: Approved - TDL
251
A BIOMECHANICAL MODEL FOR THE UPPER EXTREMITY USING OPTIMIZATION TECHNIQUES by MAHMOUD A. AYOUB, Bo in C6Eo, MoSo in I.E. A DISSFRTATION IN INDUSTRIAL 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, 1971
Transcript of : Approved - TDL
A BIOMECHANICAL MODEL FOR THE UPPER EXTREMITY
USING OPTIMIZATION TECHNIQUES
by
MAHMOUD A. AYOUB, Bo in C6Eo, MoSo in I.E.
A DISSFRTATION
IN
INDUSTRIAL 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, 1971
A~ SO/ T3 197( /vQ. 11-~o;;J Z
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my
committee chairman, Dr. M. M, Ayoub, I am also deeply
indebted to Dr. R. A. Dudek, Professor w. Sandel, Dr. J. D.
Ramsey, Dr. A. Walvekar, Dr. c. Halcomb, and Dr. c. Waid,
the other members or my advisory committee, for their
helpful advice and constructive criticism throughout the
entire study.
11
TABLE OF CONTENTS
ACKNOWLEDGMENTS o 0 0 • 0 0 0 • 0 Q 0 • 0 0 0 0 0 " 0 0
LIST OF TABLES • o • •
LIST OF ILLUSTRATIONS o
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I.
III.
INTRODUCTION 0. 0 0. 0 0 0 0 0 0 0 0 0 0 0 0
Biomechanics • • • • o • • o o o o o
Techniques of Human Motion Analysis o
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Statement of the Problem
Purpose and Scope o
THE MODEL o o o • o o •
Assumptions •
Notation • o
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Dynamic Analysis • • 0
Performance Criteria 0
The Model o
MODEL SOLUTION
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Suboptimization o 0 0 0
Dynamic Programming o
Simulation • • o • o
MODEL IMPLEMENTATION o o
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Planar Motion Problem •
Dynamics of the Arm o
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Model Algorithms 0 0 • 0 0 0 0 0 0 0 0 0 0 71
Choice of Model Algorithm 0 0 0 0 0 0 0 0 0 96
Vo MODEL TESTING 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 105
The Task 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 105
Experimental Variables 0 0 0 0 0 0 0 0 0 0 106
Equipment 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 113
Experimental Procedure 0 0 0 0 0 0 0 0 0 0 116
Hand Path of Motion 0 0 0 0 0 0 0 0 0 0 0 0 117
Results and Interpretations 0 0 0 0 0 0 0 0 122
VI. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 0 0 0 148
Summary 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 148
Conclusions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 150
Recommendations for Further Research 0 0 0 153
LIST OF REFERENCES 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 156
APPENDIX 0 0 0 0 0 0 0 • 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 169
LIST OF TABLES
Table
lo Anthropometric Characteristics 0 0 0 0 0 0 0 0 0
Anthropometric Characteristics for the Subjects o o o o o o o o o o o o o o 0 0 0
Correlation Coefficients between Experimental Paths and the Theoretical Ones o o o o o o
Percentage Differences in Areas between Experimental Paths and the Theoretical Ones
Percentage Differences of Motion Path Coordinates • • o • • • • o o o o o o o o o
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LIST OF ILLUSTRATIONS
Figure
1" Biomechanics • 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2o Techniques of Human Motion Analysis 0 0 0 0 0 0 0
3o Structural Equivalent of Muscles' Actions 0 0 0 0
4o Displacement-Time Curve • 0 0 • 0 0 0 0 0 a a a 0
5o Reactive Forces and Moments Acting upon a Link at Any Instant (t) during Its Motion a 0 a
6o Reactive Forces and Moments Acting upon the ith Cross Section •• 0 0 0 0 0 0 0 a a 0 0 a a
Feasible Region for the Hand Path of Motion under Suboptimization Approach • • a o o o a o
Arm Motion for Dynamic Programming Approach o 0 0
Stages of Dynamic Programming Approach 0 0 0 0 0
lOo Arm Motion under Simulation Approach, Assuming Sine, Ellipse, Parab~la As Possible Shapes
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for the Hand Path of Motion a o o o o o o o o o 56
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Arm Motion under Simulation Approach, Using Enumeration o o o o o o o o a a • a o o o o 0 0
Arm Configuration at an Instant t during the Motion o o o o o o o a o o o o o o o o o 0 0 0
Free Body Diagram of the Forearm-Hand Link 0 0 0
Free Body Diagram of the Upper Arm 0 0 0 0 0 0 0
Stages of Dynamic Programming Approach
Dynamic Programming Iteration Scheme
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Simulation Iteration Scheme o o o a o o 0 0
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18o Simulation--Sine and Ellipse Functions • • 0 • 0
19o Simulation--Parabola 0 0 0 0 0 0 0 0 • • 0 0 0 0
Task Configurations • 0 • 0 0 0 0 0 0 0 0 0 0 0 0
Hand Path of Motion--Task I 0 0 0 0 0 0 0 0 0 0 0
22o Hand Path of Motion--Task II 0 0 0 0 0 0 0 0 0 0
Hand Path of Motion--Task III o • • 0 0 0 0 0 0 0
Hand Path of Motion--Task IV • 0 0 0 0 0 0 0 Q 0
25o Task Configuration 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Arm Segment Masses and Their Locations Expressed as Percentages of Arm Lengths and Total Body Mass • o o o o o o • o o o o o o
Typical Effect of 40% Variations in the Anthropometric Coefficients upon the Optimum Path of Motion o o o o o o • 0 0 0
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28o Experimental Equipment • 0 0 0 0 0 0 0 0 0 0 0 0
29o Typical Examples of the Photographic Records Obtained during the Experiment o o o • o o
Correlation Analysis for Two Paths of Motion
Motion Performed at Elbow Height of 6 Inches above the Work Surface and Distance of 9 Inches o o o o • o • o o o o o o o o • o
32o Motion Performed at Elbow Height of 6 Inches above the Work Surface and Distance of 12 Inches o o o o o o o o o o o o o 0 0 0 0
33o Motion Performed at Elbow Height of 6 Inches above the Work Surface and Distance of 15 Inches o o o o • ~ ~ o o o o o o 0 g 0 g
Motion Performed at Elbow Height of 3 Inches above the Work Surface and Distance of 9 Inches g o o o o o o o o o a o a o o o o
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Figure Page
35o Motion Performed at Elbow Height of 3 Inches above the Work Surface and Distance of 12 Inches 0 0 0 0 0 II 0 0 0 0 0 0 0 a • 0 0 0 0 127
36o Motion Performed at Elbow Height of 3 Inches above the Work Surface and Distance of 15 Inches 0 0 • • 0 • • 0 • 0 0 0 0 0 • 0 0 0 0 128
37o Motion Performed at Elbow Height of 0 Inches above the Work Surface and Distance of 9 Inches 0 0 0 0 • 0 0 0 IJ 0 • 0 0 0 0 0 0 0 e 129
38o Motion Performed at Elbow Height of 0 Inches afiove the Work Surface and Distance of 12 Inches 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 130
39a Motion Performed at Elbow Height of 0 Inches above the Work Surface and Distance of 15 Inches 0 • 0 • 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 131
40o Effect of Work Surface Height upon Accuracy of Prediction 0 0 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 142
4lo Effect of Motion Distance upon Accuracy of Prediction 0 0 0 • • 0 0 0 0 0 0 0 0 0 0 0 0 0 143
42o Effect of Subject upon Accuracy of Prediction 0 0 144
CHAPTER I
INTRODUCTION
Biomechanics
In a society which has witnessed space missions
and highly sophisticated automatic systems, it still is
man who has to fly supers9nic jets, guide spacecraft and
operate high speed computerso Without mant the decision
maker, the prime mover, the controller, many of our
sophisticated systems would not function0 Over the
years, the advancements in science and technology have
made it possible to learn about both man's capacity and
limitations under various envrionmentso Biomechanics
(Figure 1), utilizing the findings of the ta~ic elhlncer
ing sciences, anatomyi physiology and psychology, is on~
of several scientific disciplines which studies man within
his environmento Biomechanics is concerned with the sci
entific study of the interaction between the human body
and the external forces resulting from the surrounding
working environmento The term environment includes all
working conditions external to the human body whether such
are on earth under normal or stressful situations, or on
the lunar surface under subgravitational effects0
1
2
Theoretical Mechanics
Dynamic Anatomy Dynamic Anthropology
\
Neuro-Muscular Psychomotorics Physiology
~ • r
BIOMECHANICS
-• • General Biomechanics Applied Biomechanics
. Biostatics Biodynamics - Industry & Trade
/~ .
I
Agriculture & Forestr;y '• -Kinematics Kinetics ~
I Medicine •, -
r-Military Work
1-Sport
1--Art
...._ Everyday Living
Fig. l.--Biomechanics (adopted from Contini, 1963]
Basically, biomechanics measures and assesses the mechan-
ical and physiological characteristics of the human body
in motion and rest. The main objective of biomechanics is
to increase the efficiency of human performance by mini
mizing or economizing the effort required to perform the
motor activities. It analyzes and specifies the functional
capabilities and limitations of the human body under dif-
ferent environments.
Applications of biomechanics appear in almost every
area where man performs an activity. Significant improve-
ment of the human body utilization in various scientific
fields has been accomplished through biomechanical analy
sis. Some of these are as follows: (1) industry [Taylor,
1912; Gilbreth, 1917, 1919; Darcus, 1954; Dempster, 1955;
Ayoub, 1966; Tichauer, 1965; Chaffin, et al~, 1967];
(2) medicine and medical rehabilitation [Contini, et al,,
1949, 1953, 1954; Eberhard and Inman, 1947, 1951; Fletcher
and Leonard, 1955; Gavagna, et al. 1 1963; Stone 1 1963];
(3) sports [Furusawa, 1928; Tarrant, 1938; Morton, 1952;
• • Hopper, 1951; Carlsoo, 1960 1 Idai and Asami 1 1961; Lloyd,
1965; Cooper, 1968; Elizabeth, et a1., 1968]; (4) music
[Hodgson, 1934; Polnauer and Marks, 1965]; (5) traffic and
motor vehicles [Dorney and McFarland. 1953 1 1955, 1963;
Severy, et al., 1954, 1955, 1956]; and (6) space research
[Kuehnegger 1 1964].
3
Techniques of Human Motion Analysis
In biomechanics, the determination of human motion
characteristics, such as displacement, velocity, and accel
eration, is a prerequisite before any objective analysis
can be performedo The scientific analysis of human motion
has been recorded for more than a centuryo Leonardo
da Vinci is generally credited with conducting the first
systematic observations on human motiono Human motion
analysis, in general, can be carried out either experiment
ally or theoretically as shown in Figure 2o
Experimental Analysis
Experimental motion analysis is the technique of
obtaining the motion characteristics from the physical
records obtained for the motiono In most of the experi
mental analyses, either displacement, velocity or accelera
tion, or their analogues versus time are measured as an
output from the recording systema In any case, motion
characteristics are obtained by either differentiating or
integrating the measured datao Experimental analysis
includes three basic techniqueso
lo Photographyo--Three methods are available for
photographic analysis of human motion~ (a) cyclography,
chronocyclography, and interrupted light photography
[Marey, 1895, 1902; Gilbreth, 1917; Popova, 1934; Polnauer,
4
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aly
sis
Vl
et alo, 1952]; (b) gliding cyclograms [Bernstein, 1928;
Drillis, 1930, 1959]; and (c) motion pictures [Eberhart
and Inman, 1947; Taylor and Blaschke, 195l]o
2o Electronic and/or electromechanical methodso--
Motion analysis by electromechanical techniques is based
upon the principle of converting the physical motion into
electrical signals which can be related to the motion char-
acteristicso These techniques can be classified into three
basic systems: (a) potentiometric systems [Karpovich,
1960; Reheja, 1966; Ramsey, 1968]; (b) radar-like systems
[Goldman, 1955; Covert, 1965]; and (c) accelerometers
[Liberson, 1936; Karger, 1958; Ayoub, 1966]o
3. Stereophotogrammetryo--Recently, human motion
analysis has been pursued by means of stereogrammetrical
techniques which include three basic methods: (a) stereo
photography, (b) stereotelevision, and (c) stereoradaro
Although the three systems vary in technical principles,
all of them make use of the very basic concepts of binocu-
lar visiono Yet, stereophotography is the leading tech-
nique over the other two as far as the principles, method-
ology, and accuracy are concernedo Stereophotography has
been used by Zeller [1953], Brewer [1962], Guetwart [1967],
Preston [1967], and Ayoub [1969] for subjective recording
of human motiono
6
Theoretical Analysis
Theoretical analysis of human motion was introduced
several decades agoo The use of theoretical mechanics for
human motion analysis has been the subject of several
investigationso The classical work of Braune [1895],
Fischer [1906], Amar [1920], and Bernstein [1926] is not-
able in this respecto Theoretical mechanics alone, how
ever, has failed to be an efficient technique for human
motion analysiso Nubar and Contini note "o 0 0 the fact
that the equations of theoretical mechanics are by them-
selves incapable of determining completely the unknowns in
human motion, a matter largely of free choice on the part
of the individual" [196l]o
Aside from theoretical mechanics, human motion
analysis based upon optimization approaches has received
attentiono It has been accepted for years that the human
body--a machine which thinks, learns, and shows a high
degree of adaptability to the external environment--selects
an optimum performance according to a certain criterion in
given circumstanceso That is to say, the human body will
perform according to the hypothesis of minimal principleso
Milsum stresses the applicability of the minimal principles
hypothesis to the human body as:
An attractive parallel arises in the nonliving physical world, namely in the "minimal" principles by which structural, electrical, hydraulic, and other networks
7
reach equilibrium when either the stored energy or dissipated power is minimized. While, therefore, we must beware of guessing rashly how nature operates, on the basis of being "logical,'' nevertheless living systems are also constrained to operate within physical laws and hence probably must use some of the same criteriao o o • There are many combinations of muscle tensions which could achieve any given desired posture, each requiring, in general, a different metabolic rate to sustain it. This condition may be compared with that of a statically indeterminate engineering structure, and, as is well known, such a problem is solved, at least in principle, by writing a stored-energy expression for the structure and by differentiating for a minimum to solve for the equations specifying the forceso It would seem plausible that the body's equilibrium posture should be deducible by a similar approach, at least in principle • o o [1968]o
Cotes and Meade [1960] have verified experimentally
that the human, indeed, follows an optimizing criterion in
walkingo For a subject walking naturally on the flat,
Cotes and Meade express the power consumed as:
where
P0 = oxygen consumption rate, 2
V = speed of walking, and
a,b = numerical constantso
Using the above formula, and their empirically determined
constants, Cotes and Meade predict the optimum walking
speed equal to 2.25 miles/hr which closely approximates
the walking speed of the average individualo On the other
8
hand, Milsum [1968] postulated a simplified model for walk
ing in which the legs are considered as cylinders, swinging
as simple pendulums with slight difference so that each leg
comes to rest on the ground at the end of each swingo Com
bining the natural frequency of the cylindrical form leg--
77 paces/min--with a reasonably normal pace length of 30
inches, the optimum walking speed for the idealized simpli
fied walking model is obtained as 2o6 miles/hro Comparing
the two optimum speeds, ioeo, the one obtained by the
simple pendulums assumption against the experimentally
determined one, should demonstrate the applicability of
the minimal principle to the human bodyo
Respiration studies by Christie [1953] and Meade
[1960] have proven without doubt that under normal condi
tions the human being does optimize his breathing fre
quencieso
Nubar and Contini have attempted to develop a
theoretical model for human locomotion by using an optimi
zation approacho They postulate the minimal principle in
biomechanics as follows: "A mentally normal individual
will, in all likelihood, move (or adjust his posture) in
such a way as to reduce his total muscular effort to mini
mum, consistent with the constraints" [196l]o Based upon
their minimal principle, Nubar and Contini propose the fol
lowing mathematical model for human motion analysiso
9
Minimize
subject to
where
Ml' M2'
ri, i=l •
•••, M 1 t) = 0 m
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
E = effort function,
• • • M = joint moments, • m c = numerical factor,
l1t = small time intervals,
• • • n - constraint equation obtained -J dynamic analysis.
(1.2)
(1.3)
by
Nubar and Contini proposed an iteration scheme for
solving the above model. In their scheme they convert the
model, the objective function and constraints, into a set
of nonlinear differential equations. Through the use of
LaGrange's multipliers and the assumption that terms con
taining second derivatives can be neglected, they feel a
solution or the model would be possible.
In spite of the detailed formulation or the model
and its mathematical analysis, Nubar and Contini's model
10
has never been tested or actually applied to human motion
analysis. That is, the model did not exceed the mathemat
ical formulation phase. However. Nubar and Contini are
considered the first to present a somewhat full mathemat-
ical treatment to the problem of human motion.
ll
Rashevsky [1962] has attempted to develop functional
relationships between the optimal speed of walking, the
optimal step size and the amount of metabolic energy for
human locomotion. He proposes the following two expres-
sions:
and
where
vm = optimal walking speed,
so = optimal length step,
m = mass of legs,
M = body mass,
1 = leg length, and
W* = metabolic energy available.
The above two expressions are obtained under extremely
simplifying assumptions. Nevertheless, Rashevsky states
12
that "in view of the crudeness of the approximation used, it
is noteworthy that we obtain plausible orders of magnitude
for Vm and S for average human walka"
Statement of the Problem
So far, since the time of Leonardo da Vinci, a fund
of knowledge has been gained from the previous studies con
cerning human motion analysiso However, there is no gen
eral model available for describing and predicting human
motion characteristics in their general form, eogo, three
dimensional motion or even under planar motionso It is very
obvious that the nature of the experimental analysis of
human motion eliminates the possibility of developing a
generalized model based upon experiments aloneo Most of
the existing motion analysis techniques require a consider
able amount of time for both recording and data reduction
phases which is, undoubtedly, beyond the capabilities of
most research activitieso Furthermore, in the previous
attempts which have been made to describe human motion
theoretically, instead of a complete analysis of the prob
lem, either a definition is given or a proposed solution
procedure is presentedo
It seems, therefore, that there is a real need for
developing a generalized biomechanical model which can be
used to simulate all possible classes of human motion
theoreticallyo
Unfortunately, developing such a model is not an
easy task, if not impossible at the present state of the
arto Furthermore, it seems very difficult to even agree
confidently upon optimization criteria which can be used
to describe human performance. The complexities multiply
at a fast rate when one discovers that the criteria are
13
not simple or necessarily always the same under different
taskso For instance, in some tasks the efficiency of per
formance is the main concerno On the other hand, maximiza
tion of the human output effort, power say, might be the
objective of the tasko It is evident that there is a need
for two optimization criteria for the two stated objectiveso
Considering the previously mentioned difficulties,
an attempt to develop a biomechanical model to study a
special class of transport movements will be of value, both
in the ultimate objective of developing a generalized model
for the human body and in developing some practical appli
cationso Indeed, developing such a model would clarify
some aspects of model building in connection with the human
bodyo For example, answers to questions concerning the
optimization criteria, assumptions, mechanics, and possible
algorithms for the model might be obtainedo This is to
say, however, that a simplified model completely developed
would eventually lead to the development of a complete model
as our knowledge about the human body increases and perfectso
Applications of such a simplified model are envi
sioned in two distinct fields. First, application could be
made in the design of a work place associated with light
manual activities. A typical example of such application
would be in designing cockpits for aircraft and spacecraft.
Second, application could be made in the field of medicine
and in medical rehabilitation for which the model could
serve as a basis for designing and evaluating artificial
limbso Also, the model could be used to simulate the per
formance of the disabled and patients with severe deformi
ties; that is, application of the model under the restric
tive conditions of those people would permit an assessment
or prediction of their performance without actually experi
menting with them.
Purpose and Scope
The primary objective of this investigation was to
develop a biomechanical model for predicting the path of
motion of the arm articulation joints which would minimize
a measure of the physical effort necessary to perform the
actual motion. The underlying principle of the model is
that the human does follow an optimizing criterion in per
forming his tasks. The use of the model is restricted to
tasks which are to be performed under normal environmental
conditions and require the maximization of performance
14
efficiency rather than the maximum possible effort output
from the bodyo
Basically, the model utilizes both theoretical
mechanics and an optimization approach for the analysis of
arm motionso Model assumptions, mechanics, and formulation
are presented for three-dimensional motionso Different
possible algorithms for the model solution were investi
gatedo These are linear and geometric programmings,
dynamic programming, and simulation analysiso
Principles of the model with its associated algo
rithms are applied in detail to analyze planar motions of
the armo Under planar motion conditions, the adequacy as
well as the accuracy of the model was investigatedo
15
CHAPTER II
THE MODEL
This chapter, dealing with the formulation of a
biomechanical model for the upper extremity, is presented
in four major sections. First, model assumptions are dis
cussed and their validity is supportedo Second, the gen
eral features of the model dynamics are explained. Third,
the model's possible performance criteria and the rationale
for selecting a specific criterion are discussed. Finally,
formulation of the optimization model in terms of the
selected objective function and constraint equations is
outlined.
Assumptions
The human body can be viewed as a structure com
posed of several links hinged together about the articula
tion joints. The stability of the structure is provided
by the action of several muscles connecting the different
links. In order to simplify the dynamic analysis and the
subsequent calculations of the model, the following assump
tions are adopted.
1. The human body can be approximated structurally
by rigid links of uniform geometrical shapes and densities.
16
Further, the anthropometric characteristics of the links
will not be affected by changes in body configurationso
According to this assumption, the arm is considered as a
system of two links; the first is the upper arm and the
second is the forearm and hando
2o The muscles' actions are represented by several
tie rods which can withstand tension. A typical joint of
the human body and its structural equivalent is shown in
Figure 3o In the structural equivalent, the hinge joint
and the muscles' actions are combined to give the effect of
a rigid joint, that is, a joint capable of withstanding
moments and reactive forces as wello It should be under
stood that the rigid joint effect is instantaneously such
that variations in the joints' links orientation during
motion are permitted. This concept can be extended to the
other articulation joints considered in this modelo
3o Rotational motions of the links around their
longitudinal axes are not permittedo That is, the class of
motions considered in this study can be performed with
translatory motions onlyo For examplei pronations or supi
nations of the hand are not permittedo
4o There are no velocity or acceleration compo
nents due to coriolis motion between the moving linkso
This assumption is validated by the findings of Pearson,
17
et al., [1963] in which they show that the displacement
between the adjacent bones is negligible.
anatomical joint1 structural joint
a. triceps b. biceps
c. brachialis d. brachioradialis e. extensor carpi radialis longus and brevis f. flexor carpi radialis and palmaris longus g, pronator, teres
Fig. 3o--Structural equivalent of muscles' actions
5o Space motion of different body members can be
treated as a two-dimensional motion performed in a plane
oriented in accordance with the direction of motiono
Kattan and Nadler support this assumption by stating:
It is possible to treat a motion of a body member in space as essentially two-dimensional, one dimension
1Adopted from Steindler [1964].
18
along the line of motion and the other along the height axise 'rhe maximum value of depth dimension, z, of motion path for any experimental condition is less than 1% of the linear movement distance. This indicates that the subject moves more or less on the X-Y plane within the experimental region, thus optimizing the motion with respect to the Z dimension [1969].
Based on this assumption, the displacement-time curve of
the free-joint, the hand in most tasks, of the body in at
least two dimensions is as shown in Figure 4. Slote and
Stone [1963] describe such a displacement-time curve by the
following functional equation:
Xt • ~{~- sin(~)) where
xt = displacement of time t,
19
XT = max displacement; the distance between initial and terminal points measured in that direction,
T • total motion time.
6. Motion time is assumed for the given task or
predetermined from similar studies or standard motion time
tables.
1. Due to the restriction on the task duration
(less than 5 sea), performance criterion is taken to be
related to the mechanical characteristics or the skeletal
muscular system rather than the physiological indices or
the supporting respiratory and cardiovascular systems.
Karvonen and Ronnholm indicate in their studies: "Purely
t
Motion Tim·.:
T
~I
Figo 4o--Displ~cement-time curve
20
mechanistic concepts (e.g., ventilation, heart rate, and
oxygen consumption) have only a limited application to
the problems of light manual work" [1964]o
Notation
X,Y,Z = global frame of reference
x,y,z = principal axes
r,J,K' = unit vectors for the XYZ frame
unit vectors for the xyz frame
= mass of link ij
= length of link ij
= distance of link ij center of mass from end i
= dimensions of link ij section
cross
(Ac)ij =
(Iij)x,(Iij)y,(Iij)z =
cross section area of link ij
principal moments of inertia of link ij at point i
• •• • ••
= Euler's angles of link ij at time t
8ij' 8iJ' 41 iJ''ij = first and second derivatives of Euler's angles
= angular velocity vector of XYZ frame attached to link ij at time t
components of wij along X,Y,Z axes
angular velocity vector of xyz frame of link ij at time t
21
-= components of nij along x,y,z axes
velocity of joint i Woroto XYZ frame at time t
ai = acceleration vector of joint i Woroto XYZ frame at time t
acceleration vector of C G of link ij Woroto XYZ frame at time t
components of velocity and acceleration of joint i at time t along X,Y,Z axes
= position vector of joint j at time t Woroto XYZ frame oriented at joint i
= position vector of link ij center of mass at time t Woroto XYZ frame oriented at joint i
reactive force vector acting at joint i at time t
components of Fi at time t along X,Y,Z and x,y,z axes respectively
moment of a reactive force or a moment vector acting upon link ij about joint i
= components of (Mi)i at time t along X,Y,Z and x,y,z axes respectively
first moment vector of link ij at time t about joint i
angular momentum of link ij at time t about joint i Woroto XYZ frame
-first derivative (Hij)i
22
Units:
d ' -(HiJ)i -dt = first derivative of (Hi~)i'
treating xyz frame fixe
Aij = matrix of transformation from the XYZ frame to xyz frame at time t
( 0i )k = normal stresses at point k of the ith cross section at time
xk,yk = coordinates of point k
AI = angular impulse
LI = linear impulse
The following system of units (MKS) is
adopted
lo mass--kilograms
2o moment of inertia--kilogram-meters squared
3o angular velocity--radians per second (rad/sec)
23
t
4o angular acceleration--radians per second squared
5o linear velocity--meters per second
6o linear acceleration--meters per second squared
7o force--newtons
8o moment--newton-meter
9o work--newton-meter
lOo power--newtons per second
llo linear impulse--newtons per second
12o angular impulse--newton-meter per second
13. stress--newtons per meter squared, and
14. stress rate--newtons per meter squared per second.
Dynamic Analysis
Consider an intermediate link ij of a system of
moving links. Let X,Y,Z be a right-hand set of orthogonal
axes rigidly attached to joint i as shown in Figure 5.
Also, let x,y,z represent the principal axes of the link
such that the z-axis is chosen along the longitudinal
axis of the link; the x-axis is perpendicular to the
z-axis and it is always in the vertical plane through the
z-axis; and the y-axis is perpendicular to the x and z
axes. At any time t during the motion, the link orienta-
tion in space is specified by the two Euler's angles eij
and tij relative to the coordinate system X,Y,Z. It is
assumed that the link motion is influenced only by the
reactive forces and moments at its ends and its weight.
Velocity and Acceleration
The angular velocity of the XYZ frame can be repre-
sented as follows:
...
where ....
= angular velocity vector at time t, and
24
25
z
y
Fig. 5.--Reactive forces and moments acting upon a link at any instant (t) during its motiono
components of the angular velocity along X,Y,Z axes, respectively
-From Figure 5 the components of the angular velocity wij
can be expressed as
(wij)X • = -eijCOScf>ij •
( CIJij )y • = e . . 0 sinct>ij and lJ '
(wij)Z • = ct> ij 0
26
Substituting the above expressions in equation (?.ol) yields
Similarly, the angular velocity oij of the xyz frame can
be written as
Linear velocity of joint j is given by
where
vj = velocity of joint j Woroto the XYZ frame,
-vi = velocity of joint i Woroto the XY_;_ frame,
- I (o\)ij = angular velocity of XYZ frame, and
.... Lij = position vector of joint j
.... ....
.... Linear acceleration aj of joint j is written as follows:
where
....
•
aj = ai + ;ijx1ij + ;ijx(:ijx1ij)
ai = linear acceleration vector of joint i w.r.t. the XYZ frame,
wij'Lij = as defined before, and
• .... = first derivative of the angular acceleration wijo
....
....
Linear acceleration (aij)G of the center of mass of link ij
can be deduced from equation (2.5) upon replacing Lij by ....
the position vector of the center of mass rijo Thus,
• (aij)G = ai + ;ijXrij + wijX(wijXrij) o
Equations of Motion
During the motion, the link is always in a dynamic
stability under the effect of the external forces and
moments;as well as the inertia forces generated by the link
motion. The equilibrium equations for the link at any
instant t are written as follows [Hagerty and Plass, 1967;
Thomson, 1961; Nelson and Loft, 1962]:
27
and
where
-Fi =
mij =
-(aij) G =
<sij) i =
ai =
Aij =
28
(2o7)
reactive force vector acting about joint i,
mass of link ij J
acceleration vector of center of mass of link ij,
moment of a reactive force or moment vector acting upon link ij about joint i,
first moment vector of link ij about joint i,
acceleration vector of joint i,
matrix of transformation from the XYZ system to xyz system, and
first derivative of the angular momentum vector for link ij written Woroto joint io
By examining Figure 5 (page 25), the above term can be
written as follows:
J -I Fi 1=1
where
- - = reactive forces vectors at joints i and j respectively,
(Fi )X' (Fi )y, (Fi )z, (Fj )X' (Fj )y, (Fj.)Z = components of the reactive forces along XYZ axes, and
g = gravitational forceo
Using equation (2.6), the above term can be written
as follows:
• mij•(aij)G = mij•ai+mij•wijXrij+mij•wijX(;ijXrij)
= M1 + MJ + m1J • gxr1J + Fjx11 j
29
= ((M1 )Xi + (M1 )yj + (M1 )zk) + ((Mj)Xi+(Mj)Yj+(Mj)Zk)
where
moment vectors at joints i and j respectively,
components of the moment vectors along the XYZ axes, and
mij'rij'Lij'Fj,g = as defined beforeo
=m • ij
where all the terms are as defined before.
In all the previous expressions, all the terms are
30
written with respect to the XYZ coordinate systemo In
order to use equation (2,8), these terms have to be written
with respect to the xyz axes. The transformation from the
XYZ to the xyz frame is obtained by using the appropriate
matrix of transformation Aijo The matrix Aij is the result
of two rotations: +ij rotation about the Z axisj followed
by eij rotation about the resulting y axiso By examining
Figure 5 (page 25), the matrix of transformation A1j can be
expressed as:
-sineij
0
The relation between the two coordinate systems is written
as:
where
X X
y = A • y
z z
angular momentum of the link ij about joint i Woro~o XYZ axes,
first derivative of the angular momentum, treating the xyz axes fixed,
angular velocity of xyz frameo
The scalar components of equation (2.9) are given by:
31
•
j
(i(Mi)i)y ~ {(Iij)y(Qij)y-((Iij)z-(Iij)x)(oij)z(oij)x}
where
components of the external moments joint i at time t along the link principal axes xyz,
principal moments of inertia of link ijo
The above scalar components are the well-known
Euler's equations of motiono - -----
Using equations (2o7) and (2o8), the reactive
forces and moments at the link's joint i, sayj can be
expressed in terms of the external forces and moments at
32
joint j as well as the link Euler's angles and their deriv-
ativeso The analysis can easily be extended to determine
the reactive forces and moments for the different joints
of a link system. The p~ocedure is best summarized in the
following steps:
lo Using equations (2ol) through (2o6), velocity
and acceleration for each joint in the system can be
obtainedo It is important to start the analysis with a
joint of known motion or with a support and proceed from
it to the other joints until all the unconnected links of
the system are reached, considering one joint at a timeo
2o Using the concepts of the free body diagram
and equations (2o7) and (2.8), the reactive forces and
moments at the different joints of the system can be
obtainedo
Following the above mentioned procedure, it can be
shown that the moments and the reactive forces at any
joint such as i of a link system of the arm are expressed
as follows:
(Mi)x y z ' '
Performance Criteria
Describing human effort during motion based upon
mechanical criteria has long been in useo Some of the
most frequently used criteria are work, power, angular
impulse, linear impulse, and stress at the articulation
jointso Generally, it is assumed that the external
mechanical function at any articulation joint is related
to the forces developed in muscles during motiono
33
34
Expressions for calculating some mechanical criteria are
given belowo
lo Worko--The work of a couple moment M acting on
a rigid body during a finite rotation is given by:
where
M = a moment vector acting on the body,
de = small angles expressed in radians through which the body rotates,
el,e2 = the initial and the final values of the angle of rotationso
2o Powero--It is defined as the rate of perform-
ing worko Power is expressed as:
Power = {2ol3)
where
• de = small angular velocity of the bodyj 0 •
e1,e2 = initial and final velocities of the bodyo
lo Angular Impulseo--Angular impulse is defined
as:
T AI = f Mdt
0 {2ol4)
where
where
M = moment vector acting on the body•
T = total motion t1meo
4~ Linear Impulseo
T_ LI = f Fdt
0
F = resultant force vector acting on the bodyo
5o Normal Stresso--The stress at a cross section
35
of a moving link of the human body, taken at the articula
tion joint, results from the moments and reactive forces
acting upon the moving link during the instant consideredo
Basically, there are two types of stresses acting upon any
cross section of the body: normal stress and shear stresso
In this study only normal stress is consideredo
Normal stress is due to normal compression
forces acting along the longitudinal axis of the limb and
bending moments acting along the axes of the cross section
(Figure 6)o The normal stress at any instant t is given
as:
(2ol6)
where
( 01 )k =
(Fi)z =
normal stress at point k of the ith cross section of link ij,
normal compression force at the ith cross section,
bending moment components along the x and y axes of the ith cross section,
(Ac)ij = area of the ith cross section,
xk,yk - coordinates of point ko
During the entire motion, the resulting total
stress is given by:
z
T Stress = f ( oi )kdt o
0
y
36
Figo 6o--Reactive forces and moments acting upon the ith cross sectiono
6. Rate of Stress.--Rate of applying stress is
obtained by:
37
Stress Rate (2.18)
Which mechanical criterion from the ones mentioned
above would best describe human effort? In other words,
which mechanical criterion is related to the muscular
forces? Extensive studies by Hill, Fenn and their associ
ates [Hill, 1960] as well as several others have revealed
that power developed at the different articulation joints
is the mechanical function which best describes human
effort.
Harrison states:
For a muscle the force is independent of displacement but dependent on the instantaneous speed. From this it may be inferred that two muscles with identical properties could contract the same distance in the same time, but with different speed-time relations, and give rise to differing amounts of work doneo Further. since the contraction times are the same, the power outputs will be different in the same ratio as the work outputs [1963].
Power and work, as any other mechanical functions,
can be either positive or negative. For instance, a posi-
tive work will result if a motion is performed against
gravity, and a negative work will occur for a motion with
gravity. As far as the human body is concerned, doing
either a positive or a negative work is an energy consumed.
Therefore, it seems very appropriate to take the absolute
values of work in order to estimate human effort [Starr,
195l]o
However 5 Hill [1960]; Abbott, et alo 1 [1952]; and
Abbott and Bigland [1953] present an interesting argument
about the contribution of positive and negative work to
human efforto Abbott, et alo 1 state~
38
When an active muscle exerts a force P and shortens a distance x it does an amount of work Px; on the other hand, if it is stretched a distance x while exerting this force, it absorbs work and is said to do an amount Px of negative worko For example, when a man climbs a vertical ladder 5 his leg extensors shorten and do positive work against gravity; when he descends, the same muscles are stretched while actively resisting the gravitational pull and may be said to do negative work [1952]o
What is the physiological cost of such negative
work? It has been shown by Abbott and his a8sociates that
doing a negative work is either as or more beneficial for
the body than doing a positive oneo
In this study the performance criteria are esti-
mated by using the absolute valueso However, the use of
positive and negative values, as will be explained later
on, will be applied in the simulation approach for solving
the modelo
For the purposes of this investigations all the
previously mentioned criteria were investigated in some
trial runso From the results obtained, it was evident
that power is the most suitable criterion for describing
the path of motion and its characteristicso Therefore,
power was chosen to be a performance criterion for the
modelo
The Model
The proposed biomechanical model, as previously
mentioned, seeks the path of motion which will minimize
a certain performance criterion and will satisfy the phys-
ical constraints imposed upon the motiono This is a
typical optimization model in which one seeks the optimi-
39
zation of an objective function subject to some constraintso
Mathematically the model can be stated as follows~
minimize~
T oo oo
J = of r(eiJ.aiJ•eiJ;+iJ•+iJ''iJ)dt
i,j=1,2
subject to~
where~
k=l oeo r j) J
J = performance criterion,
fk = constraint function,
r = integer denoting the number of constraint functionso
(2o19)
(2o20)
Solving the above model would yield the necessary
information about the motion and its characteristics such
as the path of motion, velocity and acceleration prof1les,
reactive forces and moments for each articulation jointo
40
Model solution has to satisfy three different
classes of constraints~ (a) physical constraints, (b) task
constraints, and (c) stress constraintso The nature of
these constraints is discussed in some detail belowo
ao Physical Constraintso--The maximum and minimum
values of the Eulervs angles for each link (segment) of
the human body are 9 more or less, fixed by the structure
of the body and its ligamentso Therefore, the purpose of
~ti2 ~l~ss of constraints is to assure that the optimum
solution of the model at any time t is possible to be
assumed by the human bodyo The maximum and minimum values
for the angles are determined from the previous work con-
cerning anthropometry of the human bodyo Generally, the
physical constraints are written as follows~
( e ij ) . < e ij < ( e ij ) m1n - ~ max (2o21)
and
i,j=l,2 0
41
bo Task Constraintso--The nature of the task to
be performed introduces some restrictions on the feasible
solutions of the modelo For example, in a certain task
the weight has to be moved between two specific pointso
This necessitates that any feasible solution should be
constrained in such a way that the resulting path of motion
of the hand will lead the weight to the desired pointo
That isj the solution should satisfy certain boundary con
ditions given as~
XH0 = Xl; YH0 =
and
XHT = x2; YHT. =
where
Yl; ZH 0
Y2; ZHT
= zl
(2o22)
= z2
= space coordinates of the hand at the initial and terminal pointso
For another example, consider the task of moving a
control stick during motiono In this class of motion, the
hand path of motion is restricted to that of the control
sticko It can be seen that in moving a stick, and in
similar tasks, the task constraints may vary considerably
from the previous caseo
Co Stress Constraintso--Stress constraints provide
a means to exclude all possible feasible solutions of the
model which might lead to excessive stresses at the joints
other than that joint at which the performance criterion
42
is minimizedo It follows that moments at the articulation
joints, taken to be equivalent to stress, should be less
than or equal to certain maximum valueso These maximum
values are obtained from the previous moment analyses con
cerning the human bodyo The nature of each stress con
straint is similar to that of the objective function except
that it is written in the inequality formo
CHAPTER III
MODEL SOLUTION
This chapter introduces some algorithms for solving
the proposed upper extremity biomechanical model presented
in Chapter IIo In the interest of obtaining a suitable as
well as an accurate algorithm for solving the model, three
different solution approaches were investigated~
lo a suboptimization approach, v
2o a dynamic programming approach» /
3o a simulation approacho
Applications of the above three approaches for the
model solution are presented in some detail in the follow
ing discussionso
Suboptimization
Suboptimization as a technique for solving the
model was chosen in the hope that some of the well-developed
algorithms such as linear programming could be usedo In the
suboptimization approach, the motion path is divided into
several points in timeo
For each point in time, minimizing the performance
criterion subject to the model constraints yields the nec
essary motion characteristics as well as the associated
43
44
arm configuration at the instant consideredo By repeatedly
solving the model several times at different time points
with different task constraints each time, the path of
motion as well as motion characteristics for each articu-
lation joint can be obtainedo The task constraints of the
model, page 41, have to be modified somewhato At any
instant, the hand Z-coordinate is expressed by two con
straint equations as follows:
where
ZHmax,min
= hand Z-coordinate, and
= maximum and minimum values for ZHt' expressed in percentage of the total motion distanceo
The above two equations present a feasible region
for the hand position which would force the optimal solu-
tion to take the hand to the desired terminal pointo By
adopting this concept for ZHtj the task constraints can
now be written as follows:
XH XH{2rrt i (2rrt)} t=2rrT'-snrr
YH YH{2rrt 1 (2rrt)} t=~-r-sn-r-
y
and
ZHmin(t) ~ ZHt ~ fHmax(t) •
At any instant, the hand position will be expressed by two
constraint equationso The two equations present the feas-
ible region for the hand positions (Figure 7)o By adopt-
ing this concept of task constraint$ it is certain that
the motion will be terminated at the desired pointo
z
~----------------~ X
Boundary for Hand Position
Z = % Distance max
t-- Distance
Figo 7o--Feasible region for the hand path of motion under suboptimization approacho
45
Solving the model as a suboptimization problem
becomes an easy task upon adopting one of the well
developed computerized algorithms for this kind of prob-
lemo It was decided to use linear and geometric program-
mings for solving the suboptimization modelo
Linear Programming
The nature of the model, as expected, is a non-
linear one which eliminates direct application of linear
programming to ito The model, however, can be linearized
46
by adopting the small angles assumptiono Under this assump
tion the following approximations can be made~
= l
, and
all cross products = 0 0
Similar relations can be obtained for 'ijo By
applying the above approximation, the model can be written
in a linear form as follows:
minimize:
2
J = 1,j=1{ciJ 6iJ + c2161J + c31 91J + c4i+iJl
subject to:
where
eli' e•o, c4ik =numerical constantso
Geometric Programming
47
k=l ••• r ' ,
A geometric programming model may be written as
follows: (A complete discussion of the model may be found
in Teske [1970]o)
minimize:
t=l oee T , J ,
where
such that
subject to:
where
Tm N a ( X) = \ o C II X mn t m= 0 1 ca ca • M
gm t~l mt mtn=l n ' ' ' »
which
omt = + 1. m=O 1 ° 0 ~ M· t=l GOe T -, • • •' • 'm
such that
C t > 0. m=O 1 oo• M· t=l ••o T m ' ' ' '' ' 'm
and
The optimization model can be transformed very
easily to a geometric programming model and solved by a
special computer program written by Blau [1969] and modi
fied by Teske [1970]o The only disadvantage of geometric
programming is that the optimum solution is obtained for
one form of constraints, ioeo, either equality or inequal-
ity but not botho This disadvantage, however, can be
removed to some extent by applying some heuristic rules
stated by both Blau and Teskeo
Dynamic Programming
In the dynamic programming approach, the model
under consideration has four state variables, namely, eij
and '· ., for i,j=l,2o As can be seen, it would be a very ~J
difficult task to solve such a model under this large num-
48
ber of state variableso The model dimensions, however, can
be reduced to just one state variable by adopting the fol-
lowing principles:
lo Angular velocity and acceleration can be
obtained by numerically differentiating angular displace
mentso That is,
ei+l + e._l - 2ei
(~t)2 0
2o The space configuration of the arm at any
instant during the motion would be completely described
upon knowing the four Euler's angles for the arm linkageso
Referring to Figure 8j the space coordinates of the hand
with respect to the shoulder joint {origin) can be written
as:
49
(3ol0)
where
= Euler's angles for the two links of the arm at time t,
= space coordinates of the hand at time t,
= lengths of the arm's two linkso
50
z
H
y
Figo 8o--Arm motion for dynamic programming approach
The XHt and YHt coordinates for equations (3o8) and (3o9)
can be obtained by using assumption 5, page 18s as:
XH = XH(2rrt i (2rrt)) t 2,.--snrr
51
YH __ YH(2rrt i (2rrt)) t 2 -r - s n -==rr-= (3ol2)
where
XH 1 YH =
T =
hand displacement in the X and Y directions at time tj
maximum displacements, and
motion timeo
Furthermorej it is assumed that the angular displacement-
time relationship for the elbow joint in the XY plane
(Figure 8, page 50) can be obtained by using the function
equation of assumption 5, page 18o That is,
where
= '2max{~ _ i (~)} 2rr T s n T
= angular displacement at time tj
= maximum displacement for the elbow joint in the XY plane,
T = motion timeo
By using equations (3o8) through (3ol3) for a given value
of ZH, one can obtain three nonlinear equations in three
unknowns, namely , 2 , e1 , e2 o
By using the Newton iteration scheme, a solution
for the nonlinear equations can be obtained which results
in defining the Euler's angles for the arm during the
instant consideredo
Therefore, the hand Z-coordinate is the only state
variable left to be determined in order to specify the arm
configuration in spaceo The hand Z-coordinate should be
determined in such a way as to minimize the performance
criterion consistent with the constraintso
3o The model is transformed from the continuous
case to a discrete oneo In doing so, the plane of motion
will be divided into a fine grido Its horizontal inter
vals represent all possible stages and the vertical
intervals represent the possible Z-coordinate of the hando
The stage intervals are taken to be equivalent to time
intervals, Ato On the other hand, the vertical interval
is expressed as a percentage of the total motion distanceo
4o Between consecutive stages, ioeo, the small
time interval considered, the velocity and acceleration
as well as other motion characteristics are assumed to be
constanto In other words, the changes in those character
istics are negligibleo
52
53
5o The performance criterion J during the motion
is computed by step integration over all stageso
6o A very fine grid is essential in order to mini-
mize the errors inherent in both numerical differentiation
and integrationo However, to reduce the core requirement
for the digital oomputerj a somewhat coarse grid can be
used, and smoothing by regression analysis should be
applied to the resulting optimum path of motiono
By adopting the above principles, one can write the
recursive equation for the dynamic programming approach
as follows (see Figure 9):
------
1 n-1 n N
Figo 9o--Stages of dynamic programming approach
and
where
54
= { fex x n' n-1
ex x + fn-1 xn-1 mxin { f j * ( ) } n n' n-1
fn(X ,X 1 ) = n n- the total power of best over-all
policy for the first n stages, given that the hand is in state Xn and Xn-1 was the previous state it occupied~
ex x n' n-1
= the cost of going from position X 1 nto X , n
min value off (X ,X 1 )o n n n-
A computer program was written to solve the dynamic
programming formulation based upon the previous discussiono
Simulation
Aside from the optimization approaches proposed to
solve the model, it was decided to investigate the possi
bility of using simulation for obtaining the path of motion
and the associated characteristics for the armo
In the simulation analysis, it is assumed that the
hand path of motion can be approximated by certain geo-
metrical shapeso Some of these possible shapes are:
lo a portion of a sine curve,
2o a portion of an ellipse,
3o a parabola, and
4o a polynomial regression curve fitted to a number of points of a grido
55
Using any geometric shape from the above-mentioned
curves along with the displacement equations presented on
pages 44 and 45 would yield the necessary information about
the motion characteristics as well as the arm configura-
tions during the entire motiono
Generally, the equations for sine, ellipse, and
parabola curves are written as a function of motion dis-
tance and the maximum height of the hand above the work
surface as shown in Figure lOo
Varying the maximum height (H) over discrete
points and keeping the motion distance constant would yield
a set of different curveso It is possible to compute the
performance criterion for each curve in this seto Of the
curves investigatedi the one which yielded the minimum cri
terion was considered to be the path for the best motiono
For the grid approach, a grid similar to the one
shown in Figure 11 was constructedo The vertical and
horizontal intervals are expressed as percentages of the
motion distanceo An enumeration procedure was used to
generate a set of points among all the grid pointso A
polynomial regression function fitted to any selected set
of points was obtainedo The resulting polynomial function
provided a path of motion and the associated motion
characteristicso The generated motion which yielded the
minimum performance criterion was selectedo
DHt =
ZHt =
ZHt =
ZHt =
I ZHt IH
I I
fxHt - XH0)2 +
DHt H*sin(rnr)--sine
DH
(YH -t
curve
H*( 1-DHt 2
( l5Ir) --ellipse
B0 + B1 G DHt + B2 o DH2 t
YH ) 2 0
--parabola
Figo lOo--Arm motion under simulation approach, assuming sine, ellipse, parabola as possible shapes for the hand path of motiono
56
57
AV, AH = % Motion Distance
Regression Curve
AV
AH / '
/ ' L._ /1-L------=-----'-----'----~ ;;-' ....
Figo llo--Arm motion under simulation approach, using enumerationo
CHAPTER IV
MODEL IMPLEMENTATION
This chapter presents and discusses the implementa
tion of the model and the algorithms presented in Chapters
II and III to study the arm under planar motionso Applica
tions of the model to analyze arm motion under different
tasks are presentedo During the course of the description
of these applications, the feasibility of the model and the
algorithms--linear and geometric programmings, dynamic pro
grammings, and simulation--were tested and evaluatedo
Planar Motion Problem
The planar motion problem can be stated in general
as follows:
A subject of known anthropometric characteristics
is required to move his hand between two previously
prescribed points carrying a known weight following
a certain paceo It is assumed that the motion is
restricted to arm movement, with the trunk remain
ing fixedo The problem is to find the path of
motion which would be assumed by the subject in
performing the actual motion under the previously
mentioned conditionso
58
Model solution of the above problem can be summar
ized in the following stepso
Step 1
ao Define the arm anthropometric characteris
tics for a chosen subjecto These charac
teristics for each arm segment include~
mass, length, distance of the center of
mass from the proximal joint, moment of
inertia, and cross sectional areao
59
bo Define task parameterso These are motion
distance, motion angle, external load,
initial arm configuration, and motion timeo
All the characteristics mentioned in a and b
above constitute the input to the modelo
Step 2
ao In accordance with Step l and the dynamic
analysis presented on page 24, Chapter II,
calculate velocities and accelerations as
well as reactive forces and moments for
each arm segmento
bo Based upon the principles presented on page
39, Chapter II, formulate the modelo
Step 3
Solve the model by using one of the proposed
algorithms discussed in Chapter IIIo
60
The resultant solution of Step 3 is the optimal
or near optimal (depending upon the algorithm)
solution of the problemo
Formulation of the optimization model and the use
of the proposed algorithms are presented in some detail in
the following sectionso
Dynamics of the Arm
The motion of the arm is equivalent to a motion of
two links: (1) the forearm and handt and (2) the upper
armo Figure 12 shows the arm's two links with their
Euler's angles, dimensions, and external forces acting upon
them at a time instant t during motiono
Velocity and Acceleration
ao Link--12
Using equations (2ol) through (2o5) at , 12=90° 1
and e12 , the following expressions can be obtained for any
instant t during motion:
1For planar motion, the ' angle will remain constant throughout the motion, ioeo, , 12=90°o
y
-Z Direction of Motion
= length of the upper arm
= distance of the upper arm center of mass from shoulder joint
L23 = length of the forearm-hand link
m12 = mass of the upper arm
= mass of the forearm-hand link
= arm Euler's angles at time t
W = weight carried by the hand
61
Fig. 12.--Arm configuration at an instant t during the motion
62
• -
•
Linear velocity at elbow is given as~
-Taking v1=o (shoulder joint is fixed) and expanding
the second term, the above expression becomes:
It follows that:
(4o7)
where
L12 = upper arm length ..
612 = upper arm angle at time t ..
• 612 = angular displacement of the upper arm, and
<v2)x• (V2)Y, (V2)Z = elbow velocity components with respect to XYZ frameo
The elbow acceleration vector can be written as:
o,o G 2 _., = (a12L12sina12 + a12L12 cose12 )i
- •2 -+ (e12L12 cosa 12 - a12L12sine 12 )k o
From the above expression, the scalar components of the
elbow acceleration become:
Similarly, the acceleration components of upper arm (link
12) center of mass are obtained as:
63
64
(al2)G .. 0 2 = rl2asinel2•el2 + rl2•cosel2•el2 (4ol4)
X
(al2)G = 0 {4ol5) y
(al2)G .. o2 (4ol6) = rl2•cosal2•al2 rl2osinal2al2 0
z
bo Link--23
Similar to link 12, the velocity and acceleration
expressions are as follows:
• .. = 9 23j
• - •• -= 0 23 = 923j o 0
Hand linear velocity vector is given as:
Its scalar components are:
65
where
123 = forearm-hand length,
923 = forearm angle at time t,
e
e23 = angular velocity of the forearm at time to
Hand linear acceleration is obtained as:
From the above expression, it follows immediately that:
Also, in a similar fashion, the linear acceleration campo-
nents of the forearm hand link center of mass can be
written as:
66
·Reactive Forces and Moments
By referring to the dynamical analysis presented
in Chapter rr. page 24, the reactive forces and moments of
the arm links can be computed upon examining Figures 13 and
ao Link--23
The equations of motion for link 23 can be written
by using its free body diagram in Figure 13 as~
where
-
+ e j
67
X
..
Y,y
-Z
J
Q)
w z
hand velocity vector
hand linear acceleration vector
forearm center of mass linear acceleration vector
-a 2 = elbow linear acceleration vector
-F2 = reactive force vector at the elbow
-M2 = moment vector at the elbow
m23•g = weight of the forearm-hand link
w = weight carried by the hand
x,y,z = principal axes of the forearm
X,Y,Z = global axes
Figo 13o--Free body diagram of the forearm-hand link
moment and reactive force resulting from the forearm
upper arm center of mass linear acceleration vector
moment vector at the shoulder joint
F1 = reactive force vector at the shoulder joint
m12 •g = weight of the upper arm
Figo 14o--Free body diagram of the upper arm
68
(iii) A23 - 0
0 1 0
0
Using the above expressions, equation (4a32) reduces to:
Notice in this particular case of arm motions that the y
and Y axes coincide with each othero Therefore all their
69
components (velocity, acceleration, forces, and moment) are
equivalent, ioeo 1 (M2 )y = (M2 )yo
Using equation (2o8), the reactive force vector at
the elbow can be obtained as:
\
0
Components of the elbow reactive force can be com-
puted with respect to the link principal axes xyz by using
the transformation matrix A23 o The resulting components
are the normal and shearing forces at the elbow jointo
bo Link--12
Similar to link 23 and by using the free body dia
gram in Figure 14 (page 68), the following expressions for
moment and reactive force at the shoulder joint can be
obtained:
and components of shoulder reactive force are
70
Model Algorithms
In accordance with the assumptions and principles
of each one of the proposed algorithms, formulation and
solution of the model can be achieved by using equations
(4ol) through (4o40) and the discussions presented on
page 43, Chapter IIIo Applications of the proposed
algorithms--linear and geometric programmings, dynamic
programming, and simulation--are presented belowo
Suboptimization--Linear Programming
71
With reference to the principles presented on page
43, Chapter III, the formulation of the model in accordance
with the linear programming algorithm should be a straight-
forward mattero However, writing the objective function
might need some commentso It was decided to'use power as
the performance criterion for the modelo Since under the
suboptimization approach, power function should be opti-
mized at a different point in time during motion, the
objective function is written as~
where
shoulder moment at time t,
angular v~looity of the upper arm at time to
Linearization of the above expression is not possible by
the small angles assumptiono Therefore, it was decided to
replace power by another performance criterion which can
be linearized by the small angles assumptiono Normal
stress at the shoulder joint was chosen to be such cri-
teriono
The linear programming model at any instant t
during the motion can be formulated as
minimize:
subject to:
Physical Constraints
gl = xl < c1o _,
g2 = xl > c2o _,
g3 = x2 < c30 _,
g4 = x2 > c4o -Stress Constraints
g5 - c52x2 + 0s3x4 + c54x8 - x9 = 0 - -
g6 = Xg !. c6o
g7 = c7lx4 - c72x8 - xlo = o
ga = - ca1x1 + c82x4 + Xg - xll - x1o = 0
g9 = xll < c91 ...,
72
73
Co Position Constraints
where
xl = 612 x9 = (M2)Y
x2 = e23 x1o = (F2)X
00
x4 = e12 xll = (Ml)Y
00
xa = 6 23 xl2 = (Fl)Z
c1o• c2o• c3o• c4o = maximum and minimum values for the two Euler's angles of the arm
c52 = m23og•r23 + WL23
053 = m23or23o 112
054 = (I23)y
c60 = maximum allowable moment at the elbow joint
c11 - m12112
c12 = ml2r23
ca1 = (M23g + W) o 112
c92 = 112
c93 = (Il2)y
c1 = maximum allowable moment at the shoulder joint
T = motion timeo
Solving the above model at different points in
time would yield the necessary information to generate the
hand path of motion and its characteristicso The model
objective function and constraints will remain the same
for all points in time along the path of motion except for
the position constraint which will vary in accordance with
the instant consideredo
Worthy of notice is that after linearizing the
model many terms completely disappeared from both the
74
objective function and the constraint equationso Further,
there is no constraint equation written for the hand motion
in the Z directiono
Suboptimization--Geometric Programming
Power at the shoulder joint was selected to be the
criterion for the geometric programming formulation as men-
tioned on page 47, Chapter IIIo
Minimize:
subject to:
75
(moment at 2)
(X-component of reactive force at 2)
(Z-component of reactive force at 2)
(moment at 1)
76
(position constraints for the hand)
(X-accel)
(X-veloc)
(law of sines and cosines)
(physical constraints)
77
where
oo
= 612t
C4 = (I23)y/m23r231 12
C5 = (M23gr23 + WL23)/m23r231 12
C6 = l/m121 12
c7 = r231112
Cg = r231112
c9 = l/r23
ClO = (m23g + W)/r23
ell = 1121r23
cl2 = 1 121r23
cl3 = l/ml2grl2
cl4 = <112>y
Cl5 = 112/m12gr12
Cl6 = 112/m12gr12
78
79
Upon solving the above geometric programming formu-
lation at different points in time with different position
constraints each time, the hand path of motion and its
characteristics would be obtainedo
Dynamic Programming Algorithm
Model solution by the dynamic programming approach
is obtained by solving the following recursive equation in
accordance with the principles presented on page 48, Chap-
ter IIIo
min * = X { If ex x I + f n-1 < xn-1)} 0
n n n-1
80
Application of the above recursive equation to
determine the optimum path of motion can be demonstrated in
the following discussiono
Consider an intermediate stage n at which the motion
time is given as t o At this stage, the costs in terms of n
power required for the hand to move from the initial motion
point to each possible position along the Z-axis (Figure
15) are assumed to have been computed previouslyo Now, the
task is to determine the next Z-coordinate of the hand
after small time interval ~t, that is, determining the hand
Z-coordinate at the beginning of stage n+l at time
tn+l(tn+~t)o Let us assume that fo~ each possible Z
coordinate for the hand at stage n+l, there are m possible
Z-coordinates at stage n for the hand to occupyo That is
to say, there are m possible links for the hand to follow
in moving from stage n to stage n+lo Next, for each pes-
sible coordinate ZHn+l'i,i=l,ooo,m, at stage n+l, what is
the corresponding Z-coordinate for the hand at stage n in
order to minimize the total cost (power) necessary to move
from the initial point to stage n+l? The answer to this
question can be obtained by using the following stepso
Step l
By using the two time values tn and tn+l' com
pute the hand X-coordinates at stages n and n+l as follows:
z
j
I ~i
Mo
tio
n ''~
~---
----
----
----
---~
----
----
----
----
--~-
----
----
----
----
---+
-~~
Tim
e
Sta
ge
Han
d C
oo
rdin
ate
s
An
gu
lar
Velo
cit
y
An
gu
lar
Acc
eler
o
Mom
ent
0 1 XH1
,zH
1
0 0 Mo
t ....._
_ ./t
n --...
..,...-
n+
l
n XH
n,Z
Hn,
j
• e •
·. ~
n ,J
.. e
n,j
Mn,
j
6t
n+
l
XH
n+
l,i
~ e n
+l,
i .. e
n+
l,i
Mn
+l,
i
Fig
o 1
5o
--S
tag
es
of
dyna
mic
pr
ogra
mm
ing
app
roac
h
T
N XH
N,Z
HN
0 0 MT
X
(X)
......,
where
x 2ntn+l 2nt = 2rf{ T -sin( ;+l)} + XH 0
X-coordinates of the hand at stages n and n+l, ioeo, at times tn and tn+l'
= X-coordinates of the hand at the initial point of motion, and
82
X = total displacement in the X directiono
Step 2
For a chosen position i,i=l,•oo,m, at stage
n+l, choose a position j,j=l,eee,m, at stage no The two
positions i and j define a possible motion link between
stages nand n+l for which the Z-coordinates (ZHn,j'ZHn+l,i)
can be obtained by examining the motion grido
Step 3
At stage n, by knowing the two coordinates XH n
and ZHn,j' the arm Euler's angles can be obtained by solv-
ing two nonlinear equations written as:
The above two equations can be solved by using Newton's
iteration scheme in which the two angles are expressed as
[Pennington, 1965]:
where
= partial derivativeso
83
Similarly for stage n+l, the two Euler's angles e12 and e23
can be obtainedo
Step 4
By knowing the arm Euler's angles at stages n
and n+l, angular velocity and acceleration are obtained by
numerical differentiations as follows (drop angle sub
scripts for simplicity, ioeo, e12 will be written as e):
where
(4o55)
= angular displacement at stages n and n+l,
e D
en• en+l = angular velocities at stages n and n+l,
~
en+l = angular acceleration at stage n+lo
Step 5
By using Euler's angles and their derivatives
at stages n and n+l as well as equations (4ol) through
(4o40), reactive forces and moments for all joints can be
obtainedo
Step 6
Power required to move from the initial point
to stage n+l following link ij between stages n and n+l is
given by~
n- m m n!l 1!1 j!l0.5{1Mn,j1 + IMn+l.il}· 16n+l,i- an,jl}
84
(4o56)
where
M . = shoulder moment at stage n when the n,J hand is at position j along the Z
axis,
M = shoulder moment at stage n+l when n+l,i the hand is at position i along the
Z axisj
angular velocity of the upper arm at stages n and n+l for motion along link ijo
Step 7
Repeating Steps 1 through 6 for all possible
values of j and keeping i fixed should yield the cost for
each possible motion from stage n to position i at stage
n+lo Upon selecting the link ij which gives the minimum
cost, the following parameters are defined for position
i at stage n+l~
ao cost necessary to move from the initial
point to stage n+l,
bo the hand position at stage n,
Co angular velocities, moments, forceso
Step 8
By varying i over all possible positions for
the hand at stage n+l and repeating Steps l through 7j the
cost and associated parameters at all possible hand posi
tions at stage n+l can be obtainedo
85
The above analysis for an intermediate stage can be
easily extended to all stages considered between the
initial and terminal points of the motiono An iteration
scheme for the dynamic programming approach is as shown in
Figure 16o A computer program was written in accordance
with the dynamic programming principles presented aboveo
The program is given in the Appendixo
ZH
,ZH
n i
n il
t X
Hn
+l'
ZH
n+
l,i
Tim
e G
en
era
tor
for
Sta
ges
n=
l 5••
oN
~
-~ __.. ,...
Per
form
ance
E
valu
ato
r
XH
n,Z
Hni
lj j=
l,m
__.
.,_,
MOD
EL
....
4
An
thro
po
met
ric
and
T
ask
C
hara
cte
rist
ics
Fig
o
16
o--
Dy
nam
ic
pro
gra
mm
ing
it
era
tio
n
sche
me
Op
tim
al
Pat
h o
f ~
Mo
tio
n .. r
•
co
0\
87
Simulation
The basic assumption for the simulation approach,
as previously discussed (page 54), is that the hand follows
a certain geometrical shape of known function during its
motiono That is, in the simulation's solution the shape
of the hand path of motion is specified beforehand and
the task is then to determine the function's parameters
which would yield the minimum performance criterion (power)
among a set of investigated criteriao Solution of the
model in accordance with the simulation approach is as
shown in Figure 17o A step-by-step description of the
simulation algorithm is given as followso
Step 1
algorithm:
Define the following characteristics for the
ao Anthropometric characteristics of the arm
which include~ masses (m12 , m23 ), lengths
(L12 , L23 ), CoGo distances (r12 , r 23 )s and
moments of inertia (I12 , I 23 ),
bo Task characteristics which include~ motion
distance (d), motion angle (a), motion time
(T), initial Euler's angles (8 12 , 823 )o
- -
~
Fu
ncti
on
G
en
era
tor
~
- Typ
e
Po
ssib
le
Fu
ncti
on
s fo
r H
and
Path
o
f M
oti
on
Per
form
ance
E
valu
ato
r
Han
d P
ath
_
of
1li
oti
on
-M
ODEL
~~
An
thro
po
met
ric
and
T
ask
C
hara
cte
rist
ics
Fig
o
17
o--
Sim
ula
tio
n i
tera
tio
n s
chem
e
' ~
Pat
h
of an
d
... eri
sti
cs
co
co
Step 2
Calculate the hand X-coordinates at different
points in time by using the following functional relation
ship~
where
XHt = hand X- coordinate at time t,
XH 0 = hand X-coordinate at the initial point, and
X = displacement in the X direction = dcosa,
d,T = as defined beforeo
Step 3
89
Choose a geometrical function to describe the
hand path of motiono There are four possible functions
from which to chooseo These are sine, ellipse, and parab
ola functions as well as a polynomial regression function
fitted through a set of grid points constructed between the
initial and terminal motion pointso Each one of the four
functions is uniquely determined by specifying some parame
ters pertinent to the hand path of motiono
For instance. the sine and ellipse functions are
defined upon specifying the maximum height of the hand above
the work surface (h) and motion distance (d)o The two func
tions can be written as follows (see Figure 18):
s X
E
X .. I..\ \ \ d --
\ 1
d = Motion Distance
a = Motion Angle
h = Maximum Height of the Hand above the Work Surface
Figo 18o--Simulation--sine and ellipse functions
90
(i) sine function
XH ZHt = ZH 0 + h~sin(2rre Xt) + XHtosin(a)
(ii)
where
XHt• ZHt =
ZH0 =
x. h, d, a =
XH -X 2 1-( t )
X + XHtesin(a)
X- and Z-coordinates of the hand at time t,
hand Z-coordinate at time zero, and
as defined beforeo
91
The parabola function is defined by fitting a second
order polynomial function through three points which include
the initial and terminal points of motiono The second point
is specified by two parameters: its position along the
X-axis (P) and its maximum height above the work surface (h)
as shown in Figure 19o By using these three points 1 the
parabola function for the hand Z-coordinate is written as~
where
= regression coefficients determined by using the three points, and
= as defined beforeo
E
z
~
X
P = Position of Point 2
h, d, a = As defined in Figure 18 (page 90)
lj 2, 3 = Parabola Points
Figo l9o--Simulation--parabola
92
On the other hand, the polynomial function is obtained by
fitting a fifth order function through a set of selected
points between the initial and terminal points of motiono
The function is given as:
93
ZHt = ZH 0+a 0+a 1 •XHt+a 2 oXH~+a 3 •XH~+s 4 ~xH~+a 5 •XH~ o
{4o61)
Step 4
For the chosen function, define some initial
values for its associated parameterso Using these values,
define the function's equation as explained in Step 3o
Step 5
Using the hand X-coordinates generated in Step
2 above and the chosen function, determine the hand Z
coordinates at different points in time during the motiono
Step 6
Upon determining the X- and Z-coordinates of
the hand during the entire motion, the corresponding Euler's
angles for the arm segments at any instant t are obtained by
solving the following two equations:
Solution of the above two equations can be obtained by
employing Newton's iteration scheme as explained before on
page 83o Repeatedly solving the above two equations for
all different values of XH and ZH, Euler 9 s angles for the
arm segments can be determined for the entire motiono
Step 7
By knowing the arm Euler's angles for all time
94
points during the motion, angular velocities and accelera
tions are obtained by numerical differentiations as follows:
By applying the above two expressions, angular velocities
and accelerations for each arm segment can be obtainedo
Step 8
By using Euler's angles, angular velocitiesj
and accelerations determined in Steps 6 and 7 as well as
equations (4ol) through (4o40), reactive forces and moments
at the different joints can be obtainedo
Step 9
Calculate total power, performance criterion,
expended during performance of the motion as~
where
Step 10
angular velocities of the upper arm at times t and t+l~ and
shoulder moment at times t and t+lo
95
Increment the initial values of the chosen func-
tion's parameters by preselected valueso
Step 11
For the new function's parameters, repeat Steps
2 through 9 above and obtain the corresponding power's
valueo
Step 12
After repeating Steps 2 through 11 over all
possible choices of the chosen function's parameters,
select the parameters which yield the minimum power value
among the parameters investigatedo Next, use these parame
ters to define the hand path of motion and its motion char-
acteristiCSo
A computer program was written for the simulation
algorithm in accordance with the above-mentioned stepso A
full description of the program is given in the Appendixo
Choice of Model Algorithm
96
In order to test the feasibility of the previous
algorithms, some examples of hypothetical motions were con
sideredo For these examples» motions of a subject of
known anthropometric characteristics (Table 1) were ana
lyzed under different taskso Four different tasks were
investigated (Figure 20)o Each of the tasks was defined by
the initial position of the hand with respect to the shoul
der joint, motion distance, motion angle 5 and motion timea
Using suboptimization algorithms--linear and geo
metric programmings--to solve the above-mentioned tasks was
not a successful attempto In the case of the linear pro
gramming approach, solution of the model became infeasible
for all cases consideredo The failure of linear programming
to provide a feasible solution can be attributed to the
gross approximations which were introduced in order to lin
earize the modela That isi the method of linearizing the
model by the small angles assumption seems to be an invalid
oneo It is worth mentioning at this point that the small
angles assumption was adopted by Nubar and Contini [1961]
in formulating their modelo
TABL
E 1
AN
THRO
POM
ETRI
C C
HA
RA
CTE
RIS
TIC
Sa
Up.
p.e.r
.A
rm
Len
gth
----
----
----
-mete
r O
o255
0
Dis
tan
ce o
f C
oG
o--
-met
er
Oo1
137
Mass
----
----
----
---k
ilo
gra
m(s
) lo
243Q
Mom
ent
of
Inert
ia--
new
ton
-mete
r sq
uar
ed
Oo2
345
Fo
rear
m
Oo3
604
Ool
599
1o11
70
Oo3
990
-aA
dopt
ed
from
F
isch
er
[19
06
],
cit
ed
Po
24
0o
by
Han
sen
and
Cor
nog
[19
58
],
\0
-..J
98 .---------------------~-----------------------
TASK--I
H
MOTION CHARACTERISTICS Distance---= Angle------= Weight-----= Time-------=
o3048 meter OoO dego OoO kgo Oo6 sec.
TASK--III
H
E
MOTION CHARACTERISTICS
Distance---= o3048 meter Angle------= 0.0 dego Weight-----= OoO kgo Time-------= Oo6 seco
TASK--II
H
MOTION CHARACTERISTICS Distance---= Angle------= Weight-----= Time-------=
o3048 meter OoO dego OoO kgo Oo6 seco
TASK--IV
E
MOTION CHARACTERISTICS
Distance---= o3048 meter Angle------= OoO dego Weight-----= 0.0 kgo Time-------= Oo4 seco
S--Shoulder o n E--Elbow Joint H--Hand
Figo 20o--Task configurations
The geometric programming approach to solve the
model seems to be a valid one, at least from the theoret
ical point of viewo However, the computer algorithm
[Teske, 1970] which was used created the following diffi
cultieso
la Convergence of the primal and dual functions
was in most oases impossible or obtained after a large _
number of iterationsa
2a Computational times were extremely largea In
some cases it took between 12 and 24 minutes of computer
99
time before an optimum solution for one point was obtaineda
In accordance with the previously mentioned diffi-
culties and because the validity of suboptimization
approaches as applied to human motion is questionable, no
further attempt was made to consider other suboptimization
algorithms a
Feasible solutions for the above examples, however,
were achieved by both dynamic programming and simulation
algorithmso Figures 21 through 24 show the hand paths of
motion obtained by the two algorithms under the four differ-
ent tasks consideredo As can be seen, there is a large
similarity between the predictions of some simulation
approaches and dynamic programmingo The oloseness.between
the dynamic programming and the enumeration approaches are •
considerably goodo However, the variability between dynamic
·~·
MOTION CHARACTERISTICS
Distance - Oo3048 meter Angle = OaO de go Weight = OoO kgo Time = o6 seco
INITIAL ARM CONFIGURATION
Upper Arm = 225 dego Forearm = 321 dego
HAND PATH OF MOTION z
Key~
I, I
---
'
-----
' ' ' ........... ' . ' ' ' • ,, ., ' ........ -, -.. ,. .
.......... __ '\. ......_,_ . ~.... ~ ..........
Dynamic Programming
Simulation--Enumeration
Simulation--Parabola
Simulation--Sine
..
Figb 2lo--Hand path of motion--task I
..
100
X
MOTION CHARACTERISTICS
Distance = Oo3048 meter Angle = 0 dego Weight = 0 kgo Time - Oo6 seoo -
INITIAL ARM CONFIGURATION
Upper Arm = 225o0 dego Forearm = 351o0 dego
HAND PATH OF MOTION
z
P ------ -/
Key~
---
---- Dynamic Programming
--- Simulation--Enumeration
Figo 22o--Hand path of motion--task II
101
.....
MOTION CHARACTERISTICS
Distance = Oo3048 meter
Angle = OoO dego Weight = OoO kgo Time = 0 6 seco
INITIAL ARM CONFIGURATION --
Upper Arm = 270 de go
Forearm = 43 dego z
'II HAND PATH OF MOTION
... X
Key~
Straight line obtained by both dynamic programming and simulation approaches as the optimum path
Figo 23o--Hand path of motion--task III
102
MOTION CHARACTERISTICS
Distance
Angle
Weight
Time
= Oo3048 meter
= CoO dego
INITIAL ARM
Upper Arm
Forearm
= OoO kgo
= Oo4 seco
CONFIGURATION
= 294oO dego
= 73o0 dego
HAND PATH OF MOTION
z ...
Key:
~~--- Dynamic Programming Simu1~ticn--Enumeration
Simulation--Parabola
Simula ~- -t ~- .-_ ~""A v -"-' • -_=- - -·-··--
Simulat~o~--Ellipse
Figo ~?~; -.-... ·-~h-~"''.;1d path of motion--tas~-\. IV
103
" II II
:I ,_
I[ 
104
programming and the rest of the simulation approaches (sine,
ellipse, and parabola) is somewhat largeo
The question which might be asked then is which
algorithm should be adopted for the model? The most
obvious answeri of course, would be the use of dynamic pro
gramming since it gives an optimum patho Next to dynamic
programming comes simulation by enumerating through a grid
pointo For the other simulation approaches (sine, ellipse,
and parabola functions) it seems there is a little evidence»
based upon the tasks analyzed, of their validity to predict
an optimum path of motion which would be the same as that of
the humano
CHAPTER V
MODEL TESTING
An experiment was conducted to test the accuracy as
well as the adequacy of the proposed model in predicting
planar motionso It is obviously impossible to do a very
thorough investigation or consider all possible parameters
affecting human motion in a short study, and at best this
can only serve as a test for the applicability of the modele
This chapter includes a discussion of the task, the vari
ables chosen for use in the experiment, the procedure~ and
a statistical analysis of the datao
The Task
The task chosen for this study was a simple trans
port movementc Examples of transport movements are plenti
ful in the military, industryj and everyday lifeo Many
industrial production processes, including many types of
assembly, require rapid movements from one position to
anothera This class of movements, therefore, is of some
importance in practical situationso The motion variables
of interest were motion distance and work surface heightu
The subject participating in this experiment was
standing at a table on the top of which a path of motion
105
for the various motion distances was marked (Figure 25).
He was asked to move his right hand between preselected
points without repetition, i.e., performing discrete
motionso In the course of the experiment, the subject's
motion was recorded by photographic means.
Experimental Variables
Motion Distance
It is well known that the motion distance has an
important effect on the motion path and the associated
motion characteristics such as velocity and acceleration
106
profiles [Ayoub, 1966; Ramsey, 1968; Ayoub, 1969]. Greene
emphasizes the necessity of including distance measurements
as an important variable when he states:
o o o distance is a necessary ingredient to any determination of a quantity of work in the physical sense a o o the lack of the distance factor is not an insurmountable problem ••• But to measure work without the distance factor is impossible [1958]o
Two levels of distance, 12 inches and 24 inches, are usu-
ally recommended. The shorter distance corresponds to
reach from the edge of the work surface to the center of
the work area, and the longer distance corresponds to reach
from the edge of the work surface to the rear of the work
areao For the purpose of this study, three levels of dis-
tance, 9, 12, and 15 inches, were selected. These were
found to be very adequate to compensate for arm motion
without causing bending of the subject's torso.
7--1/2"
I I + O" Elbow Heit::::.'lt
13" I 6"
I r· 9" 1• 12" ~I
I· 15" __ ___,..,...,.
I Motion Distance
~; .. ) ·..:. View .... _ .... ""' . .#
} __,__., ______ - - -------· . --
Top V1ew
107
108
Work Surface Height
Work surface height is one of the important factors
which affects the shape of the path of motion to a large
extento Several studies have been conducted in the inter
est of determining the best work surface height under dif
ferent industrial tasks [Gilbreth, 1917; Warner, 1920;
Knowle, 1946; Ellis, 1951]. It has been shown by Ellis
[1951] and verified by Konz [1967] through extensive
experimentation that the distance between the elbow and
the work surface is the best criterion for determining the
work surface height. Konz recommends a surface height one
inch below the elbow as the optimum height in contrast to a
three-inch height suggested by Ellis. For this study three
surface heights were selected. They measure 6, 3, and
0 inches from the elbow joint.
Subjects
Ten male subjects were selected for this experi
mento The only limiting factors concerning the subjects
were that they be of similar age, race, with average phys
ical build, ioe., excluding the athletic type individual,
and with an average degree of performance skills,
The anthropometric characteristics for each subject
were determined by using the method of coefficients intro
duced by Fischer [Drillis, et al., 1964]. By using this
109
method, the arm segments' masses, distances of center of
masses, and radius of gyrations are determined by using
three coefficients c1 , c2 , c3 respectivelyo The first
coefficient represents the ratio of the segment mass to the
total body masso The second coefficient is the ratio of
the distance of the center of mass from the proximal joint
to the total segment lengtho The third coefficient is the
ratio of the radius of gyration of the segment about the
mediolateral axis to the total segment lengtho
Therefore, upon knowing the total body mass and
the arm dimensions, the mass, distance of center of massj
and radius of gyration for each segment can be obtaineda
In this study the values for c1 and c2 were obtained from
Dempster's estimates rather than from those of Fischer
(Figure 26)o The value proposed by Fischer for c3
was
adoptedo It was equal to Oo3o
Table 2 shows subjects 9 anthropometric character-
istics for the arm as obtained by the method of coeffi-
cients discussed aboveo It should be appreciated that the
method of coefficients is an approximate method with some
inherent errorso However, through sensitivity analysis,
variations in the three coefficients c1 , c2 , and c3 by as
much as + 20% from the original values proposed by Dempster ~
[1955] resulted in minor changes for the model 9 s predic-
tionsa In most casesj variation in the total power
43.0%
57.0%
50.0% 50.0%
110
Mass = lo55%
Mass = 0.60%
Fig. 26.--Arm segment masses and their locations expressed as percentages of arm lengths and total body mass [adopted from Dempster, 1955].
TABL
E 2
AN
THRO
POM
ETRI
C C
HA
RA
CTE
RIS
TIC
S FO
R TH
E SU
BJE
CTS
.S.U.
B.J E
C.T
1 2
3 4
5 6
7
Upp
er
Arm
Len
gth
----
----
-mete
r o3
65
o36Q
.3
55
o3
43
o381
.3
57
.37
1
CoG
o d
ista
nce--
mete
r o1
60
o156
9 .,1
55
.,150
.,1
66
o155
.,1
63
Mass
----
----
---k
gs.
, 2o
0 1o
85
2.,0
0 1.
,91
2.3
6
1o84
2o
1
Rad
ius
of
.10
9
.,108
.,1
07
o103
o1
14
.,102
.,1
06
gy
rati
on
----
--m
ete
r
Fo
rear
m
Len
gth
----
----
-mete
r o4
60
o457
.,4
50
.,445
o5
08
.44
9
.,464
CoG
. d
ista
nce--
mete
r ol
878
.,198
.,1
94
o191
o2
18
.193
5 o1
978
Mass
----
----
---k
gso
lo
7
lo5
1o
62
1o55
1.
,91
1.,6
1o
85
Rad
ius
of
ol46
.,1
43
o135
o1
34
o152
o1
33
o138
g
yra
tio
n--
----
mete
r
8 9
.,345
.,3
61
.15
2
o155
1.,8
1o
95
.,103
.1
05
.,45
o465
.,193
o1
97
1.6
1.
,75
o134
.1
40
10
0 34
5
o152
1.,8
3
o103
o452
o194
1o45
.,135
~ ~ ~
112
required to perform the motion was within + 1.0% of the .... total power computed with the original coefficients. On
the other hand, varying the three coefficients by + 40% of .... the original values resulted in a definite change in the
predictions of the model. For instance, total power
changed by as much as 10%. Also, considerable change in
the shape of the optimum path or motion occurred as shown
in Figure 27.
Key:
optimum path of motion obtained with the original values of the coefficients (cl. c2. c3>
------optimum path or motion obtained with+ 40% variations in the coefficient values
Fig. 27.--Typical effect of 40% variations in the anthropometric coefficients upon the optimum path or motion. . . .
113
It can be concluded from the sensitivity analysis
that variations in the three coefficients by as much as 40%
from the original values are necessary before any appreci
able effects upon the model's prediction can be observedo
This phenomenon could be attributed to the small magnitude
of the coefficients and consequently the values resulting
from their useo
§quipment
Camera
A photographic camera was used to record the sub
jects' motionso The camera used in this study was a 4 x 5
Crown Graphic camera with a focal length f=l35 mmo The
Crown Graphic camera is designed to use Polaroid films as
well as sheet filmso The distortion curve and the princi
pal points of the camera's lens were previously determined
in another study [Ayoub, 1969]o The lensv distortion error
was proved to be small and can be disregarded without
affecting the accuracy of measurementso The camera was
supported by a special adjustable base as shown in Figure
28o By means of this base the camera can be completely
leveled and oriented parallel to the vertical plane of the
work tableo Polaroid films (Type 52-4x5) were used for all
recordingso
114
'
Figo 28o--Exper1mental equipment
115
Lighting during Record_~~
For motion recording a General Radio Strobotac type
1531 was used to record multiple exposures on the camera
filmsa This necessitated performing the experiment under
total darknesso The disadvantage in using the strobe for
human motion recording is its effect under certain condi
tions of frequency and intensity upon the subjectvs per
formanceo This did not constitute a serious drawback in
this study, howeverj simply because of the very short dura
tion of the experimental task and the relatively low fre
quency usedo Also~ during the recording process, the
strobe was positioned behind the subject to eliminate
direct exposure of subject to the strobe flashing lighto
~~justab~EJ.= ~e~~e
The table selected for this experiment is shown in
Figure 28 (page 114)o It was adjustable as to heighti
rotation, and tilting angleo This was accomplished by
mounting a board on a converted base of a dental chairo A
protractor was fixed to the table 9 s board to indicate the
tilting angle, thus making adjustment of the board particu
larly direct and simpleo On top of the table~ three scales
were positioned in such a way that they formed a vertical
planec This plane was used to position the camera parallel
to the subjects' sagittal planeso Also, six control
targets were attached to the three scales for the purpose
of obtaining the photographic scales from the pictureso
Experimental Procedure
Each of the subjects participating in this study
was instructed and trained thoroughly with regard to his
duties in the experimento This was done essentially to
familiarize each subject with the equipment and with the
motions he would be required to makeo
116
Following the training phase, photographic records
for all subjects' motions under all different experimental
combinations were takeno From these records a path of
motion for each subject under each experimental combination
was generatedo
At each experimental session the subject was asked
to wear a black glove, and a light-reflective tape was
placed at the arm segmentso Through an adjustment in table
heighti the subject was standing at the work table in such
a way that the predetermined distance between the elbow
and work surface could be obtainedo Each subject was
instructed to move his hand between two preselected termi
nal points at his normal pace, ioeo, a pace he could main
tain for extended periodso After photographing each
motion, the subject was given a two-minute rest period and
he was allowed to move from his positiono A total of 180
motions was recorded for the ten subjects under all levels
117
of distance and elbow heights considered in the experiment.
Figure 29 shows typical examples of some of the motions
obtained during the course of the experiment.
Hand Path of Motion
The hand paths of motion for all subjects under
different motions investigated in this study were measured
from the photographic records obtained for the motionso
The measurements were obtained by retracing upon graph
paper the hand paths of motion as well as the positions of
the six control targets appearing on each photographo
Next, for each motion, the path of motion coordinates as
well as three measurements for the photographic scale (dis
tances between each pair of the control targets) was read
from the graph paper. By computing the photographic scale
and using the measured coordinates, the actual paths of
motion for each subject under each experimental condition
were obtained. The time of each motion was obtained
directly by counting the number of exposures recorded on
the photographic film.
For each experimental motion an optimum path of
motion was generated by using the proposed model and the
dynamic programming algorithm under the same conditions
as those of the experimental one. That is, under the same
motion distance, initial arm configuration, subject's
anthropometric characteristics, and motion time an optimum
Figo 29o--Typical examples of the photographic records obtained during the experimento
118
)
path of motion as predicted by the model was obtained.
Therefore, for each motion investigated in this study two
paths were obtained: an experimental one and an optimum
one predicted by the model. To test the relationship
between the two paths the following quantitative analyses
were usedo
119
lo A correlation analysis was conducted to test
the relationship between the two paths of motion, that is,
to test for similarity between the paths as far as loca
tions of the peak points and linearity are concernedo It
should be understood that correlation does not test for the
closeness between the two paths of motiono An example
might be useful in explaining this point. Let us consider
the two paths shown in Figure 30. The two paths are simi
lar to each other in such a way that they increase and
decrease together and the location of their peak points is
the sameo Computing the correlation coefficient for the
two paths presented in Figure 30a would be equal to or very
close to 1.0. On the other hand, shifting the two paths'
peak points along the X-axis as shown in Figure 30b would
result in lowering the correlation coefficient to approxi
mately o.6o. It follows, therefore, that correlation can
be used to measure the similarity between the two paths of
motion but not the closeness between themo That is to say,
a perfect correlation (coefficient=!) between the two paths
120
X
o.o 1.0 o.o Correlation Coeffi-
o.o .10 o.o cient = 1.0
ao--Paths with peak points at the same position along the X-axis and with different height along the Z-axis.
z
o.o 1.0 • 5 o.o Correlation Coeffi-cient = .6
o.o .667 1.0 OoO
bo--Paths with peak points shifted from each other along the X-axis and with the same height at the Z-axis.
Fig. 30.--Correlation analysis for two paths of motion
121
of motion would indicate that they follow the same trend
but do not necessarily coincide on each other or even come
close to coincidinga
2o To test for the closeness between the two paths
of motion, the following quantitative measures were used:
(i) The difference at each point in time between
the coordinates of the experimental and the theoretical
paths, expressed as:
where
ZH t - ZH t P m X lOOoO
ZHmt
Et = percentage error at time t,
= predicted hand Z-coordinate at time t by the model, and
= measured hand Z-coordinate at time t by photographic meanso
(ii) The difference between the total areas under
the two paths of motion, expressed as:
A - A EA = P m X lOOoO
Am
where
EA = percentage errors of the two areas,
AP = area under predicted path by the model, and
Am - area under the experimental path.
Results and Interpretations
Figures 31 through 39 show typical examples for
122
some plots of all different paths of motion which were
recorded experimentally versus those predicted by the model
using the dynamic programming algorithm under the same task
conditions. The sequence number given on each figure indi
cates a particular combination of the experimental vari
ables under which the motion was recorded. The numbers in
each sequence indicate the levels of subjects, replication,
distance, and work surface height respectively. For
example, the sequence 31096 indicates that a first trial
for motion was performed by subject 3 at a distance of 9
inches with the elbow 6 inches above the work surface. It
can be seen that the paths predicted by the model are very
close to those obtained by experimental analysis for the
subjects. Also, the plots of motion paths show little vari
ation among different motions investigated for subjects'
performances. This could be attributed to the training
phase of the experiment.
Tables 3, 4, and 5 show the correlation coeffi
cients and measures of difference for all subjects under
different levels of distance and work surface height.
••••••••••••••••••••••••••••••••••••••••••••••••••••
MOTION SEQUENCE• 11096 ---~--~-----~---------~----------------·-----------. PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -o.zt AVERAGE- -1.11952
STANDARD ERROR• 1.63631
o.o
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 1.40061 ---~------------------~------------------. ---------PlOTS OF THE HAND PATH OF MOTION
• THEORETICAl X EXPERMINTAl
• • • • • • • • • •
T .x I • • " • •• E • X
• X
••
·········=·········································· MOTION SEQUENCE• 81096 ---------------------~------~-~------------------
PERCENT DIFFERENCE Of MOTION PATH COOR • .
o.o -0.64 -0.01 o.o AVERAGE• -0.16356
STANDARD ERROR• 0.27635 -------------------------~------------------------
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 0.21656 --~~-----------~--~---~------~---------~-------
PlOTS OF THE HAND PATH OF MOTION
• THEORETICAL X EXPERMINTAl
• • • • • • • • • •
T .x
I • X
.. • X
E ••
123
Fig. 31.--Motion performed at elbow height of 6 inches above the work surface and distance of 9 inches.
MOTION SEQUENCE• 11126 ----------------------------------------------------PERCENT DIFFERENCE OF MOTION PATH tOOR.
o.o -2.08 -1.32 -0.22
AVERAGEz -0~93719 STANDARD ERROR• 1.29950
o.o o.o
----------------------------------------------------DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 1.10922 ---------------------~-----------------------------
PLOTS OF THE HAND PATH OF MOTION
* THEORETICAL X EXPERfiiiNTAL
• • • • • • • • • • T .x
I • X
M • •x E • X
• X
• x
MOTION SEQUENCE• 62126 -------------------------------..--------PERCENT DIFFERENCE OF "OTION PATH tOOR.
o.o -4.79 -4.56 -0.40 -1.53 -2.66
AVERAGE• -1.99315 STANDARD ERROR= 1.91184
o.o
-----~-----------------------------~--------------
DIFFERENCE BETWEEN tURVES AREAS IN PERCENT• 0.91377
---~-----------------------------------------------
PLOTS OF THE HAND PATH OF fiiOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T .x
I .•x
" • •• E • X
• • • • • ••
Fig. 32.--Motion performed at elbow height of 6 inches above the work surface and distance of 12 inches.
124
as••••••••••aaasaaaaaww.-aaa•••••••••••••••••••ra•••
MOTION SEQUENCE• CJ2156
--------------------------------------------·--------PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -2.60 -1.82 -0.85 -2.CJ8
AVERAGE• -1.17685 STANDARD ERROR• 1.17CJCJ1
o.o
------------------------------------------------DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 0.11055
--------~-------------------------------------------
PLOTS OF THE HAND PATH OF MOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T •• I ••• M • X
E • X
• X
• x
===•=•••••••••••••=••••••••••=m••••=••••••••••••••••
MOTICN SEQUENCE• 52156 ---------------------~------------------
PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -1. tl -3.04 -1.46 -1.0CJ -1.02
AVERAGE• -1~39015 STANDARD ERROR• 1.26l6CJ
o.o
-------~-------------- ----------~-------------
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• -1.24745
'-------~---------------------------------------
PLOTS OF THE HAND PATH OF MOTION
• THE ORE TICAL I EXPERMINTAL
• • • • .. • • • • •
T •• I ••• " • •• E • •
• x•
• • ••
Fig. 33.--Motion performed at elbow height of
125
6 inches above the work surface and distance of 15 inches.
••••••••••••••••••••••••••••••••••••••••••••••••••••
MOTION SEQUENCE• 82091
PERCENT DIFFERENCE Of MOTION PATH COOR.
o.o -0.25 -0.16 -0.26
AVERAGE• -0•17391 STANDARD ERROR• 0.14782
o.o
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• -0.08937
PLOTS Of THE HAND PATH OF MOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T .x I .x M • X
E • X
• x
MOTION SEQUENCE• 102123
PERCENT DIFFERENCE Of MOTION PATH COOR.
o.o -1.84 -1.45 -0.18 ~1.37 -2.45
AVERAGE• -1.04219 STANDARD ERROR• 0.91108
o.o
----~-----~-----------4-------------------------
DIFFERENCE BETMEEN CURVES AREAS IN PERCENT• -0.04349 -----~--------.-...---------- ----PLOTS Of THE HANO PATH Of MOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T •• I ••• .. • I
I • I
• • • • • • I
Fig. 34.--Motion performed at elbow height of
126
3 inches above the work surface and distance of 9 inches.
HOT10N SEQUENCE= 3212 J
PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -0.97 -0.74 -1.~5
AVERAGE= -0. 6'5182 STANDARD ERROR= 0.593~2
o.o
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT= -0.32187
PLOTS OF THE HAND PATH OF MOTION
• THEORETICAL X EXPE~ .. lNTAl
• • • • • • • • • •
T .x
I .x
M • X
E • X
=========••=•=zaza::%::======•========~===•=========
MOTION SEQUENCE= 22123
----------------------------------------------------PERCENT DIFFERENCE OF ~OltON PATH COOR.
o. 0 o.o o.o o.o AVERAGE= 0.0
STANDARD ERROR= 0.0 ----------------------------------------------~-----
DIFFERENCE BETWEEN CURVES AREAS IH PERCENT= 0.0 ----------------------------------------------------PLOTS OF THE HAND PATH CF HOTION
• THEORETICAl )( EXPERHINTAL
• • • . • • • • • •
"T .J(
1 • X
M • Jl
E .x
127
Fig. 35.--Motion performed at elbow height of 3 inches above the work surface and distance of 12 inches.
.................................................... MOTION SEQUENCE• 71151 -------~-------------------------------------------
PERCENT DIFFERENCE OF MOTION PATH CDOR.
o.o -0.99 -0.01 -1.54 -1.46
AVERAGE• -0~67170 STANDARD ERROR• 0.68256
o.o
---------------------~----------------------------
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 0.20162 --------------------------------------~-----------
PLOTS OF THE HAND PATH OF MOTION
• THE ORE Tl CAl X EXPERJIIJNTAl
• • • • • • • • • •
T •• I •• M • X
E • X
• JC
••
·=··········=······································· MOTION SEQUENCE• 91153
PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -3.12 -3.00 -0.66 -1.89
AVERAGE• -leS4S64 STANDARD ERROR• 1.44431
o.o
-------~-~-------------------- ------------------DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 0.83566 ----------------------------------------------------PLOTS OF THE HAND PATH OF MOTION
• THEORETICAl X EXPERMINTAl
• • • • • • • • • •
T .x I .•x M • •• E • X
• X
• x
128
Fig. 36.--Motion performed at elbow height of 3 inches above the work surface and distance of 15 inches.
•••••••••••••••••••••••••••••••••••••••••••••••••••• MOTION SEQUENCE• 510410 ---------------------------------------------------PEMCENT DIFFERENCE OF MOll~ PATH COOR.
o.o -5.08 -2.11 -0.04
AVE It AGE • -1. ""532 STANDARD ERROR• 1.99121
o.o
----------------------------------------------------DIFFERENCE BET~EEN CURVES AREAS IN PERCENT• 1.75584 ----------------------------------------------------PLOTS OF THE HAND PATH OF MOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T .X
I •* X
" ... E • X
.x
·=·································--··············· MOll ON SEQUENCE• 61090 ----------------------·--------~------------------PERCENT DIFFERENCE OF MOTION PATH ODOR.
o.o -2.51 -o.st -0.1~ -1.~2
AVERAGE• -0.86317 STANDARD ERROR• 0.87882
o.o
-----------~----------------------------------------DIFFERENCE 8STWEEN CURVES AAEAS IW PERCENT• 0.17890 ----------------------------------------------------PLOTS OF THE HAND PATH OF MOTION
• THEORETICn X EXPEIUUNTAL
• • • • • • • • • •
T •• I ••• M • X
E • • • • • x
129
Fig. 37.--Motion performed at elbow height of o inches above the work surface and distance of 9 inches.
MOTION SEQUENCEa
--------~-------------------------------------------
PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -1.07 -1.~~ -2.00 -2.61
AVERAGE• -1.51107 STANDARD ERROR• 1.18~11
o.o
---------------------~-----------------------------
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 0.00952
----------------------------------------------------PLOTS OF THE HAND PATH OF ~OTION
* THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T .x
I ••• M • X
E • X
• x•
MOTION SEQUENCE= 51120
-----------------~----~-----------------------------
PERCENT DIFFERENCE OF ~OTION PATH COOR.
o.o -1.14 -1.69 -0.56
AVERAGE• -1.19112 STANDARD ERROR• 1.~1122
.
o.o
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• 1.48145
---~----------~-------~-----------------------------
PLOTS OF THE HAND PATH OF MOTION
• THEORETICAL X EXPERMINTAL
• • • • • • • • • •
T •• I .•x
" • X
E • • • I
Fig. 38.--Motion performed at elbow height or 0 inches above the work surface and distance or 12 inches.
l30
.................................................... 81150
--~-------------------------------------------------
PERCENT DIFFERENCE OF MOTION PATH COOR.
o.o -3.27 -1.34 -3.20 ~2.79
AVE-AGE• -1.76653 STANDARD ERROR• 1.40132
o.o
-----------------~----------------------------------
DIFFERENCE BETWEEN CURVES AREAS IN PERCENT• -0.25613
PLOTS OF THE HAND PATH OF MOTION
• THEORETICAL
• • • • • • • • • •
T •• I ••• " • X
E • •• • X
• x
.................................................... MOTION SEQUENCE• 42150 ----------------~----------------------------------
PERCENT DIFFERENCE Of ~OTION PATH CODA.
o.o -3~32 -1.25 -3.11 -2.96
AVERAGE• -1.77566 STANDARD ERROR• 1.42424
o.o
---------~-------------------------------------~~---DIFFERENCE 8ITWEEN CURVES AREAS IW PERCENT• 2.09392
-------------------------------------~----~------~ PLOTS OF THE MANO PATH CF "OTION
• THEOREtiCAL X EXPERMINfAL
• • • • • • • • • •
' .x I ••• " • •• ( • •
• X
••
131
Fig. 39.--Motion performed at elbow height of 0 inches above the work surface and distance of 15 inches.
TABL
E 3
CORR
ELA
TIO
N
CO
EFFI
CIE
NTS
BE
TWEE
N
EXPE
RIM
ENTA
L PA
THS
AND
THE
THEO
RET
ICA
L O
NES
Elb
ow
Su
bje
ct
1 2
3 4
5 6
7 H
eig
ht
Dis
tan
ce 9
o730
o9
54
.,876
o8
51
c579
o8
73
o999
o8
99
o997
o9
82
o912
.7
83
o793
o5
43
6 12
o9
25
o993
.7
62
.745
o7
45
o60Q
o9
76
o908
7 o9
99
o887
o8
49
o751
o6
97
o907
15
o914
o9
67
o884
.5
89
o747
o6
87
o984
o8
56
o998
o9
28
o832
o9
64
o766
o9
1Q
9 o8
72
o893
o7
36
o788
o7
46
o873
o9
98
o93Q
o9
93
o789
.9
92
o869
o7
3Q
o747
3 12
o8
95
o832
o9
34
o713
o9
37
o913
o9
22
o925
o8
95
o947
o8
25
o847
o8
92
o90Q
15
o778
o8
87
o886
o8
93
o90Q
o8
76
o977
o9
15
o989
o8
96
o925
o8
76
o767
o8
92
9 o9
57
o775
o7
86
o80
o813
o8
77
o987
o9
37
o939
o8
55
o876
o7
95
o894
o9
89
0 12
o8
95
o765
o7
18
o788
o7
28
o794
o9
59
o903
o9
45
o792
o8
15
o776
o8
59
o998
15
o861
o8
40
o677
o8
05
o787
o8
52
o971
o8
85
o869
o8
34
o845
o8
56
o618
o9
38
8 9
o994
c7
59
o805
.9~4
o804
o7
92
o863
.7
13
.969
o8
83
o979
o9
05
o917
o9
53
o992
o8
53
0 94
2 0 91
2 0 93
2 0 84
2 0 82
2 o8
84
o796
o8
05
0 86
4 o9
15
o781
0 77
6 o9
13
o547
o8
32
o819
o8
63
o911
o8
14
o90Q
10
o531
o6~7
o923
.8
57
.672
o7
38
o828
.9
13
o671
0 89
9 o6
91
o845
o8
83
o944
o8
16
o675
o9
16
o714
......
w
I'\)
TABL
E 4
PERC
ENTA
GE
DIF
FER
ENC
ES
IN
ARE
AS
BETW
EEN
EXPE
RIM
ENTA
L PA
THS
AND
THE
THEO
RET
ICA
L O
NES
Elb
ow
Su
bje
ct
~~
A
Heig
ht ~
1 2
3 4
5 6
7 8
9 10
9 lo
46
O
ol4
2o03
O
o92
Oo2
7 O
o82
lo6
0
Oo2
2 2o
00
Oo7
3 lo
07
O
o64
Oo2
2 1o
09
2o60
O
o65
1o04
O
o04
1o
01
O
o95
6 12
1o
10
Oo4
9 0~92
Oo4
7 O
o93
Oo3
5 O
o63
1o49
O
a71
Oo5
0 2o
11
2c59
O
o74
Oc6
9 O
o39
Oo9
1 O
o45
Oo0
7 lo
09
lo
24
15
Oo5
9 lo
15
O
o78
1o83
O
a63
Oc1
9 lo
73
2o
16
Oa7
6 2o
27
2o~6
lo4
0
1o33
O
o40
lo2
4
Oo9
6 lo
35
lc
98
O
ol3
2o80
9 1o
72
lo2
9
2o
l7
lol2
O
o39
Oo4
8 2c
00
lo0
8
Oo6
4 lo
38
O
o91
Oo6
9 O
o75
2o52
2o
86
Oo3
4 O
ol8
Oo0
9 1
o5
0
Oo0
4
3 12
lo
61
2
Oo?
.l O
o43
Ool
6 O
ol4
Oo2
6 lo
63
lo
16
2o
0 lo
l6
1o31
O
oO
Oo3
2 5o
07
Oo7
4 O
o60
Oc9
3 1o
21
Oo7
8 1
c18
15
lo3
2
2o
l26
O
o85
Dol
O
Oo1
3 O
o76
Oo2
0 lo
46
O
o84
2o01
lo
83
O
o79
Oo1
3 1o
24
1o33
O
o24
Oo0
4 1o
23
Oo9
6 lo
57
9 2o
49
Oo9
5 2
ol2
2o
75
1o76
1o
23
1o70
O
o09
Oo8
3 1o
42
Oo8
0 1o
10
lo7
6
2o75
2
ol2
O
ol8
Oo4
0 O
o79
Oo4
3 O
o50
2 2o
02
Oo6
7 O
o42
lo9
1
1~48
Oo3
2o
32
2o03
O
a01
Oo7
3 Q
1
2o02
lo
l9
1o48
lo
91
lo
61
O
o97
2o12
lo
30
O
o89
lo3
2
15
2o79
2o
00
3o56
O
o35
lo6
6
Oo0
6 O
o97
Oa2
6 O
o23
Oo9
3 2o
79
Oo9
5 O
ol49
2o
09
3c28
O
o09
2c91
1o
74
Oo4
6 1o
09
......,
w
w
TABL
E 5
PERC
ENTA
GE
DIF
FER
ENC
E O
F M
OTI
ON
PA
TH
CO
OR
DIN
ATE
S
E1a
Ese
d T
ime
(sec
on
ds)
Mo
tio
n S
equ
ence
0
o1
0 2
0 3
0 4
0 5
o6
o7
-11
096
OoO
-2o
18
-1
o5
6
-1o
54
-O
o25
DoD
12D
96
DoD
-2o
94
-3
o9
8
DoD
-Do2
1 Do
D 11
126
DoD
-2oD
8 -3
o3
2
-Oo2
2 Da
D Da
D 12
126
OoO
-1o
01
-5
o9
7
-3o
19
-O
o71
OoO
1115
6 Oo
O -2
a08
-1
o3
3
-2o
43
-4
o0
6
OoO
1215
6 Oo
O -3
o2
5
-4o
95
-1
o2U
Da
D 11
093
OoO
-lo
24
-2
o1
3
-3o
26
-2
o1
3
DoD
12D
93
OoD
-1o
89
-3
o8
1
-2o
33
Do
O 11
123
OaO
-2
ol3
-4
a44
-l
o 7
7 Oo
O Oo
D 12
123
DoD
-2o
08
-2
o1
7
-1o
09
Do
D 11
153
OaO
-1
o0
3
-laO
S
-3o
33
Oo
O 12
153
DoD
-2o
19
-4
o2
3
-4o
74
-1
a51
Oo
D 11
090
OoD
-lo
89
-4
o0
8
-4a3
6
OoD
12D
90
OoO
-Do4
3 -3
o8
3
OaD
DoD
1112
0 Do
O -3
o0
3
-3o
ll
-2o
13
Do
D 12
12D
Do
O -3
o0
3
-3o
11
-2
o1
3
DoO
1115
D
DoD
-2o8
D
-=6a
29
-3o8
D
-1o
63
Do
D 12
150
OoO
-2o8
D
-6o
29
-3
a8D
-l
o6
3
DaD
2109
6 Do
D -D
o63
-Do2
2 Do
D 22
D96
Oo
O -O
a79
-1o
25
-2
o1
2
DoD
2112
6 Oo
O Oo
O -J
.o53
Oo
O 22
126
OoO
-3o
69
-4
o3
1
OoO
2115
6 Oo
O -4
o5
5
..,Q
o90
OoO
2215
6 Oo
O -2
o1
3
-2o
22
Oo
O I-
' w
J:=
"
TABL
E 5
--C
on
tin
ued
Ela
pse
d
Tim
e (s
eco
nd
s)
Mo
tio
n S
equ
ence
0
ol
0 2
0 3
0 4
o5
o6
o7
2109
3 O
aO
-2o
02
-l
o9
5
OoO
2209
3 Oo
O Oo
O -2
o2
2
OoO
2112
3 Oo
O -O
o53
-Oo
ll
OoO
2212
3 Oo
O Oo
O Oo
O Oo
O 21
153
OoO
-2o
91
-7
o4
3
-2o
14
Oo
O 2215~
OoO
-1o
03
-1
o1
0
-1o
10
Oo
O 21
090
0 o-0
-3
o1
2
-2o
41
-2
o0
5
OoO
2209
0 Oo
O -4
o0
8
-2o
11
-O
o04
OoO
2112
0 Oo
O -3
o7
5
-1o
69
-D
o89
OoO
2212
0 Oo
O -2
o3
5
-1o
90
-3
o5
2
OoO
2115
0 O
aO
-3o
32
-1
o2
5
-3o
11
Do
O
2215
0 O
oO
-1o
72
-3
o3
8
-2o
l2
OoO
3109
6 Oo
O -4
o1
4
-2o
25
-1
o8
2
DoD
3209
6 Oo
O -O
o80
-Do
l3
DoD
3112
6 Oo
O -4
o4
1
-1o
01
-O
o55
-Oo3
2 Oo
D
3212
6 Oo
O -l
o3
4
-1o
97
-D
o32
DoO
3115
6 Oo
O -2
o7
9
-Oo2
0 -2
o9
4
-3o
66
Do
D
3215
6 Oo
O -2
o0
6
-2o
95
-D
o40
DoO
3109
3 Oo
D -l
o8
1
-2o
74
-4
o2
6
DoO
3209
3 Oa
O -l
o9
3
-Oo2
1 -2
o1
8
-3o
77
Oo
O
3112
3 Oo
O -2
o3
5
-lo
36
-l
o9
3
-lo
34
Oo
O
3212
3 O
aO
-Oo9
7 -O
o74
-lo
55
Oo
O
3115
3 Oo
O -3
o5
9
-1o
24
-3
o4
6
-3o
34
Oa
O
3215
3 Oa
O -4
o3
8
-1o
38
-2
o4
4
-2o
75
Oa
O
3109
0 Oo
O -4
o1
2
-2o
41
-2
o0
5
OoO
3209
0 O
aO
-5o
08
-2
o1
1
-Oo0
4 Oo
O .....
.
3112
0 O
aO
-3o
59
-2
o0
3
-Oo8
9 -2
o6
3
OoO
w
\}'1
TABL
E 5
--C
ort
tirt
Ued
Ela
pse
d T
ime
(sec
on
ds)
Mo
tio
n S
equ
ence
0
ol
o2
0 3
0 4
o5
0 6
0 7
3212
0 O
aO
-3o
74
-l
o6
9
-Oo5
6 Oo
O 31
150
OoO
-2o
59
-6
o7
6
-6o
65
-2
o0
9
OoO
3215
0 Oo
O -3
o2
6
-2o
61
-O
o02
-lo
33
-3
o6
3
OoO
4109
6 0"
"0
-2o
88
-O
o05
OoO
4209
6 Oo
O -3
o8
1
-Oo4
3 Oo
O 41
126
OaO
-2
o5
8
-Oo5
9 -l
o2
7
OoO
4212
6 O
aO
-3o
43
-l
o2
6
-lo
88
Oo
O 41
156
OoO
-3o
43
-3
o5
1
-Oo4
9 Oo
O 42
156
OoO
-lo
28
-3
o2
1
-2o
74
Oo
O 41
093
OaO
-2
ol0
-l
o3
3
OoO
4209
3 O
aO
-3o
08
-4
o5
4
-2o
56
Oo
O 41
123
OoO
-Oo9
6 -l
o2
2
-Oo8
8 -2
o2
6
OoO
4212
3 Oo
O -2
o3
3
-8o
47
-1
0o
38
-4
o9
1
OoO
4115
3 O
aO
-lo
ll
-lo
28
-2
o1
2
-Oo8
4 -5
o7
8
OoO
4215
3 Oo
O -3
o5
1
-2o
91
-4
o6
1
-4o
41
Oo
O 41
090
OoO
-1o
72
-3
o3
8
-6o
13
Oo
O 42
090
OoO
-1o
72
-3
o3
8
-6o
13
Oo
O 41
120
OoO
-3o
52
-1
o9
0
-2o
35
Oo
O 42
120
OoO
-2o
35
-l
o9
0
-3o
52
Oo
O 41
150
OoO
-5o
49
-l
o8
5
-2o
48
Oo
O 42
150
OoO
-3o
32
-1
o2
5
-3o
11
-2
o9
6
OoO
5109
6 Oo
O -2
o9
1
-Oo
l7
-lo
94
Oo
O 52
096
OoO
-4o
68
-4
o2
9
-lo
60
Oo
O 51
126
OoO
-3o
78
-3
o5
1
-lo
90
-O
o66
OoO
5212
6 Oo
O -3
o7
8
-1 ..
61
-Oo2
6 -3
ol2
Oo
O 51
156
OoO
-3o
23
-l
o4
8
-2o
38
-O
o35
-3o
66
Oo
O 52
156
OoO
-lo
l2
-3v
04
-3
o4
6
-lo
09
-l
o0
2
OoO
1--'
w
0\
TABL
E 5
--C
on
tin
ued
Ela
pse
d T
ime
(sec
on
ds)
Mot
ion
Seq
uen
ce
0 o
l o2
0
3 o4
o5
o6
o7
--
-
5109
3 0
0 -l
a9
2
-lo
72
-O
o07
-lo
61
Oo
O 52
093
0 0
-2o
09
-4
o8
5
-4o
67
Oo
O 51
123
0 0
-3a
26
-2
o6
1
-Oo0
2 -l
o3
3
-3o
63
Oo
O 52
123
0,0
-1
-o 3
4 -l
o9
7
-Oo3
2 Oo
O 51
153
OeO
-4~38
-1o
38
-2
o4
4
-2~75
OaO
52
153
OoO
-2
o0
6
-2o
95
-0
(140
0,
.0
5109
0 0-1
1"0
-5o
08
-2
o1
1
-Oo0
4 O
aO
5209
0 0-c
rO
-4o
12
-2
-o 4
1 -2
o0
5
OaO
51
120
0-11"0
-3
o7
4
-1o
69
-O
o56
OaO
52
120
0,-0
-2
o59
-1o
48
-2
o24
-2a
09
Oo
O 51
150
OoO
-5o
34
-6
o0
1
-3o
88
-1
o3
6
OoO
5215
0 Oo
O -2
o1
3
-2o
01
-3
o0
6
-Oo9
9 O
aO
6109
6 Oo
O -2
o3
2
-1.9
9
-Oo8
0 -O
o09
-1o
32
O
aO
6209
6 Oo
O -2
o1
8
-2o
53
-O
o26
OoO
6112
6 Oo
O -2
o6
0
-Oo8
2 -2
o3
4
-2o
65
O
aO
6212
6 Oe
O -2
o7
8
-3o
05
-1
o3
5
-1o9
2 -3
o1
0
OaO
61
156
OoO
-3o
31
-1
o5
2
-Oo9
2 O
aO
6215
6 O
aO
-2o
51
-O
o51
-Oo7
4 -1
o42
OaO
61
093
OaO
-6
o0
5
-1o
94
-3
.,20
-4
o26
OaO
62
093
OaO
-2
o59
-3.,
45
-1o
l6
-Oo9
7 -4
o7
7
OaO
61
123
OoO
-2o
40
-O
o71
-2o
l5
-2a6
5
OoO
6212
3 O
aO
-2o
51
-O
o51
-2o
06
-2
o7
0
OoO
6115
3 O
aO
-1o
65
-l
o4
9
-Oo5
9 -2
o6
8
-4o
57
O
aO
6215
3 O
aO
-2o
22
-O
o82
-lo
77
-2
o6
5
OoO
6109
0 O
aO
-2o
51
-1
o4
8
-Oa5
5 -1
o04
OaO
62
090
OaO
-2
o9
3
-3o
55
-2
o06
-2o
58
-5
o3
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22
Oo
O 61
120
OoO
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79
-4
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6
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0 -l
o5
3
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66
O
aO
......,
6212
0 Oo
O -3
o4
7
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06
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o36
-1o
04
O
aO
w
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TABL
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--C
on
tin
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Ela
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d T
ime
(sec
on
ds)
M
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Seq
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0 o
l o2
o3
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0 5
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6115
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l7
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76
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6215
0 0~0
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28
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5
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65
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36
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7109
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o88
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l9
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7209
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126
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52
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156
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87
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7109
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58
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153
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84
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Oo
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OoO
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Oo
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120
OoO
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8109
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8115
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87
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093
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70
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(X
)
TABL
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tin
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Ela
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d T
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(sec
on
ds)
Mot
ion
Seq
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0 o
l o2
0 3
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93
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l123
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53
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92
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2153
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33
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1090
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52
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8112
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20
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31
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45
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8215
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ol8
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96
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O 92
096
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126
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156
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~
0
141
Figures 40 through 42 present the correlation coefficients
and measures of difference grouped by motion distances,
elbow height and the over-all average for each subjecto
Based upon the quantitative analyses conducted for
all motions, the following remarks can be made:
lo There is a remarkable similarity between the
experimental (actual) paths and the corresponding ones
obtained by the modele Correlation coefficients as high as
o92 and as low as o78 were obtained for different subjectso
This high correlation implies that the paths of motion pre
dicted by the model have the same trend as the experimental
ones, ioeo 5 decrease or increase in the same fashiono
Furthermore, the closeness of the location of the paths'
peak points along the X-axis does prove that the model
predicts very closely the point at which the human starts
to decelerate in order to terminate his motiono
2o The measures of difference both for the dif
ferent points along the path of motion as well as for the
area under it for all different motions investigated
showed little discrepancy between the experimental and the
predicted pathso In most cases the maximum difference for
the motion points was obtained as 6oO percent, which is
reasonably good and falls within the variational limits of
average individualso Also, a similar statement can be made
with regard to the difference between the areaso
t/)
ro (1)
H ~
s:: (1) (1)
~ +l~ (1)
.OS:: or-i
(1) ()
s:: (1)
H (1)
~ ~ or-i 0
+l s:: (1)
or-i ()
or-i 1+--1 ~ QJ 0 0
s:: 0
or-t +l ro
...... (1)
H H 0
0
2o0
lo
0 5
o9
c 8
o7
g
6 3 0
Elbow Height above Work Surface (inches)
Figo 40o--Effect of work surface height upon accuracy of predictiono
142
143
en ro Cl)
~ ex: s:: Q) Q)
0~ EJ ~ D .. +-> ls!l Q)
.OS:: or-t
Q) ()
s:: Q)
~ 0 Q)
ct-. ct-. .,-t . - _./
Q
+> s:: Q)
..-1 ()
.,-t ct-. ~ Q)
0 (.)
s:: o9 0
or-t 0 0 0 +> ro oB r-i Q)
~ M o7 0 (.)
9 12 15 Distance (inches)
Figo 4lo--Effect of motion distance upon accuracy of predictiono
s:::::(1)~ Q)
~ s::: +>orf Cl>...._ .a Cl> C)
s::: Q)
H Q)
CH CH orf 0
+> s::: CIJ orf C)
orf CH CH Q)
0 0
s::: 0 ~ .-1-)
ro rl Q)
~ ~ 0
0
144
1o0
o9
o8
o7
0 1 2 3 4 5 6 7 8 9 10
(Subject)
Figo 42o--Effect of subject upon accuracy of prediction
145
3o The model's consistency in predictions was not
affected by the variations in the motion configurationso
That is, for all combinations of motion distance, work
surface height, and subject, the paths predicted by the
model described very well the actual motions as obtained
by experimental analysiso
4o As motion time decreases, the model's accuracy
in prediction increaseso That is, the discrepancy between
the experimental path and the corresponding theoretical
one decreases as the motion time decreaseso This can be
attributed to the fact that for short motion time, the
initiation and execution of motion are accomplished with
fewer decisions and feed-back corrections on the subject's
parto On the other hand, for longer motion times, the
subject would have time to evaluate his path and might
introduce some corrections in order to terminate the
motion at the desired point, ioeo, more decision and more
variations along the path of motiono
5o For most motions the model consistently under
estimated the paths of motion followed by the subjectso
That is, the experimental paths of motion occurred higher
than the corresponding predicted oneso The large errors
(as.high as 5%) between the experimental paths and the
predicted ones occurred very close to the initial motion
pointo This may have resulted from the uncertainty of
146
the subjects at the beginning of the motiono In other
wordsi at the beginning of the motion the reaction time
phase and the movement phase overlapped and resulted in
some errorso The effect of the strobe and the lighting
conditions during the course of the experiment might cause
greater change in subject's reaction time than would be
expected under normal lighting conditionso
In generali it should be appreciated that exact
prediction of human motion by the proposed model is diffi
cult to aohieveo The reasons rest with human natureo It
is known that human beings possess some variations in per
forming their daily activitieso Therefore, a deterministic
model such as the proposed one will show some discrepancy
in predicting and quantifying a stochastic process such as
human motiono The degree of discrepancy will depend to a
large extent upon the nature of the motion to be performed
and the familiarity of individuals with ito It seems rea
sonable to expect that the paths of motions would be made
with minimum variations as far as well-learned body motions
such as walking and running are concernedo This is exactly
what happened in this investigationo All subjects showed
little variations in performing their motionso The magni
tude of variations was within reasonable values and could
be attributed to individual variations and did not warrant
any alteration of the model's formulation or assumptiono
147
In short, it can be hypothesized, and the data sub
stantiate, that the proposed biomechanical model does
predict with a high degree of accuracy human motions which
fall under its assumptionso
CHAPTER VI
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
Summary
This investigation was undertaken in an attempt
to develop a biomeohanical model for the upper extremityo
A model for predicting paths of motion and the associated
motion characteristics for the arm articulation joints was
developedo The underlying principle of the model is that
the average individual does follow an optimizing criterion
in performing his taskso The use of the model is restricted
to the analysis of transport movements which are to be per
formed under normal environmental conditions and require
the maximization of performance efficiencyo Anthropometric
characteristics such as links' masses~ dimensions~ moments
of inertia as well as motion distance, angle, and time com
prise the information required as an input to the modelo
Basically, the model utilizes both theoretical
mechanics and an optimization approach for the analysis of
arm motionso A detailed description of the model assump
tions, mechanics, and formulation is presented for three
dimensional motionso The model treats the arm as a system
of two links~ the first is the upper arm and the second is
148
the forearm and hando A physical measure for performance
is selected as the criterion for the modelo Minimizing
this performance criterion under physical and stress con
straints imposed by the nature of the human body as well
149
as under task characteristics is the key idea of the modelo
Three algorithms for the model solution are presentedo
These are linear and geometric programmings, dynamic pro
gramming, and simulationo
Principles of the model and its associated algo
rithms are applied in detail to analyze hypothetical
examples of planar motions for the armo Through the
analysis of the hypothetical motions, the feasibility of
the model algorithms was tested and evaluatedo
The accuracy as well as the adequacy of the model
in predicting human motion was testedo For testing the
model's accuracy, an experiment was conducted to record
the paths of motion for ten subjects under three levels
of work surface height, 6, 3, 0 inches measured from
the elbow, and motion distances, 9, 12, and 15 incheso
Measurements from the photographic records obtained for
different motions were used to generate the path of
motion assumed by each subject for each motion consideredo
For each experimental motion, a corresponding motion
determined by, the model was obtained. A correlation
analysis as well as two measures for difference was
conducted between each two paths of motion, the experi
mental one and the predicted oneo
Conclusions
150
The conclusions which can be drawn from this inves
tigation in regard to the model, its associated algorithms,
and verification experiment will be summarized in the
following points:
lo It is possible to construct a model for accu
rately analyzing and quantifying motions of the upper
extremityo Paths of motion and the associated motion char
acteristics for the arm articulation joints predicted by
the model were very close to those measured experimentally
under the same task characteristicso Correlation coeffi
cients as high as o98 and as low as o78 were obtained for
some motionso The difference in percentages between the
measured path of motion and the predicted one was below
5% and averaged below 2% for all motions investigatedo
Also, the differences between the coordinates of the meas
ured paths of motion and the predicted ones were as high
as 6% and as low as 0% for most motionso
2o As far as describing human effort is con
cerned, total power required for the motion has been found
to be the most suitable mechanical criterion which is sen
sitive to motion characteristicso This may be due to the
151
fact that work, impulses and stress (pages 34 and 35) lack
a velocity term in their expressionso
3c Linear and geometric programmings seem to be
insufficient for providing a solution for the modele
Linearization of the model by the small angles assumption
resulted in an infeasible solution for the linear pro
gramming approacha On the other hand, in spite of large
degrees of difficulty associated with the geometric pro
gramming formulation, it seems that it is a valid approach
at least from the theoretical point of viewo However, the
geometric programming algorithm which was used failed to
provide a solution in most oaseso Thereforei the use of
geometric programming with the model is still question
ableo
Generally$ it seems very appropriate to believe
that the validity and the use of suboptimization approaches
for model building in connection with human motion are not
encouraging, if not infeasibleo This is especially true
when the dynamic forces generated during the motion are
considerably larger than static foroeso
4o Dynamic programming algorithms have yielded
paths of motion very close to those measured experiment
ally for human subjects~ Based upon the analysis, page
96, Chapter IV, it is evident that dynamic programming is
the most favorable algorithm to be used in connection with
152
the modelo This is simply true because it is an optimiza
tion approach and it is assumed that the human body opti
mizes its performanceo
5o The simulation approach by means of enumeration
has provided feasible solutions for the motions investi
gated for the hypothetical exampleso In most oases,
enumeration has yielded curves very close to those of
dynamic programmingo The assumption that the hand path
of motion can be approximated by certain geometric shapes,
namely sine, ellipse, and parabola, seems to be inaccu
rateo The sine, ellipse, and parabola functions yielded
low paths of large discrepancies in comparison to those
of dynamic programming in most of the motions investigatedo
6o Although the model formulation and algorithms
were tested only for two-dimensional motions, it is
believed that the above remarks can be extended to apply
to the three-dimensional motionso
In short, it can be concluded that the model
developed in this study might be a useful tool for the
ultimate objective of developing a generalized biomechan
ical model which can be used to simulate all possible
classes of human motion theoreticallyo Indeed, the model
developed in this investigation has clarified some aspects
of model building in connection with the human bodyo For
example, answers to the questions concerning the
153
optimization criteria, assumptions, mechanics, and suitable
algorithms have been obtainedo
Unquestionably, the proposed model in this investi
gation might open a new era in the theoretical analysis of
human motiono
Recommendations for Further Research
Attempts at modeling the human body are worthy of
careful consideration from researchers in the field of
biomechanics and related areaso The following research
pertaining to modifying and refining the proposed
model in this study merit further investigationo The rec
ommendations are presented chronologically according to
their degrees of difficulty~
lo Modify the existing computer algorithms to
handle three-dimensional motions in accordance with the
model principles and assumptionso
2o Eliminate the restrictions which prevent the
study of tasks with rotational motions about the longi
tudinal axes of body linkso This could be accomplished by
modifying the model dynamics in order to compensate for
rotational motionso
3o Refine the model dynamics by using the actual
forces developed by the muscles at the different articula
tion joints instead of grouping their actions under a
single moment as presented on page 18o This could be
154
accomplished by treating the body as a statically indeter
minate space structure which consists of several links and
tie rodso Determining the forces and moments for this space
structure might be possible through the use of advanced
techniques such as structural analysis and optimization in
conjunction with high-speed computerso
4a Examine the model performance criterion for
possible usages of physiological indices with or without
the addition of mechanical criteria to express human
effort over an extended period of task durationa Thee~
retical expressions for physiological indices such as
oxygen consumption and heart rate could be written in
terms of motion characteristics such as displacement,
velocity, and accelerations as well as muscle tensionso
If developing such theoretical expressions would be diffi
cult, and it seems soj empirical expressions might be used
instead of the theoretical oneso These empirical expres
sions might be obtained through experimental analysiso
5o So far, the model developed in this study as
well as the above suggested modifications is concerned
only with the armo However, once a somewhat general model
for the arm is developed, extending it to cover the entire
body would not be a difficult tasko In modeling the
entire body, careful consideration should be given to
modeling the spinal columna With regard to the spine, the
following questions might be raised: Could the spine be
considered as one link or two curved links? Would elas
ticity of the spine be considered, or would it be treated
as a solid link?
155
6o Stochastic approaches to model building in
connection with human motion are worthy of investigationo
Probabilistic analysis of human motions seems to be very
relevant, since man can be viewed as a machine, so to speak,
which operates to some extent in accordance with the law of
chanceo That is, variations in human performance on a daily
basis and even instantaneously are known to occuro There
fore, by using stochastic approaches in developing the
models for human motion, variations for individuals as well
as between individuals would be considered in the modelo
It is hoped that modeling of human motion will
attract wide interest among the specialists in biomechanics
and that the future will produce more attempts at modeling
the human body in general under a variety of tasks and
environmental conditionso
LIST OF REFERENCES
Abbott, Bo Co 1 and Bigland, Bo The effects of force and speed changes on the rate of oxygen consumption during negative worko Jo Physiolo 1 1953, 120, 319-325o
Abbott, Bo Co 1 Bigland, Bo Ao, and Ritchie, Jo Mo The physiological cost of negative worko Jo Physiolo
1 1952, 117, 380-390o
Allbrook, Do Movements of the lumbar spinal columno Jo Bone & Joint Surgery, 1957, 39o
Amar, Jo The human motoro New York: Eo Po Dutton, 1920o
Anderson, To Mo Human kinetics in schools, hospitals, and industryo British Jo Physo Medo 1 1955 1 18o
y/Ayoub, Mo Ao Quantification of human motion by stereophotogrammetric techniqueo Unpublished master's thesis, Texas Technological College, May 1969o
Ayoub, Mo Mo Effect of weight and distance traveled on v body member acceleration and velocity for three
dimensional moveso The International Jo of Production Research, 1966, 5, 3-2io
Barnes, Ro Mo Motion and time studyo New York: John Wiley & Sons, Inco 1 1968o
Basmajian. Jo Vo Muscles aliveo Baltimore: Williams & Wilkins Coo 1 1962o
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156
Bell, Go Application of engineering techniques to the physiology of bones, Proceedinfsa Symlosium on Biomechanics, 15-18 September, 1~6 1 Roya College of Science and Technology, Glasgow, Scotlando
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158
Biomechanics in orthopaedic surgeryo Pro-S m osium on Biomechanics, 15-18 September,
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APPENDIX
COMPUTER PROGRAM DOCUMENTATION .·
169
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BY
M.A. AYOUB
TEXAS TECH UNIVERSITY
DECEMBER, 1970
I'IAPOOO~O
MAPOlJOt>O MAP00010 1'4AP00080 1'4AP00090 MAPCOlOO MAPOOllO MA.-00120 MAPOOllO MAPOOl40 MAPOOlSO
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l. PRO&RA~MER--M.A. AYOUB l. ADVISOR-- OR. M.M. AYOUB J. ~AC~INE-- IBM 360 MODEl 50 ~- LANGUAGf -- FORTRAN IV 5. COMPILER -- FORTRAN G COMPIL~R,O/S OPERATING SYSTEM 6. CATE COMPLETED -- DECEMBER 1970 J. APPROXII'IATE COMPILE TIME-- 2.0 MINUTES 8. C~PUTATION TIME-- VARIABLE,OEPENDS UPON THE ANALYSIS
CONDUCTED. IN MOST CASES, IT IS 5 MIN. 9.
lG. LI~S OF OUTPUT -- APPROXIMATELY= 1000 LINES/ MOTION APPROXIMAfE CORE SPACE REQUIRED-- 150,000 BYTES
I~E FOLLOWING CHANGES ARE SUGGESTED TO REDUCE CORE SPACE - FOR SIMULATION ANALYSIS,REPLACE SUBROUTINE OYNAMC BY STATE~ENTS 15470,17150,17160 AND DELETE
SUBROUTINES NEWTON AND MINIM - FOR DYNAMIC PROGRAMMING,REPLACE STATEMENTS03610-036BOBY NEW STATEMENTS WHICH USE INTEGER 2 INSTEAD
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THE PROGRAM IS WRITTEN TO ANALYZE PLANAR MOTIONS Of THE UPPER EXIER~ITY BY USING BOTH THEORETICAL AS WEll AS EXPERIMENTAL APPROAC~ES. THE PROGRAM USES THE FOLLOWI~G FEATURES TO DETER~INE THE PATH Of MOTION AND THE ASSOCIAfED CHARACTERISTICS FOR A GIVEN TASK.
l. SIMULATION ANALYSIS 2. DYNAMIC PROGRAMMING 3. EXPERIMENTAL DATA
••••••••••••••••••••••~••••••••••••••••••••••••••••••••••••••••••••••~AP00470 MAPOOit80
HISTORY AND BACKGROUND MAP00490 DISCUSSIONS CONCERNING THE USE OF SIMULATION AND DYNAMIC MAP00500 PROGRA~MING ARE PRESENfED IN M.A.AVOUB,PH.O. DISSERfAfiON MAP00510 TEXAS rECH 1 1971. MAP00520
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP00530 RESTRICTia-S MAPQO~O
IF OIRENSIONS LARGER THAN THOSE APPEARING IN THE PROGRAM LISTING AAE REQIRED,THE DIMENSION STATEMENTS MUST BE MODIFIED.
MAP00550 MAP00560 MAP00570
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INPUT SEE PROGRAM LISTING
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PROGRAM fLO~ CHARTS MAP00680 PAGE MAPOObqO
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NUMBER--SUBJECT'S COOE NUMBER ll2--LENGTH OF UPPERARM L21--LENGTH OF FOREARM Rl2--C.G. DISTANCE OF UPPERARM R21--C.G. DISTANCE OF FOREARM Ml2--MA SS OF UPPERARM M2l--MASS OF FOREARM Kl2--RACIUS OF GYRATION :UPPER ARM K23--RADIUS OF GYRATIO~: FOREARM 112,123--MOMENTS OF INERTIA OF THE TWO ARM SEGMENTS Al2--CROSS SECTION AREA AT THE SHOULDER Rll2--RADIUS OF THE CROSS SECTION AT THE SHOULDER ICODE--MOTIO~ COOE NUMBER ITIME--MOTION TIME IN SECONDS MULTIPLIED RY 100 TI~E--TIME INCREMENT WEIGHT--EXTERNAL LOAD TO BE CARRIED BY HAND MOTDIS--MOTION DISTANCE MOTANG--MOTION ANGLE XIT, YINT--INITIAL ANGLES·OF THE ARM SEGMENTS AT TIME 0 N--NUM8ER OF POINTS TO BE GENERATED FOR THE PATH OF ~OTION
USING TIME INEREMENT=.05 MASTER--CONTROL VARIABLE TO iNDICATE THE TYPE OF ANALYSIS REQUIRED
MASTERaO EXPERIMENTAL DATA FOR THE ARM EULER ANGLES IS EXPECTEn.
MASTER>O MOTION DATA HAS TO BE GENERATED THEORETICALL1. NOWl--CONTROl VARIABLE FOR PRINTING THE ANALYSIS FOR
EACH ITERATION NOWl=l ANALYSIS OF EACH ITERATION WILL BE PRINTED NOWl=O NO ANALYSIS Will BE PRINTED
NOW--CONTROL VARIABLE FOR PLOTTING UATA FOR EACH ITERATICN NOW=O RESULTS WILL 8E PLOTTED FOR EACH ITERATION NOWs 1 NO GRAPHS.
ITYPE--FUNCTION TYPE ASSUMED FOR THE HAND PATH Of MOliON lTYPE=O--NUMERICAl DATA FOR HAND PATH OF MOTION IS GIVE~ ITYPE•l--SINE CURVE FUNCTION IS TO DESCRIBE HAND
PATH OF MOT IONS ITYPE=2--ELLIPSE FUNCTION IS USED TO DESCRI~E HAND PATH
OF MOTION ITYPE•l--PARABOLA FUNCTION IS USED TO DESCRIBE
~AND PATH Of MOTION ITYPE=~--PATH OF ~OliON IS GENERATED
BY DIRECT ENUMERATIONS PERlNT, PER~AX--INITIAl AND MAXIMUM VALUES FOR VERTICAL
STEP SIZE--TO BE USED WITH SINE, ELLIPSE, PARABOLA FUNCTIONS
MAP007~0
MAP00760 MAP00710 MAP00780 MAP00790 MAP00800 HAP008l0 ,.AP00820 ~APOOS"\0
MAP00840 ~AP008SO
MAP008b0 MAP00870 MAP00880 MAPOOtJqQ MAP00900 MAP00910 MAP00920 MAP00930 MAP00940 MAP009SO MAP009b0 MAP0Qq70 MAP00980 MAP00990 MAPO 1000 MAPO 1010 ~AP01020
MAP 01030 MAP01040 MAP010SO MAP01060 ~AP01070
MAP01080 MAP01090 MAPO 1100 MAPOlllO MAP01l20 MAPOllJO ~APO 1140 MAPO 11 SO MAP01160 MAPO 1170 MAPOlltiO MAP01190 MAP01200
171
C 'fJNt--INITIAL POSITION FOR HI'"EST POINT OF THE PARABOlA MAP012l0 t AlONG THE X-AXIS MAP OUZO ( OfltA-- INCREMTAL VALUE FOR PARABOLA'S HIGHEST POINT POSITION MAPOlllO C POINT--POSITION OF THE HIGHEST POINT OF THE PARABOLA MAPOlZ~O C PlN,P2N,PlN,P4N,PSN,PlM,P2M,PlM 1 P~M,P5M--INITIAl AND MAP01250 C MAXIMUM VALUES FO~ DEFINING GRID TO BE USED WITH THE UNUMRATIDN MAP0l260 C POINTS MAP01270 C .Pt,P2,Pl,P4,P5--GRID POINTS FOR ENUMERATION MAP01280 C KOUNT-- ITERATION NUMBER MAP01290 C PERCNT--MAX HANO l-COORD. IN PERCENT OF THE MOTION DISTANCE FOR AN MAPOllOO C ITERATION MAPOlllO C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP0ll20 C VARIABLES TO BE USED WITH DYNAMIC ANALYSIS MAPOlllO C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP0l340 C ANGl--UPPER ARM ANGLE MAP0l150 C ANGZ--FOREARM AN.GLE MAP0ll60 C ANGVl--ANGULAA VELOCITY OF UPPER ARM MAP01370 C ANGVZ--ANGULAR VELOCITY Of FOREARM MAP01180 C ANGU--ANGULAR ACCELERATION OF UPPER ARM MAP01390 C ANGAl--ANGUlAR ACCELERATION Of FOREARM MAP01400 C AV2MlX--MAXIMUM ANGULAR VELOCITY OF FOREARM MAP01410 C AVlMAX--MAXIMUM ANGULAR VELOCITY OF UPPERARM MAP01420 C AAlMAl--MAXUMU" ANGULAR ACCELERATION Of UPPERARM MAP01430 C AAZMAX--MAXIMUM ANGULAR ACCELERATION OF FOREARM MAP01~40 C XA2--El80W LINEAR ACCELERATION IN THE X DIRECTION MAP01450 C ZA2--ELBOW liNEAR ACCELERATION IN THE l DIRECTION MAP01460 C XAl--HAND LINEAR ACCELERATION IN THE X DIRECTION MAP01470 C ZAl--HAND liNEAR ACCELERATION IN THE l DIRECTION MAP01480 C XGAZ--X LINEAR ACCELERATION OF THE UPPER ARM C.G. MAP01490 C ZGAZ--l LINEAR ACCELERATION OF THE UPPER ARM C.G. MAP01500 C XGAJ--l liNEAR ACCELERA liON Of THE FOREARM C. G. MAPO 1510 C ZGAl--Z liNEAR ACCELERATION Of THE FOREARM C.G. MAPOl~ZO C YMOM2--ELBOW MOMENT · MAP01530 C XF2--X-REACTIVE FORCE AT THE ElBOW MAP01540 C ZG2--l-REACTIVE FORCE AT THE ElBOW MAP01550 C F2--RESULTANT REACTIVE FORCE AT THE ELBOW MAP015&0 C YMOMl-~SHOUlDER MOMENT MAP01570 C Xfl--X-REACTIVE FORCE AT THE SHOULDER MAPOl580 C Zfl--l-REACTIVE FORCE AT THE SHOULDER MAP01590 C Fl--RESULTAHT REACTIVE FORCE AT THE SHOULDER MAP01600 C TOR2--ElBOW MOMENT DUE TO STATIC FORCES MAPOl&lO C SXFl--X-COMPONENT OF STATIC REACTIVE FORCE AT ELBOW MAP01620 C SZFZ--l-COMPONENT Of STATIC REACTIVE FORCE AT ELBOW MAP01610 C SF2--RESULTANT STATIC REACTIVE FORCE AT ELBO~ MAP01640 t TORI--SHOULDER MOMENT DUE TO STATIC FORCES MAP01650 C SXFl--X COMPONENT OF STATIC REACfiVE FORCE AT SHOUlDER MAP0l&60 C SZFl--l COMPONENT OF STATIC REACTtVE FORCE AT SHOULDER MAP01670 t SFl--RESULTANT STATIC REACTIVE FORCE AT ELBOW MAP01680 C FZMAI--MAXIMUM RESULTANT FORCE AT THE ElBOW MAP01&90 C FlMAX--MAXIMUM RESULTANT FORCE AT THE SHOULDER MAP01700 C YMOMMZ--MAXIMUM MOMENT AT THE ELBOW MAP01710 C Y"O~"l--MAXIMU~ MOMENT AT THE SHOULDER MAP017ZO C TOR2M--MAXIMU" STATIC MOMENT AT THE ELBOW MAP01710 C TORlM--MAXIMUM STATIC MOMENT AT THE SHOULDER MAP017~0 C SFZMAX--MAXIMUM STATIC REACTIVE FORCE AT ELBOW MAP017~0 C Sfl~X--MAXIMUM STATICREACTIVE FORCE AT SHOULDER MAP017&0 C RATIGl--RATIO OF MAXIMUM DYNAMIC MOMENT TO MAXIMUM STATIC MAP01770 C ~OMENT AT ELBOW . MAP01780 C RATIOZ--RATIO OF MAXIMUM DYNAMIC MOMENT TO MAXIMUM MAP01790 C SJATIC MOMENT AT SHOULDER MAP01800
172
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 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
AATI01--RATIO OF MAXIMUM DYNAMIC FORCE TO MAXIMUM STATIC FORCE Af ELION
AATI04--AATIO OF MAXIMUM DYNAMIC FORCE TO MAXIMUM STATIC FORCE AT SHOULDER
STRESS--NOI~Al STRESS AT SHOULDER TOTSTR--TOTAL STRESS NORK--TOTAL NORK COMPUTED AT SHOULDER POWER--TOTAL PONER COMPUTED AT SHOULDER SSRATE--STRESS RATE TANCIM--TOTAl ANGULAR IMPULSE TFJMP--TOTAL LINEAR IMPULSE FCTl,FCT2,FCTl,FCT~tfCT~,FCT6,--CRITERION FUNCTIONS REMARKS
MAP01810 IUP 01820 MAP018l0 MAP018~0
MAP 018tSO MAP01860 MAP01870 MAP01880 MAP01890 MAPO 1900 MAP01910 MAPO 1920 MAP01930
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••~APOlQ40 THE FCLLOWING SUBPROGRAMS ARE REQUIRED
---SOLVER --PARAB ---SI .. UL ---RGRSS ---MAJEQS ---PLOT ---SfiCK ---M INTIR ---DYNAMC ---NEWTON ---fiiNIM --BIG
MAP019~0
MAP01960 MAPOl<HQ MAP01980 MAP01990 MAPOZOOO MAP02010 MAP02020 MAP020l0 MAP02040 MAP020'i0 MAP02060 MAP02070
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02080 INPUT DATA MAP02090
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02100 FIRST CARD --PROBLEM TITLE (ALPHANUMERIC CHARACTERS CULUM~l-80)MAP02110
MAP02120 SECOND CARD--ANTHROPOMETRIC DATA FOR THE UPPER ARM;Ll2,Rl2,~ll MAP02ll0
,Kl2 1 Al2,Ril2 C FORMAT 6Fl0.5) MAP021~0 MAP02150
THIRD CARD--ANTHROPOMETRIC DATA FOR THE FOREARM;L23,R23,1423, K23 IFORMAT 5Fl0.5)
MAP02160 MAP02170 MAP02180 MAP02190 MAP02200 MAP02210
FOURTH CARD--MOTION CHARACTERISTICS;MOTION NUMBER,MOTION TIME, TIME INCRMENT,WEIGHT,MOTION DISTANCE,MOTION ANGLE, INITIAL ARM ANGLES CFORMAT 21b,6Fl0.5)
FIFTH
SIXTH
CARD--CONTROL CARO;MASTER,NOWl,NOW,ITYPE,PERINT,PERMAX (fORMAT ~12,2X,2F5.2)
MASTER=O EXPERMINTAl DATA
MAP02220 MAP 02230 MAP02240 MAP02250 MAP02260 MASTER=2 THEORETICAL ANALYSIS
NOWl=l NO INTERMIDIATE RESULTS WilL NOWl=O All RESULTS WILL BE PRINTEO NOW=O NO GRAPHS WILL BE PRINTED
BE PRINTED MAP02270 MAP02280 MAP02290 MAP02300 NOW=l GRAPHS WILL BE PRINTED
ITYPE=O,l 1 2 1 3,4 FUNCTION TYPE FOR THE OF MOT ION
HAND PATHMAP02310 MAP02320
CARD-- .THIS CARD WILL BE VARIED IN ACCORDANCE WITH THE ANALYSIS REQUIRED
FOR SINE AND ELLIPSE IT IS NOT REQIRED FOR PARABOLA;PTINT,DELTA (fORMAT 2F5.2) FOR ENUMERATION;PlN, •••• P5M CFORMAT lOF5.2) FOR GIVEN PATH OF MOTION;KONT=O IFORMAT 16 ) FOR DYNAMIC PROGRA~MING;KONT=l IFORMAT 16)
MAP02HO MAP02340 MAP02350 MAPC2l60 MAP02110 IUPOl380 14AP02390 MAP02400
173
C MAP02410
C SEVENTH CA.D--DELTA,Y211NTtY2liNC1 Y1liNT 1 YlliNC;STAGE MAP02420
C PA.AMETERS FO• DYNAMIC PROGRAMMING IFORMAT 5FS.2aMAP02430
C THIS CARD IS REQUIRED ONLY FOR DYNAMIC PROG. MAP02440
C MAP02450
C EIGHTTH CA.D--ANGULA• DISPLACE~ENT IN DEG.FOR THE ARMIANG1,A~G2MAP02460
C CFORMAT 2F1D.5a MAP02470
C OR MAP02480
C X AND l COORDINATES Of THE HAND IFORMAT 2F10.5a MAP02490
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02500 C REPEAT CARD 4 THROUGH 1 AS MANY TIMES AS REQUIRED FOR THE SA~E ~AP02510
C SUBJECT--- AT THE END USE BLANK CARD ~AP02520
C IF '"E~E IS ANOTHER SUBJECT TO BE ANALYlED REPEAT CARO 2THROUGH 1 MAP02510
C OTHE.WISE USE A BLANK CARD MAP02540
c c c c c
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02550 INPUT DATA WILL BE GIVEN IN ACCORDANCE WITH MAP02560 ALL POSSIBLE USAGES OF THE PROGRAM. MAP02570
•••••••••••••••••••••••~•••••••••••••••••••••••••••••••••••••••••••••MAP02580
c••••••••• c c c c c c c
1
.255
.1604
BIO~ECHANICAL ANALYSIS OF THE AR" ••••••••••••••
.1117 1. 241 .0195 .009 .06
.1599" 1.117 .• 1041
C EXAMPLE1.--EXPERIMENTAL.DATA
MAP02590 M.A.AYOUB••MAP02600
MAP02610 MAP02620 MAP02610 MAP02640 MAP02650 MAP02660 MAP02670 MAP02680
c •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02690
c Cl5600 c
40 0.06
co 0 1 2 o.o 40.0 c 9 c c c c c c c c c c c
242.11 241. l7 263.90 294.0 l 325.11
. 346.1J' 360.76 316.12 H9~~7
257 .oo 274.16 31!.96 16!5.41 407.57 448.31 482.34 504.06 1)11.16
0.1811 242.1 257.0 MAP02700 MAP02710 111AP02720 MAP02110 MAP02740 MAP02750 MAP02760 MAP02170 MAP02180 MAP02790 MAP02800 MAP028l0 MlP02820 MAP02810 MAP02840 MAP02850
c c
111AP02860
EXA,PLE2.--~AND PATH OF MOTION IS GIVEN 111AP02870
c •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP02880
c Cl34U 0.05 C2 0 1 0 o.o 40.0 c 0 C0.2059147 C0.2097Z89 C0.2115996 C0.2858142 C0.3582572 CO.lf306856 co. 4810694 C0.5070494
0.1124793 0.1226171 0.1381476 0.1521008 0.1581993 0.1521185 0.1387471 0.1224728
0.3048 73.5 MAP02890 MAP02900 MAP02910 MAP02920 M.AP02930 MAP02940 MAP02950 MAP029b0 MAP02970 MAP02980 MAP02990 MAP03000
17~
C0.5l07147 0.1124793 MAP03010 CC MAP01020 EXAMPlE 1.--SIMULATION--SINE CURVE MAPOl030 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP03040 C MAP01050 C1141, 40 0.0~ 0.3048 294.0 73.5 MAP03060 C MAP03D70 C2 1 0 l 0.0 40.0 MAP03080 C MAP03090 C EXAMPLE5.--SIMULATION--ELLIPSE CURVE MAP03100 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP01110 C11415 40 0.05 0.3048 2~4.0 73., MAP03120 C MAP03110 C2 l 0 2 0.0 40.0 MAP01140 C MAP03150 C EXAMPLE6.--SIMULATION--PARABOLA CURVE MAP03160 c •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP03170 c
Cl3415 c
40 0.05 0.1048 294.0 73.5 MAP03180 MAP03190 . MAP01200 MAP01210 MAP03220 MAP03230 MAP01240 "AP03250
C2 1 0 1 0.0 teo.o c c o. 0 10.0 c c EXA,.PLE7.--SIMULATION--ENUMERATION APPROACHE c ••••••••••••••••••••••••••••••••••••••••••••••••~••••••••••••••••••••MAP03260 c Cl3415 c
40 o. 304 8 294.0 13.5 MAP03270 MAP03280 MAP03290 MAPOHOO MAP033l0 MAP03320 MAPOH30 MAPOH40
C2 1 0 4 o. 0 40.0 c c o.o 10.0 o.o 15.0 o.o 20.0 o.o 15.0 o.o 10.0 c C EXAMPLE 8.-- DYNAMIC PROGRA~MING c c
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP01350 c 1341, c
ItO 0.05 0.3048 294.0 73.5 MAPOH60 MAP03370 MAP03180 MAP03390 MAPOHOO MAP03410 MAP03420 MAP01430 MAP034lt0
C2 0 1 0 0.0 40.0 c c c
1
c 5.0 -1.0 1.0 -1.0 1.0 c c c c
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP03450 PROGRAM LISTINC MAP03460 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP03470 REAL l12tL23,M12,M23,112tl23tK12,K23 MAP03480 REAL MUTCIS,MOTANG,MOTIME MAP0l490 OAJA ISTARilH•I MAP03500 CIMENSION TITLEI80) 1 LINEf80J MAP03510 DIMENSION ANGlf201, ANG2f201t ANGV11201t ANGV2C2C), ANGA11201, MAP03520 1ANGA2(20) MAP03530 DIMENSION XA2120J, ZA2C201, XA3120J, ZA3C20), XGA2C20J,ZGA2C20J, MAP015lt0 1XGA3C201, ZGA3120) MAP03550 DIMENSION Yfi0MlC20J, Yfi0M2f20J, XF2'C20J, ZF2120it XflC20), MAP03S60 1Zf1ClOI,FlC201,F21201 MAP03570 DIMENSION STRESSC201, WORKC20t, POWERI20) MAP03580 ~IMENSION SXF2120I,SZF2C20I,SF2C20),TOR2C20J MAP03590 DIMENSION SXF1120t,SZF1f20J,SF1120t,TOR1120J MAP03600
175
DIMENSION AV2MAXI400t,AV1MAXI~00t 1 AA2MAX(400t 1 AA1MAXC4COI MAP01610 DI~ENSION F2MAX(400I,F1MAXI~OOI 1 YMOMM2(4001 1 YMOMM1(400) MAP01620 DIMENSION TOR2MI4001,TOR1MI4001 1 SF2MAXI4001 1 SF1MAXI400t MAP01610 DIMENSION RAT1011400I,RATI02(400) 1 RATI0114001 1 RATI041400l MAP03640 DIMENSICN FCT11400teFCT21400) 1 FCT314001 1 FCT41400) 1 FCT51400) MAP036~0 DIMENSION SSRATEI20),FCT61400J MAP01660 CIMENSICN POINT11400I,P111400) 1 P211400t,P3114001 1 P411400t,P5114001MAP03670 DI~ENSICN ~~HI400t MAP03680
C REAO PROBLE~ TITLE---80 COLUMNS' MAP03690 REACC5,4t ITITLEIIltl•1,801 MAPOHOO
4 FORMATC80A11 MAP03710 NUM•O MAP03720
C REAC SUBJECT'S INDENTIFICATION NUMBER MAP03710 1000 REAC 15,11 NUMBER MAP03740
1 FOR~AT 1161 MAP03750 IF INUMBER.EQ. 01 GO TO 1002 MAP03760
C REAC SUBJECT'S ANTHROPOMETRIC DATA FOR UPPER ARM AND THEN FOREARM MAP01770 READI5,21 Ll2,Rl2tM12,Kl2,A12,RI12 MAP01780 READ 15,21 L21, R23 1 M23, K23 MAP01790
2 FORMATI6Fl0.51 MAP01800 C CCMPUTE MOMENTS OF INERTIA FOR BOTH UPPER ARM AND FOREARM--METHOD OF MAP03810 C COEFFICIENTS IS USED MAP01820
112 • Ml2 • CR12 •• 2 + K12 •• 21 • 9.80 MAP03830 121 z M23 • IR23 •• 2 + K23 •• 2l • 9.80 MAP03840
C PMINT PROBLEM TITLE ANO SUBJECT'S IDENTIFICATION MAP03850 WRITEC6,61J ITITLECIItl"'lt80J MAP01860
61 FORMAT I 1~ 1 1 BOA 1 I MAP03870 WRITE (6 ,60 t NUMBER MAP01880
60 FORMATC1H0,10X 1 7HSUBJECT 1 161 MAP03890 00 611 1•1, 70 MAPO 1900
611 LINECil2JSTAR MAP03910 WRITEC6,62tiLINECil 1 1•1~70l MAP03920 WRITEI6 1 69t MAP01910
62 FOAMATI1~0,20X,70A1t MAP03940 C PRINT TABlE OF ANTHROPOMETRIC DATA MAP03950
WRITEI6,63J MAP01960 61 FORMATI1H0 1 20X 1 1H*,23X,20HANTHROPOMETRJC DATA,25Xe1H*I MAP03970
WRITEI6 169J MAP03980 ~RITEC6 1 64J MAP03990
64 FORMAT C 1HO, 20Xe1H* 1 38X, 9HUPPER ARM, 5X 1 7HFOREARM,9X ,1H* J MAP04000 W~ITEC6 1651 L12,L23,Rl2,R23,Mt2,M23,112ti23,A12 MAP04010
6~ FORMATClH 1 20X 1 1H*t27H LENGTH•••••••••••••••METER,10X,F8.4,5X,F8.4MAP04020 11 10X 1 1H•I21X 1 1H* 1 21H DISTANCE Of C.G ••••• METERelOX,F8.4,5X,F8.4, MAP040l0 210X 1 1H*/21X 1 1H•1 32H MASS••••••••••••••••KILOGRAMCSJ,5X,F8.4,5X, MAP04040 lf8.4 1 lOX 1 1H•I21X 1 1H•,17H MOMENT Of INERTIA •••• NEWTON-METER••2,F8.4MAP04050 41 5X 1 F8.4 1 lOX 1 1H•/21XtlH•,JOH CROSS SECTION AREA ••• METER••Z,7X,F8.4MAP04060 5 1 5X 1 F8.4,10X,1H•I21X,lH*,68X,lH*l MAP04070
WRITEC6 1 621CliNECJJ,I=1,70J MAP04080 69 FORMATC1H ,20X,lH•,6&X.lH*) MAP04090
C REAC MOTION CODE AND PARAMETERS MAP04100 1001 READC5.31 ICODE,ITJME.TIME.WEIGHT,MOTDIS,MOTANG,XINT,YINT MAP04110
1 FORMATC216 1 6Fl0.5) MAP04120 IFCITIME.EQ.OJ GO TO 1000 MAP04130 WEIGHT•WEIG~T*9.80 MAP04140 MOTIM1•1TI"E MAP04150 MOTIME•MOTIM1/100.0 MAP04160
C CCMPUTE NUMBER Of TIME POINTS BY USING.05 AS A TIME INCREMENT MAP04110 N=ITIME/5+1 MAP04180 WRITEC6.731 MAP04l90
71 FOAMATClHll MAP04200
176
~RI1EC6,6611LINEIII,I=l,221 WRITEC6t1ll
66 FOR~ATI1~0,42X,22Alt lilA I Jf C 6, b 71
67 FORMATClH0,42X22H• DYNAMICAL ANALYSIS •t •RITEC6,711 ~RI1EC6,6611LINEIII,I=l,221
C PRINJ MOJION PARAMETERS WRIJEC6,681 ICOOE,MOTDIS,MOTANG,WEIGHT,MOTIME
68 FORMATil~O,l5HMOTION NUMBER ,16//2X,21HMOTIO~ CHARACTERISTICS•// 12~X,llHOIS1ANCE •• =,F9.~,~HMETER// 22SX,llHANGLE ••••• ~,F9.5,4HDEG.// 32~X,1l~WEIG~T •••• =,F9.5,4HKGS.// 42~X,l1HTIME •••••• =,F9.~,4HSEC.)
C PRINT INITIAL ARM CONFIGURATION WRITEC6,68ll
6Rl FOR~ATilHO,lX,25HINTIAL ARM CONFIGURATION WRITEC6,6821 XINT,YINT
682 FORMATI1H0,25X,l6HUPPER ARM ANGLE=,Fl0.5,5H DEC.///26X, 116HFOREARM ANGLE =eF10.~,5H DEG.)
71 FORMATClH ,42XelH•I KOUNT=l
C READ CCNTROL NUMBERS FOR THE COMPUTATIONAL PROCEDURE IN THE PROGRAM READC5,5) MASTER,NOWl,NOW,ITYPE,PERINT,PERMAX
5 FORMATI412,2X,2F5.21 C INITIALIZE VARIABLES FOR PARABOLA AND ENUMERATION
DEL TA=25.0 PTINT=O.O Pl=O.O it2~o.o Pl=O.O Plt•O.O P5•0.0 P1Ma10.0 P2M=l5.0 PlM=20.0 P4M=l'5.0 P5M= 10.0 IFIMASTER.EQ.OI GO TO 1004
C ·READ PARABOLA PARAMETERS IFCITYPE.EQ.ll READI5,9991 PTINT,DELTA
C READ ENUMERATION PARAMETERS IFCITYPE.EQ.Itl READI5,9991PlN,P1M,P2N,P2M,P3N,P3M,P4N,PitM,P5N,P5M
999 FORMATClOF5.2) PC INT•PT INT
2003 POINT•POINT+OELTA PERCNT=PE Rl NT P1a:PlN
1003 H=MOTDIS•PERCNT/100.0 2009 Pl•P1+5.0
P2•P2N 2008 P2•P2+5.0
PJ•PlN 2007 P1•Pl+5. 0
P4•P<\N 2006 P4•P<\ t5 .0
P5•P'5N 2005 PS.P5+5.0
IFIITYPE.EQ.O.OR.ITYPE.EQ.<\1 GO TO 1009 IFfNOWL.EQ.OI WRITEI6,l4) KOUNT,PERCNT
MAP04210 MAP04220 MAP0<\230 MAP0lt240 MAP04250 MAP04260 MAP04270 MAP04280 MAP04290 MAP04300 MAP04ll0 MAP04320 MAP04HO MAP04140 MAP04350 MAP04360 MAP04370 MAP04380 . MAP04390 MAP04<\00 MAP04<\10 MAP0<\420 MAP04430 MAP04440 MAP0<\450 MAP04460 MAP04470 MAP01t480 MAP04490 MAP04500 MAP04510 MAP04520 MAP04510 MAP01t540 MAP01t550 MAP04560 MAP04570 MAP04580 MAP04590 MAP04600 MAP04610 MAP04620 MAP04630 MAP04640 MAP04650 MAP04660 MAP04670 MAP04680 MAP04690 MAP0<\700 MAP0<\110 MAPO<\lZO MAP041l0 MAP0<\740 MAP0<\150 MAP0<\760 MAPO<\UO MAP0<\780 MAP0<\790 MAP0<\800
177
74 FORMATI1~1,18~1TERATION NUMBER ,15,5.,27HPERCENT OF MAXIMUM HEIGHMAP04810 ll tfl0.51 MAP04820
C CALL SUBROUTINE SOLVER TO DETERIME ANGULAR DISPLACEMENTS MAP04830 C FOR BOT~ UPPER ARM AND fOREARM. MAP04840
1009 CAll SOlYERIL12rL2l,MOTOIS,MOTANG,ITIME,H,N,ANGl,ANG2,1TYPE, MAP048~0 lXINJ,YINT,POINT,NOWl,Pl,P2,P3,P4,P5,Ml2,M23,112ti21,Rl2,R23, MAP04860 2WEIGHTI MAP04870
IFIITYPE.EQ.OI PERCNTzPERMAX MAP04880 IFCITYPE.EQ.O.OR.ITYPE.EQ.l.OR.ITYPE.EQ.21 POINT•lOO.O MAP04890 GO TO 1005 MAP04900
1004 PERCNT•PERMAX MAP04910 POINf•lOO.O MAP0.920
C READ EXPERI~ENTAL DATA MAP04930 REACI5,31 N MAP04940 00 10 I a lt N MAP04950 READ 15,2t ANGlCII, ANGZCI) MAP04960 ANGlllt•ANGlCJteO.Ol7453 MAP04970 ANG2Cit•ANG2Cit*D.Ol7453 MAP04980
10 CONTINUE MAP04990 C SMOOTH THE EXPERIMENTAL DATA WHICH HAS BEEN READ.BY MAPOSOOO. C POLYNOMIAL REGRESSION MAP05010
CALL SMOOTHIANGlrANG2,N,TIMEI MAPOS020 GO TO 1005 MAPOS030
1006 NUM•NUM+l MAP05040 PERCNT•PERMAX MAP05050 POINT•lOO.O. MAP05060 Pl•PlM MAP05070 P2•P2M MAP05080 P3•P3M MAP05090 P4-P4M MAP05100 P5•P5M MAP05ll0 KOUNT•l MAP05120
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP05130 C DYNAMIC ANALYSIS STATEMENTS 05150 THROUGH 6490 MAP05140 t •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP05l50 t 1. VELOCITY AND ACCELERATION MAP05160 t •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP05170 t INITIALIZE ANGULAR VELOCITIES AND ACCELERATIONS NAP05180
1005 ANGYlll)•O.O MAP05l90 ANGVZCll • 0.0 MAP05200 ANGAlCll ~ 0.0 MAP05210 ANGAZil)•O.O MAP05220 ANGVliNI•O.O MAP05210 ANGV21Nt•O.O MAP05240 ANGAlCNI•O.O MAP05250 ANGA2CNI•O.O MAP05260 NMl•N-1 MAP05270
t COMPUTE ANGULAR VELOCITY AND ACCELERATION ARRAYS MAP05280 C NUMERICAL DIFFERENTATION IS EMPLOYED MAP05290
DO 20 1•2,NM1 MAP05300 ANGVllll • CANGlCI+lJ - ANGlCI- 11)/ C2e • TIMEt MAP05310 ANGV2Cit • CANG2CI + 11- ANG2CI- 1 tl/12. • TIMEt MAP05320 ANGAlllt•IANGlll+lt+ANGlii-1J-2.0*ANGlCIIJ/CTIME••zt MAP05330 ANGAZCIJ•IANG2CI+1J+ANG2CI-11-2.o•ANG211)1/CTIME••zt MAP05340
20 CONTINUE MAP05350 C FIND MAXIMUM VELOCITY AND ACCELERATION FOR BOTH UPPER ARM AND MAP05360 C FOREARM FOR T~E CURRENT ITERATION MAP05370
AV2MAXIKOUNTI•BIGIANGV2,NI MAP05380 AVlMAXCKOUNTI•BIGCANGVl,Nt MAP05390 AA2MAXIKOUNTJ•BJGCANGA2,NI MAP05400
178
AAlMAX CKOUNT 1•8 IGUNGA ltN I MAPOS410 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP0~420 t•••••CAlCUlATION OF liNEAR ACCElERATIONS MAPOS430 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP05440 C INITIAliZE liNEAR ACCELERATIONS MAP05450
XA21lt • 0.0 MAP0~460 ZAZilt • 0.0 MAPOS470 XAlllt • 0.0 MAPOS480 lAlllt = 0.0 MAP05490 XGA2llt•O.O MAPOSSOO ZGA21lt•O.O MAP05510 XGA311t•O.O MAP05520 ZGAlllt•O.O MAP05530
C COMPUTE L llltEAR ACCElERATIONS FOR THE ARM JOINTS MAP0551t0 DO 30 I • z, N MAP05550 XA211t•- ll2•SINIANGllltt•ANGAllll-ll2•COSIANGllltJ•IANGVlllt••ztMAP05560 ZA211t•ll2•COSIANGliiJJ•ANGAlllt-Ll2•SINIANGliiJJ•IANGVliiJ••zt MAP05570 XAllltzXA2lll-l23•SINIANG2lltt•ANGA211J-l2l•COSIANG2Citt•lANGV2ll)MAPOSS80
1 .. 21 . MAP05590 ZA1CII•ZA2llt+l2l•COSIANG211tt•ANGA2111-L23•SINIANG21IIJ•IANGY21l1MAP05600
1••21 MAP05610 XGA2llt~Mll•XA21lt/ll2 MAP0~620 ZGA2llt•Rl2.ZA211J/ll2 MAP05630 XGAliii•XA211t-R23•SJNIANG211tt•ANGA2llt-R21•COS1ANG2IIIJ•IAhGV2 MAP05640
lllt••2t MAPOS650 ZGAliii=ZA2llt+R21•tOSIANG211tt•ANGA2111-R21•SINIANG21llt•IANGY2 MAP05660
1111••21 MAP05670 30 CONTINUE MAP05680
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••··~··•••MAP05690 C 11. REACTIVE FORCES AND MOMENTS MAP05700 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAPC5710 C CALCULATE DYNAMIC FORCES AND MOMENTS STARTINS ~ITH ELBOW JOINT HAP05720
00 ItO l=l 1 N MAP05730 YM0,2111=123•ANGA211t-M23•R23•SINIANG2liJJ•XGAlCII+ M23*R23•COS MAP0571t0
liANG211tt•ZGA3111-M23•9.8•R21•coscANG2CIJt-WEIGHT•L23•COSlANG21111MAP05750 llf21 I t=Mll•XGA 11 It MAP05760 ZF2Cit•-WEIGHT-M23•9.8+M2l•ZGA3Cit MAP05770 F211t=SQRTIXF21lt••z+zF211t••zt MAP05780 YMOMliii•YMOM2Cit+ll2•ANGAlllt-Ml2•R12•SINIANGllllleXGA211J+M12• MAP05790
lR12•COSIANGllltt•ZGA2CIJ-Ml2•9.8•Rl2•COSlANGlliii-XF21II•Ll2*SIN MAPOS800 21lNGlll))+lF2Cit•ll2•tOSlANGllll) MAP058l0
XF1Cli=XF2111+Ml2•XGA2111 MAPOS820 lFliii=ZF2CI)+Ml2•lGA2111-Ml2•9.8 MAPOS830 FllllzSQRTIXFlii)••2•ZFlllt••2t MAP05840
C CALCULATE STATIC FORCES AND MOMENTS DURING THE MOTION MAPOS850 JOR2111•-M23•9.&•R2l•COSIANG211Jt-WEIGHT•L2l*COSIANG21 IJ) MAP05860 SXF211J:Q.O MAPOS810 SZF2CIJ•-WEIGHT-M23•9.8 MAP05880 SF211J=SQRTISXF211t••2+SZF2111••2J MAP05890 TORlii)=TOR211t-Ml2•9.8•Rl2•COSIANGliiJ)+SZF2111•Ll2•COSIANGlCIJ) MAP0~900 SXFlllt•O.O MAP05910 SZFliii•SlF2111-Ml2•9.8 MAPQ5q2Q SFliii=SQRTISXFlii)••I+SlFICI1••2t MAPQ~qlo
40 CONTINUE MAP0591t0 C COMPUTE MAXIMUM VALUES fOR FORCES AND MO~ENTS MAP05950 C BCTH FOR DYNAMIC AND STATIC VAlUES MAPQ5q6Q
F2MAXlKOUNTI•81GIF2,N) ' MAPQ5q7Q FlMAXCKOUNTt•BIGIFl,NI MAP05980 YMOMM2(KOUNTI=BIGIYMOM2,NJ MAP05990 YMOM~l(KOUNTtzBIGIYMOMl,N) MAP06000
179
TOR2MIKOUNTI•BIGCTOR2 1 NI MAP06010 TORlMIKOUNTI•BIGCTORl,N) MAP06020' SF2MAXIKOUNTI•BIGCSF2 1 NI MAP06030 SFlMAXIKOUNTI•BIGISF1 1 NI MAP060~0 AATI01CKOUNTI•YMOMM2CKOUNTI/TOR2MCKOUNTJ MAP060SO RATI02CKOUNfi=YMOMMliKOUNTJ/TORlMIKOUNTI MAP06060 RAT 1031KOUNT J •F2MA XI KOUN J I /SF 2MAX IKOUNTt MAP06070 AATIO~(KOUNTJ•FlMAXCKOUNTI/SFlMAXIKOUNTI MAP06080
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06090 C Ill. PERFORMANCE CRITERIA MAP06l00 C tMETHOO OF NUMERICAL INTEGRATION-TRAPEZOIDAL RULE--IS EMPLOYEDI , MAP06ll0 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06120 C 1. STRESS MAP06l30 C CALCUlATE NORMAL STRESS MAP061~0
DO 8S l•l,N MAP06150 STRESS I II=YMOMlll I*Ril2 .. XFlll I•COSUNGlC I I I+ZI 111 t•SINUNGliiU IMAP06160
1/Al2 MAP06170 8S CONTINUE MAP06180
Nl•N-1 MAP06190 C CALCULATE TOTAL STRESS MAP06200
SUM•O.O MAP062l0 DO 86 1•2,N2 MAP06220
86 SUM• SUM+ STRESS Cl I MAP06230 TOTSTA•TIME/2.0•ISTRESSI11+2.0•SUM+STRESSINII MAP062~0
C 2. WORK, POWER, STRESS RATE MAP06250 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06260 C INITIALIZE WORK, POWER, STRESS RATE MAP06270
WORKClJ•O.O MAP06280 POWERCllsO.O MAP06290 SSRATECli•O.O MAP06300
C CALCULATE TOTAL WORK, POWER, AND STRESS RATE MAP06310 DO ~1 1•2 1 N .MAP06320 SSRATE II I •SSRATE Cl-1 I +0.5*UNGY1C I 1-ANGYlii-U t"•t S TRESSI I) MAP06HO
1+STRESSII-111 MAP063~0 WORKIIJ•WORKCI-lt+0.5•CANGlCII-ANG1CI-lii*IYMOMliii+YMOMlll-lll MAP063SO POWERCII•POWERCI-lt+O.~eCANGYliii-ANGYlll-lii*IYMOMlCII+VMOMlll- MAP06360
llJ I MAP06HO ~~ CONTINUE MAP06380
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06190 C 3. LINEAR AND ANGULAR IMPULSES MAP06~00 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06~10
ANGIMP•O.O MAP06~20 FIMP•O.O MAP06~10 DO ~2 1•2,N2 MAP06~~0 ANGIMP•ANGIMP+YMOMlCII MAP06~50 FIMP•FIMP+fllll MAP06~60
~2 CONTINUE MAP06~70 TANGIM•TIME/2.0*CYMOM1Cli+2.0*ANGIMP+VMOM11Nit MAP06~80 TFIMP•TIME/2.0*CflllJ+2.0tFIMP+FlCNJI MAP06~90
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••**MAP06500 C STORE PERFORMANCE CRITERIA VALUES COMPUTED FOR ITERATION MAP06510 C ••••••••••••••••• .. •••••••••••••••••••••••••••••••••••••••••••••••••*MAP06520
FCTllKOUNTI•WORKCNt MAP06510 FCT2CKOUNTt•POWERCNt MAP065~0 FCTlCKOUHTI•TANGIM MAP06550 Fcl4CKOUNTI•TFIMP MAP06560 FCT5CKOUNTI•TOTSTR MAP06570 FCT6CKOUNTI•SSRATECNt MAP06580 HHHCKOUNTiaH MAP06590 POINTlCKOUNTI•POINT MAP06600
180
c c c
PllCKOUNTI=Pl P211KOUNTI=P2 Pl1CKOUNTJ=P3 P41IKOUNTI~P4 P~IIKOUNTI=P~
MAP06610 MAP06620 MAP06630 MAP066~0 MAP06650
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP06660 PRINT AND PLOT MOTION CHARACTERISTICS MAP06670
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••tttMAP06680 IFINOWl.EQ.ll GO TO 813 MAP06690
C•••••••••••••••••••••••••••••••••••••••••••• MAP06700 C PLOTTING STICK DIAGRAM FOR THE MOTION MAP06710 Ct••••••••••••••••••••••••••••••••••·~··••••• MAP06720
CALL STICK CANGl,ANG2,N,Ll2 1 L2ll MAP06730 C PRI~T AND PLOT HAND DATA MAP06740
WRITEI6,83) MAP06750 83 FORMATI1Hl,50X,9HHAND DATAl MAP06760
WRITEI6,9511 MAP06770 951 FORMAT11HO,l5H X-ACCELERATION 9 3X,1~H Z-ACCELERATION/1X 1 MAP06780
115H METER/SEC/SEC,3X,15H METER/SEC/SECI MAP06790 CO 57 l=l,N MAP06800
57 WRITEC6,70) XA3CII,ZA3CII MAP06810 WRITEC6,577) MAP06820
577 FORMATC1H0,20X,'PLOTS OF LINEAR ACCELERATIONS 1 //lX 91 X-ACCELERATIDNMAP06830
1 SYMBOL IS ••,zox,•z-ACCELERATION SYMBOL IS X\) MAP06840 CALL PLOTCXA3,ZA3,N,OI MAP06850
C PRINT ADO PLOT ELBOW DATA FOR BOTH STATIC MAP06860 C AND DYNAMIC COMCITJONS MAP06870
WRITEC6,801 MAP06880 80 FORMATC1H1,50X 1 10HELBOW DATA) MAP06890
WRITEI6,9011 MAP06900 WRITEC6,9521 MAP06910
952 FORMATilH0,15H X-FORCE 1 15H l-FORCE ,15HRESULTANT-FORCMAP06920 1E1 15H MOMENT /lX 1 15H NEWTON ,15H NEWTON , MAP06930 215H NEWTON tl5H NEWTON-METER ) MAP06940
DO 92 1=1 1 N MAP06950 92 WRITEC6,701 SXF2CII,SZF2111,SF2111,TOR2CII MAP06960
WRITEC6,9021 MAP06970 WRITEI6,9531 MAP06980
953 FORMAT(lH0,15H ANGULAR DISPL. 1 15H ANGULAR VEL. ,l5H ANGULAR ACCELMAP06990 1.,15H X~ACCEL. ,15H Z-ACCEL. /1X 1 MAPOJOOO 215H RADIANS ,15H RAD/SEC ,15H RAD/SEC/SEC, MAP07010 315H METER/SEC/SEC ,15H METER/SEC/SEC ) MAP07020
CO 50 1=1 1 N MAP07030 50 WRITEC6,701 ANG2CII,ANGV2CII,ANGA2CIItXA2CII,ZA2CII MAPOlO~O 70 FORMATI1H0 1 8F15.5) MAP07050
WRITEI6,952J MAP07060 DO 501 1=1 1 N MAP07070
~01 WRITEC6,70) XF2111,ZF21II,F2111,YMOM2111 MAP07080 IFCNOW.EQ.OI GO TO 811 MAP07090
c••••••••••••••••••••••••••••••••••••••••••••••••••••••• MAP07100 C PLOTTING RESULTANT FORCE AND MOMENT AT THE ELBOW MAP07110 c••••••••••••••••••••••••••••••••••••••••••••••••••••••••• MAP07120 WRITEC6,7011 MAP07130
701 FOR~ATI1H1,20X,35HPLOTS OF RESULTANT FORCE AND MOMENT) MAP07l40 WAITEC6,llll MAP07150
711 FORMATC1HO,l8HMOMENT SYMBOl IS t,20X,27HRESULTANT FORCE SYMBOL IS MAP07160 1XI MAPOlllO
CALL PLOTCY~OM2,F2,N,O) MAP07180 C PRINT AND PLOT SHOULDER DATA MAP07190
811 WAITEC6,Bll MAP07200
181
81 FOR~ATfl~l,~OX,llHSHOULDER DATAt -.RITEC6,90U WRITEI6,952) DO 93 1=1 ,N
91 WRITEI6,70t SXFlflt,SZFICiteSFlCit,TORlCit WRITE (6, 902 t WA If E I 6, 95~ t
95~ FOR~ATflHO,l5H ANGULAR DISPL.,I5H ANGULAR VEL. , IISH A~GULAR ACCEL./IX,lSH RADIANS ,ISH RAO/SEC Z1SH RAD/SEC/SEC )
DO 55 l=l,N ~~ -.RITEC6,70) ANG1CI),ANGV1CIJ,ANGAICI)
WRITEI6,952J 00 551 I•I,N
551 -.RJTEC6,70t Xflllt,ZFlllt,FlCIJ,YMOMIII) IFI~OW.EQ.Ot GO TO 812
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••• C PLOTTING RESULTANT FORCE AND MOMENT AT THE SHOULDER c••••••••••••••••••••••••••••••••••••••••••••••••••••••• WRITEI6,70lt
WRITEC6,1lll CALL PLOTCYMOMl,F1,Nt0t
8 12 WR IT E I 6 t 8 2 I 82 FORMATI1Hl,SOX,JOHWORK AND POWER AT THE SHOULOERt
WRITEC6,955t 955 FORMATClHOel5H
ll5H NEWTON DO 56 1=1eN
WORK ,15H
, ISH POWER NEWTON/SEC.)
56 WRITEI6,lOtWORKCIIePOWERCIJ WRITEC6,81tt
/1 x,
81t FORMATilHO,l5HANGULAR-IMPULSEt15H LINEAR-IMPULSEe15H 1 • ' WRITEI6,l0t TANGIM,TFIMP,TOTSTR
STRESS
MAP07210 MAP07220 MAP07230 MAP0721t0 MAP07250 MAP07260 MAP07270 MAP07280 MAP07290 MAP01300 MAPOHlO MAP01320 MAP07HO MAP0731t0 MAP01350 MAP07360 MAP013 70 MAP07380 MAP07390 MAP07400 MAP071tl0 MAP07420 MAP07430 MAP0l41t0 MAP07450 MAP0llt60 MAP0l470 MAP071t80 MAP071t90 MAP07500 MAP0l510 MAP0l520 MAPOlSlO MAP0l51t0 813 KOUNT=KOUNT+l
IFCITYPE.EQ.O.OR.JTYPE.EQ.1.0R.ITYPE.EQ.2.0R.ITYPE.EQ.3JGO IFtP5.LT.PSMt GO TO 2005
TO 3003MAP07550
IFIPit.LT.P~M) GO TO 2006 IFCP3.LT.P3MJ GO TO 2007 lftP2.LT.P2M) GO TO 2008 IFCP1.LT.P1Mt GO TO 2009 GO TO 3002
3001 PERCNT•PERCNT+5.0 IFtPERCNT.LE.PERMAXt GO TO 1003 IFIPOINT.LE.BO.OJ GO TO 2001
1002 IEND•KOUNT-1 IFIIEND.LE.1) GO TO 1007
C IN CASE OF EXPERIMENTAL VALUES SKIP THE NEXT SEGMENT C PRINT ITERATIONS SUMMARY IF SIMULATION ANAL.YSIS IS USED.
WRITEI6,871 81 FORMAT11Hle20X,39HS U M M A R Y 0 F I T E R A T I 0 N S/21X,
119H---------------------------------------· WRITEt6,68t ICODE,MOTOIS,MOTANG,WEIGHT,MOTIME WRI TEI6,68ll WRITEC6e682t XINT,YINT WR IT E 16 t 90 t
90 FORMATt1H0,48HMAXIMUM VALUES WRITE 16,9011
OF YEL.,ACCEL.,FORCES,ANO MOMENJSt
901 FORMATCIH0,20X,l5HSTATIC ANALYSIS) WRITEC6e956) •
MAP0l5b0 MAP07570 MAP07580 MAP07590 MAP07600 MAP07610 MAP0l620 MAP0l630 MAP0l640 MAP07650 MAP07660 MAP07670 MAP07680 MAP07690 MAPOHOO MAP07710 MAP01720 MAP077JO MAP01740 MAP07750 MAP07160 MAP07170 MAP07180 MAP07790
956 FORMATI1H0,15H ELBOW FORCE ,15H SHOULOER-FORCE,lSH ELBOW-MOMENTMAP07800
182
c
c
c
c
c
c
1 el~HSHOULOER-MOMENTI DO 911 l=l,IEND
911 ~RITEC6,9211Sf2MAXIIJ,SF1MAXCit 1 TOR2MClt 1 TOR1MCII 921 FORMATC1H0 1 8Fl5.61
WRITE 16,9021 902 FORMATC1H0,20X 1 16HDYNAMIC ANALYSISt
WRITEC6,9571 957 FORMATC1~0,15H FOREARM-VEL. 1 15H UPPER ARM-VEL. 1 15H
l.,lSHUPPER ARM-ACCELI 00 912 I •1 ,IE NO
912 WRITEC6,92ltAV2MAXCit,AVlMAXCit 1 AA2MAMI11 1 AAlMAXCtt WR ITEC6 ,9561 DO 915 1•1 1 1ENO
915 WRITE C 6, 9211 F2MAX( It 1 F lMA XCI t, YMOMM211 t, YMOMMlC II wR nE c6 ,CJOl t
903 FORMATClHOe20X 1 23HRATIO OF DYNAMIC/STATICI WRITE C6 ,9581
958 FORMATClHOel5H ELBOW-FORCE 1 l5H SHOULDER-FDRCE 1 1SH 1 elSHSHOULDER-MOMENJ)
DO 913 l•ltiEND 913 WRITEC6,9211RATI03CIJ,RATI04Cit 1 RATI01IIt 1 RATID2CII
WRITEC6,901tl 904 FORMATC1H0,20Xe19HCRITERION FUNCTIONSt
WRITEI6 1 959t .
MAP07tH 0 MAP07820 MAP07830 MAP07840 MAP07850 MAP07860 MAP07870
FOREARM-ACCELMAP07880 MAP07890 MAP07900 MAP07910 MAP07920 MAP07930 MAP07940 MAP079tr;O MAP07960 MAP07970
ELBOw-MOMENTMAP07980 . MAP07990 MAP08000 MAP08010 MAP08020 MAP08030
959 FOR~ATilH0,9HITERATIONe15H WORK 1 l5H 11SHANGULAR-IMPULSEelSH LINEAR-IMPULSE 1 15H 2ESS RATE t
POWER STRESS
• ,ISH
MAP08040 MAP08050
DO 89 l•l,tEND WRITEC6,88) 1 1 FCTlii1 1 FCT2111 1 FCT3CIJ,FCT4111eFCT5111eFCT6CII
88 FORMATC1HO,I6,6Fl5.5t
STRMAP08060 MAP08070 MAP08080 MAP08090 MAP08100
89 CONTINUE PLOT TOTAL POWER AND ANGULAR IMPULSE FOR ALL ITERATIONS
WRITE 16, 914)
MAP08110 MAPOtll20 MAP08130
914 FORMATC1H0 1 34HPLOTS OF POWER AND ANGULAR IMPULSE//lX,llHPOWER lOl IS •,20X,27HANGULAR IMPULSE SYMBOL IS XI
SYMBMAP08140
CAll PLOTCFCT2,FCT3,1ENO,ll DETERMINE THE ITERATION WHICH GIVES THE MINIMUM POWER.
CAll MINTIRCFCT2,1END,SMAlLeKKt DEFINE PARAMETERS FOR THE OPTIMUM MOTION
POJNT•PO INTI CKK t Pl•Plll KIC t P2•P21 CKKI PJ-Pll CKK t P-\•P41 CKK t P5•PSICKKt HH-Ht'HCKK)
PRINT THE OPTIMAl MOTION WRITEC6,9U
91 FORMATC1Hle20X 1 14HOPTIMAl MOTION/21X,l4HXXXXXXXXXXXXXXt WRITEC6 1 68t ICOOEeMOTDIS,MOTANG,WEIGHT,MOTIME WRI TEl 6,681 t WRITEC6e682t XINT,YINT NOW•l NOW1•0
CALl SOLVER TO OBTAIN ANGULAR DISPLACEMENTS FOR THE OPTIMUM MOTION CAll SOLYERCll2,L231 MOTDIS 1 MOTANG,ITIME 1 HH,N 1 ANGl,ANG2,1TYPE,
·111NT 1 YINT 1 POINT,NOW1,Pl,P2,Pl,P4,PS,M12eM2lei12,123,Rl2,R2l, 2WE IGHT t
GO AND PERFORM DYNAMIC ANALYSIS FOR THE OPTIMUM MOTION IFINUM.EQ.OI GO TO 1006
MAP08150 MAP08160 MAP08170 MAP08180 MAP08190 MAP08200 MAP08210 MAP08220 MAP08230 MAP08240 MAP08250 MAP08260 MAP08270 MAP08280 MAP08290 MAP08300 MAP08ll0 MAP08320 MAP08330 MAP08340 MAP08350 MAP08360 MAP08370 MAP08380 MAP08390 MAP08400
183
1007 NU"=O MAP08410 GO TO 1001 MAP08420
lC02 CALL EXIT MAP08430 E~D MAP08440
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP084SO C SUBROUTI~E SOLVER MAP08460 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP08470 C MAP08480 C PURPOSE: MAP08490 C MAP08500 C TC COMPUTE EULER'S A~GLES FOR THE UPPER ARM AND FOREARM MAP08510 C DURING THE "OTION. MAP08520 C MAP08530 C DESCRIPTIONS OF PARAMETERS: MAP08540 C MAP08550 C XL12, XL23--ARM DIMENSIONS MAP085b0 C DIS, TANGle ITIME--MOTION PARAMETERS ISEE MAIN PROGRAMI MAP08570 C XINT, YINT--INITIAL ARM ANGLES. MAP08580 C ITYPE--TYPE OF THE HAND PATH OF MOTION FUNCTION ISEE MAl~ PROGRAM) MAP08590 C EPS--THE ALLOWABLE ERROR I~ DETERMINING EULER'SANGLES MAP08600 C lEND--MAXIMUM ~UMBER OF ILERATIONS FOR SOLVING THE MAP08610 C NONLINEAR EQUATIONS MAP08620 C Cl, C2--X AND l COORDINATES Of THE HA~D MAP08630 C C11t C22--X AND l DISPLACEME~TS OF THE HAND MAP08640 C X~AX--MAXIMUM DISPLACEMENT IN THE X DIRECTION MAP08650 C 8-- VECTOR OF REGRESSION COEFFICIENTS MAP08660 C KONT--CONTROL VARIABLE FOR DETERMINING T~E TYPE OF INPUT VALUES MAP08670 C KONT•O GIVEN HAND PATH OF MOllO~ MAP08680 C KONT•1 GIVEN HAND PATH OF MOTION OBTAINED MAP08690 C BY DYNAMIC PROGRAMMING MAP08700 C N--NUMBER OF DATA POl NTS MAP08 710 C X,Y--OUTPUT VECTOR FOR EULER 1 SANGLES MAP08720 C ERRX, ERRY--ERRORS RESULT FROM SOLVING THE MAP08730 C NONLINEAR EQUATIONS MAP08740 C XPLOT, YPLOT--ANGULAR DISPLACEMENTS RECTORS FOR THE ARM MAP08750 C TIM•TOTAL MOTION TIME MAP08760 C REMARKS: MAPOB770 C THE FOLLOWING SUBROUTINES ARE REQUIRED MAP08780 C -PARAS MAP08790 C -SIMUL MAP08800 C -PLOT MAP088l0 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP08820
SUBROUTINE SDLVERCXL12,XL231 DIS 1 TANG1,1TIME,H,N,X,Y,ITYPE,XINT MAP08830 leYINT,POINT,NOW1eP1eP2,P3,P4,P5,XM12,XM23eXI12eXI23eR12eR23, MAP08840 2WEIGHTI MAP08850
REAL Ll2,L21 MAP08860 DATA EPS 1 1END/0.0001 1 50/ MAP08870 DIMENSION ClC20I,C21201 1 Clll20t,C22C201 MAP08880 DIMENSION XI20),YI201 MAP08890 OIMENS ION XPLOTI20t, YPLOH201 MAP08900 DIMENSION ERRXI20t,ERRYI201 MAP089l0 DIMENSION 8120 1 201 MAP08920 L 12-XL 12 MAP08930 L23•XL23 MAP08940 llli•XINT MAP08950 Ylli•YINT MAP08960 TANG•TANG1•0.01745 MAP08970 Xlli•XIli•O.Ol745 MAP08980 Ylli•YI1t•0.01745 MAP08990
C IF ITYPE EQUAL ZERO, GO AND READ DATA FOR THE HAND PATH OF MOTION MAP09000
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IFCITYPE.EQ.Ot GO TO 35 MAPOCJOlO ltMAX=DIS•COSCTAHGt MAP09020 lt~Aitl=0.5•JtMAX MAP09030
COMPUTE THE INTIAl x,z OF JME HAND MAP09040 Cl11 t=ll2•COSI Xl11 )+lZJ•COSI YIU I MAP0'1050 C2 11 I = ll2 • S IN I X I 1 ) hl 21• S IN I Y I 11 I 11'1 A P 09 06 0 IFIITYPE.Et.ll CAll PARABIXMAX,M,POIHT,TANG,BI MAP09070 IFCITYPE.EQ.4t CAll SIMUlCXMAX,TAHG,P1,P2,Pl,P4,P5,BI MAP09080
USE TME DISPLACEMENT FUNCTION TO GENERATE X-COORDINATE MAP09090 I= 1 MAP09100 TUU•ITIME MAP09ll0 TIM•TIMl/100.0 MAP091ZO ITIME1•1TI~E-1 MAP09130 DO 10 K=5,1TIME1,5 MAP09140 l=l+l MAP09150 TIM1=K MAP09160 TIM2=TIM1/100.0 MAP09170 TERM 1=6.28U 1M2/TIM MAP09180 TERM2=TERM1-SINITERM1l MAP09190 Cliii=C11li+XMAX•TERM2/6.28 MAP09200 C 111 II =C 1 Cl ) -C 1111 MAPOCJ210 lERMl=CllCll•l.l4/XMAX MAP09220
ACCORDING TO TME FUNCTION REQUIRED COMPUTE Z-CODROINATE MAP09210 IFIITYPE.EQ.ll C22CII=H*SINITERM3)+Clliii•TANCTANGI MAPOCJ240 IFCITYPE.EQ.2) C221 ll=H•SQRTil.O-CIA8SICllCIII-XMAX11••21/XMAX1*•2MAP09250
1l+C1lllt•TANCTANGt MAP09260 IFC ITYPE.EQ.lJ C22111=-8Cl,U+BIZ,li•Cll CII+8B,iJ•CllCII .. 2 MAP09270 IFIITYPE.EQ.41 CZZCII=BI1,11+8C2,lt•Cl1CII+BI1,1)•Cl111)••z+ MAP09280
18C4,11•tl111)••l+6(5,li*C11CI)•e4+616,11•C11CI)••5 MAP09290 C2CII•C211 )+C22 C II MAP09100
10 CONTINUE MAP09310 C11NI=C111)+XMAX MAP09l20 C21~J•C2111+XMAX•TANITANGI MAP09330 GO TO 16 MAP09340
35 REAOC5,2) KONT MAP09350 2 FORMAT 116) MAP09360
READ X AND Y COOROJNATES OF THE HAND MAP09370 IFIKONT.EQ.OI REA015,28) IC1CII,C2Cit,l=l,N) MAP09380 IFIKONT.EQ.lt CALl OYNMCIL12,L23,XM12,XM23,XIl2,XI23,Rl2,RZ3,XMAX,MAPOCJ390
!ITIME,XINT,YlNT,WEIGHT,Cl,C2,N,TANG11 MAP09400 28 FORMATI2F10.51 MAP09410 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP09420
SOLVE TWO NONLINEAR EQUATIONS BY NEWTON'S METHOD MAP09430 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP09440 36 ERRXC11s0.0 MAP09450
ERRYI11•0.0 MAP09460 START NEWTON ITERATION SCHEME MAP09470
00 30 1•2,N MAP09480 XIT•XII-11 MAP09490 YIT•VII-11 MAP09500
FX, FY, GX, GY--OERIVATIVES OF THE TWO EQUATIONS MAP09510 00 20 J•l,IENO MAP09520 FX•-L12•SINIXITI MAP09510 FYa-L23•SINCYITI MAP09S40 GX•Ll2tCOSIXITI MAP09550 GY•L23•tOSCYITI MAP09560 f•ll2•COSCXIT)+L2l*COSIYITI-Clll) MAP09570 Gall2•SINCXITI+L21*SINIYIT)-C211t MAP09580 OIV•FX•GY-GX•FY+O.OOOOOl MAP09590 XITl•XIT+CG•FY-F•GY)/OIV MAP09600
YIT1•YITHF•GX-G•FU/DIV MAP09610 Z1•ABSCFt MAP09620 Z2•ABSCG) MAP09630
C IF fHE ERRORS ARE SMALL TERMINATE THE ITERATION PROCESS MAP09640 lFCZl.LE.EPS.ANO.Z2.LE.EPS) GO TO 25 MAP09650 XIT•XIT1 MAP09660 YIT•YIT1 MAP09670
20 CONTINUE MAP09680 25 XCII•XITl MAP09690
YCI)•YIT1 MAP09700 ERRXCII•Zl MAP09l10 ERRYCII•Z2 MAP09720
30 CONTINUE MAP09730 IFCN0Wl.EQ.1) GO TO 100 MAP09740
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP097SO C PRINT DISPlACEMENT DATA AND THE BASIS FOR GENERATIONG IT MAP09760 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP09l70
WRITEC6,l11 MAP09780 ll FORMATClH0,25X,1lHDISPLACEMENT DATA) MAP09790
IFCITYPE.EQ.O.AND.KONT.EQ.O) WRITEC6 1 4) MAP09800 IFCITYPE.EQ.O.AND.KONT.EQ.ll WRITEC6 9 28ll MIP09810 IFCITYPE.EQ.1) WRITEC6,51 MAP09820 IFC ITYPE.E0 •. 21 WRITEC6 961 MAP09830 IFCITYPE.EQ.ll WRITEC6,12J MAP09840 IFCITYPE.EQ.41 WRITEC6,141 MAP09850
4 FORMITC1HO,JX,44HDATA FOR THE HAND PATH OF MOTION IS GIVEN I MAP09860 5 FORMATC1HO,JX,88H SINE CURVE ASSUMPTION IS USED AS A BASIS FOR GMAP09870
lENERATING A PATH OF MOTION FOR THE HANOI MAP09880 6 FORMATC1HO,JX,88HELLIPSE CURVE ASSUMPTION IS USED AS A BASIS FOR GMAP09890
1ENERATING A PATH OF MOTION FOR THE HINDI MAP09900 12 FORMATC1~0,JX,'SECOND ORDER POlYNOMIAl IS USED AS A BASIS FOR GENEMAP09910
1RATING A PATH OF MOTION FOR THE HAND'I MAP09920 14 FORMATClHO,lXt' SIMULATION ANALYSIS 'I MAP09930
287 FORMATC1H0 9 7X 9 ' DYNAMIC PROGRAMMING 'I MAP09940 WRITEC6,71 MAP099SO
1 FORMATC1H0,1SH X-COOR. HAND,l6H Z-tOOR. HAND , MAP09960 115HUPPER ARM ANGLE9 15~ ERROR 1 15H FOREARM ANGLE, MAP09970 215H ERROR I MAP09980
c· PRINT TABLE OF ANGUlAR DISPLACEMENTS MAP09990 DO 40 1•1tN MAPlOOOO
40 WRITEC6,31 C1CII 1 C2CI1 1 XCII 9 ERRXCII,YCII 1 ERRYCI) MAP10010 l FORMATC1H0,6F15.ll MAP,10020
DO 50 1•1,N MAP10030 XPLOT U I •X C II-XC 1 I MAP 10040
50 YPLOTCII•YCII-YC11 MAP10050 C PLOT DISPLACEMENT DATA MAP10060
WRITEC6,81 MAPlOOlO 8 FORMATC1H0,10X 9 29HPLOTS OF ANGULAR DISPLACEMENT) MAP10080
WRITEC6,9t MAP10090 9 FORMATC1HO,Z8H UPPER ARM ANGLE SYMBOL IS •,zox, MAitOlOO
1Z6HFOREARM ANGLE SYMBOL IS x· t MAP10l10 CALL PLOTCXPLOT,YPLOT,N,OJ MAP10120
100 RETURN MAP10130 ENO MAP 10140
C •••••••••••••••••••• .. ••• .. •••••••••••••••••••••••••••••••••••••••••*MAP10150 C SlBROUTINE PARAB MAP10 160 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10ll0 C PUR..OSEI MAP 10180 C GENERATE A PARABOLA--SECOND ORDER POLYNOMIAL--BY USING MAP10190 C THREE POINTS. FIRST AND THIRD POINT ARE TAKEN TO IE THE ~APlOZOO
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C INITIAl AND THE TER~INAl POINTS OF THE MOTION. MAP10210 C MAP10220 C DESCRIPTION OF PARAMETERS MAP102JO C MAP10240 C XMAX--MOTION DISTANCE MAP10250 C H--HEIGHT OF THE SECOND POINT, EXPRESSED AS A PERCENTAGE MAP10260 C Of THE ~OTION DISTANCE MAP10l70 C POINT--POSITION OF THE SECOND POINT, ON THE X-AXIS; MAP10280 C EXPRESSED AS A PERCENTAGE OF THE MOTION DISTANCE MAP10290 C TANG--MOTION ANGLE MAPlOJOO C 8--VECTOR Of REGRESSION COEFFICIENTS MAPlOllO C X,Y--INPUT VECTORS FOR REGRESSION SUBROUTI~E MAP10320 C REMARKS MAPlOllO C SUBROUTINE REGRSS IS REQUIRED MAP10340 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10350
SUBROUTINE PARABIXMAX,H,POINT,TANG,BJ MAP10360 DIMENSION XC20J,YC20,4J,BC20,20J MAP10370 XClJ=O.O MAP10380 YCl,lJ=O.O MAP10390 Xl2J=POINT•XMAX/lOO.O MAP10400 YC2,1J=H+XC2J•TANCTANG) MAP10410 XI3J=XMAX MAP10420 YIJ,lJ=XMAX•TANCTANGI MAP10430 CAll REGRSSCX,Y,3,2,1,BI MAP10440 RETURN MAP10450 END . MAP10460
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10470 C SUBROUTINE SIMUl MAP10480 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10490 C MAP10500 C PURPOSE: MAP10510 C MAP10520 C TO FIT A FIFTH ORDER POlYNOMIAl REGRESSION FUNCTION TO SEVEN MAP10530 C GRID POINTS. THE FIRST AND THE lAST POINTS ARE THE MAP10540 C INITIAl AND TERMINAl POINTS OF THE MOTION. MAP10550 C MAP10560 C DESCRIPTION Of PARAMETERS MAP10570 C MAP10580 c· XMAX--MOTION DISTANCE MAP10S90 C TANG--ANGLE OF MOTION MAP10600 C Pl,P2,P3,P4,P5--HEIGHT OF GRID POINTS EXPRESSED AS PERCENTAGES MAP10610 C Of THE MOTION DISTANCE MAP10620 C X,Y--INPUT VECTORS FOR REGRESSION SUBROUTINE MAP10630 C a--VECTOR OF REGRESSION COEFFICIENTS MAP10640 C REMARKS MAP10650 C 1. HORIZONTAl COORDINATES OF GRID POINTS ARE OBTAINED BY MAP10660 C USING Oe20,40,50,60,80,1001 OF THE MOJION DISTANCE MAP10670 C 2. SUBROUTINE REGRESS IS REQUIRED MAP10680 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10690
SUBROUTINE SIMUlCXMAX 1 TANG,PleP2,P3,P4,P5,8) MAPlOlOO DIMENSION XC20),YC20,4t,8C20,20J MAPlOllO ICli•O.O MAP10720 YCleli•O.O MAP10730 Xl21•20.0•XMAX/lOO.O MAP10740 YC2,1J.Pl•XMAX/lOO.O+Xl21•TANITANGJ MAP10750 Xlli•40.0•XMAX/lOO.O MAP10760 YC3eli•P2•XMAX/100.0+Xlli•TANCTANGI MAPlOllO IC41•50.0•IMAX/lOO.O MAP10l80 YC4,li•P3•1MAX/lOO.O+XC41•TANITANGI MAP10l90 IC51•60.0•xMAX/lOO.O MAP10800
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YC5eli•P4•XMAX/lOO.O+XC5t•TANITANGt MAP10810 XC61•80.0•XMAX/lOO.O MAP10820 YC6e11•P5•XMAX/100.0+XC6t•TANCTANGI MAP10830 Xlli•XMAX MAP10840 YCleli•O.O+XClt•TANCTANGt MAP10850 CALL REGRSSCX,Y,l,5,1,8l MAP10860 RETUR~ MAP10870 END MAP10880
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10890 C SUBROUJINE SMOOTH MAP10900 C ••••••••••••••~••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP10910 C MAP 10920 C PURPOSE: MAP 10910 C THE SUBROUTINE HAS TWO OBJECTIVES MAP10940 C 1. FITTING POLYNOMIAL REGRESSION FUNCTIONS TO TWO SETS MAPl0950-C OF EXPERIMENTAL DATA. MAP 10960 C 2. USING THE RESUlTING POLYNOMIAL FUNCTIONS, TWO NEW SETS MAP10970 C OF DATA CSMOOTHEOl ARE OBTAINED. MAP10980 C MAP10990 C DESCRIPTION OF PARAMETERS MAPllOOO C MAPll010 C XX--INPUT VECTOR OF THE FIRST VARIABLE CSETl MAPll020 C Y--INPUT VECTOR OF THE SECOND VARIABLE C SEll MAPllOlO C N--NUMBER OF ELEMENTS IN EACH INPUT VECTOR; XXeY• MAPll040 C Y1eXXl--TWO ARRAYS TO STORE THE INPUT VECTORS MAP11050 C Y,xx--PREDICTED VECTORS WHICH REPLACES THE INPUT VECTORS MAP11D60 C XXO,YO--DISPLACEMENT VECTORS OBTAINED FROM EXPERIMENTAL DATA MAP11070 C XXlOeYlO--DISPLACEMENT VECTORS OBTAINED BY REGRESSION MAP11080 C TIME--TIME INCREMENT MAPll090 C T--TIME ARRAY MAP11100 C Bl,B2--REGRESSION COEFFICIENTS MAPllllO C ERRORleERROR2--VECTORS OF DIFFERENCES BETWEEN PREDICTED MAP11120 C VALUES AND THE EXPERIMENTAL ONES MAPllUO C REMARKS: MAPlll40 C SUBROUTINES REGRESS AND PLOT ARE REQUIRED MAP11150 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP11160 SUBROUTINE SMOOTHCXX 1 Y,N,TIMEI MAP11170
DIMENSION 81C20,20le82C20e201 MAP11180 DIMENSION TC20leXXC20l,YI20) MAP11190 DIMENSION XXlC201 1 XXlOC201 1 XXOC201 MAP11200 DIMENSION YlC20leY10(201eYOC201 MAP11210 DIMENSION ERROR1120) 1 ERROR2C20) MAP11220 DO 51•1eN MAP112l0 XXlCII•XXCII MAP11240 YlCil•YCII MAP11250 5 CONTINUE MAP 11260
C GENERATE A TIME TABLE MAP11270 TCli•OeO MAP11280 DO 10 1•2 1 N MAP11290
10 TCII•TCI-l,.TIME MAPl1300 C FIT A POLYNOMIAL REGRESSION FUNCTION CFIFTH OROERI TO EACH VECTOR, MAP11310 C THA J IS XX AND Y MAP 11320 CALL REGRSSCT,XX,N,5,l,lll MAP113l0 CALL REGRSSCT,Y,N,S,l,B2l MAP11340 C EVALUATE THE RESULTING REGRESSION FUNCTIONS AT THE PREVIOUSLY MAP11350
C CALCULATED TIME POINTS MAP11360 DO 20 1•1 1 N MAP11370 XXCII•B1C1 1 li•Bll2 1 li•TIIt•B1C3,ll•TCI1••2•8114eli•TCil••J• MAP11380
lllCS,lt•TIIt••4+BlC6,lt•TCII**5 MAPll390 YCII•82Cl 1 lt+82C21 1l•TCII+82Cl,lt•TCit••2+B214eli•TCit••J+ MAPll400
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282C5,lt•TCII••~•BZC6,li•TCII••5 MAPll~lO 20 CONTI!IIUE MAP11420 CALCULATE DISPLACEMENT DATA CTWO ANGLESI AS WELL AS THE ASSOCIATED MAP11430 ERRORS MAP 11440
00 10 l=l,N MAP11450 XXOCit=XXCU-XXCU MAP11460 XXlOCII•XXlCII-XXlCll MAPll470 YOCII•YCII-YCll MAPll480 YlOC lt•YlCI 1-YlC U MAP11490 ERRORliii=XXCII-XXlCII NAP11500
10 ERROR2111=YIIt-YlCII MAP11510 PRINT A TABLE FOR DISPLACEMENTS MAPl1520
WRITEC6ell MAP11530 1 FORMATCl~lt20Xe 1 ANGULAR DISPLACEMENTS---- EXPERIMENTAL DATA 1 1 NAPll540
WRITEC6,21 MAPll550 2 FORMATCl~O,•MEASURED ANGLE U.ARM PREDICTED ANGLE U.ARM ERRORMAP11560
l . MEASURED ANGLE F.ARM PREDICTED ANGLE F.ARM ERRORNAP11510 2' I MAP11580
00 40 l•l,N MAP11590 40 WRITEC6,11 XXlCit,XXCIJ,ERRORlCI1 1 YlllltYIIItERROR2CII MAP11600
1 FORMATe lH0,6F20.1J MAP11610 PLOT DISPLACEMENTS MAP11620
WRITEC6,41 NAP11630 ~ FORMATClHO,'PLOT OF UPPER ARM ANGLE'I MAP11640
WRITEC6,8) MAP11650 8 FORMATCl~O,'MEASUREO ANGLE SYMBOL IS • PREDICTED ANGLE MAP11660
I SYMBOL IS X' I NAP 11670 CAll PLOTCXXlO,XXO,N,OI MAP11680 WRITEC6,61 MAPll690
6 FORMATC1H0 1 1 PLOT OF FOREARM ANGLE'I MAP11700 WRITEC6,81 MAP1171~ CALL PLOTCYlO,YO,N,Ot MAP11720 RETURN MAP11730 END MAP11740
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP11750 SUBROUTINE REGRSS MAPll760
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP117JO
PURPOSE: TO FIT POLYNOMIALS OF SEVERAL DIFFERENT DEGREES TO A GIVEN
OF N DATA POINTS
DESCRIPTION OF PARAMETERS
MAP11710 MAPll 790
SEJMAP 11100 MAPll810 MAP118ZO MAP11810 MAP 11840
N--NUMBER OF DATA POINTS MAP11850 M--OEGREES OF THE POLYNOMIAL TO BE FITTED MAP11860 X,Y--INPUT VECTORS MAPll8JO L--NUMBER OF Y-VALUES WHICH CORRESPOND TO EACH XCIIt IN MOST MAP11880
CASES L•l• MAP11890 a--VECTOR Of REGRESSION COEFFICIENTS MAP11900 A--cOEFFICIENTS MATRIX. MAP11910
REMARKS MAP11920 SUBROUTINE MATEQS IS REQUIRED MAP119l0 •••••••••••••••••••• .. ••••••••••••••••••••••••••••••••••••••••••••••tMAPll940
SUBROUTINE REGRSSCX1 Y1 N,MtLtBJ MAP11950 DIMENSION XCZOJ,YC20,4J MAPll960 DIMENSION AC20tZOitBC20t20),CC20t201 MAP119JO DO lO 1•1 1 N MAP11980
JO CCI 1 11•1.0 MAP11990 .. , 1• ... 1 MAP 12000
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DO 35 J•2,MPl MAP12010 DO 35 l•leN MAPlZOZO 15 CCI,Jt•CCI,J-lJ•XCit MAP12010 DO 40 l•leMPl MAP12040 DO 40 J•l,MPl MAP12050 ACI,JI•O.O MAPl2060 DO 40 K•ltN MAP12070 40 ACI,Jt•ACI,Jt•CCK,It~IK,JI MAP12080 DO 45 J•leL MAP12090 DO 4~ l•l,MPl MAP12100 BCI,Jt•O.O MAP12ll0 DO 45 K•l,N MAP12120 45 BCI,Jl•BCI,Jt•CCK,IttYCK,Jt MAP12130 CALL MATEQSCA,MPltBtL 1 DETt MAP12140 RETUR~ MAP1Zl50 END MAP12160 ttttt•••••••••••••••tttttt•ttt•t•tttt•ttttt•••••••••••••••••••••••••tMAP12170 ' SUBROUTINE MATEQS MAP 12110 ttttttttttttttt•tttttttttttttttttttt•ttttt•••••••-••••••••••••••••tttMAP12190
C PURPOSE I MAP 12200 MAP12210 MAP12220 MAP12Zl0 MAP12240 MAP12250
C TO PERFORM MATRIX INVERSION AND SIMULT LINEAR EQUATIONS. c c c c c c c c c c c c c c
DESCRIPTION OF PARAMETERS:
A--T~E GIVEN COEFFICIENT MATRIX MAP12260 N--ORDER Of A;N>,l MAP12270 I--MATRIX OF CONSTANT VECTOR MAPlZZIO M--THE NUMBER OF COLUMN VECTOR IN THE METRIX OF CONSTANT MAP12290 VECTORS. MAP12300 OET--VALUE OF THE DETERMINANT IAI MAP1Z110 REMARKSI MAP12320 THIS SUBROUTINE IS OBTAINED FROM NUMERICAL METHODS AND MAP12130 COMPUTERS, SHAN S KUO, P.168. MAP12140 ttttttt••ttttt•tttttttttttttttttttttttttttttttttttttt•t••ttttttttttttMAP12150 tttttttttt•ttttttttttttttttttttttttttttttttttttttttttttttttttttttttttMAPl2360 SUBROUTINE MATEQSCA,N,B,M,DETI MAP12370 DIMENSION AC20t20t,8CZ0t20t,IPVOT(20J,INDEXI20t2t,PIVOTCZ0t MAP12310 EQUIVALENCE CIRJW,JROWt,CICOLtJCOLt MAP12390 '' OET•leO MAP12400 DO 17 J•l 1 N MAP12410 17 IPVOTCJt•O.O MAP12420 DO 135 1•1tN MAP12410 T•O.O MAP12440 DO 9 J•1 1 N MAP12450 IFCIPVOTCJI-11 131 91 13 MAP12460 13 00 23 K•ltN MAP12470 IFCIPVOT(Kt-1J 43 1 23 1 11 MAP12480 43 IFCABSCTt-AISCACJtKitt 81t21t23 MAP12490 81 IROW•J MAP12500 ICOL•K MAP12510 T•ACJ 1 Kt MAP12,20 21 CONTINll: MAP12510 9 CONTI~UE MAP12540 IPVOTCICOLt•IPVOTCICOLt•l MAP12550 IFCIRON-ICOLJ 7Je109,73 MAP12560 71 DET•-DET MAP12570 00 12 L•leN MAP12580 T•ACIROW 1 LJ MAP12590 ACI~OW 1LiaACICOLeLJ MAP12600
12 AIICOL,li•T MAP12610 IFf~) 109,109,11 MAP12620
11 00 2 l•l,M MAP12610 T•BC IROWtl I MAP12640 BCIAOW,li•BCICOL,l) MAP12650
2 BCICOL,LJ=T MAPl2660 109 INDEXCI,11•1ROW MAP12670
INOEXCle21=1COl MAP12680 PIYOTCII=ACICOL,ICOlJ MAP12690 DET=DET•PIYOTCII MAP12700 AC ICOl, ICOL 1•1.0 MAP127l0 00 205 L•1,N MAP12720
205 ACICOLtli•ACICOL,l)/PIYOJCII MAP12710 IFCMt llt7,347,66 MAP12740
66 DO 52 l•l,M MAP127SO 52 BCICOL,LI•BCICOL,LI/PIVOJCII MAP12760
147 DO 115 li•1,N MAP12770 IFILI-ICOU 21, 115, 21 MAP12780
21 T•AILI,ICOL) MAP12790 AC ll, ICOl 1•0 •. 0 MAP12800 DO 89 l•1,N MAP12810
89 Allleli•AilleLI-AIICOleLJ•T MAP12820 IFI M I 115,115,18 MAP 12830
18 DO 68 L•l,M MAP12840 68 Blli,Lt•BilleLI-BIICOLtlJ•T MAP12850
115 CONTINUE MAP12860 222 DO 1 1•1,N MAP12870
l=N-1+1 MAP12880 IF IINOEXCL,li-INDEXCL,21J 19,3,19 MAP12890
19 JR OW• INDEX llt 11 MAP 12900 JCOL•INDEXCL,2J MAP12910 DO 549 K•1,N MAP12920 I•AIK,JROWI MAP12930 A(l,JROWI•ACK,JCOLI MAP12940 ACK,JCOLI•T MAP12950
549 CONTINUE MAP12960 3 CONTINI£ MAP 12970
81 REIUkN MAP12980 END MAP12990
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAPllOOO C SUBROUIINE STICK MAP13010 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP13020 C MAP13030 C PURPOSE: MAPll040 C MAP13050 C TO PLOT A STICK DIAGRAM FOR THE +RM DURING THE MOTION MAP13060
MAPUOJO C DESCRIPTION OF PARAMETERS MAP13080 C MAP13090 C A--INPUT VECTOR OF THE UPPER ARM ANGLES MAP13100 C I--INPUT VECTOR OF THE FOREARM ANGLES MAPllllO C N--NUM8ER OF DATA POINTS FOR EACH INPUT VECTOR MAP13120 C X--LENGTH OF THE UPPER ARM MAP13110 C l--LENGTH OF THE FOREARM MAP13140 C XE, lEt XV, IN--X AND l COORDINATES OF THE HAND AND ELBOW MAP13150 C WITH RESPECT TO. SHOULDER JOINT "AP13160 C XMAXl--VECTOR OF THE LARGEST ELEMENTS OF THE ELBOW AND MAP11170 C HAND COORDINATES MAPUliO C X"AX--LARGEST ELEMENT IN VECTOR XMAXl "AP13190 C ILANit DOlt IX, S SY,.BOL--PLOTTING SYMBOLS "AP13200
l.9l
192
C GRAP~-A TWO DIMENSIONAl ARRAY FOR PLOTTING PURPOSES MAP13210 C REfiARICS MAP 13220 C FUITION BIG IS REQUIRED MAP11230 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP13Z•O
SUBROUTI~E STICICCA,B,N,x,zt MAP13250 DIMENSION SY .. BOLC201 MAP13260
C DEFINE PLOTT lNG SYMBOLS MAPU2JO DATA 8LAN~tDOT,XX,S,E,W/1H ,1H.,1HI,1HS,1HE,lHW/ MAP13280 DATA SYMBOL/1H0,1H1,1H2,1H3 1 1H. 1 1H5 1 1H61 1H7 1 1H8,1H91 MAP13290
11H0,1H1,1H2,1H3,1H,,1H5,1H6,1H71 1H8 1 1H9/ MAP1ll00 DIMENSION AC20I,BC201 MAP11310 DI .. ENSION XE1201,ZEI20I,XWC201 1 ZWC201 MAP13320 DIMENSION XMAXlC 201 MAPUHO DIMENSION GRAPHC61,611 MAP1Jl•o
C CALCULATE THE ELBOW AND HAND X AND Z-COOROINATES MAP13350 DO 10 1•1,N MAP13l60 XECit•X•COSIACIII MAP111JO ZE lii•X•SINUC I I I MAP UJIO XWCit•XEIII+Z•CDSIBCIII MAP1ll90 ZWIII•ZEIII+l*SINCBCitt MAP13400
10 CONTINUE MAP13410 C FINO .THE LARGEST VALUE AMONG THE HAND AND ELBOVCOOROINATES MAP13420
XMAXlC21•81GlZE 1 NI MAP13430 XMAX1llt•81GCXE,NI MAP13440 XMAXlC li•BIGCXW 1 NI MAPU450 XMAX1141•BIGIZW,NI MAP13460 XMAX•BIGCXMAX1 1 41 MAPU470
C CLEAR ARRAY FOR GRAPH MAP13480 00 15 1•1,61 MAP13490 DO 15 J•lt61 MAP11500
1~ GRAPHII 1 JI•BLANIC MAP13510 C GENERATE GRAPH AXIS AND FRAME MAP13520
DO 16 J•1t61 MAP11530 GRAPHlltJI•DOT MAP11540 GRAPHll1,Jt•XX MAP11550 GRAPtH61 1 J I•OOT MAPU560
16 CONTINUE MAP13570 DO 17 1•1 1 61 MAP13580 GRAPHll 1 1t•OOT MAP11590 GRAPHI1 1 lli•XX MAP13600 GRAPHII 1 611•00T MAP 13610
17 CONTINUE MAP13620 GRAPHI11 1 31t•S MAP13610 C CONVERT THE COORDINATES TO THE:EQUIVALENT PLOTTING POSITIONS MAP13640 DO 20 1•1.1 N MAPU650 J•IIEIII/XMAX+1.0t•J0.0+1.5 MAP13660 IC•IABSIZECit/XMAX-1.0ittJO.O+l.5 MAP13670 L•CX~Cit/IMAX+1.0t•J0.0+1.5 MAP11680 M-CA8SIZ~IIt/XMAX-l.Otl•l0.0+1.5 MAP13690 GRAPHCK 1 Jt•SYM80LCII MAP13JOO GAAPHIM 1 LI•SYM80LIIt MAP13710
20 CONTINUE MAPlll20 C PLOT GRAPH TITLE AND THE MOTION STICK DIAGRAM MAP11730
~ITEC6 1 21t MAP1Jl40 21 FORMATI1Hlt20Xt27HSTICIC DIAGRAM OF THE MOTIONI MAP11750
DO JO l•lt6l MAP1l760 ~ITEC6 1 22t CGRAPHCI 1 J),J•1 16lt MAPllllO
22 FORMATC lH 1 JX 1 611A1 1 1H II MAPlJliO JO CONTINUE MAP1Jl90
RETURN MAPlliOO
c c c c c c c c c c c c c c c c c c c c c
c
c
END MAP13810 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAPll820
SUBROUTINE PLOT MAPllBlO ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••tMAPll840 PURPOSE: MAPlll50
TO PlOT TWO DEPENDENT VARIABLES VERSUS A COMMON BASE VARIABLE. THE DEPENDENT VARIABLES ARE PRESENTED AT EQUAL INTERVALS OF THE BASE VARIABLE.
DESCRIPTION OF PARAMETERS:
MAP13860 MAP111l0 MAPll880 MAP11890 MAP13900 MAP11910 MAP13920
A--INNPUT VECTOR FOR THE FIRST DEPENDENT VARIABLE MAP13930 I--INPUT VECTOR FOR THE SECOND DEPENDENT VARIABLE MAP11940 TIME--A POSSIBLE BASE VARIABLE MAP11950 ITER--A POSSIBLE BASE VARIABLE MAP11960 GRAPH--A ONE DIMENSIONAL ARRAY FOR THE PLOT MAP139l0 BLANK, DOT, STAR, XX,--PLOTTING SYMBOLS MAP11980 AMAX, AMAXl, BMAX, 8MAX1--LARGEST NUMBERS FOR ARRAYS A AND I MAP13990
RE~ARKSI MAP14000 FUNCTION BIG IS REQUIRED MAP14010 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP14020
SUBROUTINE PLOTtA,&,N,NNI MAP14030 DI~ENSION AlNI,BtNI MAP14040 DIMENSION GRAPHC100) MAP14050 DIMENSION TIMEtl61 MAP14060 OI~ENSION ITERC161 MAP14070
DEFINE PLOTTING SYMBOLS MAP140BO DATA BLANK,OQT,STAR,XX/lH ,lH.,lH•,lHX/ MAP14090 DATA TIME/lHT,lH ,lHI,lH ,lHM,~H tlHEelH el~ elH tlH elH ,lH elH eMAP14lOO
llH 1 1H I MAP14ll0 DATA ITER/lH1 1 1HT 1 1HE,lHR 1 1HAtlHTe1HielH1,1HNelH elHN,lHU,lHM,lHB,MAP14120
11HE 1 1HR/ MAP14130 FINO THE LARGEST NUMBER FOR ARRAYS A AND 8 MAP14l40
AMAX•BIGCA 1 NI MAP14150 BMAX•BIGC8 1 N) MAP14160 AMAXl•-AMAX MAP14ll0 BMAXl•-BMAX MAP141BO
C . PRINT GRAPH SCALE MAP14l90 VRITEC6,2) AMAX1 1 AMAX 1 STAR MAP14200 WRITEC6,2) BMAXltBMAX 1 XX MAP14210
2 FORMATClH0 1 20X,Fl0.5,26X 1 1H0,26X,Fl0.5,2XeAll MAP14220 DO 10 1•1 1 63 MAP14230
10 GRAPHCII•OOT MAP14240 WRITEC6 1 1J IGRAPHIIItl•l,6J) MAP14250
1 FORMATI1H0 1 25X 1 63Al) MAP14260 DO 20 1•21 62 MAP14270
20 GRAPHCII•BLANK MAP14280 GRAPHC321•00T MAP14290
COMPUTE PLOTTING POSITIONS MAP14100 00 40 l•l,N MAP14310 JaCAllt/IAMAX+OeOOOOOlt+l.OI*J0.0+2.5 MAP14120 K•CICIJ/CBMAX+O.OOOOOlt+l.OI*10.0+2.5 MAP14J10
c
GRAPHCJI•STAR MAP14J40 GRAPHCKJ•XX MAP14J50
C ·PIINT GRAPH MAP14360 IFCNN.EQ.Ot WRJTEI61 1J TIMECIJ,CGAAPHILttL•le631 MAP14J70 IFCNN.EO.lJ WRITEC6 1 31 ITERCJt,CGRAPHCLitL•lt61J MAP14110
3 FO~ATClH01 22XtAlt2lt61Alt MAP14J90 00 30 L•2e62 MAP14400
193
10 GRAPHILI•BLANK MAP1~~10
GRAPHI12)•00T MAPl~~ZO
~0 CONTINUE MAP1~~10
DO ~0 1•1,61 MAP1~~40
50 GAAPHIII•OOT MAP14~50
waJTEC6,l) IGRAPHCI1tl•1,61) MAP14~60
RETURN MAP1~4l0
END MAP14~80
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP14490 C FUNCTION BIG MAP1~500
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP145iO C PURPOSE--FINO THE lARGEST NUMBER FOR A ONE DIMENSIONAl ARRAY MAP14520
C MAP1~510
C DESCRIPTION OF PARAMETERS MAP1~540
C MAP14550 C X--AR~AY NAME MAP1~560
C M-NUfiBER OF ElEMENTS IN THE ARRAY MAP141JJO C REMARKS: MAP1~1J80
C NONE MAP1~590
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP14600 FUNCTION BIGU,MI MAP14610 DIMENSION XIMI MAP14620
81G•A8SIXI111 MAP14610 DO 10 1•1tM MAP14640 lffABSCXfiiJ.LT.BIG) GO TO 10 MAP14650
II G-ABS lXIII I MAP14660
10 CONTINUE MAP146JO RETURN MAPl4610 END MAP14690
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP14700 C SUBROUTINE MINTIR MAP1~710
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAPl~l20
C • MAP14ll0
C PURPOSE--FINO THE SMAllEST NUMBER AS WELL AS ITS POSITION MAP14740
C Of ONE DIMENSIONAl ARRAY MAP 14750
C MAP14760
C DESCRIPTION OF PARAMETERS MAP14770
C MAP1~780
C A--ARRAY NAME MAP14790
C N--NUMIER OF ElEMENTS IN THE ARRJY MAP14800
C SMALL--SMALLEST NUMBER MAP1~810
C J--POSITION OF THE SMAllEST NUMBER MAP14820
C REMARKS MAP14810
C NONE MAP14840
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP14150 SUBROUTINE MINTIRCA 1 N1 SMAlltJI MAP14860
DIMENSION ACNI MAP148l0 SMALL•AISCAClll MAP1~880
Jel MAP14890
DO 10 1•2 1 N MAP14900 IFCA8SCAtltt.5T.SMALLI GO TO 10 MAP14910 SMALL•ABSfAtltt MAP14920 J•l MAP149JO
10 CONTINUE MAP14940
RETURN MAP1 .. 50
END MAP14960
C ·••••••••••• .. •••••••••••••••••••••••••••••••••••••••••••••••••••••••*MAP14970 C SUIROUT litE OYNAMC MAP 14~110
C •••••••••••••••••••• .. •••••••• .. •••• .. ••••••••••••••••••••••••••••••tMAP14990 C . MAP15000
194
195
C PURPOSE MAPl5010 C MAPl5020 C TO FINO THE OPTIMAL PATH OF MOTION BY USING DYNAMIC MAP15030 C PROGRA~MING MAP15040 C MAP15050 C DESCRIPTION OF PARAMETERS MAP15060 C MAP1S070 C L12eL2l,M12,M23,Rl2,R2lell2,123--ARM ANTHROPOMETRIC DATA MAP15080 C XO,YO--INITIAL HAND COORDINATES MAP15090 C TOR2--INITIAL ELBOW MOMENT MAP15100 C SZF2--INITIAL ELBOW FORCE MAP15ll0 C ITIME--MOTION TIME UOO MAPl5120 C VEL--ANGULAR VELOCITY VECTOR FOR THE SHOULDER MAP151JO C YELl--ANGULAR VELOCITY VECTOR FOR THE ARM MAP15140 C fORINT--INITIAL SHOULDER MOMENT MAP15l50 C TOR--MOMENT VECTOR FOR THE SHOULDER MAP15160 C FCT--COST VECTOR IPOWER) MAP15170 C KT--T IME INDEX MAP 15180 C K--STAGE INDEX MAP15190 C X1,X2--HAND X-COORDINATES AT STAGES K AND K+1 MAP15200 C J--Z--COOROINATE INDEX AT STAGE K+1 . MAP15210 C YZ1-~RATIO OF HAND I-COORDINATE TO TOTAL MOTION DISTANCE MAP15220 C AT STAGE KH MAPU210 C I--INDEX OF I-COORDINATE AT STAGE K MAP15Z40 C Y2--HANO Z-COORD INA TE AT STAGE K+l MAP15Z50 C Yll--RATIO OF HAND Z-COOROINATE TO TOTAL MOTION DISTANCE MAPl5260 C AT STAGE K MAP152JO C Yl--HAND z--COORDJNATE AT STAGE K MAP15280 C A,B--ARM EULER'S ANGLES AT STAGE K MAPU290 C t,D--ARM EULER'S ANGLES AT STAGE K+l MAPl5300 C ANGV--ANGULAR VELOCITY VECTOR FOR THE SHOULDER MAP15110 C ANGV1--ANGULAR VELOCITY VECTOR FOR THE ELBOW MAP15320 C XA2,ZA2--COMPONENTS OF ELBOW LINEAR ACCELERATION MAP15330 C XGA3,ZGA!--COMPONENTS Of FOREARM C.G. LINEAR ACCELERATION MAP15340 C YMOM2--ELBOW MOMENT MAP15350 C XF2elF2--COMPONENTS OF ELBOW REACTIVE FORCE MAP15360 C POWER--TOTAL POWER COMPUTED AT THE SHOULDER MAP153JO C NODE--VECTOR OF GRID POSITIONS ALONG THE L-AXIS MAP15180 C XOPT,YOPT--HAND COORDINATES FOR THE OPTIMAL PATH OF MOTION MAP15390 C FCTOPT--COST VECTOR IPOWERI FOR THE OPTIMAL PATH OF MOTION MAP15400 C VELOPT--SHOULDER ANGULAR VELOCITY VECTOR FOR THE OPTIMAL PATH MAP15410 C OF MOTION MAP15420 C TDROPT--SHOULDER MOMENT VECTOR FOR THE OPTIMAL PATH OF MOTION MAP15430 C XC,ZC--X AND Z COORDINATES OF THE SMOOTHED OPTIMAL PATH OF MAP15440 C ttOTI ON "" 15450 C REMARKS MAP15460 C THE FOLLOWING SUBROUTINES ARE REQUIRED MAP154JO C -NEWTON MAP15480 C -MINIM MAP15490 C -REGRSS MAP15500 c ....................................................................... AP15510
SUBROUTINE DYNAMCI XL, YL, IM, YM, XI, YI,Rl2eR231eXMAX,I TIME ,ANG1 ,ANG2, MAP15520
lWEIGHT,XCelCeNM,TANGJ MAPL5530 DIMENSION ICI20J,ZCI20t,TC201eWC20e201 MAPL5540 DIMENSION FCTIZ01 201 1 TORI20 1 20I,NODEC20,201 MAP15550 DIMENSION POWERC20eZOe20I,VEllC20t201tVELI20t201 MAPl5560 DIMENSION FCTOPTI20J 1 POINTC20t,XOPTC20J,YOPTC20J,TIMEC201 MAP15570 DIMENSION ANGVC201 1 ANGV3120) 1 YMOMC201 1 VELDPTI20t,TOROPTI201 MAP15580 DIMENSION ZCRGI20 1 101 MAP15590 DIMENSION YMOMEC20JeTOREC20e201 MAPL5600
M E AL L 12 , L 2 3 , M 12 , M 2 3 , l 12 , l 2 3 REAL KT
C READ GRID PARAMETERS REAOC5,11 CELTA,Y211NT,Y211NC,Yl1lNT 1 Y111NC FOR,.ATC5f5.21 OEL=OEL TA/100.0 L12= XL L2l=YL 11'1l=XM "'2l=YM I 12= X I 121=Y I A=At1G1•0.0175 B=ANG2•0.0175
C COMPUTE INITIAL COORDINATES Of THE HAND XO=L12•COSIAI+L23•COSCBI YO=Ll2•SlNCAI+L23•Sit1C8J
C COMPUTE MOMENT AND REACTI~E FORCE AT THE INITIAL POINT JOR2=-M23•9.8•R23•COSlBI-WElGHT•L23•COSIBI SFZ2=-WE IGHT-M23*9. 8 TURINT=TOR2-Ml2•9.8•R12•COSIAI+SFZZ•L12•COSCAI TIJ'l=ITIIIE TII"=JI,.l/100.0
C INITIALIZE VALUES FOR STAGE 1 DO 5 I= 1, 2C VELil,II=O.O VElll 1, 11=0. 0 lURE I 1, II-=TOR2 TORiltii=TORINT
o; fC H 1, II =999999. 9 FCTC l, 11=0.0 KT=O
C SET TII"E INDEX EQUAL ZERO K=O WRITEC6,6311
631 FORII'ATI1~1 1 ' •••••• STAGES SUMMARY •••••••••••'I 1C JIMl=KT
K-=K+l TIM2=T IMl/100.0 Tl MJ= Tl 11'2 +DEL TERM1=6.28*TIM2/TIM TERM2=TERM1-SINCTERM11 TER113=6.28•TIM3/TIM TERM4=TERMl-SINCTERMJI
C COMPUTE X-COORDINATE Of THE HAND AT STAGE K AND 1<+1 X1=XMAX•TERM2/6.28+XO X2=XMAX•TERM4/6.28+XO XOPTIKI=K1 Tl ME I K I= T I ,.2 J.: 0
C INITIALIZE I-COORDINATE Of THE HAND AT STAGE K+1 Y21=Y211NT
20 J=J+1 Y2l=Y21+Y211NC Y2=Y2l•XMAK/100.0+YO+XZ*TANITANGI 1=0
C INITIALIZE Z-COOROINATE OF THE HAND AT STAGE K Y ll=Y lliNJ
30 1=1+1 Y 1 1 = Y 1 1 + Y 1 11 NC
l'lAPl5610 MAP1";620 MAP1'>630 MAP15640 MAP15650 MAP15660 MAP15670 MAP15680 MAP15690 MAP 15700 MAP151l0 MAP15720 MAP1'H30 MAP15740 MAP15750 MAP1'>760 MAP 15770 MAP15780 MAP15790 MAP 15800 MAP15810 MAP15820 MAP15830 MAPl5840 MAP15850 MAP 15860 MAP15870 MAP15880 MAP15890 MAP 15900 MAP15910 MAP15920 MAP15930 MAP15940 MAP15950 MAP1";960 MAPl5970 MAP15980 MAP15990 MAP16000 MAP160l0 MAP16020 MAP16030 MAP16040 MAP16050 MAP16060 MAP16070 MAP16080 MAP16090 MAP16100 MAP16110 MAPl6120 MAP16130 MAP16140 MAP16150 MAP16160 MAP16170 MAP16180 MAP16190 MAPl6200
196
·. c
c
c
c
c
c
c c
c
Yl=Yll•XMAX/lOO.O+YO+Xl•TANITANGI CALL NE~TON TO COMPUTE EULER'S ANGLES
CALL ~EWTONCL12,L21,Xl,Yl,A,81 C•A D•B
CALL NEWTON TO COMPUTE EULER'S ANGLES CALL NEWTONCLl2eL21,X2,Y2,C 1 DI AC=C
FOR THE ARM AT STAGE K
fOR THE ARM AT STAGE K+l
MAP16210 MAP 16220 MAP16210 MAP162't0 MAP l62'l0 MAPlb260 MAPlb270 MAP16280
80:0 MAP16290 COMPUTE ANGULAR VELOCITIES AND ACCELERATIONS MAP16300
ANGYl•CC-AI/OEL MAPlbllO ANGVCII=ANGVl MAP16320 ANGV2•1D-81/0EL MAP16130 ANGV3CII•ANGY2 MAP163it0 ANGAl•CANGVl-YELCK,III/DEL MAP163SO ANGA2= UNGV2-YELU K, I I I /DEL MAP16360
CO~PUTE LINEAR ACCELERATIONS MAP16370 XA z .. -uz·•sl NCACI *ANGAl-ll2•COS CAt I • CANGV1 .. 2 I MAP 16180 ZA2•Ll2•COSCACI*ANGAl-Ll2*SINCAti•CANGV1•*21 MAP16390 XGA2•Rl2•XA2/ll2 . MAP16it00 ZGA2•Rl2*ZA2/Ll2 MAP16itl0 XG~~·~A2-R23*SINCBDI•ANGA2-R2l•COSCBDI•CANGV2••zt MAP16420 ZGA3•ZA2+R23•COSCBDI*ANGA2-R23*SINCBDI•CANGY2*•21 MAP16430
COMPUTE ELBOW MOMENT AND REAC Tl VE FORCE COMPONENTS . MAP16440 YMOM2•12l•ANGA2-M23*R23•SINCBDI•XGA3+M23•R23*COSCBDI*ZGA3-M23*9·8*MAPl6450
1R21•COSCBOI-WEIGHT•l23*COSCBOI MAP16460 YMOMEC I I•YMOMZ MAPl6470 XF2•M23*XGA3 MAP16480 ZF2•-WEIGHT-M23*9.8+M2l•ZGA3 MAP16490
COMPUTE SHOULDER MOMENT MAPl6500 YMOHl•YMOM2+112•ANGAl-Ml2•Rl2•SINCAC)•XGA2+Ml2•Rl2*COSCACI•ZGA2 MAP16Sl0
2-Ml2•9.8*Rl2*COSCACI-XF2•Ll2*SINCACI+ZF2•Ll2•COSCAC) MAP16520 YMOM C I I•YMOH l MAP16530
COMPUTE TOTAL POWER EXPENDED FOR MOVING FROM THE INITIAL POINT TO MAP16540 STAGE K+l AND FOLLOWING LIND IJ BETWEEN STAGE K AND K+l HAP16550
POWERCK+ltJtii•FCTCKtii+0.5•CABSCTORCK,I))+ABSIYMOHllt*ABSCANGYl MAP16560 1-VELCK,IIt MAP16570 IFII.LT.201 GO TO 10 MAP161580 KK•K+l MAP16590
FIND LINK IJ WHICH GIVES THE MINIMUM TOTAL POWER MAP16600 CALL HINIMCPOWER,KK,J,I,SMALLelll HAP16610 FCTCK+l,JI•ABSCSHALL) MAP16620 NODECK+leJt•ll MAP16630 YELCK+l,Jt•ANGVCIIt HAP16640 VEL1CK+l,JI•ANGY3CIII MAP16650 TOREIK+ltJI•YMOMECIII MAPl6660 TORCK+l,JI•YMOHCIII MAP16670 WRITEC6e21 KKeJeSMALL,VELCKK,JJ,TORCKK,J),K,II MAP16680
2 fORMATClH0,216elF20.5,2161 HAPl6690 IFCJ.LT.201 GO TO 20 MAP16700 KT•KT+OELTA MAP16710 lXTIME•ITIME MAP16720 IFCKT.LT.XXTIMEI GO TO 10 MAP16710 TIMECK+li•TIM MAP16740 XOPTCK+li•XMAX+XO MAP16750 YOPTIK+li•YO MAP16760 JOR.OPTU+ia•TORCK+l,ll MAP16770 FCTOPTCK+l)•FCTCK+l,ll RAP16780 POINTCK•li•O.O MAP16790 YELCPTCK•lt•O.O MAP16800
197
C TRACE THE OPTIMAL PATH OF MOTION MAP16810
Kl=K+1 pi!APl6820
J1 =1 MAP 16830 100 J1~NODECK1,J1J MAP168~0
FCTOPT CIC.1-1 J•FCTC K 1-l.J 1 l MAP168SO
XJ1•J1 MAP16860 POINTCK1-11•XJ1•1.0-1.0 MAP16870
VELOPTCK1-1J•YELCK1-1,J1J MAP16880 TOROPTCIC.1-li•TORCIC.l-1,J1J MAP16890
YOPTIK1-1J=X~AX•POINTCK1-1J/100.0+YO MAP16900
K1=1C.1-1 MAP16910 IFCK1.GT.1J GO TO 100 MAP16920 YOPTC1J=YO MAPl69lO
FCTOPT11J•O.O MAP16940
POINTC11=0.0 MAP169SO TOROPTCli•TORINT MAP16960
KM=IC.+1 MAP16970
DO 90 1•1,KM MAP16980
90 WRITEI6,1) TIMECIJ,XOPTCIJ,YOPTCil,POINTCIJ,VE~OPTCIJ,IOROPTCIJ, MAP16990 lFCTOPTCIJ MAP17000
1 FORMA TC 1HO ,7F 15. 5) MAP 17010
C SMOOTH THE HAND PATH OF MOTION BY USI~G REGRESSION ANALYSIS MAP17020 KM~1=KM-1 MAP17030
CALL REGRSSCXDPT,YOPT,KM,IC.MM1,1,WJ MAP170~0
NM=ITIME/10+1 MAP170SO
TCli=O.O MAP17060
DO 50 1•2,N~ MAP17070
SO TC II•TCI-11+0.10 MAPl1080
00 70 1•1,NM MAP17090
TER1•6.28•H It/TIM MAP17100 TER2•TER1-SI NCTERlJ MAP 17110
XCC I J=XMAUTER216.28+XO MAP11l20
ZCRGCI,lt=WI1,1J MAP17130 00 77 Ls2,KM MAP17140
77 ZCRGft,ll:aZCRGC I,L-lt+IWIL,lJ•XCflJ••CL-111 MAP171SO
ZCCIJ=ZCRGC I,IC.MJ MAP11160
70 CONTI ~I£ MAP 17170
ZCCNMJ=YOPTCKMJ MAP11l80 ZCC1t=YOPTC1J MAP17190 RETURN MAP17200 END MAP 11210
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17220 C SUBROUTINE MINIM MAP17230
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17240 C MAP172SO
C PURPOSE MAP17260 C MAP17270
C TO FINO THE SMALLEST NUMBER FOR THREE DIMENSIONAL ARRAY MAP17280
C MAP17290
C DESCRIPTION OF PARAMETERS MAP17300 C MAP17110
C APOW.-ARRAY NAME MA~l7120
C K,J 1 11 -ARRAY INDICES MAP17330
C SMALL-SMAllEST NUMBER MAP 111~0
C II--POSITION OF THE SMALLEST NUMBER MAP17l50
C REMARKS MAP17360
C NONE MAP17l70
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17l80
SUBROUTINE MINIMCAPOWtKtJtleSMALLtllt MAP17l90 DIMENSION APOWCZO,Z0,201 MAPll~OO
198
SIIIALL•APOWIKeJell MAP11410 11•1 MAP17420 00 10 ll•1tl MAPllltlO IFIAPOWIK,J,Ilt.GT.SMAlll GO TO 10 MAP17440 SMALL•APOWIK,J,I U MAP11450 11•11 MAPll460
10 CONTINUE MAP17470 RETURN MAP17480 ENO MAP17490 .
C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17500 C SUBROUTINE NEWTON MAPI7510 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17S20 C MAP17510 C PURPOSE MAP17S40 C MAP17550 C TD SOLVE TWO NONLINEAR EQUATIONS MAP17S60 C MAPl 1570 C REMARKS MAP17580 C MAP11S90 C All THE PARAMETERS HAVE BEEN EXPLAINED BEFORE FOR SUBROUTINE MAP17600 C SOLVER MAP17610 C •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAP17620
SUBROUTINE NEWTONIXL12,Xl21,Cl,C2,THETA1,THETA2t MAP17630 ~f~l ll2tl2l · MAP17640 EPS•O.OOOI MAPI76SO ll2•XL12 MAP17660 l2l•Xl21 MAP17670 XIT•THETAl MAP17680 Yll•THETA2 MAP17690 DO 20 J•1,50 MAP11700 FX•-LIZ•SINCXITI MAPllllO FY•-l21*SINIYIU MAP 17120 GX•ll2*CO SIX IT l MAP 17710 GY•l2l*COS IY IT I MAP11740 F•l12*COSCXITl•L21*COSfYIT)-C1 MAP17750 Gall2*SINCXITt•L2l*SINCYill-t2 MAPlll60 OIV•FX*GY-GX*FY+O.OOOOOOl MAP17770 XlTl•XIT+CG•FY-F*GYl/OIV MAP17780 YITl•YJT+Cf*GX-G•FXt/OIV MAP1Tl90 Zl•AISCFI MAP17800 Z2•AIS1Gl MAP17810 IFCll.LE.EPS.ANO.l2.LE.EP$) GO TO 25 MAP17820 XIT•XITl MAPl7830 YIT•YITl MAP17140
20 CONTINUE MAP17850 25 THETAl•XITl MAP17860
THETA2•YIT1 MAPlliJO IETUR~ MAPlTIIO END MAP17890
NUMIEI OF CAIOS IS 1791
MAPI 1900 MAPIJCI10
199
PROGRAM FLOW CHARTS:
FLOW CHART--MAIN PROGRAM
2
Start Main Program
DIMENSION Variables
~--------~READ Problem Title
READ Subject Code-
NO
1
200
2 y s
1
READ and PRINT Subject's Anthro
pometric Data
READ Motion Parameters
PRINT Motion Parameters
KOUNT=l
11
1-----c:
201
11
READ Control Numbers
Initialize Iteration Values
3 YES
READ Parabola Parameters ~e--Y_E_s_~
READ Grid YES Parameters ------1
NO
NO
202
./
4
10 >-----~POINT = POINT + DELTA
Initialize PERCENT and Pl
30 >---------1 Compute H
40>-------------~ P1 = P1 + 5o0 P2 = P2N
P2 = P2 + 5o0 50>--------1 P3- P3N ...__,
6 )'-------...1 P4 = P4 + 5o0 P5 = P5N
,, 12
203
204
12
70 P5 = P5 + 5o0
20 YES 1------<
NO
WRITE KOUNT and PERCNT 1--...... AW..-<
NO
CALL SOLVER
PERCNT=PERMAX YES
POINT=lOO ._.._YE_s_~
3
10
PERCENT=PERMAX POINT=lOOoO
,, \READ N 1
\READ Armj \Angles
,, CALL SMOOTH
CHART
NUM=NUM + 1
Set Iteration Parameters to their Maximum
Limits
,, 15
205
Initialize Angular Velocity and
Acceleration Arrays
,, Compute Velocities
and .Accelerations Angular as well as Linear
.,, Compute Reactive
Forces and Moments
,, Compute .Performance
Functions
.,, 16
206
5
\ I
6
6
16
Print and Plot Dynamic Data for Different
Joints
KOUNT=KOUNT+l
17
207
208
17
18
...
" ,., ,r
'
80
209
18
PERCNT=PERCNT+5o0
~-----t10
IEND=KOUNT-1
r
11
19
Print Iteration Summary
CALL MINTTR
Obtain Parameters for Optimum Motion
CALL SOLVER
NUM=O
21
210
10
211
21
GO TO 100
CALL EXIT
FLOW CHART SOLVER
3 YES
DIMENSION Variables
DATA Variables
Initialize Arm Angles
1
\
212
CALL PARAB CHART
CALL SIMUL
YES
1
Compute Initial Hand Coordinates
NO
CHART ~--------~
DO 10 K=5,ITIME1,5
Compute Hand X-Coordinate By Using Displacement
Function
I 2
213
/
Use Sine Function to Compute z-coordinate
Use Ellipse Function ZCoordinate
Use Parabola Function to Compute Z-Coordo
Use Enumeration Aproach to Compute Z-Coordo
YES
YES
YES
YES
214
2
CONTINUE
3
2
RETURN YES
3
READ Hand X and Z Coordinates
Initialize Arrays
/ I
Compute Arm Angles For All Time Points
By Using Newton Iteration Scheme
4
215
4
Print table of Angles During
Motion
RETURN
216
FLOW CHART PARAB
DIMENSION Variables
Define X and Y Coordinates for three points of
the Parabola
CALL REGRSS
RETURN
217
FLOW CHART SIMUL
DIMENSION Variables
Define X and Y Coordinates for
The Grid
CALL REGRSS
RETURN
I
218
FLOW CHART SMOOTH
Start SMOOTH
DIMENSION Variables
Save X and Y Arrays into XXl and
Yl
Generate Time Table
CALL REGRSS
~,
1
219
1
Compute Smoothed Values as well as
The Associated Residual Errors
Displacement Table
CALL PLOT
RETURN
220
FLOW CHART STICK
Start STICK)
,, DIMENSION Variables
DATA Variables
Compute X and Z Coordinates for Hand as well as
Elbow
CALL BIG
~,
1
221
1
Clear GRAPH Array
Define GRAPH Axes and Frame
Compute plotting Positions for the
motion
Print GRAPH with title
RETURN
222
FLOW CHART PLOT
Start PLOT
DIMENSION Variables
DATA ·Variables .__......., __ _,~.-:
CALL BIG
Print GRAPH Scales
1
223
Print GRAPH
DO 40 I=l,N
Compute Plotting Positions For Graph
with Time ~Y~E~S~----, as a Base Variable
Print GRAPH with Itera-L-~~----~ tion Nurno as a Base Variable
NO
NO
3
224
2
3
~--------~ CONTINUE
Compute and Print GRAPH
and Line
RETURN
225
\
FLOW CHART BIG
2
2 YES
('
Start BIG
DIMENSION Variables
Initialize BIG
DO 10 I•2,N
BIG•ABS(X(I))
l
226
227
1
CONTINUE
FLOW CHART DYNMC
10
( Start DYNMC)
DIMENSION Variables
\ READ grid/
Parameters
Compute forces and moments at initial
point
Initialize Stage Parameters
TIMl=KT
Compute Hand X Coordinate at stages
K and K+l
~,
1
228
20
30
1
Initialize Z coordinate at Stage K+l
~----------~ J=J+l
Compute Z coordinate at Stage K+l
,, Initialize Z coordinate
at Stage K i
:)..----------! I= I+ 1 I ,,
Compute .z .coordinate at Stage K
CALL NEWTON
,, CALL NEWTON
~· 2
229
2
Compute velocities, accelerations, forces,
and moments
Compute Total Power
YES 30 ....,__ ____ .__.~
CALL MINIM Chart
save Stage Parameters
Print Stage summary
3
230
3
KT=KT+DELTA
10 1-----<
Initialize Parameters For Optimal Path
Trace the optimal· path of motion
231
··'
I
4
CALL REGRSS
Obtain the smoothest coordinates for the
path of motion
232
FLOW CHART MIN·TIR
Start MINTIR
DIMENSION Variable
Initialize SMALL
1 ~--------. DO 10 I=2, N
1 YES
NO
SMALL=ABS(4(I))
RETURN
233
SAMPLE OUTPUT
234
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~TATIC ANALYSIS
X-FURCE 1-FORt:E ~ESULJA~T-FO~CE MOMENT ~E WTU!'4 NE~TCt-. NEWTON NE:WTON-Ml:Tt:R
o. n -10.q4660 10.94660 -0.49711)4
u.o -10.946()0 10.94660 -0.47692
o.o -10 •. ..,466\.l 10.94660 -0.46414
o.o -10.94660 10.94660 -0.61496
o.u -10.94f\'>u 10.94660 -0.84767
o.u -10.94660 10.'14660 -1.08361
o.o -10.')4660 10.94660 -1.27440
o.o -10.941,60 10.94660 -1.35723
o.o -10.94660 10.94660 -1.'36844
DYNAMIC A~ALYSIS
ANGULAR OISPL. A"4GlJLAR VEL. A~GULAR ACCEL. X-ACCEL. Z-ACCEL. RADIANS RAO/SEC ' RAO/SEC/SEC METER/SEC/SEC METE~/ SEC/SEC
1.20157 o.o o.o o.o o.o
1.294H4 C.l9R41 -1.87340 7.48982 4.315'12
1.10242 -O.H103S -19.27727 -2.11771 0.12924
1.21180 -2.17161 -22.37358 1. 324 35 2.82404
1.0~t'lll) -1. 08'l77 -6.19240 -0.23456 2.11)67011)
0.90122 -l.o9<Ha;, 5.6'3278 -lt.26782 -3.41255
0. 7552 R -2.1'1774 30.44739 -2.44384 -3.04548
0.68145 -o. 82042 24.6411)61 -2.17607 -4.03627
0.67124 o.o o.o o.o o.o
X-FORCE l-FORCE RESULTANT-FORCE MOMENT NEWTON NEWTON NEWTON NEWTON-METER
. o.o -10.94660 10.9lt660 -0.49754
H.6R615 -6.22165 10.68565 -2.3549'3
4.36596 -12.78121 13.50633 -16.88528
lt.86770 -10.11666 11.24484 -10.22445
-o.l1795 -10.1027'1 10.10'348 -3.23637
-6.61961 -15.1t941t9 16.84930 1.54472
-7.081i68 -10.98034 13.06808 11.64576
-5.30157 -12.11180 13.22758 8.86581
o.o -10.94660 10.94660 -1.36841t
241
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