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Analysis of the linear displacement of the head and trunk during walking at different speeds
0021
J Biomechonic Vol 14. No h. pp 411 425. 1981. Prmted I” Great Rntam
ANALYSIS OF THE LINEAR DISPLACEMENT THE HEAD AND TRUNK DURING WALKING AT DIFFERENT SPEEDS*
Y2Y0’RI’06Wl1 I5 SO?.@O~O Pergamon Prey I rd
OF
AURELIO CAPPOZZO Laboratory
of Biomechanics.
Istituto
di Fisiologia
Umana,
Universita
degh Studi, Roma,
Italy
Abstract-The
displacement of points lying on the longitudinal axis of the upper part of the human body at head, shoulder and pelvis level was estimated in the three dimensions of space during level walking on a straight line at speeds ranging from 0.99 to 2.79 m s- ‘. A stereophotogrammetric technique was used. Fortyone walking trials of five young male subjects with no apparent abnormalities of the locomotor system, were recorded. Harmonic analysis of the displacement functions was carried out. This permitted the identification of two superposed components of the motion pattern of the upper part of the body. One, described as inrrirtsic to the locomotor act in its essential form. showed characteristics that were remarkably constant both within and between subjects. The other, described as extrinsic. showed erratic features, possibly due to functional or anatomical asymmetries of the locomotor apparatus or to environmental disturbances. The identification of these two components of the motion pattern formed the basis for the definition of a standard average normal walking. Modifications of the motion pattern arising from variations in the speed of progression are identified and discussed also with the help of Lissajous’s figures. Emphasis was given to the sources and magnitude of the inaccuracies with which the experimental and analytical results were obtained.
IYTRODUCTION
(2) the motion pattern of the upper part of the body is no less important to the motor strategy of human locomotion than that of the lower limbs. Few published articles focus their attention on these two aspects of the mechanics of walking. Grieve ( 1968) and Cappozzo et al. (1978b) stressed the importance of what the former author calls ‘whole range’ studies of walking in the perspective of the first consideration. Paul (1970) studied the relationship of the mechanics of walking to speed of progression with reference to muscular moments and to forces at the hip and knee joints. Andriacchi et al. (1977) made time-distance and foot-ground reaction force measurements during normal and abnormal gait and related them to speed of progression. One of the most exhaustive studies on the kinesiology of human locomotion ever carried out is still the one by Braune and Fisher (1895) and Fisher (189991904). The movements ofboth the upper part of the body and the lower limbs of one subject walking at the so-called natural speed were investigated. Most essential aspects of kinesiology of walking relevant to the present study were tackled by these authors. The only shortcoming of their work is the limited number of experiments carried out. But, of course, this is a shortcoming when seen from the angle of modern computer technology. At the turn of the century. the work conducted by Braune and Fisher was a gigantic achievement. Waters et al. (1973) reported a quantitative description of the linear motion of the head and trunk during walking at speeds up to 1.7 m s- i. Information on the three-dimensional displacement of points on the lower trunk and on the head with respect to time was supplied. Vertical and forward accelerations were also measured by means of accelerometers attached to the same points. Murray rf ul.
For purposes of analysing man’s locomotion the human body can be effectively divided into two subsystems, the lower limbs and the upper part of the body. The former support and transport the latter. As lucidly described by Saunders et al. (1953) the pattern of motion of the lower limbs is such as to minimize mechanical energy exchange between the two subsystems. But, as emphasized by Cappozzo et al. (1978a). the movements of the trunk and head also contribute to the reduction of this energy exchange. These body segments move, with respect to the pelvis, in a coordinated fashion so that their total mechanical energy variation during the walking cycle has a lower magnitude than it would have if trunk and head moved rigidly with the pelvis. The higher the speed of progression the more evnident is the relevance of this observation. Cappozzo et al. (1978a) also point out that the motion of the head is smoother than that of the pelvis. They relate this fact to the necessity of protecting from excessive mechanical stimulus sensory organs such as the eyes and the labyrinth which play a fundamental role in controlling the movement being performed. Protection of the brain could be another relevant factor. The higher the speed of progression. the more these protective needs seem to influence the pattern of movement of the upper part of the body. Two considerations emerge from the foregoing: (1) the study of walking throughout the range of possible speeds permits identification of motor mechanisms and a better insight into their nature which otherwise would be difficult to reveal ;
* Rrcric rtl 27 Juntiar~
1981.
411
AURELIOCAPPOZZO
412
(1966) presented measures of angular and linear displacement of the lower limb segments and of the upper part of the body segments at two walking speeds (approximately 1.53 and 2.2 ms-‘). As far as trunk and head were concerned, data about the sagittal and transverse rotations of the pelvis, the transverse rotation of the thorax, the lateral displacement of the head and the forward and vertical displacement of the neck were presented in the form of functions of time. Lamoreux (1971) studied the movement of body segments during walking at six different speeds ranging from 0.8 to 2.0 m s- ‘. Data relevant to the present topic reported by Lamoreux are the three dimensional displacement of the pelvis and the vertical displacement of the head. As can be inferred from the foregoing references, data about both rotational and linear movements of the pelvis are available in abundant detail. By contrast, information concerning the movement of higher levels of the trunk and of the head are somewhat fragmentary. The purpose of this paper is to offer quantitative information about the linear displacement of the head and trunk during the process of walking in a straight line at different speeds. Working from this information, it is proposed to analyse the intimate structure of the target motion pattern in order to elucidate the nature of each single element of it and, eventually, assess such cause-effect relationships as may exist among these and the ‘boundary conditions’ such as those related to the status of the locomotor system or of the environment. Measurements were made by means of a photogrammetric technique. Speeds of progression ranged from 0.99 to 2.79 m s-l. Because of the ‘oscillating character’ of the motion patterns being studied, harmonic analysis of the displacement functions was used in order to obtain a better insight into their characteristics. Representation in the frequency domain allowed an effective analysis of the modifications that the motion of the upper partcof the body undergoes with changes of walking speed. A functional characterization of each harmonic component of the displacement function also emerged from this analysis. Discrimination of the random from the regular within the motion pattern was achieved both between and within subjects. This formed the basis for the definition of a standard of average normal walking: a ‘stereotype’ pattern of movement. Some propositions, possibly of general utility, about the accuracy of the experimental method adopted were made based on the results of this study.
MATERIALS
technique. For this purpose the following anatomical landmarks on both the left and right sides, were chosen : zygomatic process, acromial process, the tubercle of the iliac crest. The displacement of the anterior-superior iliac spine was also measured and findings taken into consideration when assessing the repeatability of the experimental results. The 3Dcoordinates of these landmarks were reconstructed from photographic observations. The coordinates of the midpoints between the landmarks on the head, shoulders and pelvis were calculated yielding the basic data required for the present study. This was done using a reference system fixed with respect to the laboratory (Fig. 1). They were then recalculated with respect to a reference system rotated about the vertical axis in order that the new x-axis should coincide with the mean direction of progression during any one particular test. In this way comparison of the results of different tests was made possible. Stereophotogrammetry
Four nonmetric 35 mm cameras {i.e.ordinary cameras not designed specifically for measurement purposes) were used. These were placed in stereoscopic pairs, with convergent optical axes (Fig. 1). All cameras were equipped with a focusing finder and a cross-hair reticule screen which made possible a precise aiming of the optical axis. The base line length was 2.9 m (N,Nr and NsN, in Fig. 1). The base lines of the two stereopairs were parallel and separated by a distance of 9.65 m. All four of the cameras were on one plane 1 m above the ground. Relevant horizontality was assessed to the nearest 1 mm. These distances were referred to the front nodal points of the camera objectives. The length and height above the ground of the stereoscopic field on the X-Z plane (Fig. 1) were about 3.7 and 2.8 m, respectively. The object space coordinate system was defined with its origin at the cross NI
N2
i i
AND METHODS
The linear movement of the trunk and head can be satisfactorily described by referring to the motion of points lying on the longitudinal axis of the trunk at pelvis and shoulder level and of a point approximating the centre of gravity of the head. These displacements were estimated by means of a stereophotogrammetric
Fig. 1. Stereophotogrammetric
set up.
Analysis
of the linear displacement
point of the camera axes as shown in Fig. 1. Four control points were defined in fixed positions within the field of view of each camera. Light emitting diodes were placed in these positions and were photographed by the stereopairs along with the walking subject (Fig. 2). The object space coordinates of the control points and of the camera front nodal points (stereosystem constants) were assessed by direct measurement. The test subjects carried light-emitting diodes (LED’s Monsanto MV5352, 45mcd brightness, yellow emission) firmly attached to the anatomical landmarks already indicated. These LED’s were driven by telecontrolled impulses the durations of which were 3 ms: the frequency ranged horn 30 to 60 impulses per second according to the duration of the walking cycle (Cappozzo et al., 1975). The laboratory in which the experiments were carried out was illuminated with green light in order to guarantee a sufficient contrast with the yellow light of the LED’s Subjects walked along a 15m pathway which was aligned with the X-axis shown in Fig. 1. While subjects were walking. LED’s were made to flash and photographs were taken with the four open-shutter cameras. Photographs obtained were of the type shown in Fig. 2. They were enlarged to about one tenth life size and printed on polyester plates. The images of the control points ro, rb and rd permitted the detinition of the image space coordinate system (Fig. 2). The equivalent principal distance of the composite optical trail -camera plus projector used for the enlargement of the photographs - was calculated from the image coordmates of the control points ro, rb and re and from their coordinates in the object space for each record. The three values so obtained were averaged. Knowledge of the coordinates of the nodal points N,, N,, N, and N, in the object space, of the equivalent principal distances of the four cameras plus projector and the measures on each photograph of the image coordinates of the LED’s made possible the calculation of the true coordinates of the LED’s by means of parallax equations (Wolf, 1974). The image coordinates were measured by means of a digitizer the resolution of which was 0.025 mm (Hewlett-Packard, mod. 983 OA). The image points to be digitized were chosen so that the walking cycle analysed was located symmetrically with respect to the origin of the object system of reference. This was done in order to exploit that portion of the stereoscopic field in which systematic experimental errors were minimal (see below). On the pathway on which the subjects walked, adhesive metal strips 5 mm wide were laid longitudinally with a space of 1 mm between them. These strips were wired to two independent electric circuits (one for each foot) in such a way that when a subject’s shoe (the sole of which was covered with similar metal strips) came in contact with the ground, shortcircuiting two adjacent strips, a signal was generated. This signal was fed to a U.V. galvanometer recorder together
.
: _.-._.
:--. .._:
-.
rb’-“....
Fig. 2. Photograph
.
_. rc . .__..
.
__..-
of the head and trunk
during
with the impulses supplied to the LED’s Temporal factors of the walking cycle could thus be measured with great accuracy and synchronization between body segment movement data and stride phases was guaranteed.
Experimental
errors
The errors in the measured coordinates of the markers were, as is always the case, of two types: systematic and random. Systematic errors were due mainly to optical distortions of both the cameras and enlarger, to the deformation of photographic material, and to the inaccuracy with which the system constants, listed previously, were measured. The errors engendered by the optical distortions, inherent in the entire optical trail, were partially corrected for by means of a compensation procedure. A grid was constructed with white threads on a black background. The intersecting threads formed squares with 200 mm sides. The overall dimensions of the grid were 2.8 x 2.0m. Each of the four cameras was placed 5 m from the grid with its optical axis orthogonal to it and aimed at its central intersection. The photographs of the grid so obtained were enlarged to one-tenth full size and the intersections digitized. The optical characteristics of the entire photographic equipment were approximately the same as during the actual experiments. The magnification of the grid image was found to have a value 1.3% larger at the periphery of the field than at the centre. This distortion could be assumed to have a radial symmetry. By means of a regression procedure using data relative to all four cameras, a third-order polynomial function was found which gave a corrected magnification factor as a function of the radius. After having applied this correction a maximal residual error on the magnification factor of the order of 0.2”,, of the mean value of this latter factor could be expected. A sensitivity analysis was carried out in order to assess the characteristics and the relative importance of the systematic errors engendered (on the calculated values of the coordinates of the markers) by the degree of inaccuracy with which the stereosystem constants have been measured. In this respect the most critical inaccuracies were found to be those affecting the values of the equivalent principal distances, These depended above all on the errors made in the measurement of the coordinates of the control points in the object space and in the image space of each camera of a stereopair. The maximal value of these latter errors could be assumed at 0.5 and 0.2 mm respectively. At the periphery of the stereoscopic field used in the experiments reported here and on the conservative hypothesis that each error source would contribute with the same sign to the build up of the overall error. a maximal
..-.-. ..__.._
: . :_.-.__.-‘..
413
walking
.--.._ .
. . rd
.
._ ......
.:- : : X
of one walking test taken with camera N 2. The image space system of reference and the images of the control points are indicated.
414
AURELIO CAPPOZZO
systematic error of approximately 2.5 mm on the x and z reconstructed coordinates and 8 mm on the y reconstructed coordinate could be expected. These maximal errors were not significantly modified by adding the effect of the inaccuracies with which the other system constants (i.e. the coordinates of each camera) were measured provided that these inaccuracies had a magnitude below 2 mm. This was true for the experimental set-up described above. The magnitude of the overall systematic error on the reconstructed coordinates of the markers could be described as varying in quadratic fashion as a function ofdistance from the centre of the stereoscopic field. The importance of this observation will be emphasized later when the effect of this error on the results of the present study is scrutinized. Essentially the random error is elicited by the digitization process of the marker positions in the image space (i.e. quantization and inaccuracy related to the digitizer cursor alignment on the marker image). An analysis of the sensitivity of the calculated coordinates of the markers to this error was also carried out. The hypothesis made was that the measured coordinates in the two object spaces of a stereopair carried random errors having a normal probability distribution. The relevant standard deviation was assessed repeating, in a number of cases, the same measurements thirty times, taking care to reset all possible error sources each time. The average value of this standard deviation was 0.05 mm. The standard deviation of the random error on the calculated coordinates in the object space was estimated to be 0.4 mm on the x and z coordinates and of 1.3 mm on the y coordinate. These values did not significantly depend on the position of the digitized point in the stereoscopic field. The foregoing concerns inaccuracies in the measurement of the markers’coordinates. A further error has to be taken into account when the position of these markers is taken as representing the position of the relevant bony landmarks allowing for the unavoidable movement of the latter relative to the former. It must be emphasized that this movement correlates very well with the main movement, i.e. it has the same type of periodicity. This fact inhibited the possibility of reducing its influence on the measurements by means of conventional smoothing methods. The relevant error could only be minimized by exercising a wise choice of the anatomical landmarks. In summary, only partial indications can be obtained on the quantitative characteristics of experimental errors. The identification of the genuine information within the measurement will, therefore, require a speculative approach.
mation automatically once a smoothing factor was provided. This factor was fixed consistently with the a priori estimation of the random error reported above. Each weighting factor was made to equal one. Care was taken in order not to oversmooth. From the spline function so obtained 31 equally spaced data-points, that exactly fitted one walking cycle, were determined. Harmonic analysis was then carried out on the discrete smoothed displacement functions, deprived of their aperiodic component. The parameters of the following function were estimated: d(r) = a,, + ;i ai sin (iwar + $Q);
1
Several procedures can be employed for adjusting measurements containing random errors. A number of papers have been published in recent years presenting these procedures with particular reference to their use in studies of biomechanics (Zernicke et al.. 1976; Winter et al., 1974; Cappozzo et al., 1975 ; Soudan and Dierckx, 1979 ; Wood and Jennings, 1979; McLaughlin et al., 1977; Lesh et al., 1979). The harmonic regression technique referred to in Cappozzo et al. (1975), appears to be, in principle, the most straightforward way for the attainment of both the estimation of the message displacement functions and of the relative harmonic analysis. It was however shown that a better estimate of the Fourier coefficients could be obtained with this technique if the empirical functions were previously submitted to smoothing by means of spline functions. This was due to two reasons. First, the smoothing reduced the power associated to the random error at all frequencies included those occupied by the message function. Second, the spline functions yielded samples, equally spaced in time, that exactly fitted one walking cycle. This latter circumstance did not necessarily occur in row photogrammetric data, and it was a fundamental prerequisite for a correct harmonic analysis. The cubic spline algorithm provided by Dierckx and Piessens (1977) was used. This algorithm determined the spline approxi-
(1)
where d(t) is the periodic component of the generic displacement function; a, is the mean value; ai and r#i are the amplitude and phase of the i-th harmonic, respectively; T is the stride period; N is the number of harmonics taken into consideration. The parameters relative to the first ten harmonics were estimated. Since great emphasis will be given in what follows, the analysis of the displacement functions in the frequency domain, it is important to report here some considerations relating to the maximal errors engendered by experimental errors on the harmonic coefficients. On the hypothesis that the smoothing procedure has freed the reconstructed generic displacement function from random error, this function can be considered as the sum of the exact displacement function plus the systematic error: d(.) = d* (.) + e, (.).
(2)
According to what has been stated above the value of the systematic error depends, as a first approximation, on the position of the digitized point on the x-z plane, and can be assumed to have the following form: e, (.) = Kr’
where r = (X2 + Zz)i’z.
(3)
In the frequency domain the harmonics of e,(.) add to the harmonics of the same order of d* (.) (see equation 2). This means that the amplitude a and phase 4 of one harmonic of d (.) have the following relationship with the amplitude and phase of the harmonic of the same order of the exact displacement function (a*, 4*) and of the systematic error (a,, A): a = (a*2 + ai + 2a*a,cos (q5* - 4.))“’ 4 = tan-’
Smoothing and harmonic analysis
00 = +
(4)
a* sin @+ + a, sin $I, a* cos 4* + a, cos c$,
(5)
In order to proceed to an approximate evaluation of the error engendered by the experimental systematic error on the harmonic coefficients, the harmonic coefficients of the estimated function d(.) can be used in equations (4) and (5) in place of the unknown harmonic parameters of the exact function d*(.). One estimate of the maximal values that the systematic error e,(.) of equation (2) can assume, was carried out for the x and z coordinates and for the y coordinate. The values assumed by the coordinates x and z of a pelvis point during one stride ofan actual test taken as reference were used for the calculation ofr as function of time. The constant K was chosen so that, for r = IOOOmm, e, had a magnitude of 2.5 mm for the x and z coordinates and of 8 mm for the y coordinate. The maximal systematic errors so obtained in sampled form underwent the stated harmonic regression. Relevant results arc given in Table 1 to the fourth harmonic. As can be inferred from equation (4) and from the fact that 4, has a value very close to n/2 rad (Table 1) the maximal error on the estimation of the amplitudes approximately equals + a, and occurs for 4 equal to + n/2 rad. The maximal error on the phase obtained from equation (5) is approximately equal to k tan-’ (aJa*) rad and occurs for Q equal to zero or * I[ rad. This error tends to zero for 4 approaching + n/2 rad.
Analysis
of the linear displacement Table
1. Harmonic
of the head and trunk
coefficients
of the systematic
a,(mm)
rn.“. I II III IV
1.50 0.80 0.20 0.10 0.06
415
walking
error
esmar = 8mm
e ImaX= 2.5 mm Harmonic
during
a,(mm)
Urad)
Mrad)
4.00 2.50 0.50 0.30 0.15
1.52 1.53 1.62 1.59
1.52 1.53 1.62 1.59
Subjects The study was conducted having similar anthropometric
on five Italian male subjects characteristics (see Table 2). Body fat mass was assessed by plicometry (Sloan and Weir, 1970).All subjects were in good health and had a normal locomotor system. They were either students or academic staff. It is well toemphasize that this constitutes a verylimited sample. Generalization of the results reported in this article to persons whose overall characteristics are not homogeneous with those reported above, would, therefore, be rather presumptuous. During the tests subjects wore minimal clothing and used their everyday shoes, care being taken to avoid extra high heels or too rigid a sole. They were asked to walk successively at what they felt to be a very slow, slow, normal, fast and very fast speed. Subjects were requested to keep their upper limbs flexed in order not to cover the markers placed on the pelvis when walking. This was one drawback of the experimental technique which could not be avoided. It was proved, nevertheless, not to affect the shoulder and head movements to any detectable extent. This was verified by means of comparative tests during which the subjects moved their upper limbs freely. Subjects were also asked to walk looking in front of them so as to avoid erratic movements of the head.
RESULTS A total of 41 tests were taken into consideration for the present study. In order to illustrate thecharacteristics of the strides analysed the relationship between stride length and mean speed of progression (both normalized with respect to stature) is reported in Fig. 3. The parameters of a relevant regression curve proposed by Grieve (1968), the equation of which is reported below, were assessed for each subject and are also reported in Fig. 3.
Table 2. Overall characteristics
Body mass Subject
04
L’
=
60y’cl-8) u.
where L’ is the stride length relative to stature; V’ is the velocity relative to stature (s-l); c( and /r are the regression parameters. These regression parameters were calculated making reference to the tests performed at speeds below approximately 2.4 m s-r. It was observed that above this speed walking traits undergo considerable modification and assume the characteristics typical of race-walking. Only two subjects walked at these high speeds. Results of ten tests relative to one subject (B.C.) only are reported here in a thorough fashion. However, during the discussion, reference will be made to the results obtained on all subjects. In Table 3 the overall characteristics of the tests performed on subject B.C. are summarized. As has been reported in the previous section, the displacement functions of the midpoints between head, shoulders and pelvis markers were determined in sampled form and underwent harmonic analysis. In what follows these midpoints will be referred to as head, shoulder and pelvis points. The results of the harmonic analysis are reported in Figs. 4,5 and 6 up to the fourth harmonic. The time origin, with respect to which harmonic phases were calculated for all the tests, coincided with the heel strike of the left foot. Harmonics of order higher than four showed amplitudes consistently less than 0.5 mm. They were deemed not separable from the experimental errors and, therefore, not significant. When these harmonics were
ofsubjects tested. Trochanteric heights reported and left trochanteric height.
are the mean values ofright
Trochanteric height (% stature)
Stature (m)
Age (years)
1.73
21
8
52.1 51.9
Fat mass (“/ body mass)
G.B.
60
B.C.
72
1.81
32
16
21
8
50.0
E.C.
65
1.76
V.D.
65
1.72
22
14
50.9
A.-f.
70
1.74
24
13
51.7
__~
AURELIOCAPPOZZO
416
alpha 62.4
beta :Z
3:.Z 60:4 59.6
.47 .52 .52
SPEED/STATURE
subject 5:
0
5: AT
0 + x A
Cl/s1
Fig. 3. Relative stride length vs relative speed for all tests. The parameters of the relevant regression curve (equation (6) in the text) are also reported for each subject.
added to the Fourier series expansion the quality of the fit of the input data did not improve, that is, the relevant standard devation assumed an approximately constant value on average at 0.2 mm (Cappozzo et al., 1975). The foregoing was valid at all walking speeds investigated. The rapidity with which a trigonometric series relative to a rhythmic human movement (walking in particular) converges has already been shown and emphasized by Bernstein (1967), Winter et al. (1974), and Lesh et al. (1979). The present results confirm their findings. In order to represent the linear displacement of the head, shoulder and pelvis points the patterns shown in Figs. 7, 8, 9 and IO, were chosen. They are the
Table 3. Characteristics of the walking trials of subject B.C. Stance phase durations are the mean values of right and left stance phase duration
Stride period (s)
Mean speed (m s-t)
Stance duration (% of stride period)
1
1.317
1.19
60.7
2
1.197
1.32
62.4
Test
3
1.048
1.61
61.7
4
1.050
1.63
61.0
5
1.034
1.69
60.8
6
0.977
1.88
59.6
7
0.954
1.99
59.6
8
0.944
2.00
59.4
9
0.889
2.31
58.8
10
0.859
2.35
58.2
trajectories of the designated points in the sagittal, transverse and frontal planes, each referred to a system of reference endowed with a translational motion the velocity of which equals the mean velocity of the relevant point. The three components of the mean velocity were calculated by subtracting the coordinates that these points had when two subsequent heel strikes of the left foot occurred and dividing the result by the stride period. The origin of the axes of the plots in Figs. 7, 8, 9 and 10 was arbitrarily chosen coincident with the average position of the relevant point. These patterns (Lissajous’s figures) were found to be more effective for the objectives of the present study than the representation of each displacement coordinate versus time. They give an immediate picture of the ranges and type of motion and the type of modifications which this latter undergoes moving from one level of the upper part of the body to another and by cause of the change of speed of progression. These same patterns were also reported by Fisher (1900) and by Lamoreux (1971) the former with reference to one speed of progression and the latter with reference to the pelvis only. The results presented in this study are remarkably consistent with those reported by those authors.
THE HARMONIC ANALYSIS
For analytical purposes a ‘stereotype’ pattern of movement of the upper part of the body can be devised. This is characterized by a perfect symmetry of each elementary displacement with respect to the anatomical planes. The longitudinal axis of the trunk and head should therefore move along the vertical and antero-posterior axes with a cycle period corresponding to one step, and along the latero-lateral axis with a cycle period corresponding to one stride (double step)
Analysis of the linear displacement of the head and trunk during walking and with a mirror-image symmetry with respect to the mid-sagittal plane. Any deviation from this behaviour is to be considered not inherent to the locomotor act in its essential form but rather ascribed to some sort of external disturbance. This latter can be an anatomical or functional asymmetry or due to environmental disturbances. The locomotor act can therefore be seen as composed of an intrinsic pure form of movement pattern, the ‘stereotype’, eventually deformed by some extrmsrc cause. The question that arises after this statement is whether the locomotor act can be considered as a superposition of two patterns of movement, i.e. an intrinsic plus an extrinsic pattern. The terms intrinsic and extrinsic are used here in reference to the theoretical. essential locomotor act. Furthermore, the question arises whether the intrinsic pattern of motion is typical of an individual or of a class of individuals. The separation and identification of two such patterns was attempted by means of the harmonic analysis of the displacement functions. The results of this analysis show that this schematization is feasible. Due to the cycles of variation of the intrinsic pattern of motion the harmonic analysis of relevant displacement along the vertical and antero-posterior axes carried out against an interval of time equal to the stride period must exhibit even order harmonics only. For the displacement along the latero-lateral axis only odd-order harmonics are exhibited. These harmonics will be referred to as intrinsic harmonics because they pertain to the intrinsic pattern of motion. Harmonics of different order will be called extrinsic harmonics.
the same type of repeatability already observed at the head and a slightly different trend with speed. At the pelvis, the amplitude exhibited values decreasing from about 20 to about 10mm with the augmentation of speed. The phase showed the same traits of repeatability as at the two upper levels and a trend with speed further modified. These observations applied to all subjects. Third harmonic. Amplitude always showed lower than 1 mm at all three levels, Neither trend with speed nor repeatability could be assessed in the phase values. No confidence could be given to the information carried by the results relative to this harmonic. Fourth harmonic. At the head the amplitude of this harmonic assumed values falling in most cases below 0.5 mm. As could be expected given the influence of the experimental errors, the phase values showed a wide scatter. At the shoulder the amplitude had a magnitude of about 2 mm and could be considered constant with respect to speed. The phase showed repeatability and a clear trend with speed. At the pelvis the amplitude ranged from about 2 mm to about 5 mm and tended to a slight increase with speed. The phase, repeatable both intra-individually and inter-individually. exhibited a well defined trend with respect to speed.
The displacement along the antero-posterior axis is characterized by a first, second and fourth harmonic. The first harmonic (extrinsic, according to the definition given previously) has characteristics that are often unforseeable both for a given subject and interindividually. However, at the pelvis level the first harmonic parameters can exhibit a degree of withinsubject consistency. By contrast, the intrinsic harmonics, i.e. the second and fourth, always exhibit repeatable characteristics both intra-individually and inter-individually. Lutero-lateral
Antero-posterior
displacement
f Fig. 4
417
displacement f Fig. 5 )
i
Firsr harmonic. At the head the amplitude fellin the range 1 to 6 mm in subjects B.C., A.T. and V.D. In subjects E.C. and G.B. it had an amplitude up to 12 mm. It did not show any clear trend with speed. The values of the relevant phase exhibited neither repeatability in similar tests nor trend with speed. This apparently random behaviour should not be ascribed to experimental errors as the amplitude of this harmonic showed relatively large. A partial validation of this statement derives from the evaluation of the error on the harmonic parameters by means of equations (4) and (5). At the shoulders there occurred values of both amplitude and phase similar within expected error to those found at head level. This was true with few exceptions. This circumstance further supports the conclusion that this first harmonic was significant, i.e. it was not an exclusive effect of the experimental error. At the pelvis the pattern changed. The amplitude assumed values that were either similar or slightly less than those found at the two upper levels. But the phase showed a degree of repeatability within similar tests and a trend. though not very coherent, with speed. Results obtained with reference to the iliac spines were readily superposed on those reported in Fig. 4 and confirmed these observations. Results relative to the different subjects exhibited nevertheless great scatter both in phase and amplitude. Second harmonic. At the head, amplitude decreased with the augmentation of speed from about 12 to about 6 mm. Values assumed by the relevant phase were repeatable both inter-individually and intra-individually and showed a constant trend with speed. The scatter of this phase, observed inter-individually, fell within 0.5 rad. At shoulder level, the amplitude was, on average, slightly greater than at the head and showed the same trend with speed. The phase exhibited
First harmonic. At the head the amplitude was in the range 15-40 mm. It showed a slightly decreasing trend with the augmentation ofspeed. The phase exhibited aclear trend with speed and was repeatable both intra-individually and interindividually. At shoulder level the amplitude was in the range 15-40 mm. On average its decrease with the augmentation of speed was more pronounced than at the head. The phase had the same characteristics of repeatability as at the head but it was shifted in all subjects as shown in Fig. 5. At the pelvis the amplitude was in the range lo-35 mm with a marked trend towards decrease as speed augmented. Phase, which showed good repeatability (also inter-individually). underwent a further shift with respect to the one found at the shoulders and the head. Second harmonic. At the head the amplitude exhibited a maximal value of 2 mm. The phase assumed rather scattered values. No sign of coherence was found among the results obtained on different subjects. At the shoulders, observations of the same type were made as at the head. At the pelvis, on average, the amplitude tended to be slightly larger than at the upper levels. The phase showed a definite trend with speed and recurrency within tests on the same subject. Comparison of the relevant values relating to the different subjects showed a wide scatter. Third harmonic. At the head the amplitude was about 3-4 mm with no significant variation with respect to speed. Phase was largely repeatable both inter and intraindividually. At shoulder level much the same observations were made as at head level. At the pelvis the amplitude tended to be lower than at the upper levels and the phase assumed more scattered values. This latter circumstance was very likely due exclusively to the effect of the experimental errors,
AURELIO CAPPOZZO
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AURELIOCAPPOZZO
Fourrh harmonic. In most tests the amplitude of this harmonic was lower than 0.5 mm at all three levels. No confidence could therefore be given to the data generated.
The displacement along the latero-lateral axis is characterized by a first harmonic significantly greater in amplitude than the higher-order harmonics. This harmonic has repeatable characteristics both interand intra-individually. Much the same can be said with respect to the third harmonic, although its amplitude is often so small as to be masked by the experimental error. With reference to latero-lateral displacement, first and third harmonics are intrinsic to the motion pattern and exhibit the same regular behaviour already observed for the intrinsic harmonics of the antero-posterior displacement. The only extrinsic harmonic which can be revealed (though with some uncertainty) with the present experimental method is the second harmonic. It has rather unpredictable characteristics. Only at the pelvis does it show withinsubject systematic characteristics.
found that would supply a categorical justification to this observation. The extrinsic harmonics exhibit large intra- and inter-individual differences. However, a degree of within-subject consistency can be found in some cases with regard to the first harmonic coefficients. Subjects G.B. and EC. exhibited this consistency. It was interesting to note that these two subjects had the largest difference between right and left trochanteric height. The former subject had the right higher than the left trochanter, while the opposite applied to the latter subject. The phases of the first harmonic of the vertical displacement relative to these two subjects were in counterphase. It would, therefore,
seem not too presumptuous to infer the existence of a causeeffect relationship between the regular behaviour of this first harmonic coefficients and the aforementioned anatomical asymmetry.
Vertical displacement (Fig, 6) First harmonic. At the head the amplitude had a maximal
value of 4 mm except in subject G.B. where it reached 8 mm. No clear trend of variation with speed was noticed. Phases exhibited a wide scatter both inter- and intra-individually. This was not true for subjects G.B. and EC. who exhibited repeatable and regular values of the phase of this harmonic although with large individual differences. At the shoulders and pelvis much the same observations were made as at the head. Second harmonic. At the head the amplitude ranged from 16 to 35 mm and exhibited a clear trend to augment with speed. However at speeds beyond about 2.4 m s-r an abrupt fall in the value of this amplitude was observed to occur. This is a further indication that at the highest speeds of progression a definite change in gait occurs. The phase was uniformly constant with respect to speed and remarkably repeatable intra-individually. At shoulder and pelvis level the amplitude tended to be slightly greater than at head level but exhibited the same trend versus speed. Phases reproduced within expected error those observed at the head. Third harmonic. At all three levels this harmonic had in most tests an amplitude below 2 mm. When the amplitude did not assume values below 0.5 mm, the occurrence of similar phase values at the three levels was observed. This phase presented a great scatter both for one subject and among different subjects. Fourth harmonic. This harmonic occurred with amplitudes and phases that were similar within expected error at all three levels and for four subiects out of five. Subiect G.B. had different values of phase”although these were consistent inter se. The amplitude, in the range 2-4 mm, did not show any clear trend of variation with speed. Phase, on the contrary, underwent a systematic shift as speed augmented.
The displacement in the vertical direction is characterized by second and fourth harmonics which are intrinsic, and first and third harmonics which are extrinsic. The intrinsic harmonics show a systematic behaviour both intra- and inter-individually similar to the antero-posterior and that observed for latero-lateral displacements. One exception was found to this statement. It concerned the fourth harmonic coefficients for subject G.B. No objective datum was
THE LISSAJOUS’SFIGURES The plots shown in Figs. 7, 8, 9 and 10 permit visualization in reasonably immediate fashion of the following aspects of the pattern of motion of trunk and head. (1) Harmoniousness of the movement. The trajectories shown in Figs. 7,8,9 and 10 have a very ‘simple’ and smooth aspect. This is an indication of the regularity and harmoniousness of the movement. The relevant evidence is obviously not complete as in these plots reference to time is lost. (2) Repeatability among successive walking steps. In the sagittal plane pelvis-, shoulder- and head-points trace loops, one during each step. The extent to which these loops differ supplies information on the repeatability of successive steps. In the transverse and frontal planes the same type of information is given by the extent to which the pathways have a mirror-image symmetry with respect to the x axis and the z axis respectively. (3) Maximal deviations of head, shoulder and pelvis points from mean position. These can be easily inferred from the plots in Figs 7,8,9 and 10. It can be observed that along the antero-posterior axis the excursions about the mean position of the head and shoulder are smaller than that of the pelvis point. Along the latero-lateral axis the contrary applies, the more so the higher the speed of progression. The vertical excursion almost identical at the three considered levels increases as speed augments to about 2.4m s-r. At higher speeds this excursion decreases considerably. (4) Relative movement between pelvis, shoulders and head. Differences in the maximal excursions about the mean position of these points have already been pointed out above. The form of the trajectories also varies going from one level to another. This phenomenon becomes more evident as speed increases.
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AURELIO CAPPOZZO
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CONCLUSIONS
The experimental results presented in this study permit the following deductions. (1) The periodic portion of the linear displacement of the longitudinal axis of the upper portion of the human body during normal level walking is described with considerable refinement, within one walking cycle, by the first four harmonics along the antero-posterior and vertical directions and by the first three harmonics along the later*lateral direction. This is true at all walking speeds investigated. (2) Harmonic analysis of the linear displacement of the longitudinal axis of the upper part of the body made possible the identification of an intrinsic and an extrinsic pattern of motion, according to the definitions given in a previous section. The qualities of these two patterns and thus their functional characterization emerge clearly from the experimental information available. The intrinsic pattern of motion, composed of second and fourth harmonics for the displacements along the
antero-posterior and vertical axes, and of first and third harmonics for the latero-lateral displacement, has, at a certain speed of progression, very steady characteristics both within- and between-subjects. From the foregoing it can be inferred that the intrinsic pattern of movement is an individual stereotype and that the same stereotype can apply to a class of individuals. Because of the small number of subjects from which experimental evidence has been adduced, it is not possible to precisely define the characteristics of this class of individuals. However, some related experiments being conducted on young women in the author’s laboratory seem to give re,rults (Lombardi, 1980) largely consistent with those reported here. This suggests, though not exclusively, that the same stereotype could apply to a quite large and nonhomogeneousclass ofindividuals and that the intrinsic pattern of movement previously described is one possible standard of average normal walking. The extrinsic pattern of motion, composed of first and third harmonics for the antero-posterior and vertical displacement and second harmonic for the
Analysis of the linear displacement of the head and trunk during walking
Bernstein’s (1967) concept, ‘biodynamical tissue’. (3) The longitudinal axis of the upper part of the body does not move rigidly with the pelvis. Along the antero-posterior axis the pelvis undergoes larger excursions with respect to an observer moving at the relevant mean speed of progression than shoulders and head. This is true at all speeds of progression. The maximal excursion of the head and shoulder points with respect to the pelvis point along the antero-posterior axis is approximately 20 mm with minor changes attributable to variations in speed. Along the latero-lateral axis head and shoulder points undergo larger excursions than the pelvis point. The higher the speed of progression the larger becomes the displacement of the higher levels with respect to the pelvis. The overall excursion of the head point with respect to the pelvis point ranges, along the later+lateral axis, from approximately 20 to 65 mm. Only a slight variation of the vertical displacement is observable going from one level to another. (4) The pattern of movement of the upper part of the body changes with speed; that is, the augmentation of speed of progression is not achieved by means of a simple shrinkage of the time axis. Nor does it cause a
iatero-lateral displacement, exhibited on the contrary a high degree of variability between subjects and in most cases within-subject. This was invariably true with reference to the head and shoulder-points along the antero-posterior and latero-lateral axes. At the pelvis along all three directions, and at the head and shoulders along the vertical direction, a regular behaviour was, on the contrary, observed in a proportion of cases. This regularity occurred strictly withinsubject. One possible explanation of this phenomenon can be recognized in the nature of the possible causes that engender the extrinsic pattern of motion, as pointed out previously. Anatomical or functional asymmetries, though in the normal range, can be expected to have repeatable effects on the pattern of motion. Within-subject regularity can therefore be ascribed to these causes. Environmental disturbances, or other subject-born disturbances are, on the contrary, likely to have irregular effects. The nonrepeatable and non-predictable part of the extrinsic pattern of motion is therefore to be ascribed to these causes. The foregoing is one possible descriptive base for investigations into the ‘morphology’ of walking as, in
Speed
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424
AIIRELIO
Speed
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simple variation of the amplitudes of the displacement functions. As was observed in analysing the relevant harmonic coefficients, the phase of some intrinsic harmonics shifted with speed. These shifts cause a modification of the actual shape of the displacement functions and, as evidenced previously with reference to Lissajous’s figures, of the trajectories as well. Understanding of the fundamental reasons for these modifications of the pattern of motion requires reference to the dynamics as well as to the kinematics of walking. Therefore no explanation of these will be attempted here. (5) Strong emphasis has been given to the experimental and analytical methods used. The results of this study permitted accurate assessment of the resolution of these methods both from the quantitative and qualitative standpoints. In synthesis, it was observed that the error on the estimation of the three coordinates of the landmarks can be substantial, i.e. in the order of several millimeters. But the larger portion of this error will so operate as to affect the coordinates of adjacent points in a similar way. This implies that the displacement functions of the landmarks are only globally deformed (their time derivatives are not
heavily affected). As emphasized previously, it was, in fact, possible for harmonics of the displacement functions to be classified as significant within an amplitude of only 1 mm.
Acknowledgements-Data reduction and computing was in part carried out at the Bioengineering Unit, Strathclyde University, Glasgow (Great Britain), with the kind permission of Prof. J. P. Paul. The author’s sojourns in Glasgow were financed by The British Council. The author expresses his gratitude to Prof. J. P. Paul, to Prof. S. Cerquiglini, and to the designated referees of this Journal for their most helpful criticism of the manuscript.
REFERENCES Andriacchi, T. P., Ogle, J. A. and Galante, G. 0. (1977) Walking speed as a basis for normal and abnormal gait measurements. J. Biomech. 10, 261. Bernstein, N. (1967) The coordination and regulationof movements. Pergamon Press Ltd. Braune, C. W. and Fisher, 0. (1895) Der Gang des Menshen I. Abh. Math. Phys. Cl. Kiln. Siichs. Ges. Wissensch. 21, 151.
Analysis
Speed
of the linear displacement
of the head and trunk during
2.31mi.s
HEAD
rhso
lhso
rton
Itoa
SHOULDERS
walking
425
PELVIS
tdown
)down
lid own
Y+left
Y left
Y left t
I
tdown
1 div=lQmm Fig. 10. Lissajous’s
figures of the displacement of head-, shoulder- and pelvis-point and frontal planes during walking at 2.31 m s-‘.
Cappozzo, A., Leo, T. and Pedotti, A. (1975) A general computing method for the analysis of human locomotion. J. Biomech. 8, 307. Cappozzo, A., Figura, F., Leo, T. and Marchetti, M. (1978a) Movements and mechanical energy changes in the upper part of the human body during walking. In: Biomechanics VI-A, (E. Asmussen and K. Jorgensen eds), pp. 272, University Park Press, Baltimore. Cappozzo, A., Figura, F., Leo, T. and Marchetti, M. (1978b) An approach to human locomotion studies. Acta Med. Rom. 16, 540. Dierckx, P. and Piessens, R. (1977) Calculation of Fourier coefficients of discrete functions using cubic splines. Report TW32, Applied Math. and Progr. Division, K. U. Leuven (Belgium), June 1977. Fisher, 0. (1899-1904) Der Gang des Menschen. Abh. Math. Phys. CI. Kiln. Siichs. Ges. Wissensch., II-25 (1900); l-130. III-26(1901); 85-170. IV-26 (1901);469-556. V-28 (1904); 319-418. VI-28 (1904); 531-617. Grieve, D. W. (1968) Gait patterns and the speed of walking. &o-M&. Eng. 3, 119. Lamoreux, L. W. (1971) Kinematic measurements in the study of human walking. Bull. Prosthetics Res. 10-15, 3. Lesh, M. D., Mansour, J. M. and Simon, S. R. (1979) A gait analysis subsystem for smoothing and differentiation of human motion data. Trans. ASME J. Biomech. Engng. 101, 205. Lombardi, D. (1980) Biomeccanica della deambulazione nelle donne. Doctorate Thesis, University of Rome, Italy.
in the sagittal, transverse
McLaughlin, T. M., Dillman, C. J. and Lardner, T. J. (1977) Biomechanical analysis with cubic splines. Res.Q, 48, 569. Murray, M. P., Kory, R. C., Clarkson, B. H. and Sepic, S. B. (1966) Comparison of free and fast speed walking patterns of normal men. Am. J. Phys. Med. 45, 8. Paul, J. P. (1970) The effect of walking speed on the force actions transmitted at the hip and kneejoints. Proc. R. Sot. Med. 63, 200. Saunders, J. B. DeCM., Inman. V. T. and Eberhart, H. D. (1953) The major determinants in normal and pathological gait. J. Bone Jnt. Surg. 35A, 543. Sloan, A. and Weir, J. (1970) Nomograms for prediction of body density and total body fat from skinfold measurements. J. appl. Physiol. 28, 221. Soudan, K. and Dierckx, P. (1979) Calculation of derivatives and Fourier coefficients of human motion data, while using spline functions. J. Biomech. 12, 21. Waters, R. L., Morris, J. and Perry, J. (1973) Translational motion of the head and trunk during normal walking. J. Biomech. 6, 167. Winter, D. A., Sidwall, H. G. and Hobson. D. A. (1974) Measurements and reduction of noise in kinematics of locomotion. J. Biomech. 7, 157. Wolf, P. R. (1974) Elements of photogrammetr,v. McGrawHill, New York. Wood, G. A. and Jennings, L. S. (1979) On the use of spline functions for data smoothing. J. Biomech. 12, 477. Zernicke, R. F., Caldwell, G. and Roberts, E. M. (1976) Fitting biomechanical data with cubic spline functions. Res.Q. 47, 9.
J Biomechonic Vol 14. No h. pp 411 425. 1981. Prmted I” Great Rntam
ANALYSIS OF THE LINEAR DISPLACEMENT THE HEAD AND TRUNK DURING WALKING AT DIFFERENT SPEEDS*
Y2Y0’RI’06Wl1 I5 SO?.@O~O Pergamon Prey I rd
OF
AURELIO CAPPOZZO Laboratory
of Biomechanics.
Istituto
di Fisiologia
Umana,
Universita
degh Studi, Roma,
Italy
Abstract-The
displacement of points lying on the longitudinal axis of the upper part of the human body at head, shoulder and pelvis level was estimated in the three dimensions of space during level walking on a straight line at speeds ranging from 0.99 to 2.79 m s- ‘. A stereophotogrammetric technique was used. Fortyone walking trials of five young male subjects with no apparent abnormalities of the locomotor system, were recorded. Harmonic analysis of the displacement functions was carried out. This permitted the identification of two superposed components of the motion pattern of the upper part of the body. One, described as inrrirtsic to the locomotor act in its essential form. showed characteristics that were remarkably constant both within and between subjects. The other, described as extrinsic. showed erratic features, possibly due to functional or anatomical asymmetries of the locomotor apparatus or to environmental disturbances. The identification of these two components of the motion pattern formed the basis for the definition of a standard average normal walking. Modifications of the motion pattern arising from variations in the speed of progression are identified and discussed also with the help of Lissajous’s figures. Emphasis was given to the sources and magnitude of the inaccuracies with which the experimental and analytical results were obtained.
IYTRODUCTION
(2) the motion pattern of the upper part of the body is no less important to the motor strategy of human locomotion than that of the lower limbs. Few published articles focus their attention on these two aspects of the mechanics of walking. Grieve ( 1968) and Cappozzo et al. (1978b) stressed the importance of what the former author calls ‘whole range’ studies of walking in the perspective of the first consideration. Paul (1970) studied the relationship of the mechanics of walking to speed of progression with reference to muscular moments and to forces at the hip and knee joints. Andriacchi et al. (1977) made time-distance and foot-ground reaction force measurements during normal and abnormal gait and related them to speed of progression. One of the most exhaustive studies on the kinesiology of human locomotion ever carried out is still the one by Braune and Fisher (1895) and Fisher (189991904). The movements ofboth the upper part of the body and the lower limbs of one subject walking at the so-called natural speed were investigated. Most essential aspects of kinesiology of walking relevant to the present study were tackled by these authors. The only shortcoming of their work is the limited number of experiments carried out. But, of course, this is a shortcoming when seen from the angle of modern computer technology. At the turn of the century. the work conducted by Braune and Fisher was a gigantic achievement. Waters et al. (1973) reported a quantitative description of the linear motion of the head and trunk during walking at speeds up to 1.7 m s- i. Information on the three-dimensional displacement of points on the lower trunk and on the head with respect to time was supplied. Vertical and forward accelerations were also measured by means of accelerometers attached to the same points. Murray rf ul.
For purposes of analysing man’s locomotion the human body can be effectively divided into two subsystems, the lower limbs and the upper part of the body. The former support and transport the latter. As lucidly described by Saunders et al. (1953) the pattern of motion of the lower limbs is such as to minimize mechanical energy exchange between the two subsystems. But, as emphasized by Cappozzo et al. (1978a). the movements of the trunk and head also contribute to the reduction of this energy exchange. These body segments move, with respect to the pelvis, in a coordinated fashion so that their total mechanical energy variation during the walking cycle has a lower magnitude than it would have if trunk and head moved rigidly with the pelvis. The higher the speed of progression the more evnident is the relevance of this observation. Cappozzo et al. (1978a) also point out that the motion of the head is smoother than that of the pelvis. They relate this fact to the necessity of protecting from excessive mechanical stimulus sensory organs such as the eyes and the labyrinth which play a fundamental role in controlling the movement being performed. Protection of the brain could be another relevant factor. The higher the speed of progression. the more these protective needs seem to influence the pattern of movement of the upper part of the body. Two considerations emerge from the foregoing: (1) the study of walking throughout the range of possible speeds permits identification of motor mechanisms and a better insight into their nature which otherwise would be difficult to reveal ;
* Rrcric rtl 27 Juntiar~
1981.
411
AURELIOCAPPOZZO
412
(1966) presented measures of angular and linear displacement of the lower limb segments and of the upper part of the body segments at two walking speeds (approximately 1.53 and 2.2 ms-‘). As far as trunk and head were concerned, data about the sagittal and transverse rotations of the pelvis, the transverse rotation of the thorax, the lateral displacement of the head and the forward and vertical displacement of the neck were presented in the form of functions of time. Lamoreux (1971) studied the movement of body segments during walking at six different speeds ranging from 0.8 to 2.0 m s- ‘. Data relevant to the present topic reported by Lamoreux are the three dimensional displacement of the pelvis and the vertical displacement of the head. As can be inferred from the foregoing references, data about both rotational and linear movements of the pelvis are available in abundant detail. By contrast, information concerning the movement of higher levels of the trunk and of the head are somewhat fragmentary. The purpose of this paper is to offer quantitative information about the linear displacement of the head and trunk during the process of walking in a straight line at different speeds. Working from this information, it is proposed to analyse the intimate structure of the target motion pattern in order to elucidate the nature of each single element of it and, eventually, assess such cause-effect relationships as may exist among these and the ‘boundary conditions’ such as those related to the status of the locomotor system or of the environment. Measurements were made by means of a photogrammetric technique. Speeds of progression ranged from 0.99 to 2.79 m s-l. Because of the ‘oscillating character’ of the motion patterns being studied, harmonic analysis of the displacement functions was used in order to obtain a better insight into their characteristics. Representation in the frequency domain allowed an effective analysis of the modifications that the motion of the upper partcof the body undergoes with changes of walking speed. A functional characterization of each harmonic component of the displacement function also emerged from this analysis. Discrimination of the random from the regular within the motion pattern was achieved both between and within subjects. This formed the basis for the definition of a standard of average normal walking: a ‘stereotype’ pattern of movement. Some propositions, possibly of general utility, about the accuracy of the experimental method adopted were made based on the results of this study.
MATERIALS
technique. For this purpose the following anatomical landmarks on both the left and right sides, were chosen : zygomatic process, acromial process, the tubercle of the iliac crest. The displacement of the anterior-superior iliac spine was also measured and findings taken into consideration when assessing the repeatability of the experimental results. The 3Dcoordinates of these landmarks were reconstructed from photographic observations. The coordinates of the midpoints between the landmarks on the head, shoulders and pelvis were calculated yielding the basic data required for the present study. This was done using a reference system fixed with respect to the laboratory (Fig. 1). They were then recalculated with respect to a reference system rotated about the vertical axis in order that the new x-axis should coincide with the mean direction of progression during any one particular test. In this way comparison of the results of different tests was made possible. Stereophotogrammetry
Four nonmetric 35 mm cameras {i.e.ordinary cameras not designed specifically for measurement purposes) were used. These were placed in stereoscopic pairs, with convergent optical axes (Fig. 1). All cameras were equipped with a focusing finder and a cross-hair reticule screen which made possible a precise aiming of the optical axis. The base line length was 2.9 m (N,Nr and NsN, in Fig. 1). The base lines of the two stereopairs were parallel and separated by a distance of 9.65 m. All four of the cameras were on one plane 1 m above the ground. Relevant horizontality was assessed to the nearest 1 mm. These distances were referred to the front nodal points of the camera objectives. The length and height above the ground of the stereoscopic field on the X-Z plane (Fig. 1) were about 3.7 and 2.8 m, respectively. The object space coordinate system was defined with its origin at the cross NI
N2
i i
AND METHODS
The linear movement of the trunk and head can be satisfactorily described by referring to the motion of points lying on the longitudinal axis of the trunk at pelvis and shoulder level and of a point approximating the centre of gravity of the head. These displacements were estimated by means of a stereophotogrammetric
Fig. 1. Stereophotogrammetric
set up.
Analysis
of the linear displacement
point of the camera axes as shown in Fig. 1. Four control points were defined in fixed positions within the field of view of each camera. Light emitting diodes were placed in these positions and were photographed by the stereopairs along with the walking subject (Fig. 2). The object space coordinates of the control points and of the camera front nodal points (stereosystem constants) were assessed by direct measurement. The test subjects carried light-emitting diodes (LED’s Monsanto MV5352, 45mcd brightness, yellow emission) firmly attached to the anatomical landmarks already indicated. These LED’s were driven by telecontrolled impulses the durations of which were 3 ms: the frequency ranged horn 30 to 60 impulses per second according to the duration of the walking cycle (Cappozzo et al., 1975). The laboratory in which the experiments were carried out was illuminated with green light in order to guarantee a sufficient contrast with the yellow light of the LED’s Subjects walked along a 15m pathway which was aligned with the X-axis shown in Fig. 1. While subjects were walking. LED’s were made to flash and photographs were taken with the four open-shutter cameras. Photographs obtained were of the type shown in Fig. 2. They were enlarged to about one tenth life size and printed on polyester plates. The images of the control points ro, rb and rd permitted the detinition of the image space coordinate system (Fig. 2). The equivalent principal distance of the composite optical trail -camera plus projector used for the enlargement of the photographs - was calculated from the image coordmates of the control points ro, rb and re and from their coordinates in the object space for each record. The three values so obtained were averaged. Knowledge of the coordinates of the nodal points N,, N,, N, and N, in the object space, of the equivalent principal distances of the four cameras plus projector and the measures on each photograph of the image coordinates of the LED’s made possible the calculation of the true coordinates of the LED’s by means of parallax equations (Wolf, 1974). The image coordinates were measured by means of a digitizer the resolution of which was 0.025 mm (Hewlett-Packard, mod. 983 OA). The image points to be digitized were chosen so that the walking cycle analysed was located symmetrically with respect to the origin of the object system of reference. This was done in order to exploit that portion of the stereoscopic field in which systematic experimental errors were minimal (see below). On the pathway on which the subjects walked, adhesive metal strips 5 mm wide were laid longitudinally with a space of 1 mm between them. These strips were wired to two independent electric circuits (one for each foot) in such a way that when a subject’s shoe (the sole of which was covered with similar metal strips) came in contact with the ground, shortcircuiting two adjacent strips, a signal was generated. This signal was fed to a U.V. galvanometer recorder together
.
: _.-._.
:--. .._:
-.
rb’-“....
Fig. 2. Photograph
.
_. rc . .__..
.
__..-
of the head and trunk
during
with the impulses supplied to the LED’s Temporal factors of the walking cycle could thus be measured with great accuracy and synchronization between body segment movement data and stride phases was guaranteed.
Experimental
errors
The errors in the measured coordinates of the markers were, as is always the case, of two types: systematic and random. Systematic errors were due mainly to optical distortions of both the cameras and enlarger, to the deformation of photographic material, and to the inaccuracy with which the system constants, listed previously, were measured. The errors engendered by the optical distortions, inherent in the entire optical trail, were partially corrected for by means of a compensation procedure. A grid was constructed with white threads on a black background. The intersecting threads formed squares with 200 mm sides. The overall dimensions of the grid were 2.8 x 2.0m. Each of the four cameras was placed 5 m from the grid with its optical axis orthogonal to it and aimed at its central intersection. The photographs of the grid so obtained were enlarged to one-tenth full size and the intersections digitized. The optical characteristics of the entire photographic equipment were approximately the same as during the actual experiments. The magnification of the grid image was found to have a value 1.3% larger at the periphery of the field than at the centre. This distortion could be assumed to have a radial symmetry. By means of a regression procedure using data relative to all four cameras, a third-order polynomial function was found which gave a corrected magnification factor as a function of the radius. After having applied this correction a maximal residual error on the magnification factor of the order of 0.2”,, of the mean value of this latter factor could be expected. A sensitivity analysis was carried out in order to assess the characteristics and the relative importance of the systematic errors engendered (on the calculated values of the coordinates of the markers) by the degree of inaccuracy with which the stereosystem constants have been measured. In this respect the most critical inaccuracies were found to be those affecting the values of the equivalent principal distances, These depended above all on the errors made in the measurement of the coordinates of the control points in the object space and in the image space of each camera of a stereopair. The maximal value of these latter errors could be assumed at 0.5 and 0.2 mm respectively. At the periphery of the stereoscopic field used in the experiments reported here and on the conservative hypothesis that each error source would contribute with the same sign to the build up of the overall error. a maximal
..-.-. ..__.._
: . :_.-.__.-‘..
413
walking
.--.._ .
. . rd
.
._ ......
.:- : : X
of one walking test taken with camera N 2. The image space system of reference and the images of the control points are indicated.
414
AURELIO CAPPOZZO
systematic error of approximately 2.5 mm on the x and z reconstructed coordinates and 8 mm on the y reconstructed coordinate could be expected. These maximal errors were not significantly modified by adding the effect of the inaccuracies with which the other system constants (i.e. the coordinates of each camera) were measured provided that these inaccuracies had a magnitude below 2 mm. This was true for the experimental set-up described above. The magnitude of the overall systematic error on the reconstructed coordinates of the markers could be described as varying in quadratic fashion as a function ofdistance from the centre of the stereoscopic field. The importance of this observation will be emphasized later when the effect of this error on the results of the present study is scrutinized. Essentially the random error is elicited by the digitization process of the marker positions in the image space (i.e. quantization and inaccuracy related to the digitizer cursor alignment on the marker image). An analysis of the sensitivity of the calculated coordinates of the markers to this error was also carried out. The hypothesis made was that the measured coordinates in the two object spaces of a stereopair carried random errors having a normal probability distribution. The relevant standard deviation was assessed repeating, in a number of cases, the same measurements thirty times, taking care to reset all possible error sources each time. The average value of this standard deviation was 0.05 mm. The standard deviation of the random error on the calculated coordinates in the object space was estimated to be 0.4 mm on the x and z coordinates and of 1.3 mm on the y coordinate. These values did not significantly depend on the position of the digitized point in the stereoscopic field. The foregoing concerns inaccuracies in the measurement of the markers’coordinates. A further error has to be taken into account when the position of these markers is taken as representing the position of the relevant bony landmarks allowing for the unavoidable movement of the latter relative to the former. It must be emphasized that this movement correlates very well with the main movement, i.e. it has the same type of periodicity. This fact inhibited the possibility of reducing its influence on the measurements by means of conventional smoothing methods. The relevant error could only be minimized by exercising a wise choice of the anatomical landmarks. In summary, only partial indications can be obtained on the quantitative characteristics of experimental errors. The identification of the genuine information within the measurement will, therefore, require a speculative approach.
mation automatically once a smoothing factor was provided. This factor was fixed consistently with the a priori estimation of the random error reported above. Each weighting factor was made to equal one. Care was taken in order not to oversmooth. From the spline function so obtained 31 equally spaced data-points, that exactly fitted one walking cycle, were determined. Harmonic analysis was then carried out on the discrete smoothed displacement functions, deprived of their aperiodic component. The parameters of the following function were estimated: d(r) = a,, + ;i ai sin (iwar + $Q);
1
Several procedures can be employed for adjusting measurements containing random errors. A number of papers have been published in recent years presenting these procedures with particular reference to their use in studies of biomechanics (Zernicke et al.. 1976; Winter et al., 1974; Cappozzo et al., 1975 ; Soudan and Dierckx, 1979 ; Wood and Jennings, 1979; McLaughlin et al., 1977; Lesh et al., 1979). The harmonic regression technique referred to in Cappozzo et al. (1975), appears to be, in principle, the most straightforward way for the attainment of both the estimation of the message displacement functions and of the relative harmonic analysis. It was however shown that a better estimate of the Fourier coefficients could be obtained with this technique if the empirical functions were previously submitted to smoothing by means of spline functions. This was due to two reasons. First, the smoothing reduced the power associated to the random error at all frequencies included those occupied by the message function. Second, the spline functions yielded samples, equally spaced in time, that exactly fitted one walking cycle. This latter circumstance did not necessarily occur in row photogrammetric data, and it was a fundamental prerequisite for a correct harmonic analysis. The cubic spline algorithm provided by Dierckx and Piessens (1977) was used. This algorithm determined the spline approxi-
(1)
where d(t) is the periodic component of the generic displacement function; a, is the mean value; ai and r#i are the amplitude and phase of the i-th harmonic, respectively; T is the stride period; N is the number of harmonics taken into consideration. The parameters relative to the first ten harmonics were estimated. Since great emphasis will be given in what follows, the analysis of the displacement functions in the frequency domain, it is important to report here some considerations relating to the maximal errors engendered by experimental errors on the harmonic coefficients. On the hypothesis that the smoothing procedure has freed the reconstructed generic displacement function from random error, this function can be considered as the sum of the exact displacement function plus the systematic error: d(.) = d* (.) + e, (.).
(2)
According to what has been stated above the value of the systematic error depends, as a first approximation, on the position of the digitized point on the x-z plane, and can be assumed to have the following form: e, (.) = Kr’
where r = (X2 + Zz)i’z.
(3)
In the frequency domain the harmonics of e,(.) add to the harmonics of the same order of d* (.) (see equation 2). This means that the amplitude a and phase 4 of one harmonic of d (.) have the following relationship with the amplitude and phase of the harmonic of the same order of the exact displacement function (a*, 4*) and of the systematic error (a,, A): a = (a*2 + ai + 2a*a,cos (q5* - 4.))“’ 4 = tan-’
Smoothing and harmonic analysis
00 = +
(4)
a* sin @+ + a, sin $I, a* cos 4* + a, cos c$,
(5)
In order to proceed to an approximate evaluation of the error engendered by the experimental systematic error on the harmonic coefficients, the harmonic coefficients of the estimated function d(.) can be used in equations (4) and (5) in place of the unknown harmonic parameters of the exact function d*(.). One estimate of the maximal values that the systematic error e,(.) of equation (2) can assume, was carried out for the x and z coordinates and for the y coordinate. The values assumed by the coordinates x and z of a pelvis point during one stride ofan actual test taken as reference were used for the calculation ofr as function of time. The constant K was chosen so that, for r = IOOOmm, e, had a magnitude of 2.5 mm for the x and z coordinates and of 8 mm for the y coordinate. The maximal systematic errors so obtained in sampled form underwent the stated harmonic regression. Relevant results arc given in Table 1 to the fourth harmonic. As can be inferred from equation (4) and from the fact that 4, has a value very close to n/2 rad (Table 1) the maximal error on the estimation of the amplitudes approximately equals + a, and occurs for 4 equal to + n/2 rad. The maximal error on the phase obtained from equation (5) is approximately equal to k tan-’ (aJa*) rad and occurs for Q equal to zero or * I[ rad. This error tends to zero for 4 approaching + n/2 rad.
Analysis
of the linear displacement Table
1. Harmonic
of the head and trunk
coefficients
of the systematic
a,(mm)
rn.“. I II III IV
1.50 0.80 0.20 0.10 0.06
415
walking
error
esmar = 8mm
e ImaX= 2.5 mm Harmonic
during
a,(mm)
Urad)
Mrad)
4.00 2.50 0.50 0.30 0.15
1.52 1.53 1.62 1.59
1.52 1.53 1.62 1.59
Subjects The study was conducted having similar anthropometric
on five Italian male subjects characteristics (see Table 2). Body fat mass was assessed by plicometry (Sloan and Weir, 1970).All subjects were in good health and had a normal locomotor system. They were either students or academic staff. It is well toemphasize that this constitutes a verylimited sample. Generalization of the results reported in this article to persons whose overall characteristics are not homogeneous with those reported above, would, therefore, be rather presumptuous. During the tests subjects wore minimal clothing and used their everyday shoes, care being taken to avoid extra high heels or too rigid a sole. They were asked to walk successively at what they felt to be a very slow, slow, normal, fast and very fast speed. Subjects were requested to keep their upper limbs flexed in order not to cover the markers placed on the pelvis when walking. This was one drawback of the experimental technique which could not be avoided. It was proved, nevertheless, not to affect the shoulder and head movements to any detectable extent. This was verified by means of comparative tests during which the subjects moved their upper limbs freely. Subjects were also asked to walk looking in front of them so as to avoid erratic movements of the head.
RESULTS A total of 41 tests were taken into consideration for the present study. In order to illustrate thecharacteristics of the strides analysed the relationship between stride length and mean speed of progression (both normalized with respect to stature) is reported in Fig. 3. The parameters of a relevant regression curve proposed by Grieve (1968), the equation of which is reported below, were assessed for each subject and are also reported in Fig. 3.
Table 2. Overall characteristics
Body mass Subject
04
L’
=
60y’cl-8) u.
where L’ is the stride length relative to stature; V’ is the velocity relative to stature (s-l); c( and /r are the regression parameters. These regression parameters were calculated making reference to the tests performed at speeds below approximately 2.4 m s-r. It was observed that above this speed walking traits undergo considerable modification and assume the characteristics typical of race-walking. Only two subjects walked at these high speeds. Results of ten tests relative to one subject (B.C.) only are reported here in a thorough fashion. However, during the discussion, reference will be made to the results obtained on all subjects. In Table 3 the overall characteristics of the tests performed on subject B.C. are summarized. As has been reported in the previous section, the displacement functions of the midpoints between head, shoulders and pelvis markers were determined in sampled form and underwent harmonic analysis. In what follows these midpoints will be referred to as head, shoulder and pelvis points. The results of the harmonic analysis are reported in Figs. 4,5 and 6 up to the fourth harmonic. The time origin, with respect to which harmonic phases were calculated for all the tests, coincided with the heel strike of the left foot. Harmonics of order higher than four showed amplitudes consistently less than 0.5 mm. They were deemed not separable from the experimental errors and, therefore, not significant. When these harmonics were
ofsubjects tested. Trochanteric heights reported and left trochanteric height.
are the mean values ofright
Trochanteric height (% stature)
Stature (m)
Age (years)
1.73
21
8
52.1 51.9
Fat mass (“/ body mass)
G.B.
60
B.C.
72
1.81
32
16
21
8
50.0
E.C.
65
1.76
V.D.
65
1.72
22
14
50.9
A.-f.
70
1.74
24
13
51.7
__~
AURELIOCAPPOZZO
416
alpha 62.4
beta :Z
3:.Z 60:4 59.6
.47 .52 .52
SPEED/STATURE
subject 5:
0
5: AT
0 + x A
Cl/s1
Fig. 3. Relative stride length vs relative speed for all tests. The parameters of the relevant regression curve (equation (6) in the text) are also reported for each subject.
added to the Fourier series expansion the quality of the fit of the input data did not improve, that is, the relevant standard devation assumed an approximately constant value on average at 0.2 mm (Cappozzo et al., 1975). The foregoing was valid at all walking speeds investigated. The rapidity with which a trigonometric series relative to a rhythmic human movement (walking in particular) converges has already been shown and emphasized by Bernstein (1967), Winter et al. (1974), and Lesh et al. (1979). The present results confirm their findings. In order to represent the linear displacement of the head, shoulder and pelvis points the patterns shown in Figs. 7, 8, 9 and IO, were chosen. They are the
Table 3. Characteristics of the walking trials of subject B.C. Stance phase durations are the mean values of right and left stance phase duration
Stride period (s)
Mean speed (m s-t)
Stance duration (% of stride period)
1
1.317
1.19
60.7
2
1.197
1.32
62.4
Test
3
1.048
1.61
61.7
4
1.050
1.63
61.0
5
1.034
1.69
60.8
6
0.977
1.88
59.6
7
0.954
1.99
59.6
8
0.944
2.00
59.4
9
0.889
2.31
58.8
10
0.859
2.35
58.2
trajectories of the designated points in the sagittal, transverse and frontal planes, each referred to a system of reference endowed with a translational motion the velocity of which equals the mean velocity of the relevant point. The three components of the mean velocity were calculated by subtracting the coordinates that these points had when two subsequent heel strikes of the left foot occurred and dividing the result by the stride period. The origin of the axes of the plots in Figs. 7, 8, 9 and 10 was arbitrarily chosen coincident with the average position of the relevant point. These patterns (Lissajous’s figures) were found to be more effective for the objectives of the present study than the representation of each displacement coordinate versus time. They give an immediate picture of the ranges and type of motion and the type of modifications which this latter undergoes moving from one level of the upper part of the body to another and by cause of the change of speed of progression. These same patterns were also reported by Fisher (1900) and by Lamoreux (1971) the former with reference to one speed of progression and the latter with reference to the pelvis only. The results presented in this study are remarkably consistent with those reported by those authors.
THE HARMONIC ANALYSIS
For analytical purposes a ‘stereotype’ pattern of movement of the upper part of the body can be devised. This is characterized by a perfect symmetry of each elementary displacement with respect to the anatomical planes. The longitudinal axis of the trunk and head should therefore move along the vertical and antero-posterior axes with a cycle period corresponding to one step, and along the latero-lateral axis with a cycle period corresponding to one stride (double step)
Analysis of the linear displacement of the head and trunk during walking and with a mirror-image symmetry with respect to the mid-sagittal plane. Any deviation from this behaviour is to be considered not inherent to the locomotor act in its essential form but rather ascribed to some sort of external disturbance. This latter can be an anatomical or functional asymmetry or due to environmental disturbances. The locomotor act can therefore be seen as composed of an intrinsic pure form of movement pattern, the ‘stereotype’, eventually deformed by some extrmsrc cause. The question that arises after this statement is whether the locomotor act can be considered as a superposition of two patterns of movement, i.e. an intrinsic plus an extrinsic pattern. The terms intrinsic and extrinsic are used here in reference to the theoretical. essential locomotor act. Furthermore, the question arises whether the intrinsic pattern of motion is typical of an individual or of a class of individuals. The separation and identification of two such patterns was attempted by means of the harmonic analysis of the displacement functions. The results of this analysis show that this schematization is feasible. Due to the cycles of variation of the intrinsic pattern of motion the harmonic analysis of relevant displacement along the vertical and antero-posterior axes carried out against an interval of time equal to the stride period must exhibit even order harmonics only. For the displacement along the latero-lateral axis only odd-order harmonics are exhibited. These harmonics will be referred to as intrinsic harmonics because they pertain to the intrinsic pattern of motion. Harmonics of different order will be called extrinsic harmonics.
the same type of repeatability already observed at the head and a slightly different trend with speed. At the pelvis, the amplitude exhibited values decreasing from about 20 to about 10mm with the augmentation of speed. The phase showed the same traits of repeatability as at the two upper levels and a trend with speed further modified. These observations applied to all subjects. Third harmonic. Amplitude always showed lower than 1 mm at all three levels, Neither trend with speed nor repeatability could be assessed in the phase values. No confidence could be given to the information carried by the results relative to this harmonic. Fourth harmonic. At the head the amplitude of this harmonic assumed values falling in most cases below 0.5 mm. As could be expected given the influence of the experimental errors, the phase values showed a wide scatter. At the shoulder the amplitude had a magnitude of about 2 mm and could be considered constant with respect to speed. The phase showed repeatability and a clear trend with speed. At the pelvis the amplitude ranged from about 2 mm to about 5 mm and tended to a slight increase with speed. The phase, repeatable both intra-individually and inter-individually. exhibited a well defined trend with respect to speed.
The displacement along the antero-posterior axis is characterized by a first, second and fourth harmonic. The first harmonic (extrinsic, according to the definition given previously) has characteristics that are often unforseeable both for a given subject and interindividually. However, at the pelvis level the first harmonic parameters can exhibit a degree of withinsubject consistency. By contrast, the intrinsic harmonics, i.e. the second and fourth, always exhibit repeatable characteristics both intra-individually and inter-individually. Lutero-lateral
Antero-posterior
displacement
f Fig. 4
417
displacement f Fig. 5 )
i
Firsr harmonic. At the head the amplitude fellin the range 1 to 6 mm in subjects B.C., A.T. and V.D. In subjects E.C. and G.B. it had an amplitude up to 12 mm. It did not show any clear trend with speed. The values of the relevant phase exhibited neither repeatability in similar tests nor trend with speed. This apparently random behaviour should not be ascribed to experimental errors as the amplitude of this harmonic showed relatively large. A partial validation of this statement derives from the evaluation of the error on the harmonic parameters by means of equations (4) and (5). At the shoulders there occurred values of both amplitude and phase similar within expected error to those found at head level. This was true with few exceptions. This circumstance further supports the conclusion that this first harmonic was significant, i.e. it was not an exclusive effect of the experimental error. At the pelvis the pattern changed. The amplitude assumed values that were either similar or slightly less than those found at the two upper levels. But the phase showed a degree of repeatability within similar tests and a trend. though not very coherent, with speed. Results obtained with reference to the iliac spines were readily superposed on those reported in Fig. 4 and confirmed these observations. Results relative to the different subjects exhibited nevertheless great scatter both in phase and amplitude. Second harmonic. At the head, amplitude decreased with the augmentation of speed from about 12 to about 6 mm. Values assumed by the relevant phase were repeatable both inter-individually and intra-individually and showed a constant trend with speed. The scatter of this phase, observed inter-individually, fell within 0.5 rad. At shoulder level, the amplitude was, on average, slightly greater than at the head and showed the same trend with speed. The phase exhibited
First harmonic. At the head the amplitude was in the range 15-40 mm. It showed a slightly decreasing trend with the augmentation ofspeed. The phase exhibited aclear trend with speed and was repeatable both intra-individually and interindividually. At shoulder level the amplitude was in the range 15-40 mm. On average its decrease with the augmentation of speed was more pronounced than at the head. The phase had the same characteristics of repeatability as at the head but it was shifted in all subjects as shown in Fig. 5. At the pelvis the amplitude was in the range lo-35 mm with a marked trend towards decrease as speed augmented. Phase, which showed good repeatability (also inter-individually). underwent a further shift with respect to the one found at the shoulders and the head. Second harmonic. At the head the amplitude exhibited a maximal value of 2 mm. The phase assumed rather scattered values. No sign of coherence was found among the results obtained on different subjects. At the shoulders, observations of the same type were made as at the head. At the pelvis, on average, the amplitude tended to be slightly larger than at the upper levels. The phase showed a definite trend with speed and recurrency within tests on the same subject. Comparison of the relevant values relating to the different subjects showed a wide scatter. Third harmonic. At the head the amplitude was about 3-4 mm with no significant variation with respect to speed. Phase was largely repeatable both inter and intraindividually. At shoulder level much the same observations were made as at head level. At the pelvis the amplitude tended to be lower than at the upper levels and the phase assumed more scattered values. This latter circumstance was very likely due exclusively to the effect of the experimental errors,
AURELIO CAPPOZZO
418
x
HARMONIC
f1
AXIS
III IV
:
:: 1,
+ X
]
2.5 3t
&
u
SPEED
1
+
Cm& I
SHOULDERS
PELVIS n
28
a
d :
0
L
d h
. .
hl
O8
cb”
E 3 18 fa
#::
5! ..a..
A
I
a-+
.
-3r. 1.8
Fig. 4. Amplitude
i-.
p-
5X
*
.
C.;
+ +
+
no 8
;
*
.
l
n
8
Yi
u
*
0 +
+
’
’
1.5
.
.
SPEED
and phase vs speed of progression
pelvis-point
along
u -4-a
2.0 +
+
2: l-
ye u -1 _ I . 0 -2 -
a:
1.5
3k r;: f!
00
*
’
’
2.8
’
.
.+a
2.1
Cm/s I
relative to the displacement the antero-posterior axis.
of head-, shoulder-
and
Analysis
of the linear displacement
of the head and trunk
II 111 IV
RXIS
419
walking
HARMONIC
I Y
during
: +
,I I. s*
X
HERD
SHOULDERS u “E 30 c “d 20
Cl
mn q
: .
0
0
0
? 7 10 P
:
=
9.0 3”
z
2- _
I
if
’
nt
.
’ 1.5
,
da.
g
L 2.0
.
.
L-
X
0
lP
’
.
+
4
X
u
-2 -
-31.0
+ i
0
-0 % -1 c I” . n
+
.
0 ’
’ 1.5
.
X
*
SPEED
’
’ 2.0
*
1
.
’
’
* 2.!
Cm/s 3
PELVIS
-3r
1.0
’
’
’
.
’
.
’
.
.
2.0 SPEED
g. 5. Amplitude
.
1.5
tm/s I
and phase vs speed of progression relative to the displacement pelvis-point along the later+lateral axis.
.
* 2.!
of head-. shoulder-
and
420
AURELIOCAPPOZZO
Fourrh harmonic. In most tests the amplitude of this harmonic was lower than 0.5 mm at all three levels. No confidence could therefore be given to the data generated.
The displacement along the latero-lateral axis is characterized by a first harmonic significantly greater in amplitude than the higher-order harmonics. This harmonic has repeatable characteristics both interand intra-individually. Much the same can be said with respect to the third harmonic, although its amplitude is often so small as to be masked by the experimental error. With reference to latero-lateral displacement, first and third harmonics are intrinsic to the motion pattern and exhibit the same regular behaviour already observed for the intrinsic harmonics of the antero-posterior displacement. The only extrinsic harmonic which can be revealed (though with some uncertainty) with the present experimental method is the second harmonic. It has rather unpredictable characteristics. Only at the pelvis does it show withinsubject systematic characteristics.
found that would supply a categorical justification to this observation. The extrinsic harmonics exhibit large intra- and inter-individual differences. However, a degree of within-subject consistency can be found in some cases with regard to the first harmonic coefficients. Subjects G.B. and EC. exhibited this consistency. It was interesting to note that these two subjects had the largest difference between right and left trochanteric height. The former subject had the right higher than the left trochanter, while the opposite applied to the latter subject. The phases of the first harmonic of the vertical displacement relative to these two subjects were in counterphase. It would, therefore,
seem not too presumptuous to infer the existence of a causeeffect relationship between the regular behaviour of this first harmonic coefficients and the aforementioned anatomical asymmetry.
Vertical displacement (Fig, 6) First harmonic. At the head the amplitude had a maximal
value of 4 mm except in subject G.B. where it reached 8 mm. No clear trend of variation with speed was noticed. Phases exhibited a wide scatter both inter- and intra-individually. This was not true for subjects G.B. and EC. who exhibited repeatable and regular values of the phase of this harmonic although with large individual differences. At the shoulders and pelvis much the same observations were made as at the head. Second harmonic. At the head the amplitude ranged from 16 to 35 mm and exhibited a clear trend to augment with speed. However at speeds beyond about 2.4 m s-r an abrupt fall in the value of this amplitude was observed to occur. This is a further indication that at the highest speeds of progression a definite change in gait occurs. The phase was uniformly constant with respect to speed and remarkably repeatable intra-individually. At shoulder and pelvis level the amplitude tended to be slightly greater than at head level but exhibited the same trend versus speed. Phases reproduced within expected error those observed at the head. Third harmonic. At all three levels this harmonic had in most tests an amplitude below 2 mm. When the amplitude did not assume values below 0.5 mm, the occurrence of similar phase values at the three levels was observed. This phase presented a great scatter both for one subject and among different subjects. Fourth harmonic. This harmonic occurred with amplitudes and phases that were similar within expected error at all three levels and for four subiects out of five. Subiect G.B. had different values of phase”although these were consistent inter se. The amplitude, in the range 2-4 mm, did not show any clear trend of variation with speed. Phase, on the contrary, underwent a systematic shift as speed augmented.
The displacement in the vertical direction is characterized by second and fourth harmonics which are intrinsic, and first and third harmonics which are extrinsic. The intrinsic harmonics show a systematic behaviour both intra- and inter-individually similar to the antero-posterior and that observed for latero-lateral displacements. One exception was found to this statement. It concerned the fourth harmonic coefficients for subject G.B. No objective datum was
THE LISSAJOUS’SFIGURES The plots shown in Figs. 7, 8, 9 and 10 permit visualization in reasonably immediate fashion of the following aspects of the pattern of motion of trunk and head. (1) Harmoniousness of the movement. The trajectories shown in Figs. 7,8,9 and 10 have a very ‘simple’ and smooth aspect. This is an indication of the regularity and harmoniousness of the movement. The relevant evidence is obviously not complete as in these plots reference to time is lost. (2) Repeatability among successive walking steps. In the sagittal plane pelvis-, shoulder- and head-points trace loops, one during each step. The extent to which these loops differ supplies information on the repeatability of successive steps. In the transverse and frontal planes the same type of information is given by the extent to which the pathways have a mirror-image symmetry with respect to the x axis and the z axis respectively. (3) Maximal deviations of head, shoulder and pelvis points from mean position. These can be easily inferred from the plots in Figs 7,8,9 and 10. It can be observed that along the antero-posterior axis the excursions about the mean position of the head and shoulder are smaller than that of the pelvis point. Along the latero-lateral axis the contrary applies, the more so the higher the speed of progression. The vertical excursion almost identical at the three considered levels increases as speed augments to about 2.4m s-r. At higher speeds this excursion decreases considerably. (4) Relative movement between pelvis, shoulders and head. Differences in the maximal excursions about the mean position of these points have already been pointed out above. The form of the trajectories also varies going from one level to another. This phenomenon becomes more evident as speed increases.
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Fig. 7. Lissajous’s figures of the displacement of head-, shoulder- and pelvis-point in the sagittal, transverse and frontal planes during walking at 1.19 m s-l. (Ihs = left heel strike; Ito = left toe off; rhs = right heel strike; rto = right toe off).
CONCLUSIONS
The experimental results presented in this study permit the following deductions. (1) The periodic portion of the linear displacement of the longitudinal axis of the upper portion of the human body during normal level walking is described with considerable refinement, within one walking cycle, by the first four harmonics along the antero-posterior and vertical directions and by the first three harmonics along the later*lateral direction. This is true at all walking speeds investigated. (2) Harmonic analysis of the linear displacement of the longitudinal axis of the upper part of the body made possible the identification of an intrinsic and an extrinsic pattern of motion, according to the definitions given in a previous section. The qualities of these two patterns and thus their functional characterization emerge clearly from the experimental information available. The intrinsic pattern of motion, composed of second and fourth harmonics for the displacements along the
antero-posterior and vertical axes, and of first and third harmonics for the latero-lateral displacement, has, at a certain speed of progression, very steady characteristics both within- and between-subjects. From the foregoing it can be inferred that the intrinsic pattern of movement is an individual stereotype and that the same stereotype can apply to a class of individuals. Because of the small number of subjects from which experimental evidence has been adduced, it is not possible to precisely define the characteristics of this class of individuals. However, some related experiments being conducted on young women in the author’s laboratory seem to give re,rults (Lombardi, 1980) largely consistent with those reported here. This suggests, though not exclusively, that the same stereotype could apply to a quite large and nonhomogeneousclass ofindividuals and that the intrinsic pattern of movement previously described is one possible standard of average normal walking. The extrinsic pattern of motion, composed of first and third harmonics for the antero-posterior and vertical displacement and second harmonic for the
Analysis of the linear displacement of the head and trunk during walking
Bernstein’s (1967) concept, ‘biodynamical tissue’. (3) The longitudinal axis of the upper part of the body does not move rigidly with the pelvis. Along the antero-posterior axis the pelvis undergoes larger excursions with respect to an observer moving at the relevant mean speed of progression than shoulders and head. This is true at all speeds of progression. The maximal excursion of the head and shoulder points with respect to the pelvis point along the antero-posterior axis is approximately 20 mm with minor changes attributable to variations in speed. Along the latero-lateral axis head and shoulder points undergo larger excursions than the pelvis point. The higher the speed of progression the larger becomes the displacement of the higher levels with respect to the pelvis. The overall excursion of the head point with respect to the pelvis point ranges, along the later+lateral axis, from approximately 20 to 65 mm. Only a slight variation of the vertical displacement is observable going from one level to another. (4) The pattern of movement of the upper part of the body changes with speed; that is, the augmentation of speed of progression is not achieved by means of a simple shrinkage of the time axis. Nor does it cause a
iatero-lateral displacement, exhibited on the contrary a high degree of variability between subjects and in most cases within-subject. This was invariably true with reference to the head and shoulder-points along the antero-posterior and latero-lateral axes. At the pelvis along all three directions, and at the head and shoulders along the vertical direction, a regular behaviour was, on the contrary, observed in a proportion of cases. This regularity occurred strictly withinsubject. One possible explanation of this phenomenon can be recognized in the nature of the possible causes that engender the extrinsic pattern of motion, as pointed out previously. Anatomical or functional asymmetries, though in the normal range, can be expected to have repeatable effects on the pattern of motion. Within-subject regularity can therefore be ascribed to these causes. Environmental disturbances, or other subject-born disturbances are, on the contrary, likely to have irregular effects. The nonrepeatable and non-predictable part of the extrinsic pattern of motion is therefore to be ascribed to these causes. The foregoing is one possible descriptive base for investigations into the ‘morphology’ of walking as, in
Speed
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Speed
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simple variation of the amplitudes of the displacement functions. As was observed in analysing the relevant harmonic coefficients, the phase of some intrinsic harmonics shifted with speed. These shifts cause a modification of the actual shape of the displacement functions and, as evidenced previously with reference to Lissajous’s figures, of the trajectories as well. Understanding of the fundamental reasons for these modifications of the pattern of motion requires reference to the dynamics as well as to the kinematics of walking. Therefore no explanation of these will be attempted here. (5) Strong emphasis has been given to the experimental and analytical methods used. The results of this study permitted accurate assessment of the resolution of these methods both from the quantitative and qualitative standpoints. In synthesis, it was observed that the error on the estimation of the three coordinates of the landmarks can be substantial, i.e. in the order of several millimeters. But the larger portion of this error will so operate as to affect the coordinates of adjacent points in a similar way. This implies that the displacement functions of the landmarks are only globally deformed (their time derivatives are not
heavily affected). As emphasized previously, it was, in fact, possible for harmonics of the displacement functions to be classified as significant within an amplitude of only 1 mm.
Acknowledgements-Data reduction and computing was in part carried out at the Bioengineering Unit, Strathclyde University, Glasgow (Great Britain), with the kind permission of Prof. J. P. Paul. The author’s sojourns in Glasgow were financed by The British Council. The author expresses his gratitude to Prof. J. P. Paul, to Prof. S. Cerquiglini, and to the designated referees of this Journal for their most helpful criticism of the manuscript.
REFERENCES Andriacchi, T. P., Ogle, J. A. and Galante, G. 0. (1977) Walking speed as a basis for normal and abnormal gait measurements. J. Biomech. 10, 261. Bernstein, N. (1967) The coordination and regulationof movements. Pergamon Press Ltd. Braune, C. W. and Fisher, 0. (1895) Der Gang des Menshen I. Abh. Math. Phys. Cl. Kiln. Siichs. Ges. Wissensch. 21, 151.
Analysis
Speed
of the linear displacement
of the head and trunk during
2.31mi.s
HEAD
rhso
lhso
rton
Itoa
SHOULDERS
walking
425
PELVIS
tdown
)down
lid own
Y+left
Y left
Y left t
I
tdown
1 div=lQmm Fig. 10. Lissajous’s
figures of the displacement of head-, shoulder- and pelvis-point and frontal planes during walking at 2.31 m s-‘.
Cappozzo, A., Leo, T. and Pedotti, A. (1975) A general computing method for the analysis of human locomotion. J. Biomech. 8, 307. Cappozzo, A., Figura, F., Leo, T. and Marchetti, M. (1978a) Movements and mechanical energy changes in the upper part of the human body during walking. In: Biomechanics VI-A, (E. Asmussen and K. Jorgensen eds), pp. 272, University Park Press, Baltimore. Cappozzo, A., Figura, F., Leo, T. and Marchetti, M. (1978b) An approach to human locomotion studies. Acta Med. Rom. 16, 540. Dierckx, P. and Piessens, R. (1977) Calculation of Fourier coefficients of discrete functions using cubic splines. Report TW32, Applied Math. and Progr. Division, K. U. Leuven (Belgium), June 1977. Fisher, 0. (1899-1904) Der Gang des Menschen. Abh. Math. Phys. CI. Kiln. Siichs. Ges. Wissensch., II-25 (1900); l-130. III-26(1901); 85-170. IV-26 (1901);469-556. V-28 (1904); 319-418. VI-28 (1904); 531-617. Grieve, D. W. (1968) Gait patterns and the speed of walking. &o-M&. Eng. 3, 119. Lamoreux, L. W. (1971) Kinematic measurements in the study of human walking. Bull. Prosthetics Res. 10-15, 3. Lesh, M. D., Mansour, J. M. and Simon, S. R. (1979) A gait analysis subsystem for smoothing and differentiation of human motion data. Trans. ASME J. Biomech. Engng. 101, 205. Lombardi, D. (1980) Biomeccanica della deambulazione nelle donne. Doctorate Thesis, University of Rome, Italy.
in the sagittal, transverse
McLaughlin, T. M., Dillman, C. J. and Lardner, T. J. (1977) Biomechanical analysis with cubic splines. Res.Q, 48, 569. Murray, M. P., Kory, R. C., Clarkson, B. H. and Sepic, S. B. (1966) Comparison of free and fast speed walking patterns of normal men. Am. J. Phys. Med. 45, 8. Paul, J. P. (1970) The effect of walking speed on the force actions transmitted at the hip and kneejoints. Proc. R. Sot. Med. 63, 200. Saunders, J. B. DeCM., Inman. V. T. and Eberhart, H. D. (1953) The major determinants in normal and pathological gait. J. Bone Jnt. Surg. 35A, 543. Sloan, A. and Weir, J. (1970) Nomograms for prediction of body density and total body fat from skinfold measurements. J. appl. Physiol. 28, 221. Soudan, K. and Dierckx, P. (1979) Calculation of derivatives and Fourier coefficients of human motion data, while using spline functions. J. Biomech. 12, 21. Waters, R. L., Morris, J. and Perry, J. (1973) Translational motion of the head and trunk during normal walking. J. Biomech. 6, 167. Winter, D. A., Sidwall, H. G. and Hobson. D. A. (1974) Measurements and reduction of noise in kinematics of locomotion. J. Biomech. 7, 157. Wolf, P. R. (1974) Elements of photogrammetr,v. McGrawHill, New York. Wood, G. A. and Jennings, L. S. (1979) On the use of spline functions for data smoothing. J. Biomech. 12, 477. Zernicke, R. F., Caldwell, G. and Roberts, E. M. (1976) Fitting biomechanical data with cubic spline functions. Res.Q. 47, 9.