Transcript D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa  Necessary to derive kinetics from kinematics (I.e., Σ F = m.

D. Gordon E. Robertson, PhD, FCSB
School of Human Kinetics
University of Ottawa
 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 mass
 segment centre of gravity
 segment moment of inertia tensor

Biomechanics Lab, University of Ottawa
2
 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
Biomechanics Lab, University of Ottawa
3
 Pthigh
= mthigh / mtotal
 Pthigh = thigh’s mass
proportion
 mtotal = total body
mass
 Therefore,
mthigh = Pthigh mtotal
 Note, Σ Pi = 1
Biomechanics Lab, University of Ottawa
4
 point
c. of gravity =
(xcg, ycg, zcg)
Biomechanics Lab, University of Ottawa
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
5
 balance
c. of g. is above
the vertical line
body on a
“knife edge”
 balance along a
different axis
 intersection is centre
of gravity
mass on one side
balances the other
Biomechanics Lab, University of Ottawa
6
suspend body from two
different points
Biomechanics Lab, University of Ottawa
 record
plumb
lines
 intersection
of plumb
lines is
centre
7
 Rp
proximal end = (xp ,yp, zp)
rp
c. of gravity =
(xcg ,ycg)
distal end = (xd , yd, zd)
Biomechanics Lab, University of Ottawa
= 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
8
Segment
P
Kcg
Rproximal
Rdistal
Hand
Forearm
Forearm and hand
Arm
0.006
0.016
0.022
0.028
0.297
0.303
0.468
0.322
0.506
0.430
0.682
0.436
0.494
0.570
0.318
0.564
Upper extremity
0.050
0.368
0.530
0.470
Foot
Leg
Leg and foot
Thigh
0.0145
0.0465
0.061
0.100
0.475
0.302
0.416
0.323
0.500
0.433
0.606
0.433
0.500
0.567
0.394
0.567
Lower extremity
0.161
0.326
0.447
0.553
Head and neck
Trunk
Trunk, head & neck
0.081
0.497
0.578
0.495
0.500
0.503
1.000
0.500
0.660
0.000
0.500
0.370
Biomechanics Lab, University of Ottawa
9
 Rp
proximal end = (xp ,yp, zp)
c. of gravity =
(xcg ,ycg)
distal end = (xd , yd , zd)
Biomechanics Lab, University of Ottawa
= distance to c.of
g. from proximal end
as proportion of seg.
length
xcg = 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
10
 weighted
average of
segment centres
xlimb = 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
Biomechanics Lab, University of Ottawa
11
 weighted
sum of
all segments’
centres
xtotal = S(Pi xi)
ytotal = S(Pi yi)
ztotal = S(Pi zi)
 (xtotal, ytotal , ztotal)
= total body
centre of gravity
 note, SPi =1
Biomechanics Lab, University of Ottawa
12
 body’s
a
resistance to
change in its
angular motion
 second moment of
mass (squared
distance)
 of a point mass
Ia = mr 2
 for
a distributed
mass
Ia =  r 2 dm
Biomechanics Lab, University of Ottawa
13
a
r
m
Biomechanics Lab, University of Ottawa
= 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
 Ia
14
 rhip
rhip
Biomechanics Lab, University of Ottawa
= distance from
thigh centre of
gravity to hip
rhip = √[rx2 + ry2 + rz2]
Ihip = Ithigh + mthigh rhip2
 Ithigh = moment of
inertia about the
thigh’s centre of mass
 mthigh = segment mass
15
repeated application
of parallel axis
theorem
Itotal = Σ Ii + Σ mi ri2
 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

Biomechanics Lab, University of Ottawa
16
 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
Biomechanics Lab, University of Ottawa
17
 all
models are
assumed to be
uniformly dense
and symmetrical
about their long
axes
 equations are
based on integral
calculus
Biomechanics Lab, University of Ottawa
18
 Newton’s

Second Law
SF =ma
 For
rotational motion of rigid bodies Euler
extended this law to:
S M  Ia
where a = (ax, ay, az)T is the angular acceleration
of the object about its centre of gravity and I is
the inertia tensor:
Ixx Ixy Ixz

I  Iyx
Iyy
Iyz
Izx
Izy
Izz
Biomechanics Lab, University of Ottawa
19
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
I  0
0
Iyy
0
0
0
0
Izz
Biomechanics Lab, University of Ottawa
= Ixx , Iyy , Izz are called
the principal moments
of inertia
20
m
= mass, r = radius
Ixx = Iyy = Izz = 2/5 mr2
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)
Biomechanics Lab, University of Ottawa
21
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)

Biomechanics Lab, University of Ottawa
22
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 larger
Biomechanics Lab, University of Ottawa
23

for Visual3D tutorials visit:
http://www.c-motion.com/v3dwiki/index.php?title=Tutorial_Typical_Processing_Session
http://www.c-motion.com/v3dwiki/index.php?title=Tutorial:_Building_a_Model
Biomechanics Lab, University of Ottawa
24



modeling begins by selecting a Vicon processed static trial
select Model | Create(Add Static Calibration File)
usually Hybrid Model from C3DFile is chosen
Biomechanics Lab, University of Ottawa
25
 from
Models
tab select
segment to be
created
 drop-down
menu offers
predefined
segments
 e.g., select
Right Thigh
Biomechanics Lab, University of Ottawa
26
 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
Biomechanics Lab, University of Ottawa
27
 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
Biomechanics Lab, University of Ottawa
28
 local
3D axes
are shown at
the proximal
joint centres
 yellow lines
join segment
endpoints
 added epee
“segment”
Biomechanics Lab, University of Ottawa
29
 skeletal
“skin”
Biomechanics Lab, University of Ottawa
30
lacrosse
gymnastics
lifting
ballet
Biomechanics Lab, University of Ottawa
31
seat and grabrail
stairs
rowing
Biomechanics Lab, University of Ottawa
obstacle
32