Anatomy of human motion
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Anatomy of Human Motion
Wangdo Kim
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How to theorize swing?
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Autopilot: a cognitive
state in which you act
without self-awareness
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2-D Displacement in terms of the simplest path
12
2B
12P
1B
1A
2A12
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Introduction: methods for describing human joint motions (continue)
Screw axis
Screw motion of a rigid body
s
Screw axis
X
Y
Z
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Introduction: methods for describing human joint motions
Euler angles and joint coordinate system (JCS)
Joint coordinate system of the knee
x’
y’, y”
x”
z”
z’
x”, x’”
y”
y’”
z”
z’”
x’
z
z’
y’
x
y
About the z-axis About the y’-axis About the x”-axis
Euler angles with sequence of z-y’-x”
cossin0
sincos0
001
cos0sin
010
sin0cos
100
0cossin
0sincos
R
A standard joint rotation convention for the knee
joint proposed by Chao (1980a)
Grood and Suntay (1983) proposed a non-
orthogonal joint coordinate system (JCS) to avoid
sequence dependency by predefining the axes of
rotation.
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DUAL NUMBER
The concept was introduced by Clifford (1873) and the name was given by Study(1903).
The dual number is defined such that
0 and 2= 0
A dual number is written as
Where symbol a represents the primary (or real) part of duplex (or dual) number and symbol a0 represents the dual component of dual number .
a εαα
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•The dual angle express the relationship between
lines in space A and B.
DUAL ANGLES
s
Line A
Line B
sεθθ
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Description of a Vector Constrained on a Line with Dual
Vectors
the primary part V called
resultant vector comprises
the magnitude and direction
of the vector.
The dual part W called
moment vector is defined
as , where r connects the
origin to any point on the
line of the vector.
X
Y
Z
V W
O
r
V̂
ˆ V V W W r V
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Screw motions with respect to coordinate axes
Dual-number transformation
where
( ) : screw motion displacement
: dual vector
: dual-number
transformation matrix
Screw motion through X-axis
0ˆ ˆ ˆˆ( )XR
V V
1 0 0
ˆ ˆ ˆ ˆ( ) 0 cos sin
ˆ ˆ0 sin cos
XR
000ˆ WVV
a ˆ02
00 VrW
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DUAL TRANSFORMATION: Description of general spatial joint
motions with dual Euler angles
Representation of a general spatial joint motion by three
successive screw motions
Resultant dual-number transformation matrix
For sequence of screw motions z-y’-x”
' "ˆˆ ˆ ˆ ˆˆ ˆ( ) ( ) ( )
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆcos cos sin sin cos cos sin cos sin cos sin sin
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆcos sin sin sin sin cos cos cos sin sin sin cos
ˆ ˆ ˆˆ ˆsin sin cos cos cos
z y xR R R R
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The analysis of golf swing as a kinematic chain using dual Euler angle algorithm
Journal of Biomechanics, In Press,
Koon Kiat Teu, Wangdo Kim, Franz Konstantin Fuss and John Tan
with the sequence-dependent Euler angles being non-vectors, it makes velocity analysis more complex to conduct and less intuitive to understand.
This is where the dual Euler angles method stands out, especially for studies involving multi-segment biomechanics because it can provide intuitive physical interpretation.
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J. of Biomech, 2005, in press
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Modeling
5 segment model
Frame {1} was attached to the rotating torso at the glenohumeral joint
{2} was attached to the upper arm at the elbow joint.
{3} was attached to the forearm at the wrist joint .
5
G
●--
reflective
marker
L 1
z 1
x 1
y 2
y 3
y 4
z 2
z 3
z 4
x 2
x 3
x 4
L 2
L 3
L 4
y 1
●
●
●
z 0
y 0
x 0
z
G
G
x
y
L 0x
L 0z
z x 5
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Modeling
{4} was attached to the
hand at the end of the
hand grip.
{5} was attached to the
center of the clubhead.
Fixed frame {0} was
attached to the fixed
lower extremity at the
waist.
5
G
L 1
z 1
x 1
y 2
y 3
y 4
z 2
z 3
z 4
x 2
x 3
x 4
L 2
L 3
L 4
y 1
●
●
●
z 0
y 0
x 0
z
G
G
x
y
L 0x
L 0z
z x 5
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Dual Euler Angles Calculation Dual Euler angle takes account of the length of arm
segment
Zy’x” dual Euler angle convention.
Five links kinematics chain
Denavit-Hartenberg parameters
The transformation matrix for link n:
1ˆ ˆ ˆ ˆ( ) ( ) ( )n
n n n nM Z y x
1
2 z 1 y' 1 x" 1 1ˆˆ ˆ ˆ ˆˆ ˆM R ( ) R ( ) R ( L )
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Dual Velocities Let the speed for the screw motion be:
V = linear speed along the screw axis
= angular speed about the screw axis
The direction and location of the screw axis can be specified by the unit screw
vector
uxOPuu ε
Where symbol represents a unit vector and the vector extends
from the origin of the coordinate system to any point on the screw
axis. These quantities can be combined into a “motor”, the dual
multiple of unit line vector
uεΩ VV
u
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Individual joint-link transformation matrices based on dual Euler angle is:
1
2 z 1 y' 1 x" 1 1ˆˆ ˆ ˆ ˆˆ ˆM R ( ) R ( ) R ( L )
2
3 z 2 y' 2 x" 2 2ˆˆ ˆ ˆ ˆˆ ˆM R ( ) R ( 0 ) R ( L )
3''x
o
3'y3Z
3
4 LεR15βRγRM
5''x4Z
4
5 LεRLεRM
10005 o
1'1
'15
'1
15
1
o
'0"0
"05
"0
o
0'0
'05
'0
o
G0
05
0
o
50
5
MMVMVMVMV VV1o
10"
33221 o
3'3
'35
'3
o
'32
35
3
o
2'2
'25
'2
o
"21
25
2
o
'1"1
"15
"1 VMVMVMVMVM
The clubhead motion is the sum of motions produced by the joints
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Attachment of Goniometers
2 EGMs attached to the acromion process and to the upper arm
1 EGM attached to the dorsolateral side of upper arm and forearm
2 EGMs were connected to the dorsal sides of hand and proximal forearm
Accurate measurement of the joint motions? Overall protocol of 2-D goniometer and a torsionmeter’s was not validated
J. of sports eng. 2005, in press
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Torsion meter:IR/ER or FL/EXT PR/SUR?
Sports Engineering (2005), 8, Using dual Euler angles for the analysis of arm
movement during the badminton smash.
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VERIFICATION (holistic)
1 0.8 0.6 0.4 0.2 025
20
15
10
5
0
5
10
15
20
25
30
35
40
45
5050
25
rvk 0
tvk 0
0.1.2 timeik
Time(s)
Point of impact
Calculated Velocity (ms-1)
Measured Velocity (ms-1)
It could be coincident: the validity of individual joint
measurement is still needed.
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The Experiment
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0.2 0.15 0.1 0.05 015
10
5
0
5
10
15
2020
15
arsk
aask
iesk
fee k
spek
few k
urwk
torso_rotk
00.25 timeik
Results & Applications (Velocity Contribution Subject 1)
0.2 0.15 0.1 0.05 015
10
5
0
5
10
15
2020
15
arsk
aask
iesk
fee k
spek
few k
urwk
torso_rotk
3.402 1013
0.25 timeik
Upper arm Retroversion/Anterversion
Upper arm Adduction/Abduction
Upper arm Internal/External Rotation
Forearm Extension/Flexion
Forearm Pronation/Supination
Hand Extension/Flexion
Hand ulnar/radial abduction
Torso Rotation
Velocity Contribution
(ms-1)
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Passive motion characteristics of the talocrural and the subtalar joint by dual Euler angles
Journal of Biomechanics, Volume 38, Issue 12, December 2005, Pages 2480-2485
Yueshuen Wong, Wangdo Kim and Ning Ying
' "ˆˆ ˆ ˆ ˆˆ ˆ( ) ( ) ( )
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆcos cos sin sin cos cos sin cos sin cos sin sin
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆcos sin sin sin sin cos cos cos sin sin sin cos
ˆ ˆ ˆˆ ˆsin sin cos cos cos
z y xR R R R
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
ˆ
r r r s s s
R R S r r r s s s
r r r s s s
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Algorithm for computing dual-number transformation matrix from point coordinates : given screw motion
Let and (i = 1, 2, …n; (n≥3)) denote coordinates of
non-collinear points measured at the initial and final joint
positions
Dual-number transformation matrix should minimize
subject to
where
2 2
1
1( )
n
i i i i
i
Jn
V V W W
R̂
0ˆ ˆ ˆi i i iR
V V W V
0 0 0 0 0 0ˆ ( ) ( )i i i V r c c r c )()(ˆ crccrWVV iiiii
n
i
in 1
00
1rc
n
i
in 1
1rc
0irir
ˆ ˆT
R R I
the constrained optimization problem using sequential quadratic
programming (SQP) methods (Fletcher, 1980). Optimization toolbox in
MATLAB (The Math Works Inc., Natick, MA, USA)
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The Combining of measurements with biomechanical models
Ill-conditioned: a situation in which the solution is extremely sensitive to the data
Smoothing Raw Coordinate Data: A time domain approach data smoothing was implemented primarily because of the uncertain characteristics of frequencies in joint motions.
the generalized cross-validation (GCV) estimate is used for the smoothing parameter.
The advantage over a conventional filter is that the GCV algorithm chooses the cutoff frequency automatically based on an evaluation of all the data.
Dohrmann and Trujillo (1988) combined this algorithm with dynamic programming and provided a method for smoothing and estimating the first and second derivatives of noisy data.
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Kinematic measurement of the ankle joint complex
Measurement device: ‘Flock of Birds’ (FOB)
electromagnetic tracking system (Ascension Technology
Inc., USA) Mean Error and
Standard Deviation
Dual angle about
z-axis
Rotation 0.410.06
Translation 0.520.07mm
Dual angle about
y-axis
Rotation 0.470.06
Translation 0.870.08mm
Dual angle about
x-axis
Rotation 0.740.05
Translation 0.380.03mm
Accuracy of dual Euler angles obtained from FOB output : J. Biomech, 2002, 35, 1647-1657
Determining dual euler angles of the ankle complex in vivo using “FOB”, J. Biomech Eng, 2005, 127, 98-107.
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Experimental rig for in vitro experiments on foot/shank specimens
1
2
3
4
5
6
7
8
9
Anterior-
posterior
direction
Medial-lateral
direction
1: Vertical stands
2: Beams supporting shank rod
3: Shank rod
4: Foot plate
5: Screw securing foot plate on
supporting bracket
6: Horizontal axis of the foot
plate
7: Supporting bracket
8: Screw securing supporting
bracket on ground plate
9: Ground plate
Clinical Bio, 2004, 19, 153-160
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Definition of coordinate systems
Anatomical coordinate system of the tibia
Origin is at the midpoint of the line joining MM and LM
Y-axis is orthogonal to the quasi-frontal plane defined by MM, LM, and HF
Z-axis is orthogonal to the quasi-sagittal plane defined by Y-axis and TT
X-axis is the cross product of Y- and Z-axis
At the neutral position, local coordinate systems of the talus and the calcaneus are coincident with that of the tibia
TT
HF
MM
LM
Y
Z
X
Left
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The Sensors
Sensors attached to
calcaneum, talar neck and
tibia
ligaments, retinacula and
tendons preserved
Accuracy of system verified
to have resolution of 1.8mm
and 0.5 degrees
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Introduction: modeling of the ankle joint complex
Hinge joint model
Sphere joint model
Four-bar linkage model
(adopted from Leardini et al., 1999) (adopted from Dul and Johnson, 1985)
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Screw motions of the foot
y
z x
y
x
x’
y’
z’(z)
dorsiflexion-
plantarflexion
shift
x’ y”(y’)
z’
z” x”
drawer eversion-
inversion
y”
z” x”
compression-
distraction
abduction-
adduction
After screw motion
through z-axis
After screw motion
through y’-axis After screw motion
through x”-axis Initial position
J. of biomech eng, 2005, 127, 98-107
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Estimation of the axis of a screw motion from noisy data—A new method based on Plücker lines
Journal of Biomechanics, In Press, Koon Kiat Teu and Wangdo Kim
Determination of screw axis based on dual
number transformation matrix (DTM) which
transforms Plücker lines.
Demonstrate the robustness and reliability of
generating transformation results from the
mapping of vectors.
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Introduction Anatomical landmarks are located
by palpation
And are then denoted by markers
fixed to the skin
They are prone to errors due to
Subjective localization
Skin movements
Objective method for the
localization of landmarks needed
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Computing DTM from Point Coordinates
Centroids of the points at the initial and final position are given by
and
respectively.
n
i
in 1
00
1rc
n
i
in 1
1rc
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Computing DTM from Point Coordinates
According to the dual
transformation relationship,
the vector at final position
is:
The same vector can also be
calculated from the
measured data as:
0ˆ ˆ ˆi i i iR
V V W V
)()(ˆ crccrWVV iiiii
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Computing DTM from Point Coordinates
Because of noise, there is difference
between and .
In the least-square error sense, the DTM
should minimize the following function:
iV~ˆ
iV̂
2 2
1
1( )
n
i i i i
i
Jn
V V W W
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Geometry of Screw Axes
(1)
where
Rewrite (1) as
and seek solution other than .
ˆ ˆ ˆ[ ] .RV V
([ ] [ ])(
[ ] ([ ] [ ] )
R S
R S R
V + εW V + εW)
V V W
ˆ ˆ[ - ] 0I R V
ˆ 0V
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Geometry of Screw Axes
Separate into primary and dual component respectively:
(2)
The first equation in (2) means that the V is simply the eigenvector of the primary component R of the DTM.
[ - ] 0
[ - ] [ ]
I R
I R D
V
W V
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Geometry of Screw Axes
Using singular value decomposition (SVD) :
Therefore the dual part in (2) can be found as follows:
The reference point for the screw axis:
1
1/ T
jR I V diag w U
1/ [ ]T
jV diag w U D W V
C =V ×W
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Application Results
Start
End
Dorsiflexion-Plantarflexion
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Application Results
Start
End
Eversion-Inversion
(a)
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Comparison
Mean error plotted against the SNR.
(a) (b) Error in direction
0
5
10
15
20
25
1 10 100 1000
SNR
De
gre
es
Error in position
0
5
10
15
20
25
30
1 10 100 1000
SNR
Err
or (c
m)
“Plücker line method”
“Schwartz method”
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Comparison
Mean error plotted against magnitude of the skin position
artefact.
(a) (b)
Error in direction
0
2
4
6
8
10
12
0 50 100 150 200
Skin movement, % of typical position artefact
De
gre
es
Error in position
0
5
10
15
20
25
0 50 100 150 200
Skin movement, % of typival position artefact
Err
or (c
m)
“Plücker line method”
“Schwartz method”
(Journal of Biomechanics, 2005)
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Foot-Surface Cushioning Mechanism during Stance Phase of Running
The purpose of the study is to develop a biomechanical
model of the foot/ground interface
The extended Kalman filter (EKF) estimators, which
were adopted as parameter identification technique for
the physiological system
The natural frequency of the foot-surface cushioning
mechanism during stance phase of running resides below
10 Hz.
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Modeling and verification
K C
y
m
L
Fig. 1 The proposed model (sagittal view)
K = spring constant of the foot/ground interface
C = damping coefficient of the foot/ground interface
0 = initial angle at the heel strike, which measured from y-axis
m = mass of the subject
L = length of the leg—hip to ankle joint
y = direction of deformation of the contact point at the foot/ground interface
0
Fig. Positions of attached markers on a
subject’s body. The line connects the hip
to the ankle joint, representing a rigid
bar in the model in Fig 1. Even though
the Fig 3. shows the subject’s wearing
shoes, this study only carries the bare
foot case.
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The state vectors of the model
2 ˆ2 ( )y y y F w t
21
2
2 2 3 4 4 1
3
4
00
ˆ12( )
0 00
0 00
xx
x Fx x x x xw t
x
x
1
2
3
4
y x
y x
x
x
2
2 3 4 4 1( ) 2 ( )F t m x x x x x v t
•state-variable estimates may in this
circumstances be even preferable to direct
measurements, because the errors
introduced by the instruments that provide
these measurement may be larger than the
errors in estimating these variables.
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extended Kalman filter/estimators
1 1ˆ ˆ( ) ( ( ))k k kx f x
1
1 1 1 1
1 2 3 4
2 2 2 2
1 2 3 4
1
3 3 3 3ˆ ( )
1 2 3 4
4 4 4 4
1 2 3 4
k
k
x x
f f f f
x x x x
f f f f
x x x xk
f f f f
x x x x
f f f f
x x x x
f
x
ˆˆ ( ( ))k k kz h x
ˆ ( )
1 2 3 4
k
k
x x
k k k k
kH
h h h h
x x x x
h
x
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The covariance values
1 1 1 1( ) ( )k k k k kP P Q
Computing the a priori covariance matrix:
Computing the Kalman gain:
Computing the a posterior covariance matrix
Conditioning the predicted estimate on the measurement:
1( ) [ ( ) ]T T
k k k k k k kK P H H P H R
( ) (1 ) ( )k k k kP K H P
ˆ ˆ ˆ( ) ( ) ( )k k k k kx x K z z
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State variables estimated by EKF in the case of running on compliant surface (Polyurethane).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.5
0
0.5
Time (sec)
Dam
p.
Facto
r
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
50
Time (sec)
om
ega
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-1000
0
1000
2000
Time (sec)
forc
e (
New
ton)
True
Est.
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State variables estimated by EKF in the case of running on non-compliant surface (Ceramic).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.2
0
0.2
Time (sec)
Dam
p.
Facto
r
0 0.02 0.04 0.06 0.08 0.1 0.12 0.1465
70
75
Time (sec)
om
ega
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
1000
2000
Time (sec)
forc
e (
New
ton)
Measured
Estimated
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the markers and the muscle surface for the
close-range stereophotogrammetry.
Tracking inhomogeneous motions of soft tissue surfaces —
A new method based on the deformation gradient at each
material point
so local
measurement is
insufficient and
a full field
measurement is
necessary.
Tracking soft
tissue motions
is always
hampered by
material
inhomogeneity,
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A group of markers from which an estimate for the F
in point P is calculated on the curved surface.
x
[ ]F
1
X
4X
3X P
P
2X
X
4x
1x
2x
3x
xF
X dx F dX
Physical significance of
F: it relates the length
and orientation of a
material fiber dX to dx
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Deformation Gradient tensor: F
; (1,2.., )i i i i i n x F X v w
1
( ) ( )n
i i i i i i
in
1
J x F X v x F X v
01
ˆ
ˆ
T 1
00
v x F X
F X X
1
1(( ) )
n
i
in
2
0 sJ X X N ( ) N N 1
*
1
1(( ) ) ( )
n
i
in
2
0 s 0J X X N λ N N 1
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Interfragmentary Motion
hard callus
soft callus
Einhorn ‘98
cortex
Intramedullary canal
Tissue bridge crossing a fracture.
Combination of hard and soft tissue.
Secondary bone healing
Callus
Callus
tibia
tibia
Intramedullary canal
Fracture
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linear stage micrometer load cell specimen
optical
work bench
Methods: Loading
unconfined axial compression
displacement: micrometer (0.25 m resolution)
load: 50 N load cell
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Optical work bench
Micrometer screw
Linear translation stage
High resolution load cell
ESPI sensor
Fiber optic
Methods: Complete setup
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Reference image +Y
Reference image -Y Reference image -X
Reference image + X
interference fringe +Y
interference fringe -X
interference fringe -Y
interference fringe +X
Phase shift
phase map
Transformation
Speckle image
+ Y
Speckle image
- Y Speckle image
- X
Speckle image
+ X Methods: Imagine algorithm
Ettemeyer AG, Nersingen Germany
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Results: Single-Step Measurement
0.00
0.02
0.04
0.06
0.08
0.10
0.12
[%]
-0.3
0.0
0.3
0.6
0.9
1.2
[m]
displacement strain εC
X
Y
X
Y
Compressive Strain
Principal Strain
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Mechanobiology: How mechanical conditions
regulate biological process
Undecalcified Histology:
light blue: connective tissue
red-brown: new callus
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The location and the shape of secondary centers of ossification
can be predicted from the distribution of hydrostatics and shear
stress calculated in finite element analyses.
A single parameter
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Polar Decomposition: Separation of stretch
and rotation
F=QS
This novel approach has considerable potential for
investigating skin movement artifacts or material modeling
of biological tissues--dichotomy.
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Principal stretches
m1
m3
m2
m1
m2
m3
1 2 2 3 1 1 2 3 3
S = m m m m m m
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Factorizing: Stretch then Rotation
50 100 150 200
20
40
60
80
100
120
140
160
180
200
220Area Change
0.9
0.92
0.94
0.96
0.98
1
1.02
ˆ F Q S
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Human Motion Mechanics with biological examples
Researchers from biology and psychology: are
not familiar with mechanics
Researchers from mechanical eng:
underexposed to biology/psychology and
disregard the complexity of living species
Learn each other’s language so we can
communicate better
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Q&A