Anatomy of Human Motion

Wangdo Kim

How to theorize swing?

Autopilot: a cognitive

state in which you act

without self-awareness

2-D Displacement in terms of the simplest path

12

2B

12P

1B

1A

2A12

Introduction: methods for describing human joint motions (continue)

Screw axis

Screw motion of a rigid body

s

Screw axis

X

Y

Z

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.

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

•The dual angle express the relationship between

lines in space A and B.

DUAL ANGLES

s

Line A

Line B

sεθθ

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

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

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

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.

J. of Biomech, 2005, in press

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

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

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 )

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

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

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

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.

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.

The Experiment

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)

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

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)

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.

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.

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

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

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

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)

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

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.

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

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

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

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

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

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

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

Application Results

Start

End

Dorsiflexion-Plantarflexion

Application Results

Start

End

Eversion-Inversion

(a)

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”

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)

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.

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.

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.

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

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

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.

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

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,

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

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

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

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

Optical work bench

Micrometer screw

Linear translation stage

High resolution load cell

ESPI sensor

Fiber optic

Methods: Complete setup

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

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

Mechanobiology: How mechanical conditions

regulate biological process

Undecalcified Histology:

light blue: connective tissue

red-brown: new callus

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

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.

Principal stretches

m1

m3

m2

m1

m2

m3

1 2 2 3 1 1 2 3 3

S = m m m m m m

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

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

Q&A