The document discusses various methods for describing and analyzing human joint motion, including Euler angles, joint coordinate systems, screw axes, and dual Euler angles. It presents a model of the golf swing as a 5-segment kinematic chain and calculates the joint-link transformation matrices using dual Euler angles. Individual joint velocities are determined and the total clubhead velocity is calculated as the sum of individual joint contributions. An experiment is described to verify the dual Euler angle analysis of golf swing motion.

1. Anatomy of Human Motion
Wangdo Kim

2. How to theorize swing?

3. Autopilot: a cognitive
state in which you act
without self-awareness

4. 2-D Displacement in terms of the
simplest path
12

2
B
12
P
1
B
1
A
2
A
12


5. Introduction: methods for describing
human joint motions (continue)
 Screw axis
Screw motion of a rigid body
s

Screw axis
X
Y
Z

6. 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”
 
























 


























cos
sin
0
sin
cos
0
0
0
1
cos
0
sin
0
1
0
sin
0
cos
1
0
0
0
cos
sin
0
sin
cos
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.

7. 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
ε
α
α 



8. •The dual angle express the relationship between
lines in space A and B.
DUAL ANGLES
s

Line A
Line B
s
ε
θ
θ 



9. 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
ˆ
V
ˆ 
 
V V W
 
W r V

10. 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
0
ˆ ˆ ˆ
ˆ
( )
X
R 
 
  
V V
1 0 0
ˆ ˆ ˆ ˆ
( ) 0 cos sin
ˆ ˆ
0 sin cos
X
R   
 
 
 
   
 
   
 
 
0
0
0
ˆ W
V
V 


a


 

ˆ
0
2


0
0 V
r
W 


11. 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”
' "
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
( ) ( ) ( )
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
cos cos sin sin cos cos sin cos sin cos sin sin
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
cos sin sin sin sin cos cos cos sin sin sin cos
ˆ ˆ ˆ
ˆ ˆ
sin sin cos cos cos
z y x
R R R R
  
           
           
    
 
     

     
 
 
 


  

 







12. 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.

13. J. of Biomech, 2005, in press

14. 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

15. 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
x0
z
G
G
x
y
L0x
L
0z
z
x
5

16. 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:
1
ˆ ˆ ˆ ˆ
( ) ( ) ( )
n
n n n n
M Z y x
  

 

1
2 z 1 y' 1 x" 1 1
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
M R ( ) R ( ) R ( L )
 
   
    
   
 

17. 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












u
x
OP
u
u 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
 



 u
ε
Ω V
V

u

18. Individual joint-link transformation
matrices based on dual Euler angle is:
1
2 z 1 y' 1 x" 1 1
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
M R ( ) R ( ) R ( L )
 
   
    
   
 
2
3 z 2 y' 2 x" 2 2
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
M R ( ) R ( 0 ) R ( L )
 
   
     
   
 
 





































3
'
'
x
o
3
'
y
3
Z
3
4 L
ε
R
15
β
R
γ
R
M
   
















5
'
'
x
4
Z
4
5 L
ε
R
L
ε
R
M
















 1
0
0
0
5 o
1
'
1
'
1
5
'
1
1
5
1
o
'
0
"
0
"
0
5
"
0
o
0
'
0
'
0
5
'
0
o
G
0
0
5
0
o
50
5
M
M
V
M
V
M
V
M
V V
V
1
o
10"
3
3
2
2
1 o
3
'
3
'
3
5
'
3
o
'
32
3
5
3
o
2
'
2
'
2
5
'
2
o
"
21
2
5
2
o
'
1
"
1
"
1
5
"
1 V
M
V
M
V
M
V
M
V
M














The clubhead motion is the sum of motions produced by the joints

19. 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

20. 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.

21. VERIFICATION (holistic)
1 0 .
8 0 .
6 0 .
4 0 .
2 0
2 5
2 0
1 5
1 0
5
0
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5
5 0
5 0
2 5

rv k
 0

tv k
 0

0 .
1 .
2
 tim
eik
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.

22. The Experiment

23. 10
5
0
5
10
15
20
20
arsk
aask
iesk
feek
spek
few k
urwk
torso_rotk
Results & Applications
(Velocity Contribution Subject 1)
0.2 0.15 0.1 0.05 0
15
10
5
0
5
10
15
20
20
15

ars k
aask
iesk
fee k
spek
few k
urwk
torso_rotk
3.402
 10
13


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)

24. 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
' "
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
( ) ( ) ( )
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
cos cos sin sin cos cos sin cos sin cos sin sin
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
cos sin sin sin sin cos cos cos sin sin sin cos
ˆ ˆ ˆ
ˆ ˆ
sin sin cos cos cos
z y x
R 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
 
   
   
     
   
     
   
   

25. 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
J
n 
   
 V V W W
R̂
 
 
0
ˆ ˆ ˆ
i i i i
R
  
    
V V W V
0 0 0 0 0 0
ˆ ( ) ( )
i i i

    
V r c c r c )
(
)
(
ˆ c
r
c
c
r
W
V
V 





 i
i
i
i
i 




n
i
i
n 1
0
0
1
r
c 


n
i
i
n 1
1
r
c
0i
r i
r
 
ˆ ˆ
T
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)

26. 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.

27. 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 0.410.06
Translation 0.520.07mm
Dual angle about
y-axis
Rotation 0.470.06
Translation 0.870.08mm
Dual angle about
x-axis
Rotation 0.740.05
Translation 0.380.03mm
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.

28. 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

29. 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

30. 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

31. 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)

32. 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

33. 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.

34. 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

35. Computing DTM from Point
Coordinates
 Centroids of the points at
the initial and final position
are given by
and
respectively.



n
i
i
n 1
0
0
1
r
c



n
i
i
n 1
1
r
c

36. 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:
0
ˆ ˆ ˆ
i i i i
R
  
    
V V W V
)
(
)
(
ˆ c
r
c
c
r
W
V
V 





 i
i
i
i
i 


37. 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:
i
V
~
ˆ
i
V
ˆ
2 2
1
1
( )
n
i i i i
i
J
n 
   
 V V W W

38. Geometry of Screw Axes
(1)
where
Rewrite (1) as
and seek solution other than .
ˆ ˆ ˆ
[ ] .
R

V V
([ ] [ ])(
[ ] ([ ] [ ] )
R S
R S R


 
  
V + εW V + εW)
V V W
ˆ ˆ
[ - ] 0
I R 
V
ˆ 0
V 

39. 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

40. 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
j
R I V diag w U

 
   
 
   
1/ [ ]
T
j
V diag w U D
 
 
 
W V
C =V ×W

41. Application Results
Start
End
Dorsiflexion-Plantarflexion

42. Application Results
Start
End
Eversion-Inversion
(a)

43. Comparison
Mean error plotted against the SNR.
(a) (b)
Error in direction
0
5
10
15
20
25
1 10 100 1000
SNR
Degrees
Error in position
0
5
10
15
20
25
30
1 10 100 1000
SNR
Error
(cm)
“Plücker line method”
“Schwartz method”

44. 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
Degrees
Error in position
0
5
10
15
20
25
0 50 100 150 200
Skin movement, % of typival position artefact
Error
(cm)
“Plücker line method”
“Schwartz method”
(Journal of Biomechanics, 2005)

45. 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.

46. 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.

47. The state vectors of the model
2 ˆ
2 ( )
y y y F w t
 
   
 
2
1
2
2 2 3 4 4 1
3
4
0
0
ˆ
1
2
( )
0 0
0
0 0
0
x
x
x F
x x x x x
w 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.

48. extended Kalman filter/estimators
1 1
ˆ ˆ
( ) ( ( ))
k k k
x f x
 
  
1
1 1 1 1
1 2 3 4
2 2 2 2
1 2 3 4
1
3 3 3 3
ˆ ( )
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 x
k
f f f f
x x x x
f f f f
x x x x
f
x 

 
   
 
 
   
 
   
 
 
   
 
  
 
   
 
   
 
 
   
 
   
 
 


ˆ
ˆ ( ( ))
k k k
z h x
 
ˆ ( )
1 2 3 4
k
k
x x
k k k k
k
H
h h h h
x x x x
h
x  

 
   
  
   
 



49. The covariance values
1 1 1 1
( ) ( )
k k k k k
P 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 k
K P H H P H R 
   
( ) (1 ) ( )
k k k k
P K H P
   
ˆ ˆ ˆ
( ) ( ) ( )
k k k k k
x x K z z
    

50. 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)
Damp.
Factor
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
50
Time (sec)
omega
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
-1000
0
1000
2000
Time (sec)
force
(Newton)
True
Est.

51. 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)
Damp.
Factor
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
65
70
75
Time (sec)
omega
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0
1000
2000
Time (sec)
force
(Newton)
Measured
Estimated

52. 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,

53. A group of markers from which an estimate for the F
in point P is calculated on the curved surface.
x
[ ]
F
 1
X
4
X
3
X P P
2
X
X
4
x
1
x
2
x
3
x



x
F
X
 
dx F dX
Physical significance of
F: it relates the length
and orientation of a
material fiber dX to dx

54. 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
i
n 
       

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
i
n 
    
 2
0 s
J X X N ( )
 
N N 1
*
1
1
(( ) ) ( )
n
i
i
n 
       
 2
0 s 0
J X X N λ N N 1

55. 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

56. 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

57. Optical work bench
Micrometer screw
Linear translation stage
High resolution load cell
ESPI sensor
Fiber optic
Methods: Complete setup

58. 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

59. 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

60. Mechanobiology: How mechanical conditions
regulate biological process
Undecalcified Histology:
light blue: connective tissue
red-brown: new callus

61. 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 

62. 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.

63. Principal stretches
m
1
m
3
m
2




m
3
1 2 2 3
  
    
1 1 2 3 3
S = m m m m m m

64. Factorizing: Stretch then Rotation
50 100 150 200
20
40
60
80
100
120
140
160
180
200
220
Area Change
0.9
0.92
0.94
0.96
0.98
1
1.02
ˆ  
F Q S

65. 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

66. Q&A