Extracting gravity line displacement from stabilographic recordings

ELSEVIER

Gait & Posture 6 (I 997) 27 38

Extracting gravity line displacement from stabilographic recordings

Received‘9 August 1995;accepted 6 August19%

Abstract

Three algorithmsfor determininggravity line (GL) location from center of pressure(COP) and horizontal force {F,) recordings are suggesled.The afgorithms are designedto study upright standing postureand are basedon the following premises:(a) the font(feet) is 2~solid body and doesnot move, (b) the axis of rotation of the anklejoint doesnot dispface,and (c) the forcesand momentsactmg at the ankle joint can be reduced to a resultant force through the axis of rotation and i? force couple. The algorithmsare: ( 1) Single-pendululum algorithm.The human body is modeledasan inverted pendulumoscillatingaround the ankle joint(s). Becausethe oscillationsare small,the equationsof movementcan be linearized and the horizontal position of the gravity line (GLP) located. (2) Trend-eradication algorithm. The second integral of the horizontal force calculated and an initial integration constant (.?, the initial horizontal GL velocity) is found from the trend of the displacementcurve, (3) Zero-poitrr-toxro-point integration. When the horizontal force is zero, the horizontal position of the gravity line (GLP) passesthrough the COP. The instantaneousGLP and its velocity are determinedby integrating F, from one zero point to another zctropomt. k7 I’N7 Elscvier ScienceB.V. ki*,r~rorc/~~: Stabilography: Posturography:Gravity line: Center of pressure;Center of gravity

Since the pioneer study of Hellebrandt [l] posturographic recordings from force platforms are commonly used for examining a person’s ability to maintain balance [2--71. As a representative measure of balance, the center of pressure (COP), or the resulting torque on the ground plane, is usually analyzed [8-13). A variety of methods have been employed to analyze COP 114-161. The methods range from specially designed techniques like the length of the path traveled by the COP [17,18] to routine procedures of signal processing such as parameters of statistical distribution (means, standard deviations [14,19,20]), ranges and areas of variation [12,13,2 I 231, velocities of COP migration [21,23,24], transfer f-unctions 1257,spectral characteristics [25-291, and autocorrelation and autoregression analysis [30-361. During the last few years several new approaches have been used, including evolutionary spectral analysis * Corresponding author. Tel.:

t I 814

8653445;

fax:

+ 1

814

X657441, 0966-6362.97, S:17.004’ 1947 Elsevier PII Sl~‘)hh-6~h?i96101 lo! -0

Science

B.V. All rightsreserved

[37], fractal dimensions [38,39], Grassberger-Procaccia coefficients developed to study dimensionality of chaotic processes [40-421, random walk analysis [40.41,43- -451. and time-to-contact measures 146,471. For review, see also Carrol and Freedman [48] and Prieto et al. [49]. Some of these procedures have been used for purely descriptive purposes, though, in many cases the researchers assumed that the COP directly portrays body displacement (‘body sway’) and :or the efficacy of the neural control of upright stance. Center of pressure.however, is an indirect measureof body sway. As early as 1959 Thomas and Whitney [SO] reported that the COP measured by the force platform and the horizontal position of the gravi[y line (GLP) did not coincide. The gravity line is 21 vertical line passing through the center of gravity. COG. of the body. This difference has been confirmed by Murray et al. [Sl] who estimated the GLP from filming. Winter and colleagues have provided a tremendous body of research on both COP and COG characteristics in o

28

D.L.

King,

V.M.

Zatsior.sk~

variety of tasks [.52-55,16]. In particular, Eng and Winter [52] compared COP and GLP displacement curves during a balance task which required raising a horizontally oriented light weight rod held between the hands and found that the two curves were very different. The COP and the GLP or COG variables can differ in amplitude, phase, and frequency [50,56-601. Only in pure static cases is the COP identical to the GLP [l]. Stabilographic recordings contain not only a static component depending solely on GLP but also a dynamic component due to inertial forces [50,62-651. The higher the frequency of body oscillation, the greater the acceleration, and the larger the contribution of inertial forces to the stabilogram. Even during quiet stance, the contribution of the acceleration terms is substantial [53,66]. According to estimates by Gurfinkel [63], at a frequency of 0.2 Hz the forces due to body acceleration contribute 10% to the stabilographic recordings; this contribution increases to about 50% at a frequency of 0.5 Hz and at a frequency larger than 1 Hz the stabilographic recordings reflect mainly inertial forces. Spectral density analysis [67] have revealed that the mean frequency during quiet standing is approximately 0.7 Hz. At this frequency, the inertial forces may determine more than 50% of the COP displacement. Furthermore, this could be important in many pathologies where the mean frequency of the COP oscillation increases to 1 Hz or even higher. For example, increased power spectrum values have been reported around a frequency of 1 Hz in patients suffering from Friedrech’s ataxia [68] and tubes dorsalis [69], as well as in patients with local lesions of the brain [70]. Similar findings have been reported for normal people whose leg muscle afferents were blocked by ischemia [27,71,72]. ‘The one-hertz phenomenon’ [73] is thought to be due to vestibular reflexes [68,69,73]. Higher frequencies - between 2 and 4 Hz - have been reported for patients with anterior lobe atrophy [68]. At such high frequencies, the COP does not represent GLP displacement. The horizontal ground reaction force may provide more useful information than the COP [7,16,52,58,75771. Horizontal force is proportional to the horizontal acceleration of the COG of the body and after normalization by body mass its second integral is equal to the displacement of the GLP. Hence, if body sway is an object of interest, analyzing horizontal force rather than COP appears attractive. While this COG-horizontal force relationship is well established it has not been used widely in practical research because the initial constants of integration (position and velocity) are usually not known. Shimba [78] proposed estimating the initial integration constants through a curve fit of two different functions. Specifically the method requires knowing S(r), which is the second integral of the horizontal

I Gait

& Posturr

6 (1997)

27-3X

ground reaction force, and E(t) = xP + zJ;oX]FOz), where x,, is the x-coordinate of the point of application of the ground reaction force, zG is the vertical component of the position vector of the COG of the body, and F,, and F,, are the horizontal and vertical components of the ground reaction force. According to Shimba [78] (page 56) ‘the curve fitting of S(t) with E(t)’ is done using ‘least squares with three unknowns’. Note that this approach requires knowledge of both the horizontal and vertical forces. Benda et al. [79] estimated the GL migration by filtering the COP time history data. This method does not require the knowledge of integration constants and is based on the presumption that the COP fluctuates about the GLP and contains higher frequency components. Though the postulations are rational, the method is unable to accommodate phase differences between the GLP and COP time histories and, thus, can only be applied when the frequency of body sway is low. Also, the outcome of this methods depends on the filtering cutoff frequency which is selected arbitrarily. The aim of this paper is to show several different ways of estimating GL location from force platform recordings.

1. Main model

For a standing person, the following assumptions are made: (1) the feet do not move and are considered solid bodies; (2) the axis of ankle joint rotation is fixed (this assumption is confirmed by the studies of Inman [80] and Bogert et al. [Sl]); and (3) all of the forces and moments acting at the ankle joint can be replaced by a resultant force acting through the joint axis of rotation and a couple. The forces and couples acting on the foot in a single-pendulum model of a standing person are illustrated in Fig. 1. The origin of the reference system is the intersection of the supporting surface and the vertical line passing through the ankle joint center. The following forces and couples are acting on the foot: Fz,,, CF,,,, and T, are the forces and couple (torque) at the ankle joint; F,,,, F.+, and T, are the external forces and external torque; and IV’ is the weight of the foot. Because the foot system is in equilibrium, the sum of the forces and the sum of the moments are zero. The equations in scalar form (sagittal projection) are: r;,,, = - Fx,, Fz,e = - ( W- + F,,,) M,=F3,exd,+T,+F,,,xh,+

Wfxdf

(1)

The horizontal ground reaction force, F& and the vertical force acting at the ankle joint, Fz,,, pass through the origin, 0, and do not contribute to the moment about this point. The moment due to the

D.L.

King,

V.M.

Zatsiorskj,

weight of the foot is constant. Thus the equation COP location can be written as:

for

(2)

During standing, ankle joint height, h,, is constant and fluctuations in the vertical forces are so insignificant that they can be confused with ballistocardiographic (501 signals. Hence, ankle joint torque and horizontal force are the primary determinants of COP location. Since F, r = - F, a the horizontal force acting at the ankle joint(s). F,.,,, is easily determined. Thus, the con-

#‘Gait & Pmture

6 f1997)

-y

,7-- 38

tribution of the ankle joint torque. I’;,. 1s &he only undetermined variable. According to classical mechanics. the sum of the moments about a fixed point A of ah external forces acting on a system is equal to the time rate of change of the total angular momentum of the system about point A. Consider rotation around the ankle ,joint center fii,, = 1 M,, = mgd,y.,, + 7:,

or

7, --: B,

- /iI,~d,..(,

(3)

where tiU is the time rate of change of the angular momentum about the ankle joint center. IV!,: is a moment taken with respect to the ankle joint, and dp,<,is the moment arm of the force of gravity applied at the COG of the system. From Eq. (3) rt follows that the ankle joint torque is an algebraic turn of a ‘static’ moment. rrz,qd+,, which is a product of the system‘s weight times the corresponding moment arm, and a ‘kinetic’ moment. I?<,, which is numerically equal to the time rate of the change of the anguiar momentum of the system, Only the static moment is of mtcrest, however. The difficulties encountered by the presence of & in Eq. (3) need to be overcome. In essence. this is the premise of the suggested algorithms.

2. Algorithms

Buckground. For this algorithm, t&o additional assumptions are made: (1) joint motion is only in the ankle joint(s) and (2) body sway is small. Thus, the body is modeled as an inverted pendulum (Fig. 1). Similar single-pendulum models have been used previously by other researchers [63,6X75,82 -841. Small magnitudes of body sway permit linearizing the equations - instead of sin B the value of (1 can he used. The model can be described by the fol?owing equation: T, = I$ - mC cos cl + (mi - mgr sin H )

(J!

where I is the moment of inertia of the ‘whole body minus the foot(feet)’ system with respect to its center of mass, I’ is the distance from the ankle to the system’s center of mass, and 0, is the angle of the longitudinal axis of the body to the vertical. Eq. (4) can be linearized and written as: T, = I.$ - mgr0

-d,-

Fzfe -c-+ Fig

1. Single-pendulum

model.

(5)

where I<, is the moment of inertia with respect to the ankle joint(s), I, - I+ mr’. Eq. (5) is identical to Eq. (3), fit, = I,(?. Since angular and linear acceleriltions are interrelated for a single pendulum and the angle (1 is small for the system under discussion, ii can be found from the known values of .i:.

30

D.L.

King.

V.M.

Zufsiorsky

Algorithm: Step 1. Determine the moment generated by the weight of the foot. According to Zatsiorsky and Seluyanov [85], the mass of the foot is 1.371 + 0.155% of the body mass, and the foot’s center of mass is located at 44.1 +_3.6% of foot length. Using these coefficients and measuring the horizontal distance of the ankle joint center to the heel, the moment arm of the weight of the foot, dr- is calculated and its moment is determined. However, the moment is small, on the order of 0.99 Nm, and can be neglected. If higher accuracy is required, the calculated value of the moment can be included in Eq. (1) and taken into account in Eq. (2). Step 2. Calculate the torque in the ankle joint(s). The moment generated by the horizontal force acting at the ankle joint(s) is subtracted from the moment of the ground reaction force (see Eq. (2)).

T, = F,!, x d, - F,+ x h,

! Gait & Posture

6 (1997)

-77-38

2.2. Second algorithm technique

(GL-2): Trend-eradication

Background. The GLP displacement, x,,,(t), is the second integral of the horizontal acceleration of the COG. Considering only the anterior-posterior component, it follows that:

Initial constants for integration, x(t,) and i(t,), are, however, not known. If an arbitrary value, for instance zero, of x(t,) were selected there would be a constant shift of the calculated GLP with respect to the actual GLP. This would affect the mean value of the GLP. Typically, though, the mean is not an important variable in posturographic studies and any value within the range of the COG travel can be used for x(t,). Arbitrary selection of i(to) will result in an erroneous linear ‘drift’ of the calculated GLP. This drift, (ia,,., &&to)). t, where &,(tJ is an arbitrary value of the initial velocity constant and Irea, is the real value of the initial velocity, would accumulate over time and bring the estimated GLP far from the real area of GLP travel. The rate of this linear trend of the GLP displacement is used as an estimation of the initial velocity, i(r,). This method is based on the assumption that the GLP migration over the period of observation, T, is stationary. It was assumed that when the linear trend is calculated over a large time interval error due to small differences between the initial and final CL positions does not affect the estimation of the initial velocity?

Step 3. Calculate the static moment in the ankle joint(s). Eq. (5) is used. The product I,8 is calculated and then subtracted from the ankle joint torque, T,. Subsrep 2. The angular acceleration # is estimated as T/r. The vertical component of the acceleration of the center of gravity is being disregarded. The distance from the ankle joint(s) to the system’s COG, Y, is estimated. According to data from Kozyrev [86], on average, the whole body COG is located at 57.1 + 0.127% (2 + m) of body height in males and 55.9 + 0.106% of body height in females. The height of the foot’s COG is approximately 40% of ankle joint height, h,, and the two feet together are approximately 2.75% of body weight. Using these values, the location of the COG of the ‘whole body minus the foot (feet)’ system is 58.71% of body height for males and 57.48% of body height for females. The difference, Y= h, - h,, is now calculated. Substep 2. The moment of inertia, I,, is estimated. Segmental moments of inertia are determined from regression equations based on body height and mass [85]. Using segment lengths determined from body height [87] and segment COG locations determined from regression equations [85], the moment of inertia of the ‘whole body minus the foot (feet)’ system is calculated. Accuracy can be improved by measuring individual body segment lengths. Substep 3. From Eq. (4). the static moment, mgr /3, is estimated as

Algorithm: Step 1. Calculate the second integral of the horizontal acceleration, i(t) = F,(t)/m, using n(t,) = 0 as the arbitrary initial velocity value. Stop integrating when t is large, not less than 30 s. To be consistent start and stop at peak values of COP. The horizontal acceleration must fluctuate about zero. If the mean of Z(t) is not zero, either due to a short period of observation or asymmetric error of the force plate, an accumulating error of the GLP during integration will accrue. Step 2. Estimate i(t,) as a regression coefficient (linear trend) of the time series. Step 3. Determine the time course of the GLP migration using the estimated i(t,) value as the integration constant.

mgrB = T, - I,#

2.3. Third algorithm integration

(7)

Since the product mgr is constant, the moment depends on the angle 8 alone and therefore represents the GLP.

i-(&J.

(GL-3): Zero-point-to-zero-point

Background. This algorithm is based on premises that when F,= 0, the COP and the GLP coincide. This is due to the assumption that at the instant when the

horizontal force, F,. is zero. the kinetic term of the ankle joint torque is zero. A/gorithrn: Step 1. Determine the location of the COP at the instant when the horizontal ground reaction force is zero. .~,,,(r~,)lF, = 0. When using digital computers. F, is almost never exactly zero. A threshold range, + fi, around zero must be used to determine the zero crossing, - rS
3. Experimental methods

Prior to the stabilographic experiments the distance from the ankle joint center (talocrural) to the ground, h,,. was measured. The axis of rotation of this joint runs just distal to the tips of the malleoli [80]. Viewing from medial to lateral. the axis is directed posteriorly (on average 6” as reported by Inman [SO]; 6.8 ~fr.8.1” according to Bogert et al. [81]) and inclined downward (8“ -Inman [80]: 7.0 * 5.4” -~- Bogert et al. [Sl]). Ankle height. Iz,, was determined as the mean height of the medial and lateral malleoli. A Bertec force platform (model 4060s Bertec Inc., Worthington, OH, USA) was used for data collection. In different experiments the subjects were asked either to stand quietly with their eyes open or closed, or to perform intentionally exaggerated body sway. The sway was performed at different frequencies around either

the ankle joints (single-pendulum movement pattern. the ankle strategy [10,74]), or both the hip and ankle joints (double-pendulum movemrpr pattern, the hip strategy [10,74]). The vertical and horizontal i;lntrrior-postcric,r! ground reaction forces, as well as COP. were usccl for the analyses. Data were collected on II -186~(36persona! computer with a 12 bit National Insiruments model AT-M lo-64F-5 data acquisition hozir-d (National instruments Corporation, Dallas. T?i. l’S,4! kit a sam-. pling frequency of 200 Hz for 30 s. Spvclai \otiware was developed using the Lab-View softwitr:: package (N:\tional Instruments Corpor;llion) ,f;lr ci;ll:i ~.~~ilti~tion ,lnd analysis.

4. Results and discussion

Example results of slow intentional ankle sway. t&t intentional hip sway. and quiet standing eyes closed are presented in Fig. 2. Fig. 3 and Fig. 4. respectively. The coefficients of correlation between the estimations from the three different algorithms are grearcr than 0.95 fol both slow ankle sway and quiet standing eyes closed. During the fast hip sway. however, the GL-i is quite different from the GL-2 and GL-3 merhodx. Due to the low frequency
32

D.L.

King,

V.M.

2atsiorsk.v

/ Gait & Posture

6 (1997)

27-38

Time (set)

z .E 80

100

-

50

--

0

Time (set) 0

-50

t-

Time (set)

Fig. 2. Slow ankle sway (frequency of sway was approximately 0.3 Hz). All of the three methods are in relatively good agreement with each other. The GL migrates strictly in phase with the COP and out of phase with the horizontal force. The coefficients of correlation are: F, versus GL-1, r = - 0.88; F, versus GL-2, r = - 0.87; F, versus GL-3, r = - 0.85; GL-1 versus GL-2, r = 0.97; GL-1 versus GL-3, r = 0.96; GL-2 versus GL-3, r = 0.95. All of the correlation coefficients in this figure and Figs. 3 and 4 are for 30-s trials. The data are from the same subject (male, height 162 cm, weight 70 kg; during the experiment the subject wore soft walking shoes).

COG is oriented vertically and is not accelerating horizontally. In general, the acceleration of the COG can be

represented as both radial (centripetal) and tangential accelerations. Since the angular velocity of the COG is

3CG

I 3

Tme (ser

Fig, 3. Fast hip sway (approximately 4.0 Hz). The correlation between GL-2 and GL-3 is I = 0.75. The correlations between COP and OLP range from ( - 0.6, COP versus GL-I, and 0.74, COP versus GL-2). The large horizontal forces influenced the high correlation between F, and GL,-1 (I = 0.99).

small, the centripetal force is negligible. Hence, F, is essentially an account of the tangential acceleration of the COG. Due to the small inclination angle, B, of the COG from vertical, the magnitude of the tangential

acceleration of the COG and its horizontal projection practically coincide. Theoretically, the analysis of GLP travel rather than COP displacement seems attractive, especially in popu-

34

10

Time (set)

Time (see)

19 -z 6 ';

0

Time (set)

r; -10

-20 20 10 z E ?

0

:__i\,j

Time (set)

ti -10 -20 20

-.

10 -z zC-Y

0

Time (set)

i; -10 -20

Fig. 4. Quiet standing eyes closed. The GL-1, GL-2, and GL-3 estimations follow similar paths, r 2 0.96, and are highly correlated with COP (r 2 0.94).

with high frequency of body sway. In these conditions, the COP trajectory does not resemble the trajectory of the GLP and cannot be used for estimating body sway. Center of pressure and GLP have not been used concurrently to study representative groups and have not been compared systematically with either lations

parametric (mean, standard deviation, etc.) or nonparametric methods (spectral power, etc.) methods. Depending on the outcome of these comparisons, the attitudes toward the analysis of the GLP instead of the COP, will be different. If analysis of GLP provides additional information that cannot be gained with more

Fig. 5. Determining zero crossings at different sampling frequencies with different thresholds. The graphs are a one tecond portion of a horizontal force time history during a 30 s quiet standing trial. The black squares represent zero crossings and the horizontal hncs short the different threshold ranges.

traditional stabilographic methods, the suggested techniques may be well received. At this time, however, the real merit of the suggested algorithms remains to be investigated.

5. Nomenclature

Vectors are printed in bold O-XYZ

an absolute system of coordinates with the origin, 0, at the intersection of the supporting surface and the frontal plane passing through the ankle joint center force and couple (torque) at the ankle joint(s) ground reaction force and torque (external force and torque) weight of the foot horizontal and vertical components of the external force horizontal and vertical components of the force acting at the ankle joint(s) the moment of the resultant Force about the origin of coordinates 0 the height of the ankle joint COP location along the X-axis, the distance from the COG of the foot to the Z-axis (moment arm) mass velocity and acceleration along the

X-axis. acceleration aLong the %aXiS

time rate of change of the angular momentum about the irnkle joint center the moment arm of thz gravity force of the system ‘the whole body except the footifeet’ u-irh respect to the ankle joint the distance from the ankle to the system’s center of mass the body inclination; the angle of the longitudinal axis of the body to the vertical :mgular acceleration moment of inertia 1ime the horizontal location t)f the GL, and it’s velocity at / = 0 lrariables of integration horizontal location of COP and second integral of horizontal acceleration

arbitrary initial velocity integration constant and real initial veLocit> constant ‘under condition that’ zero range threshold

Acknowledgements

The authors thank Drs. R.(.‘. Nelson.

X4.1,. L,atash

36

D.L.

King.

V.M.

Zarsiorsk~

D.A. Rosenbaum and B.V. Lazareva for helpful comments on an early draft of the manuscript.

Appendix A. Determining

zero crossing points

With digital signals the horzontal force is rarely exactly zero. To determine instances when F, = 0, it is necessary to use a threshold range around zero and estimate F, = 0 by 6 < F, < 6, where + b is the threshold. The threshold must be chosen considering both sampling frequency and frequency of oscillaion. Improper selection of the zero range threshold can cause true zero values to be missed or non zero values to be included. For example, if the threshold is too large the GLP and COP will be forced to coincide in places when J’, is truly not zero. During swaying tasks, F, clearly passes through zero and an appropriate threshold is easily determined. During quiet standing, though, F, hovers about zero and is rarely greater than + 1 N. Fig. 5 illustrates that it is possible to determine zero crossings during quiet standing by adjusting the threshold to the sampling frequency. At a frequency of 120 Hz and a threshold hold of + 0.2 N, there are many false zero crossings. In some instances, the signal quickly changes from positive to negative (or vice versa) so that F, is less than - 0.05 N at sampling time ti and is greater than 0.5 N at the next sampling ti+ I. As a result, these zero crossing points would not be recognized by the computer. At 30 Hz and a threshold of k 0.2 N, the zero crossings have been reasonably determined. At a threshold of k 0.05 N, however, all zero crossings have been missed. These examples represent extremes, however, it is apparent that by adjusting the zero-crossing threshold to the sampling frequency it is possible to determine zero crossings during quiet standing. References [II Hellebrandt FA. Standing

as a geotropic reflex. AM J Physiol 1938;121:471-474. PI Ishida A, Imai S. Responses of the posture-control system to pseudorandom acceleration disturbances. Med Bio Eng Comput 1980;18:433-438. 131 Bourassa P, Therrien R. A force platform for the evaluation of human locomotion and posture. In: Matsui H and Kobayashi K, eds. Biomechanics VIII-A. Champaign, IL: Human Kinetics, 1983:582-588. [41 Peeters H, Breslau E, Mol J, Caberg H. Analysis of posturographic measurements on children. Med Bio Eng Comput 1984;22:317-321. [51 Anderson D, Reschke M, Homick J, Werness S. Dynamic posture analysis of Spacelab- crew members. Exp Brain Res 1986;64:380-391. F51Johansson R, Magnusson M, Akesson M. Identification of human postural dynamics. IEEE Trans Biomed Eng 1988;BME35:858-869.

Gait

6; Posture

6 (1997)

27-38

Goldie PA, Bach TM. Evans OM. Force platform measures for evaluating postural control: reliability and validity. Arch Phys Med Rehab 1989;70:510-517. PI Seidel H, Brauer D, Bastek R, Issel I. On the quantitative characterization of human body sway in experiments with longterm performance. Acta Biol Med Germ 1978;37:1551-1561. [91 Koozekanani SH, Stockwell CW, McCritie RB, Firoozmand F. On the role of dynamics models in quantitative posturography. IEEE Trans Biomed Eng 1980;BME-27:605-609. UOI Nashner LM. Analysis of stance posture in humans. In: Towe A and Luschei E, eds. Handbook of Behavioral Neurobiology, v. 5, Motor Coordination. New York: Plenum Press, 1981:521-561. Werness A, Anderson D. Parametric analysis of dynamic postural response. Biol Cybern 1984;51:155-168. Slobounov S, Newell KM. Postural dynamics as a function of skill level and task constraints. Gait Posture 1994,2:85-93. Slobounov S. Newell KM. Dynamics of posture in 3- and 5-year-aid children as a function of task constraints. Human Movement Sci 1994;13:861-875. s41 Murray MP, Seireg A, Sepic SB. Normal postural stability and steadiness: quantitative assessment. J Bone Joint Surg 1975;57A:510-516. II51 Hufschmidt A, Dichgans J, Mauritz KH, Hufschmidt M. Some methods and parameters of body sway quantification and their neurological application. Arch Psychiat Nervenkr 1980;228:135[71

150.

[I61 Winter DA, Patla AE, Frank JS. Assessment of balance coptrol in humans. Med Prog Tech 1990;16:31-51. Norre M, Forrez G, Beckers A. Posturography measuring instability in vestibular dysfunction in the elderly. Age Aging 1987;16:89-93. [I81 Norre M, Forrez G, Beckers A. Posturographic findings in two common peripheral vestibular disorders. J Otolaryngol 1987;16:340-344. [I91 Lucy SD, Hayes KC. Postural sway profiles: normal subjects and subjects with cerebellar ataxia. Physiother Canada 1985;37: 140148. PO1 Paulus WM, Straube A, Brandt T. Visual stabilization of posture. Brain 1984;107:1143-1163. t211 Hasan SS, Lichtenstein MJ, Shiavi RG. Effect of loss of balance on biomechanics platform measures of sway: influence of stance and a method for adjustment. J Biomech 1990;23:783-789. [221 Riach CL, Starkes JL. Stability limits of quiet standing postural control in children and adults. Gait Posture 1993;1:105-111. ~231Starkes JL, Riach JC, Clarke B. The effect of eye closure on postural sway: converging evidence from children and a parksonian patient. In: Proteau L and Elliott D, eds. Vision and Motor Control. Amsterdam: Elsevier, 1992:353-373. 1241 Riach CL, Starkes JL. Velocity of centre of pressure excursions as an indicator of postural control systems in children. Gait Posture 1994;2:167-172. 1251 Peeters HPM, Caberg CH, Mol JMF. Evaluation of biomechanical models in posturogaphy. Med Bio Eng Comput 1985;23:469-473. [261 Aggashyan RV. On spectral and correlation characteristics of human stabilograms. Agressologie 1972;13-D:63-69. [271 Aggashyan RV, Gurfinkel VS, Mamasakhlisov GV, Elner AM. Changes in spectral and correlation characteristics of human stabilograms at muscle afferentation disturbance. Agressologie 1973: l4-D:5-9. [28] Soames RW, Atha J, Harding RH. Temporal changes in the pattern of sway as reflected in power spectral density analysis. Agressologie 1976;17-8:15-20. [29] Yoneda S, Tokumasu K. Frequency analysis of body sway in the upright posture. Statistical study in cases of peripheral vestibular disease. Acta Otolaryngol 1986;102:87-92. [I71

D.L.

King,

V.M.

ZatsiorskJ,

1301 Tokita T, Miyata, Matuoka T, Taguchi T, Shimada R. Correlation analysis of the body sway in standing posture. Agressologie 1976:17-B:7 -14. 1311 Brauer D, Seidel H. The autoregressive time series modeling of stabilograms. Acta Biol Med Germ 1978;37: 1221- 1227. [32] Brsuer D, Seidel H. Time series analysis of postural sway. Aggressologie 1979;20-B: 1I I- 112. [3?] Brguer D, Seidel H. The autoregressive structure of postural sway. Agressologie 1980:2 I-E:101 - 104. [341 Brguer D, Seidel H. The autoregressive structure of postural sway. In:. Morecki A, Fidelius K, Kedzior K and Wit A, eds. Biomechatucs VII-A. Baltimore, MD: University Park Press. 1981:1550-1560. [35] Takata K. Kakeno H, Watanabe Y. Time series analysis of postural sway and respiration using an autoregressive model. In: Matsui II and Kobayashi K. eds. Biomechanics VIII-A. Champaign, IL: Human Kinetics. 1983:591-596. wd Tokumasu K, Ikegami A, Tashiro N. Bre M, Yoneda S. Frequency analysis of the body sway in different standing postures. Agressologie 1983:24-B:89 -90. L371 Schumann Y, Redfern MS Furman JM. El-Jaroudi A, Chaparro 1. F. Time-frequency analysis of postural sway. J. Biomech 144~;S:601 --Id?. [381 FirsoL VI. Rosenbum MG. Chaotic dynamics of the upright human posture control. In: Proc 1st World Congress on Biomechanics. L.a Jolla. CA. 1990. WI Prieto TE. Myklebust JB, Myklebust BM, Kreis DU. Intergroup rensitivity in measures of postural steadiness. In: Woolacott M .ind Hordk F. eds. Posture and Gait: Control Mechanisms (XIth Int Sytnp. J’ostural Gait Res) Portland, OR: University of Oregon Books. 1992122 1135. [401 Collins JJ, De Luca CJ. Open-loop and closed-loop control of posture: A random walk analysis of center-of-pressure trajectories. E\p Bram Res 1993:95:308-318. [41] Collins 53. De Luca CJ. Random walk during quiet standing. Phys Rev Lett 1994;73:764 767. [42] Newell KM, van Emmerik REA, Lee D. Sprague RL. On postural stability and variability. Gait Posture 1993;4:225-230. 1431 Roy SH. Ladm Z, De Luca CJ. Experimental evidence for a r;mdom process model of postural sway. In Proc 9th Ann. Int. Conf. IEEE ESMB 1987:9:759. [44] Collins JJ, De Luca CJ. The effects of visual input on open-loop and closed-loop postural control mechanisms. Exp Brain Res 19%;10!~151 163. [45] Collins JJ. De Luca CJ. Upright, correlated random walks: A st
Guit & Posture

6 II 997) 27

38

i7

[53] Winter DA. Biomechanics and Motor Cilntroi of Ifuman Movement. 2nd edn. New York:Wiley, 1990 [54] Yang JF. Winter DA, Wells RP. Posrurai dynamics m the standing human. Biol Cybern J990;62:?09--320. [55] Winter DA. Balance and posture in human walking. Eng Med Biol 1987:6:8-11. [56] Kapteyn TS. Afterthought about the pilysics and mechanics ot’ postural sway. Agressologie 1973:14-C:?.-. 35. (571 Massen CH. Kodde LA. Model for the &scription of Jeti-right stabilograms. Agressologie 1979;20-B:107 IO8 [58] van de Mortel PJ, Massen CH. Kodde L. Stabilograms and horizontal forces. Agressologie 1979:20-H. iO5 106 [59] Spaepen AJ. Portuin JM. Wiilems EJ. Contpanson 01’ the movements of the center of gravity and of ihc center ot pressure III stabilometric studies. Comparison with 1-s~utm a;~.~l~~~~ qgrcssologie 197920-B: I1 S- I 16. [60] Spaepen AJ, Peeraer L, Willems EJ. Cent:;- of gravity and center of pressure in stabrlometric studies. .,Z ,~lmp~ri:!:c tcch~~~~~ucs. sologie 1971;13-C:75-~78. [63! Gurtinkei EV. Physical foundation:\ cf ~t;tblloer;:ph!. Agreisc>logie 1973:X-C:9- 11 [641 Valk-Fai T. .Analysis of the dynarmc:al bchavioul- &’ the hod! whilst ‘Standing Still’. Agressologte 1973:il 21 7-S. [651 Geurssen JB. Altena D. Massen CH, Verdun M. 4 Imodel of a standing man for description of hit tivr~1~~ hchsv:our 4gressolog~e 1976; 1j-B:63 69. WI Riley PO. Mann RW. Hodge AW. Modellri;! oi the hiomechanits of posture and balance. J Biomech 1Y)O 27 503 >06. 1671 Era P, Heikkinen E. Postural sway during iranding and unexpected disturbances of balance in randon ,amples of’ men o: different age,. J Gerontol 19X5:40:187 29:; w Diener HC. Dichgans J, Ciuschlbauer I). BaL,her M. Quantitication of postural sway in normals and patients hith cerebellar diseases. Electroencephalogr clin Neurophy’ 1984:57, I 34 142. 1691Mauritr KH. Dietz V. Charactertstic; o nohtural instability Induced by ischemic blocking of les :~tt’ercr!t\. !+p Brain R.es 1980:38:1 17 ! !4. 1701 [7’1 Mamasakhlisoc GV. Elner AM. Gurfnkei \. C; Particlpalion of lariouh types ot’ afferentation in Gontroi :;r !hc orthogrude posture in man Agressologie 1973:11-A:37 ii [721 Mauritz KH, Dietz V. Hailer M. Balancing ~1~‘1 clm& tesl in the dtfferential diagnosis of sensory-motor t!lsor:iers .I Yeurol Neurosury Psychiatry 1980;43:407 4 i 2. [731 Ciagey PM, Baron JB. Ushio h. Introductlor. [o &~nical posturologq. Agressologie 1980:?1 -E: 1 19 133 [741 Nashner LM, McCollum G. T‘he organlzafi~m ol human PO&tural movements: a formal basis .md ~cp~:r~n~en~;~i\ynthcsis. Behav Bram SU 1985:X: 135~.I?.?. [751 Soames RW. Atha J. ‘The validity of‘ phys!cruc-~UX~ inyertL:d pendulum models ot’ postural sway !?ch‘lviolir 4nn tlum B~ol 1980;7:13T 15.3. [761 Mizrdtli J. Susak %. Bi-lateral reactlvt! tclrcr pitterns in postural suay activity of normal subiects. Rio1 Cybern f989:60:297 3115 1771 Mizrahi I, Solzl P. Ring H. Nisell R. Postural &bility in strokr patients: vectorial expression 01 asymmetr). ‘.ua? actl\ity and relative sequence of reactive florces. Ma1 R?ol lrce pIa:form data. J Biomech t984; 17:5% 60. [791 Benda BJ, Riley PO, Krebs DE. Biomechanlcai relationship between the center of gravity and center c,t pressurc~ during standins IEEE Trans Rchah Erie 199*:?:1 !I.

38

D.L.

King,

V.M.

ZatsiorskJ

[80] lnman VT. The Joints of the Ankle. New York: Williams and Wilkins, 1976. [Sl] van den Bogert AJ, Smith GD, Nigg BM. In vivo determination of the anatomical axes of the ankle joint complex: An optimization approach. J Biomech 1994;27:1477-1488. [82] Fomin SV, Stillkind TI. On the dynamic equilibrium of the pedatar systems.In: Gydikov A, Tankov N and Kosavov D. eds. Motor Control. New York: Plenum Press, 1973:235-249. [83] Kodde L, Caberg HB, Mel JMF, Massen CH. An application of mathematical models in posturography. J Biomed Eng 1982:4:44-48. [84] Hayashi R, Miyake A, Watanabe S, Umemoto K. A model of bending man for the description of his motor performance of

/‘Gait

& Posture

6 (1997)

27-38

body sway. In: Matsui H and Kobayashi K, eds. Biomechanics VIII-B. Champaign, IL: Human Kinetics, I983:971-977. [85] Zatsiorsky VM, Seluyanov VN. The mass and inertia characteristics of main segments of human body. In: Matsui H and Kobayashi K. eds. Biomechanics VIII-B. Champaign, IL: Human Kinetics. 1983: 1152- 1159. [86] Kozyrev GS. Center of Gravity of Human Body in Norm and Pathology. PhD thesis, Charkov Medical Institute, Charkov. 1962. (In Russian) [87] Drillis R, Contini R. Body Segment Parameters. New York: Office of Vocational Rehabilitation, Department of Health, Education and Welfare, 1966.