# D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa.

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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 M cg = I , a is acceleration of centre of gravity,  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

 P thigh = m thigh / m total  P thigh = thigh’s mass proportion  m total = total body mass  Therefore, m thigh = P thigh m total  Note, Σ P i = 1 Biomechanics Lab, University of Ottawa 4

 point at which a body can be balanced  (x cg, y cg, z cg ) = 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 Biomechanics Lab, University of Ottawa 5

 balance body on a “knife edge”  balance along a different axis  intersection is centre of gravity Biomechanics Lab, University of Ottawa 6

 record plumb lines  intersection of plumb lines is centre Biomechanics Lab, University of Ottawa 7

 R p = r p / seg.length  r p = distance from centre of gravity to proximal end  need table of proportions derived from a population similar to subject  for many segments R p is approximately 43% of segment length Biomechanics Lab, University of Ottawa 8

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 R p = distance to c.of g. from proximal end as proportion of seg. length x cg = x p + R p (x d – x p ) y cg = y p + R p (y d – y p ) z cg = z p + R p (z d – z p )  (x cg, y cg, z cg ) = centre of gravity  (x p, y p, z p ) = proximal end  (x d, y d, z d ) = distal end Biomechanics Lab, University of Ottawa 10

 weighted average of segment centres x limb =  (P i x i ) ∕  P i y limb =  (P i y i ) ∕  P i z limb =  (P i z i ) ∕  P i  (x i, y i, z i ) = mass centre of segment “i”  P i = mass proportion of segment “i”  usually,  P i  1 Biomechanics Lab, University of Ottawa 11

 weighted sum of all segments’ centres x total =  P i x i  y total =  P i y i  z total =  P i z i   (x total, y total, z total ) = total body centre of gravity  note,  P i =1 Biomechanics Lab, University of Ottawa 12

 body’s resistance to change in its angular motion  second moment of mass (squared distance)  of a point mass I a = mr 2  for a distributed mass I a =  r 2 dm Biomechanics Lab, University of Ottawa 13

 I a = mgrt 2 / 4  2  m = mass  r = radius of pendulum  g = 9.81 m/s 2  t = period of oscillation (time 20 oscillations then ÷ 20)  oscillations must be less than ±5 degrees Biomechanics Lab, University of Ottawa 14

 r hip = distance from thigh centre of gravity to hip r hip = √[r x 2 + r y 2 + r z 2 ] I hip = I thigh + m thigh r hip 2  I thigh = moment of inertia about the thigh’s centre of mass  m thigh = segment mass Biomechanics Lab, University of Ottawa 15

 repeated application of parallel axis theorem I total = Σ I i + Σ m i r i 2  I i = segment moments of inertia about each segment’s centre of gravity  m i = segment masses  r i = 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   F = m a  For rotational motion of rigid bodies Euler extended this law to:   where  = (  x,  y,  z ) T is the angular acceleration of the object about its centre of gravity and is the inertia tensor: 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: = I xx, I yy, I zz are called the principal moments of inertia Biomechanics Lab, University of Ottawa 20

 m = mass, r = radius I xx = I yy = I zz = 2/5 mr 2 Biomechanics Lab, University of Ottawa 21  a = depth (x), b = height (y), c = width (z) I xx = 1/5 m (b 2 +c 2 ) I yy = 1/5 m (a 2 +c 2 ) I zz = 1/5 m (a 2 +b 2 )

 m = mass, l = length of cylinder, r = radius I xx = 1/2 mr 2 I yy = 1/12 m (3r 2 +l 2 ) I zz = 1/12 m (3r 2 +l 2 )  l = length, b = height/2 (y), c = width/2 (z) I xx = 1/4 m (b 2 +c 2 ) I yy = 1/12 m (3c 2 +l 2 ) I zz = 1/12 m (3b 2 +l 2 ) Biomechanics Lab, University of Ottawa 22

 m = mass, l = length of cone, r = radius at base I xx = 3/10 mr 2 I yy = 3/5 m (¼ r 2 + l 2 ) I zz = 3/5 m (¼ r 2 + l 2 )  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.m 2  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 rowingobstacle Biomechanics Lab, University of Ottawa 32

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