American Journal of Primatology 12299-308 (1987)

TECHNICAL NOTE

Measurernent of Body Segment Mass, Center of Gravity, and Determination of Moments of Inertia by Double Pendulum in Lemur fulvus

JAMElS P. WELLS AND DANIEL F. DEMENTHON West Virginia School of Osteopathic Medicine, Department of Anatomy, Lewisburg, West Virginia, and University of Maryland, Center for Automation Research, College Park, Maryland The collection of data on physical parameters of body segments is a prelim- inary critical step in studying the biomechanics of locomotion. Little data on nonhuman body segment parameters has been published. The lack of standardization of techniques for data collection and presentation has made the comparative use of these data difficult and at times impossible. This study offers an approach for collecting data on center of gravity and mo- ments of inertia for standardized body segments. The double swing pendu- lum approach is proposed as a solution for difficulties previously encountered in calculating moments of inertia for body segments. A format for prompting a computer to perform these calculations is offered, and the resulting seg- ment mass data for Lemur fuluus is presented.

Key words: Lemur fuluus, segment mass moment of inertia, biomechanics, positional behavior, segment mass

INTRODUCTION Any study that attempts to determine the kinetics of locomotion necessarily

involves, as a prerequisite, the calculation of the kinematic pattern of segment movement and the pattern of distribution of body segment mass. This study presents body segment mass data collected from specimens of Lemur fulvus and offers a discussion of the methods employed to determine segment moments of inertia by using a double pendulum. The method for determination of moment of inertia in Wells [1975] has proven no longer to be accurate enough for use with present-day kinetic analysis. In that earlier approach, the segment was swung directly from a horizontal bar supported at both ends. The periodicity observed was extremely rapid and, therefore, difficult to measure. The mounting of the segment to the bar was, a t best, difficult. The double pendulum approach permits simple recovery of much more precise data.

Received April 21, 1986; revision accepted November 2, 1986.

Address reprint requests to Dr. James P. Wells, West Virginia School of Osteopathic Medicine, 400 North Lee Street, Lewisburg, WV 24901.

0 1987 Alan El. Liss, Inc.

300 / Wells and DeMenthon

Body segment data for nonhuman primates is relatively unavailable. One of the reasons for inavailability is the scarcity of unfixed and unautopsied cadaver mate- rial for many primate species. In addition, when such data are presented, they represent such a wide variety of collection techniques that comparison is not possi- ble. It is, therefore, a further goal of this study to offer a format for standardizing data collection. Literature which does present segment mass data and a pertinent discussion of methodology includes the following: Vilensky, 1979; Reynolds, 1977; Reynolds et al, 1978; Grand, 1977a,b; Jouffroy and Gasc, 1974; Wells, 1974; Plagen- hoef, 1962, 1971; Winter, 1979; Wells et al, 1977.

Link System Determination

Prior to determining segment mass parameters, a link system must be estab- lished that is appropriate for the species under investigation. In accomplishing this preliminary step, the investigator should strive to satisfy the two following criteria. Data collection should be standardized so that it may contribute to comparative studies. Care should be exercised in selecting segment combinations that will have some universal applicability. For example, in a study of quadrupedal walking (where the trunk is relatively stable), the pelvis, abdomen, and thorax may be treated as a single unit; however, if information is sought on running (where vertebral flexion becomes important), these segment combinations are inappropriate.

To establish an appropriate link system, it is advantageous to study locomotion on a natural or seminatural substrate with the subject performing all typical gaits. Radiographs, taken under anesthesia showing body segments manipulated through their complete range of motion, are also helpful in that accurate segment lengths may be measured and joint ranges of motion may be noted. Anesthesia which produces muscle spasm or rigidity (such as Ketamine Hydrochloride), should be avoided. Sodium pentathol provides a suitable alternative.

The link system chosen for Lemur fulvus is based upon the format suggested above. Whenever appropriate the authors have included notes justifying segment combinations and offering practical hints which may prove helpful in collecting data from different tissue structures in different body segments and combinations thereof.

The Link System

The link system chosen is as follows:

A. Foot-from metatarsal-phalangeal joint to the talocrural joint (the metatar- sal-phalangeal joint was chosen as it best represents the weight-bearing contact with the base of support during locomotion).

B. Crus-from the talocrural joint to the knee joint. C. Thigh-from the knee joint to the hip joint. D. Pelvis, abdomen, and tail-hip joint to the lumbo-thoracic vertebral joint

(very little mobility was found at the sacroiliac joint and the tail was held in most locomotor instances in a relatively extended fashion).

E. Thorax and head-from the lumbo-thoracic joint to the supraorbital ridge (head movement is minimal as is flexibility of the cervical spine).

F. Arm-from the glenohumeral joint to the radioulnar joint. G. Forearm and hand-from the radioulnar joint to the metacarpal-phalangeal

joint.

Lemur fulvus Body Segment Biomechanics I 301

METHODS Segment Preparations

Cadavers were frozen and then allowed to thaw only partially. All segments were isolated at the joints described above. Joints were disarticulated, rather than cut, through apparent joint centers. Tissue relationships were maintained by the freeze-thaw approach. The abdomen shape was approximated manually. The caudal end of the thorax and cranial abdominal aperture were sealed with a plastic bag. Tare weights were collected for these expendables. All tissue cuts were made follow- ing natural body creases. The validity of this approach is supported by the finding of very little difference between left and right sides of appendages and segment to segment ratios from cadaver to cadaver.

Measurement of Lengths, Mass, and Center of Gravity Segment lengths were measured by using an anthropometric board with one

fixed vertica.1 end, one sliding end, and measurement scale between. Segment mass data were measured with a Metler Balance. It was found to be critical to collect all masses in the same session, as dehydration can account for a relatively large segment weiight loss.

Location of center of gravity was determined by using a simple device as illustrated in Figure 1. The device consisted of an aluminum tray suspended by a pin. Holes in the tray sides allowed the pin to be adjusted to accommodate different segment thicknesses and were large and smooth to allow the pan to rotate freely. With the pan empty, the vertical pointer which approximated a mark on a piece of paper taped to the bench beneath the instrument was also aligned with a scribed

Fig. 1. Device for collection of data on segment center of gravity. This device consists of an aluminum tray with vertical sides; a pin that may be aligned with a mark on the surface beneath the instrument demonstrating achievement of balance and two supporting stands.

302 I Wells and DeMenthon

mark on the tray bottom. The segment to be measured was then placed perpendicu- lar to the scribed line on the tray bottom and moved back and forth until the pointed mark alignment was attained. The distance from the proximal end of the segment to the line on the tray bottom identified the segment center of gravity.

Measurement of Moments of Inertia With the Double Pendulum The double pendulum apparatus system is composed of (Fig. 2a):

1. A tray, mass m, inertia i, length 2 R. The vertical sides, cut in the shape of

2. Two parallel suspension wires, length L. Their mass was negligible compared

3. An object on the tray, mass M, inertia I.

triangles, had holes from which the suspension wires were attached.

to the mass of the tray.

Mode of Operation This system offered several potential modes of swinging segments. Only the

twisting mode, in which the tray oscillated angularly around a vertical axis and behaved much like a torsion pendulum, was used and is presented here. Oscillation was obtained by taking a suspension line with each hand at the point where the line connected to the tray and displacing both hands horizontally in opposite directions while trying to keep the center of the tray at a fixed position. Both hands released the tray at the same time. A sample of 10 periodic motions were timed. The inertia of the oscillating system was calculated as described below from the period of one oscillation cycle. The moment of inertia of the object placed on the tray was calcu- lated by subtracting the moment of inertia of the empty tray from the total moment of inertia of both tray and object.

Period-to-Inertia Formula Once the period of one oscillation cycle was known, the moment of inertia was

calculated. The whole system (2 suspension lines and the oscillating tray-plus-object ensemble) has inertia Ie = i + I. It was assumed that the sum of the potential energy and the kinetic energy stayed constant during the oscillation. Derivation of the formula is described below.

When the tray was displaced horizontally, the lines holding the pan made an angle a, assumed small, with the vertical, and an angle 8 representing the angle between rest and displaced positions (see Fig. 2b). The relationship between a and 0 was found by expressing the displacement of the attachment point either by La or by R8; therefore

L ( j - Ra

The potential energy of the system was increased because the tray was mini- mally elevated due to the displaced wires. The increase in height is L - L cosa; since a was small, cosa was approximated to 1 -a212 and the increase in height was

thus approximated by L - . a2 2

The potential energy of the oscillating system was

aL P = Prest + (M + m) g L 2

Lemur fulvus Body Segment Biomechanics I 303

Mass m Mass M Inertia i Inertia I

- 1 Body Segment I

a R R

b

Fig. 2. a: Device for calculating segment moments of inertia by double swing pendulum (note: left figure represents a head-on view and right figure a lateral view). The device consists of an aluminum tray suspended by delicate wires.

Fig. 2. b: Device for calculating segment moments of inertia by double swing pendulum. The tray in this view has been rotated to a position preparatory for release, to begin an oscillation angle. The angles a and 6' are demonstrated.

304 / Wells and DeMenthon

where Prest was the potential energy of the system at rest, and g was the acceleration of gravity. The kinetic energy for the motion downward was small compared to the kinetic energy for the angular oscillation and thus can be ignored. The kinetic energy of oscillation was one half the product of the moment of inertia times the square of the angular velocity d8/dt:

(3) 1 K = 2 (I + i) (d0/dtl2

Using the relation between a and 8 this also may be written as:

1 L , da 2 R dt

K = - ( I + i) (-1 (--I2 (4)

Thus the total energy of the oscillating system is:

(5) a2 1 E = Prest + (M + m) g L- + - (I + i) (LEO2 (da/dt)' 2 2

The energy of the system remains constant with time, assuming there is no friction; therefore

(6) a2 (M + m) g L- + 1/2 (I + i) (L/R)2 (da/dtI2 = constant 2

or

This differential equation could be solved by classical techniques. We chose, however, to try a periodic sinusoidal solution and then to verify whether it satisfied the equation. A periodic sinusoidal solution of the form

where A was the unknown amplitude of the angular oscillation and T was the yet unknown period of the oscillation, was attempted. The angular velocity was the time derivative of a:

da 2* t - = -A - sin (2r-I dt T T

da dt Substituting these expressions of a and - into equation (7) and solving for T yielded

the expression for our solution:

(I + i) L (M + m) gR2 T2 = 4n2 (10)

Lemur fulvus Body Segment Biomechanics I 305

Indeed, for this value of T the group

(11) t t T T

cos2 (2n-1 + sin2 (sa-)

can be factored, and since this group is constant and equal to 1, the left side of the equation is a constant. Therefore, the expression for cy (eq. 9) was a solution. In practice, T was measured and I was the unknown. Equation 10 was first used to calculate i of the empty tray from its measured oscillation period To and then to calculate the inertia of the loaded tray I from its oscillation period; the difference between these (I - i) yielded the inertia of the segment (I).

Analysis and Quantification A flow chart for programing a computer to perform necessary computations is

offered below. The program used in this study was written in PDP Basic running under the RSTS operating system.

1. Input geometric and mechanical constants: g = acceleration of gravity = 9.81. R = 112 distance between wires, meters. L = wire length, meters. m = mass of tray, kg.

2.1. Ti.me 10 oscillations five times and calculate the average period of one oscillation cycle. 2.2. Enter the average period of one oscillation cycle into the program. 2.3. Calculate moment of inertia of empty tray.

3.1. Enter the mass of the object into the program. 3.2. Time 10 oscillations of the loaded tray five times and calculate the average period of one oscillation cycle. 3.3. Enter the average period of one oscillation cycle for the loaded tray into the program. 3.4. Calculate the moment of inertia of loaded tray. 3.5. I = I - i 3.6. Loop on paragraph 3 for more object moments of inertia

2. Find moment of inertia of empty tray:

3. Find moment of inertia of objects:

RESULTS Body segment data presented in Table I were taken from an adult male cadaver.

Limb segment results were compared with those obtained for one juvenile cadaver and two other adult cadavers, one male and one female. All three cadavers had been previously autopsied, and, therefore, only incomplete data were obtainable. When the juvenile cadaver was normalized for size difference, the segment mass findings were not statistically different from the complete adult male. Data from the two other adult cadavers were compared and were found to be virtually identical to that of the presented case.

The double swing pendulum method was implemented because the earlier single pendulum approach was determined to be too imprecise. The effect of this impreci- sion wiis discovered during kinematic and kinetic analysis of the Lemur fulvus leap. In a computer simulation of the leap, the animal left the ground later than actually occurred, and the ground reaction forces in the model were in error. Substituting

TABL

E I.

Bod

y Se

gmen

t Mas

s of O

ne A

dult

Mal

e Lem

ur fu

lvus

Cad

aver

Cen

ter

Ave

rage

M

omen

t

Len

gtha

W

eigh

tb

Mas

s

mas

sd

RP%

e RD

%f

perio

d in

ertia

g of

1

segm

ent

of

Foot

5.

95

.147

935

15.0

8 1.

50

.252

1 .7

479

2.31

06

.960

0123

-5

Leg

12.3

5 57

0157

58

.12

7.40

,5

992

,400

8 2.

1362

,3

7611

73-3

Th

igh

13.1

0 1.

7421

6 17

7.59

7.

25

.553

4 .4

466

2.02

48

.115

1663

-2

Abd

omed

pelv

is/ta

il 15

.00

6.71

102

684.

10

5.87

.3

913

.608

67

5.69

5 .1

1051

5E-l

A

rm

8.50

.5

0217

4 51

.19

4.10

.4

824

5176

2.

0178

.1

1275

23-3

Fo

rear

m a

nd h

and

13.2

0 .4

5989

3 46

.88

6.10

.4

621

,537

9 2.

3032

.2

4171

83-3

Th

orax

and

hea

d 16

.00

7.34

279

748.

5 8.

75

5469

.4

531

2.95

5 .8

1794

33-2

aS

egm

ent l

engt

h in

cen

timet

er.

bSeg

men

t wei

ght i

n ne

wto

ns.

Seg

men

t m

ass i

n gr

ams.

dS

egm

ent c

ente

r of

mas

s in

cen

timet

ers

from

pro

xim

al e

nd a

s de

fined

by

prox

imity

to

met

acar

poph

alan

geal

joi

nt o

f fo

ot w

ith g

roun

d.

(Seg

men

t per

iods

in

seco

nds.

) R

P%, p

erce

nt o

f dis

tanc

e fr

om p

roxi

mal

end

of s

egm

ent t

o ce

nter

of m

ass.

fR

D%

, per

cent

of d

ista

nce

from

dis

tal e

nd o

f seg

men

t to

cent

er o

f mas

s.

gMom

ent o

f ine

rtia

, kilo

gram

X m

eter

X m

eter

(kg

X m

).

Lemur fulvus Body Segment Biomechanics I 307

the more precise inertial data (determined using the double pendulum technique) into the model produced the correct values.

DISCUSSION If one is to completely understand relationships between morphology and the

positional behavior which it both limits and permits, it is necessary to conduct detailed biomechanical studies of all living mate groups. A preliminary step which must be taken in order to realize this goal is the collection of a variety of standard- ized body segment measures for each species to be studied. Body segment data in and of itself is enormously valuable to those investigators wishing to conduct bio- mechanical comparisons of positional behavior in different primate species. Stan- dardization of collection techniques and format of presentation is therefore most critical.

Many calculations require moment of inertia with respect to the center of gravity. The double pendulum method gives the moment directly. Positioning of object is simple. Error introduced by misplacing an object on the tray is smaIl if the tray is much lighter than the object.

The formula for computation of segment moment of inertia is accurate only if a! is small. A number of practical considerations must be employed for optimal use of the system. The value of a may be minimized in practice by suspending the device from the ceiling and having the tray close to the floor.

Ideally the vertical axis of oscillation should pass through the center of gravity of the tray plus object, and what is obtained in this case is the moment of inertia of the object with respect to this axis. It is important, therefore, to place the center of gravity of the object above the center of gravity of the tray. This concern becomes less important, however, if the tray is very light compared to the object. When such a tray is loaded, the center of gravity is largely determined by that of the much heavier object. Therefore, it is not necessary to be absolutely precise in registering the center of gravity of the object with that of the tray, but for accuracy these should be reasonably close. Light rigid trays can be cut easily from a thin aluminum sheet.

Parasite motions, like small lateral swinging, superimpose the angular twisting motion with their own different periods. If such motions are small compared to the motion of interest, they remain uncoupled from it and do not interfere with the measurement.

CONCLUSIONS 1. The double swing pendulum approach offers a distinct advantage over earlier

methods used to estimate moments of inertia for body segments for the following reasons:

a. The moment of inertia can be directly determined from the isolated segment; b. Positioning the segment with respect to the axis of rotation does not have to

c. Because the tray is light, error introduced by misplacing the segment is

d. Because the period of rotation is slow, timing becomes more accurate.

be extremely precise and is, therefore, easy to perform;

minimal;

2. It is important that body segment data be gathered and published in a standardized manner to achieve the following goals:

a. Researchers performing kinematickinetic studies of primate locomotion may employ and obtain comparable data and results to greatly facilitate the compar- ative studies needed to develop an understanding of the evolution of locomotor patterns within the primates;

308 I Wells and DeMenthon

b. The problem of rare, unavailable, or unsuitable cadaver material may be lessened by the availability of published data.

ACKNOWLEDGMENTS We thank Dr. Elwyn Simons and Ms. Andrea Katz of the Duke University

Primate Center for their assistance in procuring and making available suitable cadaver material for this study. We also thank Ms. Eleanor Tomasulo for her help with illustrations and Dr. John Mugaas for his helpful comments on the manuscript. Partial funding was provided by grants from the Bureau of Research of the Ameri- can Osteopathic Association and from the West Virginia School of Osteopathic Medicine Research Foundation to Dr. Wells.

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Plagenhoef, S. PATTERNS OF HUMAN MO- TION: A Cinematographic Analysis. New Jersey, Prentice-Hall Inc., 1971.

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