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

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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

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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

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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

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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

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 )

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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

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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

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for Visual3D tutorials visit: Biomechanics Lab, University of Ottawa 24

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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

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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

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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

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segment mass is × 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

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local 3D axes are shown at the proximal joint centres yellow lines join segment endpoints added epee “segment” Biomechanics Lab, University of Ottawa 29

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skeletal “skin” Biomechanics Lab, University of Ottawa 30

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lacrosse gymnastics lifting ballet Biomechanics Lab, University of Ottawa 31

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seat and grabrail stairs rowingobstacle Biomechanics Lab, University of Ottawa 32

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