oO?l-9290/8O/lCQ-0833

J. Biomcchonic.~ Vol. 13. pp. 833-843. @ Pergamon Press Ltd. 1980. Printed m Great Britain

SOZ.OO,Q

A MATHEMATICAL MODEL FOR THE COMPUTATIONAL DETERMINATION OF PARAMETER VALUES OF ANTHROPOMORPHIC SEGMENTS* H. HATZE National Research Institute for Mathematical CSIR, Pretoria,

Sciences,

South Africa

Abstract - A model is presented for determining by computation the parameter values of anthropomorphic segments from a battery of 242 anthropometric measurements taken directly from the subject. The model consists of 17 segments, includes the shoulders as separate entities, and offers the following advantages over previous models: it subdivides segments into small mass elements of different geometrical structures, thus allowing the shape and density fluctuations of a segment to be modelled in detail; in general, no assumptions are made on segmental symmetry, and principal axes transformations are performed whenever necessary; the model differentiates between male and female subjects (exomorphic differences, different density functions and mass distributions), adjusts the densities of certain segmental parts according to the value of a special subcutaneous-fat indicator, and fully accounts for the specificities of pregnancy and obesity; the input data errors are drastically reduced by performing direct anthropometric measurements rather than indirect measurements from photo images ; the overall accuracy of the model is better than 3% with a maximum error of about 5%. The features listed above are confirmed by comparing experimentally determined parameter values (volumes, masses, coordinates of mass centroids, principal moments of inertia, coordinates of segmental origins) with model predictions for four different subjects. The execution time of the computer program executing the model is 0.515 seconds on a CDC Cyber 174 digital computer. A sample print-out is listed in the Appendix.

INTRODUCTION

To simulate on the computer a model of the complete neuro-musculoskeletal control system of a given individual, it is necessary to determine a set of numerical input parameter values which characterize that individual. This subject-specific parameter set consists of the subsets of segmental and articular parameters, myodynamic parameters, and myocybernetic parameters, which respectively relate to the executor (skeletal), myoactuator (muscular), and controller (neural) subsystems ofthe total neuro-musculoskeletal system. In this paper we shall be concerned only with the description of a computational method for determining segmental parameter values for a given subject. The problem of defining segmented models of the human body, and appropriately identifying the morphology of the anthropomorphic segments, has received wide-spread attention (Fischer, 1906 ; Weinbath, 1938; Dempster, 1955; Kulwicki and Schlei, 1962; Hanavan, 1964; Huston and Passerello, 1971; Vukobratovic and Stepanenko, 1973 ; Bugyi, 1976 ; Hatze, 1977; etc.). Proposed models range in complexity from a single, unsegmented rigid body (Hemami et al., 1973; Hemami and Golliday, 1977) to 15link models with simple geometrical segment shapes (Hanavan, 1964), and 16-link models in which the segments are sectioned into elliptical zones (Jensen, 1978). In all cases, the segments are modelled as rigid * Received 19 April 1979; 3L

13,io

3

in rerisedform 18 September 1979.

bodies with uniform density. The degree of approximation permissible in the construction of a segmented model of the human body is determind by the purpose of the model. While some authors consider it adequate for investigations into the stability of biped locomotion (Hemami and Golliday, 1977) to represent the actual multi-link system by a single rigid body with only one degree of freedom, it is generally accepted that for realistic simulations of gross body motions a fragmentation of the body into a minimum of 10 segments (trunk, head, arms, forearms plus hands, thighs, legs plus feet) is essential. More refined models further subdivide the trunk into a lower and upper part, and include hands and feet (e.g. the 15segment model of Hanavan, 1964), and the neck (16segment model of Jensen, 1978) as separate segments. Strangely however, the shoulder segments, which clearly constitute dynamically separate entities, are almost always considered parts of the upper trunk. Miller and Morrison (1975), and more recent]> Jensen (1978), have commented on some of the shortcomings and inaccuracies associated with the identification of anthropomorphic segments with simple geometrical bodies (such as ellipsoids, right elliptical cylinders, etc.), as is the case with the Hanavan model. However, apart from these obvious over-simplifications additional inconsistencies which introduce severe inaccuracies are common to practically all existing models. These inaccuracies are difficult to detect because they manifest themselves mainly in the computed values of the principal moments of inertia, which are not easily verifiable

833

834

H. HATZE

experimentally. Firstly, if segments are sectioned into elliptical discs of 20mm width (Jensen, 1978) these discs can no longer be regarded as infinitely thin and the correct equation for the computation of the principal moment of inertia about the equatorial x-axis (and similarly for the y-axis) is I$ = (&/4)

+ (mhZ/12),

(1)

where h (= 20 mm) denotes the width of the discs, and hi is the length of one semi-axis of the i-th disc. For a boy’s forearm, b, may be as little as 20mm. Hence neglect of the second term in (I), which is implied by the thin-disc assumption, introduces an error of up to 33%. Secondly, the sectioning per se of some segments (such as the feet or the abdomino-pelvic segment) into elliptical discs must be regarded a highly questionable procedure. Indeed, we shall demonstrate below that such an approach, when applied to the abdominopelvic segment, leads to an error (over estimation) of about 8.5% for the predicted mass, about 31% (maximum error) for the principal moments of inertia, and about 19% for the estimated z-coordinate of the mass centroid. In addition, the assumption for this segment of symmetry about the coronal plane is not justifiable, and a proper treatment necessitates a principal axes transformation. Finally, the assumption of uniform density across an elemental zone of a segment, as well as along the segmental longitudinal axis, introduces additional inaccuracies in the computed values of the principal moments of inertia. The magnitude of these errors is in the region of 4-7x, depending on the segment in question. Moreover, most models do not account for sexual differences of subjects, both as far as sex-specific mass distributions and segment densities, and exomorphic differences are concerned. In this paper we describe an anthropomorphic model which is suitable for simulations of gross body motions and which : (1) consists of 17 segments, including the shoulders as separate entities ; (2) differentiates between male and female subjects (exomorphic differences, mass distributions, density functions, etc.); (3) takes into account the actual shape fluctuations of all the individual segments; (4) accounts for varying densities, both across the cross-section of a segment (where necessary) and along its longitudinal axis; (5) adjusts the densities of some segmental elements (buttocks, lateral sections of upper thighs, etc.) according to the value of a special subcutaneous-fat indicator ; (6) in general, makes no assumptions about segment symmetries, and hence principal axes transformations are performed for all asymmetric segments (abdomino-thoracic segment, abdomino-pelvic seg-

ment, hands, feet, etc.); (7) takes into account all changes in body morphology due to obesity, pregnancy and other abnormal states, and is valid for children also (one model parameter is, in fact, a function of the subject’s age); (8) achieves an overall accuracy of better than 3% when computed parameter values are compared with corresponding values as the determined experimentally ; (9) computes all values of the parameter sets (2) and (5) below from a battery of 242 anthropometric measurements taken directly from the subject under investigation. These direct measurements obviate the introduction of errors to which photo-image techniques are subjected. The mathematical details of this model are rather involved, and their derivation is very lengthy and space-consuming. For this reason these details have not been included in this text but can be found in the various Appendices of Hatze (1979). THE SEGMENTAL

PARAMETER

SET

The subset of segmental parameters contained in the left-hand side of the differential system governing the dynamic behaviour of the neuro-musculoskeletal system model described in (Hatze, 1977), is given by

i = 1,. . ., 17,j = i(c, S)}

(2)

where the symbols denote respectively the segmental mass, the three principal moments of inertia with respect to principal axes passing through the mass centroid, the three components of the position vector locating the mass centroid relative to the local, segment-fixed Cartesian-coordinate system, and the three components of the vector locating the origin of the i-th segment relative to the coordinate system of the proximal (generally (i-1)-th) segment. For an exact definition of the integer function i(s, 6) see Hatze (1977) (this function is of no particular significance in the present context). All ordered quantities refer to right-handed systems of Cartesian coordinates (x,, Yip zi), and in all cases the local (segment-fixed) coordinate system is defined to be located at the origin of the segment with axes parallel to the principal axes passing through the mass centroid of the respective segment. It should be noted that the quantities (x7, yp. zj”)are as important as the remaining quantities in (2) since (in the notation of Hatze (1977)) p&,6) = s + ‘:::‘c;, 2 fi A’“. I 6)(wdlh,,

(3)

i.e. the position vector pi if the i-th mass centroid relative to inertial space contains the sum of appropriately transformed origin position vectors h, = (~7, y?,

zi”)’ , j = 1, . . .. i(e,6)-

1,

(4)

Model for computational determination of parameter values of anthropomorphic

segments

835

Nelson (1973), Dempster (1955), Drillis and Contini (1966), Hatze (1975), Hayes and Hatze (1977), etc. Although some of these experimental methods are quite reliable, they do require fairly elaborate instrumentation and are difficult to apply to central segments such as the abdomino-thoracic and abdomino-pelvic segments. Therefore, if a satisfactory computational method based on a battery of anthropometric measurements could be devised, such a method would greatly increase the efficiency of obtaining parameter values. It is the purpose of this report to present such a technique. DEFINITION

Fig. 1. Lateral and anterior view of 17-segment anthropomorphic model. The shapes of the segments, as depicted here, accurately reflect the morphologies of the model segments. The local (segment-fixed) coordinate systems are also shown.

OF SEGMENTAL

MODEL

MORPHOLOGIES

The body is fragmented into 17 segments as described in (Hatze, 1977) and displayed in Fig. 1 of this report. The segments are : the abdomino-thoraclc segment, the head-neck segment, the shoulder segments, the (upper) arms, the forearms, the hands, the abdomino-pelvic segment, the thighs, the legs, and the feet. The morphologies and boundaries of these model segments, as well as their local coordinate systems, are as defined in Fig. 1. A rather detailed representation of the individual segments is given in Appendix 2 of Hatze (1979). The most severe but necessary approximation made in defining the segment morphologies is the assumption that Jr1- r,J = cl, = constant,

(6)

whose components are precisely the quantities referred

to above. It is unfortunate that the significance of this latter set of segmental parameters has been overlooked in virtually all works dealing with this subject. Note also that, in general, the origin position vectors hj coincide with the average joint centres of the respective segments. Also required for simulation purposes, in addition to the set (2), are the components of the position vectors (measured relative to local segment coordinate systems) pk = (x;, y;, z;)‘,

k = 1,. . ., C,

of the C possible environmental

(9

contact points on terminal segments (head, hands, feet). For instance, the computer simulation of locomotion requires the position vectors of the heel and foot-ball support contact point of both feet. The values of the parameters defined by (2) can be determined experimentally (direct method) or computationalIy (indirect method). Experimental methods include the immersion method for determining segmental volumes (from which masses can be obtained), the reaction-change method for estimating segmental masses and locations of centroids, and several techniques (quick-release technique, oscillation method, suspension method, etc.) for determining moments of inertia about segmental joint axes. Detailed descrig tions of these methods can be found in Miller and

i.e. that in the system of point masses constituting a segment the distance between any two point masses, I and m, remains constant (rigidity assumption). If this assumption is not made, the configuration manifold of the system becomes 3n-dimensional (n is the number of point masses in the system) rather than sixdimensional as for the rigid segment. The tremendous reduction in complexity of the system due to the rigidity assumption is thus obvious. However, convenient as this approximation may be, it does not conform to reality. It is a well-known fact that parts of segments (muscles, organs, blood, etc.) execute movements relative to the (moving) segmental coordinate systems. In addition, other factors such as breathing, non-stationary joint centres ofrotation, changes in the distribution of body liquids, and the positiondependent mass distribution of soft tissues also violate the rigidity assumption. However, the variations in the computed parametric values induced by these in.fluences are estimated to be maximally 6%. In view of this, the rigidity assumption (6) appears acceptable. Another problem confronting the modeller is the definition of segment boundaries. In reality, these boundaries are fuzzy where segments join, but must be defined as being sharp for modelling purposes. The criterion adopted in this paper for determining in.. tersegmental boundaries is the dynamic behaviour of the fuzzy set of boundary particles. In a heuristic

836

H. HATZE

manner, the intersegmental boundary between the i-th and (i-1)-th segment is defined to be a two-dimensional surface satisfying 4itxi9 Yi, $1 = O,

(7)

where the vector rf = (xi, y,, z:)= of the i-th segment traces the ‘average’ layer of the fuzzy set of boundary particles. A more precise definition need not be given, since only comparatively crude estimates of the surfaces (7) are possible. The segmental boundaries at non-interfacing regions are obviously determined by the visible segmental shapes. Having defined the boundaries and internal structures of the model segments in this way, we now proceed to a more detailed description of the individual segment models. However, the detailed mathematical treatment will not be presented here as it is rather involved and lengthy, but it can be found in Appendices 1 and 2 of Hatze (1979). The general modelling procedure is to decompose segments into finite elements of known geometrical structure and to obtain volume, mass, coordinates of the mass centroid, the principal moments of inertia, and the orientation of the principal axes (through the mass centroid) relative to ‘original’ segment axes, of a segment model by triple integration over element boundaries and subsequent summation of integrals. Each individual geometrical element is assigned its own density, and in this way the varying mass distributions across and along segments are taken into account (Dempster, 1955). Densities of specific tissues are listed in Clauser et al. (1969). Some of the integrals involved cannot be solved analytically, and a special subroutine performing the numerical quadrature (automatic Patterson quadrature) is supplied in the computer program for this purpose. Finally, coordinates of segment origins (4) and environmental contact points (5) on terminal segments are obtained by the appropriate coordinate transformations from original axes to a local coordinate system whose axes are parallel to the principal axes through the mass centroid. The morphologies of the individual segment models and their fragmentations into ‘elements are depicted and described in detail in Appendix 2 of Hatze (1979). Therefore only a brief description will be given here with reference to Fig. 1. The model of the abdomino-thorocic segment is as displayed in Fig. 1. Its origin is located at point O,,ii which is situated in the horizontal cross-section at the height of the Omphalion (Umbilicus). Note that only the sagittal plane is a plane of symmetry, but not the coronal plane, i.e. the point Oi ,11 is shifted towards the posterior side of the horizontal cross-section. The reason for this approach and the exact definition of the locus of oi.ir are given in Appendix 2 of Hatze (1979). Note also that this segment does not include the shoulders, but is limited in its superior part by the outlines of the thorax proper, as indicated by the interrupted lines in Fig. 1. The segment model ter-

minates at the horizontal cross-section at the height of point OZ. Horizontal plates, with cross-sections composed of (generally) unequal semi-ellipses into which parabolas of varying size and low density (to model the lungs) are inserted, constitute the elements of which the whole segment is made up. Note that the model is asymmetrical (necessitating a principal axes transformation) and that female breasts (not shown in Fig. 1) are also included. The model of the head-neck segment originates at point O2 (Fig. 1) and is composed of an elliptical cylinder (neck) from which the part of the penetrating head has been removed, and a fairly general body of revolution (head). It was found that the modelling of the head as an ellipsoid is inappropriate since it underestimates the segment mass by 23% when the computed value is compared with the result obtained by the immersion method with subsequent correction for head density. The shoulder segments are extremely complicated to model. Their shapes are as indicated in Fig. 1. They consist of parabolic plates from which oblique-axial ellipto-parabolic sections (to account for the thorax) and a hemi-sphere (head of humerus) have been removed. The shoulder segments are, of course, asymmetrical, so that principal axes transformations are necessary. The origin of their coordinate systems is denoted by O,,, in Fig. 1. The models of the left and right (upper) arms originate at O4 and 0s respectively, and are each composed of a hemi-sphere (head of humerus) and ten elliptical cylinders of equal height but different densities. Similarly, the models of the forearms are composed of ten elliptical cylinders of equal height and different densities. The origins of the models are located at 0, and 0, for the left and right forearm respectively. The hands are modelled in a grip position, as shown in Fig. 1. Their model consists of a prism to which a hollow half-cylinder and an arched rectangular cuboid (thumb) are attached. The segment model is highly asymmetrical and again necessitates a principal axes transformation. The origins of the hand segments are 0, and O,, for the left and right hands respectively. The model of the abdomino-pelvic segment is as shown in Fig. 1. Its origin O,,,, is the same as that for the abdomino-thoracic segment, and shape tluctuations of this segment are fully taken into account. The model is composed of two elliptic paraboloids (buttocks), three horizontal plates comprising two semi-elliptical sections with varying densities, seven trapezoidal plates, seven semi-elliptic plates on the anterior-superior side of the segment, and three plates of general shape on the anterior-inferior side. The thickness of the plates is l/10 of the defined segment length. In addition, ellipto-parabolic sections are removed at the origins of the thighs (see Fig. 1) to account for the moving parts of the thighs which penetrate the pelvic region. The comparatively large differences between the various densities in this seg-

1. Comparison

Abdomino-thoracic Head-neck Left shoulder Right shoulder Left arm Right arm Left forearm Right forearm Left hand Right hand Abdomino-pelvic Left thigh Right thigh Left leg Right leg Left foot Right foot

Segment

Table

19.111 4.475 1.438 1.890 2.110 2.02 1 1.023 1.190 0.453 0.446 a.543 8.258 8.278 3.628 3.686 0.887 0.923

V

u 0.439 0.517 0.727 0.711 0.432 0.437 0.417 0.404 0.515 0.531 0.368 0.479 0.480 0.412 0.417 -

R

Subject

- 10.6 - 10.9 4.87 9.63 2.24 6.98

0.84 -0.14 2.96 6.05 - 1.7 - 4.96

8.75

-

I

r:

i,

V 2.83 4.08 -0.14 - 0.86 4.45 2.74 0.95 0.71 5.17 4.88 3.26 -4.19 -0.98 0.67 -2.53

v

-9.23 - 7.78 3.03 3.69 4.77 0.55

-0.30 1.81 4.05 4.10 -5.37 - 3.65

r

C.P. (male, 26)

0.3302 0.0337 0.0080 0.0084 0.0203 0.0229 0.0086 0.0093 0.0010 0.0010 0.0541 0.1653 0.1702 0.0798 0.0747 0.0051 0.0051

Bx

‘s system, volumes in litre)

0.444 0.516 0.706 0.699 0.437 0.428 0.413 0.412 0.533 0.524 0.395 0.473 0.466 0.420 0.417

R

Subject

values (in the m kg

19.803 4.537 2.042 2.121 2.123 2.340 1.223 1.313 0.416 0.417 9.614 a.744 8.729 3.856 3.798 1.032 1.055

parameter

3.14 -4.81 0.43 1.22

segmental

0.3117 0.0303 0.0047 0.0071 0.0196 0.0168 0.0067 0.0079 0.0011 0.0011 0.0399 0.1475 0.1415 0.0615 0.0663 0.0041 0.0042

determined

F.B. (male, 23)

and experimentally

-0.85 4.99 - 3.49 -2.07 3.49 2.46 -5.15 0.89 3.03 2.04 4.52 - 1.06 -4.72 3.59 3.85

of computed

5.03 1.44 2.98 2.63 -

1.x

Abdomino-thoracic Head-neck Left shoulder Right shoulder Left arm Right arm Left forearm Right forearm Left hand Right hand Abdomino-pelvic Left thigh Right thigh Left leg Right leg Left foot Right foot

Segment - 1.39 3.87 -

14.531 3.595 1.110 1.146 1.616 1.505 0.835 0.809 0.285 0.288 11.208 9.166 8.955 3.310 3.487 0.842 0.887 1.71 0.46 4.33 - 1.61 -0.11 0.43 4.08 2.32 3.09 0.12 -1.04 1.13 2.22

u

V

-

0.429 0.542 0.722 0.709 0.445 0.443 0.429 0.436 0.498 0.499 0.438 0.441 0.436 0.429 0.437 - 1.85 -0.69 0.92 - 0.92 0.41 4.01

2.06 1.61 0.23 - 1.39 1.58 1.38

5.58

I

R Q 0.2064 0.0209 0.0027 0.0029 0.0137 0.0119 0.0046 0.0048 0.0005 o.ooo5 0.0689 0.1563 0.1492 0.0453 0.0505 0.0036 0.0027

Subject R.M. (female, 31)

-

4.18 4.03 1.18 2.46

-

-

4 8.384 2.774 0.778 0.182 0.748 0.757 0.511 0.516 0.209 0.204 4.183 3.260 3.338 1.641 1.634 0.566 0.544

V

-2.00 0.37 3.53 2.94 -3.48 - 2.54 3.12 3.59 4.18 -0.45 0.33 1.43 -0.87

2.05 4.41 -

v 0.422 0.536 0.696 0.706 0.442 0.438 0.425 0.426 0.525 0.526 0.415 0.469 0.461 0.430 0.433 -

R

- 8.32 -6.49 0.74 0.00 1.04 2.08

- 1.49 0.54 1.06 0.93 - 3.86 - 3.94 -

5.52 -

-

r

0.0832 0.0141 0.0016 0.0016 0.0039 0.0038 0.0024 0.0023 0.0003 0.0003 0.0131 0.0335 0.0361 0.0180 0.0173 0.0019 0.0018

F$

s system, volumes in litre)

Subject G. R. (male, 12)

Table 2. Comparison of computed and experimentally determined segmental parameter values (in the m . kg.

2.87 -3.14 -0.88 - 1.01 -

i,

8 00

Model for computational determination of parameter values of anthropomorphic segments ment which distinguish male and female subjects are also taken into consideration in the model. It goes without saying that this segment is highly asymmetrical about the coronal plane and hence necessitates a principal axes transformation. It is worth noting that improper modelling of this segment by horizontal elliptical cylinders would have introduced a maximum error of 31% in the predicted principal moments of inertia. The left and right thighs have their origins at Oi2 and O,, respectively. Their models comprise the elliptoparabolic sections mentioned above and ten elliptical cylinders with horizontal cross-sections. Similarly, the legs (origins Oi3 and 0i6) consist of ten elliptical cylinders with horizontal cross-sections. In both thighs and legs the densities vary nonlinearly along the segment z-axes for each element, and are also sexdependent. The models of thefeet (origins 0i4 and Ot,) comprise 103 unequal trapezoidal plates of nonlinearly varying densities each. The segments are asymmetrical about the coronal plane, thus necessitating a principal axes transformation. COMPUTATIONAL

RESULTS

The equations used to compute the segmental parameter values are given in Appendices 1 and 2 of Hatze (1979). They were coded in FORTRAN IV and combined in program SEMCI. This program is selfcontained and includes all auxiliary subroutines necessary for the execution of the main program. The execution time for one subject is 0.515 set on a CDC Cyber 174 digital computer. A self-explanatory sample print-out is given in the Appendix. The sequence of anthropometric measurements described in Appendix 3 of Hatze (1979) was taken on four subjects (two young male athletes, one female tennis player, and one 12-year-old boy). The data were coded on punched cards and program SEMCI was executed for each of the four subjects. Segmental parameter values which could be compared either with corresponding values determined experimentally, or with values available in the literature, have been collected in Tables 1 and 2. The symbols have the following meaning: V is the computed segment volume in litres; 11denotes the relative volume error (in %) defined by 2,= lOO(l-~/‘lv*),

(8)

where V* is the value of the segment volume as determined experimentally by the immersion method ; R is the computed ratio between the centroid coordinate value in the direction of the main axis of the segment, and the length of the segment; r is defined similarly to u (see (8)); r: denotes the computed principal moment of inertia about the x-axis passing through the centroid ; and i, is again the relative error of rc, defined analogously to (8). Values for I$* and R*

839

respectively were obtained by the suspension method (Hatze, 1975) and from comparable cadaver data of Dempster (1955, vol. 2, pp. 186-200). For the feet, the value of r is defined by r = lOO[l - /Z]/(O.1721)]

((9)

where I is the length of the foot segment model and Z is measured relative to the original local coordinate system. Equation (9) permits a comparison between the prediction of the present model and the experimental results of Dempster (1955, vol. 2, p. 194). The values for the measured total mass (kg), the computed total mass, and the corresponding relative error respectively for each of the four subjects are as follows. Subject F.B. : 70.6, 70.792, - 0.272% ; subject C.P.: 76.1,75.701,0.524%; subject R.M.: 64.7,64.625, 0.116%; subject G.R.: 31.95, 31.987, - 0.1 IS?/,. DISCUSSION

A thorough statistical analysis of the data presented in Tables 1 and 2 will not be attempted, since for such an analysis the individual distributions need to be known of all the experimental variables as well as the distributions of the corresponding computed variables resulting from the individual distributions of the anthropometric input variables. Obviously, the construction of these functions is a major project itself. It is, however, possible relatively simply to extract from Tables 1 and 2 sufficient information for the evaluation of the present model. Dealing first with the maximum absolute values of the relative discrepancies (errors), we see that these are 5.17% for the volumes, 10.9% for the centroid-tosegment-length ratios, and 5.03% for the &-values. Since these values are considerably above the corresponding means (see below) they require an explanation. The value of v = 5.17% corresponds to the abdomino-pelvic segment of subject C.P. The experimental record for this subject reveals that during the immersion experiments he wore a large swimming trunk which probably trapped a considerable amount of air (despite precautions taken), and accounts for the higher experimental value. This hypothesis is supported by the fact that the corresponding value in all other subjects is also positive and comparatively large. Another but less likely source of error could have been the subtraction, from the pelvic volume, of those thigh parts which move inside the pelvic region. The value of r = - 10.9y0 appears at the right thigh of subject F.B. This value does not, in fact, indicate a discrepancy between computed and experimental value but is due to the different definitions of the segment thigh in the present model (see Fig. 1) and that of Dempster (1955, vol. 2, p. 194) from which the experimental value of R* = 0.433 was taken. This explanation is also clearly supported by the fact that the r-values for the thighs are consistently negative and large in all male subjects, The r-values for the female subject are smaller but still negative because the comparatively large subcu-

840

H. HATZE

taneous fat deposits in the upper thigh and hip regions decrease the densities in these parts and therefore tend to shift the mass centroid distally, thus increasing r. Finally, the value of i, = 5.03% for the subject C.P. is a result of the inability of this individual to relax completely his left hip muscles during the oscillation experiments (see Hayes and Hatze, 1977). The means of the absolute values of the relative errors for the four subjects are as follows. Subject F.B. : V,,= 3.09%, r;, = 4.46x, ?_ = 2.40%; subject C.P.: V. = 2.56x, f. = 3.66x, ?= = 3.02%; subject R.M.: V, = 1.860/ ?,, = 1.83x, r,, = 2.96% ; subject G.R. : U, = 2.35x, f. = 1.93x, iL,, = 1.98%. Owing to incomparibility, the values for the thighs have been excluded from the computation of ?a, for all subjects. The mean of the relative errors for measured and computed total masses is 0.260/ with a maximum error of 0.524%. The frequent changes in the signs of the relative errors in Tables 1 and 2 indicate that no significant systematic errors are present, except in the cases discussed above. A question still to be settled is the validity of the computed values of the principal moments of inertia. Except for those values which were directly confirmed by measurement (see 7; and i, in Tables 1 and 2), no direct method is available for the verification of the computed values. However, great confidence in the model can be derived from the fact that the computed values of 7: correspond closely to those obtained by direct measurements on cadaver parts. The anthropometry of cadaver no. 15097 of Dempster (1955, vol. 2, p. 198) resembles in many respects that of our subject C.P. Hence the values of r{ should also be roughly comparable for segments defined equivalently in both models. This is indeed the case. Corresponding values (in kg .m2) for subject C.P. and those listed by Dempster (in brackets) are: head-neck segment 0.0337 (0.0310), left arm 0.0203 (0.0222), right arm 0.0229 (0.0220), left forearm 0.0086 (O.OOSS),right forearm 0.0093 (0.0072), left hand 0.0010 (0.0009), right hand 0.0010 (0.001 1), left leg 0.0798 (0.0650), right leg 0.0747 (0.0620), left foot 0.0051 (0.0037), right foot 0.0051 (0.0040). It should be noted that parameter values for the abdomino-thoracic, the abdomino-pelvic, and the shoulder segments cannot be compared in the two models owing to entirely different definitions of the segment morphologies. Indeed, it is interesting to note that the abdomino-pelvic segment, as defined by Dempster, has a mass of 19.187 kg (Dempster, 1955, vol. 2, p. 186) for the cadaver in question. This mass almost equals the 18.283 kg of the abdomino-thoracic segment (as defined in the present model) for our subject C.P. Owing to the comparable shape of the two fragments, the corresponding rg values should not differ much. This is confirmed by the corresponding values of 0.3240 kg. m2 for Dempster’s cadaver and 0.3307 kg. m2 as computed for our subject C.P. These data provide additional evidence for the credibility of the present model. Assuming that the experimental methods used for

data collection were such as to justify the assumption that the variances of the probability distributions of the measured variables may be considered small, it is meaningful to pool the various relative discrepancy measures in order to obtain an overall indicator for the quality of the model. One such indicator is the mean of the absolute values of all the relative errors. Its value is 2.68%. If the errors for the measured and computed total masses are included, this value reduces to 2.49%.

CONCLUSION

For determining by calculation the set of segmental parameter values, a model has been presented which is a useful tool for the bioengineer and the biomechanist. Although the model is mathematically comparatively complicated the user does not have to be concerned with these internal complexities since the computer program executing the model is self-contained. All that is required is the collection of the anthropometric input data for a given subject, a procedure which can be carried out in less than 80 min. The program then computes the following data, with an overall accuracy of about 3%, and subject to a maximum error of about 5x, for each of the 17 segments: volume, mass, the three coordinates of the mass centroid, the three principal moments of inertia about the mass centroid, the three (non-principal) moments of inertia about segment axes through the origin, the three coordinates of the segment origin relative to the system of the proximal segment, the orientations of the principal axes relative to the original segment axes, and the coordinates of environmental contact points. The sample print-out in the Appendix illustrates the format and sequence of data presentation. One of the special features of the model is a variable which measures the subcutaneous fat content at the subject’s hips and accordingly adjusts, in a nonlinear fashion, the densities of specific parts of certain segments. The model also detects obesity and the exomorphic attributes of pregnancy, takes them into account in the computations and gives a corresponding message in the program output list. All these features are possible because no u priori assumptions are made on the coronal symmetry of the abdominopelvic or abdomino-thoracic segments. Therefore any severe non-symmetry, such as is introduced by large female breasts or by pregnancy, is automatically accounted for. This property makes the model rather useful for clinical applications. Owing to its accuracy, versatility and easy implementation the model provides a convenient means of generating segmental parameter values which otherwise would have to be gained experimentally by means of comparatively laborious procedures. Note: The CSIR Techn. Report TWISK 79 (Reference Hatze, 1979)which contains all technical details of the model, is obtainable

from the author

free of charge.

Model for computational

determination

of parameter

Bugyi, B. (1976) On the valuation of athletes’ physique by photogrammetric means. Int. Arch. Photogrammetry 21, 218. Clauser, C. E., McConville, J. T. and Young, J. W. (1969) Weight, volume and center of mass of segments of the human body. AMRL-TR X19-70, Wright-Patterson Air Force Base, Ohio, p. 96. Dempster, W. T. (1955) Space requirements of the seated operator. WADC-Techn. Rpt.-55-159 (2 vol.), WrightPatterson Air Force Base, Ohio. Drillis, R. and Contini, R. (1966) Body segment parameters. Techn. Report No. 1166.03. PB 174945, School of Eng. and Sci., New York University. Fisher, 0. (1906) Theorerische Grundlagenftir eine Mechanik der iebenden Kiirper, pp. 52-55, Teubner, Berlin. Hanavan, E. P. (1964) A mathematical model of the human body. AMRL-Techn. Report-64-102, Wright-Patterson Air Force Base, Ohio. Hatze, H. (1975) A new method for the simultaneous measurement of the moment of inertia, the damping coefficient and the location of the centre of mass of a body segment in situ. Europ. J. appl. Physiol. 34, 217-226. Hatze, H. (1977) A complete set of control equations for the human musculo-skeletal system. J. Eiomechanics 10, 799-805. Ha&e, H. (1979) A model for the computational determination of parameter values of anthropomorphic segments, CSIR Techn. Report TWISK 79, Pretoria. Hayes. K. C. and Hatze, H. (1977) Passive visco-elastic properties of the structures spanning the human elbow joint. Europ. .I. appl. Physiol. 37, 265-274. Hemami, H., Weimer, F. C. and Koozekanani, S. H. (1973) Some aspects of the inverted pendulum problem for modelhng of locomotion systems. IEEE Trans. Automat, Contr. AC-18, 6588661. Hemami, H. and Golliday, C. L. (1977) The inverted pendulum and biped stability. Math. Biosci. 34, 95-110. Huston, R. L. and Passerello, C. E. (1971) On the dynamics of a human body model J. Biomechanics 4, 369-378. Jensen, R. K. (1978) Estimation of the biomechanical properties of three body types using a photogrammetric method, J. Biomechanics 11,349-358. Kulwicki, P. V. and Schlei, E. J. (1962) Weightless man : self-

SUBJECT'S

IMALE 0

NINE&

LEFT SHOVLOEri .,40 .17E RIGHT SHOULDEL .182

LEFT Aan ,085 ,R" .OBS i(l6HT .093

.091

PRINT-OUT

SEXIF

."83

SEG"EFIT .264 .251

.CPO

.02:

.086

.025

.245

.262

.178

.08r, .c95

*l

.OEO

.077

.073

.0?2

.072

.072

.372

.083

.303

.202

.2BO

.267

,260

.256

.250

.243

.235

.237

.294

.o)h5

.cl87

.085

.078

.074

.070

.070

.076

.290

.279

.268

,260

.257

.251

.2'16

.236

.*23

.217

.2Pl

.082

.O75

.070

.ObS

.062

,058

.055

.235

,239

.239

.233

.219

.201

.185

.171

.162

.160

.252

.076

.OLP

.066

.066

.Obl

.061

.056

.231

.22P

.225

.221

.211

.lP‘J

.I86

.174

.165

.161

.257

LEFT FOREARM .085 FUrlEAR" .080 .083 RIGHT .079

MAk‘A

NOMENCLATURE mass of the i-th segment; unit : kg the three components ofthe position vector locating the mass centroid of the i-th segment relative to the local coordinate system ; unit : m the three principal moments of inertia of the i-th segment, relative to principal axes through the mass centroid; unit: kg. m* the three components of the position vector locating the origin of the i-th segment relative to the coordinate system of the proximal (generally (i-1)-th) segment; unit: m right-handed system of Cartesian coordinates fixed to the origin of the i-th segment the three components (relative to the local coordinate system) of the position vector locating the k-th environmental contact point on a terminal segment; unit: m positton vectors of the I-th and m-th point masses relative to a local coordinate system computed segment volume; unit: litre relative volume error in “/, computed ratio between the centroid coordinate value in the direction of the main axis of the segment, and the segment length relative centroid position error in O,, relative error of rc in O0

ACE 31.41

ABDORINO-THURAClC .101 ,273 .275

.145

A.

SAMPLE

841

segments

rotation techniques. AMRL-TDR-62-129, WrightPatterson Air Force Base, Ohio. Miller, D. I. and Morrison, W. E. (1975) Prediction of segmental parameters using the Hanavan human body model. Med. Sci. Sports 7, 207-212. Miller, D. I. and Nelson, R. C. (1973) Eiomechanics ofsport, pp. 88-109. Lea and Febiger, Philadelphia. VukobrataviC, M. and Stepanenko, J. (1973) Mathematical models ofgeneral anthropomorphic systems. Math. Biosci. 17, 191-242. Weinbach, A. P. (1938) Contour maps, center of gravity, moment of inertia and surface area of the human body. Human Biol. 10, 356-371.

REFERENCES

APPENDIX

values of anthropomorphic

.078

842

H. HATZE LEFT HAND .063 .071 RIGWT hAhD .0+5 .476

ABDOhINO-PELVIC ,267

SEGMENT

.301

.31G

LEFT TklGH ,191 ,191 RIGHT THIGH .136 .lb7 LEFT

LEFT

.341

.3CC

.359

.332

.326

,260

,119

,190

.lUO

.162

,166

.132

,126

,119

.139

.132

.171

.176

,163

.I36

.I30 .126

.767

.797

.344

.a01

.P40

.G54

lPlS

.097

.60*

.591

.565

.536

.500

.653

.,13

,336

.373

.,39

.364

.602

.592

.563

.523

.W5

,661

.+17

.333

.377

.+45

.353

,109

.116

.117

.105

.OPl

,079

.065

.06+

,353

.335

.362

,375

.370

.333

,293

.264

.233

,231) .CDb

.D32

.lOP

.116

.117

.lOb

.035

.07P

.066 .062

.351

.332

,353

.375

,374

.351

.309

.279

.2t5

.239

.03,

.053

.035

.D72

.216

.158

,056

.035

.077

.212

.1*7

COhPUlED IN X6.

SEGRENT COORDINATES

VOLURP

IN

L‘TREF

MASS

PARAMETER IN II,

FORhAl AhD SEPUEWCE OF DATA PRESENTATIONI SEGhENl NAME RASSr COORDINATES OF CENTROIDlX,Y,ZI VOLUME, PR‘hC. RORENlS OF INERTIA Y.R.1. CENlROIC~IX.‘Yr‘ZlrANO COORD‘NAlESlOX,UY,OZl OF OC‘GIN OF DlSlAL SEGR. SPECIAL SEGhENl PARARETERS

VALUES hORkNlS

OF

‘NERlIA

IN

XG*“1*2,

LOCAL SYSlEhS ORIGIN,IOX,‘OY,~OZ, 10 LOCAL COORO.-SISTER OF

RELATIVE

,ROXI”AL

NOTE THAT COORDINATES OF CENTROIDS ARE GIVEN Y.R.1. LOCAL LSEGhENl-FIXED1 COORD. SYSTEM, ERANATt FROM lht SEGMENT ORIGIN, AN0 YHICH ARE PARALLEL 10 THE SECtlENT PRINCIPAL AXES. PRINCIPAL AXES DIFFER FROh ORIGINAL SEGMENT 4XES (SEE MANUAL), THE CENTROIO CDOROINATES THESE AXES ARE ALSO GIVEN AS *SPECIAL SEGMENT PARAMETERS*.

ABDORINO-THORACIC 14.537 .206372 IRADIANS)

COORDINATES

0.000 .032346 0.000 PRINCIPAL

BETYEEN -.OOl IkrY~Zl OF CtNlROID 0.000

LEFT ShDULDEk NOTE THAT NO OCC,,RS ABOLl

3.593

0.003 Z-AXIS REL.

o.oco .Ol2160

0.000 VERTEX o*ooo

SEGMENT VALUES ARE CORPUTED ThEIR Z-AXES.

OF

-.OOl HEAD REL. 0.000

FOR

‘Z

OF

SYSlEh

INCLINATION

ORIG‘NIPEL.

ANGLE

(RADIANS1

AND

,132 .090123 .452 hELO-NECK .2SI.

FOR

‘DZ

VnICn WERE Y.R.T.

ABD.-THOR. -.DDl

SEGR.1 .362 Z-AXIS

SEGhEHT

OF

.08293~

TNE

SEGREHTI

SE6hENl

AXES:

.036722

.012165

COORO.-SISTER:

THE

.140 .024941

DF

.718701 Z-AXIS

fW;;PRINC‘PALl

0.000

1.14* 0.000 .002172 -.c31 .400

COORD.

AND

*I94 .693646 O.OUO ORIGINAL

lD.;;;GINAL

0.000 .017553 OF

1.110 .002725 0.000

RESTIHG

,003

.231513

COORDINAlES~VX,VY,VZT

Z

SHOULDER

SEGhENlS

SINCE

NO

RDlAl‘ON

.024437 FOR 10

2-DIM.

MODEL

hOVING

IN

SAG‘TTAL

hOR1ZOhlAL~

.lW RIGHT

SNDULDER l.lCb .OOZP22 0.000 ,RD Z COORO.

RESllN‘

LGFl

SEGRENT 0.000 1.130 .002239 .38G -.00X CF SYSlE” ORIGINLREL.

INCL‘hATIOh

AR” I.616 .013677

ANGLf -.132

1.713

0.000

A~“;,;“““.

IRADIANS,

OF

0.000

NOTES

LEF,

*RR 1.505 .011917 FOR

1.595

.“01526

O.POO

G.440

RDOEL

.001325

0.404 ORIGINS

OF

*RI

0.400 SECRENTS

,141 .026213

.02553,

SE;:;;

FOR

Z-AXIS

10

2-DIN.

-.131 .063253 .19+

-.129 .436495 .I99 ;OlhC1DE

MODEL

“OVIhG

IN

.696

0.000 .001712

FOREARh .BOP .004765

.lbG

0.440 .000523

O.ODD

0.000

0.000 .004(155

0.000 .000+72

4.000

0.000

-.103 .016956 -.2PS

.0&3017

.001523

.036516 V‘TH

ORIGINS

.001325 OF

SHOULDER

.015071

.000523

.015665

.000472

-.I12

l015595 -.291

SACITTAL

HORIZONTALI

FOREAR”

,635 .001596

RIGHT

0.000

.011937 Z-01”.

SEG”EN1

0.000

1013635 0.000

R‘GHT

StG”.

SEGMENT 12.950

hEAD-NECK SEGhENl 3.595 .02OP53

Y

.453

.bLP

FOOT

tUNIlSI

y AND

.223

,620

FOOT

,090

ANGLE

.081

LEC

.llD .I03 RIGHT LE6 ,109 .103

.ow RIGHT

.331

SEW.

,LANE,

PLANE,

.,lP

Model for computational

ANGLE

LLFT

IRAOIAkSI

l”lG” 9.166 .156305

determination

0.000

BtluEtN

P(IlH‘IPIL

.152523 _.OPI

values of anthropomorphic

C.000 0,OCO &NO Lll1‘LH.L L-AXIS

z-AXIS

o*ooo

9.593

of parameter

0.000

.035853 .I66

-.19* .516023

OF

THL

SEPPICNT‘

.512241

.035853

.C96333

.034605

.153933

.005106

.17718C

.005138

.005343

.001863

-.078

-.lPI

.500b03 -.075

LEFT

LEG 3.310 .0+5315

3.602

o*ooo .04C973

RIG”,

LEG 3,,87 .050531

3.796

5.000 .050114 0.000

.

B,Z

.00361” ANGLE

IIIAD1AhS,

DIOJO

.Pbl

.003781 DLTdEEH _.“R”

0.000 .005156

0.000

0.000 PRINCIPAL

0.000

0.000 .005$38

c.cco

-*034 .000763 0.000 I-AXIS AHJ

-.I74

.15*27Jl -.‘a9

-.183

.177L07 -.5*5

-.0*0

.006213 _.,“D

OII‘IMAL

L-AXIS

OF THE

SEC”ENT8

segments