Description of joint movements in human and non-human primate locomotion using Fourier analysis

ORIGINAL ARTICLE

Description of joint movements in human and non-human primatelocomotion using Fourier analysis

David Webb Æ William Anthony Sparrow

Received: 29 May 2006 / Accepted: 31 January 2007 / Published online: 22 May 2007

� Japan Monkey Centre and Springer 2007

Abstract To describe and help interpret joint movements

in various forms of primate locomotion, we explored the

use of Fourier analysis to represent changing joint angles as

a series of sine and cosine curves added together to

approximate the raw angular data. Results are presented

for four joints (shoulder, elbow, hip and knee) with

emphasis on the shoulder, and for five types of locomotion

(catarhine primate quadrupedal walking, human hands-

and-feet creeping and hands-and-knees creeping, and

human walking and running). Fourier analysis facilitates

functional interpretation of the angles of all four joints, by

providing average joint angles and an indication of the

number of peaks and troughs in the angular data. The

description of limb movements also afforded us the

opportunity to compare human and other catarhine joint

angles, and we interpret the Fourier results in terms of

locomotor posture and type. In addition, the shoulder data

are useful for determination of some aspects of interlimb

coordination. Non-human primates walking quadrupedally

and humans creeping on hands and knees generally evince

diagonal couplets interlimb coordination, in which the hand

on one side strikes the substrate at about the same time as

the contralateral foot or knee. Furthermore, human walking

and running seem to follow a similar pattern, as indicated

by Fourier analysis. From our data it is concluded that

human bipedal gaits are qualitatively similar to diagonal

couplets gaits in other primates, but quite different from the

lateral couplets gaits used by many non-primate mammals.

A number of other benefits of Fourier analysis in primate

locomotion studies are also discussed. These include the

ability to make statistical comparisons among various types

of limb movements in a wide variety of species, a simple

archival technique for limb movement data, and a greater

understanding of the variability of locomotor movements.

Keywords Joint angle � Interlimb coordination �Locomotion � Fourier analysis

Introduction

Fourier analysis

The purpose of this study was to explore the technique of

Fourier analysis to describe, in quantifiable terms, the joint

movement patterns of primates, including humans. A wide

variety of techniques exists for the quantitative description

of various aspects of gait. Some examples are: duty factor

(from footfall diagrams); speed (both absolute and relative

to body size); ground reaction force; cadence; various

Froude numbers; and foot angle (from footprints). In the

case of joint movements, many authors have used body

segment models and joint angle data (see, for example,

D’Aout et al. 2002; Isler 2005). Angular data analysis has

suffered from the fact that a series of joint angles is difficult

to reduce to a few simple numbers for statistical compar-

ison. It is easy enough to describe the angles through which

any joint passes, using a graph of angle versus time. Fur-

thermore, some standardization of the data can be achieved

by replacing time with a relative measure of time, such as

percent of stride cycle (see Figs. 1, 3, 5), and data can be

D. Webb (&)

Department of Anthropology and Sociology,

Kutztown University, Kutztown, PA 19530, USA

e-mail: [email protected]

W. A. Sparrow

School of Health Sciences,

Faculty of Health and Behavioural Sciences,

Deakin University, Burwood, Victoria 3125, Australia

123

Primates (2007) 48:277–292

DOI 10.1007/s10329-007-0043-4

further abstracted by dividing the stride cycle into stance

and support phases and plotting joint angle versus percent

of each phase (Fleagle et al. 1981; Hirasaki et al. 2000;

Isler 2005). However, the overall distribution of the data,

the ‘‘shape of the graph’’, cannot be quantified by these

means, leaving scientists to compare joint movement pat-

terns only in qualitative terms or only at specific points in

the stride.

To move beyond qualitative descriptions and compari-

sons, to quantify the overall distribution of gonial data, we

chose to investigate Fourier analysis. In 1822, Jean-Bap-

tiste-Joseph Fourier, a French mathematician and physicist,

published Theorie analytique de la chaleur (Analytical

Theory of Heat). Fourier’s book defines the series of sines

and cosines today known as Fourier series, and describes

their use in studying heat conductance (Bell 1937). While

there were difficulties with Fourier’s proof of his new

theorem, these were subsequently overcome by other

mathematicians and the use of Fourier analysis is wide-

spread today in such fields as electronics and the physics of

sound where transformations of the temporal distribution of

data are desirable (Richmond 1972).

With the methods of Fourier analysis, we hoped to be

able to describe and interpret joint movement patterns in

primates in quantitative terms and more thoroughly than

has been done previously. Quantitative descriptions of ‘‘the

shape of the graph’’ should allow the use of statistical

comparisons among individuals and among species, and

should lead to a greater understanding of the variability of

limb movements. In this way, Fourier analysis should help

elucidate several areas of joint analysis, including intra-

and inter-limb coordination and substrate accommodation.

We also hoped to develop a simple method of archiving

primate movement patterns, using many fewer descriptors

than would be necessary with traditional joint angle data-

points.

In order to describe joint angles and test for joint angle

similarity between humans and other primates, Fourier

analysis was used in this study to analyze the joint angle

data obtained from videotape records. Fourier analysis has

previously been used in physical anthropology to describe

the cross-sectional shapes of hominoid femora and maca-

que skulls and human skulls (Lestrel et al. 1977, 1993,

2005), to analyze sagittal movements of the head and trunk

of walking humans (Cappozzo 1981), and to describe the

extent and type of ‘‘meandering’’ while trying to walk in a

straight line (Uetake 1992). Fourier analysis allows the data

to be represented, not as measurements over time, but as

the sum of sine and cosine curves of various frequencies

and magnitudes. We can therefore approximate any cur-

vilinear data plot by means of a Fast Fourier Transform

(FFT). The analysis technique, as applied to the joint angle

data presented below, begins with a series of angles mea-

sured at regular intervals during the stride cycle. The FFT

program then calculates the magnitudes of a number of sine

and cosine curves which, when summed, approximate the

distribution of the data (Fig. 2). The magnitude of each

wave is expressed as a coefficient by which the height of

the wave is multiplied. These Fourier coefficients, properly

applied, will produce a curve of the correct shape. Then,

the average angle must be added to the resultant curve in

order to match the scale of the original data.

For the purposes of the present study, one advantage of

the technique is that FFTs allow comparison of the upper

limb movements in walking humans with the forelimbs of

quadrupedal primates. This is not possible using footfall

analysis such as revealed in ground contact diagrams

(Burnside 1927; Hildebrand 1967; Sparrow 1989), since

human upper limbs never contact the ground during normal

walking. However, Fourier analysis can be applied to bi-

pedal locomotion, if a standard for the beginning and end

of each stride is used. Thus, the most commonly applied

temporal standard, contact of the right hind limb/lower

limb, can be used in either bipedal or quadrupedal forms

of primate terrestrial locomotion. Fourier transforms also

describe each joint’s contribution to limb movement, rather

than just the results of those joints’ movements on ground

contact, and they allow the production of datasets of

average joint movements for many strides and many

individuals. In addition, quantitative ontogenetic data can

be analyzed, allowing greater elucidation of locomotor

development in various primate species (e.g., Lasko-

Macarthey et al. 1990; Kimura et al. 2005; Shapiro and

Raichlen 2006). Finally, and most importantly, FFTs

quantify angular data in a standardized form, allowing the

use of statistical procedures to compare joint movements

Fig. 1a, b Footfall diagrams illustrating coupling and sequence. LHLeft hind limb, LF left forelimb, RH right hind limb, RF right

forelimb. Thick lines indicate the time during which a given limb is in

contact with the substrate. a Diagonal couplets/diagonal sequence

gait. b Lateral couplets/lateral sequence gait

278 Primates (2007) 48:277–292

123

among different strides of the same individual, among

different individuals, and even among different species.

Fourier analysis is therefore an appropriate method of

assessing the similarities and differences among human

and non-human primate modes of locomotion.

Interlimb coordination

Both within and between species, the timing of the char-

acteristic regular movements of the limbs in walking can be

categorized and compared using two footfall parameters,

sequence and coupling. Sequence refers to the order in

which the forefeet follow the hindfeet in contacting the

ground. A lateral sequence pattern is when the right or left

forefoot strikes the ground immediately following the

hindfoot on the same side. Diagonal sequence gait occurs

when forefoot contact follows contralateral hindfoot con-

tact. Coupling refers to the footfalls of a forefoot and a hind

foot being related in time as a pair. Hildebrand (1967)

defined two types of coupling: lateral couplets and diagonal

couplets, as follows: ‘‘Lateral-couplets gaits have the

footfalls of fore and hindfeet on the same side of the body

related in time as a pair. Diagonal-couplets gaits have the

footfalls of fore and hindfeet on the opposite sides of the

body related in time as a pair’’ (Hildebrand 1967, p 119).

Sequence and coupling are illustrated in Fig. 1 using

footfall diagrams.

In a review of non-human primate gaits, it becomes

clear that the general pattern among great apes (Hildebrand

1967; Vilensky and Larson 1989), Old World monkeys

Fig. 2a–d Process of approximating angular data with a Fast Fourier

Transform (FFT). In each part of the figure, the original data are

represented by the open circles connected by the thickest line. As the

approximation progresses, the current approximation in each part of

the figure is represented by open squares connected by a medium line;

the previous approximation is shown with open triangles; and the

newly added curve is shown with open diamonds. a The original data,

plotted as joint angle over time, with the first approximation made

using a simple sine curve with one peak and one trough within the

cycle (0–350 on the Time scale). The height of the peak and the depth

of the trough are determined by the Fourier coefficient of the first sine

(5.13 in this case). b The approximation is refined by adding a simple

cosine curve (open diamonds), multiplied by the appropriate

coefficient (0.63). This cosine curve is added to the sine curve

in part a (now shown as open triangles), resulting in a new

approximation as indicated by the open squares. c In the interests of

saving time and space, two new curves are simultaneously added to

the previous approximation. Thus the current approximation is the

sum of the first sine and cosine curves of part b and the second sine

and cosine curves combined (open diamonds). Note that the combined

second sine and cosine curve has twice the frequency of the first order

curves, thereby producing a curve with two peaks and two troughs.

This permits the FFT to add finer details to the approximation. d The

final approximation, using fourth order sine and cosine curves,

produces a very good match to the original angular data. Subse-

quently, even higher order sine and cosine curves can be added to

match the original data very precisely. Note that this figure was made

using imaginary, simplified data that are not from primate locomo-

tion. Therefore, ‘‘time’’ and ‘‘angle’’ are not real units

Primates (2007) 48:277–292 279

123

(Hildebrand 1967; Prost 1969; Grand 1976; Mittermeier

and Fleagle 1976; Mittermeier 1978; Dunbar 1989; Hira-

saki et al. 1993), New World monkeys (Prost 1965; Hira-

saki et al. 1993; Vilensky and Patrick 1985; Rosenberger

and Stafford 1994; Schmitt 2003), and strepsirhines

(Vilensky and Larson 1989) is a diagonal couplets gait.

These diagonal couplets are usually combined with a

diagonal sequence, although there is some variability in

both coupling and gait. Therefore, the primitive condition

among primates is most likely a diagonal sequence/diago-

nal couplets gait that may be related to an arboreal habitat

and travelling on small branches (Cartmill et al. 2002;

Lemelin et al. 2003; Schmitt 2003). (But, see Dunbar and

Badam 2000; Shapiro and Raichlen 2006.) Furthermore,

the variability in gait patterns, especially within species,

has often been viewed as normal and adaptive to substrate

and other conditions, rather than as errors in neural pro-

gramming (Jungers and Anapol 1985; Shapiro et al. 1997;

Dunbar and Badam 2000). In contrast, variability among

species has often been explained as resulting from phylo-

genetic constraints on neurology or anatomical differences,

which may in turn be related to ecological adaptations

(Rosenberger and Stafford 1994; Shapiro et al. 1997;

Schmitt 2003).

As in the non-human species described above, human

walking and running show diagonal couplets coordination,

in the sense that the upper limbs move forward and back at

nearly the same time as the contralateral lower limbs.

However, this cannot be shown with footfall patterns such

as those used by Hildebrand and others, since modern hu-

man hands do not normally contact the ground. There is

evidence that we use diagonal couplets, in that contempo-

rary human upper limbs have been shown to move similarly

to the forelimbs of quadrupedal primates (Webb 1989) as

shown by an overlay of shoulder angles over a stride cycle

(Fig. 3). This, however, is a qualitative observation that

might be further elucidated by Fourier analysis.

To the extent that FFTs can separate different types of

coupling, we may be able to show that human bipedal gaits

are quantitatively similar to the gaits of other primates, in

terms of interlimb coordination. Fourier analysis may be

used to show the type of coupling employed by various

primates, if the stride cycle is always defined by the same

hind limb event (e.g., right hind limb strike). An FFT of the

shoulder, in particular, should help to identify interlimb

coordination, since the overall shape of the graph of

shoulder angle against stride cycle should be similar across

all animals that use similar interlimb coordination. This is

because the relative timing of shoulder movements, as

compared to right hind limb contact, is likely to be quite

similar among those animals, and the Fourier coefficients

should therefore have similar values. This notion is tested

below.

Methods

Subjects

All human subjects were healthy adults, as follows: hands-

and-feet creeping (n = 5 males, 20–27 years—from Spar-

row and Irizarry-Lopez 1987); hands-and-knees creeping

(n = 3 females 22–28 years, n = 7 males 19–29 years);

walking (n = 7 females 22–37 years, n = 12 males 19–

64 years); running (n = 5 females 20–32 years, n = 9

males 19–33 years). Other primates by genus were: Cer-

copithecus mitis (n = 2, Philadelphia Zoo), Gorilla (n = 1,

Cincinnati Zoo), Pan (n = 2, Columbus Zoo), Macaques

(n = 1 Macaca silenus, n = 4 M. mulatta, Cincinnati Zoo).

For comparative purposes, a leopard (Pantera pardus) at

Philadelphia Zoo was also included.

Apparatus and data analysis

For all trials of all subjects except hands-and-feet creeping,

a standard VHS videocamera was used to record movement

at 30 frames per second (fps; 30 Hz). Subjects in human

walking, running and creeping on hands and feet were on

treadmills, with a camera positioned to view them in norma

lateralis. For hands-and-feet creeping subjects, the camera

used was a film camera operating at 15 fps. Humans

creeping on hands and knees used a padded trackway, the

camera being positioned at right angles to the trackway.

The camera was placed as far from the subjects as possible

and only one stride (that in the middle of the field of view)

was used, in order to minimize distortion due to viewing

angle. Non-human subjects were videotaped at zoos, using

features of their enclosures (e.g., fences, pools of water) to

limit the direction of movement wherever possible. In this

way, the camera was kept perpendicular to their lines of

Fig. 3 Comparison of the shoulder angles of primates walking

quadrupedally (open circles) and humans walking bipedally (solidsquares). Greater angles indicate protraction. Both curves are the

results of averaging one stride each, for several individuals

280 Primates (2007) 48:277–292

123

progression. From the videos, it was determined that hands-

and-feet creeping subjects used lateral couplets gaits, while

hands-and-knees subjects and non-human primates used

diagonal couplets.

Using a computer with a video ‘‘frame-grabber’’

(ComputerEyes/RT, Digital Vision, Dedham, MA) and a

VHS video player, each frame of each stride was opened to

the computer screen and the x-y coordinates of the elbow,

shoulder, hip, knee, wrist and ankle were digitized using a

program (MouseSpot) developed in-house. Using additional

customized software the x-y coordinate data were smoothed

according to an algorithm wherein the coordinates for any

given frame were averaged with those of the two preceding

and two following frames, weighted as follows: 1-2-10-2-1.

The hands-and-feet creeping data were obtained by manual

digitizing of films taken at 15 fps as described by Sparrow

and Irizarry-Lopez (1987). Subsequently, the angles at the

hip, knee, shoulder and elbow were calculated for each

frame and written to a file for Fourier analysis. The methods

for measuring the angles are shown in Fig. 4. FFTs were

then performed by Mathcad� 3.1, and the resulting Fourier

coefficients analyzed using the statistical package, Stat-

view� 4.02 (www.statview.com).

The cosine and sine coefficients from the Fourier anal-

ysis are the magnitudes by which each cosine or sine curve

must be multiplied, so that the sum of all the curves

approximates the original distribution of angle versus

percent of stride cycle. The coefficients presented in the

Results are named according to the frequency of the sine or

cosine by which they are multiplied. Hence, the coefficient,

cos2, is multiplied by a cosine curve with two peaks and

two troughs during the stride cycle. This second order

cosine curve is therefore one with twice the frequency of a

normal cosine curve. The coefficient, sin3, is multiplied by

a sine curve of three times the normal frequency, and hence

three peaks and three troughs in one cycle. Higher order

coefficients, e.g., sin4 and cos5, describe more detailed

fluctuations in the original angular data, whereas lower

order coefficients (e.g., sin1, cos2) describe only the gen-

eral shape of the data. Fourier coefficients for the various

modes of locomotion were compared using Student’s t-test,

after it was confirmed that the data were normally dis-

tributed. Specifically, the resulting Fourier coefficients

were not skewed, being evenly distributed about the mean.

This was tested not only within groups of subjects, but also

among 20 strides of the same subject.

Results

Tables 1, 2, 3 and 4 present the coefficients of the FFTs for

each joint. The number of subjects in each group (n) is

followed by a0, the joint angle averaged over the entire

stride cycle for each individual and then for all individuals

in the group. From the data in Tables 1–4 an average curve

for the shoulder angles of the five groups was reconstructed

(Fig. 5). The shoulder angle is the best indicator of overall

forelimb movement, and therefore provides a qualitative

comparison of interlimb coordination among the five

groups. The angles in Fig. 5 were obtained by adding all

the sines and cosines indicated, and adding a0, the average

measured joint angle. For each point along the x-axis

(percentage of stride cycle) the reconstructed joint angle (y-

value) was obtained using the formula:

X½cosðX � 1Þ � cos 1þ cosðX � 2Þ � cos 2

þ cosðX � 3Þ � cos 3þ cosðX � 4Þ � cos 4

þ cosðX � 5Þ � cos 5þ sinðX � 1Þ � sin 1

þ sinðX � 2Þ � sin 2þ sinðX � 3Þ � sin 3

þ sinðX � 4Þ � sin 4þ sinðX � 5Þ � sin 5þ a0�

where ‘‘X’’ is the value on the x-axis, cos2 is the second

cosine coefficient (from Table 3), sin3 is the third sine

coefficient, etc. Thus, the second term in the summation,

cos(X*2)*cos2, would read: ‘‘the cosine of twice the x-

value, multiplied by the second cosine coefficient’’.

Each graph in Fig. 5 has only one peak and one trough,

indicating the relatively simple protraction and retraction

of the limb that occurs in each stride cycle. In human

walking and running, hands-and-feet creeping, and cata-

rhine primate walking, the peak that denotes maximum

protraction occurs about 40–50% after the trough denoting

maximum retraction. Maximum protraction and retraction

Fig. 4 Stick figures generated from videotaped quadrupedal and

bipedal subjects. Digitized joints: 1 wrist, 2 elbow, 3 shoulder, 4 hip,

5 knee, 6 ankle. Only the side of the body facing the camera was

digitized. Angles calculated were as shown: E elbow, S shoulder, Hhip and K knee. Note that, by the definition used here, the shoulder

angle of the bipedal figure would be negative, positive values being

ventral to the torso line, 34

Primates (2007) 48:277–292 281

123

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(8.9

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(2.1

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(2.7

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4.4

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97)

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(0.4

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0.8

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(1.3

24)

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(1.1

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61

(0.9

33)

0.0

85

(0.5

96)

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(0.1

68)

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anw

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(6.0

31)

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81)

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64

(0.9

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(0.4

28)

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69

(0.2

93)

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07)

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88)

–0.0

81

(0.1

84)

0.0

70

(0.2

25)

Hum

anru

nnin

g14

121.1

55

(23.9

93)

1.8

40

(40.8

02)

–1.3

87

(10.6

56)

0.0

87

(0.5

52)

–0.1

26

(0.3

29)

0.1

04

(0.1

70)

–2.1

52

(3.4

25)

2.0

93

(1.2

25)

–0.0

66

(0.4

85)

–0.2

87

(0.4

87)

0.1

57

(0.3

36)

aS

tan

dar

dd

evia

tio

ns

are

giv

enin

par

enth

eses

282 Primates (2007) 48:277–292

123

Ta

ble

3A

ver

age

Fo

uri

erco

effi

cien

tsfo

rth

eh

ipan

gle

so

fn

on

-hu

man

pri

mat

esan

dh

um

ans

Gro

up

na0

cos

1co

s2

cos

3co

s4

cos

5si

n1

sin

2si

n3

sin

4si

n5

Pri

mat

ew

alkin

g9

83.0

01

(8.0

51)a

–11.9

10

(2.5

98)

0.2

69

(1.6

67)

0.8

89

(0.7

05)

0.0

72

(0.4

93)

0.1

20

(0.3

37)

–2.3

35

(1.8

13)

3.4

00

(0.8

74)

0.2

05

(0.3

80)

–0.3

82

(0.4

98)

0.0

93

(0.3

99)

Hum

ancr

eepin

g

(han

ds-

and-k

nee

s)

10

103.5

52

(7.6

76)

–11.1

34

(1.9

98)

–0.3

06

(1.3

70)

–0.0

89

(0.6

04)

0.1

05

(0.2

30)

0.0

43

(0.1

95)

–0.4

04

(1.7

68)

3.0

66

(0.8

98)

0.1

75

(0.5

14)

0.0

84

(0.4

21)

0.0

08

(0.3

26)

Hum

ancr

eepin

g

(han

ds-

and-f

eet)

577.5

59

(11.2

72)

–13.7

90

(2.3

43)

–1.2

51

(1.3

47)

0.6

29

(0.9

24)

–0.2

13

(0.3

03)

–0.0

66

(0.4

36)

–5.4

11

(2.7

97)

1.8

58

(1.5

85)

0.9

35

(0.7

14)

0.4

06

(0.6

04)

0.2

48

(0.2

72)

Hum

anw

alkin

g19

170.7

56

(5.2

19)

–7.5

81

(1.5

02)

1.5

03

(0.7

09)

–0.3

98

(0.3

55)

–0.1

39

(0.3

07)

0.0

72

(0.2

65)

2.7

14

(1.2

03)

0.6

27

(0.9

40)

–0.3

53

(0.4

44)

–0.0

10

(0.2

67)

–0.2

48

(0.2

99)

Hum

anru

nnin

g14

164.6

73

(4.2

65)

–5.7

39

(1.3

63)

1.8

17

(0.7

90)

0.1

81

(0.5

17)

–0.0

02

(0.3

09)

–0.0

76

(0.1

44)

4.0

08

(1.1

74)

–1.1

83

(1.2

56)

–0.2

82

(0.3

98)

0.1

13

(0.1

76)

–0.1

94

(0.2

98)

aS

tan

dar

dd

evia

tio

ns

are

giv

enin

par

enth

eses

Ta

ble

4A

ver

age

Fo

uri

erco

effi

cien

tsfo

rth

ek

nee

ang

les

of

no

n-h

um

anp

rim

ates

and

hu

man

s

Gro

up

na0

cos

1co

s2

cos

3co

s4

cos

5si

n1

sin

2si

n3

sin

4si

n5

Pri

mat

ew

alkin

g10

136.6

13

(10.7

12)a

1.0

73

(3.6

12)

6.5

50

(1.7

97)

2.3

69

(1.4

09)

0.1

56

(0.8

87)

–0.0

03

(0.3

56)

5.9

93

(2.1

28)

1.2

26

(1.7

80)

–2.1

82

(1.0

98)

–1.2

27

(0.9

90)

–0.3

29

(0.5

35)

Hum

ancr

eepin

g

(han

ds-

and-k

nee

s)

10

82.8

66

(6.7

21)

–12.3

08

(2.6

13)

0.2

71

(1.0

32)

0.2

15

(0.8

56)

–0.1

12

(0.5

10)

0.3

15

(0.2

91)

0.5

37

(1.4

59)

3.3

09

(1.4

99)

–0.1

42

(0.6

53)

0.1

37

(0.4

52)

0.2

38

(0.2

25)

Hum

ancr

eepin

g

(han

ds-

and-f

eet)

5120.9

32

(5.1

87)

–8.0

60

(3.5

10)

2.8

18

(1.4

44)

1.9

21

(1.7

84)

0.2

07

(0.6

05)

0.1

71

(0.5

32)

–3.8

80

(3.0

85)

0.5

92

(1.9

22)

–0.2

96

(0.7

10)

–1.0

08

(1.0

38)

–0.1

35

(0.2

04)

Hum

anw

alkin

g19

152.9

88

(3.6

36)

2.3

63

(1.5

40)

6.6

48

(1.5

86)

–0.2

19

(0.7

06)

0.2

26

(0.3

08)

0.0

74

(0.3

55)

10.2

32

(1.5

61)

–4.5

26

(1.8

47)

–1.7

21

(0.4

81)

–0.5

24

(0.3

90)

–0.4

52

(0.3

33)

Hum

anru

nnin

g14

136.6

88

(3.4

43)

5.5

25

(2.2

72)

3.4

33

(2.2

75)

0.0

39

(0.5

79)

–0.2

42

(0.3

30)

–0.2

20

(0.2

27)

7.2

05

(1.5

52)

–7.4

05

(1.5

72)

–1.0

32

(0.4

38)

–0.2

64

(0.3

36)

–0.3

08

(0.3

05)

aS

tan

dar

dd

evia

tio

ns

are

giv

enin

par

enth

eses

Primates (2007) 48:277–292 283

123

are therefore fairly evenly spaced in time, and the forelimb/

upper limb oscillates smoothly. Human creeping on hands

and feet, in addition to being out of phase with the other

modes, differs slightly from this pattern, in that the

shoulder is held at maximum retraction for a moment be-

fore the upper limb swings forward. Human creeping on

hands and knees is quite different from the other modes of

locomotion, since maximum protraction follows maximum

retraction by only about 30%. Hence, we see a steep slope

between those two points, followed by a gentler slope after

maximum protraction is reached. This is an indication of

the relatively short swing phase and long support phase of

the upper limb.

In Tables 1, 2, 3 and 4, there is a general decrease in

magnitude (independent of sign) as we move to higher order

coefficients. Table 5 shows where significant (P £ 0.01)

differences in shoulder angle exist among the various

modes of locomotion. Among the 11 coefficients, average

joint angle and the first and second order coefficients are

frequently different, but the fourth and fifth are usually not;

the third order components are intermediate. With respect to

the other joints, the smallest number of significant differ-

ences is found among elbow angle comparisons, while the

greatest number is found among knee comparisons.

For the shoulder, there are a large number of significant

differences, especially in the average shoulder angle (a0),

the third cosine coefficient, and the first and second sine

coefficients. The fourth and fifth order components, both

sine and cosine, show either few or no differences among

the various types of locomotion. Average shoulder angles

(a0) are significantly different among nearly all forms of

locomotion, except between non-human primate walking

and hands-and-knees creeping, a reflection of the similar

torso positions in these gaits.

Since the values of the fourth and fifth order components

are small (Table 1), and their differences are generally not

significant (Table 5), the Fourier coefficients were reana-

lyzed to determine each coefficient’s contribution to the

Fig. 5a–e Average curves for

shoulder angle versus percent of

stride cycle for the five types of

locomotion studied. Each graph

begins with right hind limb

strike (RHS) at 0%, and greater

angles indicate protraction.

a Primate quadrupedal walking.

b Human creeping on hands

and knees. c Human creeping

on hands and feet. d Human

bipedal walking. e Human

bipedal running

284 Primates (2007) 48:277–292

123

overall estimate of the shoulder data. Table 6 shows the

contribution of each of the first, second and third order

coefficients, expressed as a percentage of the total estimate

of the data, for each mode of locomotion. In this analysis,

only the first five sine and cosine coefficients were con-

sidered. From the last row of Table 6, it is clear that almost

all (approximately 96%) of the information in each FFT is

accounted for by the first three sine and cosine coefficients.

These first, second and third order coefficients are therefore

sufficient to describe all but the fine details of the data, and

the higher order coefficients (i.e., fourth and fifth) are of

relatively little importance to our estimate. Hands-and-feet

creeping is somewhat unusual in this set, in having the

lowest cos1 coefficient (by a factor of ten in most cases),

and the highest sin1 (by at least 20%).

Similar results were obtained in comparisons among

Fourier coefficients of the elbow, hip and knee. For all

joints, the magnitudes of first and second order coefficients

were generally the greatest, the fourth and fifth were

smallest, and the third order coefficients were intermediate.

Accordingly, significant differences appeared most often

among first, second and third sine and cosine components.

Comparisons among knee movements were notable, in that

they show relatively more differences involving hands-and-

knees creeping than other joints and modes of locomotion.

Hands-and-knees creeping is the only mode in which the

knee is placed on the substrate, nearly eliminating the leg,

ankle and foot contributions to locomotion and changing

the function of the knee joint itself. For this reason, it is not

surprising that knee coefficients for hands-and-knees

creeping differ greatly from the others.

In Tables 1, 2, 3 and 4, standard deviations of Fourier

coefficients follow a pattern similar to that seen among the

absolute values of the coefficients; namely, standard devia-

tions decrease as the order of the coefficient increases. Thus,

higher order coefficients have smaller standard deviations.

Discussion

Average joint angles

Differences in the average angles of the four joints (a0) can

be interpreted with reference to body proportions and trunk

positions in various types of locomotion (Tables 1–4;

Fig. 4). During normal walking, the limbs of quadrupedal

primates oscillate in the sagittal plane under a nearly hor-

izontal torso and both hip and shoulder angles will average

approximately 90�. Thus, the quadrupedal forms of loco-

motion show average shoulder angles of 70�–100� and hip

angles of 75–105� (Tables 1, 3). However, increased or-

thogrady will tend to decrease shoulder angle and increase

hip angle. The extreme case is seen among bipedal humans

(Fig. 4), with slightly negative shoulder angles and hip

angles near 170�. Human hands-and-feet creeping, wherein

the shoulder is lower than the hip, is unusual in that it

evinces shoulder angles higher than 90� and hip angles

lower than 90�.

The fully extended elbow or knee has a maximum angle

of approximately 180�, and the average joint angle must

therefore be less than 180� (Tables 2, 4). Hence, average

elbow angles are generally between 155� and 165�, the

only exception being found in human running when the

elbow is usually held in a flexed position. Average knee

angles are generally between 120� and 155�, except for

human hands-and-knees creeping, which is somewhat be-

low 90� since the knee is flexed 90� to allow it to be placed

on the substrate.

Joint movement and Fourier coefficients

A major focus of this study is the shoulder, since Fourier

analysis of shoulder movements can elucidate similarities

in interlimb coordination when footfall diagrams are

irrelevant, as they are when comparing bipeds with quad-

rupeds. As noted above, this is possible when the stride

cycle is defined by hind limb actions such as the time when

the right hind limb strikes the substrate (right hind limb

strike; RHS). If all the datasets begin with the same hind

limb action, similar forelimb joint movements will produce

similar Fourier analysis results.

From Table 6, we see that the greatest percent contri-

bution of any component to the overall estimate of the

shoulder data ranges from 35%, for both first sine and first

cosine in primate quadrupedal walking, to 79% for first

cosine in human walking. Furthermore, in combination, the

sums of the first sine and cosine components range from

64 to 88% of the overall estimate, with higher order

Table 5 Significant differences (P £ 0.01) among shoulder angle coefficients

Hands-and-feet Human walking Human running Primate walking

Hands-and-knees a0, cos1, cos3, sin1, sin2 a0, cos3, sin1, sin2, sin4 a0, cos3, cos5, sin1, sin2 cos3, sin1, sin2

Hands-and-feet a0, cos1, cos3, sin1, sin2 a0, cos1, cos3, sin2 a0, cos1, cos2, sin1, sin4

Human walking a0, cos2, sin1, sin2 a0, cos2, cos3, sin1, sin2, sin4

Human running a0, cos2, cos3, sin1

Primates (2007) 48:277–292 285

123

coefficients adding relatively little to the estimate. These

results highlight the fact that there is only one peak and one

trough in the graphed data of shoulder angle versus percent

stride (Fig. 3), corresponding to the relatively simple

oscillation of the forelimb, as the limb is protracted during

swing phase and retracted during stance phase. Further-

more, the functional differences between human upper

limbs and non-human forelimbs are seen in the combined

contributions of the first sine and cosine coefficients in

Table 6. Specifically, the two modes of human bipedalism

evince the highest contributions of first sine and cosine

components (87–88%), well above those of the other

modes (64–70%). This tells us that the upper limbs in

human bipedalism oscillate fairly smoothly, with a single

peak and trough evenly spaced in time. In contrast, the

somewhat lower combined first sine and cosine contribu-

tions of the quadrupedal modes indicate a skewed distri-

bution of the data, in which the peak and trough are not

evenly spaced. The functional significance of this dis-

crepancy lies in the fact that, in bipedalism, the upper limbs

are not weight-bearing and hence are free to swing much

like pendulums. However, the support phase of the fore-

limb in quadrupeds (including creeping humans) is longer

than the swing phase, since the torso must vault over the

supporting limb. The swing phase is then relatively short,

as the limb is swung forward quickly to prevent the torso

from pitching forward during progression. While compar-

isons among forelimb duty factors could have shown the

disparity between stance and swing phase durations, com-

parison of Fourier coefficients is necessary when bipeds are

included in the analysis, since there is no duty factor for

limbs that do not contact the ground.

Higher order coefficients (second sine, third cosine, etc.)

are better suited to describing data with two or three or

more peaks and troughs and, for shoulder and hip angle,

will generally be most useful only in refining the overall

shape of the estimate. Tables 2, 3 and 4 support this con-

clusion. Table 3 for the hip shows clearly that the most

prominent coefficients are first cosine and first sine com-

ponents, results that agree well with those of Grasso et al.

(2000),who used Fourier analysis to study the hip, knee and

ankle in human walking. Table 2 (elbow) and Table 4

(knee), however, show very strong contributions from the

second sine and cosine components, reflecting the fact that

the elbow and knee undergo two flexion and extension

phases in each stride cycle. The first flexion phase of the

elbow or knee occurs when the limb flexes as the body

vaults over it, while the second flexion phase helps the

cheiridium clear the ground during the forward swing.

The third, fourth and fifth sine and cosine components

are quite small in comparison to the first three, as indicated

in Tables 1–6. In Table 6, for the shoulder only, the first

three sine and cosine components together account for at

least 95% of the shape of the shoulder graph, leaving only

5% or less to be covered by the fourth and fifth order

components. Even the third sine and cosine components

account for relatively little (from less than 0.1% to about

8%). In Tables 1–4, clearly the largest coefficients are

those for the first and second components, not only for the

shoulder, but also for the other joints studied. This is be-

cause none of the limb segments in any mode of locomo-

tion oscillates back and forth four or five times and, again,

these higher order components merely refine the overall

patterns of joint movements. Also, since the fourth and fifth

coefficients account not for the general shapes of the curves

but for the details in the data, they are more likely to show

individual variation. Hence, in Table 5 the variance of

fourth and fifth order components for the shoulder is such

that there are often no significant differences among the

five modes of locomotion. This is evidence that there are

no hidden, higher frequency oscillations in the function

of the limbs. Therefore, the use of high-speed film (e.g.,

120 fps) will generally be unnecessary to understand the

functions of these joints in medium- and large-bodied

primates. Perhaps in studying small primates researchers

will need to use high-speed data collection techniques for

the major limb joints, since their movements will generally

be faster. However, Fourier analysis shows that, in most

cases, the locomotor movements of larger primates can be

studied with standard (30 fps) video.

Table 6 Percent contribution

of the first three sine and cosine

coefficients to the overall

estimate of the shoulder data

Coefficient Primate

walking

Hands-and-knees

creeping

Hands-and-feet

creeping

Human

walking

Human

running

cos1 35.098 53.033 4.450 78.807 49.052

cos2 11.636 1.164 10.917 2.360 5.998

cos3 4.052 5.416 8.365 0.069 0.179

sin1 35.434 11.934 59.296 8.246 39.264

sin2 7.704 23.500 10.430 5.876 2.392

sin3 2.560 1.603 2.150 1.498 0.907

Total % 96.484 96.650 95.608 96.865 97.792

286 Primates (2007) 48:277–292

123

Variance and variability

Standard deviations of the Fourier coefficients can tell us

about the experimental procedures used and the composi-

tions of the groups studied. Hence, different study groups

can show mardedly different standard deviations. From

Tables 1–4, it is clear that in human walking and running

there is somewhat less variation in higher order coefficients

than there is among the other types of locomotion. In

Tables 1 and 3, for the shoulder and hip, respectively, the

standard deviations of first order coefficients (cos1 and

sin1) in human walking and running are similar to those

found among other types of locomotion. However, second

order coefficients are generally much lower for both forms

of human bipedalism than for any of the other modes of

locomotion. This trend continues as we look at progres-

sively higher order coefficients. Figure 6 shows the rela-

tively low standard deviations of shoulder coefficients for

human walking and running, as compared to the other

modes of locomotion studied. Elbow, hip and knee stan-

dard deviations evince a similar pattern, such that the

second and higher order coefficients for human walking

and running are generally smaller than those for primate

walking and human quadrupedal movement.

Lower standard deviations for human bipedalism tell us

about the conditions of the experiment. Since both types of

human bipedal locomotion (walking and running) were

performed on a treadmill, speed and direction were held

nearly constant. Therefore, minor fluctuations in speed

and direction that might normally occur, due to slight

unevenness of the substrate or minor asymmetries in

muscle contractions, are reduced by the experimental

technique. The non-human primates walked on level, but

sometimes slightly irregular, substrates in zoos, adding

variability to their limb movements. Furthermore, although

humans creeping on hands and knees moved across a

smooth, padded floor, small variations in muscle use, from

one stride to the next or between the left and right sides of

the body, would also cause increased variation in joint

movements. The treadmill used by bipedal subjects con-

strains such movements. Indeed, similar results have been

observed in other vertebrates. For example, Webb (1993)

found that fish swimming in artificial flumes (the fish

equivalent of a treadmill) showed less variance in their

movements than when they swam in perfectly still ponds.

The steadily moving water, even with its turbulence, con-

strained the movements of the fish. Fourier analysis thus

demonstrates the effect of constraining (laboratory) con-

ditions on movement variability and serves as a caveat to

those comparing results from laboratory settings with re-

sults from more naturalistic ones.

One might reasonably ask then, why the hands-and-feet

creeping subjects did not show standard deviations as low as

those for bipedal subjects, since they too were on a treadmill.

The answer certainly lies in the fact that creeping on hands

and feet is an unusual mode of locomotion for an adult hu-

man. Indeed, the original study from which those data were

drawn was intended to investigate the effects of learning a

novel task (Sparrow and Irizarry-Lopez 1987). With con-

siderable practice, one would expect subjects creeping on

hands and feet to show less variation in their movements and,

therefore, less variance in their Fourier coefficients.

It is unlikely that the different film speed (15 vs 30 fps)

is a significant factor in the hands-and-feet creeping study,

because the Fourier transform is useful for a variety of

sampling rates. The limiting factor is the total number

of datapoints (frames in this case), expressed in powers of

two. Thus, two datapoints limit research to the first sine and

cosine components, four datapoints limit us to the first- and

second-order components, eight points limit us to third-

order or less, and so forth. Fifteen and 30 fps for a duration

of stride of a little more than 1 s produce more than 16

datapoints, limiting us to the fourth-order components

(16 = 24), or the fifth order when stride period is well over

1 s (‡32 frames). Most of the strides in this study provided

more than 32 frames of data but fewer than 64 frames, so

we do not report any Fourier coefficients beyond the fifth

order. Specifically, those taken from Sparrow and Irizarry-

Lopez’s (1987) hands-and-feet creeping data generally

covered about 40 frames. Also, the vast majority of the

information about joint movement is covered by the first

three components (as discussed above). Hence, Fourier

analysis of primate locomotion is fairly tolerant of differ-

ences in camera speed.

Fig. 6 Standard deviations of the first five coefficients (both cosine

and sine). The two thick lines represent human bipedal locomotion

(triangles indicate running, diamonds indicate walking). Open circlesrepresent hands-and-feet creeping; open squares represent hands-and-

knees creeping; crosses represent primate walking. Standard devia-

tions of second through fourth order coefficients are markedly lower

for human bipedalism than for the other modes of locomotion

Primates (2007) 48:277–292 287

123

Fourier coefficients and mode of locomotion

As indicated in Table 6, the first and second sine and co-

sine coefficients account for most of the descriptive data in

the FFTs of shoulder angle. Hence these components were

further analyzed in Fig. 7. The box plots in Fig. 7 also

show that the first cosine component (Fig. 7a) can clearly

distinguish between lateral couplets gaits (e.g., hands-and-

feet creeping) and diagonal couplets gaits, but the other

coefficients generally did not show such clear separations.

Hence, although various modes of locomotion can be dis-

tinguished on statistical grounds (Table 5), there are few

clear discontinuities and considerable overlap among the

other coefficients, when considered individually. Thus,

human walking, running, and hands-and-knees creeping

appear in many respects to lie on a continuum with primate

quadrupedalism, a continuum that shows behavioral and

neurological similarity among diagonal couplets gaits.

These modes are, however, quite distinct from human

hands-and-feet creeping, which more closely resembles the

quadrupedalism of some other mammals. An example of a

non-primate mammal with a lateral couplets gait is the

leopard (Panthera pardus), indicated by P in Fig. 7. For all

four coefficients (first and second sine and cosine), the

leopard is indistinguishable from the lateral-couplets,

hands-and-feet creeping of humans, most notably for the

first cosine coefficient, which clearly separates the two

types of interlimb coordination. Hence, despite the obvious

mechanical and morphological differences, human biped-

alism and creeping on hands and knees resemble primate

quadrupedalism more than they do the lateral couplets gaits

of other mammals and human hands-and-feet creeping.

Furthermore, the informal observation that human walking

seems to evince diagonal-couplets interlimb coordination is

validated in Fig. 7, without the need for support diagrams.

Bivariate plots and ‘‘Kinesiological Space’’

Although it is impossible to separate all modes of loco-

motion on the basis of a single Fourier coefficient, pairs

of coefficients can distinguish among them fairly clearly.

The major components of an FFT can be plotted against

one another, thereby potentially separating different groups

along two axes; Fig. 8 shows bivariate graphs of pairs of

coefficients. Where different modes of locomotion are

grouped separately on these graphs, a kind of ‘‘kinesio-

logical space’’ is defined. Thus, kinesiological spaces are

defined by consistent association of individuals, species or

types in particular areas of the graph. In this case, FFT

coefficients describe the movement characteristics of a

particular joint in a particular mode of locomotion.

In Fig. 8a, the separation of diagonal couplets from

lateral couplets gaits is made even more obvious than in

Fig. 7a by the combination of the first sine component and

the first cosine component. But Fig. 8a also shows a clear

separation of quadrupedal walking (open diamonds) from

bipedal running (crosses), whereas there is still much over-

lap among human walking, running and hands-and-knees

creeping. Second and third cosine coefficients, by their

nature, can help to refine these distinctions (Fig. 8b). In this

Fig. 7a–c Box plots showing the distribution of first and second

cosine and first sine coefficients for human hands-and-feet creeping

(white), hands-and-knees creeping (diagonal lines), bipedal walking

(vertical lines), primate walking (stipling), and bipedal running (solidblack). In each graph, Pantera pardus is indicated by the letter

P. a cos1, b cos2, c sin1

288 Primates (2007) 48:277–292

123

case, the kinesiological spaces occupied by creeping on the

hands and knees and by human bipedalism are more clearly

separated from one another, indicating that the details of

shoulder movement in these modes are where the differ-

ences lie. Shoulder movements in human walking and

running are very similar and so are not easily distinguished

with most Fourier coefficients. However, a graph of the

first and second sine components helps to separate them,

even though they appear to lie on a continuum of the same

type of motion (Fig. 8c). (Human walking and running are

quite clearly separated by the combination of second cosine

and first sine, and both are very different from quadrupedal

primates, but in the interests of space, not all possible

graphs are shown.) Figure 8c also quite clearly separates

hands-and-knees creeping from all other modes of loco-

motion.

Conclusions and prospectus

In summary, the results of Fourier analysis indicate con-

siderable similarity among primate quadrupedalism, human

bipedalism and human creeping on hands and knees.

Without the use of Fourier analysis, some of this similarity

would be unquantifiable, since footfall analysis cannot be

applied to bipedal humans as it can to quadrupeds. A

number of benefits accrue to the use of FFTs in the study of

primate locomotion. Some of these have been realized

here, while others will be the subject of further study.

Realized benefits of Fourier analysis

In particular, we have used Fourier analysis in the fol-

lowing ways:

1. FFTs are designed to describe cyclical phenomena,

including stride cycles, and are therefore well-suited

to descriptions of joint angles in locomotion research.

Also, they are specifically designed to describe data

in terms of their shape on curvilinear graphs—exactly

what is needed in joint movement studies.

2. FFTs allow us to compare joint movements of the

upper limbs in walking humans with those of the

forelimbs of quadrupedal primates, something that is

not possible with footfall analysis (ground contact

diagrams), since the upper limbs never contact the

ground. Thus, the casual observation, that walking

and running humans seem to use diagonal couplets

gaits like those of other primates, can be shown with

some degree of precision.

3. FFTs are an improvement over the analysis of footfall

patterns to describe interlimb coordination, since they

describe each joint’s contribution to limb movement,

rather than just the results of those joints’ movements

on ground contact. We have focused on the shoulder

joint for the most part—and the shoulder and hip in

quadrupeds make fairly simple movements whose

Fig. 8a–c Bivariate plots of pairs of Fourier coefficients. Solid circlesHands-and-feet creeping, open squares hands-and-knees creeping,

open triangles human walking, crosses human running, opendiamonds non-human primate walking, P Pantera pardus. Note that

hands-and-feet creeping and the leopard generally occupy the same

kinesiological space. a cos1 vs sin1, b cos2 vs cos3, c sin1 vs sin2

Primates (2007) 48:277–292 289

123

relationship to footfall patterns is relatively clear—

but other joints can as easily be studied. Indeed, the

knee and elbow, with two peaks and troughs in their

goniometric data, are not readily understood with

footfall diagrams. Their function is best elucidated by

goniological techniques like those described here.

4. Strides of different duration can be compared as

easily as those of identical duration, since it is the

overall shape of the (graphed) data, not the stride

period, which determines the Fourier coefficients. In

that sense, FFTs act similarly to data transformations

such as expression of the raw data in percent of stride

cycle—both techniques standardize the data with re-

spect to time. In the case of Fourier analysis, how-

ever, the data are removed from the ‘‘time domain’’

entirely, and placed in the ‘‘frequency domain.’’ This

means that sine and cosine waves of various fre-

quencies are added together to produce the same

distribution of data as if a measurement were plotted

against time.

5. FFTs can produce angular data graphs with many

fewer data points than traditional ‘‘time domain’’

graphs (i.e., angle vs time). In this study, datasets of

approximately 50 angles were very well-character-

ized using only 11 Fourier descriptors (five sines and

cosines, and the average angle). Indeed, as noted

above (and in Table 6), 95% of the information can

be described with only seven numbers (three sines

and cosines, and the average angle).

6. Data collected with cameras of different speeds will

usually be compatible, since the number of frames for

each stride is irrelevant, because Fourier analysis will

give very similar results if the same stride is de-

scribed with 24 data points or 48 data points. In this

study, we used frame rates of 15 and 30 fps and the

data were very well-described in both cases.

7. FFTs quantify angular data in a standardized form,

allowing us to use statistical analyses to compare

joint movements among different strides of the same

individual, among different individuals, and even

among different species. Here, we averaged single

strides of several different individuals to arrive at a

general pattern for a particular type of locomotion in

a particular species (or group of species).

8. Fourier coefficients seem to be normally distributed,

so simple statistical tests (like Student’s t-test) can be

used in comparisons. The comparisons in Table 5, for

example, were based on t-tests.

9. The standardization of data by FFT does not distort

the original angular data. This is an advantage over

the technique of dividing the stride cycle into stance

and support phases and plotting joint angle versus

percent of each phase.

10. FFTs provide a simple method of archiving data,

using many fewer datapoints than the original raw

data. It is our hope that other researchers will benefit

from our presentation of the Fourier coefficients in

Tables 1–4. By reversing the Fourier transform, as

described above, others will have joint angle data for

use in comparative studies.

11. The number of flexions/extensions shows up fairly

clearly among the Fourier coefficients, since simple

protraction/retraction movements will have high first

order coefficients, while more complex movements

with two flexions/extensions will have high second

order coefficients.

12. FFTs have allowed us to produce graphs of average

joint movements for many strides and many indi-

viduals.

13. Analysis of the statistical variation of the coefficients

tells us about the variance in movements among

different strides of the same individual and among

different individuals, different species and different

experimental conditions. Variance helps elucidate

mechanical differences in locomotion on a treadmill

versus that on the ground where minute accommo-

dations to irregular substrates are expected.

14. Comparison of coefficients helps us understand the

differing functions of the various limb segments and

joints (in shock absorption, directional control, etc.).

For example, Table 4, for the knee joint, shows the

very large contribution of the first cosine component

to the description of hands-and-knees creeping, as

distinct from all other modes studied.

15. Since FFTs generate average angle, we can easily

investigate differences in average angle due to dif-

ferences in limb proportions and posture.

16. Fourier analysis describes the distribution of the

dataset as a whole, rather than focusing on a few

points in the stride cycle (e.g., a limb striking the

substrate or the timing of maximum extension of a

joint). Therefore, differences between these points are

also quantified for further study. The data for shoulder

angle highlight the smooth, symmetrical oscillation of

human upper limbs during bipedal locomotion, as

opposed to the uneven motion evinced when the upper

limb/forelimb is used for support or propulsion.

Future research

Not all of the benefits of Fourier analysis have been real-

ized in this project. Additional data collection would be

necessary to prove the value of FFTs for other types of

analyses and to answer other questions about primate

locomotion. Future research will undoubtedly begin

with an expansion of the small database provided here.

290 Primates (2007) 48:277–292

123

Thereafter, it may include the study of the effects of sub-

strates on joint movement, and variance may be analyzed

with respect to the level of expertise or experience that

subjects have with a particular mode of locomotion. Some

of these effects may be found most easily using Fourier

analysis. Therefore, several as yet unrealized benefits

should come from the use of Fourier analysis:

1. Because of the nature of Fourier transforms, large

datasets of angular measurement against time can be

reduced to very few datapoints. In theory, we can de-

scribe the shape of a curve which is based on 2n da-

tapoints, using n sines and n cosines. Thus, very high-

speed (120 fps) videorecordings of primate movements

can be fully described with approximately seven sines

and cosines and the average angle. [This is based on a

stride of a little more than 1 s, producing about 128

(=27) angular measurements.] Furthermore, even with

very high frame rates, the vast majority of the distri-

bution of the data can be described with only three

sines and cosines and the average angle.

2. FFTs provide a simple method of archiving data, using

many fewer datapoints than the original raw data. In

the future, even high-speed video of primate locomo-

tion can be reduced to only a few numbers and a large

database can be developed. The database can be used

in biomechanical research, even when the original

primates themselves are unavailable. Unfortunately, it

is quite possible that the greatest value of such a

database will be in archiving the movements of extinct

primates in the decades to come.

3. Since FFTs generate average angle, we can easily

investigate changes in average angle that may arise

from changes in speed or inclination of the substrate,

and we can investigate differences in average angle

due to differences in limb proportions and posture.

Average angles can be produced without Fourier

analysis but are a by-product of FFTs so no additional

work must be performed to obtain that information

while also deriving the other benefits of Fourier anal-

ysis. In addition, differently angled branches or terrain,

or more irregular substrates, may cause kinesiological

changes that do not show up in EMG studies, since

similar muscle contractions may result in different

movements under different conditions. These changes

would be found most easily with Fourier analysis.

4. Study of variance should help us understand the

learning of a new task or the difference between an

expert and a novice. Here, we noted differences in the

variance of Fourier coefficients for a novel task

(hands-and-feet creeping) and walking and running.

Further research will reveal the rate of improvement in

the performance of new locomotor tasks.

5. Analysis of variance will likely have clinical uses. The

degree of deviance from average human FFTs and

therefore average curves of joint angle will help define

rates of post-operative and post-traumatic recovery.

6. Studies of variance can be used to investigate learning

of all types of locomotor behaviors in all primates. For

this reason, analysis of variation in Fourier coefficients

should help us understand the acquisition and refine-

ment of locomotor patterns in young primates.

7. Biomechanical modelling will benefit from Fourier

analysis, since average curves of joint movements can

be used to constrain a model’s output to fit the

parameters of an FFT. Better still, several FFTs of

several joint angles can be combined with body seg-

ment lengths to produce stick figures of primates in

motion. The paths of body points in space (e.g., center

of mass, base of spine, back of head) can also be

described with Fourier analysis, providing additional

constraints on biomechanical models.

8. Biomechanical models based on FFTs can benefit the

entertainment industry in the area of animated film and

special effects. Thus, the representations of primates,

including humans, in motion pictures can be made

more realistic.

Acknowledgments The authors would like to thank Andy Baker,

Philadelphia Zoo, for organizational assistance and permission to

videotape several primates; Stephanie Silfies Donner and Sheela

Davis of the Department of Anthropology and Sociology, Kutztown

University, for their assistance in data entry; Dave Abel and Barry

Peterson, for video equipment loan and technical assistance; and

Dr. Robert Root, Department of Mathematics, Lafayette College, for

mathematical advice. The authors also wish to thank the Kutztown

University Research Committee for their financial support.

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Description of joint movements in human and non-human primate locomotion using Fourier analysis

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