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Determination of Euler rotation series on shoulder
Conference Paper · December 2008
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TOPICS ON COMPUTATIONAL BIOLOGY AND CHEMISTRY
DETERMINATION OF EULER ROTATION SERIES ON SHOULDER
AZMIN SHAM RAMBELY & MASLINA
DARUS
Universiti Kebangsaan Malaysia
School of Mathematical Sciences
Faculty of Science & Technology
43600 UKM Bangi, Selangor
MALAYSIA
.
FADIAH HIRZA MOHAMMAD ARIFF
Universiti Kebangsaan Malaysia
Fundamental Engineering Unit
Faculty of Engineering & Built Environment
43600 UKM Bangi, Selangor
MALAYSIA
Abstract: Euler rotation series is determined to describe the rotation of the shoulder joint which can be
represented through direction cosines tables used to construct the kinematic equation of shoulder movement.
Key-Words: Euler rotation, Direction cosine, Shoulder joint.
Any desired orientation of a body can be
obtained by performing rotations about three axes
in sequence. There is, however, a multitude of
possible ways of performing three such rotations.
The famous mathematician Euler recognized that
a series of three rotations could be used to
uniquely define the orientation of a rigid body in
three-dimensional space. Though there are many
sets of three sequential rotations which lead to a
unique orientation of a rigid body, the most
commonly used are Euler rotations [6]. With
Euler rotations, the axes of rotation are fixed in
the object to be rotated. This means that as the
object rotates, the orthogonal axes of rotation also
rotate. Therefore, the objective of the current
study is to determine the Euler rotation series on
shoulder joint which then used to construct the
kinematic equation for the shoulder joint.
1 Introduction
Many body movements involve turning and
rotating motions. All linear movements originated
from the lever actions of joint articulations. The
shoulder is one of the most mobile joints in a
human body and moves in a complex 3dimensional pattern. Shoulder joint is divided into
four separate articulations namely glenohumeral
joint, sternoclavicular joint, acromioclavicular
joint
and
coracoclavicular
joint.
The
glenohumeral joint is an articulation between the
humerus head and the glenoid fossa of the
scapula. It is typically considered as the major
part of the shoulder joint. It is called a ball-socket
joint and it has three rotations and no translation
in terms of movement. Clinically, the shoulder
motions are defined as flexion/extension,
abduction/adduction,
horizontal
abduction/
adduction and external rotation/internal rotation.
The first three rotations are not independent as
only two of them are needed to determine the
rotation of humerus about its long axis. Shoulder
abduction can be defined as the angle between the
humerus and the inferior direction of the trunk in
the trunk’s frontal plane. The shoulder horizontal
abduction can be defined as the angle between the
humerus and a line connecting two shoulder
marker in the trunk’s transverse plane while
shoulder external rotation can be described as
rotation of the upper arm about its own long axis
[2].
ISSN: 1790-5125
2
Literature Review
A few researches have been done either in
biomechanics field or other fields using Euler
angle rotations in determining orientation of an
object. Allen & Hahn performed a study in
determining the kinematic and joint coordination
changes necessary to cast lines of different length.
Documenting the movement patterns needed to
cast varying lengths of line should allow for
greater understanding of the underlying
mechanisms of upper extremity pathologies. It
was hypothesized that kinematic casting
parameters would increase in order to cast lines of
90
ISBN: 978-960-474-036-9
TOPICS ON COMPUTATIONAL BIOLOGY AND CHEMISTRY
workspace volume, while workspace evaluation is
carried out based on the shoulder joint motion
limits and the cable-tension analysis. An effective
cable-tension analysis method was also proposed
based on the duality between force-closed multifingered grasping and cable-driven mechanisms.
Finally, the workspace of the mechanism was
optimized to match with that of the human
shoulder and to obtain a set of dimensions for the
shoulder module prototype development [4].
different length, and that time between peak joint
velocities would decrease as length of line
increased. The shoulder motion was calculated
with respect to the trunk using Euler angle
rotation in sagittal, frontal and transverse plane
[1]. A recent study performed by McCue et al.
Revealed that shoulder and elbow pathologies
associated with repetitive, high velocity, overhand
movements common to fly-casting [3].
Another research on Euler angle rotation,
which focused on satellite orientation, has been
done by Ramachandran. He determined two-axes
Euler rotation on satellite orientation that is
usually expressed in terms of Euler rotation about
the three principle axes. According to him, the
orientation is determined using the measurement
of at least two known vectors. The Euler rotations
are with respect to the three principal axes namely
x, y, and z. This is used in control of the satellite
where the known vectors are observed by a vector
sensor like star sensor. He has constructed the
algorithm from the mathematical derivation of
Euler rotation by knowing only one pair of
vectors and one angle. The input vector may be
from the sun-sensor or vector-magnetometer
while one axis rotation may be available from an
earth sensor. This algorithm enables the
knowledge of the three axes rotation to control the
satellite [5].
3
Mathematical Formulation
The first step to model a movement of a shoulder
is to construct direction cosines tables which then
used to construct the kinematic equation. The
direction cosine table for the shoulder is
constructed based on the Euler rotation series that
determine the orientation of a rigid body in threedimensional space. The rotation of the humerus
relative to a fixed scapula is considered as shown
in Figure 1.
Mustafa et al. studied a biologicallyinspired
anthropocentric
shoulder
joint
rehabilitator: workspace analysis & optimization.
They presented the design of a biologically
inspired anthropocentric 7-DOF wearable robotic
arm for the purpose of stroke rehabilitation. The
proposed arm rehabilitator utilizes the human arm
structure to combine with under deterministic
cable-driven parallel mechanisms so as to form
completely-deterministic structures. It adopted an
anthropocentric design concept, thereby offering
the advantages of being lightweight, having high
dexterity and conforming to the human
anatomical structure. The paper mainly focused
on the workspace analysis of the 3-DOF shoulder
module, with respect to the shoulder joint motion
range. Workspace parameterization is based on a
modified Z-Y-Z Euler Angles approach and
utilizing cylindrical coordinates to determine the
ISSN: 1790-5125
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TOPICS ON COMPUTATIONAL BIOLOGY AND CHEMISTRY
Figure 1: Euler rotation of the humerus (A)
relative to a fixed scapula (N).
Initially, the scapula reference frame N is
considered to be coincident with the right humeral
reference frame A, oriented such that â1 points
anteriorly,
points laterally and
points
inferiorly. This orientation is chosen because the
resulting Euler rotations will roughly correspond
to physiological rotation angles. Each rotation is a
simple rotation, and brings the N frame into an
intermediate orientation. Figure 1b shows the
scapula in its final orientation after the three
rotations have been performed. These three
rotations are described in sequential order through
the representation of direction cosines tables (i, ii,
iii) below.
(i)
Rotation about the common n̂ 1 â ′1 ,
axis.
Table 1. Direction cosines table for reference
frame N with respect to first orientation A'.
â ′1
â ′2
â ′3
1
0
0
n̂ 2
0
c1
-s1
n̂ 3
0
s1
c1
N R A′
n̂ 1
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TOPICS ON COMPUTATIONAL BIOLOGY AND CHEMISTRY
(ii)
Table 4. Direction cosines table for reference
frame of scapula (N) with respect to humerus (A).
Rotation about the common â ′2 â ′′2 ,
axis.
Table 2. Direction cosines table for orientation A'
with respect to second orientation A".
A′ R A′′
â ′′1
â ′′2
â′′3
c2
0
s2
â ′2
0
1
0
â ′3
-s2
0
c2
â ′1
4
(iii)
Rotation about the common â ′′ â 3 ,
3
axis.
â 1
â 2
-s3
0
â ′′2
s3
c3
0
â′′3
0
0
1
R A = ( N R A′ )( A′ R A′′ )( A′′ R A )
(1)
which yields the rotation matrix N R A and its
corresponding table of direction cosines, Table 4.
ISSN: 1790-5125
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â 3
n̂ 1
c2 c3
-c2c3
s2
n̂ 2
s1 s2c3 + c1c3
-s1s2s3 + c1c3
- s1 c2
n̂ 3
- c1 s2c3 + s1s3
c1s2s3 + s1c3
c1 c 2
Conclusions
[1] Allen, J. R. & Hahn, M. E., Casting for
Distance: Effect of Line Length on Kinematic
of
Fly
Casting,
www.asbweb.org/
conferences/2006/pdfs/66.pdf, 2006.
[2] Hung, G. K. & Pallis, J. M. Biomedical
Engineering Principles In Sports. New York:
Springer. 2004.
[3] McCue, T. J., Guse, C. E. & Dempsey, R. L.,
Upper Extremity Pain Seen with Fly-Casting
Technique: A Survey of Fly-Casting
Instructors. Wilderness and Environmental
Medicine, 15, 267-273, 2004.
[4] Mustafa, S. K., Yeo, S. H., Pham, C. B., Yang,
G. & Wei Lin, A Biologically-Inspired
Anthropocentric Shoulder Joint Rehabilitator:
Workspace Analysis & Optimization,
International Conference on Mechatronics &
Automation. 2005.
[5] Ramachandran, M. P., Two-axis euler rotation
determination, Applied Mathematics and
Computation, 175, No. 2, 2006, pp. 980-983.
[6] Yamaguchi, G. T. Dynamic Modelling of
Musculoskeletal
Motion:A
Vectorized
Approach for Biomechanical Analysis in
Three Dimensions. New York: Springer.
2001.
The table of direction cosines relating the scapula
(N) and humerus (A) reference frames is obtained
most simply via matrix multiplication,
N
â 2
â 3
c3
â ′′1
â 1
Direction cosines for virtually any compound
rotation can be found easily using Euler rotation
series. One needs only to define the sequence of
simple rotations comprising the compound
rotation and to perform the matrix multiplication
correctly.
References:
Table 3. Direction cosines table for orientation A"
with respect to humerus orientation A.
A′′ R A
N RA
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ISBN: 978-960-474-036-9