The specifications for the building of a stabilometry platform aimed at clinical use planned a precision of measurement in the order of 10-3 of the measurement range, that is to say in the order of the millimeter for measures of length (Bizzo et al., 1985). That precision had been evaluated, and showed, coherent with the cadence of acquisition of the signal set at 5 Hz during 51.2 seconds - 256 positions of the center of pressure sampled, spread on average over 500 mm (AFP, 1985). Given that this precision, which does not represent a great metrological demand, is guaranteed for the force sensors required by the specifications, it had not seemed necessary to develop a calibration protocol of that type of platform.
     Two new facts now question that position. First, many authors directly or indirectly criticize that absence of calibration protocol of our platforms (Barrett et al., 1987; Bobbert & Schamhardt, 1990; Brown & O'Hare, 2000; Gill & O'Connor, 1997; Hall et al., 1996; Mita et al., 1993; Starck et al., 1993). Secondly, many authors criticize the sampling cadence at 5 Hz and propose acquisition chains sampling at 40 Hz, which requires the precision of measurement for those new platforms to be in the order of 10-4 of the measurement range (Gagey et al., 1999). Therefore it becomes desirable, and even necessary for that new type of platform, to check that our instruments actually and faithfully measure what they are reputed able to measure.

Legal checking

    The text of reference (in France) is currently decree n° 91-330 of the 27th of March, 1991. The first article of that decree enumerates all the uses of the weighing instruments with a non-automatic functioning that require a regulated instrument. Paragraph d) of that article provides for use in the medical field when the instrument is used with a view to "determine mass in medical practice concerning the weighing of patients for reasons of supervision, diagnosis and medical treatments."
     As the stabilometry platforms do not edit measures expressed in kilograms, they escape the protocol of legal checking stipulated by that decree.
     The responsibility of the checking of the platforms is not on any account assumed by the legislator, it lies exclusively on the user and/or the builder.
(That statutory text and its interpretation in the case of the stabilometry platform have been provided by the Laboratoire National d'Essai).

Label

     The builder can define with an independent metrology laboratory some metrological characteristics of the measuring chain that the builder commits himself to respect and the laboratory to check and guarantee. That procedure of "Label" can be started by the builder, at the Laboratoire National d'Essai for instance, but it cannot be enforced on the builder either by the user or by a group of users.

Checking by the user

     In the absence of such Label, the responsibility of the checking of the measuring instrument therefore exclusively lies on the user. The builder cannot be the judge in his own case.
     As such a task obviously goes beyond the competences acquired in a medical and/or paramedical education, the settlement of a calibration protocol of the platform has to be entrusted to one/some engineer(s) competent on the subject. The organisation of such a protocol will certainly arise important difficulties for most users, if only for lack of time and adequate material. An easy solution would then consist in asking the builders' help, but would that be the best solution? Would it not be wise to take into account the easiness of organisation of the calibration protocol while we are trying to define it?
     The principle of calibration is very simple: compare a supposedly known input signal to the output signal given by the acquisition chain. All the problem therefore lies in generating a known input signal with enough precision for it to be used as a standard, in a way.

     This project aims at defining a calibration method by specifying for each phase the methods used and the nature of the checked metrological characteristics.
In the following text, the term "platform", also called "measuring apparatus", refers to the whole acquisition chain, notably composed of a digitizer associated to a calculator.
The represented curves are schematic and only aim at facilitating the understanding of the text.
     The notions of metrology defined and explained in the text allow in conclusion to show the necessity of the calibration of the new type of platform.

    The actual platform is composed of a metallic sheet lying on three force sensors. It has two axes of work, as shown in figure 1, axis Ox for the study of lateral displacements, axis Oy for the study of antero-posterior displacements.

Figure 1: Presentation of the platform

    In the following lines, the described tests only correspond to one axis, Oy in our example, which only realizes half the calibration. Similar tests will therefore have to be made on axis Ox.

A) Experimentation

     On the platform, a 30 kg mass M is displaced from Pø to P1, which are 10 centimeters apart, gradually, in successive stages that are 1 centimeter apart, then back to Pø still in stages separated by 1 cm increments.
    That elementary operation is repeated three times. At each stage after each increment, the position of mass M relatively to O will be measured ten times.
If we call P the relative position of mass M and L the value measured by the platform, the obtained results could correspond to the diagram on figure 2.

B) Treatment of the results

a) Measurement range

    The measurement range [MR on figure] was set at 10 centimeters following the results of stabilometric examinations of over a thousand subjects.

b) Standardization curves

     On the ten values acquired at each stage of the progression of mass M, a calculation of the mean value is made so as not to retain any aberrant value.
    We then obtain three times ten mean values, mi, in the increasing way and as many in the decreasing one [those mi are represented with a * on figure 2]. Those values are first memorized, then used to calculate at each stage the mean of the six means. This way we obtain a series of 10 mean values, li, from which we realize a linear regression, based on the method of the minor squares.
     The equation of the line of the minor squares [SP on figure 2] is of the form:

Where
a: sensitivity expressed in mm/mm, therefore without dimensions
D: initial deviation, expressed in mm.

     Note that, if the platform was ideal, relation (1) would become:

     From relations (1) and (2) it is possible to calculate two uncertainties.

c) Uncertainty on sensitivity

     Expressed in percentage of the perfect sensitivity (a = 1).

d) Initial deviation

     Expressed in percentage of the measurement range, MR.

e) Uncertainty of linearity

FIG. 3 - Deviation of linearity. The mean li that is the most distant from the line of the minor squares, here l3, allows to evaluate Dl, the biggest absolute value of the distances | Li-li |.

     The value of the coefficients of the line of the minor squares, SP, being known, it is possible, for each stage Pi, to compare mean point li to point Li of the line of the minor squares corresponding to that stage. Let Dl be the biggest absolute value of distances Li-li [L3-l3 on figure 3], it is possible to define a new uncertainty which is the deviation of linearity:

     Expressed in percentage of the measurement range.

f) Uncertainty of reversibility (Former hysteresis)

     If we now consider the twice ten mean values, mi, that have been memorized, it is possible to realize a diagram with it (fig. 4).

     Thanks to that diagram, and as for the deviations of linearity, we search for what value of P the difference between the "increasing" means and the "decreasing" ones is the biggest [m5-m15 on figure 4].
     If we refer to that difference as Dh, it is possible to define the uncertainty of reversibility (formerly hysteresis), the value of which is:

     Expressed in percentage of the measurement range.

g) Uncertainty of resolution

     The resolution is defined as being the smallest variation of the measurand (P) that provokes a variation of the result (L). That value is refered to as Dr and the uncertainty of resolution will be written as:

     Expressed in percentage of the measurement range.

h) Uncertainty of repeatability

     The repeatability characterizes the ability of the measuring apparatus to give the same indication when the same magnitude is applied several times in a row (in theory ten times in less than five minutes).
    If we refer as Dr to the biggest variation of indication, the uncertainty of repeatability will be written as:

     Expressed in percentage of the measurement range.

i) Uncertainty of noise

     The noise is characterized by the maximum fluctuations observed on the real value whereas the measured value stays constant.
     If we refer as Db to the biggest variation observed, the associated uncertainty will be written as:

     Expressed in percentage of the measurement range.

j) Fineness

   That magnitude characterizes the ability of a measuring apparatus not to interfere with the measurand. In the case of a stabilometry platform, that magnitude is given by the expression of the rigidity of the sheet expressed in Newtons by meters, force N deforms the sheet by provoking arrow m: N/m.

     The perfect fineness has an infinite value.

k) Uniformity

     The uniformity of a platform expresses its quality to show the same sensitivity on all the parts of its measurement range.

     That operation consists in determining the rapidity of the measuring apparatus by the study of response S(t) to a force grade E(t), of amplitude A. Figure 5 shows a possible example of signals.

    The stabilometry platform not being a very damped apparatus, it will be possible to determine the two following parameters:

A) Time of establishment

     The time of establishment, Te, is the time necessary for the output signal to stay equal to the input signal in a bracket of ± 10% [on figure 5, Te = t1-tø ].

B) Natural frequency

     The natural frequency, nF, is the pseudo frequency of the output signal S(t) [on figure 5, nF = 1/(t2-t1)Hz].
   Remark: For that type of test, appropriate treatments allow to determine the transfer function of the measuring apparatus and therefore to know the passing band, the cutting frequency and generally (not very damped system) the coefficient of resonance and the absorption. That method, which dates back to a quarter of century, will not be described here, but it can be given to those who ask for it.

    We call influence magnitude any physical magnitude other than the measurand likely to influence the result of a given measure. Those magnitudes are numerous, so we will only give the major ones:

• Time,
• Ambient temperature,
• Feed,
• Ambient humidity,
• Vibrations,
• Accelerations...

     Concerning the stabilometry platform we can consider that those magnitudes are limited to time.
    The measures made in function of time mostly carry on the study of the variations of the "zero" for periods that can stretch up to a few hours, stabilization or "warming up" time excluded. The variations are then expressed in absolute value (mm/h).

    The limiting values of the measurand, beyond the measurement range, are three:

Admissible maximum value

    Momentary overload at the entry of the apparatus not leading to a new calibration after disappearing.

Nondeterioration value

    Overload requiring a new calibration . For instance the sensitivity can be modified.

Nondestruction value

    Overload not to be exceeded or the measuring apparatus will be destroyed, even if partially.

    Normally defined by the builder and checked a posteriori by the user, the mean time of good functioning specifies the mean temporal interval separating two successive calibrations. The measuring apparatus is supposed to respect all its metrological characteristics during that period.
     The most important risks of variations come from phenomenas of flowing of the force sensors relatively to the repetition of their deformations imposed by the applied forces.
     We generally agree on a one year time of good functioning.
    But it is wise to make regularly simple tests such as:
- Read the values read by the pressure gauges when the platform is "without load".
- Draw a triangle by successively applying the thumbs on the three gauges. The recording software being launched, when the transfer of the pressures from one gauge to the other is realized in a progressive way, we see the appearance on the screen of a triangle whose vertices correspond to the position of the pressure gauges and whose sides are straight and regular if the acquisition chain works correctly.

    The different uncertainties determined during the calibration not having apparent physical links, it is usual to consider as the resultant uncertainty the quadratic value of the sum of their squares.
    In the case of a stabilometry platform, if we refer to this value as eT, we then have:

     The absolute uncertainty and the relative uncertainty exactly correspond to the former terms of absolute error and relative error. If we refer as D to the difference between the read value Vr and the true value Vt of the measurand, we can write:
D = | Vr - Vt | Absolute uncertainty
er = D/Vt % Relative uncertainty

    If a platform uses three pressure gauges in the following conditions:

• Measurement range of the forces, MRc = 600 N
• Feeding voltage: 5 or 10 Vcc
• Sensitivity for the Full Scale: about 2 mV/V
• Zero: + 0.2 mV/V
• Impedance + / - 1000 Ohms
• Error Linearity / Hysteresis: 0.2 %
• Temperature of Use: 20 / 60° C
• Admissible Overload: 150 % of the Measurement Range

    The uncertainty of linearity of a sensor is:

    The biggest reading variation of a force by that sensor is therefore:

     That is to say

     We know that a Newton roughly corresponds to 0.5 mm, so the biggest reading variation of length by a sensor Dmm = ± 0.6 mm, which corresponds to an uncertainty

That is

As the platform uses three sensors, the total uncertainty, eT, will be of the form

Therefore:

     The platforms built with that type of sensors show an uncertainty in the order of the millimeter in the reading of the displacements of the center of pressure. So if they are expected to work at the closest tenth of millimeter by sampling at 40 Hz, it is essential to prove that the sensors they use are better than announced.

     The stabilometry platform is a measuring apparatus. As any measuring apparatus it has limits of uncertainty one must take care not to forget at the risk of exploiting results polluted with aleatory values.
     That remark also holds for the realization of the static calibration operations - the calibration uncertainty has to be much smaller than that of the platform.

    We would like to thank the Bureau National de Métrologie, mister Laudrel from the Comité Français d'Accréditation who helped us in the writing of that article and more particularly mister André Gosset from the Laboratoire National d'Essai for the documentation given and the careful revision of the article.

Bibliography

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