A MATHEMATICAL ANALYSIS OF THE MUSCULO-SKELETAL
SYSTEM OF THE HUMAN SHOULDER JOINT
by
Young-Pil Park, B. of Engr., M. S. in M. E.
A DISSERTATION
IN
MECHANICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Accepted
May, 1977
ACKNOWLEDGEMENTS
The author wishes to express grateful acknowledgement for the
devoted help of the committee members, Dr. Clarence A. Bell, Dr.
Donald J. Helmers of the Mechanical Engineering Department, Dr.
Mohamed M. Ayoub of the Industrial Engineering Department and Dr.
William G. Seliger of the School of Medicine.
Their guidance, sug-
gestions and consultations proved indispensable sources of inspiration.
Their limitless help and keen criticism helped in overcoming
the numerous difficulties that the author faced during this study.
Thanks are extended to Dr. James H. Strickland for his helpful
advice and constructive criticism in the final examination and to
Dr. James H. Lawrence, Jr., Chairman of the Mechanical Engineering
Department, for his encouragement and interest in this study.
thanks to Mrs. Sue Haynes for the typing of the manuscript.
n
And
ABSTRACT
The purpose of this study has been to formulate a mathematical
model capable of predicting muscular tension characteristics for muscles in the human shoulder joint.
This was done by using the data
that were collected through dissection of a cadaver and through physiological information about human skeletal muscles and anatomical
characteristics of the human shoulder joint.
By using this method,
the explicit characterization of the shoulder joint was described in
terms of a three dimensional coordinate system.
The mathematical
equations for the relationships between the electrical signal intensities that are generated from the muscles, and muscular tensions
that are exerted by muscles at various postures during abduction of
the upper extremity were investigated.
General equations that can
be applied to various individual persons who have different anthropometric dimensions were developed by using scale factors.
Computer programs were developed to determine the muscular tension in muscles in the shoulder joint of various persons and to predict the linear coefficients between electromyographic electrical
signal intensities and the muscular tensions of the skeletal muscles.
According to the results and the techniques of this study, it
was determined that most of the complicated human musculo-skeletal
systems can be analyzed mathematically without invasion of the
living body.
m
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
ii
ABSTRACT
iii
LIST OF TABLES
vi
LIST OF FIGURES
ix
I.
INTRODUCTION
1
1.1. Introduction
1
1.2. Review of Previous Work
5
Biomechanical Aspects of Muscle
II.
5
Electromyogram
10
Mathematical Analysis of Human Motion
12
Mechanism of Shoulder Joint
15
1.3. Purpose of Scope
17
CONCEPTUAL MODEL
20
2.1. Anatomical and Functional Aspects of the
Shoulder Joint
20
Skeletal System and Joints
20
Muscular System
24
Functional Aspect
26
Assumptions
34
2.2. Conceptual Model Postulated
Formulated Consideration
III.
36
42
MATHEMATICAL ANALYSIS
46
3.1. Anatomical Consideration
3.2. Equilibrium
46
48
IV
Page
IV.
V.
3.3. Minimal Principle
53
3.4. Solution Technique
58
EXPERIMENTAL PROCEDURE
61
4.1. Anthropometric Data Characteristics
62
4.2. E.M.G. Experiment
70
RESULTS OF THE THEORETICAL ANALYSIS
75
5.1. Functional Equations for the Muscular Tension
76
General Equation Form (Standard Coefficients)
84
A.
Abduction Case
84
B.
Adduction Case
85
5.2. Simplified Functional Equations for the
Muscular Tension
VI.
VII.
EXPERIMENTAL VERIFICATION OF THE THEORETICAL ANALYSIS ...
86
92
SUMMARY, CONCLUSION AND RECOMMENDATION
102
7.1. Summary
102
7.2. Conclusion
103
7.3. Recommendation
104
LIST OF REFERENCES
105
APPENDIX
111
Appendix (I). Anatomical Basic Data Table for Muscles ..
Appendix (II). Theoretical and Experimental ResultsThree Parts of Deltoid Muscle
Appendix (III). Coefficients of Theoretical Solution
of Muscular Tension
Appendix (IV). Documentation of Computer Program
112
127
133
154
LIST OF TABLES
Table
Page
4.1.
Anthropometric Characteristics of the Subject
64
4.2.
Anthropometirc Basic Data of the Subject
69
5.1.
Generalized Equation Result
91
6.1.
Linear Coefficient Values
100
6.2.
Statistical Results of Curve Fitting
101
1.1.
Anatomical Basic Data Table for Muscles Deltoid Anterior
112
Anatomical Basic Data Table for Muscles Deltoid Middle
113
Anatomical Basic Data Table for Muscles Deltoid Posterior
114
Anatomical Basic Data Table for Muscles Supraspinatus
115
Anatomical Basic Data Table for Muscles Infraspinatus
116
Anatomical Basic Data Table for Muscles Teres Major
117
1.7. Anatomical Basic Data Table for Muscles Teres Mi nor
118
1.8. Anatomical Basic Data Table for Muscles Subscapularis
119
1.2.
1.3.
1.4.
1.5.
1.6.
1.9.
1.10.
1.11.
1.12.
Anatomical Basic Data Table for Muscles Pectoralis Major (S)
120
Anatomical Basic Data Table for Muscles Pectoralis Major (C)
121
Anatomical Basic Data Table for Muscles Biceps (Long)
122
Anatomical Basic Data Table for Muscles Biceps (Short)
123
VI
vn
Page
1.13.
Anatomical Basic Data Table for Muscles Triceps
124
Anatomical Basic Data Table for Muscles Coracobi^achialis
125
Anatomical Basic Data Table for Muscles Latissimus Dorsi
126
Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (1)
127
Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (2)
128
2.3. Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (3)
129
1.14.
1.15.
2.1.
2.2.
2.4.
2.5.
2.6.
Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (4)
130
Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (5)
131
Theoretical and Experimental Results Three Parts of Deltoid Muscle - Subject (6)
132
3.1. Coefficients of Theoretical Solution of
Muscular Tension - Subject: Model, Weight:
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
0 Ibs
133
5 Ibs
134
Coefficients of Theoretical Solution of
Muscular Tension - Subject: Model, Weight: 10 Ibs
135
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (1), Weight:
0 Ibs
136
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (1), Weight:
5 Ibs
137
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (1), Weight:
10 Ibs
138
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (2), Weight:
0 Ibs
139
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (2), Weight:
5 Ibs
140
Coefficients of Theoretical Solution of
Muscular Tension - Subject: Model, Weight:
1
vm
Page
3.9.
3.10.
3.11.
3.12.
3.13.
3.14.
3.15.
3.16.
3.17.
3.18.
3.19.
3.20.
3.21.
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (2), Weight:
10 Ibs
141
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (3), Weight:
Q Ibs
142
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (3), Weight:
5 Ibs
143
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (3), Weight:
10 Ibs
144
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (4), Weight:
0 Ibs
145
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (4), Weight:
5 Ibs
146
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (4), Weight:
10 Ibs
147
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (5), Weight:
0 Ibs
148
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (5), Weight:
5 Ibs
149
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (5), Weight:
10 Ibs
150
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (6), Weight:
0 Ibs
151
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (6), Weight:
5 Ibs
152
Coefficients of Theoretical Solution of
Muscular Tension - Subject: (6), Weight:
10 Ibs
153
LIST OF FIGURES
Figure
Rage
1.1.
Process of Muscle Shortening
6
1.2.
Ideal Muscular Filament Contraction
8
1.3.
Mechanical Model of Muscle Performance
9
1.4.
Chaffin's Gross Body Action Model
14
2.1.
Bones and Joints of the Shoulder
22
2.2.
The Head of Humerus
23
2.3. Axes of Shoul der Movement
28
2.4.
Length-Tension Diagram of Skeletal Muscle
29
2.5.
Schematic Diagram of Shoulder Muscles Deltoid, Teres Minor, Teres Major, Supraspinatus
31
Schematic Diagram of Shoulder Muscles Infraspinatus, Subscapularia, Latissimus Dorsi, Biceps
32
2.6.
2.7.
Schematic Diagram of Shoulder Muscles Pectoralis Major, Coracobrachialis, Triceps
33
2.8.
Direction of Reaction Forces
35
2.9.
Conceptual Model
38
2.10.
Location of Shoulder Muscles (Anterior View)
39
2.11.
Location of Shoulder Muscles (Lateral View)
40
2.12.
Location of Shoulder Muscles (Posterior View)
41
2.13.
Changes of Insertion Point
42
3.1.
Change of Posture
46
3.2.
Schematic Diagram of Sensitivity Test
47
3.3.
Analyzing System
49
4.1.
Anthropometric Data
63
4.2.
Upper Extremity Model
65
IX
Figure
4.3.
Experimental Procedure
70
4.4.
Summation Area of E.M.G. Signal
72
5.1.
Muscular Tension Diagram - Subject (1)
77
5.2.
Muscular Tension Diagram - Subject (2)
78
5.3.
Muscular Tension Diagram - Subject (3)
79
5.4.
Muscular Tension Diagram - Subject (4)
80
5.5.
Muscular Tension Diagram - Subject (5)
81
5.6.
Muscular Tension Diagram - Subject (6)
82
6.1.
Muscular Tension vs. E.M.G. Intensity - Subject (1)
94
6.2.
Muscular Tension vs. E.M.G. Intensity - Subject (2)
95
6.3.
Muscular Tension vs. E.M.G. Intensity - Subject (3)
96
6.4.
Muscular Tension vs. E.M.G. Intensity - Subject (4)
97
6.5.
Muscular Tension vs. E.M.G. Intensity - Subject (5)
98
6.6.
Muscular Tension vs. E.M.G. Intensity - Subject (6)
99
CHAPTER I
INTRODUCTION
1.1. Introduction
In recent years, the interest in the engineering approach to enhance the effectiveness of human activities such as exercise, to develop
clinical techniques, and to improve safety in industry, has been considered very important.
Several methods of approach have been developed
to meet these problems more practically by the engineers who study biomechanics.
Even though the complexity of the human body with its nerves,
muscles and bones, which exist and coordinate to produce complicated
human activities, has been the subject of study by many researchers
ever since antiquity, no one could ever create any device which is
able to match the superiority and versatility of human activity to perform innumerable and profound activities.
With the development of more
sophisticated means of studying human activities modern man has been
able to apply scientific analysis methods to this study for the physical well-being of human beings.
Recently, many researchers have been applying human activity analysis techniques to many fields such as (1) industry, with the emphasis on
the effectiveness of work and on safety problems, (2) medicine and medical rehabilitation including the design of prosthetic devices, (3)
sports, particularly in the analysis of techniques, and (4) space research.
Previous investigations, directed toward mathematical and descriptive analysis of the developed tensile forces and electromyographic
electrical intensities of human muscles, can be divided into two classes
1
according to their methods of approach.
One group aimed at isolating
the muscles as much as possible, and was directed toward a physiological approach that would fit the mechanical and electrical phenomena of
muscle fibers, and chemical components of muscle cells.
This group of
research was started with an elementary consideration of the mechanical,
chemical and electrical theories of the muscular system and the nerve
system separately, and progressed towards an increasingly more complex
consideration of the integrated neuromuscular system.
Most of the
work in this group involved at least some theoretical analyses of the
mechanical characteristics of muscle fibers and the cell components,
and their relation to the nerve signals.
The best known physiological
descriptions resulting from this method of approach are the lengthtension relationship of human skeletal muscle, and the membrane potential theory of living cells.
The investigations of the next group were functional rather than
physiological, and sought to explain the external performance of human
beings, such as motion and effectiveness in controlling complicated
machines, and heavy and skilled work.
Here, the position of the sub-
ject and the magnitude of the subject's weight were usually considered
as the system inputs, and the electrical phenomena of muscles, kinetic
responses and fatigue characteristics were generally considered as the
system outputs.
The objective was to model the total task performance,
and the behavior of muscle entered only indirectly as a modifying function which indicated, in some cases, the quality of response.
The ex-
periments, however, usually represented only the qualitative data involving actions by muscle groups.
Recently, in addition to the qualitative characteristics of a
given task, the quantitative responses of muscle effort have been the
subjects of research using electromoyographic and mathematical techniques.
The quantitative methods of approach to muscular and electri-
cal responses of a given task are extremely valuable because they can
indicate the importance of the individual muscles and the magnitude of
the applied forces to muscle groups in an intact, nonnally operating
biological system.
Consequently, dissection is not needed.
However,
because of structural differences and differences in shape, these studies do not allow specific and functional elements to be unequivocally
localized or associated with specific anatomical structures.
The second group is more suitable for the investigation of human
performance research because the resulting analysis can be associated
with the actual behavior of the muscle instead of the properties of
muscle.
Therefore, this method of approach was adapted for this study.
In particular, the possibility that a mathematical description of
human skeletal muscles which cross the gleno-humeral joint can be developed by vector analysis and electromyographic studies, was also investigated in this study.
The idea of this investigation was that both
electromyographic and vector analyses of human skeletal muscle can be
combined to solve the complicated problems that are faced so many times
in human motion research.
The only reliable results which were obtained with the same motivation as this study, have been for clinical and medical purposes.
Con-
sequently, they considered only the qualitative analysis rather than
the quantitative analysis.
However, quantitative analysis is necessary
for the application of biomechanics to the study of the human body for
purposes such as physical training, artificial limb design and safety
problems.
Most of the former experiments were to verify the existence
of the electrical signals in a certain motion, or just to compare the
changes in the magnitude of the electrical signals which arise from
the changes of the effort of muscles.
This study involved the formulation of a musculo-skeletal
model of the human shoulder joint that can be described by mathematical
vector methods, using the data collected through dissection of cadavers
and physiological informations of human skeletal muscle.
Using this
model, the explicit characterization of the mathematical equation for
the postulated mechanism of the shoulder joint was formulated in order
to describe shoulder motion in terms of a three dimensional system,
without using any other methods.
The theoretical procedure of this
study was based upon the mathematical analysis of the shoulder muscles
and the analysis of anatomical and physiological characteristics of the
muscles.
The experimental procedure consisted of the recording of the
electromyographic signals of the shoulder muscles via surface electrodes
during application of external forces.
The work of this study also in-
cluded the analysis of the recorded results in order to formulate the
relationship between the electrical signals and muscular tensions that
are generated from the muscles at a certain motion or posture.
The ex-
ternal force was applied to the arm by means of weights that varied not
only in magnitude but also in the position of the weights.
1.2. Review of Previous VJork
The scientific research in the field of biomechanics, which is
the study of the structure and function of biological systems by means
of methods of mechanics, began with the growth of science.
Biomechanical Aspect of Muscles
The muscle itself has been an object of intense scientific interest for some time.
There is a general agreement that it is a very com-
plex biological system.
Its chemical, electrical and mechanical pro-
perties are still vague, and contraction, which is the essential physiological function of the muscle, remains as a perplexing phenomena.
In recent years, the research in contraction of muscle has been
almost entirely directed at the microscopic structure of muscular filaments with a tendency toward finer scrutiny at the submicroscopic level.
Outstanding research for the characteristics of the contraction mechanism of the skeletal muscle was conducted in the middle of the twentieth
century owing to the advanced electron microscope and its allied techniques.
In order to explain the mechanism of contraction, Huxley (1958,
1965, 1969) has determined, in his Sliding Theory, that during contraction two kinds of filaments in the voluntary muscle [thick elements
(myosin) and thin ones (actin)] slide past each other so as to produce
changes in the length of muscle.
1.1.
This effect is illustrated in Figure
After his hypothetical description of the contraction mechanism
of muscle, more detailed levels of research have been conduced by many
researchers.
This research has succeeded in verifying his theory.
And,
as a result, this concept was accepted and is now used for the understanding of the physiological phenomena of the contraction mechanism
of human skeletal muscle.
Whatever may be the molecular organization of the contractile elements in muscle, it is necessary to have a working description of the
overall mechanical behavior of muscle, in both the passive and stimulated states, if one is to understand its performance in the organism.
In fact, human muscle is very different from the solid materials with
which we are familiar in engineering fields.
Perhaps the most fundamental mechanical information concerning a
muscle is given by the length-tension relationship, in which the exerted tensile force is plotted against the length of muscle.
Tension-
elongation experiments, performed in large number in the past (Dubisson
and Monnier 1943; Bull 1945, 1946; Guth 1947; Wilkie 1958), were in the
nature of function versus shape studies.
However, by describing the
tension only in terms of change in length alone, these experiments
Sarcomere
Sarcomere
A band
A band
j
Myosin
Actin
i
«
Sarcoplasmic reticulum
Calcium
A.
At rest
Figure 1.1.
O
O
B. Contraction
Process of Muscle Shortening
1,
7
gave an incomplete picture of muscle behavior and contradicting results. Gutstein (1956) developed a generalized form of Hooke's Law
for muscle elasticity. He considered purely mechanical characteristics
of skeletal muscle without reference to the thermodynamic and myographic
properties of muscle. A direct determination of stress-strain relations
in skeletal muscle was studied by Nubar (1962-A).
In his paper he em-
ployed the concept of theoretical filaments which are in close contact
and continue from one end of the muscle to the other, as illustrated
in Figure 1.2. He considered muscle tissue to be a nonlinear material
which has a Hooke's Law property, and established a single mathematical
expression which is applicable to the passive as well as to the stimulated muscle. He related unit tension (stress) to unit change in the
length and the thickness of the human skeletal muscle (strain) as
follows,
^
= E^ (^)
+E2
(^)^
+ E3 (^)3
where
f is the filament tension,
w is cross section area,
E,, Ep and E^ are generalized Young's moduli,
L is initial length of the filament, and
dL is the elongatation of the filament.
(1.1)
8
Filamen
(length L, thickness w)
a.
DI
Muscle before stress
Figure 1.2.
Filament
(length L+dL,
thickness w')
.Plane of
Symmetry
•-—t
0
T
Muscle
after
stress
Ideal Muscular Filament Contraction
Numerous investigations of the total tension that a skeletal muscle
is capable of developing under isometric condition at various length
have been made in the past with somewhat conflicting results (Fenn
1938; Hill 1956, 1960; Bigland and Lippold 1954; Holubar 1969).
Most
of the researchers have tried to determine the relationship between
the exerted tension and biological factors such as arrangement, shape
of muscle fibers, electrical activity, and lengthening and shortening
velocities.
Although they could not determine the exact relationship,
they could predict that there must be some relationship between them.
Mechanical models of muscular actions have been the subject of the
investigations of many researchers for a long time (Levin 1927; Hill
1938).
These researchers applied the theoretical approach to living
î
e(t)
K,
^l
I
-^
•^mno pp-
f(t)
R
Resting
Length
Figure 1.3.
x(t)
Mechanical Model of Muscle
Performance.
subjects without using dissection.
Recently, Parnely and Sonmebloc
(1970) have developed a mechanical model of muscle that represents
most of the mechanical properties of skeletal muscle by using springs
and dashpots to simulate the elastic property of muscle.
They added
an ideal force generator to simulate the actively contractile part of
muscle, as can be seen in Figure 1.3.
In the figure, K, and K^ are springs and R is a dashpot.
"block box," labeled f
The
is an ideal force generator of the actively
contractile part of muscle.
Its output is proportional to the neural
input e(t) and the activity of the muscle's motor neuron.
Expenditure of energy in several simultaneous forms, such as mechanical, chemical and electrical, is associated with all muscular activities.
Based on principles of theoretical mechanics, some researchers
10
(Nubar 1962-B; Ayoub 1971; Petruno 1972; Park 1975) characterized some
described motion and discussed stresses at certain regions in the body
in order to provide fundamental understanding and to predict patterns
of significant characteristics of human motion.
Nubar and Contini (1961)
T
developed a minimal principle in order to solve the equations of the
theoretical mechanics that are, by themselves, incapable of determining
the unknown functions completely.
However, most of the problems of the biomechanical aspect of the
human skeletal muscle cannot be solved independently by purely mechanical methods.
This is because the muscle is a complicated system which
is associated with the structure of protein, the action of enzymes, and
the energy transfer in the biological system.
Most of these problems
are still not well understood and are vague and contradictory.
Electromyography
Electromyography, which involves the measurement of electrical
signals that are generated while the muscle is working, has become
accepted as a useful tool for investigating muscle actions.
It has
led to indirect determination of muscle participation at a particular
posture or at a certain movement without dissection, which is impossible
for a living human body.
The goal of the electromyographic research
was to examine and to establish the relationship between electromyographic intensities and the magnitude of the muscular tension in the
muscle.
The basic principles and the experimental procedure of electromography were explained by Basmajian (1967), and MaConaill and Basmajian
(1970).
They also investigated electromyographical properties of some
11
muscles involving important human movements.
Many investigators (Cooper
and Eccles 1930; Inman, et al 1952; Scheving and Panly 1952; Bearn 1954;
Zuniga and Simons 1969; Messier, et al 1971) have demonstrated that the
active force generated by the muscle contractile mechanism of a specific muscle depends on the level of neural activation.
By means of multi-
ple channels of electromyography, one can determine not only which specific muscle is in action, but also the extent to which it is participating
with other muscles in the performance of a certain movement.
Several experiments under several conditions have been reported
(Hill 1939; Lippold 1952; Bigland and Lippold 1954) demonstrating the
relationships between electromyograms and muscular tension, contracting
velocity, energy and the level of neural activation.
The major short-
coming of those results is that they always considered only the case of
maximally activated (tetanized) muscle contraction.
Recently, an experimental investigation of the relations among
force, velocity and electromography of partially activated human skeletal muscle was reported by Zahalak, et al (1976) for steady motion.
Also, there have been extensive studies on the electromyographic
signals from important muscles crossing several human joints such as
the knee, hip and shoulder joints (Inman 1947, 1952; Houtz and Walsh
1959; Keasy, et al 1966; Long and Brown 1964; Sutherland 1966).
A de-
tailed electromographic and morphological study of the shoulder joint
muscles has been made by Inman, Saunder and Abbott (1944).
However,
they did not investigate the role played by these muscles in supporting
either the shoulder girdle or the gleno-humeral joint during static and
dynamic loading of the limb.
A technique was developed by Cnockaert,
12
et al (1975) to calculate the torque generated by the individual muscles
that contribute to the isometric flexion of the elbow by using integrated and rectified surface electromyographic sugnal intensities.
By
using a surface stimulation technique, the dynamic characteristics of
the human skeletal muscle model was investigated by Tennant (1971).
He
developed a mathematical joint model of the behavior of the muscle group
comprised of the biceps and brachialis, by taking into account the
changes in muscle tension due to the inertia of the moving masses under
various surface stimulation conditions.
Since the type of measurement and the method of coUecting data in
electromyographic studies inevitably involve considerable variation and
uncertainty, the generalized results, which are necessary for analyzing
human motion, have not to date been established unequivocally.
Mathematical Analysis of Human Motion
The main ultimate objective of the study of biomechanics is to increase the efficiency of human performance by minimizing the effort required to perform the motor activities.
Even in the Renaissance period of the sixteenth and seventeenth centuries, such men as Leonardo da Vinci (1500) and Borelli (1685) began
to apply scientific principles to the study of human motion.
Beginning
in the nineteenth century human motion analysis, in general, has been
carried out either experimentally or theoretically as the result of
the progress of scientific research techniques (Sherrington 1893;
Braune and Fischer 1889; Ducheme 1867).
The kinematic and experimental
analysis techniques of obtaining motion characteristics by using the
physical records obtained from a motion, are widely used in current
13
research in kinesiology (Pearson. et al 1963; Dempster 1955; Engen and
Spencer 1968; Karas and Stapleton 1967; Bouisset and Pertuzon 1967).
Generally, the techniques consist of the recording of human motion
by high speed cinematography.
With this tool the successive positions
of the joints of the body and the orientation of its segments are plotted as a function of time.
These films furnish velocities and accelera-
tions which are used to evaluate, with the anthropometric data of the
segments, the forces and moments at the joints, and the kinetic and
potential energies of the segments.
The work which was done by the mus-
cle is obtained from the variation of these energies or, equally, from
the products of the moments by the rate of the joint rotations, integrated with respect to time.
Kinematic analysis is also used for the
study of cycling, cranking, walking and lifting problems under specific
conditions and speeds.
To describe the design and the application of a spatial motion
for a detailed study of the relative motions of body segments, many
researchers (Passerello and Huston 1971; Van Sickle and Harvey 1972;
Jensen and Bellow 1976; Kinzel, et al 1972; Yagoda 1974) employed the
idea of a spatial linkage which is capable of measuring biomechanical
motion.
By using this linkage system analysis method, Chaffin (1969)
developed a computer model which treats the human body as a series of
seven links articulated at the ankle, knee, shoulder, hip and wrist for
a certain gross body action as can be seen in Figure 1.4.
The model
was specially designed to investigate body movements that occur during
the lifting and carrying of weights.
However, in most of this research,
several constraints and assumptions,
that were employed in the devel-
opment of the linkage model, limit their application in practice.
14
A : ankle
K : knee
H : hip
S : shoulder
E : elbow
W : wrist
Figure 1.4.
Chaffin's Gross Body Action Model
A comprehensive, statical and dynamical analysis of human body
motion requires a set of governing equations applicable to a wide
variety of stiuations.
A principal source of difficulty in developing
such equations is the complex geometry due to the shape of the body
with its abundant possible motions.
Because of these reasons, a rela-
tively small number of researchers (MacLeish and Charley 1964; Merchant
1965; Morecki 1966; Thomas 1968; Troup and Chapman 1969; Choa, et al
1976) made an attempt to develop a mathematical model of the musculoskeletal system.
Most of the contributions apparently have been in
the area of electromyographic recording of muscular electrical signals
to explain the participation ratio of the various muscles, to observe
the types of movement, or to calculate the joint forces by means of
conventional mathematics.
Recently, a mathematical model for the
evaluation of the forces of the musulo-skeletal system in the lower
extremities was developed by Seireg and Arvikar (1973, 1975).
These
15
papers contributed greatly in advancing the study to develop a mathematical model of the musculo-skeletal system of the human body which
is capable of evaluating muscle forces and joint reactions at different
static postures.
Mechanism of Shoulder Joint
The shoulder, which is the proximal joint of the upper limb, is the
most mobile joint in the human body.
Although the shoulder joint is
the one most commonly used in human activities, it is surprising that
it has been the subject of only a few studies.
Furthermore, most of
the previous investigations have dealt with the magnitude of the forces
in terms of electromyographic and kinematic analyses of its movement.
The reason for this is that the mechanism of shoulder movement is much
more complicated than that of any other joint in the human body.
The
general mechanism of the shoulder joint movement has been studied by
Dempster (1965), who used living subjects and ligament preparations
of cadaver material.
He treated the shoulder joint as a complex com-
bination of three distinct joints, the sternoclavicular, claviscapular
and glenohumeral joints.
Each joint was discussed functionally both
in terms of its range of movement and in terms of the action of associated ligaments in restraining movement.
To determine the biomechani-
cal performance which are decisive for the determination of the magnitude of the applied actual moment of force developed by individual muscles coordinating in the upper extremities, and to establish the degrees
of their participation in the given movement, Fidelus (1967) found out
the relationship between exerted tension and the length of muscles.
Using a mirror and 35-mm motion picture camera, Engen and Spencer (1968)
16
developed two techniques for manual analysis of computer processing of
shoulder motion.
In their paper, accurate diagrams of upper extremity
movement were made from photographs of a normal person.
The points in
the diagram were connected to identify the patterns of movement, and
the angular velocity and acceleration of the points.
As mentioned earlier in this chapter, Inman, Saunder and Abbott
(1944) investigated shoulder movement in several ways, such as comparative anatomy and roentgenographic analysis of the motion.
The theore-
tical force required in shoulder motion and the action current potential
were derived from the living muscles in motion.
From the data so ob-
tained, they attempted to resynthesize the whole shoulder motion.
After
measuring the precise relationships of the body parts among each other,
and the relative positons which they occupy during a motion, they were
able to set up the equations and to calculate the force requirement
for the maintenance of the upper extremities during flexion and abduction in terms of electrical potentials.
The functions of individual muscles associated with the shoulder
joint were studied by several researchers (Basmajian and Latif 1957;
Wright 1962; Shevlin and Lucci 1969) to determine the role of a specific
muscle for a given motion such as swimming, golf, climbing and the swing
of the arm.
An interesting method of the kinematic analysis of the motion of the
shoulder, arm and the hand complex was first investigated by Taylor and
Blaschke (1945).
In their paper, in order to analyze the axes and angles
of the idealized kinematic system, several steps were involved.
These
steps were the measurement of anthropometric data, fitting the subject
17
with visual landmarks taking cinematographic pictures of the subject
performing the activities under study, and using the cartesian coordinates of visual landmarks.
Recently, de Duca and Forrest (1973) developed a technique for
calculating the forces generated by the individual muscles which contribute to isometric abduction of the upper limb in the coronal plane
when the humerus is rotated medially.
Also, mathematical relation-
ships of the forces of the individual muscle were obtained as a function of respective effect, and that of physiological cross sectional
areas.
Park (1975) analyzed the forces in every muscle crossing the
gleno-humeral joint.
He analyzed the shoulder joint by using mathe-
matical methods and vector methods based upon anatomical and physiological characteristics.
Also, anatomical and physiological proper-
ties of the shoulder joint, and the muscles associated with the joint
movements, were analyzed functionally.
The application of these methods
to other joints in the human body was also discussed.
1.3. Purpose and Scope
The principal objectives of this investigation were:
(1) to seek
a mathematical and descriptive analysis of electromyographic characteristics and muscular tensile force distribution of human skeletal muscles crossing the gleno-humeral joint, and (2) to gain a better understanding of the actual neuro-muscular activities of human skeletal muscles and their actual mechanism.
The mathematical and descriptive equa-
tions for the relationships between the electrical signal intensities
that are generated from the muscles, and the muscular tension that is
exerted by muscles at various postures during abduction and adduction
of the upper extremities, were also investigated.
This study involved the formulation of a musculo-skeletal model
of the human gleno-humeral joint that can be described by mathematical vector methods. This was done by using the data that were collected through dissections of cadavers and through physiological information about human skeletal muscles, and anatomical characteristics
of the shoulder joint. By using this model, the explicit characterization of the mathematical equations for the postulated mechanism
of the gleno-humeral joint was formulated. Shoulder joint motion was
described in terms of a three dimensional coordinate system. This
research consisted of theoretical and experimental parts. The theoretical part of this study consisted of formulation of the model. The
experimental part consisted of the recording and analyzing of the
<
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electromyograms of the shoulder surface muscles (deltoid anterior,
T
middle and posterior parts) via surface electrodes during the application of external loads. The external loads were applied to the upper
;
extremities by means of weights that varied not only in magnitude but
î
also in the position of the weights which, in turn, depended upon the
l
angles of abduction and adduction. The methods used for determining
the unknown functions and the parameters entering the equations were
adapted from the experimental results of other researches. Also, the
validity of the minimal principle as applied to human skeletal muscles
in the static case was investigated by using the theoretical and experimental results.
General equations that can be applied to different persons, who
have different anthropometric dimensions, were also developed.
Com-
19
puter programs were developed to determine the muscular tension in
the muscles crossing the gleno-humeral joint of different persons
and to predict the linear coefficient between electromyographic electrical signal intensities and the muscular tension of the skeletal
muscles.
These were developed from the results of the theoretical
and experimental procedures. According to the results and the techniques of this study, it was concluded that most of the complicated
human musculo-skeletal joints can be analyzed mathematically without
dissecting bodies.
n
r
CHAPTER II
CONCEPTUAL MODEL
The shoulder joint provides man with a unique mechanism to interact with his environment.
In comparison with other joints in the human
body, it is endowed with nearly limitless positioning ability to suit
the requirements of the environment, and the ability of adjusting the
environment to suit the requirements of the body.
The fact that this
mechanism is attached to the upper-lateral part of the trunk segment
illustrates its critical positioning in the body to gain the best advantage for building on the accumulated movement of the lower limb and
the trunk.
In addition, its position with respect to the head allows
for a visual, sighting and aiming control which cannot be duplicated
elsewhere in the body.
The shoulder complex is an integrated portion
of the upper limb, and this joint is the place where the most important motions can occur between trunk and arm.
2.1. Anatomical and Functional Aspects of the Shoulder Joint
Over the past hundred years, the anatomical and functional aspects
of the shoulder joint have been explored in detail because of its important role in human activities.
In this section, a simplified con-
cept, based on the anatomy of human body, was employed to clarify
the functions of the shoulder motions.
Skeletal System and Joints
For the purpose of studying the skeletal structure of the shoulder
joint, the joint can be divided into several segments according to
their functional purposes.
As can be seen in Figure 2.1. a pair of
20
21
clavicle and scapular bones join with the sternum at its superior edges
to form the shoulder girdle.
The shoulder motions are related with the
relative motions of these bones and the humerus bones of the upper arm.
Their functional and morphological characteristics are as follows:
(1)
Clavicle: The clavicle is shaped like an elongated S, extended
from the sternum to the acromion.
At its medial end, it artic-
ulates with the sternum and the first rib.
This articular sur-
face, which is about 2.5cm in diameter, is the only bony attachment between trunk and the upper extremity.
(2)
Scapula:
The scapula is a flat, triangular bone overlying the
upper portion of the back.
It is the site of the attachment
of the superficial muscles of the back.
Its glenoid fossa,
at the lateral angle of the scapula, is modified to articulate with the head of the humerus, and is the only joint between scapula and humerus.
(3)
Humerus:
The humerus, the bone of the upper arm, is articu-
lated with the scapula at the glenoid fossa, and with the
radius and ulna at the elbow joint.
The most important mo-
tion of the upper extremity occurs at this bone.
(4)
Sternum:
The sternum consists of three segments, which are
the manubrium, body and xiphoid process.
At the side of the
sternum, articular facets are present for the clavicle at
the upper end of the manubrium, and for the upper six ribs
along the length of the sternum.
22
The clavicle, scapula and humerus form a smoothly coordinated
system in which each bone has been allotted a share in shoulder motion.
This reflects the synergic action of the muscles that act upon
the bones.
The union between the glenoid fossa of the scapula and the head
of humerus is an example of a ball and socket type joint.
In contrast
to the hip joint, the shoulder joint sacrifices its stability for a remarkable degree of mobility.
As can be seen in Figure 2.2, the ana-
tomical axis of the shaft of the humerus forms an angle with the true
axis of the flexion and extension of about 130 degrees.
r.
X
>
I
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i
>
íl
Bones
A
B
C
D
1. Gleno-Humeral
Scapula
Humerus
Clavicle
Sternum
Figure 2.1.
2.
Sub-Deltoid
3. Scapulo-Thoracic
4.
Acromio-Clavicular
5. Sterno-Clavicular
Bones and Joints of the Shoulder
23
/ ^
^0130
/45^
'
1
Figure 2.2.
The Head of Humerus
Although the shoulder girdle consists of several joints of four
bones as can be seen in Figure 2.1, according to their anatomical positions and functional roles, we can divide them into five joints as
follows:
(1) Gleno-Humeral Joint:
This is a diarthrodial joint, anatomi-
cally, and the articular surfaces consist of hyaline cartilage
It is the most important joint in the shoulder mechanism;
therefore, in this study, only this joint was considered.
(2)
Sub-Deltoid Joint:
This is an amphiarthrodial joint.
How-
ever, it is mechanically linked to the gleno-humeral joint
because any movement in the latter brings about slight movement in the former.
24
(3)
Scapulo-Thoracic Joint:
This is an amphiarthodial joint
which does not produce any significant motion, but slight
relative motion occurs between trunk and scapula at this
joint.
(4) Acromio-Calvicular Joint:
This is a diarthrodial joint
located at the acromial end of the clavicle.
Even though
it is a diarthrodial joint, it does not produce any important motion during shoulder motions.
Therefore, it can be
regarded as a amphiarthrodial joint.
(5)
Sterno-Clavicular Joint:
This is also a diarthrodial joint,
located at the sterno end of clavicle.
But, as in the case
of the acromio-clavicular joint, it does not produce any
significant motion.
The motions of the shoulder girdle occur at all of the five joints
simultaneously, each contributing its share to the accomplishment of
the movement.
To maintain the rythmn of smooth and coordinated motions,
the shoulder requires that all the five intact joints and all the proper
forces in the muscles move the bones.
However, rectangular abduction,
which is the subject of this study, takes place mostly at the glenohumeral joint; therefore, the motion of the other joints were neglected
in this study.
Muscular System
The bones and joints of the human body are not committed to any
strictly predictable pattern of motion, so they permit an infinite
variety of motions.
Muscles can momentarily constrain a joint mechanism
25
to hold body segments in static positions, or to elicit a motion by
changing forces in one direction while eliminating freedom of the
system in the other direction.
Innumerable postures and various
patterns of motion are possible for human beings but all of them involve some degree of muscular constraint at the joint.
The muscles which act upon the mechanism of shoulder motion can
be divided into four anatomical groups as follows:
(1) Gleno-Humeral Muscles:
Those passing from the scapula
to the humerus.
(2) Axio-Humeral Muscles:
Those passing from the trunk to
the humerus.
(3) Axio-Scapular Muscles:
Those passing from the trunk to the
scapula.
(4) Others:
Those passing from the scapula to
the ulna or radius of the lower arm.
The muscles belonging to these groups can be tablulated as follows:
(i) Gleno-Humeral Group:
(1)
Supraspinatus
(2)
(4)
Coracobrachialis
(5) Teres Minor
(6) Teres Major
(7)
Deltoid
(b) middle
(c)
(a)
(ii)
anterior
Infraspinatus
(3)
Subscapularis
posterior
Axio-Humeral Group:
(1)
Pectoralis Major - (a)
(2)
Latissimus Dorsi
sterno
(b) clavicle
26
(iii)
(iv)
Axio-Scapular Group:
(1)
Trapezius
(2) Serratus Anterior
(4)
Levator Scapula
(5)
Pectoralis Minor
(a)
long head
(3) Rhomboids
Other Group:
(1) Biceps (2)
(b) short head
Triceps
Functional Aspect
The movements of the shoulder joint can be divided into two main
types:
the major movement and the minor movement.
The major movement
and the minor movement refer to the humeral and scapular movement,
respectively.
The humeral movement can be regarded as a combination
of abduction, adduction, medial rotation, lateral rotation, flexion
and extension of the forearm.
The scapular movement can be regarded
as the combination of forward movement, backward movement, upward movement, downward movement, and the rotation of the scapula.
The scapu-
lar movement can be done with the help of the flexion of the spinal
>
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i
cord.
>
Therefore, the gleno-humeral, axio-humeral, and other groups of
muscles, as described in the former section, influence the humeral
movement, and the axio-scapular group of muscles influences the scapular movement, respectively.
In recent years, studies of the shoulder joint motions proved that
the complete elevation of the arm, in either the coronal or the frontal
plane, is the combination of the free motions of all the joints of the
shoulder complex.
Although the concept may be incorrect that rectangu-
lar abduction takes place entirely at the gleno-humeral joint, and that
27
full elevation is completed by the motion of the scapula on the chest
wall, the contribution of the scapular movement to the whole movement of the shoulder is small enough to neglect in the region from
0 to 90 degrees.
Since the problem of abduction from 0 to only 90
degrees was considered in this study, the scapular movement was neglected.
Once the muscle origins and insertions had been identified from
pictures of a dissected cadaver, the next task was to obtain their
proper coordinates with respect to a suitable spatial set of coordinateaxes.
This knowledge was, of course, necessary for the mathematical
analysis.
The shoulder joint has three degrees of freedom which al-
low the upper limb movements with respect to three planes in space.
A brief description of the axes follows:
(1)
Transverse Axis:
q
Lying in a f r o n t a l plane, i t controls the
movement of f l e x i o n and extention in a s i g i t t a l plane.
(2)
Anterior-Posterior Axis:
Lying in a s a g i t t a l plane, i t
controls the movement of abduction (the upper limb moves
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>
away from the body) and adduction (the upper limb moves
toward the body) which are performed in a frontal plane.
(3) v'ertical Axis: Lyirig through the intersection of the sagittal and frontal planes, it corresponds to the third axis in
space.
It controls the movements of flexion and extension
performed in a horizontal plane while the arm is abducted
to 90 degrees.
(4)
Longitudinal Axis of the Humerus:
This controls the move-
ments of lateral and medial rotation of the arm.
J
28
The position of the reference line was selected as the line of the
upper arm hanging vertically at the side of the trunk.
These axes
are shown in Figure 2.3.
In this study, only abduction and adduction movements of the
upper limb were considered; therefore, the fourth axis was not
considered.
R
1.
2.
i
Transverse
Anterior-Posterior
i
n
3.
4.
Figure 2.3.
Vertical
Longitudinal
I
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5
Axes of Shoulder Movement
As mentioned in the discussion of the muscle model, if there is
no electrical signal present during a motion, the muscle does not exert
any active force and, there is only passive force in the muscle.
But,
according to the length-tension diagram of Ramsey and Street (1940),
for a skeletal muscle the passive force is so small that it can be
29
neglected compared to the active force.
According to their paper,
as shown in Figure 2.4, the elastic component force is only 2 percent
of the total force at 150 percent of resting length.
Even at 200 per-
cent of the resting length, it is only 47 percent of total force.
The shoulder joint muscles can be divided into three groups according to their physiological actions.
Each group consists of the
abduction muscles, adduction muscles and cuff muscles.
The abduction
— — — — —
Total
Rela tiv e Tension
to
rassive
X
/
^y
*:—-•
/
^^^^I
r
••
100
200
Percent of Resting Length
Figure 2.4.
Length-Tension Diagram of Skeletal Muscle
muscles and adduction muscles exert forces during only abduction and
adduction, respectively, and the cuff muscles exert forces during both
abduction and adduction.
Because of the small shoulder movement assumed in this study,
only 15 muscles are considered.
During the abduction from 0 to 90
degrees, there are electrical signals only in the following muscles:
r
;D
>
30
(1) Supraspinatus
(2) Deltoid Anterior
(3) Deltoid Middle
(4) Deltoid Posterior
(5) Infraspinatus
(6) Teres Major
(7) Teres Minor
(8) Subscapularis,
and during the adduction, there are electrical signals in the following
muscles:
(1) Pectoralis Major Sternal Part
(2) Pectoralis Major Clavicular Part
(3) Latissimus Dorsi
(4) Biceps Long
(5) Biceps Short
(6) Triceps
(7) Coracobrachialis
(8) Infraspinatus
(10) Teres Minor
(9) Teres Major
q
(11) Subscapularis
J
(0
Consequently, following the classification system previously discussed,
H
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T
muscles were divided into three functional groups as follows:
"^
i
(1) Abduction Muscles: Supraspinatus, Deltoid Anterior,
Deltoid Middle, Deltoid Posterior,
(2) Adduction Muscles: Pectoralis Major Sternal Part,
Pectoralis Major Clavicular Part,
Latissimus Dorsi, Biceps Long,
Biceps Short, Coracobrachialis,
Triceps,
(3) Cuff Muscles:
Infraspinatus, Subscapularis,
Teres Minor, Teres Major.
>
^
31
A : Anterior
B : Middle
C : Posterior
Teres Minor
Deltoid
r
X
>
n
I
r
S
>
Teres Major
Figure 2.5.
Supraspinatus
Schematic Diagram of Shoulder Muscles
32
Infraspinatus
Subscapularis
X
>
A : Long
B : Short
n
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i
>
Latissimus Dorsi
Figure 2.6.
Biceps
Schematic Diagram of Shoulder Muscles
33
Clacular
Sternal
Pectoralis Major
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IQ
>
Triceps
Figure 2.7.
Coracobrachialis
Schematic Diagram of Shoulder Muscles
34
According to the above analysis, it can be said that during abduction all of the muscles considered, except the adduction muscles, exert
forces and during adduction all of the muscles considered, except the
abduction muscles, exert forces.
However, cuff muscles exert force
during both abduction and adduction.
Assumptions
In this study, in order to develop a suitable mathematical model
for the musculo-skeletal system of the shoulder joint, several assumptions are needed.
(1)
These are listed below:
It was assumed that the stability of the skeletal structure
r
in any posture is maintained by the static equilibrium of
muscular entsion and reaction force at the joint.
(2) Muscles were assumed to have distinctive origin and insertion points and the tensile forces which are produced by
muscles were assumed to be directed along the lines joining
the origin and insertion points.
This assumption resulted
q
r
^
in
^
5
u.
C
in considerable difficulty in constructing the model because
^
the muscles have innumerable shapes and do not originate or
^
insert in a straight line fashion.
However, the origin and
insertion points were chosen and the lines of force were
drawn judiciously as possible to represent the model.
The
origin and insertion points chosen and force lines drawn for
each muscle can be seen in Figures 2.5 to 2.7.
35
(3)
It was assumed that the only bone in the shoulder mechanism
to have movement was the humerus.
This is because only the
relatively small amounts of abduction and adduction were
considered in this study.
Consideration of scapular move-
ment would improve the model somewhat, but it was neglected
because it would result in considerable complication in the
analysis of the problem.
However, in the case of abduction
over 90 degrees, the participation of the scapular movement
would become large and probably could not be neglected.
=1
Ul
n
0 : Center of Rotation
R : Reaction Force
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>
Figure 2.8.
Direction of Reaction Force
36
(4) The head of the humerus was assumed to be round as illustrated in Figure 2.8. Also, it was assumed that there was
no friction between the glenoid fossa and the head of the
humerus. With these assumptions the resultant reaction
force would go through the center of rotation.
(5) It was assumed that the center of rotation of the upper
limb was a stationary point which could be determined by
dissection films. However, it is known that the center of
rotation does displace slightly during the motion. As a
result, the sensitivity of the result of this analysis to
slight changes in the location of the center of rotation
was considered by referring to a previous study by Park
(1975). The study was conducted by performing a number of
r.
analyses with different centers of rotation.
>
(6) The whole set of the upper extremity, upper arm, lower arm
and hand were assumed to be a single rigid link.
X
tn
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ti
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2.2 The Conceptual Model Postulated
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>
In addition to the usual problems facing scientists and engineers
when they attempt to model a man-made system which is qualitatively
known (that is,'when all the parts comprising the mechanism to be
modeled are identifiable and their functions understood), in this
research. it was necessary to be content with a system whose operation
at present is not completely known. The actual mechanism by which a
shoulder can move is not fully understood and even less understood
are functional characteristics and logics that activate the systems
of nerves and muscles of the shoulder joint.
^
37
The observation of force and moment equilibrium of the human body
at a certain particular posture suggests that a muscle can be modeled
as a force generator producing a certain force vector which produces
a movement.
This model is the classical one that was used by the
author (1975) for a former study.
The force distribution used in the
model was not unique and, in fact, some assumptions were used to solve
the indeterminate problem.
Also, the model required complicated ana-
tomical data for the mathematical analysis.
However, this method of
approach is quite useful and general equations of motion can be developed by using the method.
Therefore, this technique was adapted in the
theoretical part of this study for the mathematical analysis of the
shoulder joint and it was also used for the formulation of the relationships between individual muscles.
R
The model for the mechanical and mathematical analysis of the
>
shoulder joint muscles, and the kinetic behavior of the muscles, is
H
briefly described below.
I
r
The model consisted of four types of basic
elements, as follow:
i
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>
(1) A force generator (muscle) whose output depends on the excitation from nerves, the length and the anatomical position
of the muscle itself, and the applied weight.
(2) An electrical signal (nerve) whose output produces muscular
tension.
This output was recorded by using surface elec-
trodes.
(3) A working media (body segment) whose output represents the
work done by the muscles.
Sometimes the weight of the body
segment was considered as an applied weight.
^
38
F
s
\
/
M//
/x/
x"
Figure 2.9.
Conceptual Model
R
(4) A weight (applied force) that controls the magnitude of
muscular tension and electrical signal intensity.
tn
fi
n
The elements of the postulated model are illustrated in Figure 2.9.
r
i
In this schematic representation of the shoulder mechanism, F represents the output of the force generator (muscle), S (which is recorded
by the electromyogram) represents the stimulating electrical signal
(nerve signal) from the nerve system, and W and B represent working
load (weight) and body segment (arm), respectively.
It should be
noted that there is a relationship between developed muscular tensile
forces and the lengths, directions, locations of origin and insertion
points, and thickness of individual muscles.
For the gross human shoulder joint, all the muscular tensile force
vectors were drawn for three directional views as can be seen in Figures
2.10 to 2.12.
>
39
>
t-:
r;
n
tD
;Q
>
1. Supraspinatus
2. Corachobrachialis
3. Pectoralis Major-Sternal
4. Pectoralis Major-Clavicular
5. Biceps-Long
6. Biceps-Short
7. Subscapularia
Figure 2.10.
Location of Shoulder Muscles (Anterior View)
40
tn
n
I
tD
;Q
>
8. Deltoid-Posterior
9. Deltoid-Middle
10.
Deltoid-Anterior
11. Triceps
Figure 2.11.
Location of Shoulder Muscles (Lateral View)
41
fr,
til
íl
n
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r
;Q
;Q
12. Latissimus Dorsi
13. Infraspinatus
14. Teres Minor
15. Teres Major
Figure 2.12.
Location of Shoulder Muscles (Posterior View)
42
Center of
Rotation
Origin
Insertion
tn
fi
n
I
Figure 2.13.
Change of Insertion Point
01
From the anatomical data the lengths, direction cosines, and
moment arms of e\/ery muscle were calculated by vector methods at ewery
rotational position from 0 to 90 degrees at 10 degree intervals.
This
situation is illustrated in Figure 2.13 for the pectoralis major clavicular muscle.
Formulated Equations
In this section the general forms of the equations for the postulated model of the shoulder joint are given briefly.
The purpose of
43
this brief description is to provide insight and guidance to the more
detailed description of the theoretical study and experiments to be
discussed later.
The objective general equation characterizing the total actual
mechanism of the shoulder joint can be represented symbolically in
the following form:
M
=
f^ (T., E., W, B., A, e)
M
is the total motion characteristic of the shoulder joint,
(2.1)
where
is muscular tension in a particular muscle,
^i
is electrical signal intensity of a muscle,
w
is applied weight,
B.
is an anthropometric factor of a segment,
A
is an antomical and physiological factor,
e
is an abduction or adduction angle, and
f.
(and any subscripted f) represents a functional relation-
1
ship.
Equation (2.1) expresses the fact that the whole motion of the shoulder
is a function of tension in the muscle, stimulation intensity, applied
weight, anthropometric factor, anatomical and physiological factors
and abduction and adduction angles of the muscles.
In the theoretical part of this study, the following generalized
equation was investigated for the mathematical vector representation of
a single muscular tension.
tn
n
I
r
s
>
íl
44
T.
=
f^ (L., D., W, B.,A, e ) .
(2.2)
where L^ and D^. refer to the length of the muscle and the direction of
its force application, respectively.
In this study D'Alembert equili-
brium equations for forces and moments were used and, in order to make
the statically indeterminate problem into a statically determinate one,
the Minimal Principle theory of Nubar and Contini (1961) was adopted
as follows for the whole shoulder joint.
-^ic.H,') dt + A
(2.3)
where
s a numerical constant,
M.
s a muscular moment of a muscle,
n
dt i s a time interval,
tn
'^o ' s an initial constant of the motion and
E
is the total muscular effort which is reduced to a
fí
n
minimal by the imposed minimal principle.
5
I
r
;Q
The purpose of the experimental part of this study was to find out
the relationships between the myographically recorded electrical signal
intensities generated by the muscles and the biomechanical factors of
the subject.
These relationships were represented by the following
equation:
E.
=
f^ (L., D., W, B., A, e)
(2.4)
Finally, from the Equations (2.2) and (2.4), the following general
equation for the relationship between the tension and electrical signal
;Q
45
intensity of the muscle formulated.
T^
= ^4 (E^.)
(2.5)
Because the experimental work was restricted to the use of surface
electrodes, electromyograms could only be obtained for surface glenohumeral muscles (deltoid anterior, dpltoid middle and deltoid posterior).
A detailed description and explanation of the equations is provided
in the following chapters.
n
I
r
S
;Q
>
:<
CHAPTER III
MATHEMATICAL ANALYSIS
3.1. Anatomical Considerations
As discussed earlier, muscles were assumed to have distinct origin
and insertion points and the tensile forces which are generated by muscles were assumed to be directed along the lines joining the origin
and insertion points.
After finding the insertion and origin points
in the reference position, the length, direction consines and moment
arms (which are necessary information for calculating mathematical solutions for every muscle) were calculated by vector methods at every position from 0 to 90 degrees in 10 degree intervals, as shown in Figure
3.1.
tfl
n
I
r
5
XI
Figure 3.1.
Change of Posture
46
47
For each of the muscles, all the information was calculated and is
tabulated in Appendix I, Tables 1.1 to 1.15.
The author (1975) calculated the effect of the change of the center of rotation on muscle force distribution by using the various positions for the center of rotation as described below:
(1) The center of rotation moved down 0.5cm,
(2) The center of rotation moved up 0.5cm,
(3) The center of rotation moved medially 0.5cm,
(4) The center of rotation moved laterally 0.5cm,
^
^
tn
n
I
r
i
>
XI
Figure 3.2. Schematic Diagram of Sensitivity Test
48
from the dissection center which was found from the films of the dissected cadaver.
3.2.
The directions of the variations are shown in Figure
The author found that the effects of these variations were small
enough to be neglected.
3.2. Equilibrium
For a system to be in equilibrium, the sum of the external and
internal forces and moments must be zero.
or static.
Equilibrium can be dynamic
However, for the case under consideration, the system con-
dition was static equilibrium because the skeletal structure was assumed to be static and the inertia forces and moments associated with
the motion of the system did not appear.
This implies that the sum
of the moments and the forces about the three space axes must be equal
to zero.
In order to analyze equilibrium, the free body diagrams of
each of the bones associated with the shoulder movement were considered
tfl
The equations of equilibrium were applied to the three reactinal for-
n
I
r
S
ces in X, Y and Z directions at the joint for each of the following
muscles:
XI
(1
Spupraspinatus
(2
Deltoid Anterior
(3
Deltoid Middle
(4
Deltoid
(5
Infraspinatus
(6
Teres Minor
(7
Teres Major
(8
Subscapularis
(9
Pectoralis Major-Sternal Part
Posterior
XI
49
^
J X ' L'^i
Figure 3.3.
Analyzing System
R
(10)
Pectoralis Major-Clavicular Part
tfl
(11)
Latissimus Dorsi
Sl
n
I
r
(12) Biceps Long
S
(13) Biceps Short
X]
>
(14)
Triceps
(15)
Corachobrachialis.
XI
As discussed earlier, for the abduction case eight muscles (muscles 1 to 8) produce muscular tension and, for the adduction case, 11
muscles (muscles 5 to 15) produce muscular tension.
Therefore, all
the mathematical equations will be developed for eight muscles in
the abduction case, and for eleven muscles in the adduction case.
T E X A 5 TEC; ; LIBRAR^Ú
50
The basic force model is shown in Figure 3.3.
Point "0" is the
center of rotation of the upper arm and point "i" is the acting point
of the muscular tension.
From force equilibrium, 3 force equations can be written.
VFX^
+
RX
= 0,
2^Fy. + Ry = Fw,
VFZ^
+
RZ
= 0,
(3.1)
where.
Fx.
is the x-directional component of muscular tension F.,
is the y-directional component of muscular tension F.,
Fz.
s the z-directional component of muscular tension F.,
Rx
s the x-directional component of reaction force,
Ry
s the y-directional component of reaction force,
Rz
s the z-directional component of reaction force at the
tfl
n
t
r
joint.
S
is the total weight of the upper arm, including the weight
>
XI
Fw
of body segments and an external weight which is applied
at the hand.
The summations are over all of the muscles involved; eight in abduction
and eleven in adduction.
Let the direction cosines of these forces "F." be called Dx., Dy.,
and Dz., respectively.
Then,
X}
51
Fx. = F.. Dx.
Fy^ = F.. Dy.
Fz. = F.. Dz.
(3.2)
By substituting Equation (3.2) into Equation (3.1) we can get,
y
F^.. Dx^ + Rx = 0
2_, ^i' Dy^- + Ry = Fw
y ^ F . . Dz. + Rz = 0
(3.3)
Three moment equilibrium equations can also be written:
^ Mx. + Mr.. = Mw
+ Mr^ = 0
VMZ^.
+ Mr^ = 0
J
U
(3.4)
Where, Mr , Mr , and Mr are the moments due to the reaction forces Rx,
^
y
^
n
^
>^
Ry, and Rz. Mw is the moment due to the weight of the segments and the
;Q
externally applied weight at the hand, and the summations are over all
^
of the appropriate muscles.
By the same procedure as for the force analysis, let the direction
cosines of moments "M." be called Bx•, By., and Bz., respectively, then,
Mx.1 = M..
Bx.1
1
My. = M.. By.., and
Mz. = Mi. Bz.
(3.5)
imi^
52
Then the Equations (3.4) become.
^ M - . B x . + Mr = Mw
X
^ M . . By. + Mr = 0
y
^ M . . Bz. + Mr^ = 0
(3.6)
Let the moment arms of the force "F." be Lx., Ly., and Lz. in X, Y, Z
1
1
"' 1
1
directions, respectively, as can be seen in Figure 3.3. Then,
Mx. = Fy^.. Lz^. - Fz.. Ly.
My.j = Fz.j. Lx. - Fx.. Lz.
Mz. = Fx.. Ly.. - Fy.. Lx.
(3.7)
By Equations (3.2) and Equations (3.7),
R
Mx. = F. (Dy.. Lz. - Dz.. Ly.)
^
My. = F. (Dz.. Lx. - Dx.. Lz.)
n
Mz. = F. (Dx.. Ly. - Dy.. Lx.)
r
5
I
I
I
1
I
(3.8)
^
I
Xi
Substituting Equations (3.8) into Equations (3.4), and using the assumption that the reaction force at the joint goes through the center of
rotation, that means Mr , Mr , and Mr are all zero, we can get the
X
y
z
following equations:
^F.
(Dy.. Lz. - Dz.. Ly.) = Mw
) F. (Dz.. Lx. - Dx.. Lz.) = 0
y
.
f 1
1
1
I I
2 , ^ (DXi' Ly. - Dy.. Lx.) = 0
(3.9)
^
53
Consequently, six equations result from this system:
three for
force equilibrium [Equations (3.3)], and three for moment equilibrium
[Equations (3.9)]. Having thus obtained the equations of equilibrium
for the system, it was evident that there were only 6 equations with
11 unknowns for the abduction case (3 reaction forces at the joint
and 8 muscular tensions) and 14 unknowns for the adduction case (3
reaction forces at the joint and 11 muscular tensions). There were
more unknowns than equations and, hence, this problem was statically
indeterminate.
This implied that there were many possible solutions
for this problem.
In order to make this problem determinate, it was
necessary to make some assumption concerning which muscles were called
into play in supporting the skeletal structure in nature. The problem
was solved by the hypothesis that the human structure adjusts itself
in such a manner so as to reduce its muscular effort to the minimum.
^
r'
'^.
tfl
f;
3.3 Minimal Principle
^
—
By using Nubar and Contini's minimal principle (1961) all the
"^
equations that are necessary to solve this problem can be determined.
^
Accordinq to their work, muscular effort is defined as the product of
;
applied moment and its duration of application.
To avoid the confu-
sion of negative and positive moments, the square of moment terms was
used in the form of c M dt as a measure of muscular effort at a joint.
For the purpose of obtaining a mathematical formulation of this theory,
they used the symbol "E" to present the sum of the muscular effort at
the joint, plus some initial constant A^ as follows:
E ^(c.M.^^dt + A^
(3.10)
54
in which the common time interval "dt" has been factored out, the subscript "i" denotes the several joint muscles, and the "c." are numerical constants.
For the specific position of the humerus, the equilibrium conditions can be considered to be constraint conditions for this system.
Therefore, Equations (3.6) can be written and used as the constraint
equations for this system:
f^ (M.) = Mw,
f^ (M^) = 0,
f^ (M.) = 0
(3.11)
where f,, f^ and f^ are the equilibrium equations in the X, Y, and Z
directions.
tfl
The result of differentiating E with respect to M.j in Equation
(3.10) is
n
I
dE = 2Y(c^. M.. dM.)dt.
(3.12)
r
S
XI
Muscular effort is minimum according to the principle; therefore,
dE = 2 ^ ( 0 . M. dM.)dt = 0.
^
(3.13)
For normal individuals, operating under normal conditions, the coefficients c
can be considered to be equal, and they will drop out, so
Equation (3.13) will be
V M . dM. = 0.
(3.14)
55
The differential elements dM.j are subject to the following conditions, obtained by differentiating Equation (3.11).
3f,
ZsT""'
1
=0
dfr,
9f 3
2,587*1 •»
(315)
The method of Lagrange's undetermined multipliers, which is a
standard technique described in many references, for example, Hildebrand (1963), can be used by multiplying each of the three equations
in Equation (3.15) by Lagrange's multipliers VI, V2, V3, respectively,
q
and by adding to Equation (3.14) to get:
^
h
in
^
H
l^
8f-|
afp
afo ^
M , + VI rr^ + V 2 vT^ + V 3 ^
dM.= 0
j 1
9M.|
gM.
^M.
1
E<
U.
Since dM.'s are not zero, and the independent in general, their coef^
ficients must be zero. Therefore,
3f
9f
9f
("i ^Vl 3 M 7 ^ V 2 ^ . V 3 ^ ) = 0
(3.16)
The number of Equation (3.16) is, in fact, the same as the number of muscles associated with the motion; one for each muscle. With
the equations of mathematical equilibrium, there are as many equations
as unknowns.
For the abduction case, there are 14 equations in 14
unknowns (8 muscular tensions, 3 reaction forces, and 3 Lagrange's
multipliers).
For the adduction case, and there are 17 equations in
r
g
XI
^
Xl
56
in the 17 unknowns (11 muscular tensions, 3 reaction forces and 3 Lagrange's multipliers) for adduction case.
According to this proce-
dure, the problem becomes a statically determinate one.
From Equations (3.6) and (3.11), the partial derivative terms
of Equation (3.16) can be obtained as follows:
^l ^^i^ ZÎ^i* ^^i - ^ = 0
f^ (M^.) ^ M ^ . By. = 0
f^ (M.) ^ M . . Bz. = 0
(3.17)
and from Equation (3.17)
9F 1
9M.
= Bx.
l
9f,
ãM" = By,
Ifl
nT
df,
W
4.
Bz.
(3.18)
r
S
X]
>
So Equation (3.16) becomes.
( M. + VI. Bx^.. + V2. By. + V3. Bz. ) = 0
X]
(3.19)
And from the phythagorean theorem,
M. = ((Mx.)^ + (My.)- + (Mz.)^)^
According to Equation (3.8),
(3.20)
57
M^ = F. ((Dy.. Lz. - Dz.. Ly.)^ +(Dz.. Lx. - Dx.. Lz.)^ +
2^ií
(Dx.. Ly. - Dy.. Lx.)^)
(3.21)
where, Dx.., Dy.., Dz.j, Lx., Ly., and Lz. are constants, so Equati on
(3.21) can be written as
M. = F . . K.
(3.22)
where
Ki^ = (Dy^. Lz. - Dz.. Ly.)^ + (Dz.. Lx. - Dx.. Lz.)^ +
(Dx.. Ly. - Dy.. Lz.)'
and
F-(Dy.. Lz. - Dz., Ly.)
Mx.
DA .
F..
"i
i
(Dy., Lz. - Dz.. Ly.)
K.
tn
Lx.
Ox^,
-
Dx. ,
Lz.
n
By^
^
Dx..
Bz.
-
•-yi
1 ~
CD
X]
Dy^. ,
Lx.
(3.23)
K
Changing notation, let us designate Bx.j, By^., and Bz.j by the constants
P., q., and r., respectively.
Then Equation (3.19) becomes.
F., K. + VI.p. + V2.q. + V3.r. = 0
where
(3.24)
aqain, K., P., q., and r, are all constants that can be calcu'
•'
i
1
1
1
lated from the anatomical and physiological data as discussed previously.
X]
58
3.4
Solution Technique
The solution to this problem can be finally resolved into the
problem of solving simultaneous equations. The problem can be stated
in matrix form as follows:
^o^
Dx-j DXp
Dx^
1
0
0
0
0
0
Dy^
Dy„
^n
0
1
0
0
0
0
Dz^ Dz^
Dz^
0
0
1
0
0
0
0
Bx, Bxp
Bx^
0
0
0
0
0
0
Mw
By^ By^
By^
•'n
0
0
0
0
0
0
0
Bz.| Bz^
Bz„n 0
0
0
0
0
0
0
K^
0
0
0
0
0
p., ql r^
0
K,
0
0
0
0
P2 ^2 ^2
0
0
DYI
^l
W
0
8
>
=<
0
tfl
íl
n
n
0
0
R
5
X]
>
X
Ry
0
Rz
0
0
O..K^_2 0 0
0
0
0
P,_2qn-2V2
VI
0
0
0,
^n-l 0
0
0
0
P,_iq,.ir^.i
V2
0
0
0
\
0
0
0
Pn %
V3
0
^n
I
r
V. J
Xl
59
where n is the number of muscles involved in the motion (n=8 for abduction and n=ll for adduction).
However, because of the minimal
energy principle of the muscular effort, the determinant of this
matrix becomes zero, and the matrix is singular.
This problem can
be solved by using Equation (3.19) to change the moment equations
as follows:
M.
= -(VI.Bx.. + V2.By^. + V3.Bz.)
Mx^. = M..BX. = -(VI.Bx."^ + V2.By..Bx. + V3.Bz..Bx.)
My. = M..By. = -(Vl.Bx..By. + V2.By.^ + V3.Bz..By.)
Mq.. = M. .Bz^. = -(Vl.Bx..Bz. + V2.By. .Bz. + V3.Bz.^)
(3.25)
For simplicity, let the following notation be introduced for some of
í
the terms on the right hand sides of these equations:
tfl
^l'-
-
=2 = -
n
2»<,'
r
S
Z'»i »«1
XI
>
^3 = -
2"'r"»i
=4 = -
2»',^
^5 = -
I!»'r"i
^6 = -
2-.^
X]
(3.26)
And instead of the moment Equations (3.6), by using Equations (3.16)
for the moment equation.
From Equations (3.25) and (3.26), the dotted
square region of the matrix will be changed as can be seen following
the matrix.
60
Dx.| Dx^
Dx
Dy^ Dy^
Dy^
n
1 0
0
1
0
0
0
0
0
0
0
0
/'_
^
0
w
Dz^ Dz^
Dz.
n
0
0
1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
K^
0
0
0
K^
0
0
0
0
0
0
s^
S2
S3
Mw
0
S2
s^
S5
0
0
0
S3
S5
s,
0
0
0
0
p^
q^
r^
0
0
0
0
P2
^2
^^2
= <
< .
n
R
X
0
0
0
I
tfl
0
0
K^_2
0
0
0
0
P,.2qn.2^n-2
R.
0
R.
0
VI
0
í
n
I
r
5
X)
0
0
0
0
K^_.,0
0
0
0
p„_-,q_-,r,
'n-rn-Vn-l
V2
K
0
0
0
p
^n
V3
n
q
^n
r
n
0
^ ^
This equation can be solved by computer methods to obtain the
muscle t e n s i l e forces, F p Fp* . . . » F^.
The results of t h i s solu-
t i o n at the various angular positions f o r d i f f e r e n t weight f o r each
subject are tabulated in Appendix I I , Tables 2.1 to 2.6 together
with the experimental r e s u l t s .
X]
CHAPTER IV
EXPERIMENTAL PROCEDURES
This chapter is devoted to the description of the experiments
needed to characterize the measured anthropometric data and the recorded electromyograms of the shoulder joint muscles.
The textbooks of anatomy describe the deltoid as a flexor, extensor, abductor, and medial and lateral rotator of the arm. The
muscle can be roughly divided into three distinct parts on the basis
of the origins and the modes of function. The anterior part produces mainly flexion and medial rotation of the humerus, the middle
part produces abduction, and the posterior part extends and laterally
rotates the humerus. The anterior and posterior parts of the del-
a
toid contain parallel fibers, while the middle part is multipennate.
These anatomical features are probably responsible for the wide range
of movements and functions which make this muscle capable of produc-
g
tfl
H
Jn
ing a variety of movements. Anatomically, these three parts of the
^
deltoid are distinctive and can easily be identified during the dis-
;Q
section of cadavers.
5
The purpose of the experimental procedure was to provide anatomical data for input to the theoretical analysis and to provide experimental verification for theoretically predicted muscle force distribution in the shoulder. However, there was a basic problem with this
verification: muscle force distribution was calculated for the cadaver but, of course, it was impossible to verify these results by electromyography on the cadaver. Electromyography was used on living
subjects but the theoretical analysis (which was to be verified) could
61
62
not be conducted on the living subjects without the detailed internal
anatomical data provided by dissection.
This problem was overcome by
making some external anatomical measurements on both the cadaver and
the living subjects so as to establish scale factors for each subject.
These scale factors, together with the internal anatomical data of the
cadaver, provided an indirect means to estimate internal anatomical
data for the living subjects.
With these internal data, theoretical
force distributions could be calculated for each subject and checked
experimentally by electromyography for the three parts of the deltoid.
4.1. Anthropometric Data Characteristics
Six male subjects were selected for this experiment.
The only
limiting factor concerning the subjects was that they were to be of
two different physical builds, i.e., three of them were of good physical build, and the rest were of average build.
As can be seen in Figure 4.1, the following anthropometric characteristics of each subject were determined by using the method of Snyder,
et al (1971):
3
tfl
d
n
X
r
Ê
5
X]
(1)
Weight (W): Weight of subject unclothed (Ibs)
(2)
Height (H): Height of the subject while maintaining an erect
standing posture (ft)
(3)
Biacromial width (BW): The horizontal distance between the
superior lateral border of the acromial process of the left
and right scapulae (ft)
(4)
Chest height:
The vertical distance from the center of the
umbilicus to the superior margin of the jugular notch of the
63
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rcJ
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s:
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to
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u
cu
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c
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> í
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ra
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tn
65
manubrium of the sternum (ft),
(5)
Upper arm length (UL): The distance from the right acromion
to the inferior head of the humerus (ft),
(6)
Lower arm length (LL): The distance between the tip of the
elbow (olecranon) and the cénter of the hand (ft).
tn
íi
n
I
r
tn
Figure 4.2.
Upper Extremity Model
Table 4.1 shows the basic anthropometric data of the model and the
six subjects.
Here, the model refers to the dissected cadavers.
All the above dimensions were necessary for the comparison of the
anthropometry of each subject.
From these anthropometric data, scale
factors with respect to the cadaver were established.
66
Figure 4.2 shows the upper extremity as it was used in this investigation for the calculation and analysis of the necessary anthropometric data. Here, all parts of the upper extremity were considered
as a single rigid body and inertia force and moment were neglected because only the static cases were considered in this study. Before the
experiments, all the basic data such as weight, height, e t c , were
gathered according to the method described earlier, and all the subjects
were checked to see whether they were in good physical condition.
By using Dempster's (1955) anthropometric data analysis method of
body action, the weight and the location of the center of gravity of
the segments were calculated as follows:
UW = W X 0.02647
LW = W X 0.02147
3
UC = UL X 0.43569
^
LC = LL X 0.5544
(4.1)
f:
n
r
where
S
UW
is the upper arm weight
LW
is the lower arm weight
UC
is the distance of the center of gravity of the upper
arm from the proximal joint
LC is the distance of the center of gravity of the lower
arm from the proximal joint.
We can calculate the applied actual weights and moments due to the
weights of segments and the abduction or adduction weight by using geometrical data and the postulated model as follows:
íi
67
TW
= UW + LW -H AW, for abduction,
TW = UW -I- LW - AW, for adduction, and
TM =
(UC X UW) + (UL X LC) x (LW + (UL + LL) x AW,
(4.2)
where
TW
is the effective weight defined as the total actual
applied weight (AW and segment weight),
TM
is the effective moment defined as the total moment due
to AW and the segment weight,
AW
is the abducting (or adducting) weight, and segment refers the whole upper extremity, i.e., the upper arm,
the lower arm and the hand.
R
In the case of abduction through the angle "ø" (See Figure 3.1), force
and moment as used in Equations (3.1) and (3.4) can be determined to
be the effective weight and effective moment as described by:
Fw = TW
Mw = TM X sin(e)
(4.3)
As mentioned earlier, for the living subject it is impossible to
get the exact geometrical data such as length, direction, and insertion
and origin points of the muscles.
In order to get the approximate
geometrical data, an anthropometric similarity method was used in this
study by employing scale factors which were defined as follows:
_ anthropometric dimensions of subject
Scaie ractor anthropometric dimensions of model
/* .^
^^^^
^
ÍH
fj
I
^
CD
>
^
68
where "anthropometric dimension of model" refers to the anthropometric
dimension of the cadaver that was measured through dissection.
The scale factors in X, Y and Z directions were calculated according to the above definition as follows:
C;FY = BW of subject
^^
BW of model
qpY
^^^
_ BW of subject (ft)
1.25 ft
= 1 /CH of subject
'^ ^CH of model
=
UL of subjectx
UL of model '
/CH of subject (ft)
1.2525 ft
h ^
. UL of subject (ft)x
1.036 ft
'
SFZ = SFX
(4.5)
where
3
SFX
is the scale factor in X direction
g
l
SFY is the scale factor in Y direction
SFZ is the scale factor in Z direction.
The last equation was due to the fact that for most humans the cross
sections of the trunk are similar to one another.
tu
g
X
r
S
>
;Q
For a specific muscle, by using the above method, necessary anatomical data of each subject were calculated as follows:
SDMX = MDMX X SFX
SDMY = MDMY x SFY
SMDZ = MDMZ X SFZ
where
(4.6)
^
69
SMDMX, SMDY and SMDZ are the subject's geometrical dimensions
in the X, Y and Z direction, respectively, and MDMX, MDMY and
MDMZ are the model's geometrical dimensions in the X, Y and Z
direction, respectively.
By using these scale factors and the mathematical analysis of
Chapter III, the computer program to get all the necessary data for
the mathematical analysis of the individual subjects was developed,
as can be seen in Appendix IV (Computer Programming, Part I).
Using this method, the anthropometric data was formulated and
is shown in Table 4.2.
With these data, the mathematical analysis
described in Chapter III was used to calculate all muscle tensions
for each subject.
Table 4.2.
tn
H
Anthropometric Basic Data of the Subjects
Subjects
Model
I
r
1
5
uw
4.62
5.29
3.78
3.39
LW
3.75
4.29
3.07
2.75
UC
0.45
0.46
0.40
LC
0.66
0.66
SFX
1.00
SFY
5.56
5.82
3.35
4.51
4.7
0.36
0.44
0.51
0.47
0.58
0.53
0.60
0.78
0.69
1.04
0.80
0.80
0.99
1.10
1.12
1.00
1.03
0.84
0.80
0.93
1.09
1.04
SFZ
1.00
1.04
0.80
0.80
0.88
1.10
1.12
TW
8.37
9.58
6.85
6.14
7.48
10.07
10.54
TM
8.45
9.78
6.04
5.00
7.16
11.77
11.11
4.13
:
70
Noise Shelter
y
Electrode
n
îi
J^Weight ^ / y
I ntegrat ing Preamplifier
Figure 4.3.
Experimental Procedure
3
tn
4.2. E.M.G. Experiment
One result of this experiment was to find out the magnitude of electromyographical signal intensities of some muscles (deltoid anterior,
deltoid middle and deltoid posterior) during abduction under the different conditions.
n
I
r
S
This experiment consisted of recording electromyographic
data during isometric contraction against the applied weight at the hand.
The basic idea of the experiment is illustrated in Figure 4.3,
which shows an idealized diagram of the upper extremity with a specific
applied weight at the hand.
Two abduction positions are displayed in
the figure.
Each electromyographic recording was made with three surface electrodes, one for each of the parts of the deltoid muscle.
These elec-
trodes were attached on the skin directly over each of the three parts
.
71
of the deltoid.
(1)
The locations of the electrodes were as follows:
Anterior Part:
(2) Middle Part:
(3)
2-inches below the lateral end of the clavicle,
2-inches below the lateral border of the acromion,
Posterior Part:
3-inches below the spine of the scapula.
Miniature electrodes, llmm in diameter (Beckman No. 650437) were
chosen because they were suitable for minimizing the interference effects
from the muscles and because, due to their small size, the exact position of each could be determined with relative ease.
Just before each electrode was positioned, the appropriate area of
the skin was rubbed with alcohol and was covered with electrode jelly
to reduce the inter-electrode resistance and skin electrical resistance.
During the recording, the action potential which is generated from
^
the skeletal muscle cells when they are in a physiologically active
state, was detected by electrodes through the skin.
The magnitude of
the action potential varied from 25 yV to 500 yV according to their
state of activation.
J
tn
^
But, due to the fact that the electrode output
was an alternating voltage signal it was necessary to integrate the ab-
'
^
r
S
>
ÎQ
solute value of the signal and to obtain the action potential as the
slope of the integrated curve.
For the integration of the electrical signal from the electrodes
on the muscles, an integrating preamplifier, a Sanborn Model 1035,
(operating in the area mode) was used to perform the operation electronically, as shown in Figure 4.4.
The output of the recording represented
the integrated voltage during a specific time interval.
in units of volt-seconds.
The output was
During the integrating process, as can be
^
72
H
Â
ii''B|Í|i«<tiiiiPi
E.M.G.
Signal
imM
Threshold Level
Area Mode
Signal
t=0
n=area of E.M.G. signal since t=0
Figure 4.4.
Summation Area of E.M.G. Signal
R
seen in Figure 4.4, the threshold triggering circuit returned the inte-
î
grated signal to a zero level each time it reached the full scale of
tn
maximum integration.
Consequently, the total integrated value calcu-
lated since the beginning of the area summation process could be found
íí
n
I
r
S
by counting the number of cycles and multiplying by the value of the
PQ
maximum displacement height of the integrator.
In this study, only the static cases of abduction (0 to 90 degrees
bylO degree intervals) were investigated.
That is, the arm and applied
weight were held fixed at one position while data was being taken.
Therefore, the slope of the integrated voltage curve could be interpreted as the intensity of the recorded electrical signal at that posture.
Or, the relative intensity (E) of the electrical signal for each
muscle could be represented in the following form.
73
E = f
(4-7)
where V is the integrated action potential curve and t is time.
The resulting relative electromyographic potential intensities of
six subjects under various conditions are shown in Appendix II, Tables
2.1 to 2.6.
Because of the low voltage of the generated electrical signal of
the muscle, it was necessary to avoid all the electrical noise effects
from lights, surrounding equipment, etc. This was achieved by doing
the experiments inside of the specially designed electrical noise shelter at Texas Tech University.
R
Each of the subjects participating in the experiment was instructed and trained thoroughly with regard to his duties in the experiment.
This was done in order to familiarize each subject with the equipment
tn
Ui
H
h
and with the motions he would be required to do. Following the training phase, the electromyographic records for each of the subjects under
r
^
>
various abduction angles for different weight were taken.
Each subject
was asked to assume an erect posture with his feet together.
The sub-
ject w a s asked to maintain a specific posture for five or six minutes
in order to get enough data.
After each experiment, the subject was
given a five minute rest period.
He was then instructed to assume an-
other posture or to use another weight.
A total of ten experiments were
done for six subjects under all the abduction angles considered in this
experiment.
Three weights (0, 5 and 10 Ibs) were tested for every level.
^
'l^
74
The relationship between the muscular tension as measured in the
experiments and muscular tension as determined by theoretical methods
is discussed in more detail in the next chapter.
R
î
tn
si
n
CD
>
î<
CHAPTER V
RESULTS OF THEORETICAL ANALYSIS
The steps taken in this investigation, which were presented in the
preceding chapters, can be summarized as follows:
Step (1): Biomechanical analysis of the musculo-skeletal system
of the human shoulder muscle.
Step (2): Collection of the geometrical data which are necessary
for the vector analysis by dissection of the cadaver.
Step (3): Mathematical description of the static equilibrium
equations of the forces and moments for the shoulder
joint model by using the vector method.
Step (4): Solution of the indeterminate problems by using the
minimal principle technique.
^
Step ( 5 ) : Formulation of the experiments that would permit the
•
^
tn
ín
verification of the application of the minimal p r i n c i -
!^
ple to the living human body.
C
CD
Step (6): Experimental procedures and collection of data to obtain muscle force distribution and electromyographic
signal intensity on the three parts of the deltoid.
In order to complete an experimental verification of the mathematical model for the muscles crossing the gleno-humeral joint, the
remaining step was the characterization of the unknown functions of
the model by using the collected data, as shown in the following
equations:
75
^
;<
76
^i " ^2 (^•' ^•' W' B.,A,e)
(5.1)
Ti = f^ (E.)
(5.2)
where the symbols are the same as those used and defined in Chapter II.
The first equation represents the results of the theoretical solutions
of the model and the six subjects as described in Chapter III and Chapter IV.
It represents the theoretical relationship between the muscu-
lar tension and the following:
the geometrical, anthropometrical and
physiological characteristics, the applied weight, and the abduction
(adduction) angle.
The theoretical solutions for the subjects were
obtained by using cadaver data and scale factors as described in Chapter IV.
The diagram of the theoretical muscular tension vs. abduction
a
angle for the three deltoid parts for each subject at various condi-
tn
tions are presented in Figures 5.1 to 5.6.
hi
n
The second equation repre-
sents the relationship between the relative electromyographic potential
intensity and the exerted muscular tension as determined by experimen-
I
r
s
>
tal data.
5.1. Functional Equations for the Muscular Tension
The first relationship of concern here is the relation of the muscular tension to the abduction angle of each subject under various conditions, as shown in Figures 5.1 to 5.6.
The relationship can be repre-
sented symbolically in the following form:
T. = fg (9)
(5.3)
77
to
100-
o
•r-
to
C
<u
s-
tn
<TJ
u
iq50
r
CD
'^
0
1
10
...
,
20
r
1
1
30
40
50
Abduction Angle
1
1
1
1
60
70
80
90
(degrees)
Figure 5.1. Muscular Tension Diagram - Subject (1)
78
: Anterior
to
JO
L.
O
Middle
D
Posterior
0 Ibs
100 .
5 Ibs
-o
IC) Ibs
-o
y
.O
c
o
to
c
cu
I—
ri
/
s-
tn
fO
/O
3 50
to
Z3
si
/
n
/
.-0'
vy
/
^
f
// ^^
/ '
/ /
/
i/ ^ ^
/jåt^
^ , , , ^ ^ ^ ^ Ty^—
10
20
30
_A———"•'^
,
40
5
'Xl
/
,
0
A
^ -
/
1
/
/
X
r
,_.., — ^
50
Abduction Angle
60
,
70
-T
80
' •
• • !
90
(degrees)
Fiqure 5.2. Muscular Tension Diagram - Subject (2)
79
to
X3
100 -
c:
o
.r—
to
c
cu
fO
3
u
>
S-
to
in
íi
50 -
n
r
E
r<
Abduction Angle
Figure 5.3.
(degrees)
Muscular Tension Diagram - Subject (3)
80
: Anterior
: Middle
Posterior
0 Ibs
J3
100-
5 Ibs
10 Ibs
to
cu
S<T3
î
:3
u
to
3
in
50-
n
CD
PQ
70
Abduction Angle
Figure 5.4.
80
90
(degrees)
Muscular Tension Diagram - Subject (4)
81
A
Anterior
O
Middle
D
Posterior
0 Ibs
to
J3
5 Ibs
100-
10 Ibs
e
o
•rtO
E
<D
fC
3
u
to
tn
50 -
U
m
n
r
fi
>
ÎQ
—T
80
Abduction Angle
Figure 5.5.
í
90
(degrees)
Muscular Tension Diagram - Subject (5)
82
: Anterior
Middle
tn
100 -
c
o
to
c:
cu
s-
01
u
to
3
50 -
I
r
æ
;Q
>
Abduction Angle
Fiqure 5.6.
Muscular Tension Diagram - Subject (6)
(degrees)
î
îrf-- -4*.-: -aMMai SaaiiMBSI
83
In this study, it was assumed that fourth order polynominals of
the form
T = a, {^) . a^ i^)'
. a3 (±)'
. a, (^)'
(5.4)
could represent this relationship. Only the fourth order polynominal
equation coefficients a^, a , a^, and a- were considered because, according to the results of polynominal regression methods, it was determined that the effect of fifth or higher polynominal terms could be
neglected.
The purpose of the polynominal equation was to simplify and to
generalize the calculating porcess. The coefficients are different
for every person and every muscle at various external conditions. After
the coefficients have been found for each case, the muscular tensions
ÍT:
can be calculated by use of Equation (5.4) directly without the comli-
iji
H
h
•L
cated calculation of all the anthropometrical and geometrical data
that are necessary for the theoretical solution. But, this process
!»
is not simple because, in order to generalize the procedure of calculation of coefficients, we must first choose coefficients for the
cadaver and then find the relãtionships between these coefficients and
the coefficients for each subject.
Detailed procedures of choosing the coefficients for the cadaver
and developing the relationships between the coefficients are explained
below.
The next step was the calculation of the coefficients of each
curve for each subject by using the Least Square curve fitting methods.
The resulting coefficient values of each subject at the different weight
^
>
"^
84
and abduction angle are tabulated in Appendix III, Tables 3.1 to 3.21
For example, for the dissected cadaver, the following nineteen
general equations for the case of 0 Ibs of lifting were developed according to the above curve fitting method.
In order to find the gen-
eralized coefficients for the various cases of each subject, the coefficients for the various cases of each subject, the coefficients of
the model for 0 Ibs of abduction were used as standard values.
General Equation Form (Standard Coefficients)
A.
Y:
Muscular Tension (Ibs)
X:
Abduction (Adduction) Angle/10
Abduction Case
1.
2.
Deltoid Anterior
*5
Y = 3.35X - 0.658X^ + O.OB^^X"^ - 0.00262X^
il
Deltoid Middle
Y = 11.20X - 2.620X^ + 0.3730X^ - 0.01880X
'5
3.
Deltoid Posterior
Y = -2.32X + 3.550X^ - 0.7110X^ + 0.04320X^
;Q
^
ts
4. Supraspinatus
Y = 19.00X - 8.880X^ + 2.0200X^ - 0.12500X^
5.
Infraspinatus
Y = 19.60X - 4.860X^ + 0.9250X^ - 0.05560X^
6.
Teres Major
Y = 3.52X - I.IBOX^ + 0.2690X^ - 0.01590X^
7.
Teres Minor
Y = 3.16X + 1.030X^ - O.IOBOX^ + 0.00334X^
85
8.
Subscapularis
Y = 31.50X - 12.600X^ + 1.9800X^ - O.IOOOOX^
B.
Adduction Case
1.
Infrespinatus
Y = 5.75X - 0.886X^ + 0.1250X^ - 0.00410X^
2.
Teres Major
Y = 1.20X - 0.163X^ + 0.0161X"^ + 0.00053X^
3.
Teres Minor
Y = 1.20X + 1.090X^ - 0.3410X^ + 0.02650X^
4.
Subscapularis
Y = 16.70X - 7.050X^ + l.O^OOX"^ - ^.^^OOOx"^
5.
Pectoralis Major-Sterno
Y = 0.65X + 0.296X^ - 0.0771X^ + 0.00664X^
'3
:<
6.
Pectoralis Major-Clavicular
>
'íî
Y = 0.97X + 0.085X^ - 0.0129X^ + 0.00032X^
H
n
7.
8.
Biceps Long
Y = 4.28X - 1.320X^ + 0.2600X^ - 0.01610X^
Biceps Short
1
ig
Q
>
tl
Y = 5.77X - 0.906X^ + 0.0777X^ + O.OOOIOX^
9.
Triceps
Y = 2.17X - 0.150X^ - 0.0195X^ + 0.00359X^
10.
Coracobrachialis
Y = 4.22X + O.OOSX^ - 0.0195X^ + 0.00359X^
11.
Latissimus Dorsi
Y = 1.41X - 0.389X^ + 0.0578X^ - 0.00203X^
ESStVXi-i)^'?!"
86
In the above equations, all the independent variables, such as
L, D, W. B, A of Equation (5.1) were already considered before, for
these variables had been used for the theoretical solution of Chapter
III.
5.2. Simplified Functional Equations for the Muscular Tension
The theoretical procedure described above was fairly complicated
and required a number of external anatomical measurements for each
living subject.
A somewhat simpler method, based on the procedure
above, was derived by making some simplifying assumptions.
This
method is described below.
From the mathematical viewpoint, the muscular tension (T) is associated with the effective moment and effective forces that were defined by Equation (4.3).
However, in the final matrix of the theore-
''HÍ
<
,i>
tical solution of Chapter III, the magnitude of the effective force
is negligible compared to the magnitude of the effective moment.
'^
i
•.í_
Furthermore, most of the effective forces are absored by the reaction
,-
forces at the joint while the effective moments are fully effective.
o
According to the definition of the moment,
T = M/L
where
T is muscular tension
M is moment, and
L is the length of the moment arm,
Also, for the model and subject:
-<
(5.5)
87
Tmodel
=
Tsubject =
Mmodel/Lmodel
(5.6)
Msubject/Lsubject
(5.7)
Dividing (5.6) by (5.7), we obtain
Tmodel
Tsubject
^ Mmodel/Lmodel
Msubject/Lsubject
^
Lsubject/Lmodel
Msubject/Mmodel
(5.8)
However, as can be seen in Table 4.2, the effective moment of the model
for the case of 0 Ibs abducting is
Mmodel =8.45
(5.9)
so.
Tmodel
Tsubject
_
Lsubject/Lmodel
Msubject/8.45
(5.10)
:rj
..<
IJl
Here, according to the definition of the scale factor in Chapter IV,
4
Lsubject/Lmodel was considered as the average of the scale factor, so
Lsubject
Lmodel
^
(SFX + SFY + SFZ) ^ ^^^
3
B
(5.11)
9
•:;<
Define the moment ratio (MR) as follows
MR = Msubject/Mmodel
=
Msubject/8.45
Then, from Equations (5.8), (5.9), (5.11), and (5.12)
Tmodel
Tsubject
Sav
MR
Let us define multification factor (MUL) as follows:
(5.12)
* < « * - t*'i-:», v - - ' T ' - 3 X : -' ••••
88
MUL
= MR/sav
Then,
Tsubject
= Tmodel x MUL
(5.13)
Therefore, in Equation (5.4) the coefficients (a. s) for the different subjects can be calculated as follows
a.(subject) = a.(model) x MUL(i=l,2,3,4).
(5.14)
Following to the process of calculating the effective moment, the
scale factors, and the multification factor (which was defined as the
ratio of the coefficients between subject and model), the following
procedures were used for all subjects:
1.
n
Collection of the anthropometric data:
(a) Height (H)
i
(b) Weight (W)
(c)
Biacromial width (BW)
(d) Chest height (CH)
(e) Upper arm length (UL)
(f) Lower arm length (LL).
2.
Calculation of the effective moment data:
(a)
Upper arm center of gravity (LC)
(UC) = 0.53469 x (UL)
(b)
Lower arm center of gravity (UC)
(LC) = 0.55440 X (LL)
^D
:<
89
(c) Upper arm weight (UW)
(UW)
(d)
= 0.02647 X (W)
Lower arm weight (LW)
(LW)
= 0.02147 X (W)
(e) Applied weight distance (AD)
(AD)
(f)
=
(UL) + (LL)
Lower arm effective distance (LAD)
(LAD) =
(UL) + (LC)
(g) Applied weight (AW).
Calculation of effective moment (M):
(M)
=
(UC) X (UW) + (LAD) X (LW) + (AD) x (AW)
Calculation of scale factors:
(a)
Scale factor in X-direction (SFX)
(SFX) =
(b)
(BW)/1.25
Scale factor in Y-direction (SFY)
'n
<
i>
.n
H
(SFY) = iá((CH)/1.2525 + (UL)/1.036)
(c)
Scale factor in Z-direction (SFZ)
(SFZ) =
(BW)/1.25
(d) Average scale factor (Sav)
(Sav) =
(SFX + SFY + SFZ)/3.0.
Calculation of moment ratio (MR):
(MR)
=
(M)/8.45.
Final calculation of the multiplication factor (MUL):
(MUL) =
(MR)/(Sav).
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ii^AttåJiÉÊMXãSíÊaMÊA •
90
By using this method, we could predict the coefficients of the
relationship between the muscular tension and the abduction ( or adduction) angle of different subjects under the different conditions
of applied weight.
In order to examine the validity of this method, the error percentage between the results of the curve fitting values and the results of this method were calculated and tabulated in Table 5.1.
From the table it can be seen that this simplified method will provide results which are almost the same as those of the more detailed
and difficult procedure described in Section 5.1.
The methods differ
slightly because the simplified method neglects the force effects of
the external load and uses averaged scale factors.
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CHAPTER VI
EXPERIMENTAL VERIFICATION OF THE THEORETICAL ANALYSIS
All of the experimental work described in Chapter IV and V which
involved external anthropometric measurements have dealt with the problem of obtaining measured data to serve as input to the theoretical
force distribution analysis.
With these data and the associated assump-
tions regarding scaling, together with the theoretical model, force versus adbuction angle (e) relations were described for each part of the
deltoid for each external weight and for each subject.
The electromyo-
graph experiments were for the purpose of verifying the theoretical
results for the three parts of the deltoid muscle.
The details of the
verification are described below.
On the basis of the well established fact that there is a linear
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relationship between the generated electromyographic potential inten-
:n
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sity and the exerted muscular tension of the muscle (Basmajian 1967;
^
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Inman, et al 1952; Bigland and Lippold 1954), Equation (5.2) can be
:D
written in the following form:
;Q
T = cE
(6.1)
The linear coefficient "c" was to be determined by experiment.
On the basis of the above fact, it was determined that a good predictor of the magnitude of the muscular tension in each muscle would be
what was recorded as the intensity of the action potential of the electromyogram.
However, for most of the muscles in the human body, it was
found that such a recording was almost impossible because of the interference of the muscles with one another during the recording, and be92
:<
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93
cause of the difficulty of recording responses from the inner muscles.
The idea of choosing the three parts of deltoid came from the fact that,
for these muscles, the recording could be done more easily without significant interference.
The necessary data for this characterization are the experimentally
recorded electromyographical signal intensities which were defined by
Equation (4.7).
In order to find the slope of the integrated curve and
because of the small mesh size of the electromyogram, a magnifying glass
was used to read accurate values of the integrated voltage curves.
The display of the data collected from the static electromyographic
recording experiments and the solution of the theoretical vector solution plots, as can be seen in Figures 6.1 to 6.6, and in Appendix II,
Tables 2.1 to 2.6 were the basis of the validity of the application of
<
the minimal principle to the human living body.
for one of the subjects.
obtained as follows:
Each of the figures is
Each of the data points in each figure was
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The subject assumed one posture (on abduction
angle) with one external weight and the corresponding electromyographic
intensity was determined at one of the three deltoid positions.
i>
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electromyographic intensity was the abscissa of the plotted point.
The
ordinate was obtained from the theoretical model and was the muscular
tension for that particular abduction angle, weight, and deltoid part.
In all, there are 81 points plotted on each curve:
nine abduction posi-
tions, three weights at each position, and at each of the three deltoid
parts.
The data points showed a remarkably linear relation between theore-
tical and experimental results and straight lines were fitted to the data
using the Least Square method.
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Fiqure 6.1.
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80
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Muscular Tension vs. E.M.G. Intensity - Subject (3)
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Figure 6.4
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99
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Figure 6.6.
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Muscular Tension vs. E.M.G. Intensity - Subject (6)
100
As can be seen in the figures and the statistical results of the
linear curve fitting of Table 6.2, the relationship between the theoretical solution, calculated according to the minimal principle, and the
experimental results, obtained from the electromyographic experiments
on living subjects, provided the basis of the validity for the application of the minimal principle to the living human body.
This, of
course, is because of the fine linear curve fitting between these values
and the negligible deviation of each subject case.
Small deviations
were expected because of the assumption involving the scale factors
that were used in the theoretical solutions and because of experimental
inaccuracies in the measurements and data.
The linear coefficients of the lines correlating the theoretical
solution and the experimental results (the slopes of the lines), which
were calculated by using the Least Square curve fitting method, are
tabulated in Table 6.1
Table 6.1.
Linear Coefficient Values
1
3
Subject
Linear Coefficient
/
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1
2.0133
2
2.2077
3
2.8851
4
2.444
5
1.758
6
1.9693
The difference in the coefficients for each subject was due to the different physical conditions of the subjects.
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CHAPTER VII
SUMMARY, CONCLUSION AND RECOMMENDATION
This chapter discusses several aspects of the theoretical and experimental procedures, and the significance of the results which were
found in this research.
Based on the results of this study, some spec-
ulation is made about the mathematical approach to human musculoskeletal problems.
Also, it is indicated how other similar investiga-
tions involving complicated and indeterminate problems could be solved
by this technique.
7.1
Summary
The purpose of this study has been to formulate a mathematical
model capable of predicting muscular tension characteristics for muscles in the human shoulder joint.
This was done by using the data
\
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that were collected through dissection of a cadaver and through physiological information about human skeletal muscles and anatomical characteristics of the shoulder joint.
By using this model, the explicit
characterization of the mathematical equations for the postulated
mechanism of the shoulder joint was described in terms of a three dimensional coordinate system.
The mathematical equations for the rela-
tionships between the electrical signal intensities that are generated
from the muscles, and muscular tensions that are exerted by muscles
at various postures during abduction of the upper extremity were investigated.
General equations that can be applied to various individual persons who have different anthropometric dimensions were developed by
102
103
using scale factors.
Computer programs were developed to determine the
muscular tension of muscles in the shoulder joint of various persons
and to predict the linear coefficients between electromyographic
electrical signal intensities and the muscular tensions of the
skeletal muscles.
These were developed from the results of the theore-
tical and experimental procedures of this study.
According to the re-
sults and the techniques of this study, it was determined that most of
the complicated human musculo-skeletal systems can be analyzed mathematically without dissecting bodies.
7.2. Conclusion
The conclusions which can be drawn from this investigation with
regard to the postulated model, the theoretical and experimental procedures, and the verification experiments, are tablulated below.
These conclusions are:
(1)
The human shoulder joint mechanism can be represented by
a mathematical vector model.
The geometrical input data
for the model can be obtained by dissection of cadavers.
The model provides muscle force distribution in the various
muscles crossing the gleno-humeral joint at various static
abduction and adduction angles of the arm.
(2)
Input data for the application of the model to living people
can be obtained by external physical measurements and scale
factors.
(3)
The Minimal Principle used in the mathematical model is valid,
as verified by electromyographic experiments.
fllll t
r.......
••
^-
104
7.3. Recommendation
There is a continuing need for generalized mathematical models to
analyze human motion characteristics.
This investigation was success-
ful in contributing to fulfilling the need by exhibiting a highly accurate prediction of the distribution of muscular tension in the shoulder
mechanism for abduction to a statically held position.
In addition,
this investigation points the way to new efforts for the fuller development of mathematical models for human musculo-skeletal system analysis.
The recommendations which should be considered in further researches include the following:
(1)
The range of possible movement should be extended past the
range of 0-90 degrees abduction and adduction.
(2)
This technique should be applied to the combination of abduction, adduction, rotation, flexion, and extension of the
upper arm.
(3)
(4)
The work should be extended to all of the human musculoskeletal system.
j
The work should be extended to include dynamic analysis of
i
joint movements.
To do this, it will be necessary to know
the dynamic characteristics of body segments.
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^
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APPENDIX
Appendix (I)
Anatomical Basis Data Tables for Muscles
Appendix (II)
Theoretical and Experimental Results
- Deltoid Three Parts Appendix (III)
Coefficients of Theoretical Solution of
Muscular Tension Tables
Appendix (IV)
Documentation of Computer Program
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APPFNDIX
SUBJECT
N^ME
(III)
TABLE 3 - 1 .
C O E F F I C I E N T S OF THEORETICAL
S n L U r i O N OF M U S C U L A R T E N S I O M
NO.:
0
ACTING
OF MUSCLE
ABDUCTION
i«/EIGHT
0
LBS
Al
A2
A3
A^
3.35
-0.658
0.0823
•0.C0262
-2.620
0.3730
•0.01880
CASE
DELTGID
ANTERIOR
DELTOID
MIDDLE
1 1.20
DELTOID
POSTERIOR
-2.32
3.550
•0.7110
0.04320
SUDRASPINATUS
19^00
-a.BBO
2.0200
•0. 12500
INFRASPINATUS
19.60
- A . 860
0.9250
• 0 . C5560
-1.130
0.2690
•0.01590
TERES
MAJOR
3.52
TERES
MINOR
3. 16
1.C30
•0.1C8C
0.C0334
31 . 5 0
•12.600
1.9800
•O^ lOCOO
INFRASPINATUS
5.75
-0.836
0.1250
-0.00410
TERES
MAJOR
1.20
- 0 . 163
0.0161
0.C0C53
TERES
MINOR
1.20
1.090
-0.3410
0.02650
16.70
-7.C50
1.02C0
-4.64C00
SUBSCAPULARIS
ADDUCTI3N
CASE
SURSCAPULARIS
PECTORALIS
MAJ.(S)
C.65
0.296
- 0 . 0 7 71
0.00664
PECTORALIS
MAJ^(C)
0.97
C.085
-0.0129
0.C0C32
BICEPS(LONG)
4.28
-1.320
0.2600
-0.01610
RICEPS (SHORT )
5.77
-0.906
O.C777
C.CCCIO
TRICFPS
2.1 7
-0.150
-0.0195
0.00359
CORACOBRACHIAL IS
4.22
C.008
-0.1140
C.C1250
L A T I SSIMUS
1^41
-0.389
0.0578
-0.00203
ORSI
133
...•-O. .-^..^.-^^.r.^..., , , ..,,>.•.,,>.-
134
APPENDIX
(III)
TABLE 3- 2.
COEFFICIENTS OF THEORETICAL
SOLUTION OF MUSCULAR TENSIO^J
SUBJECT NO. : 0
MAME OF
ACTING
MUSCLE
ABDUCTION
WEIGHT
5
LRS
Al
A2
A3
A4
7.08
-1.52 0
0 . 1910
-0.C0605
CASE
DELTOID
ANTERÎOR
DELTOID
MIDDLE
25.90
-6.070
0.8650
-0.04360
DELTOID
POSTERIOR
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8.240
-1.65C0
0 . lOCOO
SUPRASPINATUS
44.10
•20.600
4.69C0
•0.29C00
INFRASPINATUS
45.50
11.3C0
2.1500
-C.12900
TERES
MAJOR
8.16
-2.740
0.6230
•0.03670
TERES
MiNiOR
7.34
2.390
-0.2510
O.C0773
73.00
-29.300
4.59C0
• 0 . 2 3 300
13.30
-2.060
0 .2920
•0.00957
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES MAJOR
2.77
- C . 379
0.0374
0.C0123
TERES
2.77
2.520
-0.7910
0.06160
38,60
•16.400
2.3700
•c.ioeoo
PECTORALIS MAJ. (S)
1.51
0.684
-0.17B0
0.01540
PECTORAL I S
2.24
0 . 199
-0.02S9
0.C0074
MINOR
SUBSCAPULARIS
MAJ . ( C)
9.93
-3.070
0 .6030
•0.03730
13^40
- 2 . 100
0.179C
0.C0027
TRICEPS
5.04
-C.350
-0.0449
0.00831
CORACOBRACHIALIS
9.79
0.019
-0.2650
C.02890
L A T I SSIMLS
3.27
-0.934
0 .1340
•0.00472
BICEPS(LONG)
RICEPS(SHORT )
DORSI
tÍitflfeÍIÉKiílMI
îfSî B^aaaisa
135
APPENDIX
SUBJECT
MAME
(III)
TABLE 3 - 3 .
C O E F F I C I E N T S OF THFORETICAL
SOLUTION OF MUSCULAR TENSIONI
NO.: 0
OF
ACTING
MUSCLE
ABDUCTION
WEIGHT
10
LBS
Al
A2
A3
A4
CASE
DELTOID
ANTERIOR
1 1 . 10
-2.390
0.2980
-0.00947
DELTOID
MIDDLE
40.70
-9.510
1 .3600
-C.06850
DELTOIl) POSTERIDR
- 8 . 44
12.900
-2.5900
0 . 15700
SUPRASPINATUS
69.10
-32.300
7.35C0
•0.45600
INFRASPINATUS
71.30
-17.700
3.3600
•0.20200
TERES
MAJOR
12.80
-4.310
0^9790
•0.05170
TERES
MÎNOR
1 1 . 50
3.780
0.3980
0.01240
115.00
-46.000
7.20C0
•0. 36600
20.90
-3.230
0.4580
•0.01500
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES
MAJOR
4.35
-0.594
0.0587
0.C0192
TERCS
MINOR
4.35
3.960
- 1 .24C0
0.09660
60.60
-25.700
3.71C0
-0. 169C0
SJRSCAPULARI S
2.38
1.070
-O.28C0
0.02410
3 . 52
C.310
-0.0468
0.C0115
BICEPS(LONG)
1 5.60
•4.820
0 . 9 4 7C
-0.05860
BICEf^S (SHORT )
21.00
- 3 . 300
0.2830
0.CC036
7.89
-0.543
-0.0714
0.01310
15.40
C.C24
-0.4150
C. C4 5 3 0
5 . 14
-1.420
0 .2110
PECTGRALIS
MAJ.(S)
PECTORAL ÍS MAJ. ( C)
TRICEPS
CORACDBRACHI AL IS
LATISSIMUS
DORSI
-0.00742
nr^i i -
~
136
APPGNDIX
(III)
TABLE 3- 4.
COEFFICIENTS OF THEGRETICAL
SOLUTION OF MUSCULAR TENSION
SUBJECT NO.: 1
NAME OF MUSCLE
ACTING WCIGHT:
0
LBS
Al
A2
A3
A4
3.41
-0.734
0.0918
-0.00292
DELTOID MÎDDLE
12.50
-2.920
0.4180
-0.02110
DELTOID POSTERIOR
-2.57
3.960
-0.7940
0.04830
SUPRASPINATUS
21.30
-9.990
2.27CC
- 0 . 14100
INFRASPINATUS
2 1.90
-5.390
1 .0300
-0.06210
TERES MAJOR
3.95
-1.330
3.C2CC
-C.01780
TF'^ES MINOR
3.55
1.130
-0^1150
0.00338
ABDUCTION CASE
DELTOID ANTERIOR
35.00
- 1 4 . 100
^•22C0
- 0 . 0 1 120
INFRASPINATUS
6.37
-0.968
0.138C
-0.00^53
TERPS MAJOR
U34
-0.184
0.0184
0.00057
TERES MINOR
1.33
1.210
-0.379C
0.02960
-7.840
1.1300
-5.16000
SUBSCAPULARIS
ADDUCTION CASE
SUBSCAPULARI S
1 8.50
PECTCRALIS MAJ.(S)
0.73
C.328
-0.0856
0.C0738
PECTORALIS MAJ.(C)
1.08
0.09 9
-0.0152
0.00040
RICEPS(LONG)
4.77
-1.470
0.29C0
-0.01800
BICFPS(SHORT)
6.43
-1.010
0.0878
0.00030
TRÎCFPS
2.42
- C . 166
-0.0217
0.00^00
COR^COBRACHIALÎS
4.70
0.017
-0^1140
©•01400
LATISSIMUS DO^SI
1.58
-C.440
0.0655
-0.00232
..,^-...,^,...••.0. .^.... - > . . ^ . - .• .....^
137
APPENDIX
SURJECT
(III )
TABLE 3 - 5 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTION OF MUSCULAR TENSICN
NO.:
1
ACTING
NAME OF MUSCLE
ABDUCTION
WEIGHT
5
LBS
Al
A2
A3
A4
7.32
-1.580
0.1970
•0.00625
CASE
DELTOID
ANTERIOR
DELTniD
MIDDLE
26.90
-6.300
0.9C00
•C.04550
DELTOID
POSTERIOR
-5.52
8.510
-1.71C0
0 . 10^00
SUPRASPINATUS
45.90
-21.500
4.89C0
•0.30300
INFRASPINATUS
47.00
-11.600
2.21C0
•0. 13300
TE^ES
MAJOR
8.50
-2.860
0.6490
•0.03830
TERFS
MiNOR
7.61
2^430
-0^2480
0.00735
75.20
-30.200
4.73C0
•0.24100
13.70
- 2 . C90
0.2970
•C. C0974
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES
MAJOR
2.89
-0.397
0.0398
0.00121
TERES
MINOR
2.84
2.6C0
•0.8150
0.06360
39.80
-16.800
2.4400
• 0 . 1 1 100
PECTORAL Î S M A J . ( S)
1.57
C.708
-0.1840
0.01590
PECTORALIS
2.32
0.212
-0.0327
0.00087
BICEPS (LONG)
10.40
- 3 . 170
0.6230
- 0 . C3860
RICEPS(SHORT)
13.80
-2.180
0.1890
0.00057
5.20
-3.580
-0.0465
0.C0860
10.10
C.034
-0.2760
0.03000
3.40
-C.942
0.14CC
-O.C0^95
SUBSCAPULARIS
MAJ.(C)
TRÎCEPS
CORAC
BRACHIALIS
LATISSIMUS
DO^SI
138
AP=>END X ( I I I )
T A B L E 3- 6.
C O E F F I C I E N T S OF T H E O R E T I C A L
S O L U T I O N GF M U S C U L A R T E N S I O N
SUPJECT NO.: 1
ACTING
NAME OF MUSCLE
ABDUCTION
WEIGHT
10
LBS
Al
A2
A3
A4
CASE
DELTOID
ANTERIOR
1 1.20
-2.420
0.30 3 0
-0.C0964
DELTOID
MIDDLE
41.20
-9.660
1.3800
-0.06980
DELTOID
POSTFRIOR
8.48
13.100
-2.6200
0 . i5<;co
SUP^ASPINATUS
7C.40
-32.900
7.5C00
• 0 . 4 6 600
INFRASPINATUS
72.00
-17.800
3 , 3 9C0
•0. 20500
TERFS
MAJOR
1 3 . 10
-4.390
0.9960
•0.05880
TERES
MINOR
1 1.70
3.690
-0.0375
O . O I 100
115.00
-46.400
7.26C0
•0.36900
21.00
3.200
0.4580
•0.01500
SUBSCAPULARIS
ADDUCTlON
CASE
INF^ASPINATUS
TERES
MAJOR
^.43
6.100
0.0611
0.00186
TERES
MINOR
4.38
3.990
-1.2500
0.09750
61.00
-25.800
3.74CC
• 0 . 17C00
SUBSCAPULARIS
M A J . ( S)
2.43
1.080
•0.2820
0.02430
PECTGRAL IS MAJ . ( C )
3.57
C.328
-0.0505
0.00135
PECTORALIS
BICEPS(LONG)
1 5.70
-4.860
0.9560
-0.05930
BICEPS (SHORT )
21.20
- 3 . 340
0.2890
0.00142
7.98
-0.547
-0.0719
0.01330
15.50
C.C51
-0.4230
0.04600
5.22
-1.450
0.2150
-0.00076
TRI:EPS
CORACOBRACHI AL IS
LATISSIMUS
DO^SI
•MHWiiaHMiMatMaM
;i£it:^^%:
139
APPENDIX
SUBJECT
(III)
T A B L E 3- 7 .
C 0 5 F F I C I E N T S OF T H E O R E T I C A L
S O L U T I O N OF M U S C U L A R T E N S I O M
NO. :
2
ACTING WEIGHT
MAME OF MUSCLE
ABDUCTION
0
LRS
Al
A2
A3
A4
CASE
DELTOID
ANTERIOR
2.70
-C.578
0.0725
•0.C0230
DELTOID
MIDDLE
9.80
-2.280
0.3160
•0.01570
DELTOID
POSTERIOR
-2.15
3.160
-0.7110
0 . 03 790
SUPRASPINATUS
1 5.90
-7.380
1.6700
•0. 10200
INFRASPINATUS
17.60
-^.490
0.8290
•C. C 4 7 8 0
TERES
MAJOR
2.97
-0.994
3.2260
•0.01320
TERES
MINOR
2.66
1.080
-0.1330
0 . C0509
28.70
-11.500
1.7800
0.09020
INFRASPINATUS
5.31
-0.904
0.1220
•0.00390
TFRFS
MAJOR
1.01
C.133
0.0120
0.C0059
TERES
MINOR
l.ll
0.974
•0.3070
0.02380
15.30
-6.450
0.9280
-4.21CC0
MAJ.(S)
0.53
0.270
-0.0701
0.00598
PFCTORAL I S MA) . ( C )
0.85
C.C54
-0.C063
0.C0020
BICEPS(LONG)
^•83
-1.200
0.2320
•0.01A20
BICEPS (SHORT )
5^ 12
-C.8C0
0.0641
O.C0040
TRICEPS
U95
-0.149
-0.0159
0.00311
CORACOBRACHIALIS
3.80
-C.C03
-0.0976
0.C1090
L A T I SSIMUS
1.21
-0.033
0 .0477
- 0 . 0 0 162
SUBSCAPULARIS
ADDUCTION
CASE
SUBSCAPULARI S
PECTORALIS
DORSI
!ii.jj.tia.--i j
ip.^Y»..^.^
r. -
.1 »
,.,
140
APPENDIX
SUBJECT
NAME
(III)
TABLE 3 - 8 .
C O E F F I C I E N T S OF TUEORFTICAL
SOLUTION OF MUSCULAR TENSION
NO.: 2
ACTING
OF MUSCLE
WEIGHT
5
LBS
Al
A2
A3
A4
7.02
-1.52 0
0 . 1880
-0.005P4
-5.960
0.8260
-C.041C0
• 62
8.240
-1.6300
0.09890
SUPRASPINATUS
41.50
-19.300
-4.36C0
-0. 26600
INF^ASPINATUS
45^90
•11.700
^•1600
•0.12700
-2.590
©•5890
•0.03A50
ABDUCTION
CASE
DELTOID
ANTERIOR
DELTOID
MIDDLE
25.60
DELTOÎD
POSTERIOR
_ R
TERES
MAJOR
7.75
TERES
MINOR
6 . 95
2.810
-©•3460
0.01330
74^80
•30.000
4.66C0
•0. 23500
13^90
-2.370
0.32CC
•0.01C30
SUBSCAPULARIS
ADDUCTION CASE
INFRASPINATUS
TERES
MAJOR
2.63
-C.341
0^0030
0.00158
TERES
MINOR
2.91
2.540
-0.801C
0.06210
39.80
•16.800
2.4200
-O.UCOO
SUBSCAPULARIS
PECTORAL IS
MAJ.(S)
1.39
C.709
- 0 . 1840
0.01560
PECTORALIS
MAJ.(C)
2.22
0 . 142
-0.0167
-0.00C03
BI CEPS(LONG)
10.00
- 3 . 120
0.6050
•0.03700
B I C F P S Í SHORT )
1 3,40
-2.090
0.1680
© • 0 0 114
TRÎCEPS
5.08
-C.391
-0.0412
0.00811
CORACOBRACHIALIS
9.90
-0.074
-0.2560
0.02840
LATISSIMUS
3.16
-0.856
DORSI
0 . 1250
-0.00426
^
.-••.-•»^..r—^.
— . > -
141
APPENDIX
SUBJECT
(III)
TABLE 3 - 9 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTION OF MUSCULAR TENSION
NO.:
2
ACTING
NAME OF MUSCLE
ABDUCTION
WEIGHT:
10
LBS
Al
A2
A3
A4
CASE
DELTOID
ANTERIOR
1 1.40
-2.430
0.3050
-0.00963
DELTOID
MIDDLE
41.40
-C.640
1.34C0
- C . 06630
DELTOID
POSTERIOR
-9.07
13.800
-2.6400
0.16000
SUPRASPINATUS
-9.07
1 3 . 300
- 2 . 64C0
0.16C00
INFRASPINATUS
67.20
-31.200
7.0500
-0.43100
TERES
MAJOR
74.20
-18.900
3.5CCC
-0.20600
TERES
MINOR
12.60
-4.200
3.9540
-0.05590
4.520
-0.0557
0.02130
121.00
- 4 8 . 500
7.54C0
-0,381C0
1 1.30
SUBSCAPULARIS
ADDUCTION
CASF
INFRASPINATUS
TERES
MAJOR
22.40
-3.820
0,5140
-0.01660
TERES
MiMOR
4.27
-0.555
0,C497
0,00253
4.69
4 . 110
-1.3000
O.IOCOO
PECTORAL IS MAJ . ( S )
64,40
- 2 7 . 200
• OECTORAL I S MAJ . ( C)
2,25
1 . 150
-0.2970
0,02530
3.60
C. 231
-0.0272
0.0
SUBSCAPULARIS
BICEPS(LONG)
3.92C0 '
- 0 , 17800
BICEPS(SHORT)
16.20
-5.050
3.9780
-0.05980
TRICFPS
21.70
-3.380
0.2720
0.C0181
CORACOBRACHIALIS
8.20
-0.626
-0.0678
0.04590
LATISSIMUS
5.11
- 1 . 380
0.2020
-0.C0687
DO^SI
r i rSiílii
142
APPENDIX
SU3JECT
(III)
TABLE 3 - 1 0 .
C O E F F I C I F N T S OF THEORETICAL
SOLUTION OF MUSCULAR TENSIOM
NO.:
3
ACTING
NAME OF MUSCLE
ABDUCTION
WEIGHT
0
LBS
Al
A2
A3
A4
CASE
DELTOID
ANTERIOR
2.25
-0.435
0,06C7
-0,00193
DELTOID
MIDDLE
8.23
-1,930
0,2740
-0.01380
DELTOID
POSTERIOR
-1,71
2,620
-0,0524
0.03180
SUPRASPINATUS
M,00
-6.520
1,4800
- 0 . 0 9 190
INFRASPINATUS
14.40
-3.580
0,68C0
-0.04C80
0 , 1980
• 0 . 0 1 170
TERES
MAJOR
2.59
-0.873
TERES
Mir^oR
2,32
0.767
-0,0812
0.00254
23.20
-9,320
1,4600
•0.07400
INFRASPINATUS
4.23
-C,656
0.0926
-0.00303
TERFS
MAJOR
0.88
-0.119
0.0117
0.00040
TERES
MINOR
0.88
0.801
-0.2510
0.01960
SURSCAPULARIS
2.20
-5.200
0.7520
-0.03420
PECTORAL ÍS MAJ . ( S)
0,48
0.219
-0.C568
0.C0489
PECTORALIS
0,71
0,063
-0,0094
0.00023
BICEPS(LONG)
3,16
-0.977
0 , 1920
-0.01190
BICEPS(SHORT)
4,25
-0.668
0,0572
o.occoa
TRICEPS
1,60
- 0 . 112
-0,0141
0.C0263
CURACOBRACHI A L I S '
3,11
0.003
-0,0840
0.00917
L A T I S S ÍMUS
1,04
-C.287
0.0425
- 0 . C0149
SUBSCAPULARIS
ADDUCTION
CASE
MAJ.(C)
DO^SI
ff^
^-^—^-——
143
APPFNDIX (III)
TARLE .3-11.
COFFFTCIENTS OF THEORETICAL
SOLUTION OF MUSCULAR TENSIOM
SUBJECT NO.: 3
NAME OF MUSCLE
ACTING WEIGHT
5
LRS
Al
A2
A3
A4
6.29
-1,360
0.17C0
-0,00540
A3DUCT10N CASE
DELTdlD
ANTERIOR
DELTOID
MIDDLE
23. 10
-5,4C0
0,7700
-0,03880
DELTOID
POSTERIOR
-4.80
7.330
1,4700
0,08910
SUP^ASPINATUS
39. 10
•18.300
•4,16C0
-0.25700
INFRASPINATUS
40.50
•IC.OOO
1,91C0
-0.115C0
-2.430
0,5520
-0.03260
TERES MAJOR
7. 24
TEPES MINOR
6.49
2.150
0,2270
0.00712
65.CO
• 2 6 . 100
4,0800
-0.20700
1 1.90
-1.840
0.26C0
-0.00852
TERES MAJOR
2.46
-C.334
0,0329
0.00111
TERES MINOR
2,47
2.240
-0,7020
0.05470
3^.40
-14.600
2, lOCO
-0,09570
SURSCAPULARIS
ADDUCTION CASE
INFRASPINATUS
SUPSCAPULARIS
PECTORALIS
M A J . ( S)
1,33
0.613
-0,1590
0,01370
PECTORALIS
MAJ.(C)
1.99
C. 177
-0.0266
0.00065
8, 83
-2.740
0,5370
-0.0 3 320
1 1,90
-1.870
0, 16C0
0,00023
TRICEPS
4,48
-0.312
-0,0398
0.00738
CORACOBRACHIALIS
8,71
C.012
-0,2350
C.02570
LATISSIMUS
2,90
-0.801
0,1180
-0.00417
BICEPS(LONG)
BICEPSISHORT )
DO^SI
^,*^
144
APPENDIX
(III)
TABLE 3-12.
COEFFICIENTS OF THEORETICAL
SOLUTION OF MUSCULAR TENSIOM
SUBJECT NO.: 3
ACTING WEIGHT:
MAME OF MUSCLE
Al
A2
A3
10
LBS
A4
ú
• í '
ABDUCTION
CASE
DELTOID
ANTERIOR
10.30
- 2 , 220
0,2780
-0,C0e82
DELTOID
MIDDLE
37.90
-8,860
1 ,2600
-0,06360
DELTOID
POSTERIOR
-7.86
12,C00
-2,41CC
0,14600
SUiíRASPI NATUS
64.20
-30.000
6,8200
-0,42200
INFRASPINATUS
66,40
-16.400
3,13C0
- 0 . 18800
TERES MAJOR
1 1,90
-4.000
0,9090
-0,05360
TERES
10,70
3.530
-0.3740
C,01170
07,00
-42.900
6,7000
-0,34100
19.50.
-3.030
0,4290
•0,01410
MINOR
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES
MAJOR
4.04
-0.550
0,0540
0.C0183
TERES
MINOR
4,07
3.680
-1,15CC
0.08990
56,50
•23.900
3,4600
•0.15700
SUBSCAPULARI S
PECTORALIS
MAJ.(S)
2,19
1.010
-0,2620
0,02250
PECTORALTS
MAJ.(C)
3.27
C.288
-0,0432
0,C0105
BICEPS(LONG)
14.50
-4.490
0,8810
•0,05^50
PICEPS(SHORT )
19.50
-3.080
0,2640
0,C0032
7.36
-0.512
-0,0656
0.01210
CPRACOBRACHIALIS
14.30
C.C22
-0,3870
0.04220
lATÍ SSIMUS DORSI
4.77
-1.320
0,195 0
•0.00686
TRICFPS
145
APPENDIX
SURJECT
NAME
(III)
TABLE 3-13,
COEFFICIENTS OF THEGRETICAL
SOLUriON OF MUSCULAR TENSION
NO,: 4
ACTING WEIGHT:
OF MUSCLE
ABDUCTION
0
LBS
Al
A2
A3
A4
2.89
-C.618
0,0775
-0.00245
CASE
DELTOID
ANTERIOR
DELTOID
MIDDLE
10.50
-2.440
0,3350
-0.01650
DELTOID
POSTERIOR
- 2 , 35
3.390
-0.6710
0.04050
SUPRASPIMATUS
1 6,70
-7.740
1.7400
- 0 . 106C0
INFRASPINATUS
18,90
-4.880
0.8930
-0.05210
TERFS
MAJOR
3,14
-1.C50
0.2380
-0.01390
TERES
MINOR
2,80
1.220
-0.1570
0.00630
-12.400
1.9300
-C.09710
SUBSCAPULARIS
3 1 .00
ADDUCTION CASE
ÎNFRASPINATUS
5.77
-I.CIO
0,134C
-O.0OA31
TERES
MAJOR
1.07
-0.138
0,0119
0,67800
TERFS
MINOR
1.21
1.C50
-0.3320
0,02530
16.60
-6.990
1 .0100
-0,04560
SUBSCAPULARIS
PECTORALIS
MAJ.(S)
0.55
C.295
-0,C763
0,00648
PECTORALIS
MAJ.(C)
0.91
C.050
-0.0050
-0,00012
RICEPS(LONG)
4.12
-1.290
0.249C
-0.01520
BICEPSÍ SHORT)
5.31
-0.859
0.0673
0.00061
TRICEPS
2.09
- C , 165
-0.0168
0,00333
CORACORRACHIALIS
4.09
-0.044
-0.1040
0.01 160
LATISSIMUS
1.29
-C,346
0.C5C0
-0,00169
DORSI
IdtfMiilUMiiMaaM
-—-*•*—•-*'«•'>»
146
APPENDIX
SUBJECT
NAME
( I 11 )
TABLE 3 - 1 4 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTÎON OF MUSCULAR TENSIGN
NO.: 4
ACTING
OF MUSCLE
ABDUCTlON
WEIGHT:
5
LBS
Al
A2
A3
A4
7.09
-1.520
3.1910
-0.00603
CASE
DELTOID
ANTERIOR
DELTOID
MIODLE
25.70
-5.980
0.8210
- C . C4050
DELTOID
POSTERIOR
-5.75
8.320
- 1 .6500
0.09940
SUPRASPINATaS
41.00
-1<9.C00
4.2800
-0.26100
INFRASPINATUS
46.50
-12.000
2.1900
-0.12800
TERES
MAJOR
7.70
-2.570
0.5840
-0.C3410
TERES
MINOR
6.89
2.980
- 0 . 3 84 0
0.01540
76.20
-3C.600
4.73C0
-0.23800
14,20
-2.490
0,3320
-0.C1070
0.00169
SURSCAPULARIS
ADDUCTION
CASE
INFRASPÎNATirS
TERES
MAJOR
2,62
-0.335
0,0289
TERES
MÎNOR
2,97
2.580
-0.8150
40,70
17.200
2.4700
-0.11200
PECTORAL I S M A J . ( S)
1,37
0.723
- 0 . 1870
0.01590
PECTORALIS
2,24
0 . 124
-0.0124
-0.00029
BICEPS(LONG)
10.10
-3.170
0.6110
-0.03730
RICrPS(SHORT)
13,50
- 2 . 110
0 . 1650
0.00152
5 . 14
-C.407
-0.0410
0.C0816
10.10
-0.112
-0.2540
0.02850
3,16
-0.849
0 . 1230
-0.C0414
SUBSCAPULARIS
MAJ.(C)
TRICEPS
CORACOBRACHI A L I S '
L A T I S S IMUS
UO^SI
0. 06290,
H
Í iattJsláHI É
147
APPFNDIX
SUBJECT
(I I )
TABLE 3 - 1 5 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTION GF MUSCULAR TENSIOM
NO.:
NAME OF
4
ACTING WEIGHT:
MUSCLE
ABDUCTION
Al
10
LBS
A2
A3
A4
CASE
DELTOID
ANTERIOR
11,30
-2.410
0,3020
-0.00954
DELTOID
MIDDLE
41,00
-9.530
1 ,3100
-©•06460
DELTOID
POSTERIOR
-9.16
13,300
-2.62C0
SUPRASPINATUS
6 5 , 50
-3C,300
6.83C0
-0,41500
INFRASPINATUS
74,00
- 1 9 , 100
3.4900
-0,20A00
TERES
MAJOR
1 2 , 30
-4,C90
0,93CO
-0,05A40
TERES
MINOR
10,90
4,770
-0,6140
0,02460
21,00
-48,700
7.5400
-0,38C00
22,60
-3,960
0,5270
-0,01690
SUBSCAPULARIS
ADDUCTION
0 , 15800
CASE
INFRASPINATUS
TERES
MAJOR
4,17
-0,532
0,0456
0,00271
TERES
MINOR
^ . 75
4 . 100
-1,3000
c'ioooo
-27,300
-3,93C0
-0,17800
SUBSCAPULARIS
64.70
PECTORAL IS
MAJ.(S)
2 . 17
1 . 150
-0,2980
0,02530
PECTORALIS
MAJ.(C)
3,58
0 . 196
-0,0194
-0,00048
BICEPS(LONG)
1 6 , 10
-5.050
0,9730
-0.00593
BICEPSISHORT)
21,50
-3.36 0
0,2630
0,00235
8 , 19
-C.647
-0,0653
0,01300
1 6,00
- 0 . 176
-0,4060
0,04540
5,02
-1.350
0,1950
-0,00657
TRICEPS
CORACOBRACHI AL I S'
LATISSIMUS
DO^SI
ntiiiBr i r - - —
tw4^M
^t?mi
148
APPFNDIX
SUBJECT
(III)
TABLE 3-16.
C O E F F I C I E N T S OF T H E O R E T I C A L
S O L U T I O N OF M U S C U L A R T E N S I O M
NO, : 5
ACTING
NAME OF MUSCLE
A3DUCTI0N
WEIGHT
0
LBS
Al
A2
A3
A4
3.83
-0.826
0 . 1030
-0.00327
-3.300
0,4740
-0,02400
CASE
DELTOID
ANTERIOR
DELTOID
MIDDLE
1 4 . 10
PELTOID
POSTERIOR
-2.86
4.460
-0.8950
0.05450
SUP^ASPINATUS
24.30
-11.400
2.5900
-0,16100
ÎNFRASPINATUS
24,60
-6.020
1, 1600
•0.07010
TERES
MAJOR
4.49
-1.510
0,^430
•0.02030
TERES
MINOR
4.02
1,230
-0,1210
0.00334
39.20
-15.700
2,4700
•0. 12600
INFRASPINATUS
7.12
-1.060
0,1530
-0,00504
TERES
MAJOR
1.52
-C.210
0,0213
0,00061
TERES
MINOR
1.48
1,360
-0,4250
0.03320
20.70
-8.770
1,27CC
-0.05780
SUBSCAPULARIS
ADDUCTION
CASE
SUBSCAPULARIS
PECTORALIS
MAJ,(S)
0,84
C.36 6
-0,0957
0.00827
PECTCRALIS
MAJ,(C)
1.21
0.117
-0,0185
0.C0054
BICEPS(LONG)
5,36
-1.650
0,3260
-0.02020
BICEPS (SHORT )
7,23
- 1 . 140
0,0995
-0.CCC03
TRICEPS
2,72
-0.1B2
-0,0249
0,00453
CORACOBRACHI ALIS'
5,27
C.029
-0.146C
0,01580
LATISSIMUS
1,79
-C,499
0.0744
-0,00264
DORSI
• Waoi !
l iiiifn n líiiiBiiii I I
149
APPENDIX
(III)
TABLE
3-17.
C O E F F I C I E N T S OF T H E O R E T I C A L
S O L U T I O N OF M U S C U L A R TENSIOM
SUBJECT NO.: 5
ACTING
N A M E OF M U S C L E
ABDUCTION
WEIGHT:
5
LBS
Al
A2
A3
A4
8.07
-1,740
0.2170
-0,00691
-6,960
1,0000
-0,05070
- 1 ,88C0
0. 11500
CASE
DELTOID
ANTERIOR
DELTOID
MIDDLE
29.70
DELTOID
POSTERIOR
-6.04
9 . 380
SUPR^SPINATUS
51,00
-23.900
-5,4500
-0,33900
INFRASPINATUS
51.70
-12.700
2,A4C0
•0. 14600
TERES
MAJOR
9.45
-3.130
0,7210
•0,04260
TERES
MINOR
8.46
2.590
-0.2550
0 , C0723
82,50
-33.100
5,2000
-0,26500
1 5,00
-2.240
0,3230
-0,01070
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES
MAJOR
3.21
-C.443
0,0450
0, C0128
TERES
MINOR
3,11
2.870
-3,9000
0,07000
43,60
-18.500
2,67CC
- 0 , 12 200
PECTORAL I S MAJ . ( S)
1 ,76
0.774
-0,2020
0,01740
PECTORALIS
2,56
C.247
-0.039C
O.CO 113
BICEPS(LONG)
11,30
-3.480
0,6850
-0,04260
BICEPS(SHORT )
15,20
-2.400
0,21C0
-0,C0CO9
5,72
-0.3B2
-0,0527
0,00955
11,10
C.C56
-0,306C
0.03320
3.77
-1.050
0,1560
-0,00554
SUBSCAPULARIS
MAJ.(C)
TRICEPS
CORACOBRACHIALIS
LATISSIMUS
DO^SI
150
APPENDIX
SUBJECT
(III )
TABLE 3 - 1 8 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTÎON OF MUSCULAR TENSIOVJ
NO.: 5
ACTING WEIGHT:
MAME OF MUSCLE
ABDUCTION
Al
10
LBS
A2
A3
A4
CASE
DELTOID
ANTERIOR
12.30
-2.650
0.3310
-0,01050
DELTOID
MIDDLE
45.20
-10.600
1,5200
-0,07720
DELTOID
POSTERIOR
- 9 . 19
14,3C0
-2,87CC
0, 17500
SUPRASPINATUS
77.80
-36,400
8.3100
-0,17500
INFRASPINATUS
78.70
-19.300
3.71C0
-0.22400
TERES
MAJOR
14.40
-4.860
1.1000
-0.06510
TERES
MINOR
12.90
3,950
-0.3880
0.01070
126.00
-50.500
7.9300
-0.04040
22,90
-3.440
0,4970
•0.01650
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
TERES
MAJOR
4,89
-0.678
0.0689
C.00193
TERES
MINOR
4,74
4 . 370
-1 .3700
0 . 10700
66.50
•28.200
4.0800
•0.18500
2.68
1. 190
-0.3090
0.02660
C,376
-0.0594
0.C0173
SUBSCAPULARIS
PECTORAUIS
MAJ.(S)
PECTORAL I S M A J . ( C)
3.91
17.20
-5,310
1 .0500
•0.06500
23.20
-3.650
0,3200
•O.COOll
fl-73
-0.588
-0 ,0794
0.01450
CORACOBRACHIALIS
16.90
0.039
-0,4670
C.C5060
L A T I SSIMUS DORSI
5.75
-1.600
0,2830
-0,00848
BICFPS(LONG)
BICEPS(SHORT )
TRICEPS
rittBBSai
151
APPENDIX
SUBJECT
NAMF
(III )
TABLE 3 - 1 9 .
C O E F F I C I E N T S OF THEORETICAL
SOLUTION OF MUSCULAR TENSIGN
NO.: 6
ACTING
OF MUSCLE
ABDUCTION
WEIGHT
0
LBS
Al
A2
A3
A4
3.66
-C,790
0.0981
•0.00310
CASE
OELTOID
ANTERIOR
DELTOID
MIDDLE
13.62
3,220
0.4770
•0.02500
DELTOID
POSTERIOR
- 2 . 52
4.200
-0,8550
0,05200
SUPRASPINATUS
24.60
•11.500
2,6500
•0, 16700
INFRASPINATUS
22.90
-5.360
1,0800
•0,06800
TERES
MAJOR
4.55
-1.530
0.34 80
•0.02070
TERES
MINOR
4.02
0.910
-0.0550
•0,00019
35,80
•14.400
2.28C0
• 0 , 1 1 720
-6,35
0.780
0,1290
-0.00^60
SUBSCAPULARIS
ADDUCTION
CASE
TNFRASPINATUS
TERES
MAJOR
1 , 54
C.270
0.0254
0,00031
TERES
MINOR
1.30
1.280
-0.3960
0.03120
1 8 . 70
7.980
1,1600
-0.05300
MAJ.(S)
0.88
0,340
-0.0880
0 . 0 0 770
PECTORAL IS MAJ . ( C )
1.17
0.150
-0.0269
0.00108
PICEPS(LONG)
5.01
1.520
0.3050
-0.01920
RICFPS(SHORT)
6.84
1.080
0 . 1020
-0.C0062
TRICEPS
2.54
0 . 143
-0.0263
0.00490
CORACOBRACHIALIS
4.91
C.C98
- 0 . 1460
0,01540
LATISSIMUS
1,78
0.510
0.0780
-0.00287
SUPSCAPULARIS
PcCTORALIS
DORSI
5gg:;g
152
APPFNDIX
(III)
TABLE 3-20.
COEFFICIENTS OF THEORETICAL
SOLUTTON OF MUSCULAR TENSION
SUBJECT NO.: 6
ACTING
NAME OF MUSCLE
ABDUCTION
WEIGHT:
5
LBS
Al
A2
A3
A4
7, 49
-1,620
0.2010
-0.00638
CASE
DELTOID
ANTERIOR
DELTOID
MIDDLE
27,90
-6.590
0,0979
-0.C5C70
DELTOID
POSTERIOR
- 5 , 15
8,600
-1 ,75C0
0. 10700
SUPRASPINATUS
50,30
-23,600
5,4300
-0.34300
INFRASPINATUS
46,90 ^
-U.CCO
2,2300
-0.13900
TERES
MAJOR
9,32
- 3 . 140
0,7130
-0.04250
TERES
MINOR
8. 25
1,860
-0,1120
-0.00043
73,40
-29.500
4,6800
-0.24CC0
13.00
-1.620
0.2650
•0.00942
TERES MAjnR
3 . 17
-C.464
0,0520
0.00064
TERES
2.65
2.630
-0,8120
0.06^00
38,50
•16.300
2,3800
•0,10900
SUBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
MINOR
SUBSCAPULARIS
PECTORALIS
MAJ,(S)
1,79
0.688
- 0 , 1810
0.01580
PECTORALIS
MAJ.(C)
2,40
C.308
-0,0555
0.00223
BICEPS(LONG)
10.30
- 3 . 110
-0,6250
-0.03940
BICEPS(SHORT )
14.00
-2.210
0,2090
- 0 . 0 0 126
5.21
- 0 . 290
-0,0545
0.00914
1 0 , 10
0.200
-0,2990
0.03150
TRICEPS
CORACOBRACHIALIS
LATISSIMUS
DORSI
3.65
-1.040
0 . 1590
-0.00585
153
APPENDIX
(III )
TABLE 3-21.
C O E F F I C I E N T S OF T H E O R E T I C A L
S O L U T I O N OF M U S C U L A R T E K S I C N
SUBJECT NO.: 6
ACTING
NAME OF MUSCLE
ABDUCTION
Al
WEIGHT:
10
LBS
A2
A3
A4
CASE
DELTOID
ANTERIOR
11.30
-2.450
0.3040
-COICOO
DELTOIO
MÎODLE
42.20
9.970
1.A8C0
-C,C7680
DELTOID
POSTERIOR
-7.80
13.000
-2.6500
0 , 16200
SUPRASPINATUS
76.10
-35.700
8.21C0
- 0 . 51900
INFRASPINATUS
70,90
-16.600
3,3600
-0.21C00
TERES
MAJOR
1 4 . 10
-4.760
1,08CC
-0.06430
TERES
MINOR
12,40
2.830
-0,1730
-0.00049
11,00
-44.600
7,07C0
-0,36300
19.80
-2.490
0,4070
-0.
SJBSCAPULARIS
ADDUCTION
CASE
INFRASPINATUS
0U50
TFRES
MAJOR
4.78
-0.700
0,0782
0.00100
TERES
MINOR
4,01
3.970
-1,23CC
0,09670
58.20
-24.700
3,6000
-0,16A00
SUBSCAPULARIS
PECTORAL IS
MAJ,(S)
2,71
1.C30
-0,2730
0,02380
PECTORALIS
MAJ.(C)
3.62
0.468
-0,0845
0,00340
BICEPS(LONG)
15.50
-4.710
0,9460
-0.05960
BICEPS(SHORT)
21.20
-3.350
0,3160
-0,00193
7.88
-0.439
-0,0822
0,01380
15.20
0.30 0
-0,4520
0,04770
5.53
-1.580
0.241C
- 0 , 0 0 687
TRICEPS
CORACOBRACHIALIS
LATISSIMUS
DO^SÎ
APPENDIX IV
DOCUMENTATION OF COMPUTER PROGRAM
Programmer:
Young-Pil Park
Advisor:
Dr. C. A. Bell
Machine Used
IBM 370/145
Language:
Fortran IV
Compiler:
Fortran G Compiler
Date Completed:
December 1976
Compile Time:
Approximately 2 minutes
Computation Time
Part I: 2 minutes
Part II: 5 minutes
Part III:
1 minute
Lines of Output:
Each 2000 lines
Purpose:
This program is designed to analyze the musculo^
skeletal system of the human should joint.
Part I (ANTHR):
Calculation of the necessary anthropometric
data of a subject and geometrical data of each
muscle that are necessary for the mathematical
analysis.
Part II,(THEOR)
Calculation of the theoretical solution of the
muscular tension in the shoulder muscles of a
subject and the relationships between muscular
tension and the abduction (adduction) angle of
the arm.
154
-.'Ví
•"im
155
Part III (COEFF):
Calculation of the linear coefficients relating muscular tension and electromyographic signal intensity of a subject.
3.
Inputs and all the necessary nomenclature used in this program
are explained and listed in the content of the program.
4.
Definition of symbols used in the flowcharts:
M
- Problem code
SA
- Anthropometric data of a subject
SF
- Scale factor of a subject
GM
- Geometrical data of muscles of the cadaver
GS
- Geometrical data of muscTes of the subjects
D
- Direction Cosines of muscular tension and moment
CG
- Center of gravity of the body segments
SW
- Weight of segments
EW
- Effective weight
EM
- Effective moment
SF
- Scale factors of a subject
N
- Case code (abduction = 14, adduction = 17)
AB
- Abduction coefficient matrix
AD
- Adduction coefficient matrix
SUM
- Moment coefficients
SUMM - Summation of moment coefficients
CE
- Coefficient matrix of muscular tension
F
- Solution vector of muscular tension
KK
- Number of coefficients for curve fitting
m
156
KD
- Number of case for curve fitting
LQ
- Least square curve fitting coefficient matrix
CFM
- Curve fitting for muscular tension
EMG
- Electromyographic signal intensity
157
Program Flowcharts
The Main Program
0
Call
ANTHR
vattf. 'ii!
158
Subroutine ANTHR
0
Read
SA
159
Compute
GS
Write
GS,D
1
Return
3
irr
iMii.HHWmill—^l
160
Subroutine THEOR
0
Read
SA
Compute
CG,SW
HMBKSíaKBB
161
^
Wri te
SA,CG,SW,
EW,EM,SF
Set
AB=0
0
^"T™™^?
162
AN=AN-100
Compute
SUM
1
Compute
Compute
CE
SUMM
0
miw.im
163
k
t
/
Write
/
'
'
/
/
!
1
Read
/
/
KK,DK
/
^'
Compute
LQ
'
f
Call
FITIT
,
/
^
Write
,
/
/ "" 1
1'
f
Return
J
Subroutine COEFF
Read
KK,KD
Write
CFM
C
Return
)
165
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
*
*
APPENDIX(IV)
*
*
*
TITLE
*
PROGRAMMER
:
-
CGMPUTER
PRCGRAMMING-
A MATHEMATICAL ANALYSIS OF THE
MUSCULO-SKELETAL SYSTEM OF THE
3LEN0-HUMERAL JOINT
:
YGUNG P I L PARK
GRADUATE STUDENT
TEXAS TECH UNI VERSI TY
*
«
*
*
OATE
:
MAY
1977
«:ít){t«j}c«5}c:;c«<t««««3îc:íts>::ec3{.«««:{c){t3{t««)>3{!«*:ít«>{c:ít:íc){c*«::tj{(«:ít3:tj0c
T H I S PRDGRAM I S MAINLY CONSISTED OF THREE
MAIN PARTS ACCORDING TO THEIR PULPOSES
READ ( 5 , 1 ) NC3DE
1 FORMAT ( 1 5 )
c
c
c
c
c
c
c
c
c
c
«*:(t:í::«c«*:0tj0ti{t;ít«;ít30c«j(c>^:íc«j{cj{tJÍ::{t««30t«j{t:{íj;ci{:!{t3}t3!t:Oíj(c){t*«:^«){c:{t«5{e
*
MAIN
*
*
îí:
*
*
*
T H I S PROGRAM DETERMINES WHICH
METHODS HAS TO BE EXECUTED FOR
THE PULPOSE OF CALCULATION
*
*
*)»:4t<ts!t:{c){c:{s:{ti>:j(tj{(3!t){t:{e:ít«3!t«:í!>îej{t«*«j5c){t)0t:{tjîc«:í!«){c)ît:{c)!t:0c*«*««*«
IF
(NCODE)
2,3,4
2 CALL ANTHR
GO TO 5
C
C
C
C
C
C
C
c
c
c
*
*
«
PART
I
SUBROUTINE
ANTHR
CALCULATION OF NECESSARY ANTHRCPGMETRIC
DATA OF I N D I V I D U A L SUBJECTS
DATA
SET
ORDER
Hl PROCESS DETERMINATION CODE ( - 1 )
H2 PERSONAL B A S I C ANTHRCPOMETRIC DATA
1^^^^——^
,^^^^^^^^„1
.- ..-—.^.-^
166
C
C
C
C
C
C
C
C
a^
*
MODFL GEOMETRICAL DATA FOR I N D I V I D U AL MUSCLES ( I N D I V I D U A L MUSCLES, 0 TO
9 0 DEGREES BY 10 DEGREES ÍNTERVAL )
THIS PROGRAM
SI X SUBJECTS
IS
PROGRANMED FOR THE
CASE
3 CALL THEGR
3 0 TO 5
C
C
C
PART
II
SUBROUTINE
THEOR
C
C
C
C
C
C A L C U L A T I O N DF T H E O R E T I C A L S O L U T I O N AND
5TH O R D E R C U R V E F I T T I N G FGR THE M U S C U L A R
TENSILE FORCE V S . ABDUCTION (ADDUCTION)
A N G L E S FOR A SINGLE S U B J E C T
C
C
C
C
C
C
C
C
C
C
DATA SET ORDER
«1 PROCESS DETERMI NATIGN CODE
( 0 )
#2 PERSONAL B A S I C ANTHRGPOMETRIC DATA
( S P E C I F I C ONE SINGLE PERSON)
H3 A N T I C I P A T I N G NUMBER OF MUSCLES
( F I R S T ABDUCTION CASE :
14)
#4 MODEL GEOMETRICAL DATA FOR A S P E C I F I C
ANGLE ( EACH ANGLE 14 MUSCLE )
H5 AN3LE •• 1 0 0 . 0 VALUE CARD
C
^6
210.0
C
C
C
C
C
C
C
*
PROCEED #4 ANC U5 UP TO 90 DEGREES
FROM 10 OEGREES BY 10 DEGREES I N T E R V .
m
A N T I C I P A T I N G NUMBER OF MUSCLES
(SECOND ADDUCTION CASE
: 17 )
PROCEED « 4 AND # 5 UP TO 90 DEGREES AS
ABDUCTION CASE PROCESS
C
C
C
C
C
C
C
C
C
C
C
*
^8 2 0 0 . 0
^9 CURVE F I T T I N G
( 5, 10 )
4 CALL COEFF
PART
ÎII
INFORNATICN
SUBROUTINE
CARD
CGEFF
C A L C J L A T I O N OF T H E L I N E A R C O E F F I C I E N T S
BETWEEN MUSCULAR TENSILE FORCES V S .
E . M . 3 . SIGNAL I N T E N S I T I E S
DATA S E T O R D E R
m l l W B r a •raini I iJiMn_L5«S^a^SBBWfe^
167
C
C
C
C
^l P R O C F S S D E T E R N I N A T I O N CODE ( +1 )
#2 C U R V E F I T T I N F I N F O R M A T I O N C A R D
( 2 , 81 )
C
C
C
C
C
C
C
^3
THEGRETICAL SOLUTION AND EXPERIMENTAL
RFSULT DATA
( THECPETICAL RESULTS GF
t>ART I I AND E . M . G . RESULTS OF THRE
DELTOIDS PARTS ( 8 1 DATA FOR THREE
PERSON, THREE WEIGHT, 9 ANGLES ) )
C
C
C
C
C
* SUBROUTINE
LINEQ
FOR THE S O L U T I O N OF LINEAR S I M U L T A N E O U S E O U A T I O N S OF T H E O R E T I C A L PART
C
C
C
*
SUBROUTINE
FIFIT
FOR THE CURVE F I T T
c
5 CALL E X I T
END
NG PROCEDURE
168
SUBROUTINE
C
C
C
C
C
C
C
C
C
c
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c
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c
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c
c
ANTHR
*)0t*4e*){t){tJ0t,»:)(c«j»:j(c4*j{tj{C){c)<t*){ct<t«+*+*4*«>»)*)»)»J^J^j)rj{.j{.**)>j(c)^
*
SUBROUTINE
ANTHR
*
*
*
*
*
*
T H I S TS THE COMPUTER PRGGRAMMING FOR
ThlE CALCULATION OF THE NECESSARY ANTHROTMETRIC DATA FOR ANALYSIS
*
*
*JÍt*JÎ:********:^;(,,ît:0,:^^3j^:^i^3^^^3^^^^^^^^^^^3^^^^j^^j^j^^
NOMENCLATURES
P
W
H
BW
CH
UAL
AR
Al
A2
A3
A4
A5
A6
A7
NUMBER OF THE SUBJECTS
WEIGHT ( L B S )
HEIGHT ( F T )
B I A C R O M I A L WIDTH ( F T )
CHEST HEIGHT ( F T )
UPPER ARM LENGTH ( F T )
LOWER ARM LENGTH (FT)
A B D U : T I O N ( A D D U C T I O N ) ANGLE (DEGREE)
X-COMPONENT OF MUSCLE LENGTH (MM)
Y-:OMPONENT OF MUSCLE LENGTH (NM)
Z-:OMPDNENT OF MUSCLE LENGTH (MM)
X-COORDINATE OF INSERTION (MM)
Y-COORDINATE OF INSERTION (MM)
Z - C 0 3 R D I N A T E OF I N S F R T I O N (MM)
DIMENSION P ( 7 ) , W ( 7 ) , H ( 7 ) , B W ( 7 ) , C H ( 7 ) ,
/ U A L ( 7 ) ,AR( 7) , S F X ( 7 ) , S F Y ( 7 ) , S F Z ( 7 ) , Y ( 7 , 1 5 , 1 0 ) ,
/ A l ( 1 5 , 10 ) , A 2 ( 1 5 , 1 0 ) , A 3 ( 1 5 , 1 0 ) , A 4 ( 1 5 , 1 0 ) ,
/ A 5 ( 1 5 , 1 0 ) , A 6 ( 1 5 , 1 0 ) , A 7 ( 1 5 , 10),X(7,15, 10),
/Z(7,15,10),XL(7,15,10),YL(7,15,ia),
/ZL(7,15, 10),TL(7,15,10),DX(7,15,10),
/DY(7,15,10),DZ(7,15,10),AX(7,15,10),
/AY(7,15,10),AZ(7, 15,10),TM(7,15,10),
/ D M X ( 7 , 1 5 , 10) ,9MY( 7 , 1 5 , 1 0 ) , D M Z ( 7 , 1 5 , 1 0 )
C
C
c
c
c
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C
C
C
c
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c
PRINTING
ORDER
I . D E L T 3 I D ANTERlOR
2 . D E L T 0 I D MIDDLE
3 . D E L T 0 I D POSTERIOR
4.SUPRASPINATUS
6 . INFRASPINATUS
7 . T E R E S MAJOR
8 . T F R E S MINOR
9 . S U B S : APULARIS
l O . P E C R O R A L I S MAJOR (STERNAL)
ll.PECTORALIS
MAJOR(CLAVICULAR)
169
C
C
C
C
1 3 . B I C E P S (SHDRT)
14,TRICEPS
15.C0RAC0BRACHIALIS
W R I T E ( 6 , 1)
FORMAT(3X,«SUBJECT',3X,•WEIGHT',5X,»HEIGHT',
/ 6 X , » B W ' , 8 X , « C H « , 7 X , « U A L ' , 8 X , « AR' , 7 X , ' S F X ' ,
/ 7 X , 'SFY' , 7 X , « S F Z ' , / / )
c
c
c
c
READ ANTHRCPOMETRIC DATA GF I N D I V I D U A L
( THESE DATA ARE COLLECTED FROM MEASURING
)
DO 4 1 = 1 , 7
R E A D ( 5 , 2 ) P ( I ) ,W( I ) , H ( I ) , B W ( I ) , C H ( I ) ,
/ U A L ( I ),AR( I)
2 FORMAT ( 7 F 1 0 . 4 )
C A L C U L A T I O N OF SCALE
C
C
C
C
C
SFX
SFY
SFZ
SCALE FACTOR
SCALE FACTOR
SCALE FACTOR
FACTCRS
IN
IN
IN
X-DIRECTION
Y-DIRECTIGN
Z-DIRECTION
SFX(I)=(BW(I)/1.25)
SFY(I)=((CH(I)/1.2525) + (UAL(I)/1.036))'!'0,5
SFZ( I ) = S F X ( I )
W R I T E ( 6 , 3 ) P ( I ),W ( I ) , H ( I ) , B W ( I ) ,CH( I ) ,
/ U A L d ) , A R ( I ) ,SFX( I ) ,SFY( I ) ,SFZ( I )
3 FORMAT( 1 0 F 1 0 . 4 , / / )
4 CONTINUE
WRITE(6,12)
C
C
C
C
READ GE3GRAPHICAL DATA OF THE MODEL
( MEASJRED FROM THE DISSECTED CADAVOR )
DO 5 1=1,15
DO 5 J = l, 10
REA0(5,6) Al (I ,J) ,A2(I ,J),A3(I,J)•A4( I,J),
/A5(I,J),A6(I ,J),A7( I,J )
5 CONTINUE
6 FORMAT(7F10.4)
WRITE(6,12)
DO 14 1=1,7
DO 13 J= l, 15
WRITE(6,7)
7 F0RMAT(6X,»SUBJECT» ,3X,«MUSCLE« ,5X,'ANG.' ,9X,
/ 'X », U X , 'YS IIX, • Z' , lOX, 'LENGTH' ,6X, 'COSXSBX,
/'COSY' ,8X, 'COSZ' , //)
170
WRITE(6,8)
8 FORMAT ( 3 9 X , « L X ' , l O X , » L Y * , l O X , ' L Z ' , 1 0 X , ' T M S ^ X ,
/ • M D O S X ' , 8 X , ' M C 0 S Y « , 7 X , » MCCSZ' , / / )
C
C
C
CALCULATION
OF GEOMETRICAL
DATA FOR
SUBJECTS
DO 11 K = l , 1 0
X( I , J , K ) = S F X ( I ) * A 2 ( J , K )
Y( I , J , K) = S F Y ( I ) * A 3 ( J , K )
Z( I , J , K ) = S F Z ( I ) * A 4 ( J , K )
XL(I,J,K)=SFX(I)*A5(J,K)
YL ( Î , J , < ) = SFY( I ) * A 6 ( J , K )
71( I , J , K ) = SFZ( I ) * A 7 ( J , K )
T L ( I , J , < ) = ( X ( I , J , K ) « « 2 +Y( I , J , K ) « * 2 +
/Z( I, J,K)«*2)**0. 5
c
c
c
c
C A L C U L A T I O N OF THE D I R E C T I O N COSINFS
FORCES &ND MOMENTS
CF
DX ( I , J , K ) = X( I , J , K ) / T L ( I , J , K )
DY(I ,J,K) =Y ( I , J , K ) / T L ( I , J , K )
DZ ( I , J , < ) = Z( I , J , K ) / T L ( T , J , K )
AX ( I , J , < ) = D Y ( I , J , K ) * Z L ( I , J , K ) - D Z ( I , J , K ) *
/YL(I,J,K)
AY( I , J , K ) = DZ( I , J , K ) * X L ( I , J , K ) - D X ( I , J , K ) *
/ZL ( I , J , < )
AZ ( I , J , K ) = D X ( I , J , K ) * Y L d , J , K ) - D Y ( I , J , K ) *
/XL(I ,J,K)
T M ( I , J , K ) = (AX( I , J , K ) * * 2 + A Y ( I , J , K ) ' í = « 2 - ^
/ A Z ( I , J , < )* * 2 ) « * 0 . 5
DMX( I , J , K ) = AX( I , J , K ) / T M ( I , J , K )
D M Y d , J,K) = A Y ( I , J , K ) / T M ( I , J , K )
DMZ( I , J , K ) = AZ( I , J , K ) / T M ( I , J , K )
WRITE(6,9) I , J , A 1 ( J , K ) , X ( I , J , K ) , Y ( I , J , K ) ,
/ Z ( I , J , K ) ,T L ( I , J , K ) , D X ( I , J , K ) , D Y ( I , J , K ) ,
/DZ(I,J,K)
9 F 0 R M A T ( 2 I 10 , 8 F 1 2 . 5 )
W R I T E ( 6 , 10 ) XL ( I , J , K ) , Y L ( I , J , K ) , Z L ( I , J , K ) ,
/TM(I ,J,<) , DMX(I,J,K),DMY(I,J,K),DMZ(I,J,K)
10 F3RMAT(32X , 7 F 1 2 . 5 , / / )
11 CONTINUE
WRITE(6,12 )
12 FORMAT(IHl , 5 X )
13
14
CONTINIUE
W R I T E ( 6 , 12
CONTINUE
RETURN
END
)
•HUB
171
SUBROUTINE
THEOR
C
C
C
**){t**){t*j|t){t*){t**)fj{c**j{t>>*)(cj{t*j»tJ^j{tj>4:*4j>*jf*jOt**««*)J«:*){tj{t
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
*
c
c
c
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c
c
c
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c
c
c
c
c
*
*
*
*
*
*
*
SUBROUTINE
THE3R
*
*
THIS I S THE COMPUTER PRCGRAMMING FOR THE
CALCULATION OF THE THECRETICAL CALCULATlON OF MUSCULAR T E N S I L E FORCES AND FOR
THE FOUNDING OF THE RELATlONSHIPS BETWEEN THESE VALUES VS. ABDUCTICN(ADDUCTIGN
ANGLES OF THE ARM
*
*
*
*
*
* * * ) { : * ){t)0t>!tj0e>!t*****>!t*jît)^j(cjî:*){t*j0c3îc*4jîrj}!)>«jt*<t*j!tj{c:{tj0tjj«*j0t*
NOMENCLATURES
P
H
LW
W
BW
CH
XUAL
XLAL
XUAC
XLAC
XUAM
XLAW
TW
XM
NUMBE R OF THE SUBJECTS
HEIGH T ( F T )
L I F T I NG WEIGHT ( L B S )
W E I GHT ( L B S )
BIACR OMIAL WIDTH ( F T )
CHEST HEIGHT ( F T )
UPPER ARM LENGTH ( F T )
LCWER ARM LENGTH ( F T )
UPPER ARM C . G . ( F T )
LOWER ARM C . G . ( F T )
UPPER ARM WEIGHT ( L B S )
LOWER ARM WEIGHT ( L B S )
EFFEC T I V E WEIGHT ( L B S )
EFFEC T I V E MOMENT ( F T - L B S )
REAL N 1 , L X , L Y , L Z , L E N G T H , K 1 , L W , J L B , P
INTEGER
ZZ
PRINTIN3
ABDUCTION
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
ORDER
CASE
REACTION FORCE I N X - D I R E C T I C N
REACTION FORCE IN Y - D I R E C T I C N
REACTION FORCE IN Z - D I R E C T I O N
LAGRANGE'S M U L T I P L I E R V I
LAGRANGE«S M U L T I P L I E R V2
LAGRANGE'S M U L T I P L I E R V3
D E L T O I D ANTERIOR
D E L T O I D MIDDLE
D E L T O I D POSTERIOR
SUPRASPINATJS
172
C
C
C
C
C
C
C
C
C
c
1 1 . INFRASPINATUS
1 2 . TERES MAJ3R
1 3 . TERES MINOR
1 4 . SUBSCAPULARI S
DIMENSION
Al( 25,25 ),D1(25),X9(25),C(16) ,XD(150),
/ Y D ( 1 5 0 ) , Y C d 5 0 ) , ABD(20,100, 10),ADD(20,1C0,10)
ADDUCTION
CASE
1.
REACTICN
FORCE
IN
X-DIRECTICN
2.
REA:TION
FORCE
IN
Y-DIRECTÍCN
C
C
C
C
C
C
C
C
C
C
C
C
c
3 . REACTION FORCE IN Z - D I R E C T I O N
4 . LAGRANGE'S M U L T I P L I E R V I
5 . LAGRANGE'S M U L T I P L I E R V2
6 . LAGRANIGE'S M U L T I P L I E R V3
7.
INFRASPINATJS
8 . TERES MAJOR
9 . TERES MINDR
1 0 . SUBSCAPULARIS
1 1 . PECTORALIS MAJOR ( STERNAL )
1 2 . PECTORALIS MAJGR ( CLAVICULAR )
1 3 . BICEPS ( LONG )
1 4 . BICFPS ( SFORT )
15,
TRI:EPS
C
C
C
16. CORACOBRACHILIS
17, LATISSIMUS D RSI
COMMON X(10, 150) ,A( 10)
C
C
C
C
READING ANTHROPDMETRIC DATA FOR SINGLE
SUBJECT AND LIFTING WE IGHT
READ(5,1) P,LW,H,W,BW,CH,XUAL,XLAL
1 F0RMAT(8F10.4)
C
C
C
C A L C U L A T I O N OF C . G
AND WEIGHT
OF
SEGMENTS
XUAC=0.43569*XUAL
XLAC=0.5544*XLAL
XUAW=0,02647*W
XLAW=0.02147*W
C
C
C
CALCULAT ON OF EFFECTIVE WEIGHT AND MCMENT
XLW = XUAL-^XLAL
XLAAC=XUAL-^XLAC
XTW=XUAW4-XLAW
mÊ*m
^
173
TW = LW4^XTW
XM=XUAC*XUAW4-XLAAC*XLAW + XLW«LW
C
C
C
DETERMINATION
OF
SCALE
FACTGRS
SFX=(BW/ 1 . 2 5 )
SFY=((CH/1.2525)-»-(XUAL/l.C36))*0,5
SFZ = SFX
W R I T E ( 6 , 2 ) P , L W , H , W , B W ,CH , X U A L , X L A L
W R I T E ( 6 , 2 ) XUÛC,XLAC,XUAW,XLAW,XLW,XLAAC
W R Î T E ( 6 , 2 ) XTW ,TW,XM
WRITE( 6 , 2) S F X , S F Y , S F Z
2 F0RMAT(8F15 . 5 )
3 FORMAT ( 1 2 )
4 FORMAT
(7F10,4,/,7F10.4,/,8F10.4,///)
5 FORMAT ( » 1 SOLUTIGN OF ' ^ I ^ , *
S IMULTANECUS,
/ L I N E A R ALGEBRIC
E O U A T I O N • / / , ' 0 COEFFICIENT
/ MATRIX
:•//)
6 FORMAT ( « 0 SOLUTION VECTOR:'//)
C
C
MAKE THE COEFFICIENT MATRIX ZERO AND
DETERMINE
LIFTING METHCDS
c
c
8
9
10
11
c
c
c
READ(5,3) M
IF (M,EO.14) GO TO 8
IF (M.EO- 17) GO TO 10
NUM = 21
DO 9 1=1,14
Dl(I)=0.
Al( I ,J)=0.
GO TO 12
DO 11 1=1,17
Al (I ,J)=0.
Dl( I ) = 0.
INITIAL DATA FGR MOMENT COEFFICIENT.
12 SUMXX=0.
SUMXY=0.
SUMXZ=0.
SUMYY=0.
SUMYZ=0.
SUMZZ=0.
c
c
c
REACTION
FORCE
1=6
Al ( 1 , 1 ) = 1 , 0
CO EF F IC I ENT S .
áhgna TiBsaiíí møm
174
Al ( 2 , 2 ) = 1.0
Al(3,3)=1.0
C
C
C A L C U L A T I O N OF L E N G T H , D I R E C T I O N C O S I N E S
FORCES AND MGMENTS FROM B A S I C DATA
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
OF
N = ABDUCTIONI OR A D D U C T I O N CER3REE
X X = X - C O M P O N E N T OF LENGTH VECTOR
Y = Y - C O M P O N E N T OF LENGTH VECTOR
Z = Z - C O M P O N E N T OF LENGTH VECTOR
LX=X-COMPONENT 3F MOMENT ARM VECTOR
L Y = Y - C O M P O N E N T OF MOMENT ARM VECTOR
L Z = Z - C O M P O N E N T OF MG^ENT ARM VECTOR
READ
GEOGRAPHICAL
13
READ
( 5 , 14)
14
FORMAT (
1 = 1+1
TF ( N . E O . 2 0 0 . )
IF(N.EQ.210.)
IF(N.GT.100.)
DATA
(MOCEL
CADAVOR)
N,XX,Y,Z,LX,LY,LZ
7F 1 0 . 3 )
CALCULATION
GO
G3
G3
TO
TO
TO
25
7
15
OF ACTUAL
ANTHRCPOMETRIC
XX=(SFX*XX)/304.8
Y=( SFY'î^Y) / 3 0 4 . 8
Z=(SFZ*Z) / 3 0 4 . 8
LX=(SFX-^^LX)/304.8
LY = ( S F Y < ' L Y ) / 3 0 4 . 8
LZ = (SFZ'!'LZ ) / 3 0 4 . 8
LENGTH = ( X X * * 2 - ^ Y « * 2 - ^ Z * * 2
)**0.5
DCOSFX = X X / L E N G T H
DCOSFY = Y / LEN3TH
DCOSFZ = Z / LENGTH
AX=DCOSFY*LZ-DCOSFZ*LY
AY=DCOSFZ«LX-0C0SFX*LZ
AZ=DCOSFX*LY-DCOSFY*LX
Kl= ( AX**2 + AY**2-»-AZ«*2)**0.5
DC0SMX=AX/K1
DC0SMY=AY/K1
DC0SMZ=AZ/K1
DCOMXX=-(DCOSMX*DCOSMX)
DCOMXY=-(DCOSMX*DCOSMY)
DCOMXZ=-(DC0SMX*DCOSMZ)
DCOMYY=-(DCOSMY*DCOSMY)
DCOMYZ=-(DC0SMY*DC0SMZ)
D:0M7Z=-(DC0SMZ*DCGSMZ)
DATA
175
SUMXX=SUMXX+D:OMXX
SUMXY=SUMXY+D:OMXY
SUMXZ=SUMXZ+D:OMXZ
SUMYY=SUMYY+D:OMYY
SUMZZ=SUMZZ+D:GMZZ
SUMYZ=SUMYZ+D:OMYZ
DE = 0 . 0 1 7 4 5 3 2 * \ i
C
C
C
SETTING
Al
Al
Al
Al
Al
Al
COEFFICIENT
^ATRIX
( 1,1 ) = DCOSFX
( 2 , 1 ) = DCOSFY
( 3 , 1 ) = DCOSFZ
( I , 4 ) = ( ABSOCOSMX) ) / 3 0 4 . 8
( 1,5 )= ( ABS(DC3SMY) ) / 3 0 4 . 8
(I,6)=(ABS(DC0SMZ) )/304.8
Al(I,1)=K1
GO TO 1 3
15
c
c
c
c
c
c
c
N=N-100.0
Al(4,4)=(SUMXX)/304.8
Al(4,5)=(SUMXY)/3C4.8
Al (4,6)=(SUMX? )/3 0 4 . 8
Al(5,5)=(SUMYY)/304.8
A1(5,6)=(SUMYZ)/304.8
Al(6,6)=(SUMZZ )/304.8
Al ( 5 , 4 ) = A 1 ( 4 , 5 )
Al (6,4)=A1 (4,6)
A1(6,5)=A1(5,6)
16 D 1 ( 2 ) = T W
Dl(4)=XM*SIN(DE)
WRITE ( 6 , 5 ) M
DO 17 1 = 1 , M
17 WRITF ( 6 , 4 ) ( A l ( I , J ) , J = 1 » M ) , D 1 ( I )
WRITE ( 6 , 1 8 )
18 FORMAT ( I H l , 5X )
WRÍTE ( 6 , 6 )
CALL FOR
EOUATION
CALL
WRITE
SGLUTION OF LINEAR SIMULTANEOUS
I N THE FORM OF MATRIX
LINE0(A1,D1,X9,f)
THE
SOLUTION
VECTOR
WRITE ( 6 , 4 )
(X9(I),1=1,M)
WRITE ( 6 , 1 8 )
I F ( L W . E 3 . 0 . 0 ) GO TO 19
NN = N
176
19
20
21
22
23
24
25
NW = LW
GO TO 20
NN=N
LW = LW-»-1.0
NW = LW
IF(M.EQ.17) GO TO 22
DO 21 1 = 7,M
ABD( I,NN|,NW) = X9( I )
GO TO 24
00 23 1=7,M
ADD( I ,NN,NW) =X9( I )
CONTINUE
IF(M.EQ.14) GG TO 8
IF (M.EQ. 17) GO TO 10
CONJTINUE
DO 26
26
27
28
29
C
C
C
C
C
C
1 = 7 , 14
DO 26 NN = 1 0 , 9 0 , 1 0
W R I T E ( 6 , 2 7 ) I , NN , ABD ( I , NN , NW)
FORMAT ( 1 4 , 1 1 5 , F 1 0 . 4 )
0 2
i0'\.j
1 . 18
>
»1—= t7, , 1
i 7
f
DO 2 8 N N = 1 0 , 9 0 , 1 0
WR
W R 1 T E ( 6 , 2 9 ) I , N N , A D D ( I ,NN,NW)
FORMAT ( 1 4 , 1 1 5 , F 1 0 . 4 )
CURVE
KK
KD
FITTING
OF
THE THEGRETICAL
SCLUTIGN
NUMBER OF ORDER ( 5TH
NUMBER OF POINTS ( 10
R E A D ( 5 , 3 0 ) KK,KD
30 F O R M A T ( 2 I 4 )
K O P l = KD + 1
K K P l = <K + 1
C
C
C
C
SCALE FACTCRS FOR X AN D Y CATA
( Y I S MUSCULAR TENSl L E , X I S ANGLE
SCY=1.0
SCX=1. 0
PP = 1 . 0
Y=0.001
X1=0.0
YD( 1 ) = Y
XD(1 )=X1
Y=SCY*Y
X1=SCX*X1
X( 1 , 1) = 1 . 0
X ( 2 , 1 )=X1
)
^^
177
X( 3 , 1 ) = X 1 * * 2
X(4,l)=x1**3
X( 5 , 1 ) = X 1 * * 4
X(KKP1,1)=Y
3 1 11 = 7
32 X l = 1 0 . 0
DO 35 1 = 2 , K D
NX=X1
NW=LW
I F ( P P . G T , 8 . 0 ) GO TO 33
Y=ABD(II,NX,NW)
GO TO 34
33 Y=ADD( I I , N X , N ^ )
34 CONTINUE
YD(I)=Y
XD(I)=X1
Y=SCY*Y
X1=SCX*X1
X( 1 , I ) = 1 . 0
X(2, I)=X1
X(3,1)=X1**2
X(4»I)=X1**3
X( 5 , I ) = X 1 * * 4
X(KKPl,1)=Y
35 X 1 = X 1 + 1 0 . 0
c
c
c
CALL CURVE
36
/
/
FITTI
CALL F I T I T
(KD,KDP1,KK,KKP1)
PRINT 3 6
FORMATI 1 H 0 , 3 0 X ,
34HT^E CALCULATED C C E F F I C I E N T S
9H F O L L O W S - / / )
ARE A S ,
C
c
c
ACTUAL
37
38
COEFFICIENTS
ARE PRINTED OUT
DO 38 J = 1,KK
WRITE(6,37) J , A ( J )
FORMAT( 1 H 0 , 4 4 X , 2 H A ( , 12 , 4 H )
CONTINUE
77 = 0
JLB = 0 , 0
SS = 0 . 0
P = 0.0
SD = 0 . 0
SUM = 0 . 0
DO 40 J = 1 ,KD
T = X ( K K - H , J)
=
,3X,E12,5)
178
G = 0.0
DO 39 K = 1 , K K
Ql = X ( K , J ) * A ( K )
39
G=G+Q1
YC(J) = G
J L f i = J L B 4- ABS( ( T - G ) / T )
T
=
T-G
IF(
SS
40
41
/
/
/
/
/
/
42
43
44
45
46
/
/
47
T . L T . O . 0)
=
SS
ZZ = ZZ -^ 1
•»• T
P = P + ABS ( T )
SD = SD -^ G*G
SUM = SUM 4- T * T
FDRMAT( IHO,20HNUMBER OF DATA P 0 I N T , I 4 /
10X,18HSQUARED D E V I A T I O N , E 1 2 , 5 /
1 0 X , 1 0 H D E V I ATION , E 1 2 , 5 /
lOX,18HA3SDLUTE D E V I A T I O N , E l 2 5 /
10X,24HSUM OF THE SQ, OF C A L , Y , E 1 2 . 5 /
10X,30HNUM3ER OF DATA PT, G T . STAND. , 1 4 /
10X,24HSUM OF THE AVG D E V I A T I O N , E 1 2 . 5 / / )
^
WRITE(6,41)
KD,SUM,SS,P,SD,ZZ,JLB
A(KK-el) = 1 .0
DO 43 K = 1 , K K P 1
AV = 0 . 0
0 0 42 J = 1,KD
AV = AV -^ X (K , J)
C(K) = AV*A(K)/FLOAT(KD)
DO 45 J = 1 , K K P 1
WRITE(6,44)
J,C(J)
FORMAT( H O , 1 0 X , 2 H C ( , I 2 , 3 H ) = , E 1 2 . 5 )
CONTINUE
PRINT 4 6
FORMAT(///17X,16HINDEPENDENT
DATA,8X,
14HDEPENDENT DATA , 8 X , 1 6 H C A L C U L A T E D VALUE,
8X,9HDEVIATI0N,8X,13HPERCENT ERROR///)
DO 49 I = 1,KD
YC(I) = 0.0
DO 47 J = 1,KK
YC(I) = YC(I) + A ( J ) * X ( J , I )
C
CHAN3E
c
c
48
49
YCd)
ONLY
IF
DATA
IS
SCALED
Y C ( I ) = YC( I ) / S C Y
DEV = Y D ( I ) YC(I)
PCE = 1 0 0 . 0 * A B S ( D E V ) / Y D ( î )
WRITE ( 6 , 4 8 ) XD( I ) , Y D ( I ) , Y C ( I ) , D E V , P C E
FnRMAT(lH0,21X,F9.2,llX,F11.6,l?X,Fll,6,
1 2 X , F 8 , 5 , 1 2 X , F 8 . 3)
CONTINUE
179
WR I T F ( 6 , 1 8 )
PP=PP+1,0
11=11+1
50
IF ( P P . 3 T , 1 9 , 0 ) 3 0 TO 5 0
I F ( P P , L E . 8 . 0 ) GD TO 32
I F ( P P . E Q . 9 . 0 ) GO TO 31
GO TO 32
CONTINUE
RETURN
END
180
SUBROUTINE
F I T IT
C
C
C
•
c
c
c
*
G
C
C
C
)0tj{t**)0cj(t*>5ts(e*4t)(t*:t<:*4cj}ejO:j0tjOc
(N,NP1,M,MPl)
««««:(t<t^j!t:ít:ít:>:{t«*:íc:Ot*«*)<f)ít)!t:íc)OcjOe>}tJÎij{í^jOcjOtj;tj{tjOtjOtJ*;jOt*jOt«)Ot3»jjc*
SUBROUT INJE F I T IT
*
T H I S PROGRAM IS FOR THE CALCULATION OF
OF INVERSE MATRIX THAT I S USED FOR THE
CURVE F Î T T I N G PRGBLEM
Ocj^«««)Ot)ítj!t«jîí:*«:O5
« « • * * * « « « « «
M IS NUMBER OF C O E F F I C I E N T S
N I S NUMBER OF DATA POINT
A ( I ) ARE THE OESIRED C O E F F I C I E N T S
c
c
c
30
40
50
60
92
70
80
COMMON X( 10 , 1 5 0 ) , A ( 1 0 )
DIMENSION Z ( 1 0 , 1 5 0 )
DO 50 I = 1,M
DO 40 J = 1 , M P 1
Z(I ,J) = 0.0
DO 30 K = 1 ,N
Z d , J ) = Z(I,J) + X(I,K)*X(J,K)
CONTI NUE
CONTI NUE
DO 110 KM = 1 , M P 1
K = M -f 2 - KM
D = 0.0
DO 92 I = 2 , K
Í F ( A B S ( Z( I - l , 1 ) ) . L E . D) GO TO 60
L = I-l
D = ABS(Z(L,1))
CONTINUE
CONTINJE
IF((L-l).E0.0)
GO TO 80
DO 70 J = 1 ,K
D = Z(L,J )
Z(L,J) = Z(1,J )
Z(1,J) = D
CONTINUE
DO 90 I = 1 fM
90
A d ) = Z( 1,1)
95
DO 100 J = 2 , K
D = Z( 1 , J ) / A ( 1 )
0 0 95 I = 2 ,M
Z(I-1,J-1)
= Z d ,J)
Z(M,J-1) = D
CONTINUE
RETURN
END
100
110
-
A(I)*D
181
SUBROUTINE
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
L I N EQ ( A , B ,X , N )
***«**««'5c*«**i;'^«í!t)<t«)}:>!f4:*jOtJOc:^3!f««j}tjOt5!fjOcjOt«)ît:{t«jOcjOc:{i:Ot«30!j(t
*
*
*
SUBROUTINE
LINEQ
*
*
*
*
«
T H I S I S THE SUBROUTINE
OF LINEAR SIMULTANEOUS
FCR THE SOLUTION
EQUATION
*
*
*
if
jOt:î:<t*«JOc4t«>!t:Oc«*««***««*){t){t)!t«){:){tJÎt)OtJ!<JÎc««j{t>ît>5i>!ejO::{t«:Ot««««'ít
FUNCTIGN
RFFERENCES
THE C O E F F I C I E M T MATRIX ( A )
THE FORCE VECTOR ( B )
THE NUMBER OF EQUATICNS ( N )
THE SUB^OUTINE W I L L RETURN THE SOLUTIGN
VECTOR ( X ) TO THE C A L L I N G PROGRAM
DIMENSION
DO 4
1
2
3
4
5
A(25,25) ,B(25) ,X(25)
I=1 , N
DO 2 K = 1 , N
F ( K . E 3 . I ) GO TO 2
CONST = - A ( K , I ) / A ( 1 , 1 )
DO 1 J = l , N
A(K,J)=A(K,J)4-C0NST*A(I,J)
I F ( J . E O . I ) A( K, J ) = 0 .
CONTINUE
B(K)=B(K)+CONST*B(I)
CONTINUE
CONST=A( 1 , 1 )
DO 3 J = 1 , N
A( I , J ) = A ( I , J ) / C O N S T
A(I,1)=1.
B ( I ) = B ( T )/C3NST
CONTINUE
DO 5 I = 1 , N
X ( I )=B( I )
RETURN
END