BODY SEGMENT PARAMETERS

D. Gordon E. Robertson, PhD, FCSBSchool of Human Kinetics

University of Ottawa

BODY SEGMENT PARAMETERSBRANCH OF ANTHROPOMETRY Necessary to derive kinetics from

kinematics (I.e., Σ F = m a, Σ Mcg = I a, a is acceleration of centre of gravity, a is ang. acceleration)

Called “inverse dynamics” Need to compute:

segment masssegment centre of gravitysegment moment of inertia tensor

Biomechanics Lab, University of Ottawa 2

SEGMENT MASS:

mass is a body’s resistance to changes in linear motion

need to measure total body mass using “balance scale”

each segment is a proportion of the total

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SEGMENT MASS:E.G., THIGH

Pthigh = mthigh / mtotal

Pthigh = thigh’s mass proportion

mtotal = total body mass

Therefore, mthigh = Pthigh mtotal

Note, Σ Pi = 1

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CENTRE OF GRAVITY:DEFINITION

point at which a body can be balanced

(xcg, ycg, zcg) = centre of gravity

also called centre of mass

first moment of mass i.e., turning effect on

one side balances turning effect of other side of centre of mass

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c. of gravity =(xcg, ycg, zcg)

CENTRE OF GRAVITY:EMPIRICAL METHOD: KNIFE EDGE

balance body on a “knife edge”

balance along a different axis

intersection is centre of gravity

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c. of g. is abovethe vertical line

mass on one sidebalances the other

CENTRE OF GRAVITY:EMPIRICAL METHOD: SUSPENSION

record plumb lines

intersection of plumb lines is centre

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suspend body from twodifferent points

SEGMENT CENTRE OF GRAVITY: PROPORTIONAL METHOD

Rp = rp / seg.length rp = distance from

centre of gravity to proximal end

need table of proportions derived from a population similar to subject

for many segments Rp is approximately 43% of segment length

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c. of gravity =(xcg ,ycg)

proximal end = (xp ,yp, zp)

distal end = (xd , yd, zd)

rp

TABLE OF PROPORTIONS: DEMPSTER (MODIFIED)

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Segment P Kcg Rproximal Rdistal

Hand 0.006 0.297 0.506 0.494Forearm 0.016 0.303 0.430 0.570Forearm and hand 0.022 0.468 0.682 0.318Arm 0.028 0.322 0.436 0.564

Upper extremity 0.050 0.368 0.530 0.470

Foot 0.0145 0.475 0.500 0.500Leg 0.0465 0.302 0.433 0.567Leg and foot 0.061 0.416 0.606 0.394Thigh 0.100 0.323 0.433 0.567

Lower extremity 0.161 0.326 0.447 0.553

Head and neck 0.081 0.495 1.000 0.000Trunk 0.497 0.500 0.500 0.500Trunk, head & neck 0.578 0.503 0.660 0.370

Foot 0.0145 0.475 0.500 0.500

Head and neck 0.081 0.495 1.000 0.000

SEGMENT CENTRE OF GRAVITY: E.G., THIGH

Rp = distance to c.of g. from proximal end as proportion of seg. lengthxcg = xp + Rp (xd – xp)ycg = yp + Rp (yd – yp)zcg = zp + Rp (zd – zp)

(xcg, ycg, zcg) = centre of gravity

(xp, yp, zp) = proximal end

(xd, yd, zd) = distal end

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c. of gravity =(xcg ,ycg)

proximal end = (xp ,yp, zp)

distal end = (xd , yd , zd)

CENTRE OF GRAVITY:LIMB OR PART OF A BODY

weighted average of segment centresxlimb = S(Pi xi) ∕ SPi

ylimb = S(Pi yi) ∕ SPi

zlimb = S(Pi zi) ∕ SPi

(xi, yi, zi) = mass centre of segment “i”

Pi = mass proportion of segment “i”

usually, SPi 1

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CENTRE OF GRAVITY:TOTAL BODY

weighted sum of all segments’ centresxtotal = S(Pi xi)ytotal = S(Pi yi)ztotal = S(Pi zi)

(xtotal, ytotal , ztotal) = total body centre of gravity

note, SPi =1

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MOMENT OF INERTIA:DEFINITION

body’s resistance to change in its angular motion

second moment of mass (squared distance)

of a point massIa = mr 2

for a distributed massIa = r 2 dm

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a

MOMENT OF INERTIA:EMPIRICAL METHOD

Ia = mgrt2 / 4p2

m = mass r = radius of

pendulum g = 9.81 m/s2

t = period of oscillation (time 20 oscillations then ÷ 20)

oscillations must be less than ±5 degrees

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a

r

m

PARALLEL AXIS THEOREM:E.G., THIGH ABOUT HIP CENTRE

rhip = distance from thigh centre of gravity to hiprhip = √[rx

2 + ry2 + rz

2]Ihip = Ithigh + mthigh rhip

2

Ithigh = moment of inertia about the thigh’s centre of mass

mthigh = segment mass

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rhip

MOMENT OF INERTIA:LIMB OR TOTAL BODY

repeated application of parallel axis theoremItotal = Σ Ii + Σ mi ri

2

I i = segment moments of inertia about each segment’s centre of gravity

m i = segment masses ri = distance of each

segment’s centre to limb or total body centre of gravity

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GEOMETRIC MODELS:HANAVAN (1965)

Hanavan developed the first 3D model of the human for biomechanical analyses

model consisted of 15 segments of ten conical frusta, two spheroids, an ellipsoid, and two elliptical cylinders

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MOMENT OF INERTIA: GEOMETRIC SOLIDS OF UNIFORM DENSITY

all models are assumed to be uniformly dense and symmetrical about their long axes

equations are based on integral calculus

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NEWTON-EULER EQUATIONS:SECOND LAW Newton’s Second Law

S F = m a For rotational motion of rigid bodies

Euler extended this law to: where a = (ax, ay, az)T is the angular

acceleration of the object about its centre of gravity and is the inertia tensor:

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II I II I II I I

xx xy xz

yx yy yz

zx zy zz

I

S M I a

MOMENT OF INERTIA IN 3D:INERTIA TENSOR it can be shown that the inertia tensor can

be reduced to a diagonal matrix for at least one specific axis

if body segments are modeled as symmetrical solids of revolution, using a local axis that places one axis (usually z) along the longitudinal axis of symmetry reduces the inertia tensor to:

= Ixx , Iyy , Izz are called the principal

momentsof inertiaBiomechanics Lab, University of Ottawa 20

II

II

xx

yy

zz

0 0

0 00 0

MOMENT OF INERTIA:SPHEROID & ELLIPSOID

m = mass, r = radius

Ixx = Iyy = Izz = 2/5 mr2

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y

xzy

xz

a = depth (x), b = height (y), c = width (z)

Ixx = 1/5 m (b2+c2)Iyy = 1/5 m (a2+c2)Izz = 1/5 m (a2+b2)

MOMENT OF INERTIA:CIRCULAR & ELLIPTICAL CYLINDERS

m = mass, l = length of cylinder, r = radius

Ixx = 1/2 mr2

Iyy = 1/12 m (3r2+l2)Izz = 1/12 m (3r2+l2)

l = length, b = height/2 (y), c = width/2 (z)

Ixx = 1/4 m (b2 +c2)Iyy = 1/12 m (3c2 +l2)Izz = 1/12 m (3b2 +l2)

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y

xz

y

xz

MOMENT OF INERTIA: RIGHT- CIRCULAR CONE AND FRUSTUM

m = mass, l = length of cone, r = radius at base

Ixx = 3/10 mr2

Iyy = 3/5 m (¼ r2 + l2)

Izz = 3/5 m (¼ r2 + l2)

subtract smaller cone from largerBiomechanics Lab, University of Ottawa 23

y

xzy

xz

VISUAL3D USES GEOMETRIC SOLIDSE.G., FENCING for Visual3D tutorials visit:http://www.c-motion.com/v3dwiki/index.php?

title=Tutorial_Typical_Processing_Sessionhttp://www.c-motion.com/v3dwiki/index.php?title=Tutorial:_Building_a_Model

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VISUAL3D - MODELS:CREATING THE MODEL modeling begins by selecting a Vicon processed static trial select Model | Create(Add Static Calibration File) usually Hybrid Model from C3DFile is chosen

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VISUAL3D - MODELS:SELECT SEGMENT FROM MENU

from Models tab select segment to be created

drop-down menu offers predefined segments

e.g., select Right Thigh

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VISUAL3D - MODELS:E.G., RIGHT THIGH SEGMENT (RTH)

define proximal lateral marker and radius of thigh

define distal lateral and medial markers

check all tracking markers for thigh or

or check box marked Use Calibration Targets for Tracking

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VISUAL3D – MODELS:SEGMENT PROPERTIES

segment mass is 0.1000 × total body mass (default)

geometry is CONE (actually conical frustum)

computed principal moments of inertia are shown in kg.m2

centre of mass’s axial location (metres) is based on thigh’s computed length

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VISUAL3D – MODELS: COMPLETED WHOLE BODY

local 3D axes are shown at the proximal joint centres

yellow lines join segment endpoints

added epee “segment”

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VISUAL3D – MODELS: COMPLETED WHOLE BODY skeletal “skin”

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EXAMPLES:GROUND LEVEL PLATES lacrosse gymnastics

liftingballet

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EXAMPLES:SPECIAL FORCE PLATESseat and grabrail stairs

rowing obstacle

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