Michigan State
‘ QUnivésicy ‘

   
     

This is to certify that the
thesis entitled

THE ACCURACY OF SCREW AXIS ANALYSIS USING POSITION
DATA FROM ANATOMICAL MOTION STUDIES

presented by

Jeffrey Howard Marcus

has been accepted towards fulfillment
of the requirements for

MASTER Guzman—degreein MECHANICA ENGINEERING

WWW

lhmnpflmuun

Date4/Zf 80
/ /

0-7639

 
 
 

{fl‘“\\\ b

 
   
 

‘1‘». 4.31,!!! .
llfillllflifllflilil

     

 

OVERDUE FINES:
25¢ per day per item

RETUMIM LIBRARY MATERIALS:

Place in book return to new
change from circulation recon

 

 

 

THE ACCURACY OF SCREW AXIS ANALYSIS USING POSITION DATA FROM
ANATOMICAL MOTION STUDIES

BY

Jeffrey Howard Marcus

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1980

ABSTRACT

THE ACCURACY OF SCREW AXIS ANALYSIS USING POSITION DATA FROM
ANATOMICAL MOTION STUDIES

BY

Jeffrey Howard Marcus

Research involving the kinematics of human joint mobil-
ity often involves screw axis analysis. As a prelude to
such research a screw axis analysis program was developed
and implemented for use in the Systems Anthropeometry
Laboratory. This thesis presents a detailed disussion of
the algorithms used to find Displacement Matricies (DM) and
screw axis parameters. Error prOpogation due to uncertainty
in DM is analytically developed and then demonstrated. The
rotation angle is found to be the most critical screw axis
parameter, and the components of a unit vector in the direc-
tion of the screw axis are the most sensitive to error.
Using the condition number of a matrix, a method is deve-
loped and presented for evaluating the error propagation

due to the matrix operations used to find DM.

This thesis is dedicated to those peOple whose contri-
bution to society did not end with their death:
Specifically, those people who willingly donated their
bodies to the medical school, and who were studied as part

of this project.

ii

ACKNOWLEDGEMENTS

One of the more educational parts of writing this
thesis has been learning how indebted I am for the help
people have given me. I would like to thank my advisor, Dr.
James Bernard, for all his help. Dr. H. M. Reynolds, who I
have worked so closely with these last two years, is the
person who introduced me to the challenges of working with
biological systems. Dr. Robert Hubbard listened to various
ideas I had for my thesis, from the flailing for a subject
stage through his helpful reviews of this thesis. These
three men, who served on my committee, are the people who
introduced me to serious technical writing, which I disco-
vered to be quite different from my previous background in
rhetorical and political writing.

When I wondered about matrix theory I consulted with
Mr. Joeseph Whitesell. His quick grasp of the mathmatics
of a problem helped me on many occasions in the preparation
of this thesis.

None of the work done for this thesis would have been
possible had not the United States Air Force funded this
project under contract AFOSR-F49620-78-C-0012. The support

given me by the Department of Biomechanics of the College of

iii

Osteopathic Medicine of Michigan State University is also
gratefully acknowledged.

Jim Freeman and Kathy Hornback worked with me on the
many mysteries of computer science, and helped me with some
of my programming needs.

To all of these people, thank you.

iv

TABLE OF CONTENTS

LIST OF TABLES. . . . . . . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . . . . . . . .
LIST OF SYMBOLS . . . . . . . . . . . . . . . .
INTRODUCTION. . . . . . . . . . . . . . . . . .

BACKGROUND AND LITERATURE REVIEW . . . . . . .

2.1 Literature Review . . . . . . . . . .
2.2 The Systems Anthropometry Laboratory.

DESCRIPTION OF THREE DIMENSIONAL MOTION . . . .

3.1 Introduction. . . . . . . . . . . .
3. 2 Displacement Matrices . . . . . . . .
3. 3 Computation of DM from Position
Data. . . . . . . . . . . . . . . .
3. 4 Determining the Screw Axis Parameters
from DM . . . . . . .

3.5 Relationship to the Eigenvalue Problem.

THREE DIMENSIONAL POSITION DESCRIPTION. . . . .

4.1 Introduction. . . . . . . . . . . . .
4.2 Algorithm for Setting Up a Coordinate
System Based on Three Data Points .
4.3 Effect of Errors on Coordinate System
Determination . . . . . . . . . . .

Page
viii

xi

QMWH

11

11
12

14
15
21
22
22
24
26

TABLE OF CONTENTS (Con't)

Page
ERROR ANALYSIS I I I I I I I I I I I I I I I I I I I I 28
5.1 How Errors in Position Effect DM. . . . . . 28
5.2 How Errors in DM Effect the Screw
AXiS I I I I I I I I I I I I I I I I I I I 33
5 I 3 conCIUSionI I I I I I I I I I I I I I I I I 40
EVALUATION OF THE ACTUAL EFFECT OF ADDING ERRORS. . . 41
6.1 Introduction and Description of
Procedure I I I I I I I I I I I I I I I I 41
6I 2 Resu1ts I I I I I I I I I I I I I I I I I I 42
6. 3 Discussion. . . . . . . . . . . . . . 49
6.4 Evaluating Cond (Pl). . . . . . . . . . . . Sl
EXAMPLE OF SCREW AXIS ANALYSIS USING ANATOMICAL
DATAI I I I I I I I I I I I I I I I I I I I I I 54
7 I 1 IntrOdUCtiono I I I I I I I I I I I I I I I 54
7 I 2 DiscuSSionI I I I I I I I I I I I I I I I I 56
SUMMARY, REVIEW, AND RECOMMENDATIONS. . . . . . . . . 59
8.1 Summary and Review. . . . . . . 59
8. 2 Is the Condition Number of a Matrix of
Any use?I I I I I I I I I I I I I I I I I 61
8.3 Recommendation for Future Work. . . . . . . 63
8 I 4 ConClUSions I I I I I I I I I I I I I I I I 67
LIST OF REFERENCES I I I I I I I I I I I I I I I I I I 69

vi

Table

5-1

LIST OF TABLES

Effect of Significant Figures on Small

Angles.
Error With
Error With
Error With
Error With
Error With
Summary of
Screw Axis

Screw Axis

¢=l°; 4 Significant Figures .

¢=l°; 5 Significant Figures .

¢=5°; 4 Significant Figures .

¢=10°; 4 Significant Figures.

¢=60°; 4 Significant Figures.

Error Analysis.

Analysis for Hip Motion . . .

Analysis for the Sacro-iliac Joint.

viii

Page

32
40
41
42
43
44
45
57
57

LIST OF FIGURES

Figure Title Page~

2-1 Systems AnthrOpometry Data Collection and
Analysis System. . . . . . . . . . . . . . . 7

2-2 Basic Stereo-Radiographic Configuration for
Systems Anthropometrv Laboratory . . . . . . 8

4-1 Data Based Coordinate System. . . . . . . . . . 25

5-1 Rotation Angle Versus Error in Rotation
Angle. I I I I I I I I I I I I I I I I I I I 35

ix

LIST OF SYMBOLS

aij - the i j th entry in DM
A - a matrix which is known in a system of linear equations
A - a three dimensional point on a rigid body
AI - point A at rigid body position 1
A5 - point A at rigid body position 2
AA - a vector from point A to point B
ARI - vector AB at rigid body position 1
ABA - vector AB at rigid body position 2
AC - vector from point A to point C
- the matrix on the right hand side of the standard
equation Axsb of a system of linear equations
A - a three dimensional point on a rigid body
5 - a three dimensional point of a rigid body

C¢ - cosine of angled

cond(A) - condition number of a matrix A; the euclidean norm
of A times the euclidean norm of A‘1

5

a three dimensional point on a rigid body
DM - Diplacement Matrix

DOF - degrees of freedom

ei - error in some parameter i; may be a matrix or a scalar
depending on context

i - unit vector in x direction

j - unit vector in y direction

k - unit vector in z direction

X

the identity matrix; a matrix which is 0 everywhere
except the diagonal which is all 1's

ISA - istantaneous screw axis

5? - origin of new coordinate system

P1 - a 4x4 matrix describing the position of a rigid bodv
at time 1

P2 - a 4x4 matrix describing the position of a rigid body
at time 2

RM - Rotation Matrix

3 - translation parallel to the screw axis

S¢ - sine of angle¢

TrRM- trace of RM; sum of the diagonal entries of RM

U - unit'vector in the direction of the screw axis

ux - x compoment of U

uy - v component of U

uz - 2 component of U

V¢ - vers of<i; l-cos ¢

0 - an angle, particularly between relative position
vectors

¢ - rotation angle in screw axis analysis

i=n

2: - summation from i=j to i=n

1-1

IABI - magnitude of some vector AB; the square root of the
sum of the squared components of a vector

”A“ - Euclidean norm of a matrix A; the square root of

the sum of squared components of a matrix

xi

CHAPTER 1

INTRODUCTION

Recently the need and capability to describe human
joint mobility has prompted studies into the kinematics of
joint motion. Particularly since the 1960's when the advent
of large digital computers made complicated three dimen-
sional kinematic analysis practical, there has been an
expansion of research aimed at creating the elements of a
model for describing human body motion.

The kinematics of body motion often involves compli-
cated three-dimensional displacement descriptions. One com-
monly used method of three-dimensional kinematic analysis is
the screw axis. By describing a displacement with a single
translation and a single rotation, screw axis analysis aids
in understanding how parts of the body move during a partic-
ular displacement.

The Systems Anthropometry Laboratory is a new facility
dedicated to the study of human body motion. Screw axis
analysis is used as a part of these studies, but how accura-
tely can a body displacement be described by the screw axis?
The displacement descriptions are derived from empirical
data such as position descriptions. Measurement errors,

which are present in empirical data, will propogate through

2
the kinematic analysis. Even if there are no measurement
errors, the number of significant figures affects the
calculations.

This thesis will examine the limitations and require-
ments for accuracy in the kinematic analysis of human body
motion. Two different but related questions will be
examined. First, how do measurement errors in position data
propogate through the computation of Displacement Matricies?
The second question is, given some error in the Displacement
Matrix, how is the screw axis affected? Along with measure-
ment errors, the number of significant figures in the
Displacement Matrix may limit the screw axis analysis.

In summary, the limitations on Displacement Matricies
due to measurement errors, and the limitations on screw axis
analysis due to the Displacement Matrix will be considered

in the following pages.

CHAPTER 2

BACKGROUND AND LITERATURE REVIEW

2.1 Literature Review

Many situations encountered in design require a
knowledge of the position and motion of the human body.
Examples include the design of vehicles for situations where
large dynamic forces are encountered, as in a crash. The
design of a chair requires knowledge of the interaction be-
tween the chair and the person sitting in it. A prosthesis,
such as an artificial hip, must accurately reproduce the
function of the replaced part. Further applications are in
workspace definition. If a driver cannot reach the controls
of a car when wearing a shoulder harness, he is not likely
to use the shoulder harness.

Traditionally, the human body is modelled using rigid
body mechanics. By dividing the body into a series of "mass
links", and connecting these links with a number of dif-
ferent types of joints, a kinematic model of the body is
created. In the late 1800's Braune and Fisher {3,4,5} laid
the foundation for this approach when they investigated the
biomechanics of the body positions assumed by German
infantrymen. The technique of using mechanical analogues of

human anatomy, such as the "mass link”, was continued into

4
the 20th century. Dempster {8,9 lachieved the most
extensive results in 1955, and much of his data is still
in use. Much of Dempster's research in body mobility con-
cerns locating a path of instantaneous joint centers of
rotation, which is a two-dimensional description of body
motion.

Dempster and other investigators prior to the 1960's
used the two-dimensional kinematics of Reuleaux {19}.
Three-dimensional kinematics existed, indeed Chasles {7}
described the screw axis theorem in 1830, but analytical
methods were cumbersome and impractical until the advent of
large digital computers in the 1960's. Potthoff{18} was
' among the first to apply the more sophisticated kinematic
analysis made possible by computers. Potthoff tried to ana-
lyze Braune and Fisher's data using the screw axis theorem,
but found that their data was not accurate enough for screw
axis analysis.

Braune and Fisher's data were taken from surface
targets on the skin of a person. Emanuel and Barter {10}
found that targets placed on human skin do not maintain a
stable position relative to the skeleton, or relative to
other targets on the skin, and thus violate the rigid body
assumption of the ”mass link" concept. Measurement tech-
niques not relying on surface targets were developed and
used in the late 1960's and early 1970's. Thompson {28} and
Kinzel {13,15} used instrumented linkages to describe the

relative motion between body segments. Both Thompson and

5
Kinzel used screw axis analysis, with Thompson reporting
that an averaging technique was necessary before accurate
results could be obtained. Kinzel optimized his linkage
and reported use of the screw axis with confidence, but he
too noted its sensitivity to error.

The use of the "mass link" concept requires knowledge
of the characteristics of the joints which connect the ”mass
links." The least constrained joint model is the spatial
joint, which allows translation in all three coordinate
directions, and rotation around each of the three coordinate
axes. A spatial joint is, therefore, a full three-
dimensional six degree of freedom* joint, and the resulting
kinematic analysis is complicated. Researchers often make
assumptions to reduce the degrees of freedom and thus
simplify the analysis. Kinzel {13} has identified five
joint models commonly reported or implied in the literature.
These five are:

1) the one DOF hinge, or revolute joint

2) the three DOF planar joint

3) the three DOF spherical, or ball and socket joint

4) the two DOF spherical joint

5) the six DOF spatial joint
Kinzel discusses each of these in detail. Kinzel also sta-
tes "all anatomical joints permit six degrees of freedom to

some extent." The implication is that use of a joint model

 

*degree of freedom, or DOF, is defined as the minimum
number of independent parameters required to completely
define the relative position or displacement of one member
relative to another {13}.

6
other than a spatial model is unduly restrictive.

While Thompson and Kinzel used instrumented linkages to
describe anatomical motion, others {6,12,17,21,26} were
investigating the use of stereo-radiography. The most nearly
rigid part of the anatomy is bone, and most single bones are
essentially rigid bodies under the forces normally encountered
in the body. Because radiography allows the determination of
a bone's position in vivo, and bones are essentially rigid
bodies, rigid body position data may be obtained. The utility
of stereo-radiography is its ability to accurately determine
the three-dimensional coordinates of a target on a bone.

Other methods such as instrumented linkages must contend with

the problems due to skin not being a rigid body.

2.2 The Systems Anthropometry Laboratory

The Systems Anthropometry Laboratory (SAL)
{20,21,22,23} of the Department of Biomechanics, College of
Osteopathic Medicine at Michigan State University is a new
facility for obtaining accurate and repeatable data relating
to the kinematics of human joint mobility. Through the use
of stereo-radiography {21,22,23} the relative movement be-
tween bones, or the absolute position of a bone with respect
to an inertial axis system may be measured. Body position
and mobility are studied from the viewpoint that the human
body is a three dimensional system composed of links con-
nected by joints with six DOF {23}.

As a bone is moved, the use of stereo-radiography allows

 

 

 

 

 

 

 

 

 

 

3056 ML mmvdamam 2.933533. cog 005830: Ba 255$? mwmflma

l-e-Stereo Base—pl

   
 

 

 

 

Focal spot 1 E.“ Focal spot
:§
I
:3?
I J."
I
I
O
I
Film plane

FIGURE 2-2 Basic Stereo-Radiographic Configuration for
Systems Anthropometry Laboratory

9
the measurement of the position of the bone. Since large
doses of X-ray radiation are required, cadavers are used for
the joint mobility studies. Thus there is no restriction,
based on radiation exposure limits, on the number of
radiographs which can be obtained. Embalming tends to make
cadavers unnaturally stiff, so only fresh, unembalmed cadavers
are used.

A joint mobility study begins by imbedding x ray
targets, tungsten-carbide balls .8mm in diameter, in a bone
near a skeletal landmark. Figure 2-1 shows the subsequent
steps involved in collecting the data. The cadaver is
placed between the X-ray tubes and the film holder, as illus-
trated in Figure 2-2.

Two X-ray tubes are mounted a fixed distance apart. At
each step in the movement of the cadaver's joint, a film is
loaded into the film holder. One of the X-ray tubes then
"fires," exposing the film. The film is changed and the other
X—ray tube ”fires,” exposing the second film from a different
angle. The two pieces of exposed film are called a stereo
pair, and contain all the information necessary to determine
the three-dimensional coordinates of the X-ray targets in the
cadaver. Large film sizes of 14" x 36' are used to allow the
imaging of an entire "mass link” with all anatomical targets,
such as the femur, on one piece of film.

Between the cadaver and the film holder is a grid of
tungsten wires which define an inertial axis system.

Devices on the film holder create images which can be used

10
to determine the geometry of the film holder relative to the
X-ray tubes. The film holder is free to move between the
two exposures of a stereo pair. To aid in imaging the
bone, the film holder may translate in a plane parallel to
the wire grid, and the film holder may rotate in the same
plane around an axis normal to the wire grid.

Once the film is devleoped it is placed on an X-Y
digitizer, which is accurate to within 1.013 cm. The output
from the digitizer, the coordinates of a digitized image,
are then processed through an algorithm which computes the
three-dimensional coordinates of the target. The coor-
dinates are reported in the inertial axis system defined by
the wire grid.

To summarize, radiopaque targets implanted on a bone
are tracked as the bone is moved through a series of finite
steps representative of the joint's mobility. Radiographs
of the targets on the bone are digitized for later use and

study.

CHAPTER 3

DESCRIPTION OF THREE-DIMENSIONAL MOTION

3.1 Introduction

 

The description of three-dimensional motion involves a
4 x 4 matrix called the Diplacement Matrix (DM). All infor-
mation necessary to describe a three-dimensional displace-
ment of a rigid body is contained in DM. Screw axis analy-
sis is an attempt to put DM into a form more easily
understood.

For a change of position of a rigid body there exists
a unique instantaneous screw axis (ISA) {7}. The ISA is
simply a line in space. All points on the rigid body may be
thought of as translating parallel to the screw axis by an
equal amount 3, and rotating around the screw axis by an
angle¢ . The screw axis analysis describes displacement in
terms of two vector quantities and two scalars. The two
scalars are the translation 5, and the rotation<b. The vec-
tor quantities are a unit vector U in the direction of the
screw axis, and the position vector of a point A which fixes
the screw axis in space. All of these parameters are found
from DM, but not all of the parameters are independent.
There are eight parameters for screw axis analysis (s,¢ ,

three in U, and three in A), but since this is six DOF

ll

12

displacement, only six are independent.

3.2 Displacement Matrices

If A is a point on a rigid body, and A is known at two
different positions 1 and 2, then there exists a linear
transformation which maps A from 1 to 2. This linear
transformation, called the Displacement Matrix by Suh
and Radcliffe {27}, is defined as follows:

DM 51:32
where:

AI is the position vector of a point A at
rigid body position 1

'A7 is the position vector of a point A at
rigid body position 2

Point A is expressed in homogeneous coordinates. The need
for homogeneous coordinates arises from the fact that in
order to fully describe an object in n dimensional space,
n+1 coordinates are needed. The subject of homogeneous
coordinates is treated in computer graphics, and a typical
text is Rogers and Adams {25}. Insight into the need for an
n+1 system for n dimensional space can be attained from the
classic text FLATLAND {l}. The fourth coordinate for a
three-dimensional system is arbitrary and usually 1. Thus

A2 in homogeneous coordinates is

l3

——A2x
AZY

{3|
ll

A22

1

—~

 

 

DM may be partitioned as follows:

"' " "”311 a12 813| 314‘
.321 322 £323] 824
DM. = (3.2)
a31 a32 a33| a34

 

 

 

 

where for homogeneous coordinates

a41=a42=a43=0.0 and a44=l.0

The upper 3 x 3 represents a rotation matrix (RM),

while the last column has information about the translation.

If there is no translation of the rigid body,

a14=a24=a34=0.0

and

RM AI = A7 (3.3)

If no rotation occurs then RM = I, the identity matrix.

Note that for equation (3.3) both AI and A2 are not in

14
homogeneous coordinates, and are simply position vectors
with x, y, and 2 components. Unless otherwise stated in
this thesis, all position vectors and position matrices are

in homogeneous coordinates.

3.3 Computation of DM for Position Data

If the coordinates of a rigid body are known at two
different poisitons, DM can be computed. Assume there are
four points A, A, U, U in homogeneous coordinates at

two different positions 1 and 2. Then the following is true

 

 

 

 

 

 

{27}:
- - F- __ __ .—
Aly 31 c1V 01 A2 32 c2v 02
DM . Y .. Y 8 y Y - Y (3.4)
A12 312 C12 012 A22 322 C22 022
L 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
or
DM P1 = P2 (3.5)

Post multiply both sides by P1 ‘1 to get:

DM pz Pl '1 (3.6)

where:

P1 is a 4 x 4 matrix describing the first
position of the rigid body

15
P2 is a 4 x 4 matrix describing the sec-
ond position of the rigid body
Thus DM may be found directly from three-dimensional
empirical data in homogeneous coordinates. In this example
it has been assumed that four points are specified. In
fact, only three non-collinear points are needed. A fourth

point may be created at each position by point C 90°

about an axis from A to A. Alternatively, a

coordinate system based on the three non-collinear data
points may be set up. Four new points are then available,
one on each axis, and the origin. Once DM is calculated the

parameters which specify the screw axis can be determined.

3.4 Determining The Screw Axis Parameters from DM

In this section an algorithm for determining the
screw axis parameters is presented. This algorithm is based
on a method presented by Sub and Radcliffe {27}. Begin with
the Rotation Matrix, which is the upper 3 x 3 partition in

equation (3.2).

ux2V¢ + co uxuyv¢ - qu¢ uxuzv¢ + uYS¢
RM . uxuyv¢ + qu¢ uy2V¢ + c¢ uyqu¢ - uxS¢ (3.7)

uxqu¢ - uYS¢ uyqu¢ + uxS¢ u22v¢ + C¢
_J

 

 

16

V¢ = l-cos
C¢ = cos
S¢ = sin

and ux, uy, and uz are components of a unit
vector U which indicates the direction of

the screw axis.

The rotation angle d is found in the following manner:

Trace = all + a22 + a33 (3.8)

where aij indicate entries in RM

Trace ux2V<b+ C<p+ uy2V¢ + co + uzzvo + C¢ (3.9)

(ux2 + uy2 + uzz) (l-cos¢ ) + 3 cos¢ (3.10)

Since U is a unit vector

1 = ux2 + uyz + u22 (3.11)

Combining (3.10) and (3.11)

 

Trace = l + 2 cos¢ (3.12)
_ (3.13)
cost = Tracef l
2
¢ = cos"1

17
Once o is calculated ux, uy, and uz are determined.

Referring to equation (3.7).

a32 = uyuzv¢ + uxS¢ (3.15)
3.23 = UVUZV¢ - ude) (3.16)
a32 - 323 = 2 uxS¢) (3.17)
__ a‘32 ' a23
“x " =—"__ (3.18)
33$¢

By similar reasoning

uy = ___.____ (3.19)
u2 = ——____ (3.20)

Note that for this algorithm the values of ux, uv, and uz
are computed using the value of'? calculated in equation

(3.14).

Now assume that two points, A and U, are on a rigid
body. A31 is a vector from point A to point A when the rigid
body is at position 1, and AB? is the same vector at rigid
body poisition 2. Since A and B are on a rigid body the

magnitude of AB cannot change, but AA will follow the rigid

body rotation. From equation (3.3):

RM ABl = 2 (3.21)

or
BI

RM( -AI)=B_-A—2 (3.22)

18
where:

ET is point 5 when the rigid body is at
position 1

52 is point 3 when the rigid body is at
position 2

Point A has similar notation

A, 5, AB are not in homogeneous coordinates
Rearrange (3.22)
RM(§T-X1)+X§=§2 (3.23)

Equation (3.23) is a 3 x 3 system of equations. Add the equa-

tion l=l to get:

132x RM : 32' - RM Ki 131x
132 Bl
Y = | y (3.24)
822 --------- Blz
1 o o o: 1 1

 

 

 

 

 

 

The 4 x 4 matrix mapping 51 to 37 is DM. The
elements of the fourth column of DM can be defined from

equation (3.24).

314 ‘3 AZX " allAlx - a12A1Y - al3Alz (3.25)

and so on.

AI is any arbitrary point on the same rigid body as E.

19
Assume A is on the screw axis so that it translates only, and

rotations do not change the position of A. Thus,

A2 = A2 + s 5.

Therefore:
a14 = sux + A1x - a11A1x - a12A1y - a13Alz (3.26)

and so on for a24 and a34

In matrix form:

'— 11—1
a14 “x (l‘all) ”a12 -a13 S

 

 

 

 

 

 

a24 = “Y -a21 (l-a22) -a23 Alx (3.27)
a34 uz ‘331 '332 (1'3331J A1y
L. _J __ A12

The translation 5, and Alx, Aly, and A12 cannot be
solved for because there are three equations and four
unknowns. However, A is of interest only because it fixes
the screw axis in space. Thus any point on the screw axis
will suffice. The point where the screw axis intersects a
coordinate plane of the inertial axis system is called a
piercing point. At a piercing point one of the coor-
dinates is 0.0 depending on which plane is intersected. For
example, the piercing point for the XY plane has a z coor-
dinate of 0.0. Setting one of the components of A equal to

0.0, reduces equation (3.27) to three equations and three

20
unknowns, which yields A and s.

Suh and Radcliffe's method {27} has been presented so
far. Kinzel, et a1. {13, 14} present a different algorithm.
As part of this research Kinzel's methods were compared to
Suh and Radcliffe's. In most cases both methods gave similar
answers, but at not time did Kinzel's method give more
accurate results, and on occasion Kinzel's method gave less
accurate results. Suh and Radcliffe's method involves less
computation, and is more straightforward.

One final note on the algorithm. In determining ux,
uy, and uz Suh and Radcliffe rely on the off diagonal ele-
ments of DM. These terms are very small for a small
rotation, and thev mav be adversely effected by round off
error. For this case one further method exists for finding

5. Recall equation (3.7).

 

all = ux2V¢+ c¢ (3.28)
all - c¢ = ux2 (1 - c¢) (3.29)
a11 ‘ °¢ (3.30)
ux 3 .1-c¢
(an - w
11,, = T7071). (3.31)

This method can be similarly applied for uy and uz.
Note that in equation (3.31) the sign of ux is not known.

The sign must be determined using Suh and Radcliffe's

21
method, but the magnitude of ux is found with equation
(3.31). In the screw axis program implemented 5 is found
using equations (3.18-3.20). If the magnitude of U is not
within some epsilon (1.01 in SAL) of 1, then equation (3.31)

is used.

3.5 Relationship to the Eigenvalue Problem

If an eigen analysis is done on RM there will be one
real eigenvalue, and one pair of complex conjugate
eigenvalues. The imaginarv part of the complex eigenvalue is
the rotation angle g in radians. The one real eigenvector will
be a vector in the direction of the screw axis. Because of
the different algorithms used to compute eigenvectors, the
eigenvector may not have a unit magnitude, but a simple scaling
of the eigenvector will give 5. The real eigenvalue also
gives a method of determining if RM is orthogonal. If the
real eigenvalue does not equal 1, RM is not orthogonal. In
computer graphics, if the size of an object is to be enlarged
or shrunk, the eigenvalue represents the magnification
factor. Orthogonal transformations preserve lengths and
thus have eigenvalues (magnification factors) of 1. The
real eigenvalue of RM provides information on how much the
lengths between the data points changed between position 1

and position 2.

CHAPTER 4

THREE DIMENSIONAL POSITION DESCRIPTION

4.1 Introduction

As discussed in Section 3.3, ”Computation of DM from
Position Data“, only three non-collinear points are needed
to describe the position of a rigid body in space. Two
algorithms for computing DM from position data were
presented, namely, 1) a fourth point may be created by
rotating the third data point about an axis between the
first two data points, or 2) a coordinate system based on
the three data points is set up, and three new points at
unit distances from the created origin, one on each axis,
together with the origin make up the four points.

If the first method, rotating the third point, is
used, the three points must maintain their position relative
to one another, from one position of the rigid body to
another. Any change in the true relative position of the
points, or an apparent change due to measurement error,
violates the rigid body assumption. Violating the rigid
body assumption results in a non-orthogonal RM. The effect
of measurement error is an apparent shrinking or stretching
of the rigid body. DM reflects this change in size even

though the object has not changed. Numerical problems mav

22

23
result. For example, in the determination of ¢ through
equation (3.14) it may be necessary to find the inverse co-
sine of a number greater than 1. Or the computed magnitude
of 5 may deviate from 1. Different values of s, the
translation parallel to the screw axis, may result from
using different piercing points.

Early in the work for this thesis the possibility was
investigated that scaling the magnitudes of the columns of
RM to 1.0 would avoid a non-orthogonal RM. This did not
work because scaling simply changes the magnitudes of the
entries in RM. The reason RM is not orthogonal is that
the ratios within the rows and columns of RM are incorrect.

The second method of computing DM from position data,
setting up a coordinate system based on the three data
points, prevents measurement errors from propagating through
the analysis. An algorithm for setting up a coordinate
system is presented in the next section. The coordinate
system approach always gives an orthogonal RM since the
coordinate systems rather than the measured data are used to
compute DM. Each point used is always a unit distance away
from the origin. Further, since the coordinate system set
up has three mutually perpendicular axes, the system is
already orthogonal. This does not mean the method is error
free. An error in any of the data points used to create the
new coordinate system will change the location and orien-

tation of the new axis system.

24

4.2 Algorithm for Setting Up a Coordinate System Based on
Three Data Points

Three points define a plane, and in the problem at
hand the three points define one of the coordinate planes.
Call the data points A, A, and 5. Create vectors from A to
A (AA), and from A to 5 (A5). This is done by subtracting

components.

A3 = vector from A to B
= (Bx - Ax) i + (By - AY) j + (82 - Az) k (4.1)

where i, j, k are unit coordinate vectors
A5 is similarly defined

The dot product of the two vectors AB and AC is defined by

AE - KE 2 ABxACx + ABYACY + ABzACz = IABIIACI case

(4.2)
Rearrange (4.2) to get:
Xfi - AC

cos<3 = __ __ (4.3)
IABI IACI

The origin of the new coordinate system is a point on
A3 which lies on a line perpendicular to AA and containing
5. The distance along AB from A to the new origin (5?) is
$6 cose . Thus the three dimensional coordinates of the

origin are determined from:

25

fi.©C10 Al.

ow 03¢ 00 010 C3.w <00w01m 01oomom.
cm x on ooaom 0C» 0* «To V000
>.m.0 01o >30w03.00_ Unmo $0.3wm

CW
01

0090 momma 0001Q_30w0 m<m~oa

26

AB

 

6? = A + IXEI cose (4.4)

|A§|

Vector AA is divided by its magnitude to form a unit
vector in the direction of AA. Once the origin's coor-
dinates (5?) are established with equation (4.4) unit vec-
tors in the direction from 5? to A (55), and from 5? to E
(55) are determined. These unit vectors are then added to
O? to determine the location of a point on each axis.
Finally, the vector cross product of the unit vectors 55 and
55 determines the direction of the third axis. Four points
have been created, 5?, and the end points of unit vectors
5E, 55, and 5E x 55. These four new points may now be used

to directly solve for DM.

4.3 Effect of Errors on Coordinate System Determination
The effect of an error in a data point used to define

a coordinate system has been analyzed by Robbins 24 . The

analysis which follows uses a different technique but simi-

lar conclusions are reached. Recall equation (4.2).
|K§| IAEI cose = ABxACx + ABYACY + ABZACZ (4.5)
The effect of an error, dx, in the x coordinate of point A

can be evaluated using the partial derivative with respect

to x {2} of equation (4.5).

27

[33] IKE] d 9 sin = Adex (4.6)

Adex
d0 = ::r—::T (4-7)
[ABIIACI sine

If the angle 0 between AA and AA is small then sine will be

small, and the error propogation will become large.

Adex

 

lim d6 = 1im __ __ =
640 9+0 |AB||AC| sine

” (4.8)
In addition, the closer point A is to point A, the smaller

the magnitude of AA.

Adex
lim d6 a 15m __ __ = m (4.9)
IAB|+o lABI+0 IABIIACI sine

 

Equations (4.8) and (4.9) show that the angle between
AA and AA should be as close to 900 as possible, and points
A and A should be located as far away from point A as
possible. At present the computer program which creates the
coordinate system for use in the screw axis analysis, does
not check for either of these conditions, and this is an

area worthy of further investigation in SAL.

CHAPTER 5

ERROR ANALYSIS

5.1 How Errors in Position Effect the DM

 

The propogation of measurement errors through the
matrix operations used to compute DM is a complicated
question to analyze. This section develops a method of
bounding this error propogation based on methods used in
numerical analysis. It is not within the scepe of this the-
sis to provide a full and complete analysis of matrix propo-
gation of measurement error. Rather, this thesis provides an

introduction to this subject. Recall equation (3.6):
DM = 22 p1 '1 (3.6)

The error in P1"1 is difficult to evaluate because each ele-
ment of P1‘1 is a function of several elements of P1. In
addition, the elements of P1 which determine an element of
131'1 are different for each element of Pl'l.

The field of numerical analysis has dealt with the
question of how error in a matrix effects the inverse.
Round off error in a computer has an effect similar to
measurement error in empirical data. To analyze the effect

of round off error, Forsythe and Moler {11} define a number

28

29
called the condition number. Assume a system of linear

equations exists,
A x = b (5.1)

where:
A is a known n x n matrix

x is an unknown n vector, or an unknown matrix,
and is to be solved for

b is a known n vector, or a known matrix

By definition:

condition number = cond(A) = IIAII [IA-1|) (5-2)

where [IAII is the euclidean norm of
the matrix A. For an n x n matrix A,

the euclidean norm is

 

.\/

The condition number will never be less than 1, but it

aijz (5.3)

"MD
"MS

1 1 j 1

may approach infinity. The greater cond(A), the greater
the error propagation in the resulting inverse and/or solu-
tion to a set of linear equations.

As an example of equations (5.1-5.3), assume

30

 

 

1 4
A:
Li 5
80
F“ .—
5 -4
A'1= '%
-3 1
__ __J

 

 

From equation (5.2)

 

“((1)2 + (3)2 + (4)2 + (5)2

 

A =
and
A-1 = ‘\](1/7)2 + (5)2 + (-3)2 + (-4)2 + (1)2.

Using equation (5.2)

cond(A) = (7.14) (1.02)

Forsythe and Moler {11} state that for

equations such as equation (5.1).

Max“ HebH

i cond(A)

b

 

 

where:

ex is a matrix containing the errors in the

solution x

7.14

1.02

a system of

(5.4)

eb is a matrix containing the errors in matrix b

31
In similar fashion, an error in A, eA, produces an error in
x bounded by

e e
..._|.._|._X.LL icond (A) I..|.__A_l_l. (5,5)

||X+exl| IIAII

Assume that ex is insignificant compared to x, then equation

(5.5) becomes

 

 

e e
:l xll _<_ com, (A) I AH (5.6)

l
lxll IIAH

Assume the worst case in equation (5.4) and (5.6) so that
the inequality is replaced by an equality.

Define exb as a matrix containing the error in the .
solution due to errors in b, and define exA as a matrix con- ‘
taining the error in a solution due to errors in A. Combine

equations (5.4) and (5.6)

e e e (5.7)
L1_§§LL. +[J_3211. = cond(A) .11_§11. + cond(Aflliilil

II x ll II x [I [| All II b II
Define ex = eXA + eXb
| |~ 5.8
He llglfxll cond(A)leAl + Heb” ( )
X ‘*————-
llAll llbll

To continue the previous example assume A and b con-

sist of measurements with a measurement error of +.01.

32

Then,
{— -—1
.01 .01
eA = eb =
.01 .01

 

 

Use equation (5.2) to find

 

 

HeAH = Hebll =.01\(4 =.02
Assume
1.5 4.25
b:
2.5 5.6
and

 

|| bl] =;fl(1.5)§ + (2.5)5 + (4.25)5 + (5.6)2A = 7.61

Equation (5.1) may be solved by inverting A and multiplying

the inverse times b.

 

 

 

 

 

 

 

 

 

x = A ‘1 b
"‘1
-5/7 4/7 1.5 4.25
x:
3/7 -1/7 2.5 5.6
.357 .164

.286 1.02

 

 

33

and using equation (5.3)

 

||x ||= «J(3.57)2 + (.206)2 + (.164)2 + (1.02)2 = 1.13;

Substitute into equation (5.8)

llex II (1.13) (7.28) .02 + .02
7.14 7.61

0.45

In computing DM, A is Pl, b is P2, and x is DM. Substitute

into equation (5.8) to get

[[eDM II = '[DM'I cond(A) llepilL+lezll (5.9)
"_PI" “§2“
II II II II
5.2 How Errors in DM Effect the Screw Axis
In determining the screw axis parameters the oppor-
tunity for error to multiply is enhanced by the fact that A
is determined fromwb, which may already be in error, and s
is determined from A. In this section it will be shown that
an error in the angle of rotation may be very large, and yet
that rotation angle is needed to determine directly, or
indirectly, all other screw axis parameters. Recall from
Chapter 3 that the first step in finding the screw axis
parameters is to find 9 using equation (3.14).

Trace -1
<0: cos-1 (5.10)

2

34
The trace is the sum of the diagonal elements of RM. If
each element is in error by an equal amount eDMr the trace

will be in error by the following

eTr = 90M '43 (5.11)

This is derived by setting the resulting error equal to the

square root of the sum of the individual errors {2}. As an

example assume eDM = 1.0008, then eTr = 1.00138.
PrOpogation of errors in the inverse cosine is

calculated by taking the derivative of the function {2}:

0 = cos-1 x (5.12)
dx
d¢=——_—
1.- x (5.13)
Tnm£-l.

 

where: x

Figure 5-1 presents the error in the rotation angle
resulting from eTr = .00138. The figure indicates that the
error term is significant for small angles. For example, if
the correct value of 9 is 0.80 the induced error is 5.60.
Further calculations using equation (5.13) show that if
rotation angles on the order of 10 are to be measured to
within 10.50, the required accuracy in the Trace is 1.0003,
or eDM must be less than .00018.

Note that eTr for Figure 5-1 is .00138 indicating only

35

 

 

 

 

 

m...
m..
m
.1
.1 Vr.
0
.1 m...
.
3 mfi
8
AI
W].
N]
.1
Q _ — h m P. n k
Q .Q Na ma Ea ma ms uo

mOwOw.OD >30_0 .3 0001oom

“.0C10 ml. mowOw.OD >Jc.o <01mcm N1101 .3 mowow_03 >30_0

36
three significant figures are available. The number of
significant figures which can be used in calculations has an
effect on the accuracy of<9. Table 5-1 illustrates the
effect of significant figures when 0 is small.

Table 5—1 was computed in the following manner. If
cos 9 has two signficant figures that means the value is be-
tween .9949 and .985. The inverse cosine of .9949 is 5.73°.
The inverse cosine of .985 is 9.94°. The resolution is

9.940 -5.73°, or 4.210.

TABLE 5-1

Effect of Significant Figures on Small Angles

 

 

 

Significant
Figures Cos¢ Resolution
2 .99 4.21
3 .999 1.33
4 .9999 0.42
5 .99999 0.13
6 .999999 0.04
7 .9999999 0.01

 

Because at small angles, sine is less affected by an
error than cosine, the possibility of using the sine to find¢
was investigated. Two different methods exist for using the
sine, but neither is more accurate than using the cosine.

The first method investigated involves substituting l - sin2¢
for c0320, and the second method utilizes the off diagonal
elements of RM. Changing the names of variables does not
avoid the basic cause of the error propogation. To

measure small rotation angles on the order of 1° accurate

37
measurements of rigid body positions are needed, and as many
significant figures as possible should be retained. To find¢
to 1.13° when the rotation angle is 1° requires that five
significant figures be available for computations.
Once ¢is known the components of A, the unit vector
in the direction of the screw axis, are computed using

equations (3.18-3.20). Consider the computation of ux

_a32 ‘ 8‘23
u = --

To find the effect of an error, take the partial derivatives

{2}, one with respect to¢?, and one with respect to
(932 - a23)

cos 0 d( - ) d(a - )
dux = .332 a23 + —3-2—:2—3- (5.14)

25fin?¢ 2 sum)

 

It may be argued that a32 and a23 are functions of ¢
(see equation 3.2 for the proof of this) and should have
been differentiated with respect to ¢. An error in 0 arises
from errors defined in equation (5.13) which is independent
of a32 and a23. The values of a32 and a23 arise from the
computation of DM, and an error in the calculated value of¢
will not effect a32 and a23.

Equation (5.14) reveals that even if 9 is known

exactly, i.e. d 0: 0, error in DM may still prOpogate into

38

the value of A. Error in (a32 - a23) is related to the

1
error in DM, and that error is multiplied by -f-§;- . If
sun
¢ is small, .l¢ is large, independent of d0. In addi-
Sln

tion to measurement error in DM, when ¢ is small RM
approaches an identity matrix causing off diagonal elements
such as a32 and a23 to be effected by round off error.

An error in (a32 - a23) is defined by finding the

square root of the sum of the individual errors squared {2}.

 

d(a32 - a23) = “4&322 + e232 (5.15)

As an example of the error progation in finding ux,

assume
¢ = 1° (.0174 rad):
d 0 = 0.250 (.004 rad)
(a32 - a23) = .001
e32 = 623 = .0008

By equation (5.15)

d(a32 - a23) = .0008 ‘d2 = .0011

To find the error in ux use equation (5.14)

(.999) (.004) (.0010) + .0011

 

dux =
2(.0174)2 2(.0174)

.038

39

The error in ux will be increased compared to the

error in DM when 9 is small, even though 9 may be known

exactly.

As with the determination of<b, small angles

require accuracy and precision to determine A.

The final step in the screw axis algorithm is to solve

for the translation 5, and a piercing point AI.

finding DM this is a matrix operation.

(3.27)

a14

a24
a34

 

Assume Alz = 0.0,

plane

a14
a24
a34

 

Set b =

 

—‘

 

_—

 

 

 

a14

a24
a34

 

 

(1'911) ~a12

-321
'331

(1'922>-a23
(1-633)

‘332

”313

— 1

 

—

 

Alx
Aly

A1z

 

As in

Recall equation

(3.27)

i.e., the piercing point lies in the XY

(1'811) a12
a21 (1-322)
a31 a32
ux (1-a11)

= uy ~a21
uz ‘331

 

 

 

“312

(l-azz)

-a32

 

 

 

L. .1:

(5.16)

Alx

A1y

 

40
A and b are known to within a certain error. The error in

x is bounded by equation (5.8)

e
llexll = leII cond(A) U—Ll-l— + M (5.8)

llAll llbll

5.3 Conclusion

In this theoretical error analysis the difficult
question of error propogation through matrix operations has
been evaluated using methods from numerical analysis.
Within a given accuracy in DM the screw axis analysis is
particularly sensitive to small rotation angles. This sen-
sitivity cannot be reduced by using sind>instead of cos¢.
The limit of accuracy in determining A is the size of<p. To
find values of 0 on the order of 1° to within 1 .5° requires
that the accuracy of DM be better than 1.0002, and that all
four significant figures be retained. The next chapter
shows how the addition of error to “perfect data” actually

changes the screw axis parameters.

CHAPTER 6

EVALUATION OF THE ACTUAL EFFECT OF ADDING ERRORS

6.1 Introduction and Description of Procedure

The results of Chapter 5 indicate that screw axis
analysis is inaccurate for small rotation angles. To
gain insight into how an actual error in the position data
effects the entire screw axis algorithm, including coordinate
system determination and computation of DM, "perfect“
error free position measurements were computed.
Selected errors were then added to the measurements and
the effect on the screw axis was evaluated. The data
were created with a set of points in a geometrical
arrangement, and at distances (in cm) typical of
targets on a human femur. The "perfect" data is created

by specifying:

1) The number of significant digits in the position
measurements

2) A point in space which fixes the screw axis

3) Three components of a unit vector in the direction
of the screw axis

4) A rotation around the screw axis

5) A translation parallel to the screw axis

The data points created are then used in a computer

41

42

program which preforms a screw axis analysis, prints out the
results, and then adds an error of +.10 cm to a single data
coordinate. The analysis then starts over again, including
the creation of the coordinate system at each position.
This is followed by removing the error added in the previous
step, and then adding an error of +.10 cm to another data
coordinate. This process is repeated until all 18 coordin-
ates specifying two rigid body positions have been perturbed.

To simulate the effect of random error, it is also
possible to add different random errors to each coordinate
of the ”perfect” position measurements. This is

accomplished by specifying:

1) The number of significant digits
2) A scale factor bounding the random error. If a

digitizer accurate to 1.01 cm is to be
modeled, the scale factor would be .01

The results of this error adding routine are then passed to
the screw axis analysis program. In this manner the effect

of random errors in position data is evaluated.

6.2 Results

In view of the sensitivity of screw axis analysis to
small rotation angles¢ , a number of different rotation
angles were analyzed. The rotations were varied from 1° to
60°. In each case the screw axis was the same, both in
location and direction. The translation 5 was specified as

0.0 in order to see how 5 varied positively and negatively.

3
4

 

 

5m.4 4o.o. c 5H. 54H.H «44.: ~4m.u m55. 506:4“ H.m

43.5 5H.¢ a me. 445. mm. 5mm.. 54. soccmH as.”

mm.m 4m.4 5 OH. mam. a 44m.1 mom. 206:4“ mo.+

44.5: 55.5 o o omo.H Nam. mmm.u H5m. 5H.+

H.4u 54.4 9 mo.: mO4.H ~44. 4H4.u 44m. 4 scaom xH.+

4m.H 44.5 o «a. 5mm. «m4. 454.: 444. NH.+

m4.4 4o.5 o mo. 44c.H mm4. 54m.u mo5. 5..+

55.4 mm.5 a we. mac.4 moH.s mc5.u cc5. m ucfiom xH.+

4~.H mo.oH o mo. 445. 54m. ~5m.- 4.4. NH.+

54.4 45.4 o mo.u 4mm. mom. 544.- 5cm. 5H.+

mm.m 44.4H o mo. 5cm. 45m. .44.- com. 4 “c.om x..+

N :Owufimom

I: a- In 4m.o 45.5 o 4o. 45. 444. ~o4.u mmm. 5H.+

4o.55 45.45 mmO4 54.4 mm.m o 5o. cum. Hm4.ou 444.: m55. m ucfiom xH.+

4o.55 Hm.45 Hmc4 5m.ou m4.m a o mmc.a 454. o4m.u 4mm. NH.+

4o.55 55.45 5504 o 4m.44 4m.4 No.) mma. mam. m~4.u 454. 5H.+

4o.55 ~m.45 NmO4 mo.mn mo.5 c 4a.: 44m.H m5m. 4mm.1 5mm. 5 ucfiom x..+

4o.55 Hm.o5 owo4 om.ou 54.5 o o ~oo.a 4mm. ~om.| mam. u..+

4o.55 mm.45 omc4 54.4- 45.5 o ac. mmo.a 4am. ~mm.- 454. 5H.+

4o.55 om.m5 m504 54.mu 45.5 o 4c.u «mo.. Hem. 544.- 54m. 4 u:.om x..+

H COfiufimOm

4o.55 44.45 HmO4 o N.OH 54.- as. 454. 4m. 5mm.u 4cm. .uouum ozc 6:62

__Nm__ _:m__ Adm.ccoo 5H4 MH<. xac m .mmn.m mm. mm mm cofiumnsuumm
oc.H "A. mm55m. .mm55m.u .mm55m. u a o .o. .o u ucdom masoumfim

 

wumcfipuooo x on» On cmppm mm: EOH.c ans» mwum0.p:. xH.+ .cEsHoo newumnsuumm :H "wuoz.

mwusmfim ucmowmwcmwm 4 .oHu..cu.3 uOuum Hum fiance

44

 

 

 

44.4: 4.4 4 44.: 444.4 444. 444.: 444. 54.+

54.4: 44.4 4 44.: 444.4 444. 444.: 444. 4 44444 x4.+

44.4 54.44 4 44. 454. 454. 444.: 444. 44.+

44.4 44.5 4 44. 454.4 444. 544.: 444. 54.+

5.4 4.4 4 44. 444. 544. 445.: 444. 4 44404 x4.+

44. 44.44 4 44. 444. 444. 444.) 444. 44.+

44.4 45.4 4 44.: 444. 44. 444.: 454. 54.+

44.4 44.44 4 44. 444. 444. 454.: 444. 4 44404 x4.+

N COMHMWOW

44.4 44.44 4 44. 454. 444. 444.: 444. 44.+

44.4 44.4 4 54. 444. 444. 444.: 445. 4 44404 44.+

44.4: 54.4 4 44.: 444.4 444. 444.) 444. 44.+

45.4: 44.44 4 44.: 444. 444. 444.: 444. 54.+

44.4: 44.5 4 44.: 454.4 444. 444.- 544. 4 44404 x4.+

)n u- I: 45.4: 44.4 4 44.: 444.4 444. 444.: 444. 44.+

44.55 44.45 4544 44.4- 44.4 4 54. 44.4 444. 444.: 444. 54.+

44.55 4.45 4544 44.4: 44.4 4 44.: 444.4 444. 444.- 544. 4 44404 x4.+

H COfiHHmom

44.55 4.45 4444 44.: 44.4 4 4 444.4 554. 554.: 554. .40444 oz. 4402

__44_. 4_44__ .4444400 4mm. 4mm 444 m .444.4 mm mm wm 40444444444
Oc.H n 9
44554. .44554.: .44554. n a

4 .44 .4 a 44404 44444444

 

mu424puooo x on» Ou cocoa no: EOH.c umcu mmumo4pcw xd.+ 4252400 cofiumnsuuwm :H

mwusm4m ucmo4u4cm4m m «0404 zufiz acuum mum mqm<a

«wuoz.

45

 

 

 

44.4 54.4 4 54. 54.4 444. 544.- 544. 404444 44.w
54.: 45.4 4 4 44.4 444. 444.- 454. 404444 44.4
44.4 44.4 4 44.: .44.4 444. 444.: 544. 404444 44.+
44.4- 44.4 4 44.: 44.4 544. 454.- 454. 44.+
44.4: 44.4 4 44.: 44.4 544. 444.- 444. 54.+
44.4 44.4 4 44. 44.4 44. 444.- 444. 4 44404 x4.+
44.4 44.44 4 54. 44.4 444. 444.: 444. 44.+
44.4 44.4 4 44. 44.4 454. 454.: 444. 44.+
44.4 45.4 4 54.: 44.4 444. 444.- 444. 4 44404 x4.+
N COwufimom

44.4 44.4 4 44. 44.4 444. 444.- 544. 44.+
44.4 44.4 4 44. 45.4 454. 444.- 444. 4 44404 x4.+
44.: 44.4 4 4 44.4 444. 444.: 454. 44.+
54.: 44.44 4 44.- 44.4 444. 444.- 544. 54.+
44.- 44.4 4 44.: 44.4 444. 444.- 544. 4 44404 44.+
44.4: 44.4 4 4 44.4 454. 444.: 454. 44.+
44.4- 44.4 4 44. 44.4 444. 444.- 544. 44.+
45.4: 44.4 4 44.: 44.4 444. 444.- 454. 4 44404 x4.+
H coflufimom

44. 44.4 4 44. 44.4 454. 554.: 444. .40440 024 4402

4mm. 44¢ 444 m .404..4 mm Mm mm 40444044444

04.4 n 4

44554. .mmnhm.l .44554. D
c .o4 .o n ucmom mcwoummm
.mumcfipuoou x on» On cmppm was eoa.o umnu mwumo4p24 x4.+ .c52400 cofiumnsuumm :4 ”muoz.

mwu=m4m ucmo4u4cm4m v “Omuavnufiz “Guam mlw Nancy

46

mo.
om.

mm.l

ma.
mm.
«m.

am.
mm.
mm.

mo.
¢m.

mo.l
MH.1
#4.!

vo.l

mm.l
mm.l

mo.

mN.m
mc.cH
oo.o~
m¢.m

vm.m
vw.m
mo.oa

mm.m
ww.m
wc.cH

mm.m
¢m.¢

m.m
MN.OH
mh.m
H¢.m
mm.m
Hm.m

GOO GOO COCO

GOO 000 CO

No.1
me.

Ha.l
hc.l

No.
mo.
mo.

Ho.
no.l
mo.

Ho.
ho.

Ho.l
~o.l
mo.l

Ho.l
mo.
wo.l

HH.cH
cm.m

mo.cH
Hm.oa

vm.a
vc.oa
ab.¢

mm.m
mm.¢
Nm.m

m.m

.b.m
no.0H

mm.m
m~.oa
Ho.cH

ao.cH
wo.oH

mm.m

.mmcvm

mmrbm.

Ham.
wmm.
«mm.
haw.

qmm.
mom.
mum.

mum.
hbm.
mbm.

mew.
mmm.

mmm.
Ham.
mum.

mnm.
«hm.
mmm.

¢hm.

ND

hhm.l vhm.
cam.l New.
ohm.l «hm.
mwm.l m¢m.
«mm.l mam.
mhm.l mam.
coc.l mew.
ohm.l owe.
vmm.l th.
mmm.l mam.
omm.l mam.
me.l mam.
vrm.l mum.
mom.l wmm.
mmm.l Hmm.
omm.l arm.
th.l mam.
ham.l mhm.
mhm.l mm.
mm mm
Oc.oa u

.mmnhm.l .mmrhm. u

c .OH

§ou

mwusmwm ucm04mficm4m v «Ocaue :uHB acuum 414 mamfla

 

404444 44.m
soccmm mc.+
>H.+

m ucfiom xH.+
NH.+

>H.+

unwom xH.+
NH.+

>H.+

ucmmm xH.+
N cofiufimom
>H.+

44404 x4.+
NH.+

>H.+

ucfiom xH.+
NH.+

>H.+

Damon xa.+

 

4 sawufimom

.40440 02. 4:02

 

yawn wcmofifim
.mumcfipuooo x on» 0» wound was an.o umnu mmumoflpcfi xH.+ 4259400 cofiumnsuumm cH

cofiumnsuuwm

«wuoz.

47

mm.Hm Hm.mh

mm.Hm Hm.wh

whov

.Hwoc

 

a_~m__ __Hm+fi Namvccou

wN.o
Ho.
No.l

mo.c
hc.c

HH.o
ac.
Ho.l

Ho.o
HH.o
No.l
c.o
c.o
Hc.I

cH.I
mo.l

o

NHC

mm.m
m¢.m
mo.oH

ma.m
oo.oa

mm.m
mm.m
HQ.¢H

mm.¢
vm.m
mm.m
vo.oa
om.m
am.m
vo.oa
He.OH

mm.m
had

COO CO COO

H
000 HOG OO
O

O

NH‘

cm. «.mm 5mm. mbm.l
No. o.cw chm. bbm.l
mo. c.0m mbm. chm.|
we. c.0w mbm. m5m.l
mo. m.am Mbm. ohm.l
mo. m.mm mbm. hhm.l
moo! m.mm bhm. mbm.l
OH. o.cm mhm. mhm.l
Ho. 9.9m chm. bhm.l
ma. h.mm Hhm. mhm.l
o 6.9m mbm. bbm.l
Nc.l m.mm ohm. mbm.l
mo.l N.om mmm. hbm.l
o H.0w hbm. bhm.l
ma. H.ow hbm. mfimoi
wo.l H.00 whm. mhm.l
HO. c.0m hbm. hbm.l
m dammfia mm mm
Oo.ow
mmwhm. .mm55m.l ~mmrhm.
6 .OH ~o

5mm.
hhm.
chm.

Ham.
Ham.

bum.
hbm.
mum.

ohm.
mam.

mum.
whm.
Ohm.

whm.
¢bm.
mum.

mum.
mm

9
D

 

Ecocmu H.W
SOCCMH Ho.H
Eoccmu mo.+
>H.+

m ucfiom xH.+
NH.+

>H.+

H unwom xa.+
N cofiufimom
>H.+

m ucfiom xH.+
NH.+

>H.+

N ucfiom xH.+
NH.+

>H.+

H ucfiom xH.+

 

H :ofiufimom

AHOuum ozv wcoz

 

:Owumnsuumm

acmom mam0umfim

 

Awumcficuooo x may Ou cwccm mmz EUH.o umnu mmumoficcfi xa.+ U55500 cofiumnsuumm :H

amusmwm unmofiuficmfim q uocoue nufiz uouum mum mamcs

”wuomu

48

 

 

mo.| ca. >.am H.oo Hum. mam. mbm.l mbm.| ohm. mam. v Com mum
50.: mo. om.m Hm.oa mmm. mmm. mmm.| ocm.| aem. mam. v Goa «no
bc.l mo. N>.e mm.m cue. mow. wmm.| vmo.l cmm. mvw. v 0m mum
mo.| mo. mom. wmv.H hmo.l mam. mmm.| ~H>.| 5mm. mob. m CH mum
mo.r mo. «mm. mcv.H moH.| mum. Hmm.l moh.l va. mph. v 0H Hum
aw: an: a“: xmz a“: N xmz cw: an: cw: mm: moummwm e wanna
e 5 >5 x: ucmufimficmfim usmcm
omusmaou
mmuhm. .mmhhm.| .mmrhm. n D
c ..cH .o u ucHom mewOumflm mmmmo Ham mom
Acmnmamcm mmmmo uOuum Ecccmu monaocw uoc mmoov

mfim>HMC£ neuum mo wumessm mum mamce

49
Tables 6-1 through 6-5 display the results. In addition to
perturbing each coordinate by +.lO cm, three random error
cases were analyzed. The digitizer used in SAL is
accurate to i.013 cm, thus the first random error
case was 1.01. The second and third random cases are of
1.05 cm and i .10 cm. Table 6-6 summarizes the five
Tables preceeding it, but the random error cases are not
included. The inputs and the maximum and minimum values

of the screw axis parameters are detailed in Table 6-6.

6.3 Discussion

Tables 6-1 and 6-2 show that small angles were com-
puted more accurately than Chapter 5 would have indicated.
For rotations as small as 1° the range of calculated values
for was i.4l°, -.ll°. It is interesting to note that
increasing the number of significant digits did not decrease
the affect of an error. It should be kept in mind however,
that the error added was .10 cm. With that large of an
error it is doubtful that 5 significant figures could be
claimed.

The real effect of adding significant figures is seen
in comparing Tables 6-1 and 6-2 for the case of no error
added (first line). In Table 6-1, which is the case of
¢=l° and 4 significant figures, there is considerable error
in ¢, and particularly 5, even though there is no error in
the input data. Increasing the accuracy of DM to 5 signifi-

cant figures, as in Table 6-2, results in no error in the

50
screw axis parameters when no error is added to the ”perfect
data.”

Table 6-6 shows that while ¢ could be determined with
some accuracy, the determination of the components of 5 was
not possible for a rotation angle of 1°. This is consistent
with the findings of Chapter 5. The problem is that the
sine of a small angle is a small number, and dividing by
that small number will magnify an error.

As would be expected, Table 6-6 reveals that as the
rotation angle ¢ increases so does the accuracy of the screw
axis parameters. Interestingly enough, errors in the
translation, s, do not change as the rotation angle
increases. In Table 6-6 it is seen that s is determined to
within 1.10 cm.

Examining Tables 6-1 through 6-5 reveals that the
worst errors occurred when point 2 was perturbed. This is
not surprising since point 2 is the closest of the three
position data points to the origin of the created axis
system. Recall that Chapter 4 indicated that the further a
data point is from the origin of the axis system created
based on the data points, the less effect an error in the
data point has on the created axis system. The first data
point is 38.9 cm from the origin, the second data point is
0.72 cm from the origin, and the third is 8.47 cm. Since
point 2 is so close to the origin, an error in one of its
coordinates will have a much greater effect on the coor-

dinate system, thus on DM and the resulting screw axis

51
parameters.

The piercing point selected for this test was an
arbitrary point. The screw axis direction was selected so
that there would be equal components in all three
dimensions. If a different screw axis orientation or loca-
tion was selected the error effects might be different.

This chapter is mainly to illustrate the effects of
error, and give some idea of their magnitude. It should
not be used as an indication of the maximum errors for a par-

ticular value ofcp.

6.4 Evaluating Cond (P1)

The first point of interest is that cond(Pl) changes
very little as error is added. It also does not change as
the rotation angle is increased, for increasing the rotation
angle does not effect the initial position, only the second
position. Adding significant figures to the data does not
change cond(Pl). It would seem on this basis that cond(Pl)
is a function of the geometry of the targets. Using
Table 6-1 and equation (5.9) it is now possible to evaluate
the effects of an error in P1 on the error in DM. Since
there are four significant figures (2 beyond the decimal

point) assume

epl = 6P2 = .005 per element

 

 

52
From equation (5.10)
||eP1|| = llepzll = .005 '112 = .0173
From equation (5.9)
r
IIeDM || = IIDM ll cond(Pl) .0173 + .0173 ‘
75751 77.04 .
L
= IlDM Il 4080 x .00045 = 1.835 ‘
In this particular example ||DM|| = 2
SO
lleDM}| = (1.835) (2) = 3.67

If it is assumed that 9DM is evenly distributed over all 16

elements of DM then the error in DM per element is

3'67 = .9175 or a 91.75% error (6.1)

 

Obviously, DM is known better than .9175. The assump-
tion that the error is evenly distributed among all of the
elements of DM may be unrealistic. It should also be kept
in mind that Forsythe and Moler {ll}, who developed the

equations which equation (5.9) is based on, were not trying

53
to define the error propogation, only to put an upper bound'
on the resulting error in a solution to a set of equations.
In view of the answer in equation (6.1), does equation (5.9)
have any real use in this error analvsis? This question

will be addressed in the final chapter.

 

CHAPTER 7

EXAMPLE OF SCREW AXIS ANALYSIS USING ANTOMICAL DATA

7.1 Introduction

This Chapter presents the results of a screw axis ana-
lysis performed using data collected from a cadaver in SAL.
The joints analyzed are the hip, and the sacro-iliac joint.
The bone movements analyzed are the femur moving relative to
the left inominate for hip motion, and the sacrum moving
relative to the inominate for the sacro—iliac joint.

The cadaver used was a Caucasian male who was 80 years
old. The primary cause of death was a brain tumor and
diagnositic radiographs revealed no abnormalties in the
lumbar/pelvis/femur linkage system. During the studied
motions the cadaver was supported by an overhead assembly
which held the subject upright in a standing position.

Table 7-1 summarizes the screw axis analysis for the
hip, and Table 7-2 summarizes the analysis for the sacro-
iliac joint. In all cases the motion was from the same ini-
tial position of a supported erect posture with both feet
on the floor. The motions analyzed were abduction, abducto-
flexion (approximately equal amounts of abduction and
flexion), and flexion. The final position was the extreme

position of the indicated movement.

54

PLEASE NOTE:

Page 55 is lacking in number
only per school. No text is
lacking. Filmed as received.

UNIVERSITY MlCROFlLMS INTERNATIONAL.

56
The motions studied were to the extreme positions.
Since no intermediate positions were analyzed it is not
possible to see the motion of the piercing point and/or
screw axis. When joint mobility is studied, motions must be
broken up into a number of small displacements. In this

manner the motion of the screw axis may be studied.

 

 

7.2 Discussion F.

A summary of the screw axis analysis for hip motion is
presented in Table 7-1. Note the large amount of transla- 3
tion which occurs. As much as 1.4 cm is seen in flexion. i
L

The analysis of Chapter 6 would indicate that these values
are accurate to within 1.10 cm. Since these motions were to
the extreme positions, the movements of the femur were
large, and the accuracy should be good. The classic model
of the hip is a ball and socket joint. If the hip is a true
ball and socket no translation should be observed. Note
also that the range of values of cond(Pl) is 7800 to 2800.
This Chapter reveals that anatomical motions, such as
those of the sacro-iliac joint, occur in the range where
screw axis analysis is most sensitive to error. Table 7-2
presents the screw axis analysis for displacements between
the extreme positions, yet the rotation angle is between
1° and 2°, and the translation is less than 1.15 cm.
Chapters 5 and 6 showed that the requirements for accuracy
and the retention of significant digits, are most stringent

for motions of the magnitide in Table 7-2.

57

 

 

 

m¢.om m>.oo Hmvm th.| omc. mam. mm.~ o.c cOMmeh
hm.vm ah.¢m Hmmm ~m.l ham. mmm. oN.H ma. :ofixoaw

leuosonc

mm.wm mv.mm mmmw mum. mvm.l mvH.I mm.H mo. cofiuospo<

IfiflMMIfi_ fifiMMflfi AdMfimmmm mm. am am dfiMMfldl 50 m mmmmmm

ucfiOH omHHH10uomm msu uOu m«m>am:¢ mfix¢ smuom Nth Nance

mm.mma H~.~h mwhm o.o mom.l Hma. w.mm «v.H cofixoam
m.moH m.bh mqam mmm. 50¢.I New. m.v¢ we.l cofixoau

nouocsnc

~.NMH H.@OH mmmh and. mmm.| qu. o.v~ hc.l cofiuospn<

3.0.: gum: a mm a m and; so. a

 

cofiuoz a“: uOu mfiw>amc¢ mwxd souom .Huh mqmce

58
Though these motions are small, they are important.
Common engineering practice when modelling the human body,
as with an anthropometric dummy, is to treat the pelvis as a
rigid body and assume there is no motion in the sacro-iliac
joint. The anatomical screw axis analysis presented in this
Chapter indicates that motion occurred in the pelvis at the

sacro-iliac joint.

 

 

CHAPTER 8

SUMMARY, REVIEW, AND RECOMMENDATIONS

8.1 Summary and Review

 

 

R‘
Present models of the human body generally make some .i

simplifying assumptions to reduce the DOF present in a

joint. However, human joints have a full six DOF, and any

assumption which reduces these DOF artificially constrains Bi

the joint model or analysis.

The Displacement Matrix fully describes a three-
dimensional displacement. DM maps a rigid body from one
position to another. Screw axis analysis is a technique of
making DM more easily understood. By specifying a vector 5
which is the screw axis, a translation, 3, parallel to the
'screw axis, and a rotation, 0 ,around the screw axis, any
three-dimensional six DOF displacement can be described.

Chapter 4 discussed the data used to compute DM. In
SAL the data consist of position coordinates of three points
on a bone at two different positions P1 and P2. A
coordinate system is set up based on the data points,
and four new points on this coordinate system are used to
compute DM. A coordinate system approach is used in order
to insure an orthogonal RM. The coordinate system approach

is effected by error, leading to the coordinate axes being

59

 

60
in the wrong location, or having the wrong orientation.
The errors in the coordinate system are minimized by
locating the data points used as equidistant from each other
as possible, and by making the angle formed at the intersec-
tion of the relative position vectors as large as possible.

Given that there will be some error in the coordinate
system, how does this error affect DM? And how does an
error in DM effect the screw axis parameters? These
questions were dealt with in Chapter 5. The problem of
error propogation through matrix operations was discussed
first. This is a complicated question, and a detailed
analysis is outside the scope of this thesis. However,
Chapter 5 developed a method for bounding the error propoga-
tion based on the condition number of a matrix.

An error in DM effects the screw axis analysis most
when the rotation angle is small. While the determination
of the rotation angle is sensitive to errors in DM, it is
the determination of the components of U, the unit vector in
the direction of the screw axis, which is most sensitive to
error.

Chapter 6 illustrates the theoretical error analysis
of Chapter 5. By creating and then perturbing "perfect"
position measurements it was possible to see how the screw
axis parameters were effected. As expected the error for
small rotation angles was large, and the error for large
rotation angles was small. In general, rotation angles were

determined with greater accuracy than Chapter 5 had

 

61
indicated, but the determination of 0 was not possible at
small angles. For small rotation angles, on the order of
1°, a minimum of four significant figures are needed to find
¢ to within an uncertainty of 10.410.

In Chapter 6 an example indicating how the condition
number could be related to expected errors in DM resulted in
an exceedingly high error bound. The use of condition num-
bers is discussed in the next section.

This thesis describes a method of analyzing anatomical
joint motion, and evaluates the limits of this method.

As an illustration of what these anatomical studies might
produce for screw axis parameters, Chapter 7 contains

some analysis of joint mobility for a cadaver studied

in SAL. The need for being able to accurately model small
rotations was illustrated by the small rotations which ocucr

in the sacro-iliac joint.

8.2 Is the Condition Number of a Matrix of Any Use?

From Chapter 6 it appeared that equation (5.9) produ-
ces an error bound which is too high to be useful. DM is
known better than the 192% computed using equation (5.9).
Does this indicate the method is of no use? The interplay
between measurement error and condition numbers is an area
worthy of further study. This section presents some ideas
which might serve as an introduction to such study, but for
now equation (5.9) is not useful.

This thesis has used Euclidean norms in computing the

condition number. This is in line with what Forsythe and

 

62
Moler {11} use, however, other norms can be used and are
discussed by Forsythe and Moler. They also discuss methods
of reducing the condition number, though they unfortunately
conclude, '...it is quite unclear to us how to program a
reasonable scaling of a general matrix.” Scaling is a
method of reducing the condition number. Two methods of
scaling are discussed in Forsythe and Moler, the first
involving pre-and post- multiplying by two different scaling
matricies. The second method attempts to equilibrate the
matrix in question.

The point of both scaling and equilibrating is to make
the norms of the columns and rows of a matrix as close in
value as possible. The closer the norms, the smaller the
condition number, and the more accurately a solution is
determined. The norms of the rows of one of the position

matricies from Chapter 6 are given below.

Row 1 norm = 15.82
Row 2 norm = 3.45
Row 3 norm = 75.23
Row 4 norm = 2.0

Because the norm of the third row is so much larger
than the other rows the condition number is large. The
third row is composed of the z coordinates of the four
points used to compute DM (refer to equation 3.4). The

coordinates of the points are related to the location of the

63
origin of the coordinate system set up based on the anatomi-
cal data. While the x and y coordinates, in the inertial
axis system, of the four points are close to the origin of
the inertial system, the z coordinate is much further away.
By relocating the origin of the anatomical axis system so
that the z coordinate is closer in value to the x and y
coordinate values, the condition number is reduced. This
indicates that the condition number is a function of the
geometry of the data used (the anatomical axis system) to

solve for DM.

8.3 Recommendations for Future Work

 

Screw axis analysis is sensitive to errors at small
rotation angles. It would seem then that whenever possible
large rotation angles should be used.‘ This is not a prac-
tical restriction for two reasons. First of all, as was
seen in Chapter 7 much of the desired data are at small
rotation angels.

The second reason might be termed the paradox of screw
axis analysis. While small rotations and translations are
prone to error, large rotations and translations yield a less
than accurate description of the motion. If an air traveler
catches a plane in New York, and is later seen in Los
Angeles, the motion description would be from New York to
Los Angeles. But what if the traveler went to Florida
first, then London, then Chicago, and finally Los Angeles.

His motion would be much different in this second case, but

 

64

if all that was available was a beginning and ending point,
the path in between is not well known. It seems that the
only way to use screw axis analysis is to acquire accurate
data for small motions and retain five significant figures.

In Chapter 5 it was observed that even with no error
in the data, when only four significant figures are
available, a rotation angle of 1° can only be found to
within 10.4°. Chapter 6 showed that for this same case the
components of‘fi were adversely affected when only four
significant figures were available, even when there was no
error in the data. Increasing to five significant figures
removed the error in the determination of the screw axis
parameters. Chapter 7 pointed out that rotation angles on
the order of 1° are to be expected in anatomical studies.

The algorithm for creating an axis system based on
anatomical data does so without regard for the most accurate
resulting coordinate system. It would be useful to create
an algorithm which would compromise between placing the
three data points as far from each other as possible, and
making the included angle between the relative position vec-
tors as close to 90° as possible. An appropriately selected
coordinate system might reduce the effect of measurement
error present in the position data.

There has been considerable discussion in this thesis
of evaluating the error propogation through matrix
operations. The ideas presented are only a beginning. The

relationship between the condition number and the coordinate

65
system may lead to a method of reducing the error propogation
through the matrix operations used to find DM. An optimiza-
tion study where the function to be minimized is the con-
dition number, might provide the optimum anatomical
coordinate system for reducing error propogation. The inde-
pendent variables would be the coordinates of the origin of
'the anatomical axis system, and two scaling matricies.

Another mathmatical technique for reducing error pro-
pogation involves the use of surfaces. There should be much
interest in screw axis surfaces. By passing a surface
through all of the screw axes produced by a series of
displacements the beginning of a joint model is created.

Two surfaces are needed, one for the fixed object (the ino—
minate in the analysis of Chapter 7) moving relative to the
moving object (the femur in the analysis of Chapter 7). The
second surface results from the motion of the moving object
relative to the fixed object. The two surfaces produced
would appear to roll in the direction normal to the axis
common to the two surfaces, and to slide along the common
axis {13,27}. If the geometry and location of the two sur-
faces is known, along with a definition of the amount of
sliding, the spatial motion is uniquely defined {l3,2fl-.

The second and initially more useful feature of screw
axis surfaces is that a surface smoothes data. An outlier
from a screw axis surface could be in error, though actual
slippage in the joint may also appear as an outlier. It

would be interesting to select a screw axis from a smoothed

66
screw axis surface, and then recontruct DM.

In a similar manner it would be useful to construct a
surface based on all of the anatomical position data for one
target. This surface would then represent a surface in
space that is the locus of motion of a target on a bone.

The smoothing effect of the surface should be an aid in

dealing with measurement error, and the possibility of r
selecting positions not measured exists. The shape of the 3
surface would also be of interest. If the hip is a ball and *

socket joint, all points of the femur should move on a

 

sphere around the hip. It would be interesting to locate L
the center of such a sphere. If, as is more likely, the hip
is not a perfect ball and socket, the shape of the surface
would contain interesting and useful information.

If three points are the minimum needed to specify
rigid body position could not many points be used? The
points would be put into a position matrix and an
overdetermined set of equations would result, allowing the
use of a least squares analysis. Lennox and Cuzzi {16}
report that this method does not improve the accuracy of
screw axis analysis, but this question is still worth exa-
mining because of possible different measurement techniques
used in SAL. Lennox and Cuzzi do report on a method of
improving position data for use in screw axis analysis.
Their method relies on photographic centers and might be
worthwhile exploring for use in SAL. The author feels that

an improved coordinate system algorithm would incorporate

67

some of Lennox and Cuzzi's ideas.

8.4 Conclusions

 

1)

2)

3)

4)

5)

In order to contain measurement error to position data
an anatomical axis system approach for computing DM

should be used.

The origin of an anatomical axis system should be
equidistant from all of the data points used to compute

the anatomical axis system.

The bound on the error propogation due to the matrix
Operations used to find DM as developed in equation
(5.9) is too high to be useful. Study of the propoga-
tion of measurement errors through matrix Operations is

an area worthy of future investigation.

If small rotation angles are to be analyzed, five signi-

ficant figures are needed in the data.

As the rotation angle decreases the error propogation
increases. This is not a linear function. If rotation
angles on the order of 1° are to be studied, the ele-
ments of DM must be accurate to within 1.0002 for an

error in i of less than 10.5°.

 

6)

7)

68
Screw axis analysis requires accurate data. Either
s0phisticated measurement techniques must be used, or
mathmatical techniques must be developed to reduce the

uncertainty in DM to less than 1.0002.

Of the two methods suggested in Conclusion 6, it is
recommended that mathmatical techniques be used in the
near future. Specifically, improved selection of an
anatomical axis system for use in computing DM, and the

use of surfaces are the most fruitful areas to explore.

 

LI ST OF REFERENCES

LIST OF REFERENCES

Abbot, E. A., Flatland, Dover Press: 1963.

Baird, D. C., Experimentation: An Introduction to
MeasurementandExperiment Design, Prentice
Hall, 1962.

Braune, W. and 0. Fisher, “Die Bewegungen des
Kniegelenkes nach einer nuen Methode am lebenden
Menschen gemessen” (The Movements of the Knee
Joint as Meaured by a New Method in Living
Humans), Abhandluggn der Mathematischen-
ghysischen Classe der Konigl: Sachsischen
Gesellscaft der Wissenschaften, 17:
75-150, 1891.

 

, 'Bestinmung der tragheitsmonents
des menschlichen korpers und seiver glieder,"

Abh. d. Math. Phys. Cl. d. K. Sachs. Gesell.,
d. Wiss., 18(8): 409-492, 1892.

 

, "The Center of Gravity of the
Human Body as Related to the German Infantryman,"
Leipzig, 1889, available from National Technical
Information Service, ATI 138 452.

 

Bullock, M. I. and I. A. Harley, ”The Measurement of
three-dimensional bodv movements by use of
photogrammetry," Ergonomics, 15(3):

309-322.

 

Chasles, M., “Note sur les propietes generalies du
systeme de deux corps," Bulletin Sci. Math.,
Ferrusac 14:321-326, 1830.

Dempster, W. T., "The Anthropometry of Body Motion,"
Annals of the New York Academy of Science,
63:559-585, 1955.

, "Space Requirements of the
Seated Operator," Wright Air Development Center,
Technical Report 55-159, Wright Air Force Base,
Ohio, 1955.

 

69

10.

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

70

Emanuel S.,and J. T. Barter, "Linear Distance Changes
Over Body Joints," Wright Air DevelOpment Center,
Technical Report 56-364, ASTIA Document #AD
1118003, 1957.

Forsythe, G. and C. B. Moler, Computation 011Linear
Alge—braic Systems, Prentice Hall, 1967.

 

Hallert, B., ”The Basic Geometric Principles of X-ray
Photogrammetrv,” Transactions of the Royal
Institute of TechnolBgy, 123:1-53, 1958.

 

Kinzel, G. L., On the Design of Instrumented Linkages
gorghe Measurement of RelatiVe Motion Between
Two Rigid Bodies, Ph.D. Thesis, Purdue University,
1973.

 

 

Kinzel, G. L., A. S. Hall, Jr., and B. M. Hillberry,
"Measurement of the Total Motion Between Two Body
Segments, Part I - Analytical Development," The
Journal of Biomechanics, 5:93-105, 1972.

 

 

, "Measurement of the
Total Motion Between Two Body Segments, Part II -
Description of Application," The Journal of
Biomechanics, 5:283-293, 1972.

 

 

Lennox, J. B. and J. R. Cuzzi, "Accurately
Characterizing a Measured Change in Configura-
tion,” ASME Paper No. 78-DET-50, 1978.

MacNeil, G. T., "X-ray Stereo Photogrammetry,"
Photogrammetic Engineering, 32:993-1004.

 

Potthoff, R. E., An Analysis of the Biomechanics of the
Human Knee, M. S. Thesis, Stanford University,
6 .

 

Reuleaux, D. F., The Kinematics of Machinery,
translated and edited by A. B. W. Kennedy, Dover
Publications, 1963.

 

Reynolds, H. M., "A Foundation for Systems
Anthropometry,” Interim Report to United States
Air Force Office of Scientific Research,

AFOSR - TR - 77 - 0911, 1977.

Reynolds, H. M., J. Freeman, and M. Bender, "A
Foundation for Systems AnthrOpometry, Phase II,“
Final Report to the Air Force, AFOSR - TR - 78 -
1160, 1978.

 

22.

23.

24.

25.

26.

27.

28.

71

Reynolds, H. M., R. P. Hubbard, J. R. Freeman, and J.
Marcus, "A Foundation for Systems AnthrOpometry,
Phase III," interim report to the Air Force, 1979.

Reynolds, H. M., R. C. Hallgreen, and J. Marcus, "A
Stereo Radiographic Measurement System for Systems
Anthrooometry," paper submitted to Journal of
Biomechanics.

 

Robbins, D. H., ”Errors in Definition of an Anatomi-
cally Based Coordinate System Using Anthropometric
Data," p. 29-34 in Reynolds, H. M. ”A Foundation
for Systems Anthropometry, Phase I,“ reference 20
above.

Rogers, D. F. and J. A. Adams, Mathematical Elements
for Computer Graphics, McGraw Hill, 1976.

 

Selvik, G., A Roentgen Sterophotogrammetric Method for
the Study offtheJ Kinematics of the Skeletai‘
System, Ph. D. Thesis, University offLund, Sweden,
1974.

Suh, C. H. and Radcliffe, C. W., Kinematics & Mechanism
Design, John Wiley & Sons, 1978.

 

Thompson, C. T., A System for Determining the Spatial
Motions of Arbitrary Mechanisms - Demonstrated on
the Human Knee, Ph. D. Thesis, Stanford University,