A great wader taken at the very moment of its landing, such is the symbol chosen by the first posturologists to express the meaning of their research. This bird shows the dynamics of stabilisation in a very strong way as its entire body stretches forward to its rest posture. On the contrary, a human body at rest secretly and almost imperceptibly continues its struggle with the forces of gravity.

     But the actual normalised platform, with its 5 Hz sampling cadence, only has 256... One cannot simply increase the recording time in order to obtain more points, for that would tire the patients too much. Nor could one increase the sampling cadence of the actual normalised platform, for the distances between the successive sampling positions of the centre of pressure would then become too small for evaluation by that measurement chain (Gagey et al., 1999). Thus, a new model of platform of Stabilometry must be studied, that would be adapted to the requirements of a faster sampling while remaining as close as possible to the building standards defined by the Association Française de Posturologie in 1985 (Bizzo et al., 1985).

     For the building of this new platform the following changes in the specifications of the normalised platform can be proposed:

• The sampling cadence would be set at 40 Hertz instead of 5 Hz.
• The analog digital transformation of the signal would be done by a 16 bits digitizer instead of a 12 bits one.
• The height between the pressure gauges plane and the superior surface of the platform would be of 13 mm instead of 65 mm. (The opportunity of the specification changes would be seized to propose this desired reduction of the platform's height).

     This model of platform will be referred to as «platform AFP40/16», considering its characteristics:

• Fidelity to the norms of the AFP,
• 40 Hz sampling,
• The analog digital transformation of the signal by a 16 bits digitizer.

     A prototype of this new platform was built in order to try and answer experimentally two questions:

• Can this new platform be assimilated to a normalised platform?
• Is the uncertainty of this new platform coherent with the expected accuracy of its measurements?

     To allow platform AFP40/16 to be assimilated to an AFP normalised platform, the one and only requirement is that the distributions of the conventionnal stabilometric parameters delivered by this platform are similar to the theoric distributions of the normalised platform.
     Such similarity is impossible for the Length parameter and for all the parameters that derive from it - LFA, speed Variance, acceleration Variance, VFY - because the values of length and of its derivatives change according to the sampling cadence (Gagey et al., 1999). Yet it is possible to reconstitute a signal sampled at 5 Hz from a signal sampled at 40 Hz, just by taking one point out of each 8. The calculation of length, and of its derivatives, from this reconstituted signal can give values similar to those of normalised platforms, if the other changes brought to the building of platform AFP40/16 do not interfere.
     As for the values of parameters X-mean, Y-mean, Area, Romberg Quotient, ANØ2X, ANØ2Y, they stay the same whatever the sampling cadence of the signal (Gagey et al., 1999). Only the other changes brought to the building of platform AFP40/16 could modify the value of these parameters by their intervention.

     The experimental study of the «normality» of platform AFP40/16 was run in two different ways:

• Comparison of consecutive recordings of identical subjects on platform AFP40/16 and on a normalised platform.
• Mean comparison of the distributions of parameters published in Normes85 and the distributions obtained from recordings carried out on an AFP40/16 platform.

Comparison of recordings
     Ten «normal» subjects - i.e. free from sensations of vertigo, instability, actual or recent back pain - have been recorded in open and closed eyes situations, successively and at short intervals, on a normalised platform (AFP5/12) and on platform AFP40/16. The parameters have been calculated out of the total number of points of each signal. The distributions of these two series of parameters have been submitted to a mean comparison under the Student's t test (table I).
     There is no significant difference between the distributions of parameters X-mean, Y-mean, Area, obtained for those 10 subjects on one platform and the other. However there is a statistically very significant difference between the distributions of length parameters, as could be expected. The latter difference is due to the different sampling cadences on one plaform and the other.

TABLE I - Comparison of the distributions of parameters obtained from recordings on platforms AFP40/16 and AFP5/12.
     Means, standard deviations and significance of the mean comparison of parameters X-mean, Y-mean, Area, Length, Length in X (L in X), Length in Y (L in Y) obtained from the recordings of ten subjects on an AFP40/16 platform and on a normalised AFP5/12 platform.

Comparison to Normes85
     From the recordings on platform AFP40/16 of the ten «normal» subjects, the parameters have been recalculated, no longer out of the total number of points, but out of the 256 points obtained by reduction to a sampling cadence of 5 Hz. The distribution of those parameters has been compared to the distribution of the Normes85 parameters, thanks to a mean comparison under the Student's t test (table II).

TABLE II - Comparison to Normes85 of recording results on platform AFP40/16.
     Student's t-test of means comparisons between the distributions of parameters published in Normes85 and the distributions observed with the new platform, after a reduction of the cadence to 5 Hz. (N=10).

     The distributions of parameters LFA, VFY, ANØ2Y and Romberg Quotient observed with the new platform are statistically different from the distributions published in Normes85. However these differences do not allow us to draw the conclusion that the new platform cannot be assimilated to a normalised platform. The number of subjects studied is limited, and above all, the abnormal values of the individual parameters have no systematic appearance (table III). On the contrary the presence of aberrant subjects is more likely to come from specificity in the subjects' recruitment than from a systematic difference due to the platform. Complementary studies are clearly indispensable in order to draw proper conclusions.

	X-mean	Area	LFA	VFY	Variance	ANØ2X	ANØ2Y	RombergQ
E.O.	-3.20 -1.37 5.86 5.60 7.39 -16.37 1.93 6.69 1.33 -0.59	100.8 113.8 99.4 287.3 109.9 123.9 111.2 56.7 102.1 71.7	1.19 0.76 0.86 0.69 0.47 0.88 1.16 0.82 0.87 0.71	0.16 0.50 0.73 0.87 -1.12 3.99 14.99 4.11 0.43 -0.56	5.43 4.33 5.40 5.66 2.46 11.23 25.78 10.46 4.46 3.37	22.20 23.05 12.54 3.88 1.65 1.87 12.81 9.59 5.45 27.36	4.95 7.51 4.86 7.01 1.63 2.20 8.34 6.60 17.50 12.02
E.C.	-4.59 -0.50 8.83 6.98 -1.58 -15.69 4.08 6.30 5.16 2.54	317.9 187.5 206.5 430.6 77.0 107.3 162.0 155.4 187.4 149.9	1.25 0.88 1.37 0.87 0.71 0.77 1.33 0.87 1.14 0.78	0.16 -0.69 2.78 0.42 -2.53 3.41 15.18 1.04 -0.07 -0.94	8.57 5.67 11.94 7.23 4.21 12.48 27.21 9.19 7.64 5.49	10.96 12.63 24.63 18.84 30.04 3.26 8.44 15.46 1.50 4.74	15.56 13.30 23.69 44.91 23.34 47.06 27.30 16.75 10.01 31.77	315.4 164.8 207.8 149.9 70.1 86.5 145.6 274.0 183.4 209.0

TABLE III - Individual values of the stabilometric parameters, calculated at 5 Hz, of the ten normal subjects.
     The presence of aberrant individual values (v) can be noticed, yet not a systematic variation of the value of all the parameters.
The subjects' order is the same in the two parts of the table (open eyes and closed eyes).

     No measurement chain is perfect. There is always a differential between the value delivered by the measurement chain and the true value. The undimensional equation between that differential, Di, and the true value of a given measurement i:

is used in metrology to define a measurement chain. That equation is called «the uncertainty» of the measuring apparatus.
     The determination of the minimum necessary level of uncertainty of a measuring apparatus obviously depends on what is expected from it. But that level of uncertainty cannot be set exactly by any data. One can just compare on the one hand the range of measurement expected from the apparatus, and on the other hand its concrete performances, in order to give the soundest foundations possible to a judgement that will have to stay cautious anyway, in answer to the question: is the uncertainty of the new platform «coherent» with the measurements expected from it?

The range of measurements at 40 Hz
     The mean length between two successive elementary positions of the centre of pressure sampled at 5 Hz by platform AFP5/12, is of 1.6 mm and 19% only of such lengths are inferior or equal to 0.2 mm (Gagey et al., 1999). But platform AFP40/16 samples eight times faster, so it has to measure much smaller distances that have to be evaluated theoretically and experimentally.
     Theoretically, the mean length of the statokinesigram for recordings of 51.2 s at 40 Hz is of 740 mm (Gagey et al., 1999), which represents a mean speed of the centre of pressure of about 14 mm/s. We could expect the elementary position change of the centre of pressure between two successive samplings, i.e. with a 0.025 s interval, to be of an average 0.36 mm.
     In reality the analysis of the distribution of the lengths of 10,655 elementary position changes of the centre of pressure of normal subjects recorded in open and closed eyes situations (fig. 1), shows that the median of these lengths, measured by platform AFP40/16, is of 0.2 mm and 15% of them are equal or inferior to a tenth of millimetre.

FIG. 1 - Histogram of the lengths of elementary position changes measured with platform AFP40/16. Median: 0.22 mm; First quartile: 0.14 mm; Third quartile: 0.31 mm; Line of 0.1 mm: 14%; N = 10,655.

The requirement for measuring accuracy in Stabilometry
     Is it necessary to demand to this platform a measurement to the nearest tenth of millimetre of the position change of the centre of pressure, as more than 10% of those changes belong to this scale of sizes? Is such accuracy useful?
     In order to calculate the means
     In order to calculate such statistical means as the X-mean or Y-mean positions, and even to calculate the Area of the confidence ellipse at 90%, such accuracy is obviously and totally useless.
     For spectral analyses
     Similarly, it seems to be an acquired fact that a fairly important number of random measurements in the obtained signal is compatible with a good spectral analysis.
     In order to calculate length and its derivatives
     For the calculation of lengths and derivative parameters, the measurement errors of each elementary interval add to one another when the distances are summed up... But there is no reason to think that the distribution of those measurement errors is systematically asymmetric. Hopefully they will more or less cancel each other out during the summing up. The accuracy to the closest tenth of millimetre therefore appears superfluous as far as length is concerned too, and anyway... what is the length of the statokinesigram? (Gagey et al., 1999)
     For the non-linear dynamic analysis
     We have the intuition, still very poorly formalised, that the non-linear dynamic analysis is more affected than the statistical analyses if too many random measurements interfere in the signal.
     We were very surprised that Collins and De Luca (1993) did not find any sign of a chaotic signature in their analyses of the stabilometric signal, but simply a «random walk». The difference between their results and that of other researchers (Martinerie & Gagey, 1992; Myklebust et al., 1995; Boker et al., 1997; Cao et al., 1998) may be explained by the fact they obtained their stabilometric signal with a 100 Hz sampling, apparently without paying attention to the performances of their chain of acquisition. When a temporal series is mainly composed of random values, is it surprising that a mathematical analysis, even thoroughly driven, ends up finding this one and only aspect: random?
     All the non-linear dynamic analyses start out by building an image of the system's dynamics - an image on which all subsequent calculations will be executed. That image is a mere trajectory connecting, in a mathematical space, the series of successive points that describe the state of the system at every instant. The position in space of each of these points is determined by its co-ordinates, which are precisely the values, at this «instant», of the different variables that govern the system's dynamics. Those variables are unnamed, if not unknown. The one and only piece of information we have on them is the temporal series of the analysed signal. That one and only temporal series delivers all the co-ordinates of the trajectory, thanks to a simple mechanism: a series of given temporal delays shifts the values of the series of the signal that have to be attributed each «instant» to every variable. Each variable that affects the system's dynamics clearly intervenes to determine, partially and in its own way, every value of the signal. The trick of the trajectory's building simply lies in diffracting in time the effect of each variable in order to apprehend it better.
     In these various operations of building of the trajectory-image of the system, one does not find the characteristic operations of statistical analyses that allow distributions of non-systematically asymmetric random errors to cancel each other out.
     All these observations and remarks lead us to the idea that the non-linear dynamic analysis could be more affected than the statistical analyses by too great a number of random measurements interfering in the signal. This intuition, still poorly formalised, leaves a doubt, which calls for some caution. However the techniques of non-linear dynamic analysis are fairly strong and experience teaches us that it is probably unnecessary to have less than 15% of random measurements in a signal for it to undergo that kind of analysis.

The concrete performances of the platform
     It is now possible to compare the concrete performances of the platform with the expected range of measurements and level of accuracy.

     The resolution of the 16 bits digitizer
     When sampling at 40 Hz, approximately half the measurements cannot be read correctly by a 12 bits digitizer(1), because of its resolution (Gagey et al., 1999). Therefore it is important to check what happens at this cadence with a 16 bits digitizer.
     If the whole scale is set at 981 N, the last bit of the digitizer will number 0.03 N (only half the voltage range, between 0 and 10 Volts, being used), which corresponds, for an adult subject, to a position change of about two hundredth of millimetres (Cf. in appendix 1, conversion of Newtons to millimetres). Therefore the resolution of the 16 bits digitizer is much higher than required.

Note 1: In reality the resolution of a digitizer depends on the full scale of measurement. A 12 bits digitizer could read correctly the measurements if only it did a double reading, by programmatically adapting the full scale between the two measurements.

     The differential with the true value of the measurement
     As a reminder, the platform measures forces and not distances.
     If the force produced by a given and constant mass set on the platform is measured several consecutive times, it appears that the measurement chain delivers the true value of that force plus or minus a certain differential with the true value. As this differential does not exclusively depend on the measurement chain but on all the measuring conditions, the equation between that differential, Di, and the true value of the measurement, i:

will not define exactly the uncertainty of the platform, but it gives a good experimental approach to it.

Statistics describing the differential with the true value
     Three aspects of the distribution of this differential with the true value have been studied: Is it normal? Is it constant? Does it depend on the force measured?
     The 95% limit of certainty of this distribution has been evaluated from these studies.
     A known mass being set near the centre of the platform, recordings are made of a series of measurements of the value of the force applied on the platform by that still mass. After checking that the mean value of these numerous measurements (N > 1,000) corresponds to the true value of the force, the differential with that true value is calculated for each measurement:

where Dji stands for the differential with the true value, i, of the jth measurement mj.
     The distribution of the differential with the true value is studied from such data.
     The histogram of the distribution of the differential with the true value (fig. 2) has been drawn up out of 2,048 measurements of a force of 87.31 N produced by a mass of 8.900 Kg set on the platform.

FIG. 2 - Histogram of the distribution of differential with the true value of a force of 87.31 Newtons. (N = 2,048)

     Then the same measurements, of the same 87.31 N force, have been repeated five times in the same conditions at time intervals varying from 3 minutes to 48 hours. The parameters describing the distributions studied are laid out in table IV.

	Force applied on the platform
	87.31	87.31	87.31	87.31	87.31
Mean	0.0000	0.0000	0.0007	0.0008	0.0000
Standard Deviation	±0.0933	±0.1033	±0.0932	±0.0780	±0.1276
Confidence Limits 95%	0.1828	0.2026	0.1827	0.1529	0.2501
N	2048	2048	999	999	2048

TABLE IV- Mean, Standard deviation, confidence limit at 95% of the distributions of the differential with the true value studied during five series of measurements of the same force at different time intervals.

Unit: Newton.

     Three series of measurements have been made in the same conditions, but with different forces applied on the platform. The parameters describing the distributions of the differential with the true value studied in these conditions are laid out in table V.

	Force applied on the platform
	4.9	87.31	785
Mean	0.0000	0.0001	0.0000
Standard Deviation	± 0.1048	± 0.08	± 0.1200
Confidence Limits 95%	0.2054	0.1629	0.2320
N	2048	1010	2048

TABLE V - Mean, Standard-deviation, confidence limit at 95% of the distributions of the differential with the true value studied during three series of measurements of different forces.

Unity: Newton.

     According to the measurement results laid out in tables IV and V, one has less than five chances out of a hundred to be wrong in assuming that a force measure delivered by this platform is true to the closest ± 0.2 N.
Now, this measurement of the forces at the closest ± 0.2 N corresponds to a measurement of the position of the centre of pressure at the closest ± 0.1 mm, for an adult subject approximately weighing 70 Kg (cf. Appendix 1).

     The present study aimed at answering two questions:

• Can the new AFP40/16 platform be assimilated to a normalised platform?
• Is the uncertainty of that new platform coherent with the expected accuracy of its measurements?

     The first results are encouraging: it seems that this platform can be assimilated to a normalised platform, yet complementary studies are indispensable to be absolutely sure.
     However, the uncertainty of the measurements executed by the new platform appear to us to be very compatible with the actual analysis of the signal, as well as with its treatment by non-linear dynamic analyses.

     The platform measures forces and not distances. It is therefore necessary to convert millimetres into Newtons and vice and versa, thanks to equations of the platform:

	FIG. 2 - Plan of the platform.

Where:
The referential Oxy has its origin in the centre of the equilateral triangle at the vertexes of which the gauges are set, the Ox axis parallel to the line connecting the right and left gauges (fig. 3)

L: Distance between the gauges; 400 millimetres according to the AFP standards,
F: Sum of the forces measured by the gauges,
Fv, Fd, Fg: Forces measured respectively by the front, the right and the left gauges.
(The developments of these formulas appear in the second lesson of posturology [Gagey et al., 1993]).

Measurement of the elementary position changes of the centre of pressure.
     From those equations of the platform it is possible to show, according to the measured variations of the forces, dF, the variations of position of the Cartesian projections, dX and dY, of the centre of pressure between two successive samplings at times t and t+1:

     Calculation of dX
     If X[t]=0, the expression of dX is simplified:

     If we assume that the centre of pressure moves parallel to the Ox axis, then the variations of force, dF, measured by the right and left gauges are equal and of opposite signs and if we assume that the centre of pressure has moved to the left, it appears that:

If we assume that F[t+1]=F[t] we then have:

Formula 1

     Calculation of dY
     When the centre of pressure moves forward, the sum of the force variations measured by the back gauges is equal to the force variation, dF, measured by the front gauge, therefore:

We then have:

     In the normal conditions of Stabilometry, we do not make an important error by assuming that F[t+1]=F[t]. If we replace L by its value, 400 mm, it then appears:

Formula 2

Bibliographie
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Aknowledgments to Mr Guy BIZZO for his precious help.