does not fully quantify the am ount of m arker m ovem ent: if the markers both m ove in the sam e direction, then the uncertainty is reduced; equally, the opposite case m ay apply. Som e of these uncertainties may be due to both random and system atic digitizer operator uncertainties and errors, and include errors due to the precision and accuracy of the m easurem ent system . T he uncertainties associated with m arker m ovem ents im pose an additional lim it on the accuracy of calculated variables; these uncertainties could not be suf® ciently quanti® ed to perm it inclusion in the present analysis. The assum ption of rigidity im plies that there is no m otion of the segm ent relative to its ow n m ember reference fram e. In the general case, this assum ption is false, as the m uscles, organs, blood and other bodily ¯ uids m ove relative to the m ember reference fram e. D enoth (1986), in a theoretical analysis of the im pact of a segm ent onto a hard surface, showed that this `wobbling mass’ was a signi® cant variable contributing to the ground reaction forces. It is still dif® cult to m easure these effects, and until they can be quanti® ed the rigid body m odel is the best approxim ation. Sum mar y T his study has endeavoured to quantify the uncertainties in the resultant joint m om ents, due to the in¯ uence of errors and uncertainties in the param eters and variables required to com pute these m om ents. For the activity exam ined, the m om ents were m ost sensitive to the uncertainties in the derivative inform ation required to com pute these m om ents. In the analysis of hum an m ovement, it is generally not possible to have knowledge of true derivative values. C om m only, derivative inform ation is obtained from the numerical differentiation of displacement data; inherent in the position data will be noise, and despite low-pass ® ltering of the data, som e noise will rem ain in the signal and be am pli® ed due to numerical differentiation. T herefore, it m ust be anticipated that errors in the derivatives will arise and in¯ uence the uncertainty in the com puted resultant joint m oments for other activities. The rem ainder of the param eters and variables did not m ake such signi® cant contributions to the uncertainty in the resultant joint m oments, although for different activities this m ay not be the case. N evertheless, use of the protocols presented here, or sim ilar ones, will perm it the quanti® cation of the uncertainty with w hich the resultant joint m oments are determined for other activities and exper-

Challis and Kerwin im ental set-ups, highlighting the contributions of each of the param eters and variables to the uncertainties in resultant joint m om ents. It is possible that such analyses would suggest revised experim ental protocols where special attention is given to a particular param eter or variable which has a large in¯ uence for a particular activity, experim ental set-up or subject.

Conclusions As resultant joint m om ents are used to exam ine hum an m ovem ent, it is im portant that the errors and uncertainties in these resultant joint m oments are quanti® ed, and this study has presented an appropriate procedure for doing this. The m ajor factor ignored was how the lack of segm ent rigidity affected the assum ptions im plicit in using a rigid body m odel for determ ining the resultant joint m oments. T he procedure has been applied to the exam ination of an elbow ¯ exion against an external load. The resultant joint mom ents for the activity examined were most sensitive to uncertainties in the kinematic data. For the activity exam ined, uncertainties in the inertial properties of the body segments were less in¯ uential than they m ay be for other activities, due to the dominating in¯ uence of an object w ith well-de® ned inertial param eters held in the hand. The uncertainties and errors reported in this study are speci® c for the activity exam ined, but for other studies the protocols used here for uncertainty estimation could be applied to the exam ination of other m ovem ents. Such uncertainty analyses should be considered part of the standard protocol for determ ining resultant joint m om ents, or other sim ilar biomechanical variables.

Acknowledgem ent Dr Challis gratefully acknowledges the support of the Nuf® eld Foundation for part of the work reported here.

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