TECNOLOGIE PER LA RIABILITAZIONE - Ingegneria … · TECNOLOGIE PER LA RIABILITAZIONE ... (three Cartesian coordinates - X g, Y g, Z ... Definition of two orthogonal axes in that

1

dismus la cattura del movimento – 1

TECNOLOGIE PER LA RIABILITAZIONE

Prof. Aurelio Cappozzo

Dipartimento di Scienze del Movimento Umano e dello SportLaboratorio di Bioingegneria

Istituto Universitario di Scienze Motorie - Roma

AA 2005-2006

Università degli Studi di Napoli "Federico II“

Facoltà di Ingegneria Corso di Laurea in Ingegneria Biomedica

lezione # 5la “cattura del movimento”


Both movement and morphology are represented

the movement requires time variant information about the pose

the morphology may be time-invariant

Let’s separate the two problems !!

Yg

XgZg

t1t2

t3 tk

. . .


movement data acquisition/estimation (variables)

anatomical data acquisition/estimation (parameters)

Data collection

Yg

XgZg

t1t2

t3 tk

. . .


Data collection

movement data acquisition/estimation (variables)

Yg

XgZg

t1t2

t3 tk

. . .

2


The term pose alludes to the location in space of a body

The description of the pose

Yg

XgZg


In order to describe the pose of the bone, we substitute a complex morphology

with a simple and time invariant morphology (rigid body hypothesis)


Yg

XgZg


In order to describe the pose of the bone, we substitute a complex morphology

with a simple and time invariant morphology (rigid body hypothesis)


Local system of reference(Local frame)

Global system of reference(Global frame)

Yg

XgZg

yl

xl

zl


Motion capture

For each bone involved in the analysis,and in each sampled instant of time,

motion capture providestwo vectors,

i.e., six scalar quantities

zl

Yg

XgZg

Ol

lg t

lg�xl

yl

orientation vector

[ ] k,...,j;lzjg

lyjg

lxjg

ljg 1== θθθ�

[ ] k,...,j;ttt lzjg

lyjg

lxjg

ljg 1==t

position vector

This is, normally, done in an indirect fashion

3


Estimate of the instantaneous bone pose using

the reconstructed instantaneous position of markers

orientation vector

[ ] kjlzjg

lyjg

lxjg

ljg ,...,1; == θθθ�

[ ] kjttt lzjg

lyjg

lxjg

ljg ,...,1; ==t

position vector

[ ] kjcippp zijg

yijg

xijg

ijg ,...,;,...,; 11 ===p

position vectors

Minimal set up: three non-aligned markers


Stereophotogrammetry

Given a point in motion in the 3-D laboratory space (marker),

stereophotogrammetry provides its position vector

(three Cartesian coordinates - Xg, Yg, Zg) in each sampled instant of time.

Yg

ZgXg



Given a point in motion in the 3-D laboratory space (marker),

stereophotogrammetry provides its position vector

(three Cartesian coordinates - Xg, Yg, Zg) in each sampled instant of time.


Camera

Nodal point (N)

Principal plane

Optical axis

4


Camera and object point

Nodal point (N)

Principal plane

Optical axis

Object point


Camera and object point

Nodal point (N)

Principal plane

Optical axis

Object point


Camera, object and image points

Nodal point (N)

Principal plane

Optical axis

Image point

Object point


Image point

chip CCD

Image pane

5



N1

P

N2


recording phase


N1

P

N2


recording phase


N1

P

N2


PN1

N2

recording phase


6


PN1

N2

reconstruction phase



PN1

N2




PN1

N2

reconstruction phase (with errors)



Object point position reconstruction has been achieved by using the following information:

• Global camera position and orientation• Local position of the nodal points

• Local position of the image points measured variable (time variant)

P

N1

N2



calibration parameters (time invariant)

7


Analytical stereophotogrammetry

Measured variables

Object point global coordinates

Calibration parameters

Mathematical model


Position and orientation of a camera

N

y

z

x

Yg

XgZg


pp

gθθθθp

N

y

z

x


Yg

XgZg

pp (3 numbers)

gθθθθp (3 numbers)

parameters


pp

gθθθθp

N

y

z

x


Yg

XgZg

d

pp (3 numbers)

gθθθθp (3 numbers)

d (1 number)

parameters

8


N

y

z

x

Image coordinates

yp

data

xp

ypxp

Yg

XgZg


Mathematical model X, Y, Z

pp1, gθθθθp1, d1, pp2, gqp2, d2

xp1, yp1, xp2, yp2

Analytical stereophotogrammetry


X, Y, Z


xp1, yp1, xp2, yp2

System calibration

Mathematical model


Calibration object

The control points (markers) are located in known positions. The system of reference with respect to which their location is given becomes the stereophotogrammetric global frame.

Y

Z

X

9


Courtesy of NIH

Calibration object


After a first approximation calibration using a simple and stationary calibration object, the markers mounted on a rigid wand are tracked while moving within the measurement volume.Calibration parameters are iteratively modified while optimizing an objective function.

Y

ZX

So named “dynamic calibration”


So named “dynamic calibration”

Courtesy of NIHdismus la cattura del movimento – 36

X, Y, Z


xp1, yp1, xp2, yp2

System calibration

Mathematical model

10


The experiment

In each sampled instant of time

X, Y, Z


xp1, yp1, xp2, yp2 Mathematical model

dismus la cattura del movimento – 38http://www.charndyn.com/Products/Products_Intro.html

Active markers


The movement analysis laboratory

Yg

Xg

Zg

pg

Marker trajectory reconstruction


Determination of the instantaneous bone pose using


?

orientation vector

[ ] kjlzjg

lyjg

lxjg

ljg ,...,1; == θθθ�

[ ] kjttt lzjg

lyjg

lxjg

ljg ,...,1; ==t

position vector

[ ] kjcippp zijg

yijg

xijg

ijg ,...,;,...,; 11 ===p

position vectors

11


Yg

Xg

ZgB

A

C

A simple example

the position vectors or three non-aligned markers are given


Yg

Xg

Zg

A

B

C

Determination of the pose of a local set of axes


Yg

Xg

Zg

A

B

C

1. Definition of a plane2. Definition of two orthogonal axes on that plane3. The third axis is orthogonal to the former two axes



Yg

Xg

Zg

A

B

C

yl

zl

1. Definition of a plane2. Definition of two orthogonal axes in that plane3. The third axis is orthogonal to the former two axes


12


Yg

Xg

Zg

A

B

C

yl

zl xl

1. Definition of a plane2. Definition of two orthogonal axes on that plane3. The third axis is orthogonal to the former two axes


Appendix


Determination of the instantaneous bone pose using


This local frame is referred to as marker-cluster technical frame

yc

zc

xc


Determination of the poses of the marker-cluster technical frames

mathematicaloperatoryg

xg

zg

orientation vector

[ ] kjczjg

cyjg

cxjg

cjg ,...,1; == θθθ�

[ ] kjttt czjg

cyjg

cxjg

cjg ,...,1; ==t

position vector

[ ] kjcippp zijg

yijg

xijg

ijg ,...,;,...,; 11 ===p

position vectors

yc

zc

yc

xc

zc

yc

xc

zcC=3

Minimal set up: three non-aligned markers

xc



yg

xg

zg

orientation vector

[ ] kjczjg

cyjg

cxjg

cjg ,...,1; == θθθ�

[ ] kjttt czjg

cyjg

cxjg

cjg ,...,1; ==t

position vector

[ ] kjcippp zijg

yijg

xijg

ijg ,...,;,...,; 11 ===p

position vectors

yc

xczc

yc

xc

zc

yc

xc

zcC>3

Redundant set up

mathematicaloperator

13



yg

xg

zg

orientation matrix

k,...,jcjg 1; =R

[ ] kjttt czjg

cyjg

cxjg

cjg ,...,1; ==t

position vector

[ ] kjcippp zijg

yijg

xijg

ijg ,...,;,...,; 11 ===p

position vectors

yc

xczc

yc

xc

zc

yc

xc

zcC>3

Redundant set up

mathematicaloperator


fine della lezione # 5


TECNOLOGIE PER LA RIABILITAZIONE

Prof. Aurelio Cappozzo

Dipartimento di Scienze del Movimento Umano e dello SportLaboratorio di Bioingegneria

Istituto Universitario di Scienze Motorie - Roma

AA 2005-2006

Università degli Studi di Napoli "Federico II“

Facoltà di Ingegneria Corso di Laurea in Ingegneria Biomedica

Appendixrigid-body pose determination


A

B

C

yg

xg

zg

The rigid body

14


A

B

C

yg

xg

zg

Apg

Bpg

Cpg

Point position vectors


ylA

B

Cj

yg

xg

zg

Bpg

Apg

( )BA

BA

ppppj

gg

gg

−−=

Frame unity vectors determination


yl

xl

A

C

yg

xg

zg

Bpg

Apg

( )( ) jpp

jppi×−×−=

BC

BCgg

ggi

B

Cpg


j


yl

zl xl

A

C

yg

xg

zg

jik ×=

Bj

ik


15


yl

zl xl

A

C

yg

xg

zg

kjiR =lg

k

Bj

i

Bpg

Frame orientation-matrix and position vector determination

cg

lg pt =


��

�

�

��

�

�

=

lglglg

lglglg

lglglg

zzyzxz

zyyyxy

zxyxxx

lg

coscoscoscoscoscoscoscoscos

θθθθθθθθθ

R

Frame orientation-matrix and direction cosines

kjiR =lg

lg

lg

lg

xz

xy

xx

lg

coscoscos

θθθ

=i

lg

lg

lg

yz

yy

yx

lg

coscoscos

θθθ

=j

lg

lg

lg

zz

zy

zx

lg

coscoscos

θθθ

=kyg

xg

zg

yl

zl xl

k

j

i


��

�

�

��

�

�

=

lglglg

lglglg

lglglg

zzyzxz

zyyyxy

zxyxxx

lg


θθθθθθθθθ

R [ ]lzg

lyg

lxg

lg θθθ=�

Frame orientation-vector determination

yg

xg

zg

yl

zl xl

k

j

i lg�


( ) ( ) ( )[ ]

( )

( )

( )baba

baba

baba

bababa

babababababa

yxxybza

xzzxbya

zyyzbxa

zzyyxx

\

yxxyxzzxzyyzb

a

coscossin

coscossin

coscossin


tn

θθθ

θθ

θθθ

θθ

θθθ

θθ

θθθθθθθθθ

−=

−=

−=

��

�

�

�

−++−+−+−

=

2

2

2

1

21222

�

��

�

�

��

�

�

=

bababa

bababa

bababa

zzyzxz

zyyyxy

zxyxxx

ba


θθθθθθθθθ

R [ ]bza

bya

bxa

ba θθθ=�

Relationship between orientation matrix and orientation vector

Given two frames “a” and “b”

16


yl

zlxl

A

B

C

yg

xg

zg

Apg

Bpg

Cpg

In summary

ApgBpg

Cpg

Given the marker position vectors in the global frame:

We were able to estimate the position and orientation vectors of the local frame (marker cluster frame):

[ ] czg

cyg

cxg

cg θθθ=�

[ ] czg

cyg

cxg

cg ttt=t

In addition, the orientation matrix has been presented and its relationship with the orientation vector found.


yl

zlxl

yg

xg

zg

In summary

The orientation of a rigid body can therefore be described by a position vector

cg�

or by an orientation matrix

lgR


The end of Appendix

Documents

TECNOLOGIE PER LA RIABILITAZIONE - Ingegneria … · TECNOLOGIE PER LA RIABILITAZIONE ... (three Cartesian coordinates - X g, Y g, Z ... Definition of two orthogonal axes in that