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This is to certify that the
thesis entitled
Development of a Data Smoothing Software Package
with Application to Smoothing Technique Comparison

presented by

Daniel J. Wilson

has been accepted towards fulfillment
of the requirements for

M.A. degree in Physical Educ.

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czhlmMuna-pd

 

 

 

 

 

 

DEVELOPMENT OF A DATA SMOOTHING CONIPUTER SOFTWARE PACKAGE
WITH APPLICATION TO SMOOTHING TECHNIQUE CONIPARISON

By

Daniel James Wilson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

MASTER OF ARTS

School of Health Education,

Counseling Psychology and Human Performance

1989

@050 505

ABSTRACT

DEVELOPMENT OF A DATA SMOOTHING COMPUTER SOFTWARE
PACKAGE WITH APPLICATION TO SMOOTHING TECHNIQUE CONIPARISON

By

Daniel James Wilson

The purpose of this investigation was to create a flexible computer software package
to assist the researcher of human motion in selecting and implementing appropriate data
smoothing techniques. This was accomplished by including a tutorial in the package to
instruct the researcher about the proper smoothing routine for each major classification of
human motion.

Validation of the smoothing routines was accomplished by calculating the mean
square error associated with differences in data values between raw and smooth numeric
data. All smoothing routines written for this investigation were found to be acceptable at a
pre-determined level of ninety-five percent mean square agreement.

The smoothing routines were compared and quantitatively ranked based upon their
mean square error. The routines which consistently produced the best results, in
descending order were: ( 1) natural cubic spline function, (2) Chebyshev polynomial, (3)
Butterworth filter, (4) least squares polynomial, and (5) fourier series. It was noted that the
natural cubic spline function produced no error.

 

m

DEDICATION

TOMYPARENTS

 

ACKNOWLEDGMENTS

While attempting to write this acknowledgment, I find the task of identifying all those
individuals who have offered their support overwhelming. So, for those not specifically mentioned
here, I offer my deepest appreciation if they should find themselves reading this manuscript.

The following individuals must be acknowledged both for their invaluable assistance in the
completion of this project, and for demonstrating that friendship and caring are always the best
teachers.

I must begin by expressing my sincere appreciation to the man who has guided my
graduate career, my committee chair, advisor, and friend, Dr. Eugene W. Brown. As always, I
must thank my long time mentor and friend, Dr. Douglas E. Hanson, who continually provides the
role model which guides so many toward achieving their goals. I offer a special thank-you to my
committee member and friend, Dr. Tyler Haynes, who shows that a love of teaching comes from a
love of students. Also, I feel I must express my gratitude to Dr. Vern Seefeldt, Dr. John
Haubenstricker and Dr. Rebecca Henry for providing me with the research assistantships which
allowed me to pursue my graduate education. Finally, thank-you to Mr. Kaveh Abani for his
assistance in the interpretation of the mathematical theory which builds the foundation of this
thesis.

TABLE OF CONTENTS

Chapter Page
LIST OF TABLES ...................................................... viii
LIST OF FIGURES .................................................... ix
1. INTRODUCTION ...................................................... 1
Statement of the Problem ......................................... 2
Purpose of the Study .............................................. 3
Need for the Study ................................................. 4
Delimitations of the Study ........................................ 5
Limitations of the Study .......................................... 5
Definitions of Terms ............................................... 6
II. REVIEW OF RELATED LITERATURE ............................ 8
An Assessment of Selected Data Smoothing Techniqes ....... 8
Least squares approximation ................................ 8
Cubic spline functions ........................................ 9
Fourier series .................................................. 10
Chebyshev polynomials ...................................... 10
Butterworth filter .............................................. 1 1
Classification System for Motion Studies ....................... 12
The Past and Present of Computer Software and
Future Considerations for Development ........................ 15

III. DEVELOPMENTAL METHODOLOGY AND

EXPERIMENTAL PROCEDURES .................................. 17
Criterion for Inclusion ............................................. 17
Structure of the Program .......................................... 19
Developmental Procedures ....................................... 20

Least squares approximation ................................. 20
Spline functions ............................................... 22

vi

Chebyshev polynomials ......................................
Fourier series ..................................................
Butterworth filter ..............................................
Validation of the Program .........................................
Comparison of Technques ........................................
Use of the Computer Program ...................................
Summary ............................................................

RESULTS AND DISCUSSION ......................................

Data Smoothing Information Questionnaire .....................
Comparison of Smoothing Methods .............................
Validation .......................................................
Comparison of techniques ...................................
Using the software package ................................

Summary .......................................................
CONCLUSIONS .......................................................
Summary ............................................................
Findings ............................................................
Implementation ....................................................
Recommendations for Further Research and Development . . .
REFERENCES ..........................................................
GENERAL REFERENCES ...........................................
APPENDIX A Data Smoothing Questionnaire ....................

APPENDD( B Numeric Data ........................................

APPENDIX C Data Smoothing Software Package ...............

vii

23
23
24
24
25
26
26

27

27
3O
30
31
35
35

40
40
4o
41
42
43
45
47
so

56

 

LIST OF TABLES

Table Page
1 Literature Review Frequencies by Motion
Classification ............................................................. 14
2 Summary of Data Smoothing Information Questionnaire ........... 28
3 Mean Square Difference for Exact Numeric Data ................... 32

4 Data Smoothing Routine Ranking by Motion
Classification in Descending Order .................................... 33

viii

Figure

LIST OF FIGURES

Page

The Trigonometric Nature of Repeating or

Oscillatory Motion ..................................................... 12
Digital Filter Black—Box Design ...................................... 13
Block Diagram of Motion Classifications ........................... 15
Flow Chart of Developmental Procedures in Creating

a Software Package .................................................... 21
Main Menu of Data Smoothing Software Package ................. 36
Tutorial Menu Screen .................................................. 36
Tutorial Screen from Least Squares Approximation ............... 37
Main Screen for Inputting Data from Disk .......................... 37
Entering Drive and Filenarne for Data Retrieval from Disk ........ 38
Main Menu for Data Smoothing Techniques ........................ 38
Smoothed Data Curves from Various Data Smoothing

Techniques .............................................................. 39

CHAPTER I
INTRODUCTION

The biomechanical analysis of human motion is a complicated and time
consuming task. Our predecessors, uu'lizing qualitative observation techniques, realized
the complexity of analyzing human motion and sought to develop quantitative
techniques (Bates, 1973; Hoffman and Warsham, 1971; and Ward, 1971). These
quantitative methods were developed to help solve the problem of reducing human
movement to a speed which could be effectively analyzed. Human movement often
occurs far too rapidly to be observed and accurately analyzed with the naked eye. This is
especially true in sports skills in which the goal of performance is to produce maximal
velocity, acceleration, and/or force.

With the advent of the high speed motion picture camera, technology gave the
researcher a new tool to assist in capturing movement. The development of this
electronic "eye" to assist researchers derives its origin from the early sequential still
photographs taken by Muybridge in order to make general statements as to the nature of
human and animal motion. The plates presented in his book, Animgs in Motion
(c1899), include brief descriptions of body segment reactions to various every-day
tasks. From this beginning, the high speed motion picture camera has evolved into an
invaluable tool for the collection of film data used in biomechanical research.

With the development of the high speed motion picture camera, a powerful new
device for the investigator of human motion, also came new problems in the analysis
and interpretation of data. Two major complications faced the investigator.

The first problem came in the form of overwhelming amounts of data which had
to be organized and interpreted. Recorded images collected by means of high speed
motion picture photography needed to be converted into cartesian coordinate values. The
advent of electronic digitzers, instruments which could electronically convert points
located on projected film images into coordinate data, effectively bridged the gap
between the acquisition of data and the subsequent transformation of that data to a
usable electronic form. From film taken at one hundred frames per second, a typical rate
used by researchers who analyze sports skills and physical activities, three-thousand or
more body segment coordinate pairs could be generated in the analysis of just ten
seconds of filming. The investigation of movement through high speed motion picture
techniques was made practical by the invention of the digital computer. The ability of the
digital computer to store, retrieve, display, and analyze vast amounts of data made it an
invaluable tool in biomechanical research. The impact of the computer on data
processing in general is best stated by Blakesley:

CHAPTER I
INTRODUCTION

The biomechanical analysis of human motion is a complicated and time
consuming task. Our predecessors, utilizing qualitative observation techniques, realized
the complexity of analyzing human motion and sought to develop quantitative
techniques (Bates, 1973; Hoffman and Warsharn, 1971; and Ward, 1971). These
quantitative methods were developed to help solve the problem of reducing human
movement to a speed which could be effectively analyzed. Human movement often
occurs far too rapidly to be observed and accurately analyzed with the naked eye. This is
especially true in sports skills in which the goal of performance is to produce maximal
velocity, acceleration, and/or force. ‘

With the advent of the high speed motion picture camera, technology gave the
researcher a new tool to assist in capturing movement. The development of this
electronic "eye" to assist researchers derives its origin from the early sequential still
photographs taken by Muybridge in order to make general statements as to the nature of
human and animal motion. The plates presented in his book, Animals in Motion
(c1899), include brief descriptions of body segment reactions to various every-day
tasks. From this beginning, the high speed motion picture camera has evolved into an
invaluable tool for the collection of film data used in biomechanical research.

With the development of the high speed motion picture camera, a powerful new
device for the investigator of human motion, also came new problems in the analysis
and interpretation of data. Two major complications faced the investigator.

The first problem came in the form of overwhelming amounts of data which had
to be organized and interpreted. Recorded images collected by means of high speed
motion picture photography needed to be converted into cartesian coordinate values. The
advent of electronic digitzers, instruments which could electronically convert points
located on projected film images into coordinate data, effectively bridged the gap
between the acquisition of data and the subsequent transformation of that data to a
usable electronic form. From film taken at one hundred frames per second, a typical rate
used by researchers who analyze sports skills and physical activities, three-thousand or
more body segment coordinate pairs could be generated in the analysis of just ten
seconds of filming. The investigation of movement through high speed motion picture
techniques was made practical by the invention of the digital computer. The ability of the
digital computer to store, retrieve, display, and analyze vast amounts of data made it an
invaluable tool in biomechanical research. The impact of the computer on data

Processing in general is best stated by Blakesley:

 

"In the broadest and simplest concepts, computers have their greatest value in
extending man 's mental capacities in two interrelated areas. Computers are
useful where: (1) significant time- c-onsuming, repetitious, and voluminous
operations of data and facts exist; and (2) where not so simple, but significantly
more complex calculations and logical determinations are to be made." (1967,
p. 185).
The second point made by Blakesley applies to the analysis of high speed motion
picture data. Kinematic and kinetic calculations of body segment parameters are
complex. Minimizing the possibility of human error by using the capabilities of the
digital computer to perform mathematical calculations further decreases the burden
placed on the researcher.

The second problem arising from cinematographical analysis of human
movement is related to the transfer of information into usable form. In order to obtain
usable data from film, the images must be transformed into a series of coordinates by
use of an electronic digitizer. This process introduces a "noise" component into the
numerical data values obtained. Noise can be defined as: "Error present in data collected
that is unrelated to the process being studied" (Rodgers and Cavanagh, 1984). Error is
introduced into the data by a variety of sources. These include the inherent vibration of
high speed motion picture cameras and noise produced through human error (e.g.,
subjective selection of reference points). In the process of electronicly digitizing points
on a projected image of the body, a certain amount of subjective judgement is needed in
order to locate body landmarks. Consistently choosing the exact location of a selected
body landmark is a demanding task for even the most experienced and persistent
investigator. The noise created by these potential sources of errors, while perhaps not
evident within the raw positional data, may become magnified when they are applied to
high-order derivatives, characteristic of certain data smoothing techniques. If
researchers are to extract meaningful kinematic and kinetic analyses from
cinematographic data, appropriate data smoothing methods which minimize inherent
errors associated with derived values is critical.

Statement of the Problem

The problem of this investigation was to create and validate a versatile general
purpose computer program for data smoothing that could also be used to compare
various smoothing techniques. The program was designed to assist researchers in
minimizing the influence of "noise" associated with the cinematographical analysis of
human movement by guiding them in analyzing the output from various data smoothing
techniques.

 

The process of carrying out this study was divided into two parts: (1) the design
and development of a computer program that could be used to smooth numeric data, and
(2) the validation of the program through comparisons of various smoothing techniques
applied to biomechanical data.

The first part of the plan was carried out through the development of a general
purpose computer software package on an IBM AT personal computer in the BASIC
programming language. BASIC was selected because of the universal familiarity among
researchers and ease of learning for those not trained in computer programming. The
package breaks down the smoothing process into component parts. The parts of the
package include: (1) a tutorial to assist the researcher in selecting a proper data
smoothing technique; (2) an input/output routine allowing for varying internal/extemal
storage formats to assure maximal flexibility in data entry; (3) a menu for selection of
different smoothing techniques and; (4) kinematic descriptions of data including graphic
displays of position, velocity, and acceleration. The package is designed in this fashion
because of the critical need for selecting the appropriate smoothing technique. Flexibility
of input and output of data are maintained in view of this program being part of a larger
proposed software package for the complete kinematic and kinetic analysis of
biomechanical information. The input/output routines link these individual packages
together and varying internal storage formats may occur through different programmers.
It is a function of this package to allow for versatility of format.

The second part of this plan was validation. Validation of the data smoothing
routines was accomplished by comparison of the mean percent error between the
generated numeric data values and the resultant smoothed data. Zernicke, Caldwell, and
Roberts (1977) utilized a similar method in determining the mean percent error when
comparing force platform data of typical human gait. In their study, a 95.25 percent
mean agreement was reached based upon these comparisons. For the purposes of the
current investigation, a ninety-five percent mean agreement has been established to

 

determine acceptability of the smoothing routines.
Purpose of the Study

The purpose of this study was to create a flexible computer software package to assist
the researcher of human motion in selecting and implementing appropriate data
smoothing techniques. This was accomplished by including a tutorial in the package to
instruct the researcher about the proper smoothing method to select for each
Classification of motion. All routines in the package are menu—driven to guide the

investigator in the use of each program. This will aid researchers, coaches, and

M L...-

4

educators in making more precise kinematic descriptions of movement from cartesian
coordinate data generated from high speed film. The program will serve as the second
phase of a three part process (data acquisition, data smoothing, and data analysis) in
biomechanical investigation utilizing high speed motion picture photography.

Need for the Study

The smoothing of biomechanical data, as a research topic, takes its origins from
the mathematical subject of numerical analysis. Since biomechanics is a relatively new
field of study, the amount of investigation specifically related to data smoothing is
limited to relatively few studies. Among the researchers who have attempted to develop
computer programs for data smoothing, the approaches have been diverse, creating
problems in generalizing results from one program to another. This lack of continuity
between software packages comes in many forms. The input/output format is the first
problem that arises in these programs. Since each investigator has his/her own style of
computer programming, the corresponding methods of input and output are unique to
that author. This can create problems in comparisons of output generated by different
programs. With a more universal approach to the design of these procedures, the
opportunity for comparison of analyses of various computer programs is created. The
actual kinematic and kinetic output from these programs is a second problem. Each
researcher develops specific routines needed in the interpretation of his/her own data.
Therefore, the ability to compare and contrast results may not be available since these
calculations may not have been performed. The problem being addressed most
prominently within this investigation is the selection of the proper smoothing technique
for the data being analyzed. This step is often taken for granted because researchers
develop smoothing routines specific to a type of data being collected. If researchers are
to make the information gained through biomechanical investigation available to coaches
and educators in the field, they must make the tools suitable for their use. A selection of
smoothing routines, together with a tutorial to aid users in their selection, is a logical
solution to this problem.

The need for the study, then, becomes the design and development of a computer
program for data smoothing which can be used by researchers, coaches, and educators.
It is to be designed to allow the user to analyze the results of the cinematographical data.
Comparisons become critical in demonstrating the selection of appropriate techniques.

 

5

Delimitations of the Study

The following delimitations refer to the development of the computer program:

H

. The program is written in the BASIC computer language specifically for an IBM
or IBM compatible microcomputer.

N

. Kinematic calculations performed within the program are for motion in one plane
at a time.

U)

. Availablility of memory size specific to a microcomputer restrict the number of
data points allowable.

.5;

. Graphic comparisons of only two curves at one time are allowable, due to
memory space availablity on some microcomputers.

Ur

. Classifications of motion given in this study are not those typically found in
physical activity, but are laboratory conditions developed to allow for testing.
The data generated for each classification contains no error, thus facilitating
direct comparison.

Limitations of the Study

Human motion occurs in a three dimensional space. Using cinematographical
techniques, it is possible to study motion either from a two or three dimensional
perspective. Because of the nature of photographic techniques, three dimensional
analysis of movement is a complicated process. However, through the use of proper
analytical tools, three dimensional investigation should be utilized when appropriate.
The problem which arises in high speed cinematography is that a single camera can only
record within a plane of motion (x,y coordinate system). It is through the use of
multiple cameras that a three dimensional view of the action being investigated can be
accomplished (x, y, z coordinate system). The single plane of action captured by each
individual camera in a study employing multiple cameras can be converted into three
dimensional coordinates by the use of a direct linear transformation routine (Shapiro,
1978). Although this process will not be used in this study, a detailed discussion of this
procedure can be found in Walton (1981).

 

Definition of Terms

Terms defined in this section are primarily based on those given by Rodgers and
Cavanagh (1984.)

Acceleration. The rate of change of velocity with respect to time, mathematically, the
second time derivative of displacement and the first time derivative of velocity.
This term applies to both linear ( straight line), and angular (curved path)
motions.

BASIC. Beginner's All-Purpose Symbolic Instructional Code.

Derivatives. Quantities obtained by the process of differentiating a given curve or
function. The most commonly used derivatives are of displacement (x) where
first derivatives dx/dt = velocity, and second derivative (PX/dt2 = acceleration.

Differentiation. A technique of calculus for finding the rate of change of a quantity.

Digitizer. A device that is capable of acquiring planar coordinates in numerical form.
In biomechanics, the most frequent use of a digitizer is to convert the location
of body markers on the projected image of a film into numbers that can be
processed by a computer.

Displacement. The change in the position (linear or angular) of a body.

Force. A vector quantity that describes the action of one body on another.

Integration.A technique of calculus for determining the area enclosed between a
curve and the x-axis. This can result in either a single value or a new curve
derived from the varying function. Integration can be performed either
graphically, mathematically, or numerically. It is the inverse process of
differentiation.

Kinematics. The description of motion.

Kinetics. The study of the forces that cause motion.

Noise. Error present in data collected that is unrelated to the process being studied.
Some noise is almost always present in data collected in biomechanics and
most other fields (e.g., vibration).

Smoothing. A class of techniques for reducing the effects of noise.

Software. Sets of instructions called programs that cause a computer to execute
certain operations.

Vector. A quantity that has both direction and magnitude.

Velocity. A measure of a body's motion (linear or angular)in a given direction.

7

Because velocity has both magnitude and direction, it is a vector quantity that
can be positive, negative, or zero.

CHAPTER H
REVIEW OF RELATED LITERATURE

This chapter has been structured to present a detailed review of smoothing methods
being applied in biomechanics research that utilizes high speed cinematography. A three
part approach has been selected to give a logical sequence to the presentation of this
information. The first section will give an assessment of each technique to be
incorporated in the computer software package, including a brief description of the
method and the main considerations for its use. Part two includes a flow diagram of
motion classifications and a frequency table based on a review of research articles in
biomechanics which incorporate the use of specific data smoothing routines. The third
part is a discussion of the past and present state of data smoothing software and future
considerations for computer application development.

An Assessment of Selected Data Smoothing Techniques

Several investigations have addressed problems associated with data smoothing.
Generally, these studies have focused on a particular smoothing method. The
development of a general purpose computer software package incorporates several of
these routines. Therefore, each data smoothing technique selected will be reviewed. A
brief description of each method will be given, followed by a discussion of its relative
merits and considerations for utilizing the procedure.

Least-squares approximation

The least-squares approach to curve fitting involves developing rn equations (the
normal equations) which minimize the sum, S, of the squares of the deviations of the
equation:

S = 2 [yr -p(ti )lz, (2-1)

where Z is the summation of i from 1 to n (n = number of coordinate data pairs). The
In equations can be solved sirnutaneously to yield the coefficients for a least-squared fit
of a polynomial of degree m to a set of empirical data (Wood, 1982). The popularity of
local least-squares approximation techniques has resulted from their simplicity of

 

9

application, and the convienience of this procedure in determining linear approximations
(Burden and Faires,1985). With this technique, smoothed data values and
corresponding derivatives can be directly obtained from experimental data. Also, if the
nature of the data is known or can be assumed to have its mean distributed in a linear
manner, the values obtained from a linear least-squares routine are unbiased estimates
for the equation that describes the line of best fit through the mean (Dodes, 1978).
Decisions made in applying the least-squares approximation technique includes: (1) the
degree (m) of the approximating polynomial, (2) the number of data points (n) to
include in the smoothing and differentiation formula, and (3) the number of repeat
applications of the smoothing algorithm prior to the use of the differentiation formula
(Wood, 1982). The degree of the approximating polynomial (m) should be less than
four according to Hershey et a1. (1971). However, larger degree polynomials may be
necessary when greater error is present within the data. Selection of the number of data
points (n ) and repetitions are at the discretion of the investigator. It should be kept in
mind, however, that over smoothing is possible with a large n (number of data points)
or number of repetitions.

Cubic spline functions

Mathematically, a spline function is a group of cubic polynomials arranged so that
adjacent curves join each other with continuous first and second derivatives (Dodes,
1978). These polynomials are "pieced" together at points in time (t ) called "knots" (xj ; j
=l,2 ....... n ). The resulting cubic spline function consists of n -1 cubic polynomials of
the form:

g<t>=pj<t>=aj+bj<t)+c,-(t)2+d,-(t)3. (2.2)

on the interval xj-l <= t < xj . This "piecewise" nature of the function allows it to adapt
quickly to changes in curvature.

A "spline" is actually a pliable strip of wood or rubber which was used by draftsmen
in patterning curves. The mathematical form was made popular as an approximation
procedure during the 1960's, when it was shown that a spline function was the
smoothest of all functions for fitting (n) data within specified limits (Greville, 1969).

One unique characteristic of the cubic spline is that it involves four constants; (a, b,
c, and (1), thus insuring that there are sufficient continuously differentiable first and
second derivatives (Burden and Faires, 1985). If all constants within equation (2.2)
were zero, taking the second derivative would eliminate c and d. This means that
velocity and acceleration values can be obtained, because they are the first and second

10

derivatives of the fitted equation respectively. Other data smoothing techniques may not
have four constants. Therefore, if velocity and acceleration values are critical, the cubic
spline offers assurance of the needed orders for analysis.

Fourier series

For a series of (n) data points with equally spaced time values, there exist (n)
orthogonal functions over these data points which are either a sine or cosine of a
multiple of t (Pizer, 1975). These functions make up what is known as the fourier
series. The general form of the equation is:

aj + 2 [aj sin(j 2m IT) + bj cos(j 21rt /T)], (2.3)

where 2 is the summation of j from 1 to n . The first sine and cosine terms from
equation (2.3) represent functions describing one cycle (21c radians) in time (T
seconds), and subsequent terms represent functions whose frequency are a multiple of
this base frequency. Functions are referred to as "orthogonal" if their first definite
integral on the range (a,b) is equal to zero (Anton,l980).

One of the purposes of using a data smoothing method is to "fit" an equation to data
obtained from the digitizing process. If the result should take the form of various
ordinary and partially differential equations, the fourier series is an extremely powerful
tool in describing the solution (Burden and Faires, 1985). This is true because of the
trigonometric nature of a fourier series which fits the approximating polynomial closely.
Since partial differential equations require considerable skill to compute (Anton, 1980),
the fourier series offers a time-saving method which eliminates costly mistakes by the
researcher.

Chebyshev polynomials.
By definition, the set of Chebyshev polynomials [Tn ] are:
Tn (x ) = cos [n arccos x ], (2.4)
for x, an element of [-1,1], and n = 0,1,2 ..... (Burden and Faires, 1985). Introducing

the substitution theta = arccos x , it can be shown that the following Chebyshev
polynomials are derived

 

T2(x ) = 2x Tl(x ) - T0(x ) = 2x2-1,
T3(x ) = 2x T2(x ) - Tl(x ) = 4x3-3x,
T4(x ) = 2x T3(x ) - T2(x ) = 8x4—8x2+1,

and so on.With the aid of the equation Tn (x ) = cos(n theta) from the previous
substitution, these equations reduce to

Tn+1 (x ) = 2x Tn (x ) - Tn-l (x ), (2.5)

for each n = 1,2,..... and x an element of [-1,1]. This form of the Chebyshev
polynomials is used to approximate a"1ine of best fit".

The Chebyshev method utilizes the trigonometric cosine to follow harmonic or
repeating forms of motion (Dodes, 1978). Harmonic motions repeat in a wave-like
pattern which closely approximate the graph of the trigonometric sine and cosine (See
Figure 1). This type of movement is common in biomechanical research with such
motions as running or cycling.

The main asset in using the Chebyshev polynomial is in its ability to remain "stable"
for higher-order equations (Dodes, 1978). A stable condition, simply stated, means that
a small error made at any step in a calculation does not cause a geometrical summation
of the error as the procedure is repeated (Pizer, 1983). Since harmonic motions are
cyclic in nature, these motions will be smoothed repetitively introducing the opportunity
for small errors to be compounded The Chebyshev technique allows the researcher to
be confident of minimal errors when working with vast amounts of data.

Butterworth filter.

The Butterworth filter is an electronic process to remove "noise" from data. The
basic design of a digital filter is quite simple. A digital filter by design is: (1) some form
of electronic input, (2) a mathematical process to remove noise, and (3) the resulting
output (See Figure 2). The digital filter which will be used in this investigation will be a
low-pass Butterworth filter, so called because the operation it performs involves passing
only the low-frequency components of a signal. A low-pass filter has been selected
based upon the results of previous investigations (Pezzack, Norman and Winter, 1977;
and Winter 1979) which concluded that application of a low-pass digital filter followed
by a first order finite difference routine produced the best approximation of
accelerometer data taken from various classifications of motion.

 

\V
/

 

Figure 1. The trigonometric nature of repeating or oscillatory motion.

A digital filter is an effective tool in data smoothing because of its ability to represent
oscillating or periodic types of motion effectively (Williams, 1986). Oscillatory motion
repeats itself along the same path. This type of harmonic movement, while present in
biomechanical research, is not reported as frequently as periodic motion which does not
repeat the same path exactly. The main use of the Butterworth digital filter is for periodic
types of movement. Since repetitive motion is so prevalent in human locomotion, the
Butterworth filter is one of the most frequently utilized data smoothing methods (See
Table 1).

Classification System for Motion Studies

In order to properly select a data smoothing technique, it is essential to develop an
understanding of the nature of motion being investigated. One such system is presented
in the flow chart of Figure 3.

Human motion can be defined as: "Change of position in space and time through
force applications" (Barham, 1978). The three general classifications used in describing

 

 

. fl SIGNAL fl X
nput PROCESSING utput

 

 

 

Figure 2. Digital filter black-box design.

human movement are: (1) linear, (2) curvilinear or angular, and (3) harmonic. Linear
motions are those actions which occur in a straight line. Cuvilinear or angular motions
follow a curved or arc-like path. This may include projectile motions or the flight paths
of implements. Harmonic movements have a repetative nature in which the directional
path is cyclic.

Further subclassifications of curvilinear motion include circular and parabolic
movements. Circular movements remain a constant distance from a fixed point. The
constant distance from a fixed point is, of course, the radius of a circle. Motion that
follows a path which is always an equal distance from a fixed point and a fixed line is
parabolic (Barham, 1978). An example of this motion is the pathway followed by a
projectile in flight in the earth's atmosphere when it is moving at an oblique angle to the
horizontal and the effect of air resistance is ignored.

Periodic motion is any movement that repeats itself. This is distinguished from
oscillatory motion by the fact that the latter repeats itself over the same path. An example
of periodic movement is the path taken by the foot when running; the motion repeats in
an eliptical form, but never exactly over the same course. If the path taken by a
particular point of a pendulum were to be traced, it would exemplify oscillatory motion,
where the exact arc is retraced on each successive swing.

Table 1 gives a detailed review of the classification system developed in Figure 3 as
applied to human motion studies. The far left column of Table 1 includes the types of
motion under investigation. Noticably absent from these descriptions are the terms
circular, parabolic, periodic, and oscillatory. Each of these subdivisions are considered
to be part of the description preceding them in the flowchart shown in Figure 3. The
labels "constant" and "varying" have been added because human motion rarely occurs as
a constant. In the second column in Table 1, the motion classifications are subdivided
into "velocity" and "acceleration". They will be considered separately since studies of

 

MOTION CLASSIFICATION

LEAST SPLINE FOUFIIER
SQUARE FUNCTION SERIES

n=2

TYPES

LINEAR ”=1

n=4
LINEAR n=4

“=6

ACCELERATDN

ACCELERATION
VARYING VELOCITY
ACCEERADON n=8 n=7

 

Table 1. Literature review frequencies by motion classification.

human movement are often concerned with only one of these subclassifications.

The headings for the frequency cells in Table 1 are the data smoothing techniques
included in the computer software package developed for this study. Numeric values in
the cells of Table 1 represent a frequency of appearance of the smoothing techniques by
type and subclassification of motion found in a review of biomechanics studies in the:
(1) International Journal of Sport Biomechanics (1985-1988), (2) Journal of
Biomechanics (1984-1987), and (3) Research Quarterly for Exercise and Sport (1977-
1986).

Referring to Table l, the frequency cells indicate the number of articles reviewed
which specifically mentioned the use of the smoothing technique heading each column.
Main considerations brought out by this table include the types of motion most
frequently under investigation by each technique and the distribution of frequencies
throughout the entire table.

Varying nonlinear or angular motion was the most common classification of
movement encountered. Because these studies usually involved some sort of electronic
signal processing, the Butterworth filter contained the highest cell numbers.

 

15

 

HUMAN MOVEMENT

Change of position in space and
time through force applications

|
I l I

 

 

 

 

 

LINEAR CURVILINEAR HARMONIC
Straight-line Curved-line Repetitive-path
motion motion motion

 

 

 

 

 

 

 

 

 

     

 

CIRCULAR PARABOLIC PERIODIC OSCILLATORY

    
    
  

Constant Equal distance Repeating
distance from fixed pt. pattern

   

Repeating
. th

 
 
 

 

 

 

 

 

 

 

Figure 3. Block diagram of motion classifications.

The relative distribution of cell frequencies is the most important characteristic of
Table 1. No specific pattern of smoothing method related to motion classification is
evident. Since one of the purposes of this investigation was to develop guidelines as to
the utilization of data smoothing techniques, the fact that routines seem to be selected at
random supports the need for this study.

The Past and Present of Computer Software and
Future Considerations for Development

In an overview of software available, Bates (1973) discussed the characteristics of
the programs being used. He was trying to solve the problem of software flexibility by
developing a program which could be used by anyone interested in the analysis of
human motion. The programs in use were highly individualized to the needs of
investigators within a particular institution. His premise, at that time, was that
researchers must take a more generalized approach to software development. Programs
available such as ANGLCALC (Hoffman and Warsham, 1971) were very efficient at

16

generating the values desired by individual investigators, but lacked the versatility and
power to meet the diverse needs of researchers within human motion studies.

Researchers were making progress during the early nineteen-seventies in simulating
movement patterns. The program MOTION (Baumgartrrer, 1971) was designed to give
a representation of segment movement, including kinematic descriptions of the limbs.
This would lead researchers into new fields such as computer graphics and simulation
of movement.

Today's software for the analysis of human motion has adapted itself to directions
being taken in biomechanics. Modeling of movement and the subsequent graphic
display give the biomechanical investigator powerful tools in the analysis of motion.
Availability of packaged software offers the researcher the technology to save vast
amounts of time in data analysis and interpretation. For those who can afford the price
of these packages, it is well worth the investment in eliminating time consuming data
management and software development.

The future of biomechanical software is in user-friendly packages that allow for the
complete analysis of movement from data acquisition to data analysis. Through a shared
user environment utilizing multiple CRT's hooked to a mainframe, users can collaborate
on research and standardize results. Programs must be written to include clear
instructions, so that investigative tools are made accessible to coaches and educators in
the field. The development of this package takes this approach in creating a user-
friendly, menu-driven program to assist in selecting and implementing appropriate
smoothing methods.

 

CHAPTER III
DEVELOPMENTAL METHODOLOGY AND EXPERIMENTAL PROCEDURES

The purpose of this investigation was to design, construct, and validate a general
purpose data smoothing computer package to be used in the comparison of selected data
smoothing techniques. The design of this program is based on current need as
determined from: (1) a review of literature that assessed and compared smoothing
methods for application in biomechanics and (2) a review of current research that
reported the application of specific smoothing routines for biomechanical analysis.
Based upon the results of a questionnaire sent to researchers in the area of
biomechanical analysis of human movement (see Appendix A), subsequent verification
and revisions of the software package have been implemented.

The first section of this chapter will consist of a discussion of the criterion used in
selecting smoothing techniques for inclusion within the software package. Section two
will detail the structure of the software package. Section three will be a discussion of the
developmental procedures undertaken, including considerations for each smoothing
routine. Section four presents the checks followed in verifying the fact that the package
carries out the functions for which it was written. Finally, section five will be a
discussion of the procedure followed in comparing the data smoothing techniques.

Criterion for Inclusion

In selecting the data smoothing methods to be included in the computer software
package, several factors had to be considered. First, the purposes of fitting equations to
data will be reviewed to develop rationale for smoothing data. According to Daniel and
Wood (1971), the main considerations for fitting equations to data are: (1) to summarize
a mass of data in order to obtain interpolation formulae or calibration curves, and (2) to
confirm or refute a theoretical relation; to compare several sets of data in terms of the
constants in their representing equations; to aid in the choice of a theoretical model.
Second, variables that determine a good method of fitting should be listed to provide
guidelines for choosing the smoothing routines. From Daniel and Wood (1971), a good
fitting technique should include the following:

(1) use all relevant data in estimating each constant,

(2) maintain reasonable economy in the number of constants required

17

18

[Equations with large numbers of constants require excessive calculations to
determine the constants involved],

(3) provide some estimate of the uncontrolled error in y [For example, in the
equation; y =mx + b, y is the dependent variable],

(4) provide some indication of the random error in each constant estimated [Too
great an error may indicate another method of smoothing would be more
appropriate],

(5) make it possible to find regions of systematic deviations ("function bias") from
the equation if any such exist [Such deviations could indicate a line of best fit
which is not truly representative of the data being given],

(6) show whether the conclusions are unduly sensitive to the results of a small
number of runs, perhaps even of one run [Converging on an answer too quickly
may introduce error due to sudden changes in each numeric value], and

(7) give some idea of how well the final equation can be expected to predict the
response-both in the overall sense and at important sets of conditions inside the
region covered by the data.

Based upon these guidelines, a list of data smoothing techniques was prepared while
conducting the review of literature for this study. From those listed, five routines were
selected as being most frequently used by researchers (see Table 1). The utility of these
methods is supported by the fact that they are applicable to the classifications of motion
given in Figure 3 (Pezzack, Norman, and Winter, 1977; Wood, 1982). These
techniques are: (l) least-squares approximation, (2) Chebyshev polynomials, (3) fourier
series, (4) spline functions, and (5) Butterworth filter.

One additional method that deserves mention is the finite difference technique. This
routine, used to calculate velocity and acceleration values is not included as a smoothing
technique because it does not generate an equation which can be used to describe the
smoothing of raw data. The method can be used following the smoothing process to
calculate kinematic values.

An additional criterion used for inclusion within the program is the results of the
questionnaire concerning existing software. Information gained from this questionnaire
was used to support or refute the selections made on the basis of literature reviewed.
Each method previously mentioned has been specifically investigated to determine
whether or not biomechanical investigators deem it worth inclusion in data smoothing

19

software packages. By surveying the experts within the field of biomechanics of human
motion, a better understanding of what is needed can be developed. The actual content
of the program is also determined in this manner. Therefore, subroutines included in
this program are the result of a thorough investigation, synthesizing the stated needs of
researchers of human motion.

Structure of the Program

The first goal in the development of the program was to assure flexibility. The
program is the second part of a three phase process in the anaysis of human motion.
This process consists of: (1) data acquisition, (2) data smoothing, and (3) data analysis.
Due to memory availability in many microcomputers, data smoothing algorithms were
developed as subroutines. A unique feature of this program is memory tracking. Each
subroutine will erase memory allocated by the previous subroutine in order to maximize
the availability of storage size.

Subroutines selected for inclusion within the package are:

(1) a tutorial to assist the user in selecting the proper smoothing technique and
instructions for running the program,

(2) an input routine to allow for data to be entered either from key-board or
diskette,

(3) a data display for immediate viewing on a monitor,

(4) a menu of smoothing routines to allow for proper techniques to be used
according to the nature of the data,

(5) a printer option for hardcopy of data, and

(6) a graphics routine allowing for plotting of curves from any chosen data set
(position, velocity, and acceleration values).

20

Developmental Procedures

The first step in developing the computer program was to decide which subroutines
were to be included. These routines have been enumerated in Table 1. The next step was
to interprete the mathematical theory of each smoothing technique into a usable
algorithm that served as the basis for writing the code (See Figure 4). When each
algorithm had been designed, the routines were written and tied together through a
menu-driven control routine to form the program. A short description of the steps taken
is provided. Each discussion will include: (1) a mathematical definition of the method,
(2) steps in developing an appropriate algorithm, and (3) considerations taken in
incorporating each routine as part of the overall package.

least-squares approximation

The method of least-squares is one of the most universal techniques used for data
smoothing. According to Daniel and Wood, the least-squares method says: "Find the
values of the constants in the chosen equation that minimize the sum of the squared
deviations of the observed values from those predicted by the equation." (1971, p. 6).
In developing an equation to "fit" the data acquired by digitizing, the researcher strives
to choose the equation which best describes the coordinates. The least-squares method
accomplishes this task by considering the equation in a "piece-wise" fashion. It
represents the true curve by a series of segments between points. When minimizing the
sum of the squared deviations, this mathematically reduces these segments to the desired
curve. In this way, the statistical variance is used to insure that the result is an unbiased
estimate of the true curve. An unwritten assumption in utilizing this and other smoothing
methods is that the data are " good". By this, it is meant that no wild points or outliers
are contained within the data.

In developing an effective algorithm for the least-squares technique the first task is to
seek constants a and b which minimize the least-squares equation:

2 [yi ‘ (axi + b)I2, (3-1)

where 2 is the summation of i from 1 to n. In order to solve for these constants, a
system of two equations must be developed which effectively solve for this process.
Once found, these equations are easily solved to give a single equation in terms of y .
The corresponding x values are the raw data points.

Problems to be considered before linking the subroutine to the overall package are:
(1) determining which variables must be transferred from the main program and (2)

21

 

SELECTED DATA SMOOTHING
TECHNIQUES

 

 

 

 

 

 

 

LEAST
SQUARES

 

 

 

CI-IEBYSI-IEV FOURIER
POLY. SERIES

 

 

 

 

 

 

I

Figure 4.

 

CUBIC
SPLINE

 

 

 

 

 

 

INTERPRET
THEORY

 

 

 

 

 

DEVELOP
ALGORITHM

 

 

 

 

 

WRITE
CODE

 

 

 

 

 

TIE EACH ROUTINE

 

TOGETHER THROUGH A MENU SYSTEM

 

 

 

 

DATA SMOOTHING
SOFTWARE PACKAGE

 

 

 

BUTTERWOR
FILTER .

 

Flow chart of developmental procedures in creating a software package.

 

 

22

what information must be saved besides the smoothed data values for later review of the
results. Raw data values, time coordinates, and indicator variables are transferred to the
least-squares subroutine. Following the smoothing process, resulting data and the
newly calculated indicator values are to be returned.

Spline functions

Spline functions are a recent mathematical tool developed in the 1960's. However,
their development is well established in engineering. A spline is actually a flexible strip
or ruler that is constrained to pass through a given set of points. When it is constrained
in this way, the spline assumes a shape that minimizes its stored strain energy.
Mathematically, a spline function is a connected group of cubic polynomials arranged so
that adjacent curves join each other with continuous first and second derivatives.This
concept is moST easily understood by listing conditions in which a function is
discontinuous. From Anton (1980), a function y =f(x ) at a point x=c is discontinuous
if:

(1) the function, f, is undefined at c; or
(2) the limit, lim f(x), as x approaches c does not exist; or

(3) the function, f, is defined at c and the limit, lim f(x), as x approaches c exists,
but the value of the function at c and the value of the limit at c are different.

Cubic spline functions are one example of the general class of continuous
functions. In order to construct a cubic spline, it is necessary to specify the coefficients
that uniquely described each cubic polynomial between given data points. Selecting an
appropriate spline for a given smoothing problem depends on the nature of the data
being manipulated, and on what type of output is desired. In selecting the least-squares
polynomial it was necessary to select the order of the equation. With higher-order
polynomials it is often better to use a spline curve. Again, a higher-order polynomial is
simply one whose highest power (highest exponential value within the equation) is
greater than a certain value as determined by the investigator.

Allocation of internal storage for all smoothing routines is strictly on a demand basis.
Arrays will be appropriately dimensioned according to the size of the data set being
utilized. Once the smoothed values have been calculated, non-essential indicator
variables will be erased. Upon subsequent runs of the smoothing routines, previously
calculated data will be overwritten.

23

Chebyshev polynomials

The nTH Chebyshev polynomial is defined by: T(x) = cos (n arccos x), where x is
on the interval -1 to l (Dodes, 1978, p. 361). Essentially, the Chebyshev method is a
trigonometric technique for fitting an equation to a given set of data. Because of the
trigonometric nature of the method, repeating periodic and oscillatory types of harmonic
motion may be represented quite well by this technique.

A unique characteristic of this method is that it remains "stable" for higher-order
polynomials, and converges more rapidly on a solution than other techniques. An
improper integral is said to "converge" if the limit exists (Anton, 1980, p. 539). A
criterion imposed upon an algorithm whenever possible is that small changes in the
initial data produce correspondingly small changes in the final results. A routine that
satisfies this property is called "stable" (Burden and Faires, 1986).

A second feature of this technique is that it has many possible orders which may be
selected to best fit the given data. For cyclic types of movement it has been shown that
little signal exists beyond the seventh harmonic (Wood, 1980). Therefore, in order to
maximize flexibility within the Chebyshev routine, a seventh-order polynomial was
constructed. Most types of human activity fall within the range of the seventh harmonic.
It has been suggested, however, that children's gait may contain more complex
components (Foley, Quanbury, and Steinke, 1979).

Fourier series

"Given N data points with equally spaced data arguments, it can be shown that there
exist N functions orthogonal over these data arguements which are either sine or cosine
of a multiple of t " (Pizer, 1975, p. 393). These functions are collectively known as the
fourier series. Their general equation is:

aj + 2, [a]- sin(j 27tt/T) + bj cos(j 21tt/I‘)], (3.2)

where 2 is the summation of j from 1 to n. "Orthogonality" refers to the first definite
integral on the range (a,b) equaling zero (Anton, 1980).

One of the most powerful of the numerical methods for its ability to approximate a
curve, the fourier series is also difficult to calculate and lengthy to process. The
constraint placed on this routine is that the data points must be equally spaced in time. If
the data points (t,y) are considered, with trepresenting the time coefficient, it is obvious
that cinematographical data is appropriate for this technique; motor driven cameras
provide a uniform time standard (t) that can be used to generate the fourier coefficients.

 

24

Zarrugh and Radcliffe (1979) have suggested that at least twenty harmonics might
be necessary to adequately describe the motion of the foot. However, for cyclic motions
it has been generally shown that little signal exists beyond the seventh harmonic (Wood,
1980). Because of this wide range necessary to study human motion, a general fourier
equation was developed applicable to the first through seventh harmonic. For
comparison purposes, only the seventh harmonic will be used because the activity under
investigation can be temporally controlled.

Butterworth filter

The Butterworth filter is a digital smoothing process designed to extract "noise"
from data collected by electronic instrumentation. The theoretical design of a digital
filtering system is given in Figure 1. According to Williams "The Butterworth filter is so
popular because its passband and stopband are without ripples" (1986, p. 332). What
the author means is that this filter represents the path of a curve without oscillating
around the curve to a great extent . The passband and stopband refer to the waves used
to represent the data curve. This can best be understood by means of an example. A
typical sine function oscillates around the x-axis, with each peak the same distance from
the ordinate. If each end of the curve could be stretched, the waves would be flattened at
the peaks and brought closer to the x-axis. This new oscillatory curve would give a
better representation of a straight line, with a correspondingly smaller passband and
stopband.

Two design problems must be considered when writing a Butterworht filter
program: (1) the order of the filter, and (2) the filter's cutoff frequency. For the
purposes of this investigation, a low-pass filter was utilized as it was found to give the
best approximation to typical human movements (Winter, 1979). The cutoff frequency
is the sensiu'vity with which the filter can represent the desired curve. Low-pass filters
contain good recursive properties, increasing their wave sensitivity (Pezzack, Norman,
and Winter, 1977).

Validation of the Program

In order to validate whether or not this computer software package carries out the
functions for which it is designed, resultant smoothed data values were compared
against the corresponding raw data through the means square error. Data calculated from
mathematical formulas contain a known error term of zero at each coordinate pair.
Therefore, the summation of the differences between the smoothed values and exact
values at their respective locations will yield a composite error term. This error term,

 

25

when divided by the number of coordinate pairs, results in the mean numeric error.

Zernicke, Caldwell, and Roberts (1977) utilized a similar method in determining
mean percent error. They achieved 95.25 percent mean agreement in comparing force
platform data of typical human gait. For the purposes of this investigation, a five percent
or less difference between generated data and smoothed values was considered
acceptable.

Comparison of Techniques

The purpose of this investigation was to design, construct, and validate the use of a
computer program which could subsequently be used to make comparisons of selected
data smoothing techniques.

Numeric data was generated from equations which was subsequently used to make
the comparisons. This process was to contrast each of the classifications of motion
enumerated in Table 1 using the respective smoothing techniques that had been selected
for inclusion within the software package. Numeric data was generated from
mathematical equations representative of the classifications of motion under
investigation (See Appendix B). These values, that contain no error terms, were
mathematically processed utilizing each smoothing routine. From the results of this
procedure a ranking of smoothing techniques by classification of motion was
constructed. The table will be an invaluable tool for researchers, coaches, and educators
in helping them make informed decisions as to the appropriate smoothing method based
upon the type of movement under investigation. This ranking system is an ideal
situation in that no error is present within the raw data coordinates. Therefore, any error
detected in the resulting smoothed values can be assumed to be the result of the
mathematical routines. This particular approach for comparison was selected because of
the problem of generating specific types of linear movement classifications to
cinematographically record. Errors which are the result of photographic methods, the
digitizing process, and subjective judgement of the researcher are unique to the
investigation in which they are used. Therefore, in order to give further validity to the
results of this study, the comparison process involved the use of exact numeric data.

Use of the Computer Program
The question of selection of smoothing routines is of increasing interest to

researchers, engineers, and mathematicians. Selection of the appropriate method is
critical in the analysis of data and has been the subject of numerous studies. This

26

investigation was developed to create a valuable tool for researchers, coaches, and
educators, as well as, make a critical study of the techniques used in the program.

The main use of the computer software package is for investigators of human
motion interested in eliminating "outliers" from their numeric data in order to gain a
better picture of what the motion under investigation truly looks like. This would
encompass a broad range of professionals from the pure research scientist interested in
extending man's understanding of locomotion, to the clinician interested in quantifying
abnormalties in the gait of patients. It is the purpose of this investigation to develop a
tool which may be valuable to all who are interested in the investigation of motion.

Summary

The purpose of this study was to design, create, and validate a computer program
which could be used to compare various data smoothing methods. The design of the
program is based on a review of current literature, and the results of a questionnaire on
software construction. Smoothing routines included are: (1) least-squares
approximation, (2) Chebyshev polynomials, (3) cubic spline functions, (4) Butterworth
digital filter, and (5) fourier series. The development of the program was on an IBM
PC, making it compatible with other microcomputers. Each routine was validated with
generated data which was known not to contain any error term. To carry out the
comparison of the smoothing methods, data generated from the smoothing process was
contrasted to the original data by the use of the mean numeric error. From these
comparisons, a table ranking each technique according to the classification of motion
being studied was constructed.

CHAPTER IV
RESULTS AND DISCUSSION

This chapter has been organized to present a detailed review of the pertinent findings
resulting from the data smoothing software package developed for this investigation.
Each of the five data smoothing routines enumerated in Table 1 were used to smooth
data which was generated from equations written to represent the classifications of
motion diagrammed in Figure 3. Subsequent numeric data values were quantitatively
ranked according to their mean numeric error, which is a measure of the deviation from
the original data. Part one of this chapter will present the results of a questionnaire sent
to professionals within the field of cinematographical analysis of human motion to
assess the present state of data smoothing software. Part two is a discussion of the
comparison process, contrasting the relative merits of the five data smoothing methods
as applied to the major classifications of human motion. Part three is a description of the
use of the smoothing package in assisting researchers. Part four is a summary of the
results.

Data Smoothing Information Questionnaire

A data smoothing information questionnaire (see Appendix A) was sent to
professionals in the field of high-speed cinematographical analysis of human motion in
order to assess the present state of data smoothing software used in research, and to
determine which peripheral support may be deemed valuable in further enhancing the
capabilities of these routines. A summary of the results of this questionnaire is presented
in Table 2.

The majority of researchers polled (86%) responded positively to the existence of
data smoothing software at their College/University. This would support the fact that
computer software is considered a valuable tool in assisting researchers with the
analysis of cinematographical data. For those who responded negatively (14%), an
available data smoothing software package may prove invaluable in saving time and
labor in the duplication of effort.

Fifty percent of researchers questioned indicated that multiple smoothing techniques
were available with the software they are currently using. This value would seem to
indicate that much of the time a single routine is applied to motion data regardless of the
nature of the motion under investigation.

27

 

28

 

 

Questionnaire Items Yes No
Data smoothing software 86* 14
available at your college? (12)" (2)
Multiple smoothing techniques 50 50
available? (7) (5)
Manual/tutorial to assist users? 44 55
(4) (5)
User view data on monitor? 83 17
(10) (2)
Computer printouts available? 100
(12)
Graphic displays be plotted? 92 8
(11) (1)
FD L S C S F S B F C P Q S
Data smoothing techniques 12 12 24 15 21 09 09
SUPPOfled? (4) (4) (8) (5) (7) (3) (3)
Key Floppy Main Digitizer Harddisk
Input Devices? 27 27 17 20 10
(8) (8) (5) (6) (3)
Raw Velocity Accel. Compar. Force
Displays available? 29 31 29 06 06
(10) (11) (10) (2) (2)
Yes No Not Certain
Current software meets 45 45 5
needs? (5) (5) (1)
Table 2 Summary of Data Smoothing Information Questionnaire.

”Numbers in parentheses represent frequency of responses.
* Numeric values without parentheses are percentages.

29

The third question addressed the problem of proper manuals/tutorials to assist the
researcher in the implementation of the smoothing routines. Fifty-five percent responded
that no manual or tutorial existed to aid them in the utilization of their software. Whether
this manual were to be in published form, or incorporated within the software, for those
researchers unfamilair with the various smoothing methods this could create great
difficulty in the implementation of their routines.

Questions four through six can be effectively considered as a group of items related
to the common peripheral devices utilized in computer usage. Specifically, these items
ask about the existence of the output devices which may be selected in order to view the
resulting data generated from the smoothing routine utilized. In each case, the majority
reSponded affirmatively. Monitors (83%), printouts (100%), and graphics displays
(92%) are used by most researchers in order to display the resultant data from their
routines. Based upon these findings, each of the output devices selected for this
software package can be deemed warrented.

Column headings for question seven are two letter abbreviations for the smoothing
methods in Table 1 (e.g. FD=FINITE DIFFERENCE). Two additional methods have
been added to this table based upon their frequency of response. CP (cubic polynomial)
and Q8 (quintic spline) are data smoothing methods similar in nature to the cubic spline
function which have been utilized by nine percent of the researchers polled in each case.
Although these routines have not been incorporated within the software package, they
can be considered for inclusion upon subsequent revisions of the program.

Among the routines which were included in the software package, the cubic spline
function was the most popular (24%) of the smoothing methods. This is followed by
the Butterworth filtering (21%), fourier series (15%), and the finite difference and least
squares routines with 12% each. Since the researchers polled in this questionnaire were
encouraged to list any smoothing method which was not specifically mentioned within
the questionnaire, the top five routines are supported by these results.

Twenty-seven percent responded positively to the use of a keyboard or floppy disk
as an input device. The remaining devices, in order, include a digitizer (20%),
mainframe computer (17%), and hard disk (10%). Each of these input devices are
supported either directly by the software package or indirectly by an intermediate device.

In order to determine what types of graphics displays would be considered valuable,
the researchers were asked to list all curves which could be plotted by their software.
Velocity curves were the most popular choice (31%), followed by the raw data and
acceleration curves (29%), and force and comparison curves (6%). Most commercially
avaliable software pakcages have multiple graphics routines that may be selected.

The last question on the questionnaire asked whether current software in general is

 

30

considered to meet the needs of researchers, coaches, and physical educators within the
field of high-speed cinematographical analysis of human motion. Forty-five percent
responded positively, 45 percent negatively, and 5 percent uncertain. Although no
majority responded either "yes" or "no" to the question, the fact that 45 percent
responded negatively indicates that further advancement of software would certainly be
welcomed by those dissatisfied with current data smoothing packages either written to
meet the specific needs of the researcher, or provided commercially.

Perhaps the most valuable information gathered by this questionnaire was the
general comments provided by the researchers. In the words of one prominent educator:
"I would like to see a program which offers a selection of data smoothing techniques,
along with guidelines (written &/or on-line) to assist in selecting the proper smoothing
technique." Similar comments related to the difficulty in locating and operating
appropriate smoothing methods were to be found in the majority of the questionnaires.
With this additional information to draw from, the following section will present a
detailed review of both the validation of the individual smoothing routines and the
comparisons made in contrasting the relative merits of each algorithm.

Comparison of Smoothing Methods

In order to establish a basis upon which the smoothing routines written for this
investigation could be compared, data was generated by pre-written equations to
represent each of the classifications of motion diagramrned in Figure 3 (See Appendix
B). The first step in this process was to validate the usefulness of each routine by
comparing the generated data values to the corresponding smoothed data values. This
was accomplished by calculating the mean square error. A five percent mean square
error or less was established as acceptable. The second step was to contrast the
generated mean square error values in order to assess the relative merits of each data
smoothing routine as applied to the generated data.

Validation

In order to assess the validity of each smoothing routine data was generated by pre-
written equations representing each of the classifications of motion diagrammed in
Figure 3 (See Appendix B). The resulting data was subsequently smoothed utilizing
each of the data smoothing methods. Software was then written to calculate the mean
square error based upon the equation:

 

31
[2(X-x1)2/nl.

where x is the raw data values, X] is the smoothed data values and n is the number of
data values. The mean square error was calculated for each of the classifications of
motion diagrammed in Figure 3. A five percent mean square error was established
acceptable as outlined by Zernicke, Caldwell, and Roberts (1977). It should be noted
that the mean square error will be greater for larger data values and so the percent error
gives a better value for comparison.

Each of the smoothing routines were found to be acceptable within the limits
established (See Table 3). However, every routine did not meet the requirements for
each classification of motion. This was to be expected as the individual smoothing
routines adapt themselves to varying types of data. Table 3 lists the mean square error
for each (x,y) coordinate data list found in Appendix B.

The natural cubic spline function was found to be a special case during the coding
phase of this investigation. Because of the nature of the natural cubic spline, the
selection of a predetermined power for the polynomial function is not possible. Rather,
the natural cubic spline adapts itself to fit through each data coordinate pair creating a
function whose power is equal to the number of coordinate pairs. For this reason, the
mean square error associated with the natural cubic spline will always be equal to zero
as denoted in Table 3. Although this makes validation of the natural cubic spline
unnecessary, it creates difficulty in the comparison of the routine as will be discussed
later.

Comparison of techniques

In order to compare the data smoothing routines written for this investigation the
mean square error was again used. Since each routine was utilized to smooth the same
data sets, the resulting mean square error values could be directly contrasted in order to
quantitatively rank the routines. The natural cubic spline function was not directly
compared to the other routines because of a mean square error of zero. However, it will
be included in the discussion.

The results of contrasting the data smoothing routines are presented in Table 4.
Motion classifications as diagrammed in Figure 3 are listed in the first row. Smoothing
routines appear in descending order below each classification based upon their mean
square difference values. The natural cubic spline function does not appear in the table
due to a constant mean square error of zero.

The Chebyshev polynomial produced the lowest mean square error for the constant
linear data, varying linear data, varying nonlinear data, constant harmonic data, and

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MOTION CLASSIFICATION DATA SMOOTHING TECHNIQUES
TYPES LEAST SPLINE“ FOURIER CHEBYSHEV BUTTERW. i
SQUARE FUNCTION SERIES POLY. FILTERING I,
x=1.08E o7 ' x=0 'x=2.314E -02 ' x=7.386E -05 x=3.29E 16
m1. y=1.08E 07 . Y=° =6.239E .02 y=1.991 E -04 y=6.03E 16
x=5863.681 " x=O x=233089.8 x=743.7868 x=918
V3233 y=5865.12 ' y=0 y=233192.7 y=744.1159 y=918
CONSTANT ' x=12.65 j x=0 x=399.6757 ' x=1.275359 ' x=0
NONLINEAR y=2.51 E 08 y=0 y=3-169E 09 y=1-0112E 07 y=60979.7
' “.496 Z x=0 ' x=17.88722 ' x=5.707E -02 #155 03
NgfilFL‘IENESR y=35912_65 =0 y=490854.8 y=1566.317 ‘ y=22,783
. — = - = . =0
CONSTANT o X=23.537 . X—o - X 925.9224 . X 2.954617 .
HARMONIC ' y=_512 y= y=.1180212 y=3.766E -04 y=4.058
VARYING x=53.541 x=0 . x=1970.269 :x=6.287112 x=16.32
HARMONIC y=.191 y=0 y=.1100592 y=3.512E -o4 y=2.519

Table 3. Mean Square Difference for Exact Numeric data.

* Denotes a mean square diference less than five percent.
** No error produced by natural cubic spline.

33

MOTION CLASSIFICATIONa‘

 

CONSTANT VARYING CONSTANT VARYING CONSTANT VARYING
LINEAR LINEAR NONLINE ONLINE

AR HARMONIC HARMONIC

CHEBYSH CHEBYSH BU'I'I'ERW CHEBYSH CHEBYSH CHEBYSH
POLY. POLY. FILTER POLY.

POLY. POLY.

 

 

FOURIER BUTTERW CHEBYSH LEAST BUTTERW BU'I'I'ERW
SERIES FILTER POLY.

SQUARE FILTER FILTER

LEAST FOURIER LEAST FOURIER
SQUARE SERIES SQUARE SERIES

BU'I'I'ERW LEAST FOURIER BUTTERW FOURIER FOURIER
FILTER SQUARE SERIES FILTER SERIES SERIES

 

LEAST LEAST
SQUARE SQUARE

 

 

 

 

 

 

 

 

 

Table 4. Data Smoothing Routine Ranking by Motion Classification in
Descending Order.

*Rankings are based upon mean square difference.

34

varying harmonic data. The natural cubic spline function produced the only consistently
lower mean square difference values. All routines used a seventh order polynomial to
smooth the data where applicable.

The Butterworth filtering routine, which is a low—pass filter, produced the next best
results. This routine must be considered slightly biased because of the nature of the
technique. Unlike the other smoothing techniques used in this investigation the
Butterworth filter does not produce a polynomial through a given discrete data set.
Rather, the data is scaled according to certain criteria set down by the user. For this
investigation, high frequency (outliers) data points were filtered out of the raw data set
in order to produce a smoother curve through the resulting data.

The least—squares routine, which was concluded the most popular of the smoothing
techniques in the review of literature, produced the next most consistent results. This
routine produced consistently good data curves for all classifications of data. Although
this technique is certainly acceptable for most types of human motion, the popularity of
this routine may need to be reexamined based upon the results of this investigation.

The fourier series was found to consistently produce the greatest mean square error.
Although these results may seem to lead to the conclusion that this is the poorest of the
routines selected, it should be noted that the fourier series did produce acceptable results
four out of six motion classifications.

The rankings produced by comparing the mean square error of each routine should
be considered with caution. The nature of smoothing routines is to produce a certain
amount of error even under exact mathematical calculation. Through the interpretation of
these routines and the subsequent coding for computer use more error is inevidably
introduced. Each computer represents data in a slightly different manner. Investigators
code their programs according to their own expertise. The error introduced by these
factors is random. Therefore, the best the prudent researcher can do is to minimize this
random error and compare the results to those previously reported.

The quantitative ranking system presented is not meant to be all-inclusive. The tables
presented here are the result of comparisons at a generally accepted power for
smoothing polynomials. This was done both to add validity to the routines developed
for this investigation and to help provide information concerning the proper smoothing
technique. The results of this study should not be extrapolated to all smoothing
packages.

It was noted in Table 3 that the natural cubic spline function developed for this study
produced no error. This routine is designed to fit a polynomial through each data
coordinate pair. The result is a function which passes through all points including
outliers. This should be considered before using the cubic spline routine in order to
avoid outliers in the final smoothed data.

35

Using the Software Package

The purpose of this investigation was to develop a computer software package
which could assist the researcher of human motion in choosing and implementing
appropriate smoothing techniques. A tutorial was written as part of the software package
to fulfill the goal of assisting in the selection of the appropriate technique. As an
example, consider the constant nonlinear (angular) data found in Appendix B.

The tutorial is useful in selecting the appropriate data smoothing technique where no
a priori knowledge of the data or technique is available. The main menu shown in figure
5 is the first screen given to the user upon booting the package. Choose selection one,
the tutorial in order to find information concerning the use of the smoothing routines and
peripheral support routines. Figure 6 shows the main tutorial selection menu. Each
subroutine, including the smoothing techniques have complete explanations of its use
and applicability.

Figure 7 is an example of a tutorial screen for least squares approximation. Similar
tutorials are provided for each smoothing routine. Once the appropriate technique has
been selected, the data can be input through the data input routine. Figure 8 shows the
main selection menu for inputting data. If the data is stored on disk, figure 9 shows the
screen for entering the appropriate drive and filename for retrieval.

In order to smooth the data, a selection of techniques is offered as shown in figure
10. After smoothing, the data may be plotted in order to evaluate the results. Figure 11
shows a graph of the nonlinear data and subsequent smoothed data generated by each of
the routines.

Summary

This chapter presented the pertinent findings resulting from the data smoothing
software package developed for this investigation. Validation of the smoothing routines
was accomplished by calculating the mean square error produced by smoothing numeric
data. All smoothing routines written for this study were found to be acceptable at a pre-
determined level of ninety-five percent mean square agreement. In contrasting the
smoothing techniques it was found that in the routines which consistently produced the
best results, in descending order were: ( 1) natural cubic spline function, (2) Chebyshev
polynomial, (3) Butterworth filter, (4) least squares polynomial, and (5) fourier series.
It was noted, however, that the natural cubic spline produced no error and must be
considered seperately.

 

36

 

DATA SMOOTHING PROGRAM
SCHOOL OF HEALTH EDUCATION, COUNSELING
PSYCHOLOGY & HUMAN PERFORMANCE
MICHIGAN STATE UNIVERSITY

TUTORIAL ................................................... 1
INPUT DATA. ................................................ 2
SAVE DATA TO DISK ....................................... 3
VIEW DATA AT TERMINAL ................................ 4
SMOOTH DATA ............................................. 5
SEND DATA TO PRINTER ................................. 6
GRAPH DATA ............................................... 7
EXIT PROGRAM ............................................ 8

 

Figure 5. Main menu of data smoothing software package.

 

DATA SMOOTHING PROGRAM
TUTORIAL

PLEASE SELECT ONE OF THE FOLLOWING:

WHAT IS DATA SMOOTHING .............................
HOW DO I INPUT DATA ...................................
HOW DO I SAVE DATA .....................................
HOW DO I VIEW DATA .....................................
HOW DO I PRINT DATA ...................................
HOW DO I GRAPH DATA ..................................
CHEBYSHEV POLYNOMIALS ............................
LEAST SQUARES APPROXIMATION ....................
FOURIER SERIES ..........................................
CUBIC SPLINE FUNCTION ...............................
BUTTERWORTH FILTERING ............................. 11
RETURN TO MAIN MENU ................................ 12

\DWQQUI-BMNH

.—
O

 

Figure 6. Tutorial menu screen.

 

37

 

********TTTTTTTTTTT* TTTTTTT “‘******

LEAST SQUARES: APPROXIMATION

*iTT TT """"""""""""" ******

Least squares approximation is based upon the ideal that every value
has a certain deviation from its true value (an error value), the least squares
technique takes into account the deviations of all data points & attempts to
minimize the error. This method is highly recommended for biomechanical
data as it is very powerful at giving a smooth curve representation of a set
of data points.

PRESS ANY KEY TO RETURN TO MENU

 

Figure 7. Tutorial screen from least squares approximation.

 

 

INPUTTING DATA FROM DISK

*1*************************

PLEASE SELECT ONE OF THE FOLLOWING:

INPUT DATA SMOOTHING DATA FROM DISK. ........ 1
INPUT DATA ACQUISITION DATA & FORMAT ......... 2
RETURN TO MAIN MENU ................................. 3

YOUR SELECTION PLEASE:

 

Figure 8. Main screen for inputting data from disk.

38

 

YOUR FILE NAME & DISK DRIVE ID
ARE ENTERED IN THE FOLLOWING FORM:
FOR DISK DRIVE: ‘B’ & FILENAME: ‘SAVE’

B:SAVE

FILENAME PLEASE:

 

Figure 9. Entering drive and filename for data retrieval from disk

 

 

DATA SMOOTHING TECHNIQUES

****¥************************

PLEASE SELECT ONE OF THE FOLLOWING:

CHEBYSHEV POLYNOMIALS ............................. l
LEAST SQUARES APPROXIMATION .................... 2
FOURIER SERIES .......................................... 3
CUBIC SPLINE FUNCTION ............................... 4
BUTTERWORTH FILTERING ............................. 5
CALCULATE VELOCITY OF RAW DATA. ................ 6
CALCULATE ACCELERATION OF RAW DATA ......... 7
RETURN TO MAIN MENU ................................. 8

 

Figure 10. Main menu for data smoothing techniques.

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ael

39

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ucl

GI

 

 

 

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IOI 3:12. .818

0'0 Osage—.9.

DID wine—.32.;

L...

325:
:5» E5

_
8%....

 

_
Nag:

as?

Figure 11. Smoothed data curves from various data smoothing techniques.

CHAPTERV

CONCLUSIONS

Summary

The purpose of this investigation was to create a flexible computer software package
to assist the researcher of human motion in selecting and implementing appropriate data
smoothing techniques. This was accomplished by including a tutorial in the package to
instruct the researcher about the proper smoothing method to select for each major
classification of human motion. The package created for this study is the second part of a
three-phase process in the analysis of human motion through high speed cinematographical
techniques. This process consists of: (1) data acquisition, (2) data smoothing, and (3) data
analysis.

The development of the computer software package took place over a two-year
period. The package was written on a IBM AT microcomputer and can be used with all
IBM and IBM compatible brand microcomputers equipped with color graphics capability.
The following is a list of the main characteristics of the package:

(1) a tutorial to assist the user in selecting the proper smoothing technique, and

instructions for running the program,

(2) an input routine to allow for data to be either entered from keyboard or diskette,

(3) a data display for immediate viewing on a monitor,

(4) a selection of smoothing routines to allow for proper techniques to be used

according to the nature of the data,

(5) a printer option for hardcopy of data, and

(6) a graphics routine allowing for plotting of curves from any chosen data set

(position, velocity, and acceleration values).

Findings

The findings of this investigation were the result of two sources. First, a data
smoothing information questionnaire was sent to professionals in the field of high-speed
cinematographical analysis of human motion in order to assess the present state of data
smoothing software used in research, and to determine which peripheral support may be
deemed valuable in further enhancing the capabilities of this package. Second, comparisons
of raw data values to smoothed values were made by the mean square error. The mean
square error was then used to quantitatively rank each smoothing routine according to the

40

 

41

major classifications of human motion.

The results of the questionnaire have been utilized in the design of the data smoothing
package. Responses to direct questions regarding the capabilities of existing software gave
insight into the needs of researchers of human movement. The responses indicated that
although the majority polled (86%) had some kind of data smoothing software available,
the capabilities of the software were diverse. The responses also indicated that 45% of
those questioned were dissatisfied with the current state of data smoothing software. The
results of the questionnaire will also be used in the recommendations for further research
and development.

The first part of the comparison process was to validate the effectiveness of the
smoothing routines. This was accomplished by determining the mean square error for each I
routine. Every routine was found to be acceptable within the limits set of ninety-five
percent mean square agreement. It was noted that for the natural cubic spline function no
error term was generated.

The second part of the comparison process was to compare each routine according to
the major classifications of human motion. The routines were quantitatively ranked
according to their mean square error. The natural cubic spline function was not directly
compared to the others. The routines which consistently produced the best results, in
descending order were: (1) natural cubic spline function, (2) Chebyshev polynomial, (3)
Butterworth filter, (4) least squares polynomial, and (5) fourier series.

Implementation

The data smoothing software package developed for this investigation was designed
to be used in conjunction with existing software at Michigan State University. However,
one of the purposes of this study was to make the software as flexible and user-friendly as
possible. For this reason the software may be utilized by anyone interested in the reduction
of numeric data. The peripheral support developed within the package is designed to allow
for input of data from various sources.

The areas of research interested in the use of this package range from the analysis of
human gait to the reduction of accelerometer data from impact testing. It is the hope of the
author that the software developed for this investigation will meet the needs of a broad
array of researchers where the reduction of numeric data to usable form is necessary.

42

Recommendations for Further Research and Development

It is the nature of software to be continuously updated and improved. As research
advances our understanding in all fields of science, the tools we use for research must also
be improved Based upon the recommendations of professionals in the field of high speed
cinematographical analysis of human motion, the following recommendations are offered.

(1) The data smoothing software package should be incorporated into a package for

the complete analysis of high speed film, including: (a) data acquisition, (b) data
smoothing, and (c) data analysis.

(2) The graphics capabilities of the package should be expanded to include

comparisons of various body segments.

(3) The data smoothing routines written for this package should be evaluated on

polynomials of varying degree.

 

REFERENCES

REFERENCES

Anton, A,. Calculus Mill anamie gm John Wiley and Sons, Inc.,
1980.

Barharn, J .N ., Meeham'eallgmmglqu. The C. V. Mosby Company,
Saint Louis, 1978.

Bates B.T., "The Development of a Computer Program with Application
to a Film Analysis: The Mechanics of Female Runners.", Ph.D.
Dissertation, Indiana University., 1973.

Baumgartner, T., "Using the Computer and the Computer Program
MOTION.", in Selected Topics on Biomechanics: Proceedings of the
C.I.C. Symposium on Biomechanics. (ed. by J .M. Cooper) Chicago:
Athletic Institute, 191-194., 1971.

Blakesley, J. F. "Administration -- Integrated Records and Proceedings",
edited by Ralph N. Gerard, McGraw Hill

W
Book Co., St. Louis, 1967, 307 pp

Burden, KL, and Faires, J .D., m Analysis, Prindle, Weber &
Schmidt, Boston, 1985 (3rd edition), 676pp.

Daniel, C., and Wood, F.S., Eitfmg m to Data. John Wiley and
Sons, Inc., 1971.

Dodes, I.A., Namerigsal Analysis for Compare; Science. Elsevier-Holl-
and, Inc., New York., 1978.

Foley, C.D., Quanbury, A.O., and Steinke,T., "Kinematics of Normal
Child Locomotion - A Stationary Study Based on TV. Data.", J.
Biomech. 12: 1—6, 1979.

Greville, T. Theory and Applieatigns 9f Spline Fungtigns. New York,

Academy Press, 1969.

Hoffman, SI, and Warsham, K., "ANGLCALC: A Computer Program
for Descriptive Analysis of Movement Used During Repeated
Performances of Skills", in Kinesiology Review Washington, DC:
American Association for Health, Physical Education and Recreation,
55—59., 1971.

Muybridge, E., Am'mals in Motion. Chapman & Hall, London. (c1899).

43

 

44

Pezzack, J.C., Norman, RW., and Winter, D.A., "An Assessment of
Derivative Determining Techniques Used for Motion Analysis. Journal
of Biomechanics, 10: 377-382, 1977.

Pizer, S.M., Numerigal mugging and Mathmfical Analysis. Science
Research Associates, Inc., 1975.

Pizer, S. Mi, and Wallace, V. C. ,ECgmpate Namsrigally, gm am
.Little, Brown and Company. 1983.

Rodgers, M.M., and Cavanagh, P.R., "Glossary of Biomechanical Terms,
Concepts, and Units." Physical Therapy. 64(12), December, 1984.

Shapiro, R., “Direct Linear Transformation Method for Three-Dimensional
Cinematography.” Research Quarterly. 49(2): 197-205, May, 197 8.

Walton, J. S. "Close- -range cine- photogrammetry: a generalized technique
for quantifying gross human motion.‘ 'Ph.D. Dissertation. University
Park, PA: The Pennsylvania State University, May, 1981.

Ward, R., "An Investigation into the Use of Computer Integration of
Kinematics, Kinetics, and Cinematography Data in Motion Analysis",
Ph.D. Dissertation, Indiana University., 1971.

Williams, C.S., Mg figltal Hugs, Prentice-Hall, Inc., Englewood
Cliffs, New Jersey, 1986.

Winter, D.A., Bigmeghanigs Qt Human Mm New York: John
Wiley and Sons, 1979.

Wood, G.A., "Data Smoothing Differentiation Procedures in
Biomechanics", Exercise and Sport Science Reviews, Volume 10,
1982.

Zarrugh, M.Y., and Radcliffe, C.W., "Computer Generation of Human
Gait Kinematics", J. Biomech., 12: 613—621, 1979.

Zemicke, R.F., Caldwell, G., and Roberts, E.M., "Fitting Biomechanical
Data with Cubic Spline Functions", Research Quarterly, 47(1): 9—19,
March., 1976.

 

GENERAL REFERENCES

GENERAL REFERENCES

Belkin, G.S. Twalvg—ngg Basis: for 1h; EM E Campag'blas, McGraw-Hill
Book Company, 1986.

Bernstein, N W New York:
Pergamon Press, 1967.

Cuthbert, D, and Wood, F. S. W. John Wiley & Sons,
Inc.,1971.

Douglas, B. Nameg’ga! Basis. Howard W. Sams & Co., Inc., Indianapolis,
Indiana, 1973.

Forsythe, G.E. “Generation and the Use of Orthogonal Polynorrrials for Data
Fitting with a Digital Computer.” Journal of the Society for Industrial
Applications of Mathematics. 5:74-88, 1957.

Garrett, R.E., Widule, C., and Garrett, G.E. “Computer-Aided Analysis of
Human Motion”, in Kinesiology Review Washington DC: American
Association for Health, Physical Education and Recreation, 55-59, 1971.

Gustavson, F.G., and Sackson, M.V. “problem Solving and BASIC. A
Modular Approach.” Science Research Associates, Inc., Chicago, 1986.

Hamming, R.W. Digitalfimgs. Prentice-Hall Inc., 1977.

Henrici, P. f P k 1 r
Damgnstrafign ns. John Wiley & Sons, Inc. 19.82

Jackson, KM. “Fitting of Mathematical Functions to Biomechanical Data.”
IEEE Transactions on Biomedical Engineering. 26(2): 122— 124, February,
1 979.

James ML, Smith, G.M, and Wolford, J..C WWW
In.ternationa1 Textbook Company,
19167.

James, S.L., and Brubaker, C.E. “Biomechanical and Neuromuscular Aspects
of Running.” In: Exercise and Sport Sciences Reviews, J .A. Wilmore
(Ed.), New York, Academic Press, 1973.

Johnston, R.L. Namgjgal Memgls A ngnyam Apprgagh. John Wiley &
Sons, Inc., 1982.

Lees, A. “An Optimized Film Analysis Method Based on Finite Difference
Techniques.” Journal of Human Movement Studies. 6: 165-180, 1980.

45

 

46

McCormick, J. M., and Salvadori, M..G Nl’l‘lll meg'gal MethglsinFQRTRAN.
Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1964.

McLaughlin, T.M., Dillman, GI, and Lardner, T.J. “Biomechanical Analysis
with Cubic Spline Functions.” Research Quarterly for Exercise and Sport.
48: 569-582, 1977.

Patrick, S.B., Widule, CB, and Hillberry, B.M. “Digital Filtering of
Cinematographical Data”, in Biomechanics: Proceedings of the C.I.C.
Symposium on Biomechanics. (ed. by J .M. Cooper, and B. Haven) The
Indiana State Board of Helath. 202—213,l980.

Petak, K.L. “Acquisition and Reduction of Biomechanical Data by Mini-
Computer.”, in Selected Topics in Biomechanics: Proceedings of the C.I.C.
Symposium on Biomechanics (ed. by J .M. Cooper) Chicago: Athletic
Institute. 201—205, 1971.

Shoup, T.E. N ameg'cal Methfls far the Persgnal Cgmpagr. Prentice-Hall
Inc., Englewood Cliffs, New Jersey, 1983.

Widule, CI, and Gossard, D.C. “Data Modeling Techniques in
Cinematographic Biomechanical Analysis.” In: Biomechanics III, E. Jokl
(Ed.) Basel: S. Karger, 1973.

Winter, D.A., Sidwall, H.G., and Hobson, D.A. “Measurement and Reduction
of Noise in Kinematics of Locomotion.” Journal of Biomechanics. 7: 157-
159, 1974.

Wold, S. “Spline Functions in Data Analysis.” Technometrics. 16: 1-11, 1974.

 

 

APPENDIX A

DATA M THIN INF RMATI N E Tl NNAIRE

DIRECTIONS

Please respond to each item by circling the letter preceding
the appropriate response. The questionnaire is designed to gather
information on up to four individual programs. Please list each
program separately and answer the corresponding questions.

Your assistance is greatly appreciated

1 .) Do you currently have data smoothing software at your College/
University? (If NO, go to question # 11.)

a. Yes
D. No
2.) Please state the name(s) of your program(s).

(SPACE IS PROVIDED FOR FOUR INDIVIDUAL PROGRAMS)

(I) _._..____ (II) ______ (Ill)

 

3.) Referring to the program(s) listed above; are multiple smoothing
techniques available to the user? (Le. more than ONE)
(it NO, go to question #5.)

(I) a. Yes (II) a. Yes (lll)a. Yes (IV) a. Yes
b. No b. No b. No b. No
4.) is there a manual or tutorial to assist users with the program(s) listed

above in selecting the proper smoothing technique?

(I) a. Yes (ll)a. Yes (lll)a. Yes (IV) a. Yes
b. No b. No b. No b. No

47

 

n
V

48

Which data smoothing technique(s) can be supported by the program(s)?
(Place ALL appropriate answers in the spaces provided.)

e.g. If program III has a finite difference technique and a butterworth
filter, place (a) and (e) in the spaces following program Ill.

Finite Difference Technique

Least Squares Approximation PROGRAM(S)
. Cubic Spline Function

Fourier Series (I)
Butterworth Filtering

Chebyshev Polynomials (ll)
Other (Please List)

79900?

(III)____ __ __ __ __ ___

 

=9

 

I (IV) _ ___ __ __ __ ___

 

Which input device(s) can be selected with the program(s)?
(Place ALL appropriate answers in the spaces provided.)

 

 

 

7)

8.)

9.)

PROGRAM(S)
a. Keyboard
b. Floppy Disk (I) ___ ___ ___ __
c. Mainframe
d. Digitizer (II) __ __ ___ ___ ___ __
Other (Please List)
e. (lll)___ __ ___ __
f.
9. (IV) ___ ___ ___ ___ __- -__
Can the user view data on a computer monitor with the program(s)?
(l) a. Yes (ll)a. Yes (lll)a. Yes (IV) a. Yes
b. No b. No b. No b. No
Are computer printouts of data available with the program(s)?
(I) a. Yes (ll)a. Yes (lll)a. Yes (IV) a. Yes
D. No b. No b. No b. No
Can graphic displays of data be plotted by the program(s)?
(If NO, 90 to question #11.)
(l) a. Yes (ll)a. Yes (lll)a. Yes (IV) a. Yes
b. No b. No b. No b. No

49

1 0.) Which displays can the program(s) plot?

(Place ALL appropriate answers in the spaces provided.)

 

 

a. Raw Data Curve PROGRAM(S)
b. Velocity Curve
c. Acceleration Curve (l) _ __ __ __ __
d. Comparison Curves of Techniques
Other (Please List) (ll) __ __ __ __ __
e.
f. (Ill)___ _________
9

 

(IV)-.. ___. __

Do you feel that current data smoothing packages In general are sufficient
to meet the needs of researchers, coaches, and educators?

a. Yes
b. No
c. Not Certain

PLEASE USE THE REST OF THIS QUESTIONNAIRE TO DISCUSS ANY
PROBLEMS, OPTIONS, OR TRENDS YOU FEEL MAY BE VALUABLE IN
DESIGNING A DATA SMOOTHING PACKAGE.

YOUR COMMENTS ARE GREATLY APPRECIATED!

 

APPENDIX B

CONSTANT LINEAR DATA

.01 ":

        

.22 .4400001

.23 .4500001

.2400001 .4600001 ~
.25 .4700001

.26 .48

.27 .4?

.28 .5

.29 .51

.3 .52 ,
.31 .53 i
.32 .54

.33 .55

.34 .56

.34????? .57

.35????? .53

.36????? .5?

.3799??? .5999???

.38????? .6099???

.3°9???? .61?????

.40????? .62?????

.41????? .63?????

.42????? .6499???

.43????? .65?????

.4499??? .6699998

.45?????
.46????3
.4799???

 

.48????3 .70????3
.4999??? .71???93
.50???93 .72????8
.5199??? .73????3
.52????8 .74????8
.53????8 .75????3

VARYING LINEAR DATA

100
115
160
175

1000
1015
1060
1075
1120
1135
1180
11?5
1240
1255
1300
1315
1360
1375
1420
1435
1480

1495

1540 '

1555
1600
1615
1660
1675

100.25
115.25
160.25

175.25

 

1675.25

51

 

CEINSTANT NONLINEAR (ANGIJLAR) DATA

 

12 1723
3 2137

14 2744
15 3375
16 4096
17 4313
13 5333
13 635?
20 3000
21 9261
22 10643
23 12167
24 13324
25 15625
26 17576
2 19633
23 21952
29 24339
30 27000
31 29791
32 32763
.3 35937
34 39304
35 42375
36 46656
37 50653
33 54372
39 59319
40 64000
41 68921
42 74033
43 79507
44 35134
45 91125
46 97336
47 103323
43 110592
49 117644
50 125000
51 132651
5: 140603
53 143377
54 157464
55 166375
56 175616
57 135193
53 195112
59 205379
60 216000
61 226931
62 23332.
63 230047
64 262144

53

VARYING NONLINEAR (ANGULAR) DATA

9.999?99E—04
-3.000003E-03
-.12
-.512

-1.331

-2.744001

I
U]

I
'10 U! l-J H

II
.-
\lJ-‘IH

I
J M M N '4 >—- H -

-4.913

 

I
\1 50¢

 

-26.99.99
-35. 3699
-46.655?9
-59.313?9
-74.037?3

-91.125

l
h-hMEJL-Jfidt
.
'L'IIJQO‘C-

 

-4.B -110.592

-5.100001 ~132.651

-5.400001 -157.4b41
-5.700001 -135.1931
-6.0QOOOI -216.0001
-6.300001 -250.0471
—é.300002 -237.4962
-6.900002 -328.5092
-7.200002 -37?.2433
-7.500002 >-421.8754
-7.800002 -474.552

-3.100003 -531.4415
—8.400002 -592.7045
-8.7000O2 -658.5036
-?.000003 -729.0003
-9.300003 -804.3578
-9.600003 -334.7369
-9.900003 -970.3001
-10.2 -1061.209
-10.5 -1157.b26
-10.S -1259.713
-11.1 -1367.633
-11.40001 -1431.S46
-11.70001 -1601.615
-12.00001 -1723.002
-12.30001 -1360.369
-12.60001 -2000.373
-12.70001 -2146.692
-13.20001 -2299.971
-13.50001 -2460.37S
-13.30001 ~2623.075
~14.10001 -2303.22

-14.40001 -2985.983
-14.70001 -3176.527
-15.00001 -3375.005
-15.30001 -3581.‘82
-15.60001 -3796.421

CONSTANT HARHONIC DATA

2 .9092975
4 -.7563026
6 -.2794155
3 9993532
10 -..5440212
12 - 536573
14 9906074
16 - 237903
13 - 7509373
2) 9129452
22 3351309503
24 - 9055734
26 .7625535
23 .2709058
30 - 9880317
32 .5514267
34. .5290327
36 -.9917739
33 2963636
40 .7451132
2 -.9165216
44 1.7701935-02
46 .9017884
43 -.7632547
50 -.2623749
52 .9866276
54 -.5537391
56 -.52155_1
53 .9928726
60 -.3043107
62 -.7391307
64 .9200261
66 -2.6551!SE-02
68 -.3979273
70 .7733907
72 .2533234
74 -.9851463
76 .5661076
73 .5139735
30 -.?938387
32 .313239
34 .7301904
36 -.9234585
:33 .0333983
90 .3939966
92 -.7794661
94 -.245252
96 9335373
93 -.5733319
100 -.5063657
102 .9948269
104 -.3216224
106 -.7271425

108 .9263136

54

55

VARYING HARHONIC DATA

 

1 -.4161469
2 -. o .4

7 -.1455

3 -.9111303

13 .1367372

14 -.7596379

19 .4030321
20 -.5477293 “—
25 .6469194 "
2» -.2921338

31 .3342224

32 4327675542
37 .9550737

33 2.66643

43 9793434

44 .525322

49 9649661

50 7421543

55 .3532202

56 .8993669

61 .6735071

62 .9858966

67 .4401431

68 .9m904

73 .1717173

74 9217512

79 -.1103872

so 776686

85 -.3836935

96 .5697504

91 -.6264445

92 .3174237

97 -.3l92883

98 3.932033342
103 -.9463631

104 -.2409591

109 -.9990209
110 -.5025443
115 -.9715922
116 -.7240973

121 -.3667671

122 -.3379639
127 -.6923953

12 -.9311056
133 -.4633289

134 -.9960879
139 -.1973136

140 —.9317224
145 3395944502
146 -.7931365

151 .3590443

15 -.5913697

157 .6055279

153 -. 3424943

APPENDIX C

56

100 OIM X1( 500),Y1(500),NP(100),PH$(100),TV(100) :
UP

200 _DIM BF1(1) ,BF2(1),C31(1), I332(1),CP1(1),CP2(1),F31(1),F32(1),L31(1),L82(1)
300 REM:

400 REM:

1000 REM: ALL OTHER PROGRAMS BRANCH TO THIS POINT.

1100 REM:

1200 REM:

1300 KEY OFF

33 00 REM:

3900 CLS

4000 LOCATE 1,15

4100 PRINT “#***************§******44*******§**************§*§*§*"

4200 LOCATE 2,15

4300 PRINT "* *"

4400 LOCATE 3,.

4500 PRINT "*" : COLOR 0,7 : LOCATE 3,30 : PRINT "DATA" : CO
5 : COLOR 0,7 : PRINT "SMOOTHING" : COLOR 7,0 : LOCATE 3,45
"PROGRAM” : COLOR 7,0 : LOCATE 3,67 : PRINT "*"

4600 LOCATE 4,15

4700 PRINT ”* *"

4800 LOCATE 5,15

4900 PRINT "*" : COLOR 0,7 : LOCATE 5,22 : PRINT "SCHOOL OF HEALTH EDUCATION, CO
UNSELINO“ : COLOR 7,0 : LOCATE 5,67 : PRINT "*"

5000 LOCATE 6,15

5100 PRINT "* *"

5200 LOCATE 7,15

5300 PRINT "*" : COLOR 0,7 : LOCATE 7,26 : PRINT ”PSYCHOLOGY & HUMAN PERFORMANCE
": COLOR 7,0 : LOCATE 7,67 : PRINT "4"

5400 LOCATE 8,15

5500 PRINT TAB(15) "*" TAB(67) "*"

5400 LOCATE 9,15

5700 PRINT "*":COLOR 0,7:LOCATE 9,29:PRINT"MICHIOAN STATE UNIVERSITY"=COLOR 7,0:
LOCATE 9,67 : PRINT "*"

5300 LOCATE 10,15

5900 PRINT “4 *"

6000 LOCATE 11,15

6100 PRINT ”§***&**********4*&************#**§*§************#***§“

6200 LOCATE 13,

6300 PRINT "TUTORIAL.. ...... ........................... 1“

6400 LOCATE 14,19

6500 PRINT "INPUT DATA................................. 2“

6600 LOCATE 15,1?’

6700 PRINT "SAVE. 9DATA TO DISK.......................... 3"

6300 LOC'ATE 16,

6900 PRINT "VIEW yOATA AT TERMINAL ....... ...... ......... 4"

L0 , O
: OR

R
C PRINT

7 : LOCATE 3, 3
UL 0 7 :

5

 

57

7000 LOCATE 17,19
7100 PRINT "SMOOTH DATA.. ...... .,...................... 5"
7200 LOCATE 13,19
7300 PRINT ”SEND DATA TO PRINTER....................... 6"
7400 LOCATE 19 19

7500 PRINT ”GRAPH DATA......... ........... ............. 7"
7600 LOCATE 20,19
7700 PRINT "EXIT PROCRAM....... ............ ............ E"

7300 LOCATE 22,27
7900 INPUT "YOUR SELECTION PLEASE: ",N$

 

 
   

8000 IF N$ = "1" THEN OOSUB 9300

3100 IF NS = "2" THEN CHAIN ” :3.INPUT",1000,ALL
3200 IF NS = “3” THEN CHAIN "A: .SAVE",1000,ALL
3300 IF NS = "4" THEN CHAIN ”A:C.VIEW",1000,ALL
8400 IF N$ = "5" THEN CHAIN "A' .3MOOTH",1000,ALL
3500 IF NS = "6" THEN CHAIN ”A . PRINT“,1000,ALL
8600 IF NS = "7" THEN CHAIN ”A:S.ORAPH",1000,ALL

3700 IF N$ = "3" THEN OOSUE 41000
8800 ERS = ”A:S.MAIN" : LN = 9000
3900 CHAIN "A=S.ERROR",1000,ALL
9000 OOTO 1400

9100 LOCATE 10,54

 

9500 REM: TUTORIAL SUBROUTINE FOR EXPLANATIONS OF HOW TO USE THE

M.
9700 REM: ROUTINES & PROCEDURES IN THIS PROGRAM.
9800 REM:
9900 LOCATE 1,25
10000 PRINT "DATA SMOOTHING PROGRAM“
10100 LOCATE 2,32
10200 PRINT "TUTORIAL"
10300 LOCATE 3,25
10400 PRINT "4*****§§§*4§**§**§***§"
10500 LOCATE 5,20
10600 PRINT “PLEASE SELECT ONE OF THE FOLLOWING:”
10700 LOCATE 7,15
10300 PRINT "NHAT IS DATA SMOOTHINO...................... 1"
10900 LOCATE 8,15

11000 PRINT “HOW DO I INPUT DATA ..... . ..... .............. 2”
11100 LOCATE 9,15

11200 PRINT “HOW DO I SAVE DATA ...... .................... 9”
11300 LOCATE 10,15

11400 PRINT ”HON DO I VIEU DATA ..... ............ ..... .... 4"
11500 LOCATE 11,15

11600 PRINT ”HOW DO I PRINT DATA ..... ... ....... ..... ..... 5”
11700 LOCATE 12,15

11300 PRINT "HOW DO I ORAPH DATA. ..... ........ ...... ..... 6

11900 LOCATE 13,15

12000
12100
12200
12300
12400
12500
12000
12700
12800
12200
13000
13100
13200
13300
13400
13500
13600
13700
13300
13900
14000
14100
14200
14300
14400
14500
14600
14700
14300
14900
15000
15100
15200
15300
15400
15500
15000
15700
15500
15900
16000
16100
10200

16300

58

PRINT "CHEBYSHEV POYNOMIALO. ..... ................. 7"
LOCATE 14,15

PRINT "LEAST SOUARES APPROXIMATION...
LOCATE 15,15

PRINT "FOURIER SERIES.............................. 9"
LOCATE 16,15

PRINT "CUBIC EPLINE FUNCTION...................... 10"
LOCATE 17,15

PRINT “BUTTERwORTH FILTERING...r ...... ....... ...... 11"
LOCATE 15,15

PRINT "RETURN TO MAIN MENU........................ 2"
LOCATE 21,25

INPUT "YOUR CELECTION PLEASE:”;N$

.
.
.
.
.
.
.
u
.
.
.
.
.
.
LII

IF N5 = "1" THEN OOSUB 14300

IF N5 = "2" THEN OOSUB 17100

IF NS = "3" THEN OOSUB 21900

IF N$ = "4" THEN OOEUB 23900

IF N5 = "5" THEN OOSUB 25900

IF N$ = ”6" THEN OOSUB 27300

IF NS = “7" THEN OOSUB 29300

IF N$ = "8" THEN OOSUB 31900

IF NS = "9” THEN OOSUB 34200

IF NS = "10” THEN OOSUB 36100
IF NS = "11" THEN OOSUB 38400
IF NS = "12" THEN GOTO 1000

CLS = REM: INPUT ERROR
OOSUB 9300
OOTO 9300

CLS
REM:
REM: *§***§*4§***#**§*******4*§§******§4*%********#*******
REM: 4 wHAT IS DATA SMOOTHING *
REM: 4%***§§4**********&§***************************§*****
REM!

LOCATE 1,25

PRINT "WHAT 15 DATA SMOOTHING?"

LOCATE 2,25

PRINT ”**§*§****##*§*§**a§****”

LOCATE 5,1

PRINT " Data SMOOthing is a may OF FiTtering Out “noise” FPOm raw data

PRINT
PRINT "The assumption made is that the data Fit a predetermined shape and"

PRINT
PRINT "that by Fitting the assumed shape t0 a “best-Fit“ with the Paw data

 

59

14400 PRINT

16500 PRINT "a smooth curve will'result.”

16600 LOCATE 22,24

16700 INPUT ”PRESS ANY KEY T0 RETURN TO MENU“,CB$ : GDTO 3300
16300 CLS

16900 LOCATE 22,24

17000 INPUT "PRESS ANY KEY TO RETURN TO MENU",ED$ = GDTD 9300
17100 CLS

17200 REM:

17300 REM: *§*****¥******************#**4**4**#e#****ee**%**#***
17400 REM: * HUN DC! I INPUT UATA‘?
17500 REM: *****************************?***********************

17600 REM:

17700 LOCATE 1,25

17300 PRINT "HOW DO I INPUT DATA?"

17900 LOCATE 2,25

13000 PRINT "****#***************"

13100 PRINT 2 PRINT

13200 PRINT " Inputting data is done by Following the instructions given an

the screen.“

13300 PRINT

13400 PRINT ”This can be done by either inputting Frdm the terminal or
5k. II

13500 PRINT

13600 PRINT ”When inputting From the terminal several important questions will b

e asked.”

13700 PRINT

13300 PRINT "First, you will be asked to input a two line header For the File. T

his will

13900 PRINT

19000 PRINT “be used to tell you which File you are using when either viewing or
printing"

19100 PRINT

19200 PRINT “the File. Second, you will be asked to input the number 0F pairs of
points”

19300 PRINT

19400 PRINT ”you will input. This is critical since the data is treated seperate
ly as the"

13500 PRINT

19600 PRINT "number 0F entries beneath each header or pair. Third, you will be a

 

+.
n

r-m a di

19700 LOCATE 22,23

19300 INPUT ”PRESS ANY KEY TO VIEW NEXT SCREEN",C;$ : CL3

19900 PRINT : PRINT

20000 PRINT ”to input the headers For the pairs. These will be placed above the

60

20100 PRINT

20200 PRINT ”later when viewing or printing the data. Fourth, you will be asked
to tell"

20300 PRINT

20400 PRINT "whether the time interval is a constant. This is very common in Bio
mechanical"

20500 PRINT

20600 PRINT "research. IF the interval is not constant you may now merely enter
the time"

20700 PRINT

20300 PRINT "value, the X-coordinate, & the Y‘coordinate For each set oF values.

20900 PRINT

21000 PRINT "indicate the end oF your input enter ”E“ as prompted."

21100 PRINT

21200 PRINT " When inputting From a disk, you will give the disk drive ID &

 

21300 PRINT
21400 PRINT "Filename as explained in the program. The program will then automat

21600 PRINT "input the File & tell you how many pairs oF data points were in the
File."

21700 LOCATE 22,23

21800 INPUT "PRESS ANY KEY TO RETURN TO MENI",CC$ : SOTO 9300

21?00 CLS

22000 REM:

22100 REM: §*§**&*****#§**§*u******§*************§********§*§***
22200 REM: * HOW DO I SAVE DATA? #
22300 REM: ******4***#*§***§*******§***********#**********§§**§*
22400 REM:

22500 LOCATE 1,25
22600 PRINT "HON DO I SAVE DATA?"
22700 LOCATE 2,25
22300 PRINT "*********4**§*§§*§§”
2900 PRINT : PRINT
23000 PRINT ” Saving data to a disk is quite simple. First, you indicate whe
ther the"
23100 PRINT
23200 PRINT "data being saved is to be used by this program or by the DATA ANALY
SIS program."
23300 PRINT
23400 PRINT “ Second, you indicate which data File is to be saved. You then mere
ly give the"
23500 PRINT
23600 PRINT "disk drive ID & Filename to be saved under 0 the rest is done autom

61

atically.”

23700 LOCATE 22,24

23300 INPUT "PRESS ANY KEY TO RETURN TO MENU",OO$ : OOTO 9300

23900 CLS

24000 REM:

24100 REM: ************#********%*******§*§*§****§#*************
24200 REM: * HON DO I VIEW DATA? *
24300 REM: ********************o#***********§*************4*§**s
24400 REM: '

24500 LOCATE 1,25

24600 PRINT ”HON DO I VIEW DATA?"

24700 LOCATE 2,; S

24300 PRINT "****§****i**§******"

24900 PRINT : PRINT

25000 PRINT " Viewing the data at the terminal is done to check the values o
F the“

25100 PRINT

25200 PRINT "indicated File. By simply indicating which File you wish to view, t
he computer"

25300 PRINT

25400 PRINT "will set up the File For you with your header, pair headings, time
values"

25500 PRINT

25600 PRINT ", and X-Y coordinates."

25700 LOCATE 22,24

25300
25900
26000
26100
26200
26300
26400
26500
26600
26700
26300
26900
27000
g the
27100

INPUT ”PRESS ANY KEY TO RETURN TO MENU",OE$ : GOTO 9300

CLS
REM:
REM: ****§*&*******§*********§*****§****#**#*****§*oooee**
REM: * HON DO I PRINT DATA? *
REM: ****§¢**§******§**************sooe**%*§*e§***§******e
REM:

LOCATE 1,25

PRINT "HOW DO I PRINT DATA?"

LOCATE 2,25

PRINT “**********o*oeooooae"

PRINT : PRINT

PRINT ' Printing out your data File is a very simple process. By givin

PRINT

27200 PRINT "computer which File you wish to print, it will automatically print
out the File"
27300 PRINT

27400

PRINT ”with the appropriate File description, pair headings, & X-Y coordin

es.
27500 LOCATE 22,24

27600
27700
27300
27900
23000
23100
23200
23-300
23400
23500

 

INPUT ”F'RESB ANY KEY TO RETURN TO MENU”, 00$ = GOTO 3300

INPUT "PRESS ANY KEY TO RETURN TO MENU",C [Izs : OOTO 1400

CLS

REM:

REM: §I*************§****#§%§***#***§§****#****%******§***
REM: * HON DO I GRAPH DATA? *
REM: ***&**seeo**aos*4aaeee«seeo#****#**********e**4******
REM:

LOCATE 1,25

PRINT "HON DO I GRAPH DATA"'“
LOCATE 2,25

PRINT “§§*§**#§#§§fi**¢**§%é”

 

29300 PRINT
23900 pRINT
You will"

29000 PRINT
29100 PRINT
u must then"

 

 

29300 PRINT

r within”
29400 PRINT
29500 PRINT
29600
29700
29300
29900
30000
30100
30200
30300
30400
30500
30600
30700
30300
ch data
30900 PRINT
31000 PRINT
irll
31100
31200
ell
31300
31400
d and"
31500 PRINT
31600 PRINT
0F the data.
31700 LOCATE
3 300 INPUT
31900 CLS
32000 REM:
32100 REM:
32200 REM:
32300 REM:
32400 LOCATE
32500 PRINT

INPUT
CLS
REM:
REM:
REM:
REM:

PRINT

PRINT
PRINT
PRINT

PRINT
PRINT

PRINT
PRINT

LOCATE

62

PRINT
Graphing

” data is done by answering the appropriate questions.

“he graphing the X-Y coordinates For a certain File against time. Yo

"tell the computer which File is to be graphed & also which data pai

"that File, as it can only graph one pair at a time."

LOCATE 22,24

"PRESS ANY KEY TO RETURN TO MENU",CO$ : OTO 9300

G

«a*ea****4§*§*¢*§***§*§***§*******e*******a*******a*#
* FINITE DIFFERENCE TECHNIQUE *
***§*a*aa4*»***§*******§********a**§§**************§*

1,24
"FINITE DIFFERENCE TECHNIQUE"

LOCATE 2,24

"l§******%*****##***§***§§**”
PRINT
The Finite diFFerence technique

is based upon the idea that ea

"point when considered against the points on either side oF it & the
"corresponding time values can be ’smoothed’ to better Fit within th
"proportion on either side oF itselF. Each value is therFore weighte

"multiplied by a constant to give a better Fit to the overall shape

22,24
"PRESS ANY KEY TO RETURN TO MENU”,CC§

: OOTO 3300

*§***§*§-§**§ii-‘l'*flnfl-‘lf-IHI'*‘INI'4-#41")?*********§****#*#********

* LEAST SQUARES APPROXIMATION *
§***§**#4*******§*****#****#***#********************i
1,24
"LEAST SQUARES APPROXIMATION”

 

63

32600 LOCATE 2,24

PRINT "*********§***§*********§***“

PRINT : PRINT

PRINT Least squares approximation is based upon the idea that every

has" ‘ ‘

PRINT

PRINT "a certain deviation From its true value (an error va1ue), the Ieast
squares“

33200 PRINT .

33300 PRINT "technique takes into account the deviations oF all data points a at
tempts to"

33400 PRINT V

33500 PRINT "minimize the error. This method is highly recommended For Biomechan

 

33600 PRINT

33700 PRINT "data as it is Very powerFuT at giving a smooth curve representation
of a set“

33300 PRINT

33900 PRINT "oF data points.“

34000 LOCATE 22 24

 

34100 INPUT ”PRESS ANY KEY TO RETURN TO MENU",CC$ : GOTO 9300

34200 CLS

34300 REM: §***********%*********#******§****§*******é*#*§****
34400 REM: * FAST-FOURIER TRANSFORM *
34500 REM: ***§u****§**4§******#****§******4********§*#*******
34600 REM:

34700 LOCATE 1,29

34800 PRINT "FAST-FOURIER TRANSFORM"

34900 LOCATE 2,29

35000 PRINT "**§*§#****#***w***é§**"

35100 PRINT : PRINT

35200 PRINT “ The Fast-Fourier transForm is a trigonemetric may to smooth da

ta based"

35300 PRINT

35400 PRINT "on the concept of harmonic ana1ysis. The sine and cosine waves when
graphed"

35500 PRINT

35600 PRINT ”are a smooth, harmonic curve. Based on this concept, the raw data r
on through"

35700 PRINT

35300 PRINT "this transForm give a smooth representation of the raw data."

35900 LOCATE 22,24

36000 INPUT ”PRESS ANY KEY TO RETURN TO MENU",CC$ : GUTU 9300

36100 0L3

39200 REM: oeeo»**o**#*****§§*o***§*******4*§4§****§******%

36300
36400
36500
36600
36700
36300

64

CUBIC SPLINE FUNCTIONS *

REM: *

REM: *eeeeeeeeeeeee*********e*ee*********%*******ee*e
REM:

LOCATE 1,2?

PRINT "CUBIC SPLINE FUNCTIONS"

LOCATE 2,29 .

36900 PRINT "**********************"
37000 PRINT : PRINT _
37100 PRINT " Cubic spline Functions attempt to represent the

een data"
37200 PRINT
37300 PRINT
gh these"
37400 PRINT
37500 PRINT
t 0F data"
37600 PRINT
37700 PRINT
echnique“
37800 PRINT
37900 PRINT
some other"
33000 PRINT
33100 PRINT
38200
33300
38400
33500
33600
33700
33800
38900

INPUT
CLS
REM:
REM:
REM:
REM:
REM:

intervals betw

“points with a Function or equation representing a smooth line throu

"points. By taking the equations and using them through an

"points we can then represent the curve through those

“does introduce more error in the terms as it

“techniques."

LOCATE 22,24

"PRESS ANY KEY TO RETURN TO MENU",CC$ : COTO 9300

*********************%*§******§**********************

* BUTTERNORTH FILTERING *
********#***e*****************§*******%**********§**e

points.

entire se

This t

is not as powerFul as

39000
39100
39200
39300
39400
39500

LOCATE 1,29

PRINT

"BUTTERNORTH FILTERING"

LOCATE 2,29

PRINT
PRINT
PRINT

" ********************* ”

: PRINT

" Butterworth Filtering is an extension or special case

1 processing"

39600
39700

PRINT
PRINT

ct ’noise’

33800
39900

PRINT
PRINT

en when "
40000 PRINT
40100 PRINT

"Extensively used in engineering, Filtering oF data is

“From the signal. This noise crmes From the natural

"using electronic signals to record data. These Filters

done to

interFerence

oF signa

extra

giv

sort through

65

the data"
40200 PRINT
40300 PRINT "and pull out the true signal by giving a smooth picture oF what the
data may"
40400 PRINT
40500 PRINT "look like. The butterworth Filter uses this concept to smooth the r
aw data"
40600 PRINT'
40700 PRINT "into what seems to be a smooth representation oF the data."
40300 LOCATE 22,24
40900 INPUT "PRESS ANY KEY TO RETURN TO MENU",CC$ = OOTO 9300
41000 CLS : REM : EXIT PROGRAM
41100 LOCATE 4,15
41200 PRINT “********************4********************************"
41300 LOCATE 5,15
41400 PRINT TAB(15) "*" TAB(67) “*"
41500 LOCATE 6,15
41600 PRINT "*" : COLOR 0,7 : LOCATE 6,30 = PRINT ”DATA SMOOTHING PROGRAM" :
COLOR 7,0 : LOCATE 6,67 : PRINT "*"
41700 LOCATE 7,15
41300 PRINT TAB(15) "*" TAB<67> “*"
41900 LOCATE 3,15
42000 PRINT “*" : COLOR 0,7 : LOCATE 3,20 : PRINT "SCHOOL OF HEALTH EDUCATION, C
OUNSELINO” : COLOR 7,0 : LOCATE 3,67 : PRINT "4"
42100 LOCATE 9,15
42200 PRINT TAB(15) "e" TAB(67) 4*"
42300 LOCATE 10,15
42400 PRINT "*" : COLOR 0,7 : LOCATE 10,26 : PRINT "PSYCHOLOGY o HUMAN PERFORMAN
CE" : COLOR 7,0 : LOCATE 10,67 : PRINT "*"
42500 LOCATE 11,15
42600 PRINT TAB(15) "*" TAB(67) "*"
42700 LOCATE 12,15
42300 PRINT "*" : COLOR 0,7 : LOCATE 12,29 = PRINT "MICHIGAN STATE UNIVERSITY" =
':0LOR 7,0 : LOCATE 12,67 : PRINT "*0.
42900 LOCATE 13,15
43000 PRINT TAB<15> "*" TAB<67> "*"
43100 LOCATE 14,15
43200 PRINT "*****************************4****e*******##**#**e***"
43300 LOCATE 16,20
43400 PRINT ”DID YOU REMEMBER TO SAVE YOUR DATA!!!"
3500 LOCATE 13,13
43600 PRINT "RETURN TO MAIN MENU.... ............ ....... 1”
43700 LOCATE 19,13
43300 PRINT "EXIT PROGRAM... ..... ... ...... . ........... . 2"
43900 LOCATE 22,27
44000 INPUT "YOUR SELECTION PLEASE: ",3$

66

44100 IF S5 = "2" THEN GOTO 44300
44200 SOTO-1400

44300 KEY ON

44400 END

1000
1100
1200
1300
1400
1500
1600
1700
1300
1900
2000
2100
2200
2300
2400
2500
2600
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3100
3200
3300
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3500
3600
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3300
3900
4000
4100
4200
4300
4400
4500
4600
4700
4300
4900
5000
5100
5200
5300
5400
5500
5600

67

CL3

REM:

REM: SUBROUTINE TO INPUT DATA (1) INPUT DATA FROM TERMINAL
REM: (2) INPUT DATA FROM A DISKETTE
REM:

REM: THE DATA POINTS ARE INPUTTED IN THE FORMAT:

REM:

REM: ._ X-COORDINATE Y-COORDINATE
REM:

LOCATE 5,30

PRINT "INPUTTING DATA"

LOCATE 6,23

PRINT "FOR DATA SMOOTHING"

LOCATE 7,23

PRINT "******************"

LOCATE 9,20

PRINT "PLEASE SELECT ONE OF THE FOLLOWING:"
LOCATE 11,15

PRINT "INPUT DATA FROM TERMINAL.... ....... ...... 1"
LOCATE 12,15
PRINT "INPUT DATA FROM A DISK.. ..................... 2"
LOCATE 13,15
PRINT "RETURN TO MAIN MENU... ......... . ......... .3"

LOCATE 15,25

INPUT "YOUR SELECTION PLEASE:";N$

IF N$ = ”1" THEN OOSUB 4200

IF N5 = ”2“ THEN OOSUB 17200

IF NS =- "3" THEN CHAIN "A:S.MAIN",1000,ALL
CLS : REM: INPUT ERROR

ER! ' "A:3.INPUT" : LN = 4100

CHAIN “ :3.ERROR“,1000,ALL

GOTO 1000

CLS

REM: ***§**************************4**********
REM: * INPUTTING DATA FROM THE TERMINAL *
REM: *********4*4*****************************

LOCATE 5,25

PRINT "INPUTTING DATA FROM TERMINAL"

LOCATE 6,25

PRINT "*******************+********"

IF DP > 1 THEN OOTO 15600

LOCATE 3,19

PRINT “ENTER PAIRS OF DATA POINTS WHEN PROMPTED"

LOCATE 10,13

PRINT "AFTER TEN PAIRS HAVE BEEN ENTERED, THEY CAN BE REVIEWED"
LOCATE 12,15

PRINT “YOU MAY ENTER A TWO LINE HEADER FOR DATA DESCRIPTION"

5700
5800
5900
6000
6100
6200
6300
6400
6500
6600
6700
6800
6900
7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
9100
9200
9300
9400
9500
9600
9700
9800
9900

68

LOCATE 17,25
INPUT "READY TO BEGIN..........(Y/N) ";st
IF YN$ = "N" THEN OOTO 4200

CLS = REM: INPUTTINC DATA DESCRIPTION
LOCATE 5,1

INPUT "ENTER FIRST LINE OF DESCRIPTION PLEASE: ”;D1$
LOCATE 7,1 .‘

INPUT "ENTER SECOND LINE OF DESCRIPTION PLEASE: "gDES
HC = 100 : REM: HEADER CHECK FOR VIEWING DATA.

CLS

LOCATE 5,25
PRINT "THIS IS HON YOUR HEADER LOOKS="
LOCATE 6,25
PRINT "******************************"
LOCATE 10,1
PRINT D13
LOCATE 11,1
PRINT 02$
LOCATE 15,23
INPUT "IS THIS CORRECT. ......... (Y/N) "gYN$
IF YNS = “N" THEN OOTO 6000
CLS : REM: INPUT PAIRS OF DATA POINTS.
LOCATE 5,14
INPUT "HON MANY PAIRS OF DATA POINTS WILL YOU HE INPUTTING ",NP
LOCATE 8,19
PRINT "ENTER DATA HEADERS FOR PAIRS NHEN PROMPTED"
LOCATE 10,24
PRINT "EX: PAIR 1=RIOHT ANKLE ’R.ANKLE’"
LOCATE 15,1
FOR I = 1 TO NP
PRINT "HEADER FOR PAIR # “:1
INPUT " =9",PH$(I) = PH$(I) = LEFT$<PH$(I),S) : PRINT
NEXT I
CLS : REM: SET UP DATA ENTRY.
LOCATE 3,17

INPUT "IS THE TIME INTERVAL A CONSTANT .......... (Y/N)”,YN$
HN a DP : REM: HN KEEPS TRACK OF CURRENT HEADER NO.
IF YN$ 8 "Y" THEN TIC = 2

I = 1 : REM: I KEEPS TRACK OF NHICH HEADER THE PAIRS ARE UNDER.
IF YNs = “Y" THEN BOTO 12500
CLS = REM: INPUT PAIRS FDR EVEN/UNEVEN TIME INTERVAL.
LOCATE 5,13 : COLOR 0,7
PRINT "ENTER ’E’ FOR FIRST VALUE FOR END-OF-DATA OF THE HEADER"

10000 COLOR 7,0
10100 LOCATE 3,20
10200 PRINT "ENTER DATA PAIR # u; : COLOR 0,7 : PRINT HN; : COLOR 7,0 : PRINT

FOR HEAOINC: u; :

10300
10400
10500
10600
10700
10800
10900

11000'

11100
11200
11300
11400
11500
11600
11700
11800
11900
12000
12100
12200
12300
12400
12500
12600
12700
12300
12900
13000

3100
13200
13300
13400
13500
13600
13700
13300
13900
14000
14100
14200
14300
14400
14500
14600
14700
14300

69

PRINT PH$(I)

IF YN$ = "N" THEN LOCATE 15,33
IF YNs = "N" THEN INPUT "TIME VALUE =:",HALT$
IF YN$ = "N" AND HALT$ = "E" THEN COTO 11300
IF YN$ = "N" THEN TV(DP) = VAL(HALT$)

OOSUB 10900

COTO 11300

LOCATE 10,32

INPUT “X-COORDINATE =:",HALT$

IF HALTs = "E" THEN OOTO 11000

X1(DP) = VAL(HALT$)

LOCATE 20,32

INPUT "Y-COORDINATE =:",Y1(DP)

CHECK = ((HN/10) - INT(HN/10))*100

IF CHECK = 0 THEN OOTO 13200

OP = DP + 1 : HN = HN + 1 : RETURN

CLS : REM: CHECK FOR END OF HEAOER.

IF HALTs = "E" AND I = NP THEN GOTO 14300

IF HALTs a “E" THEN I=I+1

IF HALT$ = "E" AND VN>HN THEN VN=VN+<HN-(10*INT(HN/10)))-1
IF HALTs = “E" AND VN<HN THEN VN=VN+<HN~VN—1>
IF HALTs = "E" THEN HN=1 : COTO 9700

OOTO 9700

CLS : REM: INPUT CONSTANT TIME INTERVAL VALUE.

LOCATE 10,22
INPUT "ENTER CONSTANT FOR TIME INTERVAL =:",TC

FOR II = 1 TO 100
TV(II) = TC
NEXT II
OOTO 9700
CLS : REM: DATA CHECK FOR PAIRS.

IF VN>HN THEN VN=VN+10 ELSE VN=VN+<HN—VN)
LOCATE 1,20

PRINT "DATA PAIRS ";VN-9;" THROUOH ";VN;" ARE: "
LOCATE 3,1

PRINT PH$(I);" DATA LISTING: “ : PRINT
PRINT TAB(5) "TIME" TAB<22) “X-COORDINATE"

FOR II = VN-9 TO VN

TAB(55)“Y—COORDINATE" : PRINT

PRINT TAB(2) TV<II) TAB(25) X1(II) TAB<55) Y1(II)
NEXT II
LOCATE 20,13
INPUT “ARE THESE OATA POINTS CORRECT.... ...... (Y/N)",NY$
IF st a "v" THEN OP=OP+1 : HN=HN+1 : OOTO 9700

LOCATE 22,14

PRINT “YOU WILL HAVE TO REENTER ALL TEN PAIRS OF DATA POINTS"
FOR II = 1 TO 10000 : NEXT II : DP = DP -9 : GOTD 9700
CLS : REM: END OF DATA FILE.

14900
15000
15100
15200
15300
15400
15500
15600
15700
15300
15900
16000
16100
16200
16300
16400
16500
16600
16700
16300

32(1),

16900
17000
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17300
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17500
17600
17700
17300
17900
13000
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13200
13300
13400
13500
18600
13700
13300
13900
19000
19100
19200

70

REM: NP N . OF DATA PAIRS.
REM: DP NO. OF DATA POINTS.
LOCATE 5,32

PRINT "END OF DATA FILE"
LOCATE 22,22

INPUT “PRESS ANY KEY TO RETURN TO MAIN MENU“,CC$

CHAIN_“A:S.MAIN",1000,ALL

CLS

LOCATE 5,22 '

PRINT "YOU HAVE A RAN DATA FILE IN MEMORY!"
LOCATE 7,18

PRINT "READING IN A NEW FILE HILL DESTROY THIS FILE”'

LOCATE 10,19

PRINT "YOU SHOULD SAVE YOUR RAN DATA FILE TO KEEP"

LOCATE 11,30
PRINT "FROM DESTROYINO IT!"

LOCATE 15,16
INPUT "DO YOU UANT TO ENTER THE NEN FILE .......... (Y/N)”,NY$
IF NYs = "N" THEN OOTO 17100

CLEAR : OIM X1(500),Y1(500),NP(100),PH$(50),TV(100),BF1(1),BF2(1),C31(1),C

CP1(1),CP2(1),FSI(1),FS2(1),LSI(1),LS2(1)
DP 3 1

COTO 4200

CHAIN "A:S.MAIN",1000,ALL

CLS

REM:

REM: **§**§*********§**********************************#**
REM: * INPUTTING DATA FROM DISK *
REM: §****************************************************
REM:

LOCATE 5,25

PRINT "INPUTTING DATA FROM DISK”

LOCATE 6,25

PRINT “**********fi***i***§*****"

IF DP } 1 THEN COTO 23800

LOCATE 10,22

PRINT "PLEASE SELECT ONE OF THE FOLLOWINO:"
LOCATE 12,16

PRINT “INPUT DATA SMOOTHING DATA FROM DISK..

LOCATE 13,16

PRINT "INPUT DATA ACQUISITION DATA 6 FORMAT .........

LOCATE 14,16

PRINT "RETURN TO MAIN MENU... .......................

LOCATE 20,28
INPUT "YOUR SELECTION PLEASE:",N$

":- N
A'-

4') n
v“.

19300
19400
19500
19600
19700
19300
19900
20000
20100
20200
20300
20400
20500
20600
20700
20300
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21300
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21900
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22700
22300
22900
23000
23100
23200
N THE
23300
23400
23500
23600
23700

71

IF N5 = "1" THEN OOSUB 20000
IF NS = "2" THEN OOSUB 22300
IF N5 = "3" THEN CHAIN "A:S.MAIN”,1000,ALL

CLS : REM: INPUT ERROR
ER$ = "A:3.INPUT" : LN = 19300
CHAIN "A:S.ERROR",1000,ALL
OOTO 17200
CLS : REM: INPUT THE DATA SMOOTHING DATA.
LOCATE 10,25
PRINT “YOUR FILE NAME 5 DISK DRIVE ID"
LOCATE 11,24
PRINT "ARE ENTERED IN THE FOLLONINO FORM:"
LOCATE 15,20
PRINT ”FOR DISK DRIVE: ’3’ & FILENAME: ”SAVE’"
LOCATE 17,37
COLOR 0,7 : PRINT "B:SAVE" : COLOR 7,0
LOCATE 20,31
INPUT "FILE NAME PLEASE: ",F$
OPEN F5 FOR INPUT A3 #1
DP = 1
INPUT #1,NP
INPUT 91,015
INPUT $1,025
FOR dd = 1 TO NP
INPUT #1,PH$(JU)
NEXT dd
IF EOF (1) THEN GOTO 22400
INPUT #1,TV(DP),X1(DP),Y1(DP)
DP = DP + 1
OOTO 21900
CLS : REM: INPUT DATA ACQUISITION DATA & FORMAT APPROPRIATELY.
CLS : REM: DATA INPUTTED.
DP = DP - 2
IF TV(1) = TV(2) THEN TIC = 2
LOCATE 5,29
PRINT ”DATA HAS BEEN INPUTTED“
LOCATE 6,29
PRINT "**********************“
LOCATE 10,24

PRINT "THERE WERE "; : COLOR 0,7 : PRINT DP; : COLOR 7,0 : PRINT " PAIRS I
FILE."

CLOSE #1

LOCATE 20,21

INPUT "PRESS ANY KEY TO RETURN TO MAIN MENU”,CC$

HC = 100 : REM: HEADER CHECK FOR VIEUINO OATA.
CHAIN "A:S.MAIN",1000,ALL

CLO

LOCATE 5,27

72

24000 PRINT “YOU HAVE A RAN DATA FILE IN MEMORY!"

24100 LOCATE 7,13

24200 PRINT "READING IN A NEW FILE WILL DESTROY THIS FILE"

24300 LOCATE 10,13

24400 INPUT "DO YOU NANT TO READ THE FILE..........(Y/N)",YN$

24500 IF YN$ = "N" THEN CHAIN "A:S.MAIN",1000,ALL

24600 CLEAR : DIM X1(500),Y1(500),NP(100),PH$(50),TV(100),BF1(1),BF2(1),C31(1),C
S2<1>,CP1(1),CP2(1),FSl(1),FS2(1),L31(1),L32(1)

24700 DP = 1

24300 OOTO 17200

 

 

1 000
1 100
1200
1300
1400
1500
1600
1700
1300
1900
2000
2100
2200
2300
2400
2500
2600
2700
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3000
3100
3200
3300
3400
3500
3600
3700
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3900
4000
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4200
4300
4400
4500
4600
4700
4rur

   

4900
5000
5100
5200
5300
5400
5500
5600
5700
5300
5900
6000
6100
é200
6300

73

CLS

REM:

REM: 4***********+***********§**§******************§*****#
REM: * SAVING DATA TO A DISK SURROUTINE

REM: ******§**************§***********+****§**************
REM:

LOCATE 5,30

PRINT "SAVING DATA TO"
LOCATE 6,32

PRINT "A DISKETTE”

LOCATE 9,20

PRINT"PLEASE SELECT ONE OF THE FOLLOWING:"

LOCATE 11,12

PRINT ”SAVE DATA TO A DISKETTE........................... 1“
LOCATE 12,12

PRINT ”SAVE DATA IN FORMAT FOR DATA ANALYSIS PROGRAM..... 2
LOCATE 13,12

PRINT “RETURN TO MAIN MENU............................... 3"
LOCATE 14,25

INPUT“YOUR SELECTION PLEASE:";NS

IF N$ = "1" THEN GOBUB 3900
IF N$ = "2" THEN BOTO 1000
IF N$ = "3" THEN CHAIN "A:S.MAIN",1000,ALL

CLS : REM: INPUT ERROR

ERS = "A:S.SAVE" : LN = 3800

CHAIN "A:3.ERROR”,1000,ALL

GOTO 1000

CLS : REM: SAVING DATA TO A DISKETTE SUBROUTINE.

LOCATE ,25

PRINT ”SAVING DATA TO A DISKETTE"

LOCATE 2,25

PRINT "*********§*******4§§§§§**"

LOCATE 4,30

PRINT "SAVE TO DISK:"

LOCATE 6,15

PRINT "BUTTERWORTH FILTERING BMOOTHED DATA..........(BF)"
LOCATE 7,15

PRINT ”CUBIC SPLINE SMOOTHED DATA........... ..... ...(CE)”
LOCAT 3,15

PRINT "CHEBYSHEV POLYNOMIAL SMOOTHED DATA...........(CP)"
LOCATE 9,15

PRINT "FOURIER SERIES SMOOTHED DATA..............
LOCATE 10,15

PRINT ”LEAST SQUARES SMOOTHED DATA..................(L3)“
LOCATE 11,15

PRINT "RAW DATA.... ..... .............. ..... ......
LOCATE 12,15

PRINT ”RETURN TO MENU.............z.................(MM)”

LOCATE 14,15

PRINT "MAKE YOUR SELECTION BY ENTERING THE TwO DIOIT CODE"
LOCATE 15,12

PRINT “IF YOU wOULD LIKE THE RESULTANT VELOCITY OR ACCELERATION”

.(FS)"

.(RO)”

 

74

6400 LOCATE 16,13

6500 PRINT "FROM THE DATA PLACE AN (A) OR (V) IN FRONT OF THE CODE”

6600 LOCATE 13,12

6700 PRINT "EX: TO 3AVE RESULTANT ACCELERATION FROM THE SMOOTHED DATA”

6300 LOCATE 19,27
6900 PRINT "BY CUBIC 3PLINE FUNCTION:“
7000 LOCATE 20,34

7100 PRINT "ENTER: "3 : COLOR 0,7 : PRINT "AC3" : COLOR 7,

7200 LOCATE 22,23
7300 INPUT "YOUR SELECTION PLEASE:",30$
7400 REM:

7500 REM: THE DATA HILL BE SAVED WITH THE TIME-VALUE BEINO

7600 REM: FOLLOWED BY X & Y COORDINATES.
7700 REM:
7800 DIM T1(DP),T2(DP)

7900 IF 30$ = "BF” THEN FOR I = 1 TO DP : T1(I) = BF1(I)
NEXT I : OOTO 9000

3000 IF 30$ = “C3" THEN FOR I = 1 TO DP : T1(I) = C31(I)
NEXT I : GOTO 9000

3100 IF 30$ = "CP" THEN FOR I = 1 TO DP : T1(I) = CP1(I)
NEXT I : GOTO 9000

3200 IF 303 = "F3" THEN FOR I = 1 TO DP : T1(I) = F31(I)
NEXT I : GOTO 9000

3300 IF 30$ = "R0" THEN FOR I = 1 TO DP : T1(I) = X1(I) :
NEXT I 3 GOTO 9000

8400 IF 305 = “L3" THEN FOR I = 1 TO OR : T1(I) = L31(I)
NEXT I : GOTO 9000

3500 IF 30$ = "ABF" OR 305 = ”AC3" OR 30$ = "AFD" OR 30$
0R 30$ = ”ARD" THEN FOR I = 1 TO DP : T1(I) = AC1(I)
NEXT I : OOTO 9000

3600 IF 305 = “VBF” OR 30$ = "VCS" OR 30$ = "VFD" OR 30$
OR 309 = "VRD" THEN FOR I = 1 TO OR : T1(I) = VC1(I)
NEXT I 3 OOTO 9000

8700 IF sns — "MM" THEN ERASE T1 : ERASE T2 : GOTO 1000
3300 REM: INPUT ERROR : 00309 700

3900 OOTO 1000

9000 REM:

9100 REM: ENTER FILE NAME & DISK DRIVE.

9200 REM:

9300 CLS

9400 LOCATE 5,20

9500 PRINT "ENTER YOUR FILE NAME & DISK DRIVE ID”

9600 LOCATE 7,21

9700 PRINT "ACCORDING TO THE FOLLONINO FORMAT”

9300 LOCATE 10,15

9900 PRINT "EX: FOR DISK DRIVE ’A’ AND FILENAME ’3AVE’="
10000 LOCATE 12,25

0

STORED FIRST

T2(I) = BF2(I) :

T2(I) = C32(I) '
T2(I) = CP2(I) :
T2(I) = F32(I) :
T2(I) = Y1(I) :

T2(I) = L32(I) :

"APE" OR 30$ = ”ALS"
: T2(I) = AC2(I)

"VFS" OR 305 = "VLS"
: T2(I) = VC2(I)

10100
10200
10300
10400
10500
10600
10700
10800
10900
11000
11100
11200
11300
1.1400
(1500
11600
11700
11300
11900
12000
12100
12200

75

PRINT "ENTER: "; : COLOR 0,7 : PRINT "A=3AVE" : COLOR 7,0
LOCATE 20,25
INPUT "FILE NAME PLEASE: “,Fs
LOCATE 22,15
INPUT uDOES THIS FILE ALREADY EXIST. .............. (Y/N)",YN$
IF YN$ = "v" THEN OPEN F$ FOR APPEND AS #2 ELSE OPEN Fs FOR OUTPUT AS #1
IF YNs = “N" THEN wRITE #1,NP : REM: NP = NO. OF HEADERS.
IF YNs = "N" THEN HRITE #1,Dl$
IF YNs = "N" THEN NRITE 31,025
FOR 00 = 1 TO NP
IF VNs = "N“ THEN wRITE #1,PH$(JJ)
NEXT 00
FOR II = 1 TO DP
IF YN$ = "v" THEN uRITE #2,TV(II),T1(II),T2(II)
IF YNs = "N" THEN wRITE #1,TV(II),T1(II),T2(II)
NEXT II
IF YN$ = "v" THEN CLOSE #2 ELSE CLOSE #1
ERASE T1 : ERASE T2
CLS : REM: DATA SAVED.

LOCATE 10,15
INPUT ”DATA SAVED, PRE33 ANY KEY TO RETURN TO MAIN MENU",CC$
CHAIN "A:3.MAIN",1000,ALL

76

1000 CL3
1200 REM:
1400 REM: 54*********************************************%*****
1600 REM: 9 VIEwINO DATA AT THE TERMINAL *
1300 REM: **********************************§******************

2000 REM:

2200 DIM T1(DP),T2(DP) : REM: TEMPORARY ARRAYS.

2400 LOCATE 1,25

2600 PRINT "VIEWING DATA AT THE TERMINAL"

2500 LOCATE 2,25

3000 PRINT "*********§******************"

3200 LOCATE 4,35

3400 PRINT "VIEW"

3600 LOCATE 6,15

3300 PRINT "BUTTERNORTH FILTERING SMOOTHED DATA..........(BF)"

4000 LOCATE 7,15

4200 PRINT "CUBIC SPLINE SMOOTHED DATA...................(C3)"

4400 LOCATE 5,15

4600 PRINT "CHEBYSHEV POLYNOMIAL SMOOTHED DATA...........(CP)"

4800 LOCATE 9,15

5000 PRINT "FOURIER SERIES SMOOTHED DATA.................(FS)"

5200 LOCATE 10,15

5400 PRINT "LEAST SQUARES SMOOTHED DATA............ ..... .(LS)"

5600 LOCATE 11,15

5800 PRINT "RAN DATA.....................................(RD)"

6000 LOCATE 12,15

6200 PRINT "RETURN TO MENU...............................(MM)"

6400 LOCATE 14,15

6600 PRINT “MAKE YOUR SELECTION BY ENTERING THE TWO DICIT CODE"
6800 LOCATE 15,10

7000 PRINT "IF YOU NOULD LIKE TO VIEN THE RESULTANT VELOCITY OR ACCELERATION"
200 LOCATE 16,10

7400 PRINT "FROM THE DATA PLACE AN (A) 0R (V) IN FRONT OF THE CODE"
7600 LOCATE 15,10

7800 PRINT "EX: TO VIEW RESULTANT ACCELERATION FROM THE DATA SMOOTHED"
5000 LOCATE 19,15

3200 PRINT "BY LEAST SQUARES APPROXIMATION"

S400 LOCATE 20,30

5600 PRINT "ENTER: ": : COLOR 0,7 : PRINT "ALS" : COLOR 7,0

3300 LOCATE 22,25

9000 INPUT "YOUR SELECTION PLEASE:”,SD$

9200 IF 30$ = “BF" THEN FOR I = 1 TO DP = T1(I) = BF1(I) : T2(I) = BF2(I) :

NEXT I : OOTO 11400

9400 IF 30$ = "CS“ THEN FOR I = 1 T0 DP : T1(I) = 031(1) : T2(I) = 532(1) :
NEXT I : GOTO 11400

9600 IF 305 = “OP" THEN FOR I = 1 TO DP : T1(I) = CP1<I> : T2(I) = CP2(I) :
NEXT I : OOTO 11400

7300 IF 30$ ”F3" THEN FOR I = 1 TO 0P : T1(I) = F31(I) 3 T2(I) F32(I) :

 

 

 

1 0000

10200

10400

10600

10800
11000
11200
11400
11600
11300
11301
12000
12200
12400
12600
12300
13000
13200
13400
13600
13300
14000
14200
14400
14600
14300
15000
15200
15400
15600
15300
16000
16200
16400
16600
16300
17000
17200

77

NEXT I : OOTO 11400
IF 80$ = "LS" THEN FOR I = 1 TO OR : T1(I) = L31(I) : T2(I) = L32(I) :
NEXT I : OOTO 11400
IF 30$ = "R0" THEN FOR I = 1 TO DP : T1(I) = X1(I) : T2(I) = Y1(I) :
NEXT I : OOTO 11400
IF 505 = "ABF" OR 305 = "ACS" OR 30$ = "AFD" OR 50$ = "AFB" OR 30$ = "ALB"
OR 305 = "ARD" THEN FOR I = 1 TO OR : T1(I) = AC1(I) : T2(I) = AC2(I) :
NEXT I : OOTO 11400
IF 305 = "VBF" OR 30$ = "VCS" OR 305 = "VFO" OR 30$ = "VF5” OR 505 = “VLS”
OR 30$ = "VRD" THEN FOR I = 1 TO DP : T1(I) = VC1(I) : T2(x) = VC2(I) :
NEXT I : OOTO 11400
IF 80$ = "MN" THEN ERASE T1 : ERASE T2 : CHAIN "A:3.MAIN",1000,ALL
REM: INPUT ERROR : COSUE 700
GOTO 1000
CLS
00300 13400
L1 = 1 : NH=DP/NP : REM: NH = NO. OF ENTRIES UNDER EACH HEADER.
PRINT "NH: "gNHg" 0P: ";0P;” NP= "gNP
IF NH >= 10 THEN L2 = 10 ELSE L2 = NH
LOCATE 1,32 : COLOR 0,7 : PRINT ”FILE DESCRIPTION” : COLOR 7,0
IF HC = 100 THEN LOCATE 3,1 : PRINT 01$
IF HC = 100 THEN LOCATE 4,1 : PRINT 02$
IF HC 4) 100 THEN LOCATE 3,1 : PRINT “PRESENT FILE HAS”
IF HC C} 100 THEN LOCATE 4,1 : PRINT "NO FILE DESCRIPTION”
LOCATE 6,1
FOR OJ = 0 TO NP—l
PRINT TAB((UU+1)*20+3) PH$(JU+1);
NEXT JO
LOCATE 7,1 : PRINT TAB(2) "TIME VALUE";
FOR dd = 0 TO NP—1
PRINT TAB((OJ+1)*20+1) "X" TAB((JU+1)*20+12) "Y";
NEXT JO
LOCATE 9,1
FOR II = L1 TO L2 : PRINT TAB(3) PT(II);
TC=0
FOR KK = II TO DP STEP NH
PRINT TAB<<TC+1)*20) T1(KK) TAB<<TC+1>R20+10) T2(KK);
TC = TC + 1
NEXT KK
NEXT II : PRINT
IF L2 = NH THEN GOTO 17600
LOCATE 22,20
INPUT "PRESS ANY KEY TO VIEN NEXT SCREEN",CC$
L1 = L1 + 10
IF NH }= L2 + 10 THEN L2 = L2 + 10 ELSE L2 = NH

78

17400 CLS 3 GOTD 12200

17600 LOCATE 22,15

17800 INPUT "END OF DATA--PRE33 ANY KEY TO RETURN TO MENU",CC$
13000 ERASE T1 : ERASE T2 : ERASE PT

13200 BOTO 1000

13400 REM:

13600 REM: SET UP TIME VALUES.

13300 REM:

19000 DIM PT (DP+1)

19200 FOR I = 1 TD UP

19400 PT(1) = 0

19600 IF TIC = 2 THEN PT(I+1) = PT(I) + TV(I) ELSE PT(I) = TV(I)
19300 IF PT(I) <} 0 THEN PT(I) = (INT(PT(I)*1000))/1000

20000 NEXT I

20200 RETURN

1000
1100
1200
1300
1400

1500
1600
1700

1300
1900
2000
2100
2200
2300
2400
2500
2600
2700
2300
2900
3000
3100
3200
3300
3400
3500
3600
3700
3300
3900
4000
4100
4200
4300
4400
4500
4600
4700
4300
4900
5000
5100
5200
5300
5400
5500
5600
5700

79

CLS

REM:

REM: ****§Rfi%*********%§**************§******************
REM: 5 DATA SMOOTHING TECHNIQUES *
REM: 5554***%********************************************
REM:

LOCATE 5,25

PRINT "DATA SMOOTHING TECHNIQUES"

LOCATE 6,25

PRINT "*******%*********%*******"

LOCATE 3,20

PRINT "PLEASE SELECT ONE OF THE FOLLOWING:”

LOCATE 10,15

PRINT "CHEBYSHEV POLYNOMIALS......... ..... . ............ 1"
LOCATE 11,15
PRINT "LEAST 550ARE5 APPROXIMATION.. ................... 2"
LOCATE 12,15
PRINT "FOURIER 5ERIE5 ....... . .................... ...... 5"
LOCATE 15,15
PRINT "CUBIC SPLINE FUNCTION ................... . ....... 4"
LOCATE 14,15
PRINT "BUTTERuORTH FILTERING... ..... . ......... ......... 5"
LOCATE 15,15
PRINT "CALCULATE VELOCITY OF RAN DATA... ............... 6"
LOCATE 16,15
PRINT "CALCULATE ACCELERATION OF RAN DATA .............. 7"
LOCATE 17,15
PRINT "RETURN TO MAIN MENU.................. ......... .. 5"

LOCATE 20,25

INPUT "YOUR SELECTION PLEASE:”;N$
IF N5 = "1" THEN GOSUB 5000

IF N5 ="2" THEN OOSUB 10300

IF NS = "3" THEN OOSUB 23000

IF N5 = "4" THEN OOSUB 23600

IF N5 = "5" THEN OOSUB 33600

IF N0 = "6" THEN VV$ = "RD” : OOSUB 50000 : OOTO 1000

IF N5 = "7" THEN AAs = "RD” : OOSUB 55000 : GOTO 1000

IF NS = "3" THEN CHAIN "A:5.MAIN",1000,ALL

REM: INPUT ERROR : OOSUB 700

GOTO 4000

CLS

REM:

REM: ********5******4*544********#****************555*5545
REM: * CHEBYSHEV POLYNOMIAL *
REM: §**%**#*****%*4**%*#4%*****%****%******#*%****%******
REM:

LOCATE 5,25
PRINT "CHEBYSHEV POLYNOMIALS”

3005
3010
3015
3020
3021
3022

80

LOCATE 6,25

PRINT "§********§******§#***“

LOCATE 10,20

PRINT "PLEASE SELECT ONE OF THE FOLLOWING:"

LOCATE 12,15

PRINT "SMOOTH WITH CHEBYSHEV POLYNOMIALS................. 1"
LOCATE 13,15

PRINT "CALCULATE VELOCITY OF SMOOTHEO OATA...............
LOCATE 14,15

PRINT ”CALCULATE ACCELERATION OF SMOOTHED DATA...
LOCATE 15,15

PRINT "RETURN TO MAIN MENU...............................
LOCATE 20,20

INPUT "YOUR SELECTION PLEASE: ",NS

M

u
a
.
u
I
u
.
s
1,1

{-

IF NS = "1” THEN OOSUB 7300

IF NS = ”2” THEN VV$ = "FD" : OOSUB 50000 : COTO 5000
IF NS = “3" THEN AA$ = "FD" : GOSUB 55000 : OOTO 5000
IF NS = ”4" THEN CHAIN "A=5.MAIN", 1000,ALL

REM: INPUT ERROR : GOSUB 700

BDTU 5000

CLS : ERASE 0P1,CP2,FSI,F32,CSI,C32,BFI,BF2,L81,L52

DIM CP1(DP),CP2(DP)

CLS : LOCATE 5,17 : INPUT "PLEASE INPUT THE DEGREE OF THE PDLYNUMIAL =:",PD

PI = 3.141592 : N = PD

DIM C1(N),C2(N),F1(N),F2(N)

CLS : LOCATE 10,25 : COLOR 0,7

PRINT ”CHEBYSHEV POLYNOMIAL ROUTINE NORKINC....." : COLOR 7,0
OOTO 9000 : REM:FLAO=9 : REM:OOTO 13100

STUPID = 1
FLAO = 9 : IF STUPID >= 3 THEN COTO 3350 ELSE OOTO 14600
FLAO = O : A = PT(1) : B = PT(DP)
BMA = .5 * (B - A)
BPA = .5 * (B + A)
FOR K 1 TO N

Y = COS(PI * (K — .5))/N
IF STUPID = 2 THEN F1(K) = C(l) + C(2) * (Y*BMA+BPA) + C(3) * (Y*BMA+BP
C(4) * (Y*BMA+BPA)

IF STUPID = 3 THEN F2(K) = C(1) + C(2) # (Y*BMA+BPA) + C(S) * (Y*BMA+BP
C(4) * (Y*BMA+BPA)

= 5UM1 + F1(K) # COS((PI * (J-1)) * ((K - .5)/N))
3UM2 = 5UM2 + F2(K) * COS((FI * (J-1)) * ((K - .5)/N))

 

81

11600 LOCATE 13,15

11700 PRINT "CALCULATE VELOCITY OF SMOOTHED DATA....... ..... .. 2“
11300 LOCATE 14,15

11900 PRINT "CALCULATE ACCELERATION OF SMOOTHED DATA.......... 3“
12000 LOCATE 15.

12100 F'RINT ”RETURN TO MAIN MENU......... ...... ............... 4"
12200 LOCATE 20,20

2300 INPUT ”YOUR SELECTION PLEA'SE: ”,NS

12400 IF N5 "1" THEN OOSUB 13000

12500 IF N$ "2" THEN VVS = COSUB 50000 : OOTO 10300
12600 IF NS "3" THEN AA$ = ”LS" OOSUB 55000 : COTO 10300
12700 IF NS — "4" THEN CHAIN "A:S.MAIN",1000,ALL

12300 REM: INPUT ERROR : OOSUE 700

12900 OOTO 10300

13000 CLS : ERASE CP1,|3P2,FS1,F'32,I-:1 C'32,BF1, BF2,L51,LS2

13100 DIM L31(DP),LS2(DP) : GD'SUB 22000

13200 NH = DP/NP : REM: NH = NO. OF ENTRIES UNDER EACH HEADER.
13300 KK = 5 : REM: NUMBER OF COEFFECIENTS.

13400 STUPID = 1

3500 DIM C(6) : REM: ARRAY OF COEFFECIENTS TO BE FOUND.

13600 DIM A(6,7): REM: TEMPORARY ARRAY.

13700 REM:

3300 REM: THIS ORDER MAY BE SET UP SO THAT IT CAN BE INPUTTED TO SET
13900 REM: THE BEST POSSIBLE FIT. WE CHOOSE A 4TH ORDER HER FOR REASON
14000 REM: OF LESS COMPUTER MEMORY.

14100 REM:

14200 REM: RESULTING POLYNOMIAL WILL BE OF THE FORMAT:

14300 REM:

14400 REM: Y = C(1) + C(2)*X + C(3)*X“2 + C(4)*X“3 + C(5)*X“4.
14500 IF STUPID >= 3 AND FLAO = 0 THEN COTO 21500

14550 IF FLAC = 9 AND STUPID } 1 THEN OOTO 3030

14600 REM:

14610 CLS : LOCATE 10,20 : COLOR 0,7

14620 PRINT "LEAST-SQUARES APPROXIMATION NORKINO.....” : COLOR 7,0
14700 REM: LOAD THE A-ARRAY.

14300 FOR II = 1 TO DP STEP NH

 

  
 

14900 FOR L = 1 TO KK

15000 FOR M = 1 TO KK

15100 51 = 0 : 32 = 0

15200 FOR I = II TO (II+NH-1)

15300 SI = SI + PT(I)“(L‘1) * PT(I)A(M—1)

15400 IF STUPID = 1 THEN ‘ 2+PT(I)“(L-1)*X1(I)
ELSE 3 S2+PT(I)A(L-1)*Y1(I)

15500 NEXT I

15600 A(L,M) = 51

15700 A(L,KK+1) S2

_-..‘__..

15300
15900
16000
16100
16200
16300
16400
16500
16600
16700
16300
16900
17000
17100
17200
17300
17400
17500
17600
17700
17800
17900
13000
13100
13200
13300
13400
13500
18600
13700
13900
13900
13000
19100
19200
19300
19400
19500
19600
19700
12-00
19900
20000
20100
20200
20300

NRON

82

NEXT M
NEXT L
REM: SOLVE THE EQUATION SYSTEM BY THE CHOLESKY METHOD
REM: USING PARTIAL PIVOTINO.

= KK : NCOL = KK +
FOR K = 1 TO NRON
PIVOT = A(K,K> : IL = K
FOR L = K+1 TO NRON ,
IF ABS(A(L,K)) 4 ABS(PIVOT) THEN OOTO 16900
PIVOT = A(L,K)
IL = L
NEXT L
IF IL = K THEN OOTO 17400
FOR LL = 1 TO NCOL
TEMP = A(K,LL)
A(K,LL) = A<IL,LL)
A(IL,LL) = TEMP
NEXT LL
NEXT K
FOR J = 2 TO NCOL
A(1,J) = A<1,O) / A(1,1)
NEXT O
FOR L = 2 TO NRON
FOR I = L TO NRON

SUM = 0
FOR K = 1 TO L - 1
SUM = SUM + A(I,K) * A(K,L)
NEXT K
A(I,L) = A(I,L) - SUM
NEXT I
FOR U = L+1 TO NCOL
SUM = 0
FOR K = 1 TO L - 1
SUM = SUM + A(L,K) * A(K,d)
NEXT K
A(L,J) = (A(L,d) - SUM) / A(L,L)
NEXT U
NEXT L

REM: CET C(I) VALUES BY BACK-SUBSTITUTION.

C(NROH) = A(NROW,NCOL)

FOR M = 1 TO NROW - 1
I = NROW - M

SLIM = 0
FOR J = I + 1 TO NRON
SUM = SUM + A(I,J) * C(J)
NEXT J

 

 

21000

2 1100

21200
21300
21400
21500
21600
21650
21700
21300
21900
22000
22100
22200
22300
22400
22500
22600
22700

22900
23000

 

23300
23400
23500
23600

 

 

_O 00
24000
24100
24200
24300
24400
24500
24600
24700
24300

83

C(I) = A(I,NCOL) - SUM
NEXT M
REM: FINIEHEO.
OLE : REM: PRINT COEFFECIENTS.
REM: CALCULATE SMOOTHED DATA POINTS.
FOR I = II TO (II+NH-1)
IF ETOFIO = 1 THEN L31(I) = 0(1) + C(2)*PT(I) + C(3)*PT(I)“2 + 0(4)
*FT(I)A3 + C(5)*PT(I)“4
IF STUPID = 2 THEN L52(I) = 0(1) + C(2)*PT(I) + C(3)*PT(I)“2 + 0(4)
*PT(I)“3 + C(S)*PT(I)“4
NEXT I
NEXT II
STUPID = STUPID + 1 : OOTO 14500
ERASE A : ERASE FT : ERASE O
LOCATE 10,23 : OOLOR 0,7 : FRINT "LEAST EOOAREO CALCULATION OOMFLETE"
COLOR 7,0 : OIM CP1(1),CP2(1),F31(1),FSZ(1),CSI(1),C32(1),BF1(1),BF2(1)
LOCATE 22,20
INPUT "PRESS ANY KEY TO RETURN TO MENOH,OOs
OOTO 10300

REM:
REM: SET UP TIME VALUES.
REM:
DIM PT(DR+1)
PT(1) = 0
FOR I = 1 TO DP
IF TIC C} 2 THEN PT(I) = TV(I) ELSE PT(I+1) = PT(I) + TV(I)
IF PT(I) C) 0 THEN PT(I) = (INT(PT(I)*1000))/1000
NEXT I
RETURN
CLS
REM:
REM: #****§***********************§*****************#*****
REM: * FAST FOURIER TRANSFORM *
REM: *****X*%%****§*******X**##****§**************4**§***§
REM:

LOCATE 5,25

PRINT "FAST FOURIER TRANSFORM"

LOCATE 6,25

PRINT "*%#*#*******§**#******"

LOCATE 10,20

PRINT ”PLEASE SELECT ONE OF THE FOLLONING=”

LOCATE 12,15

PRINT ”SMOOTH WITH FAST FOURIER TRANSFORM.................... 1"
LOCATE 13,15

PRINT ”CALCULATE VELOCITY OF SMOOTHED DATA......... ...... .... 2”
LOCATE 14,15

PRINT ”CALCULATE ACCELERATION OF SMOOTHEO OATA............... 3'
LOCATE 15,15

27300
27400
27500
27600
27700
27800
27900
23000
23010
23020
23030
23040
23050
23060
23070
23100
23200
23300
23400
23600
23700
23300
23900
29000
29100
29200
29300
29400
29500
29600
29700
29300
29900
30000
30100
30200
30300
30400
30500
30600
30700
30300
30900
31000
23:1 1()(3
31200

84

CLS : OIM EF1<11,OF2(1),OO1<1),032(1),OF1<1),0F2<1), 31(1),L32(1)
LOCATE 10,20

COLOR 0,7 : PRINT ”FAST-FOURIER TRANSFORM COMPLETE" : COLOR 7,0
LOCATE 22,20

INPUT "PRESS ANY KEY TO RETURN TO MENU”,CC$

OOTO 23000

REM: SET TIME VALUES.

DIM PT(UP+1)

PT(1)=O
FOR I = 1 TO OP
IF TIC C} 2 THEN PT(I) = TV(I) ELSE PT(I+1) = PT(I) + TV(I)
IF PT(I) C} 0 THEN PT(I) = (INT(PT(I)*1000))/1000

 

NEXT I
RETURN

FOR I = 1 TO OP

F31(I) = X1(I)*.94 : F32(I) = Y1(I)*.91

NEXT

GOTO 27200
CLS
REM:
REM: *****§*************************4*******§§*§******
REM: * CUBIC SPLINE FUNCTION *
REM: * *
REM: * THIS SUBROUTINE FINDS A NATURAL CUBIC SPLINE *
REM: * PASSING THROUOH (OP) POINTS *
REM: *********************************§************X*§
REM:
REM:

LOCATE 5,25

PRINT "CUBIC SPLINE FUNCTIONS”

LOCATE 6,25

PRINT "*fi**%**********§***i**"

LOCATE 10,20

PRINT ”PLEASE SELECT ONE OF THE FOLLONINO:"

LOCATE 12,15

PRINT "SMOOTH WITH CUBIC SPLINES............................. 1”
LOCATE 13,15

PRINT "CALCULATE VELOCITY OF SMOOTHED DATA...... ....... ...... 2"
LOCATE 14,15

PRINT "CALCULATE ACCELERATION OF SMOOTHEO DATA............... 3"
LOCATE 15,15

PRINT "RETURN TO MAIN MENU......................... ..... ..... 4"
LOCATE 20,20

INPUT "YOUR SELECTION PLEASE: ",N$

IF NS = ”1” THEN OOSUE 31300

31300 IF NS
31400 IF NS
31500 IF NS
31600 REM:
33 1 70C) GUTC‘
31300 CLS =
31350 OOTO
3 1 900 IF ST
32000 FOR I
32100

32200

2300
32400

32500

32600
32700

85

”2” THEN VV$ "C3" 'UB 50000 GDTQ

"3" THEN AA$ =

 
  
 

      

= G -B 55000 BOTO'

= "4" THEN CHAIN A.0.MAIN ,1000,ALL

INPUT ERROR : GDEILIE 700
20000

ERAEE 0R1,0P2,F31,F32,051,Cs2,BF1,EF2,L31,Lsz : 00309 37100
33510

UPID }= 3 THEN OOTO 36400
I = 1 TO DP STEP NH : D(II)=1 : C(II)=0 = Z(II) =0
FOR I = II+1 TO II+NH-2
D(I) = 2*(PT(I+1)-PT(I-1))
C(I) = PT(I+1) - PT(I)
IF STUPID = 1 THEN TEMP (X1(I+1)—X1(I))/(PT(I+1)-PT(I))
ELSE TEMP (Y1(I+1)-Y1(I))/(PT(I+1)-PT(I))
IF STUFID=1 THEN Z(I)=6*(TEMP-(X1(I)-X1(I-1))/(PT(I)-PT(I-1)))
ELEE Z(I)=6*(TEMP-(Y1(I)-Y1(I-1))/(PT(I)-PT(I-1)))

NEXT I

D(II+NH-1) = 1

C(II+NH-2) = 0

Z(II+NH-1) = 0
FOR I = II+1 TO II+NH-1
XMULT = C(I-1)/D(I-1)
D(I) = 0(1) - XMULT*C(I-1)
Z(I) = Z(I) — XMULT*Z(I-1)

I

XT
Z(II+NH-1) Z(II+NH-1)/D(II+NH-1)
FOR I — II TO II+NH-2
Z(II+NH-1-I) = (Z(II+NH-1-I)-C(II+NH-1'I)*Z(II+NH-1-I+1))/
D(II+NH-1-I)
NEXT I
FOR J = II TO II+NH-1
FOR dd = II TO II+NH-3
I = II+NH-1-Jd
TEMP = PT(JJ)
IF TEMP )= 0 THEN GOTO 34700
NEXT dd
I

= 1
TEMP = PT(JJ) - PT(II)
H = PT(I+1) - PT(I)
A = TEMP*(Z(I+1)-Z(I))/(6*H)+.5*Z(I)
I STUPID=1 THEN B=TEMP*A+(X1(I+1)—X1(I))/H-H*(2*Z(I)+Z(I+1))/6
ELSE B=TEMP*A+(Y1(I+1)—Y1(I))/H-H*(2*Z(I)+Z(I+1))/6
IF STUPID =1 THEN C31(d) = B*X1(J)
ELEE CS2(J) = B*V1(J)

"'1

NEXT J

 

40000
40100
40200
40300
40400
40500
40600
40700
40300
40900
41000
41100
41200
41300
41400
41500
41600
41700
41710
41720
41800
41900
42000
42100
42200
42300
42400
42500
42600
42700
42800
42900
43000
43100
43200
43300

3400
43500
43600

43700
43300
43900
44000
44100
44200
44300

86

LOCATE 13,15

PRINT ”CALCULATE VELOCITY OF SMOOTHED DATA................... 2"
LOCATE 14,15

PRINT "CALCULATE ACCELERATION OF SMOOTHED DATA............... 3”
LOCATE 15,15

PRINT ”RETURN TO MAIN MENU................................... 4”
LOCATE 20,23

INPUT ”YOUR SELECTION PLEASE: “,N$

IF NS = ”1" THEN OOSUB 41400

IF N$ = "2“ THEN VVS = ”BF" : OOSUB 50000 : OOTO 33600
IF NS = ”3".THEN AA$ = "BF“ : GOSUB 55000 = OOTO 33600
IF N$ = ”4” THEN CHAIN "A:S.MAIN",1000,ALL

REM: INPUT ERROR : OOSUB 700
GOTO 33600
CLS : ERASE CPI,CP2,CS1,CS2,F51,FSZ,BF1,BF2, SI,LS2
DIM BF1(DP),BF2(DP) : REM: X-Y COORDINATES FOR SMOOTHED DATA.
DIM T1(DP),T2(DP) : REM: TEMPORARY ARRAYS.
NH = DP/NP : REM: NH = NO. OF ENTRIES UNDER EACH HEADER. a
CLS : LOCATE 10,25 : COLOR 0,7
PRINT “BUTTERWORTH FILTER WORKING....." : COLOR 7,0

BUTT1 = 1.41412

BUTT2 = .1

PI = 4 * ATAN(1)

TEMP1 = PI*BUTT2

TEMP2 = SIN(TEMP1) / COS(TEMP1)

TEMP1 = TEMP2 * BUTTI

TEMP2 = TEMP2 * TEMP2

BUTT3 = TEMP1 + TEMP2 + 1

BUTT4 = TEMP2 / BUTT3

BUTTS = BUTT4 * 2

BUTT6 = 2 * (1 - TEMP2) / BUTT3

BUTT7 = (TEMP1 - TEMP2 - 1) / BUTT3
FOR K = 1 TO DP STEP NH
T1(K) = X1(K) : T1(K+1) = X1(K+1)
T2(K) = Y1(K) : T2(K+1) = Y1(K+1)

NEXT K
FOR MM = 1 TO DP STEP NH

FOR II = MM+2 TO NH+MM—1

T1(II) = BUTT4*X1(II)+BUTT5*X1(II-1)+BUTT4*X1(II-2)+BUTT6*T1(II-1)
+BUTT7*T1(II-2)

NEXT II

FOR KK = MM TO NH+MM-1

d = (MM+NH-1) + 1 -KK

T2(d) = T1(KK)

NE T KK
BF1(MM) = T1(MM) : BF1(MM+1) = T1(MM+1)
BF2(MM) = T2(MM) : BF2(MM+1) = T2(MM+1)

44400
44500

44/500

44700
44300
44000
45000
45100
45200
45300
45400
45500
45600
45700
45800
0

45900
46000
46100
46200
50000
50100
50200
50300
50400
50500
50600
50700
50800
50900
51000
51100
51200
51300
51400
51_:00
51600
51700
51300
51900
52000

52100

52200

FOR I = MM+2’ TO NH+MM-1
BF1(I) = BUTT4*T1(I)+BUTT5I*T1(I-1)+BUTT4*T1(I-2 )+BUTT6*X1(I-1)+
BUTT7§X1(I-2)
BF2(I) = BUTT4§T2<I)+BUTT5*T2(I—1)+BUTT4*T2(I-2)+BUTTb*Y1(I—1)+
BUTT7§Y1(I—2)
NEXT I
NZ = (MM+NH—1)/2
REM: FOR J = 1 TO NZ
REM: dd = (MM+NH-1)+1 — d .
REM: XTEMP = X1(J) : YTEMP = Y1(U)
REM: X1(d) = X1(Ud) = Y1(J) = Y1(JJ)
REM: X1(dd) = XTEMP : Y1(dd) = YTEMP
REM: NEXT J
NEXT MM
CLS
LOCATE 10,13
COLOR 0,7 : PRINT "BUTTERHORTH FILTERING CALCULATIONS COMPLETE" : COLOR 7,
LOCATE 22,20
INPUT "PRESS ANY KEY TO RETURN TO MENI",CC$
ERASE T1,T2 : DIM CP1(1),CP2(1),Fs1(1),F32(1),I:31(1),C32(1),L31(1), L32(1)
OOTO 38600
CLS
REM:
REM : fi********§**é§*§§*§***§*******#*§#§#*§**§******{NH-{H}!-
REM: * VELOCITY CALCULATION *
REM: * *
REM: * THIS BUBROUTINE NILL CALCULATE VELOE ITY. IT *
REM: 4 FINDS THE DERIVATIVES OF A SET OF DATA VALUES 4
REM: * BY MEANS OF THE FIRST FORNARD——FIR3T CENTRAL *
REM: 4 —-ANO FIRST BACKwARD FORMULAS FOR EITHER *
REM: * UNIFORMLY OR NON-UNIFORMLY SPACEO VALUES. *
REM: III41’II-**§§#****#§*********************§*§****§*#*******
REM:
REM:
REM: WRITE ROUTINE TO USE THE PROPER ARRAY NHEN CALCULATING VELOCITY.
REM:
DIM VCl<DP),VC2(DP)
DIM V1(DP),V2(OP) : REM: TEMPORARY ARRAYS.
REM:
REM: CALCULATE VELOCITY.
REM:
IF VVS = "BF" THEN FOR I = 1 TO OF : V1(I) = BF1(I) : V2(I) = BF2(I) =
NEXT I : GOTO 52000
IF VVS = ”CS” THEN FOR I = 1 TO DP = V1(I) = CSI(I) : V2(I) = C52(I) :
NEXT I = SOTO 52é00
IF VV$ = "CF" THEN FOR I = 1 TO OP : V1(I) = CP1(I) : V2(I) = CP2(I) :

87

 

88

OOTO 52600

"F5" THEN FOR I = 1 TO DP : V1(I) = F51(I) : ¥2(I) = F32(I) :
BDTD 52600 f

52400 IF VV$ “' THEN FOR I = 1 TO DP : V1(I) = LSl(I) : V2(I) = L's-Z(I) :

52300 IF VVS =
NEXT I : OOTO .4 2600

 

52500 IF VVS ”RD" THEN FOR I = 1 TO DP : V1(I) = X1(I) : V2(I) = Y1(I) :

NEXT. I GOTO 52600 .
52 .600 VC1(1) (V1(2) — V1(I)) / TV(I) : REM: FIRE --T FORUARD- DIFFERENIZE (X)
52700 VC2(1) — (V2(2) - V2(1)) / TV(I) : REM: FIRST FORWARD- DIFFERENCE (Y)
52800 FOR I = 2 TO DP -1 : REM: FIRST CENTRAL—DIFFERENCE ARRAY.
52900 VC1(I5 = (V1(I+1) - V1(I-1)) / (2*TV(I))
5'3000 VOE(I) = (V2(I+1) - V2(I-1)) / (2*TV(I))
53100 NEXT I

53200 VC1(DP) = (V1(DP) - V1(DP-1)) / TV(DP) : REM: FIRST BACKNARD-DIFFERENCE
3300 VC2(DP) = (V2(DP) - V2(DP-1)) / TV(DP) : REM: FIRST BACKUARD-DIFFERENCE

53400 REM: VELOCITY CALCULATION COMPLETE.

53500 ERASE V1 : ERASE V2

53600 CLS

5 3700 LOCATE 10,20

53800 COLOR 0, 7 : PRINT "VELOCITY CALCULATION COMPLETE" : COLOR 7,0

53900 LOCATE 22, 20

54000 INPUT "PRES ANY KEY TO RETURN TO MENU",CC$

54100 RETURN

5

 

.5000 CLS
55100 REM:
55200 REM: *****§§*******§f§§§***********§*§*********§*§§******
55300 REM: * ACCELERATION CALCULATION *
55400 REM: ****§**§***§******4**§**************§***************

55500 REM:

55600 DIM AC1(DP),AC2(DP)

55700 DIM A1(DP) A2(DP) : REM: TEMPORARY ARRAYS.

55800 IF AAS "B THEN FOR I = 1 TO DP : A1(I) = BF1(I) : A2(I) = BF2(I) :

56400 AC1(1) (A1(3) — 2*A1(2) + A1(1)) / TV(I)”2 : REM: FIRST FORWARD- DIFF.
56500 AC2(1) (A2(3) - 2*A2(2) + A2(1)) / V(1)“2 : EM: FIRCT FORWARD- DIFF.
56600 FOR I = 2 TO DP - 1 : REM: FIRST TCENTRAL— DIFFERENIIE ARRAY.

NEXT I : OOTO 56400

55900 IF AAS = "C3" THEN FOR I = 1 TO DP : A1(I) = C31(I) : A2(I) = C32(I) :
NEXT I : OOTO 56400

56000 IF AAS = ”CP" THEN FOR I = 1 TO DP : A1(I) = CP1(I) : A2(I) = CP2(I) :
NEXT I : SOTO 56400

56100 IF AAS = "F5" THEN FOR I = 1 TO DP : A1(I) = F31(I) : A2(I) = F32(I) :
NEXT I : OOTO 56400

56200 IF AA$ = ”LS" THEN FOR I = 1 TO DP : A1(I) = 51(I) : A2(I) = L52(I) :
NEXT I : OOTO 56400

56300 IF AAQ = ”RD” THEN FOR I = 1 TO DP : A1(I) = X1(I) : A2(I) = Y1(I) :
NEXT I : OOTO 56400

56700
56800
56900
57000
57 100
57200
E- 300
57400
57500
57600
57700
57800

89

AC1(I) = (A1(I+1) -2*A1(I) + A1(I-1)) / TV(I)“2
AC2(I) = (A2(I+1) -2*A2(I) + A2(I-1)) / TV(I)“2
NEXT I

AC1(DP) = (A1(DP) -2*A1(DP-1) + A1(DR-2)) / TV(DR)“2

AC2(DP) = (A2(DP) -2*A2(DP-1) + A2(DP-2)) / TV(DP)“2

REM: ACCELERATION CALCULATION COMPLETE.

ERASE A1 : ERASE A2 .

CLS : LOCATE 10,20

COLOR 0,7 : PRINT “ACCELERATION CALCULATION COMPLETE"

LOCATE 22,20

INPUT “PRESS ANY KEY TO RETURN TO MENU“,CC$

RETURN

REM: FIRST BACK.
REM: FIRST BACK.

COLOR 7,0

 

 

6000
6200
6400
6600
6800

0000
0200
0400
0600
0300
1000
1200

90

CLS
REM:
REM: ******§*#**#fi***§************§§****fi*§***§*§****§***§
REM: 4* F‘RINTINO DATA SUBROUTINE 4-
REM: **#*****é§*****éé*****§§§*******§*#***********§***##*
REM:

LOCATE 5,30

PRINT "PRINTING DATA"

LOCATE 6,30

PRINT "*************"

LOCATE 10,20

PRINT "PLEASE SELECT ONE OF THE FOLLONINO:"

LOCATE 12,15

PRINT "PRINT DATA................................. 1"
LOCATE 13,15

PRINT "RETURN TO MAIN MENU........................ 2“
LOCATE 15,25

INPUT “YOUR SELECTION PLEASE:”,N$

IF NS = "1" THEN OOSUE 5300

IF Ns = "2“ THEN CHAIN "A:S.MAIN",1000,ALL

CLS : REM: INPUT ERROR

ERS = "A:S.PRINT“ : LN = 5600

CHAIN ”A:3.ERROR",1000,ALL

OOTO 1000

CLS : REM: PRINTING DATA SUEROUTINE.

LOCATE 1,30

PRINT "PRINTING DATA"

LOCATE 2,30

PRINT "§***#********"

LOCATE 4,35

PRINT "PRINT:“

LOCATE 6,15

PRINT "BUTTERWORTH FILTERING SMOOTHED DATA..........(BF)"
LOCATE 7,15

PRINT ”CUBIC SPLINE SMOOTHED DATA...................(C3)"

LOCATE 3,15

PRINT ”CHEBYSHEV POLYNOMIAL SMOOTHED UATA...........(CP)”
LOOATE 9,1

PRINT ”FOURIER SERIES SMOOTHED OATA.................(FO>"
LOCATE 10,15

PRINT "LEAST OOOAREO SMOOTHEU OATA..................(LO>"
LOCATE 11, 15

PRINT "RAN OATA.....................................(RO)"

LOOATE 12,15

PRINT ”RETURN TO MENU...............................(MM)”
mmME14J5

PRINT ”MAKE YOUR SELECTION BY ENTERINO THE THO OIOIT CODE"
LOOATE 15,12

PRINT "IF YOU NOULO LIHE THE RESULTANT VELOCITY OR AOOELERATION"
OxAm m,w '

PRINT "FROM THE DATA PLACE AN (A) OR (V) IN FRONT OF THE CODE”
LOOATE 13,12

 

 

91

    

1400 PRINT "EX: TO PRINT RESULTANT ACCELERATION FROM THE SMOOTHED DATA"
1600 LOCATE 1?,27
1300 PRINT “BY CUBIC SPLINE FUNCTION"
2000 LOCATE 20,34
2200 PRINT "ENTER: “; : COLOR 0,7 : PRINT "AC3" : COLOR 7,0
2400 LOCATE 22,23
2600 INPUT "YOUR SELECTION PLEASE:”,3D$
2300 REM:
3000 REM: THE DATA WILL BE PRINTED WITH THE X—COORDINATE BEINO SAVED & THE
3200 REM: Y-COORDINATE WILL BE REPLACED WITH THE SMOOTHED VALUE.
3400 REM: I
3600 DIM T1(DP),T2(DP) : REM: TEMPORARY ARRAYS. »
3300 IF SOS = "BF" THEN FOR I = 1 TO DP : T1(I) = BF1(I) : T2(I) = EF2(I) :
NEXT I : OOTO 16000
-4000 IF SDS = ”CS" THEN FOR I = 1 TO DP : T1(I) = C31(I) : T2(I) = C32(I) :
NEXT I : GOTO 16000
14200 IF EDS = "CP" THEN FOR I = 1 TO DP : T1(I) = CP1(I) : T2(I) = CP2(I) :
NEXT I : GOTO 16000
14400 IF EDS = "F3” THEN FOR I = 1 TO DP : T1(I) = F81(I) : T2(I) = F32(I) :
EXT I : GOTO 16000
14600 IF ODS = "L3" THEN FOR I = 1 TO DF : T1(I) = L31(I) : T2(I) = L32(I) :
EXT I : OOTO 16000
14300 IF SD$ = "RD" THEN FOR I = 1 TO DP : T1(I) = X1(I) : T2(I) = Y1(I) :
EXT I : GOTO 16000
15000 IF SD$ = "ABF" OR SOS = "AC5” OR 5D$ = "AFD" OR 3D$ = "AFS" OR SOS = "ALB“
OR SDS = "ARD" THEN FOR = 1 TO DP : T1(I) = AC1(I) : 2(1) = AC2(I) :
NEXT I : GOTO 16000
15200 IF 30$ = ”VBF" OR 5D$ = "VCS" OR ODS = “VFD” OR SD$ = "VFS" OR 3D$ = ”VL3"
OR 506 = ”VRD” THEN FOR I = 1 TO DP : T1(I) = VC1(I) : T2(I) = VC2(I) :
NEXT I : OOTO 16000
15400 IF SOS = "MM” THEN ERA3E T1 : ERASE T2 : OOTO 1000
15600 REM: INPUT ERROR : OOSUB 700
15300 OOTO 1000
16000 REM:
16200 REM: SEND ARRAY TO PRINTER.
16400 REM:
16600 CL3
1%300 DIM PT(DP+1) : PT(1) = 0
17000 NH = DP/NP : REM: NH = NO. OF ENTRIES UNDER EACH HEADER.
17200 LOCATE 10,15
17400 COLOR 0,7 : PRINT ”REMEMBER TO ALICN PAPER TO TOP OF A SHEET" : COLOR 7,0
17600 LOCATE 10,53
17300 COLOR 0,7 : PRINT "!!'" : COLOR 7,0
13000 LOCATE 22,20
13200 INPUT ”PRE'J ANY KEY TO BEGIN PRINTING”,CC$
13400 LOCATE 1,32 : LPRINT "FILE DESCRIPTION" : LPRINT
13600 IF HC = 100 THEN LOCATE 3,1 : LPRINT 01$

 

92

IF HC = 100 THEN LOCATE 4,1 : LPRINT 02$
IF HC i} 100 THEN LOCATE 3,1 LPRINT "PRESENT FILE HA3"
IF HC 5} 100 THEN LOCATE 4,1 LPRINT "NO FILE DESCRIPTION"
LOCATE 6,1
FOR OJ = 0 TO NP - 1
LPRINT TAB<<JJ+1)*20+3) PH$(JJ+1);
NEXT dd
LOCATE 7,1 = LPRINT TAB(2) "TIME VALUE";
FOR dd = 0 TO NP — 1 ‘
LPRINT TAB((JJ+1)*20+1) "X" TAB((JJ+1)*EO+12) "Y";
20800 NEXT dd -
21000 LPRINT
21200 LOCATE 9,1 '
21400 FOR II = 1 TO NH = PT(II+1) = PT(II)+TV(II) = PT(II+1) = (INT(PT(II+1

:-
u .-

 

)*1000))/1000 : LPRINT TAB(3) PT(II);

21600 TC = 0

21300 FOR KK = II TO DP STEP NH

22000 LPRINT TAB((TC+1)*20) T1(KK) TAB<<TC+1)*20+10) T2(KK);
22200 TC = TC + 1

22400 NEXT KK

22600 NEXT

22300 LPRINT : LPRINT : LPRINT : LPRINT

33000 LPRINT TAB<20); "END OF PRINTOUT*****"

{3200 CLS

23400 LOCATE 10,15

29600 INPUT "END OF DATA--PRESS ANY KEY TO RETURN TO MENU",CC$
23300 ERASE T1 : ERASE T2 : ERASE PT

T4000 OOTO 1000

 

1000

 

2300
2900
3000
3100
3200
3900
3400
3500
3600
3700
3300
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
S400
5500
5600
S700
,300
5900

     

93

CLS
REM:
REM: **********#**§§&*§**§******************§***§§*§******
REM: * ORAPHING DATA SUBROUTINE *
REM: **§*§*§*********************§****+*******************
REM:
LOCATE 5,20

PRINT "ORAPHINO DATA"

LOCATE 6,30

PRINT "**#*********4"

LOCATE 9,20

PRINT "PLEASE SELECT ONE OF THE FOLLONINO:“

LOCATE 11,15 '
PRINT “GRAPH DATA................................... 1"
LOCATE 12,15

PRINT "SEND ORAPH T0 PRINTER........................ 2“
LOCATE 13,15

PRINT "RETURN TO MAIN MENU.......................... 3“
LOCATE 15,25

INPUT "YOUR SELECTION PLEASE:",N$

1F NS = “1" THEN GOSUB 3700

IF N3 = "2” THEN GOTO 1000

IF N3 = "3" THEN CHAIN ”A:S.MAIN",1000,ALL

CLS : REM: INPUT ERROR

ER$ = “A:3.0RAPH" : LN = 3600

CHAIN " :S.ERROR",1000,ALL

GOTO 1000

CLS : REM: ORAPHINO DATA

LOCATE 1,25

PRINT “ORAPHINO DATA SUBROUTINE"

LOCATE 2,25

PRINT "********§*§*§§*§********"

LOCATE 4,35

PRINT "GRAPH:"

LOCATE 6,15

PRINT "BUTTERWORTH FILTERING SMOOTHED DATA..........(BF)"
LOCATE 7,15

PRINT ”CUBIC SPLINE SMOOTHEO DATA...................(03)"
LOCATE 8,15

PRINT ”CHEBYSHEV POYNOMIAL OATA..................

...(CP)“
LOCATE 9,15
PRINT "FOURIER SERIES SMOOTHED DATA.................(FS)"
LOCATE 10,15
PRINT "LEAST SQUARES SMOOTHED DATA.............. ...(LS)”

LOCATE 11,15
PRINT "RAN DATA.....................................(RD)”
LOCATE 12,15
PRINT "RETURN TO MENU.......................... ..... (MM)"
LOCATE 14,15
PRINT ”MAKE YOUR SELECTION BY ENTERING THE TNO DIOIT CODE”

 

6000
6100
6200
6300
6400
6500
6600
6700
6300
6900
7000
7100
7200
7300
7400
7500
7600
7700

7300

7900

8000

3100

9600

 

94

LOCATE 15,12
PRINT "IF YOU wOULD LIKE THE RESULTANT VELOCITY OR ACCELERATION"
LOCATE 6,13

PRINT "FROM THE DATA PLACE AN (A) OR (V) IN FRONT OF THE CODE”

LOCATE 13,12

PRINT "EX: TO ORAPH RESULTANT ACCELERATION FROM THE SMOOTHEU OATA”
LOCATE 1?,27 .

PRINT ”BY CUBIC SPLINE FUNCTION"

LOCATE 20,34 '

PRINT "ENTE': "; : COLOR 0,7 : PRINT "AC3" : COLOR 7,0

LOCATE 22,25

INPUT "YOUR SELECTION PLEA3E=",30$

REM:

REM: THE DATA WILL BE ORAPHED wITH THE x—COOROINATE BEING SAVED & THE
REM: Y-COORUINATE HILL BE REPLACED WITH THE SMOOTHED VALUES.

REM:

DIM T1(0P),T2(0P) : REM: TEMPORARY ARRAYS.

IF 30$ "BF” THEN FOR I = TO DP : T1(I) = BF1(I) : T2(I) = BF2(I) :

NEXT I : GOTO 3300

IF EDS = "CS" THEN FOR I = 1 TO OR : T1(I) = 031(I) = T2(I) = 032(I) =
NEXT I = GOTO 3300

IF 30$ = ”DP“ THEN FOR I = 1 TO OR = T1(I) = CP1(I) = T2(I) = CP2(I) :
NEXT I : GOTO 3300

IF 30$ = ”F3" THEN FOR I = 1 TO OR : T1(I) = F31(I) : T2(I) = F32(I) =
NEXT I : GOTO 3300

IF 305 = "LS" THEN FOR I = 1 TO OP : T1(I) = L81(I) : T2(I) = L32(I) :
NEXT I = GOTO 3300

IF 305 = "R0" THEN FOR I = 1 TO DP = T1(I) = x1(I) : T2(I) = Y1(I) :
NEXT I = GOTO 3300

IF 30$ = "ABF" OR 30$ = "AC3" OR 30$ = "APO" OR 305 = "AFB” OR 30$ = ”ALB"
OR 30$ = ”ARO" THEN FOR = TO DP = T1(I) = ACl(I) = T2(I) = ACEtI) :
NEXT I = OOTO BPOO

IF 30$ = "VBF" OR 30$ = ”V03" OR 30% = "VFD" OR 305 = "VF3" OR 30$ = "VLS"
OR 30$ = ”VRO" THEN FOR I = 1 TO OR : T1(I) = VC1(I) = T2(I) = VC2(I) 2
NEXT I = OOTO 3300

IF 505 = ”MM” THEN ERASE T1 : ERASE T2 = GOTO 1000

REM: INPUT ERROR = OOSUB 700

OOTO 1000

REM:

REM : in}'fifinfl’i-it41"}#:0104301?!-fl-fl-#1-fiat-##4##*{-46-}*{r-fl-i-fl-#§************§******

REM: 4 GRAPHINO DATA 3UBROUTINE

REM: ****§*************#al-4H-*4!-4i»********#*********-fl-*****fl-****

REM:

CLS : ERAOE BF1,BF2,C31 CC‘,CP1,CP2,F31,F32,L31,L32
DIM OI<DP),O2<DP),O3(DP),O4(DP)

REM: ARRAYC FOR DETERMINING SCALING.

IF PT(1) 3: 0 THEN ERASE PT

 

 

 

95

9700 DIM PT(DP+1),TP(0P+1)

9300 LOCATE 5, 13

9900 PRINT "WHICH OF THE FOLLOWING DO YOU WISH TO ORAPH=”
10000 PRINT : PRINT : PRINT

10100 FOR I = 1 TO NP

10200 PRINT TAB(5) PH$(I) TAB(20) ”.......... ”;I
10300 I

10400 PRINT TAB(5) "OTHER" TAB(20) ”...... .... “-NP+1
10500 LOCATE 20,29 = INPUT "YOUR SELECTION PLEAE E: ",IS
10600 IF 13 < NP+1 THEN GOTO 12100

10700 CLO

10300 LOCATE 5,10 »
10900 PRINT ”YOU MAY GRAPH TUO BODY SEGMENTS AGAINST EACH OTHER AT A TIME"
11000 LOCATE 7,22

11100 PRINT ”PLEASE SELECT THO OF THE FOLLONINO: "

11200 PRINT : PRINT : PRINT

 

11300 FOR I = 1 TO NP

11400 PRINT TAB(5) PH$(I) TAB(20) ".......... ”:I

11500 NEXT I

11600 PRINT TAB(5) "RETURN TO MENU" TAB(20) ”.. " ”NP 1

11700 LOCATE 20,26 : COLOR 0,7 : INPUT "YOUR FIRST SELECTION PLEASE: ",13 :
COLOR 7,0

11500 IF 13 = NP+1 OOTO 15200

11900 LOCATE 22,26 : COLOR 0,7 : INPUT "YOUR SECOND SELECTION PLEAOE: ",51 :
COLOR 7,0

12000 CHECK = 2 =|3O51JB 15500 = OOTO 12200
12100 CHECK = 1 : OOSIJB 15500
12200 REM: SET UP ~Y COORDINATE AXIS.

12300 SCREEN 1,0,0X

12400 LINE (20,150)-(20,0)

12500 LINE (20,150)-(320,150)

12600 REM: GRAPH X-Y COORDINATES -VS- TIME
12700 IF CHECK = 2 THEN OOTO 13700

12300 FOR I = (NH*IS-(NH-1)) TO NH*IS-

12900 LINE (PT(I),OI(I))-(PT(I+1),CI(I+1)) : REM: X—GRAPH
13000 LINE (PT(I),O2(I))-(PT(I+1),O2(I+1)) : REM: Y‘GRAPH
13100 NEXT I

15 200 LOCATE 20,11
3300 PRINT ”X-OREEN Y—YELLON”

13400 LOCATE 22,5

13500 INPUT "PRESS ANY KEY TO RETURN TO MENU”,CC$
13600 GOTO 15000

13700 REM: ORAPH TNO :EOMENTO.

13500 FOR I = (NH*I'--(NH-1)) TO NH*IS-1

13900 LINE (F'T(I),OI(I))-(PT(I+1),Ol(I+1))
14000 LINE (PT(I),OZ(I))-(PT(I+1),OZ(I+2))
14100 NEXT I

14200 FOR J = (NH*31-(NH-1)) TO NHfiSI-I
14300 LINE (TP(J),O3(J))-(TP(J+1),OB(U+1))

14400 LINE (TP(J),O4(J))-(TP(J+1),O4(J+1))

14500
14600
14700
14800
14900
15000
15100
15200

15300'

15400
15500
15600
15700
15800
15900
16000
16100
16200
16300
16400
16500
16600
16700
16800
16900
17000
17100
17200
17300
17400
17500
17600
17700
17800
17900
18000
18100
18200
18300
18400
18500
18600
18700
18800
18900
19000
19100

96

NEXT 0
LOCATE 20,5
PRINT PH$(I8);"-OREEN "gPH$(8I);"—YELLOH"
LOCATE 22,5
INPUT “PRESS ANY KEY TO RETURN TO MENU",CC$
SCREEN 0,1;0,0 : COLOR 7,0
uIOTH,30
ERA8E 01,02,08,G4,T1,T2,PT,TP
DIM BF1(1),BF2(1),081(1),882(1);OP1(1),OP2(1),F81(1),F82(1),L81(1),L82(1)
OOTO 1000
REM:
REM: SUBROUTINE TO DETERMINE SCALING.
REM:
NH = DP/NP : REM: NH = NO. OF HEADERS.
FOR I = 1 TO DP
OI<I>=T1<I) : G2(I)=T2(I) = 03(I)=T1(I) : C4<I>=T2<I>
NEXT I
FOR I = (NHiIS-(NH-1)) TO NH*18-1
FOR J = 1+1 TO NH*18
IF T1(I) >= T1(d) THEN COTO 16800
TEMP = T1(d)
Tl(d) = T1(I)
T1(I) = TEMP
NEXT 4
NEXT I
FOR I = (NH*18-(NH-1)) T0 NH*IS
FOR J = 1+1 TO NHRIS
IF T2(I) >= T2(J) THEN OOTO 17600
TEMP = T2(J)
T2(J) = T2(I)
T2(I) = TEMP
NEXT 8
NEXT I
REM: SET MIN-MAX VALUES.

XMIN = T1(NH*I3)
XMAX = T1(NH*I3-(NH-1)) 3 XRANOE = XMAX - XMIN
YMIN = T2(NH*I3)
YMAX = T2(NH*IS-(NH-1)) = YRANOE = YMAX — YMIN

REM: SCALE COORDINATES.
IF XMIN }= 0 THEN 18900

FOR I = (NH*I3-(NH-1)) TO NH*I5
01(I) = Ol(I)+ABS(XMIN)
NEXT I

OOTO 19200
FOR I = (NH*IS-(NH-1)) TO NH*IS
01(1) = Ol(I)-XMIN
NEXT I

19200
19300
19400
19500
19600
19700
19800
19900

20000'

20100
20200
20300
20400
20500
20600
20700
20800
20900
21000
21100
21200
21300
21400
21500
21600
21700
21300
21900
22000
22100
22200
22300
22400
22500
22600
22700
22800
22900
23000
23100
23200
23300
23400
23500
23600
23700
23800
23900

IF YM

OOTO

IN >=
FOR I
62(1)
NEXT
20000
FOR I

VJ2(I)

XOON
YCON

1F CH

REM:
XMIN2
XMAX2
YMIN2
YMAXZ
REM:

NEXT
= 150
= 150
FOR 1
01(1)
01(1)
02(1)
02(1)
NEXT

ECK =
FOR 1

NEXT
FOR I

NEXT
SET M

= T1
T1
T2
T2
SCALE

97

0 THEN OOTO 19700
(NH*IS-(NH-1)) TO NH*18
02(I)+A88(YMIN)

I

(NH*IS-(NH-1)) TO NH*IS
:2(I)-YMIN

I

(XMAX-XMIN)

(YMAX-YMIN)
(NH*IS-(NH-1)) TO NH*IS
61(I)*XCON

150 - 61(1)

GZ(I)*YCON

150 - 02(1)

II II II II ll\\

I
1 THEN GOTO 25500
= (NH*SI-(NH-1)) TO NH*8I-1
FOR J = I+1 TO NH*SI
IF T1(I) >= T1(J) THEN GOTO 21500
TEMP=T1(J)
T1(J)=T1(I)
T1(I)=TEMP
NEXT 4
I
= (NH*SI-(NH-1)) TO NH*81
FOR J = I+1 T0 NH*SI
IF T2(I) >= T2(J) THEN GOTO 22800
TEMP=T2(J)
T2(d)=T2(I)
T2(I)=TEMP
NEXT 8
I
IN-MAX VALUES
(NH*SI)
(NH*81-(NH—1))
(NH*SI)
(NH*8I-(NH-1))
CODRDINATES.

1F XNINZ }= 0 THEN OOTO 23600

GOTO

FOR I
03(1)
NEXT
23900
FOR I
03(1)
NEXT

(NH*31-(NH-1)) TO NH*SI
83(1) + ABS(XMIN2)

I

(NH*SI-(NH-1)) TO NH*SI
03(1) - XMIN2

1

IF YMIN2 }= 0 THEN GOTO 24400

24000
24100
24200
24300
24400
24500
24000
24700
24000
24900

25200
25300
25400
25500
25600
25700
25800
25900
26000
26100

26200
26300
26400
26500
26600
26700
26300
26900
27000
27100
27200
27300
27400
27500
27600
27700
27300
27900
23000
23100

23200

98

PM I=(MfiM-MH%Y)TDNMSI
04(1) = 04(1) + AEE<YM1N2)
NEXT 1

OOTO 24700
FOR 1 = (NH*SI—(NH—1)) TO NH*SI
04(1) = 04(1) — YMIN2
NEXT 1

XCON2 = 150 / (XMsz—XM1N2)

YOONz = 150 / (YMAX2-YM1N2)

FOR 1 = (NH*SI—(NH-1)) TO NH*SI
63(1) = B3(1)*XCDN2

03(1) = 150 - 03(1)

O4(I) = G4(1)*YCON2

64(1) = 150 - 04(1)

NEXT 1

0L0
LOCATE 5,17

PRINT "GRAPHING IS OONE ONLY FOR CONSTANT TIME VALUES"
LOCATE 15,20

INPUT ”DDNTINUE..........(Y/N)",YN$

IF YNs = "v" THEN GDTD 20300

ERASE T1,T2,GI,O2 = DIM BF1(1),BF2(1),DSI(1),CS2(1),FD1(1),FO2(1),F31(1),

F32(1),L81(1), 32(1)

GOTO 1000

REM: SET TIME VALUES.
FOR I = (NH*13-(NH-1)) TO NH*13
PT(I+1) = PT(1) + TV(I)
NEXT 1

TCON = 300/PT(NH*13)

FOR 1 = (NH*IS—(NH-1)) TO NH*13
PT(1) = PT(I)*TCON

PT(1) = PT(1) + 20

NEXT 1

IF CHECK = 1 THEN GOTO 23200
TP(NH*BI-(NH-1)) = 0
FOR I = (NH*SI-(NH-1)) TO NH*$I
TP(1+1) = TP(I)+TV(I)
NEXT 1
TCON2 = 300/TP(NH*SI)

FOR I = (NH*31—(NH-1)) TO NH*SI
TF(I) = TP(1)*TCON2
TP(I) = TP(I) + 20
NEXT 1
RETURN

 

 

     

 

 

 
           
         
   

   

         

 

   

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