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DATA SMOOTHING AND

DIFFERENTIA TION PROCEDURES

IN BIOMECHANICS

Graeme A. Wood

Department of Human Movement and Recreation Studies

University of Western Australia

Nedlands, Australia

"When you can measure what you are speaking about. and express it in

numbers. you know something about it."

(William Thomson, Lord Kelvin.

Popular Lectures and Addresses. 1891-1894)

INTRODUCTION

The Problem

In many scientific fields a continuous process is measured at discreteoints and, on the basis of these observations. the scientist attempts to

xplain the nature of the underlying process. These observations are, haw-

ver, usually prone to error. and the scientist-can ~ faced with the dilemma

epicted in Figure I. . .

Clearly in this example additional observations, either at new points in t ,

'r at the same points (thereby obtaining an estimate of the magnitude of the

'he author gratefully acknowledges the valuable advice of Dr. Les Jennings and Dr.ohn Henstridge of the Departments of Mathematics and Biometrics, The Univer-

it y of Western Australia. during the preparation of this paper.



Data Smoothing and Differentiation 309

y

. . . ".

t

o an error?

OR

fa damped oscillation?

\\ f'

I I ~ t

Figure I. Problems in fitting an empirical equation to discrete data points.

measurement error), would help to resolve the matter. Or, if some a priori

knowledge were available concerning the nature of the underlying function,

the problem would be simplified, but certain model parameters would still

need to be determined.

Fitting an empirical equation to data is usually done as a basis for theory

development, or as a means of obtaining new measures. Examples of thefirst instance would be the identification of components in a muscular

fatigue curve (44), establishment of a force-velocity relationship (38), or

analysis of human growth curves (73), while prediction of future perform-



ance records and determination of velocity and acceleration from posi-

310 G.A. ~Vood

tion-tirne data would be examples of the latter instance. In all instances. sci-

entists attempt to obtain the simplest representation of the data that ade-

quately describes the underlying process. while eliminating from considera-

tion data scatter arising from experimental errors. This process is usuallyreferred to as "curve fitting" or "data smoothing."

Measurement of internal and external forces and the motions arising

from them is fundamental to the study of biomechanics. While in many

instances a biomechanist may only be concerned with the measurement and

description of motion, this is not an easy task, for animal and human move-

ment patterns can be very complex and the measurement process is quite

error-prone. Typically. body position is recorded at discrete points in time,

and other kinematic variables (velocity and acceleration) are obtained by

numerical differentiation. When measures of the causal variables (the mus-

cle and joint forces) or of the segmental and whole-body energetics are

required, they cannot be determined by direct means and must therefore be

derived from measures of body motion. This inverse dynamics approach

evolves from the Newtonian expressions

2:F = rn : a

and2 = I'a

That is, the resultant force (F) and moment of force (M) acting on a body of

known mass (rn) and moment of inertia (1) can be indirectly determined

from its acceleration behavior (a and a). This calculation requires not only

accurate measures of body motion. but also estimates of body segment

parameters (15). ;:'

Instantaneous acceleration of a body can be produced by direct methodsthrough the use of electromechanical devices (accelerometers). or, as men-

tioned above. by double differentiation of displacement-time records. The

direct approach suffers from the disadvantage that absolute body position

cannot be readily determined (62) and fixation to the body is difficult. while

numerical differentiation amplifies small errors in displacement-time data

to an extent that is alarming. as the error can occur with relatively high fre-

quency (5, 14,49. 95).

To illustrate the differentiation problem. consider a sinusoidal motion offrequency f as depicted in Figure 2a, with some added measurement error in

the form of another sinusoid of frequency lOx f but of one-tenth the ampli-

tude. In communications-engineering parlance. the signal-to-noise ratio is

10: I. Upon differentiation, the noise is found to.be of equal amplitude to

the signal (Figure Zb), and when differentiated twice, it becomes 100 times

the signal value (Figure 2c). That is, the amplitude of the first derivative is

proportional to I, and that of the second derivative is proportional to fl.

* Information on body segment parameters can be found elsewhere (61).




Since the process of differentiation preferentially amplifies higher-

frequency components, the motion analyst is presented with a problem, for

human movements are generally.of low frequency (8, 95}, while measure-t·

ment errors span the whole frequency spectrum (see Figure 2d). Therefore,the biomechanist not only seeks an empirical function that adequately

describes the processes underlying displacement-time data. but also tries to

minimize the effect of measurement errors that will destroy the validity of

derived measures. To this end various procedures have been employed in

biomechanics. as this paper will show. While this work was written from a

biomechanist's perspective. the material is equally applicable to any area of

the exercise and sport sciences where data smoothing/curve fitting and

differentiation/integration procedures are required.Before proceeding. it is useful to consider the nature and sources of error.

for much of the problem lies in the manner in which experimental data arc

gathered.

Sources of Error

The sources of error associated with the gathering of displacement-time

data are numerous and often mentioned by investigators, but there have

been few reported systematic studies of measurement errors. The following

list. compiled from several references (39, 66, 86, 95, I07). details some of

the potential sources of error associated with displacement-time data

obtained by the methodology most commonly used in biomechanics. i.e.,

cinematography. Error sources include misalignment of the camera; per-

spective error due to objects (subject or scale) out of the photographicplane; stretching of film or imperfect registration of film in camera or pro-

jector; movement of camera or projector; distortion due to optical system

of camera or projector; graininess of film; movement of body segment mar-

kers in relation to joint axes of rotation. either through skin movement or

axial rotation of the segment; precision limits in the digitization process;

errors in recording temporal and spatial scales; and operator errors of

judgement and parallax in locating joint axes of rotation.

Some errors arising from these sources can be described as systematic inthat they introduce consistent biases into the data. Such errors include

image distortion. inaccurate scales. and placement of body markers. Sys-

tematic errors can also result from drift in a body marker's position with

respect to an anatomical landmark. or from an analyst's faulty perception

of where an anatomical point or joint center lies (these factors co-vary with

movement). Such errors can only be eliminated by adherence to sound cine-

matographic procedures (61, 72), adoption of 3-D filming techniques (61,

87), or adequate training of personnel. Data smoothing will seldom help inthis regard. for the apparent trends introduced into displacement-time data

through systematic errors are usually of a lower frequency than the real

motion. i.e., less than 10 hertz (Hz).



312 G.A. Wood

18

7.5 - SIGNAL • NOISEZ ----. SIGNAL [Y.SINCl»ER 50

TH 2.5

I)

E8

RIV

A -2.5

T

IV -6E

-7.5

-18

e

28

16

F "

I 18RST

5

DER aIvA

-6TIVE -18

-16

-28

TIHE

II ~(8)

- SIGNAL + NOISE----. SIGNAL CDY-COSCl )]

n" r· - ",

" ", II , , II, .. ." . ,, •, •

",

v "

", ,

It, ,,

, , •. - .•

• ., '.•.

ii'..

-, - .. , I'

V

V V1 1 1 1 1 ·1 1 I I

• 7 II 18•

Figure 2. The differentiation problem: (A) a noisy sinusoidal signal; (8) first

derivative of pure and noisy signal; (0 second derivative of pure andnoisy signal; (0) typical frequency spectrum of displacement signal (solid

line) and measurement noise (broken line). Higher frequencies are

preferentially amplified with differentiation. which presents a problem in



Data Smoothing and Differentiation 31J

Figure 2. Continued.

158

(C)- SIGNAL + NOISE----- SI6NAL CD2Y-S <T 1

1885

EcoN s aI)

I)

E

R

IVAT

IV

E

8 - - . ~ _ - .. . . . .

- . -... . _

. . - _ . .- - -

-&8

-188

-168

8 I.

T.I t1E

D .•

fRECUENCY SPECTRUM Of SIGNAL AND NOISE

(0)

p

o'"' -D ••

R

0.0

1-1 ••

L

oG1-1•4

" ' - .,~

'-.' v;---,

-4.0;-------~------~------~------~------r_------r_------r_---_,o 4 • 11 1. fO

fREQUENCY I H Z Ifll

that higher frequencies in displacement data are predominantly measure-

ment errors.



314 G.A. Wood

Displacement-time data are also susceptible to random errors arising pri-

marily from the digitization process. Random errors are usually assumed to

be associated with the dependent variable (displacement) and to be normal-

ly distributed with a mean of zero and variance M12. 51). Considering the

various sources of random error listed above, it seems reasonable to assume

that the additive result would be normally distributed (central limit

theorem) and independent of the "signal." which is assumed to be some

unknown deterministic process.

In one of the few reported studies of measurement errors associated with

cinematographic analysis, McLaughlin et al. (56) concluded on the basis of

repeated measures of known distances that a point could be located to the

nearest ±3.0 mm (the average absolute error) in real life. Of this measure,

± 1 mm was the random error introduced by the precision limits of thedigitizer. and the differential (±2.0 mm) was assumed to be caused by

image distortion. When the additional uncertainty associated with the loca-

tion of a subject's joint centers is introduced, this error becomes much

bigger.

Such inaccuracy. coupled with the time-consuming nature of film analy-

sis, has encouraged some biomechanists to explore alternative methods of

motion analysis. However, even with on-line opto-electronic and TV sys-

tems (32,96). or electronic displacement transducers directly attached tothe body (57). the data are still susceptible to those errors listed here, and to

some others as well."

NUMERICAL METHODS FOR DATA SMOOTHINGAND D IFFERENTIA TION

Various methods for data smoothing and differentiation have been used

in biomechanics. and have become increasingly sophisticated with the

advent of high-speed digital computers and a growing demand for detailed

information in the human engineering and medical sciences. Sports bio-

mechanics has also emerged as an area demanding accurate measurement to

permit subtle distinctions between high-level performers.

Historically, kinematic data were obtained by graphic means. but the

extreme computational time involved subsequently led to the application of

numerical methods (59). although initial attempts were not at all promising

(23). These attempts were based on finite-differences calculus and gave the

derivates directly while providing a degree of smoothing. More recently,

polynomial and trigonometric functions have been fined to displacement-

time data and then explicitly differentiated. The dependence on high-speed

digital computers evidenced here has, however. led some investigators to

the use of simple moving averages which define a smoothed point in terms

of a weighted sum of itself and near neighbors. These procedures have

"Electrical interference and analogue-digital conversion software noise (50).



Data Smoothing and Differentia/ion 315

recently evolved into the more sophisticated form of low-pass digital filter-

ing in which the degree of smoothing can be explicitly stated in terms of a

frequency cut-off for the separation of signal from noise. A detailed

account of the mathematicarbasis underlying these procedures follows. to-

gether with a commentary on their use in biomechanics.

Later sections of this paper will compare the various procedures and

make suggestions for the choice of an appropriate data-smoothing proce-

dure.

G raphica l M ethods

The derivative at some point Q on a curve y (y = f(t». for some unknown

function f (of time) is represented by the slope of a tangent at that point (see

Figure 3). Thus. the time-derivative of positional data can be obtained by

manually plotting the data, drawing a smooth curve through the plotted

points (line of best fit), and measuring the slope of tangents to the curve at

the points of interest. A second application of this process, involving suc-

cessive determinations of the slope of a smooth representation of a velocity-

time plot. will yield acceleration-time data.While these procedures are outlined in several biomechanics monographs

(31,61,72) and are undoubtedly used in noncalculus approaches to the

teaching of kinematics. there have been few reported instances (9,23, 43)

of their application since the advent of high-speed digital computers.

Clearly. this method is tremendously time-consuming. highly error-prone

(78). and very subjective.

Finite D ifference T echniques

The derivative f' (t) of the function y = I(t) at some specified point (t.) can

be defined as

d y = f' ( U = l i m ! _ U . _ ± _ h ) _ _ - : : - _ l _ ( l _ )

dt Ht h

(I)

In terms of Figure 3, this expression represents the slope of the chord QR.

and simply states that as h approaches zero. the slope of the chord QR more

closely approximates the slope of the tangent at Q (t = t.). For finite time

intervals, however. a better approximation to [he derivative at t . is found in

the slope of the chord PR o which is given by the formula

dy , f(t, + h) - f(t; - h)

-- ~ f (tJ=---~----------~---dt 2h

(2)

where f'(t. + h)-f(t,-h) is referred to as a first central difference.



316 G.A. J;f/ood

y

y = f(t)

Slope of tangent

- -2L- dt

I- h h h_.

I I

I I

t

ti-2 t. I t. ti+l ti+2.- l.

Figure 3. Graphic representation of a derivative as the slope of a tangent to a

curve. As h approaches zero. the slope of the cord QR more closely ap-proximates the true derivative value. but in reality a better approximation

is given by the slope of the cord PRo

The second derivative. ["(t). at point t. can be similarly obtained using

central differences

(3)



Data Smoothing and Differentiation 3/7

These approximations are equivalent to passing a second order

(quadratic) interpolating polynomial through the three points P. Q. and R .

To obtain an approximation for ...the m'" derivative of f(t). at least an rn "

order interpolating polynomial is necessary. To demonstrate the equiv-

alence of this procedure to that outlined above. while laying a basis for the

understanding of finite difference formulae used by biomechanists. con-

sider an interpolating polynomial p(t) = ao + alt + - a1t1 + ... + a",l '",

which. when written as a Taylor series expansion with finite differences

replacing successive derivatives. becomes

p(t) = f(t,) + df_(!J u + Q_J1 ! J u' + _d...:TI~J(u - 1 )(u + I) + - . . . H)

I! 2! 3!(Stirling's formula)

The first three central differences 6 kf(t,); k = I . 2 . 3 .... can be definedas ':,

f(t, + h) - r(C +h ldf(t,) =- -- -- ·- r- -- -- -

d1f(U = f(C + h) - 2f(t.) + f(t, + h]

and. for simplicity. we let

u=t _ - t,

h

In order to differentiate p(t) it is necessary to use the relation

p'(t) = p Tu) du =! p'(u)- dt h

Thus.

1 { d f ( l ; ) 2d1f(t,) d.lf(t,) }p , (t) =------ + . .u +--- [u(u-I ) - + - u(u t I ) + - (u- 1)(u + - I) t··· (5)

hI! 2! 3!

and

p"(O=_I{2d2

f(t,) + d.lfi_!i)[u+(u-I)+Ut-{u-t-I)+(u-I)-t-(u + I) + ... } (6)

h2 2! 3!

*Strictly speaking. the first and third central differences shown here are averaged

differences. See (103) for a more formal approach.



318 G.A. f;f,'()od

Now . a t po in t t = I, (i.e .. u =0)

p ' (I,) = - L [ d f ( I , ) + 61

f(t,) (0)h t?~-U(-l) + ... J3!

" I [ z r d1 f(t,) ]p (t,)= d I(t) + ------- f- '"hl ' 3!

(7)

( ; o j )

By substitu ting for the first and second centra l d ifference term s in Equa-

tions 7 and 8. expressions for the deriva tives a t t. from a second-order in ter-

pola ting polynom ia l passing through poin ts P . Q. and R in F igure 3 can befound:

p Tt.) = ~ [ df(t;) + 0 ]~i,[f(L_~ hl~ f(t,~_ht]

p " (t.) =~-~ [ dlf(C)]

= I [f(t,+ h) - 2f(t, )+f(C - h)]hl

T hese expressions a re identica l to those obta ined in Equa tions 2 and 3. B y

reta ining the first three cen tra l d ifference term s in an expression . a third-

order (cubic) in terpola ting polynom ia l is obta ined . while a fourth- order

(qua rtic) interpola ting polynom ia l is obta ined by reta in ing the first four

cen tra l d ifference term s in the deriva tive expression. T he quadra tic in terpo-

la ting polynom ia l provided the second-order centra l d ifference expressions

eva lua ted by Felkel (23) in his ea rly resea rch on methods of determ ining

accelera tion . Expressions for a ll deriva tives up to the fourth . based on cen-

tra l difference interpola ting polynom ia ls up to the fourth order. can be

found in the work of Southworth and Deleeuw (84).

T he error in the deriva tive of an m" degree interpola ting polynom ia l is of

the order h"'. im ply ing tha t a fourth-order polynom ia l passing through a

poin t and its four nea rest neighbors w ill y ield a more exact va lue for the

deriva tive a t tha t poin t than a quadra tic passing through the sam e poin t and

its immedia te neighbors w ill.

H owever. it m ust be rem embered tha t the in terpola ting polynom ia l pro-

vides an exact fit. and when some error,~ , . isjrssociated with each observed

da tum . y" better approxim a tions a re ava ilable through the use of sim ila r

formulae based on a la rger tim e in terva l. T ha t is ,

(tJ)

(I())

y , = f(t,) + ~,

and thus Equa tion 9 takes the form

r u . + h) - f(t, ~ 11 ) t., - ~'-l

+)h Ih



Data Smoothing and Differentiation 3/9

where the left side is the actua l deriva tive ca lcu la ted. I t can be seen tha t

sm all va lues of h can produce a large error term (sm a ll denom ina tor). Ihere-

by decreasing the accuracyof th e ca lcula ted derivative.' For th is reason . the

tim e span over which ca lcu la tions a re based (va lue of h) should be commen-

sura te w ith the error m agnitude (23).

Genera lly . finite d ifference expressions have been found to behave poorly

when applied directly to noisy da ta (23. 70. 71.92, 93) and better deriva -

tive estim a tes a re usua lly obta ined by smooth ing the da ta first (27. 69). For

example. Felkel (23) found tha t smooth ing da ta graphica lly and then apply -

ing finite difference formulae to obta in time-deriva tives provided reason-

able estim a tes of known accelera tions. whereas the d irect applica tion of

numerica l m ethods (B essel's and New ton's formulae) had to be rejected.

M ore recently P ez z ack et a !. (70, 71) advoca ted the use of first centra l d if-

ference formulae follow ing the applica tion of a smoothing digita l filter.

T hese au thors a lso suggested tha t a posteriori smooth ing of fin ite d ifference

deriva tes could produce acceptable resu lts a lthough it has been trad itiona l

to smooth da ta first.

Least Squares P olynom ia l A pproxima tions

As mentioned ea rlier. m ethods for the interpola tion of exact da ta and th e

differentia tion formulae so obta ined a re not readily applicable to experi-

m enta l da ta where

f(t) = p(t) + ~

T herefore. ra ther than fit a polynom ia l pet) of degree m through n data

po ints where m= n -1 (the interpola tive case outlined above). a lower-orderpolynom ia l can be chosen to approxim ate the underly ing biologica l func-

tion. ignoring the error ~ in the empirica l da ta .

Furthermore. an expression for th is low -order polynom ia l can be derived

in such a way tha t it represents an ana ly tica l line of best fit. T his procedure.

ca lled the least squa res m ethod . requires tha t the slim . S. of the squa res of

the devia tions be m inim ized . tha t is

n2

S =L (y, - p(U]i= }

IS a rrnrumum .

For a polynom ia l of degree rn, th is expression can be w ritten as

n

S "[ 1 ,'J1 - I J l=.:;.. y, - au - a.t, - a2t, - ... - am! '

i= 1

and for this function to be a m in imum . the pa rtia l deriva tives a s/ a a .,as /aal •... a s / a a " , must be zero. T hus. m equa tions can be obta ined (the

norm a l equa tions) which . when solved simultaneously . yield th e c oe ffic ie nts



320 G.A. Wood

au. a" ...• am for a least-squared fit of an polynomial of degree m to a set of

n empirical data. Once an expression for the polynomial is found. time-

derivates can be easily obtained by analytical differentiation of the approx-

imating function.These procedures have been used by numerous researchers (10,16,29,

58, 67, 106) to fit low-order polynomials (m = 1 -7). either to N total data

in the series (n = N). or to a small number (n = 3 -9) of data points at a

time (see Figure 4).

The latter is a local approximation technique. and is often referred to as

the moving average or moving arc procedure. in that a polynomial is fitted

successively to n data points (I. 2. 3..... n ; 2. 3.4 ..... n + I ; etc.) in the

same manner as the finite difference formulae are applied. A simple illus-

tration of the least-squares local approximation technique is seen in the

three-point moving average formula used by Smith (82) to obtain smooth

displacement-time data. Y i • for a study at forces in jumping. namely

rv

y, = Y J (y.,, + Y i + y,+I)

This formula is derived from a first-order polynomial (straight line) approx-

imation to three consecutive data points having the form

yet) =a, + a, t (11 )

The least-squares function that is minimized is

3S = I ( Y i - a, - a.t.) '

i=l

which. with respect to coefficie nts a, and a., has the partial derivatives

.~-~-=-2

a a u

3I (yi-aU - a.U(1) =0

i=1and

as 3_- = -2 I ( v . - a, - a.U(t,) =0a a . i=l

rearranging these equations. and. for convenience. scaling the ordinatevalues to 1;= -1.0.1

3ao.l.. t +i=I

3 3+ a. I L= I y,

i= I i=l··~ " < f

3 3a. L t~ = Ley,i=l i=l

that is

Ja, = y,-. + Y i + y,tl

andz a , =Y i> J -Y i-I



Data Smoothing and Differentiation 32!

Finally, substituting for au and a, in Equation IIand evaluating at the mid-

point (C =0). we have

rv

y, = au + all;

= 1/, (y.., + y, + Yi+I)-t-O

The above expression cannot. however. be differentiated twice. and for

this reason second-order polynomial least-squares fits have been more wide-

-ly used in biomechanics (16,17,58,63-65,68.81,91). The equation for a

second-order least-squares approximation to five data points is

Y I =- 3 'y - : : _ 2 _ _ ± _ J _ ~ y ' - _ I_ ± _ _lb:~ 1 ) ~ ~ - = _ 3 _ X _ . + _ 1 ( 1 2 )

3 5

while the smoothed value of the first and second derivatives at the midpoint

are given by

and

N -2Yi-2 -Y,_I+ Yi+1+ 2Y,tly i = ---I 6h--- - --- ---- (13 )

rv '~ 2Y,-2 - Yi-l - 2y, - y,+1+ 2Yi+2y, - -----7~---- -----

The derivation of these formulae can be found in Wylie (103) and LanCLOS

(48). Both authors, however. point out that a smoother second derivative

can be obtained by a reapplication of Equation 13, namely

(14)

which. in terms of displacement data values. becomes the nine-point

formula

y," =~ l : : i _ 4 _ _ _ -±-_4Y'.:L_±-Y i-2_-=-4Y ,-cl9i '!-=-1Y '·_I_±__Y i~_:_t_41'~.1_±~Y~+~ (15)

100h1

Only N - n + I smoothed values are obtainable from central formulae.

and auxiliary formulae must be derived to obtain smooth values at the end-

points. In the case of the first-order least-squares polynomial derived

above. this process simply requires that Equation IIbe evaluated at 1; =-I

orl.thus

rv

y(C-l) = au + a.i.,

= Y'(Yi_1 + Yi + y,.d + I~(y, 1 Yi+l) (-I)

= 1/(,(5Y"1 + 2y, - y,_I).



, _0

0,. o;l

, . . 0..0

= '- 0

:J0..-

J:c. . !:lIJ

rjo;l

. . : : : ~CQJ_

'-

~ o;l ~. . : : : oI J (',I

C

>. > => J 0: : ; , . . 0

=;l o;l

. . : : : :::l

'Jr;J xr

::J

E L L JV"

-=0

c v:

<l J~ =J 0 0: c. .

N <l J c. .. . . . ::J c;: j

'" : : : r o;lQ) n :! uI)() + <l J

0 C > Jn :! , J J1

'1:J~ r oQ)

'" r o - <> + c

' "0 V < lJ

III<l) 'J

e oE c: r;J> ::

'rl 0 - : :J:>

. , . . . . .o;l

0 . . . . Vr E = < lJr >.

V l-. ;;-. . . . wE u .:

E. . . . .

<l JiF) V"\

8 . E uV 0;. . ~~

l-.

0 C > J. . . . c, ::J c: II. . . . '"C J " 0f)V> E- : : : l 0

V":U

o;l rjc: <l J

.~ V ::J V

0..:::-. . . . . . . . .

~~Eu <l)

uC > J ::J r o !:lI)

0 E . . . . c"D

UC r o ~. . . . . . . .

(J

E 0- ~;.-

< = rj ("l

ir:cr 0

;;,/Jc

J: c: cr :

0 c:r o 0u::J'- ~:: - : : : l

::J ';;:j »E'- V ..0. . . . ;..:r o <l) "0 0

E. . : : : V l-

t: : 0.

, . , 0'- '~ C.t: : 0 r oc-,

<Jl ..0

0<l)0

m. . . .

0. m c E:J 0

o;l C J " . : : 2 c:- ..0 VJ . . . . c-,

< t : 0 Vr o

E 0Q . . : : : c.. . .

'7

<l Jl-.

IN]'J3JVldS Ia

:l00

u,

322 G.A. J;f/ood



Data Smoothing and Differentiation

LI1

. . . .<f,

rn+ ~-" . . . .

. . . . ~t-

. . . raIII ++ ""

. . . ,

" " ' . . . . . . . .. . . . " " ' " " ':: "I M ro rn0 . . . . -o + +. . . .

N + 1::"1'". . . III 1::"1 0 . . . . . . . .

I:: U + 0 . . . . . . . .N I:: N

0 I:: . . . . . , . . .N . . . m 0 ra. . . . . . . . Z . . . . ra o + . . . . . +

. . . . ra u + I:: . . . . . . . . . . . .u rn + c : . . . :l Uc : + u 0 :l . . . . r a c ::l 0 . , . . . r a . . . . . r u + ~

. . . . . r a . . . . + o 0 . . . .r a u o .... r u

~"

s, . . . . r u . . . ."

Unl '0 . . . . " ~ .. .II) r a . . . . c:~ r a . . . .

~. . . .

c : . . . .6 - '-'

. , . . . . . . . ;J.r< : : . . . : :>~ a >, u c-,...:l : : . . . c- >-

e

/i

·l,I't •I, .i

i,I

(I

, \, I

"" ~,,,

..

..\\\. .

. . . . . . . . .'.".-,. . . . .

-,

. . . . .

' . .... . . . . .. . . . . . .

" ......

".

..

-II

I N 3 W 3 J V l d S I G

'..



324 G.A. Wood

andrv

y(t,tl)= 1/1 (y,.1 f- - y, + y.i.) + I:(y, 1- y, •.) (I)

= l/d5y, I + - Zy, - y,+I)

Similar auxiliary formulae can also be derived for the interpolating poly-

nomials outlined earlier based on forward and backward finite differences

(84), but, unlike the above formulae, a loss in accuracy results. While a

least-squares approximation polynomial provides the same degree of

accuracy over all data points, it should be noted that outside this interval

the function departs to plus or minus infinity. For this reason it is often bet-

ter to record additional data points outside the time interval of interest (56).

A central formula can then be applied, following which the additional

points are discarded.

The popularity of local least-squares approximation techniques has

undoubtedly resulted from their simplicity of application. Both smoothed

data values and time-derivatives can be obtained directly from experimental

data, without recourse to large digital computers. Decisions must be made,

however. as to the degree (m) of the approximating polynomial. the number

of data points (n) to include in the smoothing and differentiating formula.

and the number of repeat applications (£) of the smoothing algorithm prior

to the use of the differentiating formulae. Hershey et al. (37), in applyingthese techniques to physical chemistry data. suggest that an appropriate

degree of polynomial is m ~ 3. and outline an analysis-of-variance

(ANOY A) approach to determine whether a polynomial of degree m + 1

provides a significantly better fit. Rationales for an appropriate choice of n

and £ are less available. A large n of 7 or 9 may be necessary when signifi-

cant error is present in the data. but will tend to over-smooth. Similarly,

repeated applications of the smoothing formula will produce still smoother

data.Gagnon and Roderique (26) applied least-squares approximation formu-

lae (using various combinations of m, n, and £) to cinematographic data

representing the horizontal displacement of a sprinter's center of gravity

during a standing start. Two force platforms were also used to obtain an

independent measure of the sprinter's horizontal acceleration. The authors

reported that a single application (£ = 1) of a cubic polynomial (m = 3) to

five data points (n = 5) provided the best basis for estimating the second

derivative. The second derivative was obtained through the application of a

second-order (m = 2) differentiating formula identical to that described in

Equation 14 above, and resulted in a mean error of 21 % when compared to

the force-platform measures. The film data series was not extended prior to

smoothing and differentiating, and most-error' was observed in the end part

of the curve.

A similar validity check was conducted by Cavanagh and Landa (17) dur-

ing biomechanical analyses of the karate chop. In this instance. the nine-

point formula (Equation 15) was used to obtain accelerations from

displacement-time data collected by means of high-speed cinematography,

and comparisons were made with measures obtained directly from an

accelerometer. Considerable discrepancy in both amplitude and phase was




noted. and the authors doubted the suitability of this local approximation

procedure. However. it is unclear whether the displacement data series was

extended, which may have -been "important in view of the rapid changes

occurring during the last 25 milliseconds (5 data points) prior to contact. Ithas also been suggested (49) that the sampling rate in this study (200 fps)

was too high for the inherent inaccuracies in film data. This problem will be

discussed later.

As mentioned previously. one of the appealing aspects of local approxi-

mation techniques is their computational simplicity. which is particularly

important in measurement and control systems when on-line differentiation

is required. For this reason Allum (2) advocated the use of a least-squares

cubic numerical differentiator that only relied on n current and previous

data values (i.e .. a recursive formulation). Appropriate values for n and the

sampling rate were based on the highest frequency for which velocity was

required. and corrections for amplitude and phase distortion were included

as part of the algorithm.

An alternative to the local approximation approach is to fit a single least-

squares polynomial to all N data points. This global approximation tech-

nique has, in the past. been widely used in biomechanics (6,2/,47.60,72,

77, 104). probably because of the generality of approach and the accessibil-

ity of computer programs for this purpose. The procedure requires onlythat the user specify the degree m of the polynomial. the solution to the set

of normal equations being readily accomplished by high-speed computer.

In fact. for equispaced data, tables of coefficient values for a small range of

m and a reasonable range of N have even been published (20).

Decisions as to the most appropriate value of m are usually made on a

trial-and-error basis (72), although when something is known of the under-

lying function. either in terms of the number and ordering of maxima and

minima (26) or its theoretical basis (92. 93), a more rational choice of m canoften be made. Again, a statistical test based on ANOY A procedures (25,

36) or an examination of the residuals (40) can be used to determine whether

a polynomial of degree m + 1 provides a significantly better fit (see Figure

4).

When computing successive polynomials for increasing values of m,

much time can be saved by expressing the approximating function in terms

of a linear combination of orthogonal polynomials. Pm(t), that is

rather than as a power series

This procedure has a distinct advantage in that when m is incremented. only

one new coefficient need be computed (rather than m + 1). and the solution

for high-order polynomials is generally more stable (/9). The Chebyshev

polynomials [where Pm(t) = cos (m CO.';-I)J are frequently used in biomechan-



326 G.A. Wood

ics (12, I04). the algorithm and FOR TRA N program presented in Kuo (45)

often being adopted. As their name implies, orthogonal polynomials also

provide a set of statistically independent coefficients which can be used for

tests of comparison between curves.

Several researchers have determined the accuracy of results obtained

through the application of global polynomial approximations to experi-

mental data. When the underlying theoretical function was parabolic (in

free-fall situations, for example), good smoothing and different iation was

achieved (52, 92, 93). Similarly, close agreement between instrumentally

recorded ground reaction forces and those predicted from numerical differ-

entiation of smoothed film data has been found by Roberts et al. (77) (mean

error for a kicking action was 6070), by Miller and East (60), and by Lamb

and Stothart (47) (mean error = 3.5% and r = .94, respectively, for vertical

jumping).

These studies, and several studies cited previously (17,23.26), highlight

some of the problems associated with validating acceleration data obtained

by the double-differentiation procedure. First, when derived values depend

not only on the accuracy of segmental displacement-time data but also on

the accuracy of body-segment parameters (segmental masses and center of

mass locations), errors are confounded and it becomes difficult to evaluate

the performance of the differentiation process per se. Second, various sta-tistics have been used to represent the amount of agreement between two

acceleration curves, including mean errors and product-moment correla-

tions. The latter are clearly inappropriate statistics in that they use a prob-

abilistic method to examine a deterministic process and furthermore,

assume independence between observations in a time series. Mean errors,

while su ffering a disadvantage in that no test of significance can be applied,

arc more appropriate, although discrepancies in amplitude versus phase are

unresolvable. Such resolution could be obtained through time-series analy-sis (e.g., cross-correlation, cross-spectral analysis), but these elaborate pro-

cedures seem unwarranted in view of the relative simplicity of human accel-

eration patterns. and many investigators have found "eye-balling" to be a

perfectly satisfactory measure.

A major limitation of global polynomial approximation is that the choice

of m (the degree of the function) predetermines the order of the derivatives

since the k " derivative of a polynomial is always of order m-k. While a qua-

dratic fit to displacement-time data is entirely appropriate when the acceler-ation is known to be constant (as with a freely falling body), such a priori

knowledge of body motion is seldom available, and it seems unwise to pre-

empt the fundamental nature of the derivati:res during the smoothing pro-

cess (13). '" .4,

Another shortcoming of fitting global-polynornial approximations to

biomechanical data is the inability of these functions to adequately fit

regions of varying complexity within the one data series. Roberts et al. (71)

noted that "a single polynomial curve does not completely describe the

whole action." while Miller and East (60) reported some loss in intra-

subject sensitivity. And in a comparative study, Gagnon and Rodrique (26)

indicate that more valid estimates of an acceleration curve containing two



Data Smoothing and Differentia/ion 327

maxima and one minimum can be achieved through the application of local

polynomial approxirnatjons. This need for an approximating function that

can fit data in different-regiens of time with varying curvature has led

several researchers to explore the utility of spline functions.

Splines

A spline function consists of a number of polynomials, all of some low

degree m. that are "pieced" together at points in t called "knots" (x.: j :;=: I.

2 ..... n) and joined in such a way as to provide a continuous function g(l)

with m -I continuous derivatives. When m equals 3 as is most common, theresulting cubic spline function consists of n-I cubic polynomials. each of

the form

spanning an interval X j_1 ~ t < x, and satisfying the continuity condition

p;(x) = P 7 + 1 (x.): (k = O. I. 2; j = 1.2 ..... n)

where p; denotes the k " derivative of the j ' l ! polynomial piece. The condition

by which the function has m - 1 continuous derivates ensures that it is

smooth in itself but. unlike a global polynomial. its "piecewise" nature

enables it to adapt quickly to changes in curvature.

The term "spline" has its origins in a pliable strip of wood or rubber used

by draftsmen in patterning curves. but the mathematical form was popular-

ized 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 (30).

Detailed descriptions of the mathematical formula of spline functions

can be found elsewhere (22,30, ,76, 99). In general. one need only specify

the degree of the spline. the required accuracy of the fit (least-squares criter-

ion). and the number and position of the knots. The last specification can

pose some problems. although strategies exist for the optimal placement of

knots (76) based on an iterative analysis of residuals. Alternatively. Wold

(99). on the basis of experiences with spline functions in the field of physical

chemistry, suggests the following rules of thumb: I. There should be as few

knots as possible. ensuring that there are at least four or five points per

interval; 2. there should not be more than one extremum point (maximum

or minimum). or one inflexion point per interval; 3. extremum points

should be centered in the intervals; and 4. in flexion points should be close

to the knots.

When there are enough data points (Wold suggests 10-15 points between

each extremum and/or inflexion point). and the accuracy of the experimen-

tal data is fairly well known. the Reinsch (74,75) method can be employed.



328 G.A. Wood

This method places knots at every t., thereby piecing together N -) poly-

nomials in a manner that ensures that a smoothness integral defined (for acubic spline) as

r,

Q = f [g II (l)Jl dtt I

( )6)

is minimized subject to a least-squares constraint

~ r g ( t,.) - y ] 2 ~ S ( 17)

i=ll dy. jwhere the dy, are standard errors of measurement and S is a parameter con-

trolling the extent of smoothing (see Figure 5).

Reinsch's spline method has the advantage of obviating the necessity of

deciding where the knots should be, and it is also the preferred method for

differentiation and integration since it treats all data points in the same

way, with one knot in each (99). However. when there are few data « 25),

or the time interval is far from equispaced, the spline method should be

used with caution (22, 99).

Experiences with spline functions in biomechanics for data smoothing

and differentiation have been rewarding (56,83, 102. 107), the Reinsch

method in particular providing very acceptable results. Using both cubicspline functions and global polynomials to model the displacement-time

data, Zernicke et al. (107) derived kinetic data for a kicking action, and

compared the results to simultaneously recorded force-platform data.

While both methods could provide an extremely accurate fit to the displace-

ment data. the inability of the best-fitting global polynomial (of degree 5) to

respond to local trends without distorting other regions resulted in an over-

smoothed approximation to the force platform record (mean percent differ-

ence of 10.6, versus 4.7 for the spline function). An important observationmade by these investigators was that the least-squares criterion (parameter S

in Equation 17 above) could be continuously varied with spline functions,

whereas it has a fixed incremental value for a global polynomial. Hence. a

spline-function fit can be more finely tuned than a global-polynomial f i t .

In the above study. the dy, were set to unity while S was given an initial

value equal to an average standard error obtained from repeated digitizing

of a film frame. The value of S was then varied on a trial-and-error basis

until the best approximation to the experimental data was found. AsZernicke et al. (107) noted. spline functions display a remarkable sensitivity

to over-fitting in that the second derivative quickly degenerates as S ap-

proaches the random-error level. This property of spline functions provides

a useful way of testing for overfit. "".~

McLaughlin et al. (56) subsequently presenteda more refined approach

in which estimates of dy. are based on film measurements of known objects

in space. and used the more appropriate value? of S = N for cubic spline

';'Reinsch (74) suggests that natural values for S lie in the range N(l-v2lN) ~ S ~ N

(I + V 2/N), while Wahba and Wold (88) present a numerical solution for the op-

timal value of S. viz .7N - .9N.




N

x-

.r>

~ X"-'.r> V

..Q+ ~. . . . .IIIII VI

~ . . . . .I

0.. X

". . . . .x

I N 3 W 3 J \ f l d S I O

329

V" . . J

t:(\j

::l

0" .•

C] J >"ir:



330 G.A. "flood

approximations to various sets of empirical data. Comparisons were made

with other smoothing and differentiation techniques (namely, finite differ-

ences and global polynomials), and again the superiority of splines was

demonstrated. It was also apparent that considerable variation in secondderivatives occurred at the endpoints. and while the cubic spline function

was the most restrained. inappropriate values were nevertheless often

obtained. An implicit feature of the Reinsch spline algorithm. however. is

that

gk(tl) = g'(t",) = 0; (k = l , .... 2£ - 2; m =2f - I; £ = 1.2 ..... etc.)

Thus, the first and last values of the second derivative (k = 2) of a cubic

spline function U = 2) will be zero and, consequently, spurious inflections

may arise in acceleration data close to the endpoints. ':' To cope with this

anomaly without compromising the otherwise excellent behavior of spline

functions, Wood and Jennings (102) suggest the use of quintic splines U =

3). Insofar as spline functions have 2£-2 continuous derivates. it can also be

argued that a quintic spline function is a better model for biomechanical

data since, unlike a cubic spline. its third derivative (jerk) is both a contin-

uous and a smooth function (102).

Recently, Soudan and Dierckx (83) gave further endorsement to the suit-

ability of spline functions for human motion analysis. and showed that

accurate Fourier analyses can be obtained following a preliminary applica-

tion of splines from which equispaced data can be generated. These authors

also presented graphic evidence of the behavior of spline functions with

excessive over- or under-smoothing. and it was evident that they adopted a

strategy similar to that of Zernick e ct al. (107) in arriving at a "best fit;"

i.e .. they visually examined the residuals g(t,)-Y(U for important trends that

still remained there while ensuring that the second derivative hadn't become

ragged. Programs were rerun with varying values for S until a good com-

promise was reached. Naturally. this procedure can be rather time-consum-

ing and attempts have been made to automate it (46, 101).

Using a runs test. Laananen and Brooks (46) examined the residuals for

randomness and proceeded on an iterative basis, varying S until a unit nor-

mal deviate corresponding to a probability of 5% (within a specified toler-

ance) was obtained. Wood and Jennings (101). on the other hand. applied aruns test to the second derivative as a test of raggedness. while the residuals

were "filtered" for remaining trends. The filtering procedure required a

least-squares approximation (0 the residuals by.a set of trigonometric poly-

nomials of frequency below a predetcrrnrhed dnDise" cut-off. The dy . were

then reduced in regions where low-frequency trends in the residuals

remained. and a new spline function was fitted. When no significant low-

"A nother end-condition can be stipula ted. i.e .. g"(t,) = g"(L} = lJ .R m .se c.-1; or the

tim e series can be expanded by a few poin ts a t ea ch end .




frequency trends remained in the residuals, or when the second derivative

began to degenerate, the "optimal" spline fit was considered found.

Alternative procedure), are ""suggested when repeated observations are

available (35), or when (he accuracy of measurement varies in some defin-

able way with time. The dy , can then be given independent values commen-

surate with the accuracy of measurement of each datum from the outset

When an exact fit is required, parameter S (and thus dy) is set to zero, re-

sulting in an interpolative spline function (see Figure 5).

In summary, spline functions can be efficiently implemented to provide

.good approximations to biomechanical data. Their extreme flexibility and

pronounced local properties make them well suited to biomechanical appli-

cations, although consideration must be given to the appropriate degree of

spline function, the choice of the least-squares constraint, and the manage-ment of end-conditions.

D ig ita l F ilter ing

A digital filter is a frequency-selective device that accepts as input a

sequence of equispaced numbers y(t), and operates on them to produce asoutput another number sequence, y (1), of limited frequency (41). Figure 6a

depicts one form of digital filter, the low-pass filter, so called because the

operation it performs involves passing only the low-frequency components

of a signal. For obvious reasons such a device has attracted the interest of

biomechanists.

The basic operation of a digital filter can be represented by a series of

mathematical operators (a.z/). where a, designates a weighting (multiplier)

coefficient and Zi designates a time delay, and where the exponent j indi-

cates the number of time interval shifts. These are shown schematically in a

block diagram (Figure 6b). Here, if a, = ",~and j = I, the output Y i of thefilter a.z: will have a value equal to one-half of Yi-l'

In a similar fashion, the simple three-point moving average presented ear-

lier can be considered a digital filter of the form shown in Figure 6c, if au =

a, = a2 = 1 /1• The series: auzu + a.z " + a.z ? is called the z-transform of the

weighting sequence and plays an important role in digital-filter design (28).

The digital filter depicted in Figure 6c, where the output is a simple

weighted combination of a finite number of present and past data samples,

is called a nonrecursive filter. and has the general form

n

Y i = LaSi-j; (i=n, n+ I, n+ 2, .... N- n)j=O

An alternative form is the recursive or autoregressive filter, whose output

depends not only on present and past samples of y(t) but also on past values

o(y(t}. Such a filter is depicted in Figure 6cl and has the general form



332 G.A. Wood

Figure 6. Schematic representations of digital filters: (A) low-pass digital filler ef-

fect; (B) block diagram of the mathematical operation of a simple filter;

(0 block diagram of a non recursive filter (equivalent to the simple three-

point moving average if au = a, = a , = Ifl); (D~ block diagram of a recur-

sive fil ter.

Recursive filters are said to have an infinite memory. or "feedback." In

that some knowledge of all previous data is retained in the process. A

specific example is the second-order recursive filter (Butterworth) advo-

cated by Winter et al. (97). i.c ..

recursive terms

Values for the filter coefficients a., b, are determined on the basis of the

desired cut-off frequency and. for convenience. have been published by

Winter (95) for various sampling and cut-off frequencies.

The behavior of a digital filter can be described in terms of its frequency

response or transfer function. which describe the mathematical relationshipbetween input and output. It is customary to decribe this relationship in

terms of the effect that a filter has on a sinusoidal input of varying frequen-

cy. The ratio of the amplitude of the output to that of the input expressed as

a function of frequency is termed the gain of the filter. while frequency-

dependent time shifts are termed the phase characteristic. For example. the

frequency-response characteristics of the simple three-point moving average

can be readily determined by considering the input function to be

yet) = sin(2nft)

where f is the frequency of the sinusoid in cycles per second and t is Lime.

Thus. for the more familiar centered form

rv_rv( _ 1 / [ ]Y i - Y t I) _ /" J Y i· I + Y , + Y i+ 1

= 1/3 (y(t,-fit) + y(C) + ytt, + fit) ]

where fit is the time interval. Now. substituting for y(t;)

y(t,) = Y l [sin(2rrf(t,-ht») + sin (2fdL;)+' sin (Z rrf'(t. + fit»]

=sin(2rrft,) (YJ( l + 2cos2nfht)J

As can be seen. the first term here is equivalent to the input function. and

since the filter gain is defined in terms of the ratio of output to input,rv

. yet)gam =--- = = 'Ii(1 + 2cos2nf M)

y(t)



Data Smoothing and Differentiation JJ3

I N P U TO U T P U T

(A) Y")v\;- LOW-PASS

DIGITAL FILTER

S I M P L E F I L T E R

(B)

r--- -l

y(t) O - " " " : ~ G 'oY, - 1 " ---z--I- .... I

~UL~LIER __ ~ME_EELAY _j

N O N - R E C U R S I V E F I L T E R

I--I

I

--,

I I(C) y(t) y(t)

I I

L _j= aOYi + 31Yi-l + 3;;Yl-2

-----

R E C U R S I V E F I L T E R

I----- ,

y(tl

I+ 30Yi -1i

(0)I

y (tl Z-I

I

I I

I I

L--- ---_j



334 G.A. Wood

By evaluating this expression at various frequencies (0 ~ f ~ I _ L where I. is

the sampling rate). the gain function can be obtained. For example. when f

= a the gain is unity, and when f = ; ; 1 f. the gain is zero (see Figure 7). This

gain characteristic is that of a low-pass (smoothing) filter, with low frequen-

cies being almost in their original form (gain ~ 1_0) while frequencies

approaching one third of f. become progressively more attenuated. While

the gain function of the three-point moving average pales in comparison to

that of the ideal low-pass filter (also shown in Figure 7), it approximates the

type of processing required for noisy displacement-time data. Gain func-

tions for other moving average formula have been presented by Lees (49).

The phase characteristic for the three-point moving average is also

depicted in Figure 7. While the phase shift is given as zero over all frequen-

cies up to one half of the sampling frequency, it can be seen that the gaincharacteristic becomes negative at one-third C. This fact indicates an ampli-

tude reversal of components above this frequency. which is often shown as

a 1800 (rr radians) phase shift.

It should be noted that only frequencies up to one half of the sampling

rate, C. are resolvable in a sampled data sequence (the Nyquist limit),

higher-frequency components being confounded ("aliased") in lower parts

of the spectrum _* For this reason it is essential that sample data always be

taken at a rate at least twice that of the highest known frequency(Shannon's sampling theorem). Given the relative low-frequency nature of

many human movements (8,89, 95, 96). it has been suggested that

displacement-time data seldom need to be sampled at rates in excess of 100

Hz. and indeed, that rates as low as 25-30 Hz often suffice (12, 49, 98). The

essential criterion here is the frequency bandwidth of the motion to be stud-

ied, which can only be determined by Fourier analysis.

When simple algorithms such as the three-point moving average are used

to smooth displacement-time data and to calculate derivatives, "optimal"sampling rates can be determined by a joint consideration of the frequency

response function and upper frequency limits of the motion. For example,

if the highest frequency component of a motion were 5 Hz, and a gain of .7

(3dB point) at this frequency were considered adequate (which from Figure

7 is seen to correspond to a frequency of approximately one-sixth f.), an

appropriate sampling rate would be 6 x 5 Hz = 30 Hz. This rate would place

the 3dB cut-off point at the appropriate frequency (N5 Hz). Lee (49) pre-

sented a similar procedure for the use of simple differentiating formulae(e.g .. Equations 13-15). but pointed out that several investigators have con-

demned these simple formulations as a result of inappropriate sampling

(and thus cut-off) frequencies.

In most applications, however, digital filters are designed in such a way

that the transfer function is independent of the isampling rate and conforms

as closely as possible to the ideal filter. Good low-pass characteristics can be

obtained from many types of filters (28, 42,53,54, 90, 94,96). the recur-

*Winter (95) gives a good illustration of aliasing.




+ T T

- T T

1.0

.7

ZH

{ 3

33.

".._Gain function of ideal

low-pass filter

,-Gain function of 3~point

movingaverage

FREQUENCY

Figure 7. Gain and phase characteristics of a simple three-point moving

average and of an ideal low-pass filter. When filtering

characteristics are invariant, a judicious choice of data sam-piing frequency (U can give the appropriate filtering effect

(3dB point at. say, 5 Hz; achieved here by sampling at approx-

imately 30 Hz).



JJ6 G.A. Wood

sive formulation often being preferred since it needs fewer terms to produce

a sharp cut-off. and hence less computer time. The poor phase characteris-

tic of a recursive filter can be overcome by filtering the data both forward

and backward.

Pezzack et al. (70, 71) compared the results of smoothing by digital filter

followed by finite difference estimation of the second derivative against

directly obtained acceleration values. An accelerometer was fastened to a

horizontally rotating arm and 16 mm film of simulated arm motions was

obtained. Digitized film data was then smoothed in several ways. different-

iated twice, and compared with the analogue record of acceleration. Appli-

cation of a second-order Butterworth digital filter followed by first-order

finite difference calculus (95) was found to give the best approximation to

the accelerometer trace, while the direct application of the finite differencestechnique or approximation by Chebyshev polynomial produced unaccept-

able results. Several equally acceptable fits to the Iilrn-datalist published by

these investigators have since been reported (26, 34, 49, 51, 83, 102) includ-

ing, contrary to the expectations of Pezzack et al. (71) the use of spline func-

tions (83, 1(2).

Lesh et al. (51) used the nonrecursive FIR (finite impulse response)

digital-filter approach of McClellan et al, (53) to formulate a filter whose

length (number of coefficients) and frequency response characteristics canbe defined by the user. This algorithm can combine several filtering features

into one operation, thereby providing a more unified approach to the prob-

lem than that of Pezzack et al., (71) which is essentially low-pass filtering

followed by high-pass differentiation. A low-pass filter with a 0-6 Hz pass

band, 6-9 Hz transition band, and 9-15 Hz (Nyquist frequency) stop band

was designed by Lesh et a!. (51) for smoothing gait displacement-time data,

while a 4 Hz low-pass differentiating filter with 59 terms was chosen when

time-derivative information was required. The same low-pass differentia-ting filter with an 8 Hz cutoff gave excellent agreement to the test data of

Pezzack et al. (71).

As can be seen. nonrecursive filters often have a large number of terms in

their formulation, and for this reason the computational costs are high.

With this fact in mind. Gustafsson and Lanshammar (32) developed an

iterative filtering procedure that used both nonrecursive and recursive dif-

ferentiating filters in its processing. The nonrecursive filter was designed to

minimize the expected value of the mean-squares error in the derivative to

be estimated, subject to the constraint that estimates of the k" order deriva-

tive were unbiased (i.e., no systematic error due to the filter) when the input

signal was a polynomial of degree m for m ~ k. Typically, 51 filter terms

were required, so a recursive formulation of the "Filter that reduced the num-

ber of arithmetic operations by a factor of five ~as developed. However.

the recursive algorithm was potentially unstable. ,;,so an iterative procedure

was implemented to monitor the error due to instability by comparing ceca-

"A grow th in trunca tion errors occurs when the a lgorithm is itera ted m any tim es.

Some au thors (32) suggest tha t th is problem i . , a lso inherent in rhc CU I3IC recursive

a lgorithm of A llum (2).




sional recursive filter output with a nonrecursive filter prediction. If the

error was within specified bounds, the recursive-filter computational pro-

cess continued; if the etr9r was too large, processing "back-tracked" to a

previous point in time when it was acceptable. non recursive-filter initial

values were provided, and recursive filtering was allowed to continue. This

procedure resulted in a 600/0 -70% saving in computer time. Equally note-

worthy, though, is the use of measurement theory principles and basic func-

tions akin to those used in the spline algorithm. ::'

Gustafsson and Lanshammar (32) evaluated the efficacy of their filtering

procedure by comparison with the Pezzack et al. (71) test data set. with

excellent results. Comparisons between ground reaction force measures

during the single-support phase of walking and simultaneous measures

obtained from a force platform were equally good. although it was appar-ent that some very low-frequency error occurred, presumably due to shifts

in location of anatomical markers.

The application of digital filters to data smoothing and differentiation

obliges the user to make value judgements about cut-off frequencies in

much the same way as a polynomial user decides on the degree of poly-

nomial. Digital filtering, however, requires that the data be equispaced. and

care must be taken to ensure an adequate sampling rate in relation to the

chosen ultimate cut-off frequency. Gustafsson and Lanshammar (32)

present a method by which suitable sampling rates can be determined on the

basis of precision limits of the measured and differentiated data. and the

bandwidth of the signal. i.e..

I d 2 • W 1k+ I

f = = - > __L__ I > - - -

, T ~ d; . r r . (2 k + I)

where (P k is the variance of the estimates of the k" order derivative, e f is thevariance of the noise of the measured data.and co, is the bandwidth of the

signal. in radians/sec. [f'(Hz) =(1)2 rr·). Thus, to estimate the second deriva-

tive (k = 2) within a maximum variance of. for instance, (O.3m/sec1)1 when

the measurement error is estimated to be (0.003m)1, and of the signal band-

width is 5 Hz (IOrr ' rad/s), a necessary sampling rate would be

f . = liT _ (O.003)lx(IOrr):i

,mm ,,"ux- (0.3)2XrrX5

::::200Hz

This frequency is much higher than that usually recommended by

researchers (12,49,98), despite the realistic estimates of error variance

(better than 5% in the second derivative). For instance, Cappozzo et al. (/2)

"This also applies to any time-domain realization of fillers which are essentially

polynomial-pass in nature in that they will pass a locally fit led polynomial o r degreem ~ n - 2 without distortion.



JJ8 G.A. U100d

recommended a sampling rate of five to six times the highest frequency of

interest (f, = 20-30 Hz. for walking). The apparent discrepancy can be

explained in terms of the theoretical interpretation of the noise spectrum

given by Gustafsson and Lanshammar (32). These authors assume that the

noise spectrum is flat up to and beyond the Nyquist frequency. which in

practice is probably not the case. Unfortunately. no details of noise spectra

are available in the literature. Also. as the authors themselves point out.

much of the signal power. at least for gait. occurs at frequencies even lower

than 5 Hz.

In some laboratory situations. real-time processing of displacement-time

data is required. as for example when immediate feedback is to be given to a

subject. or a subject's on-going movement is to be perturbed in a motor-

control experiment. In this context. Woltring (lOO) found adaptive-filtering

(Kalman) techniques to be useful. Here systematic discrepancies between

incoming data and model predictions are used to adjust the filter parameters

-a procedure that has found wide application in the aeronautical industry.

Such elegant realizations of a filter are. however, computationally demand-

ing and the advantages have yet to be demonstrated.

In summary. digital filtering affords an efficient means of smoothing and

differentiating displacement-time data in a way that is intuitively appealing.

Some knowledge of the frequency spectrum of signal and noise is required.however. although residual analysis can be a useful guide to optimal cut-

off. The major drawback is that the procedure does not provide a smooth

analytical function for future computations. although an interpolative

spline could be used to achieve this after smoothing. Some sort of interpola-

tive procedure is also necessary prior to smoothing if the data is not equi-

spaced.

Fourier A naly sis

Just as a function f(t) can be expressed as a weighted sum in termsof t, t '.

t'. etc. (Equation 4), with weightings based on the values of the derivatives

of the function (the Taylor series), the Fourier series provides a means of

expressing a periodic function as a weighted sum of sine and cosine terms of

increasing frequency:

f(t) =ao + a,sin(2rrt/T) -j- a.sinf-lnr/T) + aJsin(6rrt/T) + ...

+ b,cos(2rrt/T) + b2cos(4rrt/T) + b1cos(6rrt/T) + ., .. .

n '" .~=a,J +.2: [a,sin( jZrrt/T') + b , cos(j2rrtJT)]

J = 1

The first sine and cosine terms represent functions describing one cycle

(2rr radians) in the total time period (T seconds). and subsequent terms rep-

resent functions whose frequency is a multiple of this fundamental frequen-

cy. These functional components are called harmonics. and their respective

coefficients a., b, (the Fourier coefficients) indicate the strength of the j,ll

( 18)




harmonic whose frequency is j2rr/T radians per unit of time. The coefficient

a, is the mean of the data series. and t is time. the independent variable. The

basic structure of these'harrnonics is shown in Figures 8a and 8b. while the

manner in which a periodic-data sequence can be approximated by a

weighted combination of harmonic terms can be seen in Figure Sc, The

respective weightings given to each term during the least-squares estimation

procedure are indicated by tabular inset.

This approach can be generalized to nonperiodic (= cyclic) data

sequences by a suitable manipulation of the data t I, 7), whereupon it is

assumed that the data does repeat. Under these circumstances the frequen-

cies at which the data are analyzed have no meaningful relationship to the

physical phenomenon being studied and are only a mathematical conveni-

ence. Often a fast Fourier transform (FFT) algorithm is used to provide thetotal frequency spectrum of a time series up to a frequency equal to one half

of the sampling rate (Nyquist frequency).

Several investigators have used this harmonic, or Fourier analysis

approach in studying human movement. in some instances to search for

simple terms in which to describe motion, and in other instances to examine

th e frequency content of human movement patterns. It can also be used as a

direct means for obtaining time-derivative information. Insofar as a com-

plete time-history can be represented by a few Fourier coefficients, harmon-ic analysis provides an economic means of storing kinematic and kinetic

data for later comparison or statistical analysis. and can provide a useful

basis for the study of normal and abnormal gait patterns (80).

Harmonic analysis has also been used to gain insight into the frequency

spectrum of displacement-time data as a forerunner to the design of digital

filters to separate signal from noise. For cyclic motions it has been generally

shown that little signal exists beyond the seventh harmonic. which for nor-

mal walking where T is approximately 1.2 seconds is equivalent to

W j = j2rr/T

7x2xn_-_._- ..-

1.2

= 36.6 radians/sec

Since one cycle equals 2n radians. this corresponds to 5.8 Hz [f'(Hz) =

w/2n]. However, Zarrugh and Radcliffe (105) suggested that at least 20 har-

monics might be necessary to adequately describe the motion of the fool.

Children's gait also appears to contain more complex components (24).

The use of Fourier analysis techniques to obtain smooth displacements.

velocities, and accelerations is a relatively recent innovation, but the results

are extremely appealing. Cappozzo et al. (f 2) computed successive harmon-

ics for sets of displacement-time data until the variance of the residuals

approached a minimum. and then obtained velocities and accelerationsfrom the Fourier coefficients up to that order. The appropriate equations

are obtained by differentiating Equation IS:



340 G.A. Wood

n

f' (t) = ,2 j2nlT [ -ajsin(j2ntlT) + bjCOS(j2nt/T)]

J=I

and

n

f" (t) = ,L (j2n1T)2 [ajcOS(j2nt/T) + b jSin(j2nt/T)]

J= 1

A generalized smoothing procedure has been described by Bar et al. (7),

where the total frequency spectrum was first computed and then "filtered"

through the application of a Hamming window (33). The lowest cut-off fre·

quency at which the deviation of the filtered signal from the original did not

exceed the known error amplitude was selected as the most suitable. Bar et

al. (7) demonstrated the validity of this approach through. successful recon-

struction of an analytic displacement signal synthesized from a 4 Hz cosine

function and a 12 Hz sine function, and showed the superiority of the

frequency-domain method over a local polynomial (time domain) method.

Naturally. though. a Fourier series approximation to a trigonometric func-

tion would be expected to be superior to a polynomial approximation.

(A) fj(t) = sin(2.jt/T) ; (j = 1. 2, ..•• 5)

" \,\

\:s

, ;'\

.....

• I:,'/

•',/ :,I

/,,-t/

.

.

, .\

\. ,\\,\

\

\, f":u-I:,:I z :

\ . ':-.:'~.:

Til'(

Figure 8. Fourier series approximation to a typical displacement-time data se-

quence: (A) the first five harmonics of the fundamental sine function; (8)

the first five harmonics of the fundamental cosine function; (0 synthesis

of first harmonics. first three harmonics. and first five harmonics. eachweighted appropriately to approximate the displacement-time data se-

quence, Tabular inset indicates respective weightings given to each Ire-

quency component and total contribution. power. being a/ + b;'.




Figure 8, Continued.

o

COS(2wjt~T t , . ; Ii :"'1, 2•. , •• 5)

J41

~< . : : >z::<C

(8) Lt))

, " " " ' , . . : 1 / " • • • • •

• I \.

I , :

I ,.

I ' .lI :\

I • ,

\

\

\,

I~I~

I:I:.,',:

I :

I :'

1 :,:, :'1 :, :1 - •I :, :I :

•I :'

1 :

I. : •

\ ',/ :. . . . _ " , -,_ ....

.:........ ,'- .. .,I \

... ..I \..

• ',I \

t , \I, ,

I: ,

I \

1,1

\

\

\ .\ :\,"

,\:\\,\

\,,\

,I,

~I:,;,:

I

I '

, I

11'I

I

(C) f(t) = ao + [a),Sin12Wjt/T) + b),CO.(2Wi/Tl]

j=1

,)

:":\

" \

" \:I

" \

.: \

:,: . \

:\:\: . \. \': \" \:,.', ,.: \

" \ ,. . ,\.'. .; I

....a._ .... _..~

fourier Coefficients

Hannonic Sine Cosine Power

1 -25 :79 -10.46 774.5~0.88 5.15 27,3z: 2 -

UJ3 3.52 1.91 16.0

:E:W 1.25 0.00 1.5w

O.BB 0.16 O. BC_ ,0...c.n

<=I

. / " " 1 - - - - ,1.' / - - - -' ~ I,,'------,

/'

THE

,,,,,,,,,,

,,II

","" \",\

" \"",

Harmonics

... :tt]"'4+~

... "} + 3

T I M E

A Fourier analysis approach to data smoothing and differentiation

requires that the time series be equispaced , with any errors in the time mea-

surement being ascribed to the dependent variable. Furthermore, the data

are assumed to be periodic, a situation which for noncyclic displacements



342

oI5P 1.5

LACE 1.r

M

ENT D .•

1.4ORIGINAL TIME SERIES

G.A. Wood

Detrend, taper

(and extend if FFT)

c:>

O . D ~ - - - - ~ - - - - - - r - - - - - ~ - - - - ~ - - - - - - r - - - - - ~ - - - - ~ - - - - - -D.D

o

I5r L.I

LACE 1.1:

MENT 0.1

f.O

~.D Z.4

0.4

D.4 D.I L ••

T I M Er.1

Reconstruct by

inverse transformand replace trend

D . D ~ - - - - ~ - - - - - - r - - - - - ~ - - - - - ' - - - - - - ~ - - - - ~ - - - - - - - - - - ~0.0

1-1

t.4 Sc-100THED ':'IME SERIES

f.O

0.4 0.1 1 . &

T I M E'.0 I.'

0.4

L . , f.4

Figure 9. Schematic representation of processing steps In generalized Fourier

analysis approach to data smoothing.




Figure I). Continued.

oI5P 1.0

LR

~ 0.5

MENT 0.0

f.D DETRENDED TIME SERIES

-D.S

-L.D~-------r------T--------~------T-~~or-----r------'-----'a .a a .• a., 1.f Lot i

T I M Ef.O 1.4 f.'

4 PERIODOGRAMTrans form to

Fourier series

and apply window

Dpol o J aER

-------,

.'.j -r.

Lo0 -

j -4

. . . . ... . ... . .'

r

- .

12 11 raF R E Q U E N C Y

n•

can be achieved by detrending the data in a way that Y l = Y \ " The essentia

steps in this Fourier analysis approach are depicted in Figure 9. To expedit



344 G.A. Wood

the process. a fast Fourier transform (79) is used, in \vhich case the time-

series needs to be extended to some power of two (N = 2"), either by adding

zeros. or as is more appropriate when derivatives are required (3). by an odd

extension of the time series.

In principle, the optimal window for filtering the periodogram (raw spec-

trum) of a noisy signal has a weighting sequence identical to the signal-to-

noise ratio. That is. at low frequencies where the periodogram contains

mainly signals. the window would have weightings close to unity, while at

higher frequencies. where the signal portion diminishes, the window weight-

ings taper to zero. Anderssen and Bloomfield (3.4) documented a proce-

dure for determining an optimal window based on the regularization proce-

dure of Callum (11). It is assumed that the signal spectra can be modelled by

a function of the form

and optimal vaues for the parameters b and a are found by the maximum-

likelihood principle. Recently, Hatze (34) adopted this approach for esti-

mates of higher-order derivatives from noisy biomechanical data. In this

implementation. an optimal window is determined for filtering the periodo-

gram of the second derivative of the time series, and lower-order derivatives

are obtained by integration. Alternatively, optimal filtering of each deriva-

tive could be obtained separately. but the first approach has the advantage

of using the same mathematical model for all derivatives. Hatze (34)

showed that the optimally regularized Fourier approach provides excellent

conformity with the data set of Pezzack et a!. (70,71), even in the presence

of added noise.

COMPARISON OF METHODS

Given the varying mathematical bases of the smoothing and differentia-

tion procedures discussed here, comparisons can often be less than mean-

ingful. In essence, most researchers opt for a procedure that produces valid

results in the context of those aspects of human movement in which they are

interested. Although no precise data are available. it appears that at pres-

ent, most laboratories use either splines or digital-filtering techniques.

Though i t would produce no definitive results, it was felt that it would be

instructive to compare some of the currently used procedures in terms of

computational cost and "filtering" efficiency .. Table 1 presents computer

CPU times for various smoothing and differenfiaring routines, all of which

were coded in FORTRAN and accessed via the same main program. The

test data set of Pezzack et at. (71) was used as input. and some additional

random error (normally distributed with a mean of zero and variance of

0.05). was added to provide a more realistic test of the procedures.Computations were performed on a DEC System-IO computer, and pro-

cessing was replicated 100 times to minimize 110 and time-sharing biases,




Appropriate values for the dy . and S parameter (Equation 17) for the spline

routines (102) were logically defined in terms of the known magnitude of

error associated with the: displacement data (dYi = .05 ; S = N). and the

Fourier analysis proceduretemploycd the optimally regularized implemen-

tation of Hatze (34). The Chebyshev polynomial approximation routine

was taken from Kuo (45), while the digital filter was the second-order

Butterworth filter plus finite-difference formulation used by Pezzack et al.

(71). In the latter instances optimal smoothing was determined by trial-and-

error. the analogue acceleration trace of Pezzack et al. (71) being used as

the criterion for' best fit.'

Tab le tComparison of Computing T imes for Various

D ata Sm oothing and D ifferen tia tion P rocedures

Routine"c.P.U. time"

(sec) Remarks------_._------_._--------- ---- ------_._---_._----_._---

Global polynomial

Cubic spline

Quintic spline

Optimally regularized

Fourier series

Second-order Butterworth

0_092

1.285

2.889

1.624

3.143

0.034

Ninth-degree Chebyshev

dy. .. 05; S = N

dy, = .05 ; S = N

Option 2

Option I

Cut-orr 3.5Hz. Time

series extended by

20 points

"Software sources are indicated in text.

hAverage of 100 replications on DEC System-IO.-.-------- --- .----------------------. _ ...._-.-

Predictably the processing time required was directly proportional to the

complexity of the numerical method involved. but it is worth noting that

spline routines need not be as inefficient as Hatze suggested (34). Indeed. in

terms of processing time. those used here compare quite favorably to the

fast Fourier transform approach advocated by that author. And even if sig-

nificant differences were noted in computing times. it could be argued that

these costs would pale in comparison to the costs of the initial data acquisi-

tion process. and therefore represent only a fraction of the total investment

in the data.

Of greater concern though is the validity of the output, evidence of

which is provided in Figures 10-14 (a and b). While all procedures except

that of the polynomial approximation provided good fits to the noisy

displacement-time data. some differences in the conformity of the second

derivative to the analogue acceleration trace were evident.The phase shifts in the polynomial function brought about by the influ-

ence of the first dominant peak in the displacement-time data are aggra-



346 G.A. Wood

vated in the second derivative. and the departures to infinity at the end-

points render this model totally unsuitable in this application. More accept-

able results were provided by the spline fits, the second derivative of the

quintic spline providing a smoother representation of the acceleration trace

with little loss in accuracy in maxima and minima.

The additional noise in the displacement-time data necessitated a cut-off

frequency of 3.5 Hz for the Butterworth filler, whereas Pezzack et al. (71)

had found a 9 Hz cut-off sufficient to produce a good facsimile of the ana-

logue acceleration trace in the absence of added noise. The approximation

nevertheless still compares favorably with the optimal spline firs. The opti-

mally regularized Fourier filter produced a slightly smoother. but neverthe-

less acceptable second derivative.

To gain some insight into the filtering efficiency of each procedure, a

time-series analysis approach was used to obtain the equivalent transfer

function of each numerical process. Both the noisy displacement-time data

series and the smooth approximations were transformed into a frequency

domain (Fourier) series, and the cross spectrum of the series pairs was

derived. The gain characteristic of the cross spectrum then defines the trans-

G L O B A L P O L Y N O M I A L [ N I N T H D E G R E E ) (A ):l.S

2.0

THE l.5

TA

/ 1.0

RAo/0.5

0.0

-0.50

- + .0---

0"T" ••----oT".,---L .,....Z---,Lr-6-~-_2".-0-;-~. -z~•--2 ....--3-'. 2'

TI M E /SEC/

Figure 10. Global polynomial approximation to noisy angulardisplacement data (dots): (A) ninth-degree Chebyshev fit;

(8) second derivative comparison to true acceleration trace

(dotted line); (0 transfer function of smoothing process.





\RA D

o/SI -40

S\

GR

~ 0.75

L20

GLOBAb POLYNOMIAL (NINTH DEGREE);, '"

(B)

_ _ .

-80

-120~----.----.----~----r---~----~----~--~

0.0 0.4 1.2 1.6 2.0

TIME \SEC\3.2.II 2.11

GLOBRL POL YNOM IAL (N INTH DEGREE) (C)

1.50

1.%5

1.00

0.50

0.2S

a .00+----=~-........,~--..,..------r"--~--...-----..-~--.c 4 12 16 20

FREQUENCY I H Z III

----------------- --------------------------

fer function of the smoothing process as a function of frequency. Gain

functions for each smoothing process reported above are shown in Figures

IOc-14c.



348 G.A. Wood

CUBIC 5FLINE (5=.355) (A)

THE 1.6

TA

Z.O

/ I.!

RAo/0.'

0.4

O.O~----~----~---'----~-----r----'-----r---~0.0 D . ' l.r 1.6 r.o

TIME /SEC/r.'

Figure 11. Cubic spline approximation to noisy angular displace-

ment data (dots): (A. optimal cubic spline fit; (m second

derivative comparison to true acceleration trace (dotted

line); (0 transfer function of smoothing process.

The signal distortion evident in the polynomial fit is seen in Figure 10c as

an amplification (gain> 1.0) of low frequencies, and the very low cut-off

point (3dB point = 2 Hz) resulted in a function containing little of the small

detail inherent in the original data. The transfer functions of the other pro-

cedures are very comparable and more closely approximate the ideal filter

previously shown in Figure 7, with those functions of the quintic spline and

optimally regularized Fourier filter being particularly impressive. The 3dB

points (300,70 attenuation) of the spline and Fourier filter-transfer functions

are approximately 3.2 Hz, which agrees closely, with the formally defined

3dB point of the Butterworth filter (2.8 Hz).'~ The roll-off slopes of the

transfer function of the quintic spline and Fourier filter are. however,

somewhat superior. indicating that a better separation of signal from noise

has been achieved.

---------

"The double (reverse) application of a second-order filter produces a fourth-order

zero-lag filter with a cut-off CO, equal to .802 of the original value (3.5Hz).




Figurell, Continued.

CUBIC SPL[NE (5= .355)ltO

A IDL :.p ..

HA 40

\RA D0/S/ -40

5\

-10

-110-t----,r-----r--~--r__---r--...,._--r__-_.,0.0 0.4 1.1 1.8 r.o

T IM E \ S EC\1.8

349

(B)

(C)

D.O+---r----.~~~----~--__,r__-_r---._--_.o

0.8 1.4

CUBIC SP L INE (5= .355)1.1

1.0

a . 1

GA

~ 0.8

0.4

0.1

4 Il 12 111 20

FREQUENCY I H Z I24 21l



350 G.A. Wood

QUINTIC SPLINE (5=.355) (A)

2.4

2. 0

THE 1.6TA

/ 1.2

RAo/0.11

0.4

0.0~----~----r-----r----'----~-----r----,,--~

0.0 0.4 0.1 1.2 1.6 2.0

TIME /SEC/2.4 2.11

Figure 12. Quintic spline approximation to noisy angular displace-

ment data (dots): (A) optimal quintic spline fit; (8) second

derivative comparison to true acceleration trace (dotted

line); (0transfer function of smoothing process.

CO NCLUD ING R EMA R KS

As has been shown, acceptable results can be obtained from several

smoothing procedures when the controlling parameters are appropriately

specified. The finite-difference technique was not evaluated here, as it was

previously shown (71) that without some prior smoothing this approach

produces unacceptable results. Similarly, the simple moving averages pre-

sented earlier would have produced unacceptable results, not because of

their inherent formulation, but by virtue of a mismatch between their filter-

ing capacity and the sampling rate used in collecting the test data sequence.For example, the simple three-point moving average would have had an

effective 3dB point at approximately 9 Hz (refer to Figure 7). These tech-

niques. together with global polynomial approximations, are fraught with

problems and should not be generally used' for the differentiation of

displacement-time data. ~

Digital-filtering techniques are naturally suited to the problem of noise

separation. but are restricted in that the time series must be equispaced and

the procedure does not provide an analytical function upon which future

computations (differentiation, integration, statistical comparison) can be

based. Fourier analysis and the use of splines fill this need, but Fourier

analysis also requires equispaced data.





Q U I N T I C S P L I N E ( 5 = . 3 5 5 )110

R I O

L . .r

.,

HR 40 . .

..\RA D0/5/ -40

5\

-10

-120 -+-----.....----.----.--__,.---.---....----y-----,0.0 0.4 o.e I.e.2 1.' 2.0

T I M E \ S E C \2.4

351

(8)

!I.2

Q U I N T I C S P L I N E ( 5 = . 3 5 5 ) (C)

L.0

0.'

GR

~ 0.6

0.4

0.2

0.04--~....-~~---- __ ----....._-~--,_--~r_-_,

o

•II 12 16 to 24

f R F . C U E N C Y I H Z IZ I I



352 G.A. Wood

2ND ORDER BUTTER. FlLTER (CO=3.5HZ) (A)t...

2.0

THE I.'TA

/ 1.2

RA

D/ 0.1

0...

O.O;-----.-----r-----r----,.----.----~----_r----~0.0 0.• D.I 1.2 1.& 2.0

TIME /SEC/t.• 2.1

Figure 13. Second-order Butterworth filter smoothing of noisy

angular displacement data (dots): (A) 3.5 Hz filtering ef-

fect; (8) second derivative (finite differences) comparison

to true acceleration trace (dotted line); (0 transfer function

of smoothing process.

It appears highly desirable that the procedure to be adopted can be imple-

mented in such a way that optimal smoothing is attainable. The Fourier

approach used by Hatze (34) is one such procedure. but Jackson (40),

Winter and Wells (98), and Cappozzo et al. (12) have also provided

approaches that can be applied to any smoothing procedure. Essentially.

these techniques involve a residual analysis in which the average error is

computed for successive increments in the smoothing parameter (including

degree of polynomial, number of harmonics, cut-off frequency. least-

squares constraint). Knowing that the trend in average residuals will beessentially flat once the random-error level is reached, a threshold value

above this level can be established as the cut-off point (see Figure 15). While

at present choices of threshold value are made on rather arbitrary bases, it

seems likely that as more details on th.e statistical nature and frequency

spectrum of signal and noise become available', such an approach will be

widely accepted by biomechanists. Indeed such information is already

incorporated in the Reinsch spline algorithm and the optimally regularized

Fourier routine.While Fourier analysis is ideally suited for the analysis of periodic data

and provides the most efficient means of storing the details of a time series

by virtue of the small number of coefficients usually necessary, it can also





R 1 1 0

LPHR 40

\

RR OoIS1-40S\

" ,2 N D O R D E R B U T T E R . F I L T E R ( C O = 3 . 5 H Z ) (8)

L2D

- 1 1 0

-120+-----r---.--__,~-_r_--..---_..,..--....._-__.0 . 0 0.4 2.4 r.• ].2. 1 1 1.2 1.6 r.D

TIME \ SEC\

2 N D O R D E R B U T T E R . f I L T E R ( 3 . 5 H Z ) (C)

lor

' . 0

0 . 1 1

GA

~ O.S

0.4

0.2

o II 12 .6 20 24

fREQUENCY I H Z I28



354 G.A. J/f/ood

OPTIMAL REG. fOURIER fILTER (OPTION=2) (A )r...

/l.r

RA

o/ 0.1

2.0

THE 1.6

TA

O.OT-----~----r---_.r_--_,----_.----_r----,_--__.0.0 0... 0.1 1.2 1.6 2.0

T IM E /S EC/2 . " 2.11

Figure 14. Optimally regularized Fourier series approximation to

noisy angular displacement data (dots): (A) filtered Fourier

series fit; (B) second derivative comparison to true accelera-

tion trace (dotted line); (0 transfer function of smoothing

process.

be used in the more general context of a periodic data. The frequency con-

tent of displacement-time also provides a useful basis for distinguishingbetween those properties of a movement pattern that arc individual as

opposed to general in nature.

Spline functions have some added features that make them extremely use-

ful analytical tools for the biomechanist. First, they are the ideal interpola-

tive function to use when a set of time histories must be synchronized in

order to process data. This need may arise when multiple cameras are used

to film a movement pattern, when different systems are used to record dif-

ferent kinematic and kinetic aspects (e.g., a camera and force platform). orwhen a multi-channel electronic system is sequentially sampled (multi-

plexed). Interpolation can also be performed on non-equispaced time series

to render them amenable to digital filtering or Fourier analysis. Second.

some useful generalizations of spline functions can be formulated through

modifications of the end-condition constraint 185). For instance a periodic

spline where gh(tl) = gk(tN). has application in gait analyses. while para-

metric or cyclic splines. being smooth curves that pass through multi-

dimensional points [x.ry.), i = I. 2•.... N], would allow the two-dimen-sional spatial coordinates of an anatomical landmark to be processed in a

uniform way that preserves the essential relationship between these vari-

ables. This approach could be further extended to the third dimension. and





AID

LPHR 40

\RR O

oI51-405\

L20

-t o

~"

OPTIMAL REG. FOURIER FILTER (OPTION=2) (8)

-120+------,--------~----r----r------._----~----,_----_,0.0' 0.4

1.0

0.1

GA

~ 0.6

D.4

0.2

0.1 l.t 1.& 2.0

T I M E \ S E C\t.1I

OPT [MAL REG. FOUR I ER F I L TER (OPT I ON=2) (C )

o.o~----~~~~----,-----r_--~----,_--------r_----_,o 12 18 20 %4

FREQUENCY I H Z I' 2II



356 G.A. Wood

I : t : :

0 A

p , : ;

p , : ;

w

w

~

<p , : ;

w

:>

<

~\ Cut-of f point chosen on b asis. of random error level

_ _ _ ' - A11-tJ--I1-I:r--~

S MO O T H I N G P AR A M ET E R

(H armonic order; P olynomial order; Cut-of f f requency;

L east-S quares co nstraint, etc.)

Figure 15. Schematic representation of the principle of optimal fit based on

residual analysis. Smoothing parameter is sequentially changed to

produce more exact fit until average error approaches level of random

noise (plateau effect),

to the imposition of constraints on the distance separating coordinate pairs

that are anatomical neighbors. .

Given the indirect analytical methods ernployed by biomechanists in the

quantification of forces and movements, the problem of data smoothing

and differentiation is likely to remain. Some years ago. however. Chao and

Rim (18) outlined an approach that permitted the determination of joint

moments in lower extremities during the pre-stance phase of gait without

recourse to double differentiation of displacement-time data. This method.

based on the mathematical theory of optimization. proceeded from

assumed joint moments to the prediction of displacement-time histories by




integration. The predicted movements were then compared with actual

displacement-time data. and the calculations were repeated with slight

changes to the input parameters (jpint torques) until a close agreement was

obtained. While there are soJ1e difficulties in generalizing this approach to

systems with more degrees of freedom. the work does serve to remind usthat alternative approaches must also be explored.

During the last decade, however. considerable progress has been made

toward improving the accuracy of velocity and acceleration data. and

several very acceptable numerical procedures have been developed. It seems

that further refinement will stem from more detailed information on the

statistical properties and frequency spectrum of biomechanical data, and

from the development of more accurate data - acquisition systems.

Essentially, though. any procedure that can be shown to produce valid re-sults within the context of the motion being analyzed is acceptable. It is

hoped that this review has provided a basis for sound decision-making in

this regard.

REFERENCES

l.Alexander. R.McN .. and A.S. Jayes. Fourier analysis of forces exerted in walk-

ing and running. J. Biomech, 13: 383-390. 1980.

2. Allum. J.H.J. A least mean squares CUBIC algorithm for on-line differential

of sampled analog signals. IEEE Trans. Biomed. Comput . c-24(b): 585-590,

1975.

3. Anderssen, R.S .. and P. Bloomfield. A time series approach to numerical dif-

ferentiation. Technometrics+St+): 69-75.1974.

4. Anderssen, R.S .. and P. Bloomfield. Numerical differentiation procedures for

non-exact data. Numer. Math. 22: 157-182, 1974.

5. Andrews, B.. and D. Jones. A note on the differentiation of human kinematic

data. In: Digest of the' , th International Conference on Medical and Biological

Engineering. Ottawa. 1976, pp. 88·-89.

6. Ayoub. M.A., M.M. Ayoub. and J.D. Ramsey. A stereometric system for mea-

suring human motion. Hum. Factors 12: 523-535. 1970.

7. Bar. A., G. Eden. G. Ishai, and R. Seliktar. A method for numerical different-

iation of kinematic data. In: Biomechanics of Sport Games and Sports Activi-

ties, edited by A. Ayalon, Natanya, Israel: Wingate Institute for Physical Edu-

cation and Sport. 1979, pp. 39-45.

8. Bernstein, N. The Co-ordination and Regulation of Movements. Oxford:

Pergamon Press. 1967.

9. Bresler. B.. and J.P. Frankel. The forces and moments in the leg during level

walking. Trans. Am. Soc. Mechanical Eng. 72: 27-36. 1950.

10. Burkett, L. Development of a computer data smoothing programme for cine-

matographic research in three dimensions. N.Z ..I. Health. Phys. Educ. Recrea-

tion6(4): 15-18. 1974.

II. Callum. J. Numerical differentiation and regularization. Siam J. Numer.

Anal. 8: 254-265.1971.

12.Cappozzo. A., T. Leo. and A. Pedotti. A generalized computing method for

the analysis of human locomotion. J. Biomech. 8: 307- 320. 1975.



358 G.A. U/ood

13. Cavanagh. P.R. Recent advances in instrumentation and methodology of bio-

mechanical studies. In: Biomechanics V-B. edited by P.V. Komi. Baltimore:

University Park Press, 1976, pp. 399-411.

14.Cavanagh, P. R. Instrumentation and methodology in biomechanics studies of

sports games. In: Biomechanics of Sports and Kinanthropometry, Miami. FI.:Symposium Specialists, lnc.. 1978, pp. 19-43.

15. Cavanagh, P.R. Instrumentation and methodology of applied and pure

research in biomechanics. In: Biomechanics of SpONS Games and Sports Actil'-

ities. Natanya, Israel: Wingate Institute for Physical Education and Sport,

1979, pp. 39-45.

16. Cavanagh, P.R .. and R . .I. Gregor. Knee joint torque during the swing phase of

normal treadmill walking. 1. Biomech, 8: 337-344, 1975.

17. Cavanagh. P.R .. and .I. Landa. A biomechanical analysis of the karate chop.

Res. Q.47: 610-618,1976.18. Chao, E. Y .. and K. Rim. Application of optimization principles in determining

the applied moments in human leg joints during gait. J, Biomech .« : 497-510.

1973.

19.Clenshaw, C.W. Curve fitting with a digital computer. Comput . 1.2: 170-173.

1960.

20. Davis. H.T. Tables of the Higher Mathematical Functions. Vol. 1.

Bloomington, IN: Principia Press, 1933.

21. Dillman, C ..I . A kinetic analysis of the recovery leg during sprint running. In:

Biomechanics Proceedings of C.I.C. Conference. Bloomington, IN: IndianaUniversity Press, 1971, pp. 137-165.

22. Dunfield. L.G., and .I.f. Read. Determination of reaction rates by the use of

cubic spline interpolation . .I. Chem . Phys. 47(5): 2178-2183, 1972.

23. Felkel. E.O. Determination of acceleration from displacement-time data.

Prosthetic Devices Research Project, Series II. Berkeley: University of

California, Issue 16, 1951.

24. foley, C.D., A.O. Quanbury, and T. Steinke. Kinematics of normal child loco-

motion -a statistical study based on T.V. data. 1. Biomech.12: 1-6, 1979.

25. Forsythe, G.E. Generation and use of orthogonal polynomials for data fit lingwith a digital computer. 1. Soc. lndust . Appl. Math. 5: 74-88, 1957.

26. Gagnon, M .. D. Rodrique. Assessment of smoothing and differentiation meth-

ods for the determination of acceleration from film data. Can . .I. App/. Sports

Sci. 3(4): 223-228. 1978.

27. Garrett. G .. W.S. Reed. C. Widule. and R.E. Garrett. Human motion:

Simulation and visualisation. Biomechanics !I. Medicine and Sport Vol. 6. Uni-

versity Park Press, Baltimore. 1971. pp. 299-303.

28. Gold, B.. and C.M. Rader. Digital Processing of Signals. New York: McGraw-

Hill. 1969.29. Gregor. R ..I.. and D. Kirkendall. Performance efficiency of world class female

marathon runners. In: Biomechanics VI-B. edited by E. Asmussen and K.

Jorgensen. Baltimore: University Park Press, 1978. pp. 40-45.

30. Greville, T.N.E. Introduction (0 spline ....uncaons. In: Theory and Application

of Spline Functions, edited by T.N.E. Greville. New York: Academic Press.

1969. pp. 1-35.

31. Grieve. D. W .. D. Miller. D. Mitchelson . .I. Paul. and A . .I. Smith. Techniques

for the Analysis of Human Movement. London: Lepus. 1975.

32. Gustafsson. L., and H. Lanshammar. ENOCH - An Integrated System forMeasurement and Analysis of Human Gait. Uppsala. Sweden: UPTEC, 1977.

33. Harris. F.J. On the use of windows for harmonic analysis with the discrete

Fourier transform. Proc. IEEE. 66: 51-53. 1978.




34. Hatzc, H. The use of optimally regularised Fourier series for estimating higher-

order derivatives of noisy biomechanical data. J. Biomech . 14: 13-18. 19R I.

35. Hayes, K.C.. K.L. Robinson. G.A. Wood. and L.S . .Ienning. Assessment of

the H-refle.x excitability'curve" using a cubic spline. Electroencephatogr. Clin.

Neurophysiol.46: 114-117.1979.36. Hayes. J.G. Numerical Approximation (0 Functions and Data. London:

Athlove Press. 1970.

37. Hershey. H.C., J.L. Zak in, and R. Shimha. Numerical differentiation of

equally spaced and not equally spaced experimental data. Ind. Eng. Chern.

Fundam.6(3): 413-421.1971.

38. Hill. A. V. The efficiency of mechanical power development during muscular

shortening. Proc. R. Soc. Lond/Biol.} 159: 139-324. 1964.

39. Hyzer, W.C. How accurate are photographic measurements? Res. Dev,

August: 43-44, 1971.40. Jackson, K.M. Fitting of mathematical functions to hiomechanical data. IEFE

Trans. Biomed. Eng. BME-26(2): 122-124,1979.

41. Kaiser. J.F. Digital filters. In: Digital Filters and the Fast Fourier Transform,

edited by B. Liu , Stroudsburg: Dowden. Hutchinson. and Ross. 1975. pp. 5-

79.

42. Kearney. R.E. Evaluation of the Wiener filter applied to evoked EMG poten-

tials Etectroencephalogr Clin. Neurophysiol.46: 475-478.1979.

43. Koniar, M. The biornechanical studies on the superimposition of angular

speeds in joints of lower extrerneties of sportsmen. In: Biomechanics III. editedby S. Cerguiglini. A. Venerando, and J. Wartenweiler. Basel: Karger. 1973. pp.

426-428.

44. Kroll. W. Logically deduced or statistically defined components in muscular

fatigue curves. Res. Q. 36(1): 113- 116, 1965.

45. Kuo. S.S. Numerical Methods and Computers. Reading. M!\: Addison-

Wesley. 1965.

46. Laananen. H., and C.M. Brooks. Determination of critical parameters for

spiked track shoe design through analysis of sprinter motion. In: Biomechanics

VI-A. edited by E. Asmussen and K. Jorgensen. Baltimore: University ParkPress. 1978. pp. 310-316.

47. Lamb. H .F .. and P. Stotharl. A comparison of cinematographic and force

platform techniques for determining take-off velocity in the vertical jump. In :

Biomechanics VI-A. edited by E. Asmussen and K. Jorgensen. Baltimore: Uni-

versity Park Press. 1978. pp. 387-391.

48. Lanclos. C. Applied Analysis. London: Pitman. 1957.

49. Lees. A. An optimised film analysis method based on finite difference tech-

niques . .I. Hum. Mov. Studies 6: 165-180. 1980.

50 . Lenz . .J .E .. and E. Kelly. A potential difficulty in A/D conversion using micro-computer systems. IEEE Trans. Biomed. Eng. 27: 668-669, 1980.

51.Lesh. M.D .. J.M. Mansour. and S.R. Simon. A gait analysis subsystem for

smoothing and differentiation of human motion data. J. Biomech. Eng. 10l:

204-212. 1979.

52. MacMillan. M.B. Determinants of the flight of the kicked football. Res. Q.4h:

48-57.1975.

53. McClellan . .I.H., and T.W. Parks. A unified approach to the design of opti-

mum FIR linear phase digital filters. IEt:E Trans Circuit Theory20(h): 697-

701. 1973.54. McClellan . .I.H .. T.W. Parks. and L.R. Rabiner. A computer program for

designing optimum FIR linear phase digital filters. IEEI:' Trans. A udio Electro-

acoustics 21(6): 506-52h. 11)73.



360 G.A. H/uod

55. McClements, J.D .. and W.H. Laverty. A mathematical model of speed skating

performance improvement for goal setting and program evaluation. Can . J,

Sport. Sci. 4(2): 116-122. 1979.

56. McLaughlin. T.M., C.l. Dillman, and T.J. Lardner. Biornechanical analysis

with cubic spline functions. Res. Q. 48: 569-582, 1977.57. McLeod , P.C., D.n. Kettelkamp, V. Srinivasan, and O.L. Henderson,

Measurements of repetitive activities of the knee. 1. Biomech.B: 369-373,

1975.

58. Mann, R., M. Adrian, and H. Sorensen. Computer techniques to investigate

complex sports skills: Application to conventional and flip long jump tech-

niques: In: Biomechanics V-B, edited by P. V. Komi. Baltimore: University

Park Press, 1976. pp. 161-166.

59. Martin. 1.. and .I.H. Meier. Improving analysis of graphical records. Machine

Design March: pp. 115-1 ]9.1949.60. Miller, 0.1., and 0.1. East. Kinematic and kinetic correlations of vertical

jumping in women. In: Biomechanics V-B, edited by P. V. Komi. Baltimore:

University Park Press, 1976. pp. 65-72.

61. Miller. D.I.. and R.C. Nelson. Biomechanics 0/ Sport. Philadelphia: Lea and

Febiger.1973.

62. Morris. 1.R. W. Accelerometry-a technique for the measurement of human

body movements. J. Biomech. 6: 729-736, 1973.

63. Morrison, 1.B. Bioengineering analysis of force actions transmitted by the knee

joint. Biomed. Eng. 3: 164-170, 1968.64. Morrison. 1.B. The function of the knee joint in various activities. Biomed.

Eng. 4: 573-580. 1969.

65. Morrison, 1.B. The mechanics of the knee joint in relation to normal walking.

J. Biomech.3: 51-61,1970.

66. Noble. M.L.. and D.L. Kelly. Accuracy of tri-axial cinematographic analysis in

determining parameters of curvilinear motion. Res. Q. 40(3): 643-645, 1969.

67. Nordeen, K.S .. and P.R. Cavanagh. Simulation of lower limb kinematics dur-

ing cycling. In: Biomechanics V-B, edited by P. V. Komi. Baltimore: University

Park Press, 1976;, pp. 26-33.68. Paul, 1.P. Forces transmitted by joints in the human body. Proc. Inst . Med,

Eng. lSI: 8-15,1967.69. Pearson, 1.R .. D.R. McGinley, and L.M. Butzel. A dynamic analysis of the

upper extremity: Planar motion. Hum. Factors 5: 59-70. 1963.

70. Pezzack, J .C .. R. W. Norman, and D.A. Winter. An assessment of derivative

determining techniques used for motion analysis. Digest 0/ the f f th Int. Con/.on Med. and BioI. Eng. Ottawa. 1976. pp. 86-87.

71. Pezzack , 1.C.• R.W. Norman, and D.A. Winter. An assessment of derivative

determining techniques used for motion analysis. 1. Biomech . 10: 377-382.1977.

72. Plagenhocf , S.c. Pal/ems 0/ HUn/an Motion: A Cinematographic Approach.

Englewood Cliffs. N.l.: Prentice-Hall. 1971 ..

73. Preece. M.A. Analysis of human growth curves. Post grad. Med, J.54(Suppl.

I): 77-89, 1978.'

74. Reinsch. C.H. Smoothing by spline functions. Numer. Math. 10:177-183.1967.

75. Reinsch. C.H. Smoothing by spline functions 11. Nunter. Math. 16: 451-454.

1971.76. Rice, J.R. The Approximation 0/ Functions II. Reading. MA: Addison-

Wesley, 1969.

77. Roberts, E.M., R.r. Zernicke, Y.Youm. and T.C. Huang. Kinetic parameters




of kicking. In: Biomechanics IV. edited by R.C. Nelson and C.A. Morehouse.

Baltimore: University Park Press. 1974. pp. 159-162.

78. Severy. D .. and P. Barbour. Acceleration accuracy: analyses of highspeed

camera film. J, Soc. M,oiion Picture Television Eng, 65: 96-99. 1956.

79. Singleton. R.C. An algorithm for computer the mixed radix fast Fourier trans-

form. IEEE Trans. Audio and Electroacoustics 17(2): 93-103. 11)69.

80. Smidt. G.L., J.S. Arora. and R.C. Johnston. Accelerographic analysis of

several types of walking. Am. J. Phys. Med. 50: 255-300. 1971.

81.Smidt. G.L., R.H. Deusinger. J. Arora. and .I.R. Albright. An automated

accelerometry system for gait analysis. J, Biomech. 10: 367-375. 1977.

82. Smith. A.J. A study of forces on the body in athletic activities with particular

reference to jumping. (Doctoral dissertation). University of Leeds. 1972.

83. Soudan. K .. and P. Dierckx. Calculation of derivatives and Fourier coefficients

of human motion data. while using spline functions . .I. Biomech. 12: 21-26.

1979.

84. Southworth. R.W .. and S.L. Deleeuw. Digital Computation and Numerical

Method. New York: McGraw-HilI. 1965.

85. Spath. H . Spline Algorithms for Curves and Surfaces. Winnipeg: Utilitas

Mathematic. 1974.

86. Sutherland. D.H .. and J.L. Hagy. Measurement of gait movements from

motion picture film. J. Bone Joint Int. Surg.54-A: 787-797.1972.

87. Van Gheluwe. B. Computerized three-dimensional cinematography for any

arbitrary camera setup. In: Biomechanics VI-A. edited by E. Asmussen and K.

Jorgensen. Baltimore: University Park Press. 1978. pp , 343-348.

88. Wahba. G .. and S. Wold. A completely automated French curve: Fitting spline

functions by cross validation. Communications in Statistics 4: 1-17. 1975.

89 . Wartenweiler. J. Status report on biomechanics. I n: Biomechanics III. edited

by S. Cerguiglini, A. Venderando. and J. Wartenweiler. Basel: Karger. 1973.

pp.65-72.

90. Wastell, D.G. The application of low-pass linear filters to evoked-potential

data: Filtering without phase distortion. Eicctroencephatogr. Clin. Neuro-

physiot. 46: 355-356. 1979.91. Wells. R.P. The kinematics and energy variations of swing through crutch gait.

i.Biomech. 12: 579-585.1979.

92. Widule. c.J., and D.C. Gossard. Data modelling techniques III cinemato-

graphic research. Res. Q. 41: I OJ-III. 1971.

93. Widule , C.J .. and D.C. Gossard. Data modelling techniques in cinernato-

graphic-biomechanical analysis. In: Biomechanics /II. edited by S.

Cerguiglini, A. Verderando , and J. Wartenweiler. Basel: Karger. 1973. pp.

108-115.

94. Wilcock. A.H .. and R.L.G. Kirsner. A digital filter for biological data. Med.Bioi. Eng. 7: 653-660. 1969.

95. Winter. D.A. Biomechanics 0/ Human Movement. New York: John Wiley and

Sons. 1979.

96. Winter. D.A.. R.K .. Greenlaw. and D.A. Hobson. Television-computer

analysis of kinematics of human gait. Comput . Biomed, Res. 5: 498-504. 1972.

97. Winter. D.A .. H.G.Sidwall, and D.A. Hobson. Measurement and reduction of

noise in kinematics of locomotion. J, Biomech. 7: 157-159. 1974.

98. Winter. D.A .. and R.P. Wells. Proper sampling and filtering frequencies in the

kinematics of human gait. In: Proceedings of 7th Canadian Medical andBiological Engineering Conference, Vancouver: 11)79. 137-138.

99. Wold. S. Spline functions in data analysis. Technometrics 16: I-II. 1<)74.

100. Woltring, H.J. Calibration and measurement in 3-dimensional monitoring of



362

human motion by optoelectronic means. II. Biotelemetry 3: 65-1.)7. I()76.

101. Wood. G.A .. and L.S. Jennings. An automatic spline for cinernarogruphic data

smoothing. In: Biomechanische Untcrsuchungs-Methoden im Sport. lnter-

national Symposium, Karl-M arx-Stadt. DDR . Sepr.. 1978. pro 2lJ3-2lJlJ.102. Wood. G.A .. and L.S. Jennings. On the use of spline functions for data

smoothing. J. Biomech . 12(6): 477-479. 1975.

103. Wylie. C.R. Advanced Engineering Mathematics. New York: McGraw-HilI.

1960.104. Yourn, Y., and Y.S. Yoon. Analytical development in investigations of wrist

kinematics. J. Biomech. 12: 613 -621. 1979.

105. Zarrugh. M. Y.. and C. W, Radcliffe. Computer generation of human gait kine-

matics. J. Biomech.12: 99-111,1979.

106. Zatziorsky. V.M .. and S. Y. Aleshinsky. Simulation of human locomotion. In:Biomechanics V-B. edited by P. V. Komi. Baltimore: University Park Press,

1976, pp. 387-394.

107. Zernicke, R.F .. G. Caldwell. and E.M. Roberts. Fitting biornechanical data

with cubic spline functions. Res. Q. 49: 569-582. 1977.

data smoothing & differentiation

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