DYNAMIC STUDY OF THE PLATFORM

(V S Gurfinkel's equation, 1973)

(As studied again by G Bizzo - Annex C)

I) Inventory of the forces and accelerations Figure 1 : Dynamic equation of the platform.                          Inventory of the forces and accelerations

II) Hypotheses

1) The system is undeformable

2) The subject's weight is positive - therefore the referential linked to the subject is submitted, relatively to the absolute referential, to a rotation of + (p+F) around G.

3)                          OH = h    HG = L

OA = a    M : Total mass of the subject

OB = b   XG = - Lsin F, abscissa of G in XOY.

4) The moments are counted as positive if they tend to straighten up the body axis, negative in the opposite case.

5) The subject is supposed to be in a situation of forward falling :

dF/dt = F ' > 0

dF/dtÓ = F " > 0

III) Fundamental equations of the dynamics

1) Equation of the forces. Which induces : When applied to referential XOY, that equation allows us to write :

R1y + R2y - Mg = Mgy     (1)

R1x + R2x = Mgx              (2)

2) Equation of the moments relatively to H : Which means : Given hypotheses 4 & 5 of ¤2 :

aR1y - bR2y - h(R1x + R2x) - MgLsinF = -J.F"       (3)

3) Value of J :

The system in rotation around H is equivalent to an inverted pendulum of length L and mass M, its moment of inertia is equal to :

J = ML2

therefore :

- JF" = - ML2F"

From (1) follows :

R1y + R2y = Mg + Mgy (4)

From (2) and (3) follows :

aR1y - bR2y = MgLsinF - ML2F" + Mhgx  (5)

IV) Transformation of the accelerations (Values of gx and gy)

G splits up in GT and GN, tangential and normal accelerations, on the reference system X'GY'.gx  and gy are deduced from GT and GN by a change of coordinate axes corresponding to a rotation of reference X'GY' of -(F+p) followed by a translatory movement along axis OG of reference XOY so as to superimpose G and O. In that case the transformation matrix is of the form : We therefore have : In the reference linked to G, G has for coordinates : GT, GN

Now                  - GN = - LF

and                    GT = LF"

therefore : therefore : gx = - L(F"cosF - F'2sinF)

gy  = - L(F"sinF + F'2cosF)

V) Calculation of absolute error e and relative error e

If we apply gx in (5) we have :

aR1y - bR2y = MgLsinF - ML2F" + Mhgx - MhL(F"cosF - F'2sinF)

if we multiply right and left by -1:

bR2y - aR1y = MhL(F"cosF - F'2sinF)+ ML2F" - MgLsinF (6)

If we apply gy in (4) we have :

R1y + R2y = Mg - ML(F"sinF + F'2cosF) (7)

If we put :

X1 = (bR2y - aR1y) / (R1y + R2y)

Applied to (6) and (7) we have : Therefore, if we simplify by M As we also know that XG = -LsinF, and if we put e = X1 - XG, we then have after reduction to a common denominator : that is to say : (8)

Relation (8) represents the absolute error committed on the determination of the projection of the centre of gravity on the plane of the support basis.

Note that if F(t) = Cte we have F'(t) = F"(t) = e, which proves that this error only appears in dynamic conditions.

In order to be more general, we will now consider relative error e, defined by :

e = (X1 - XG ) / D

with D : peak by peak value of the displacement

VI) Numerical values of e

1) Definitions and conventions :

In what follows we consider F under the form :

F (t) = F0 + F1 sin wt     (10)

F' (t) = F1 w cos wt         (11)

F'2 (t) = F12 w2 cos2wt  (12)

F"(t) = - F1 w2sinwt     (13)

2) Value of the different parameters.

The subject being animated by a movement of form

F = F0 + F1 sin wt

error e max appears for the extreme values of sin wt, that is for

w t = ± 90¡.

The values of the different parameters or variables first are the following :

 F 0 ¡ F 0 ¡ t ¡ F (Hertz) 0 1 ± 90¡ 0,1  2 0 10 ± 90¡ 0,1  2 -10 1 ± 90¡ 0,1  2 -10 10 ± 90¡ 0,1  2

Calculation of D

D : peak to peak value of the displacement.

D = XC1 - XC2

XC2 = - Lsin(F0 + F1)

= - L(sinF0cosF1 + sinF1cosF0)

XC1 = -Lsin(F0 - F1)

= - L(sinF0 cosF1 -sinF1cosF0) Figure 2 : Conventional notations used for the calculation of the relative error.

3) Starting equations

D = 2L sinF1 cosF0                                                         (1) (2)

Knowing that 1-sin2 F = cos2 F, we have : F = F0 + F1 sin wt     (3)

F ' = F1 w cos wt        (4)

F " = - F1 w2 sin wt    (5)

erelatif = er = e / D (6)

er % = 100 er (7)

4) Particular numerical values

L = 1 m

h = 0.10 m

g = 10 m/s2

t = ± 90¡

sin w t = ± 1

cos w t = 0

F = F0 ± F1

F' = 0

F" = ± F1 w2

5) Equations used

D = 2 sinF1cosF0                          (1) (2)

F = F0 + F1 pour wt = + 90¡       (3)

F = F0 - F1 pour wt = - 90¡          (3')

F" = - F1 w2 pour wt = + 90¡      (4)

F" = F1 w2 pour wt = - 90¡          (4')

er = er = e / D                                  (5)

er % = 100 er                                  (6)

6) Results Figure 3 : Relative Error as a function of frequencies and of amplitude of oscillations. Relative error depends sharply on the frequency, but not of the amplitude of studied oscillations (2 ¡ and 10 ¡).

The results are presented in the form of curves in figure 3. The scale of relative errors is logarithmic. The 'white squares' curve : F= ± 10¡, the 'black squares' curve : F= ± 2¡. The importance of the relative error practically only depends on frequency. Figure 4 : Left-right Stabilogram ( A ) and FFT ( B ) of a subject presenting an important reduction of his(her) oscillations according to frequencies

If the fundamental of the signal is very low (0.05 Hz on figure 4) and the attenuation of the sway is important in function of the frequencies, we observe that the centre of pressure is led to its extreme positions,which determine the value of the area of the statokinesigram, by a big sway of an 18 seconds period (Figure 4 A) for which the position of the centre of pressure coincides with the projection of the centre of gravity to within a 0.4% error. The small sway added on top, of around 0.4 Hertz, measures the sway of the centre of gravity to within a considerable error, of around 30% peak to peak, that is to say 15% of the amplitude that intervenes to modify the value of the area, but it represents only 10% of the amplitude of the fundamental, therefore it only introduces an error of around 1.5%. Which is all the more negligible that the algorithm of the calculation of the area eliminates 10% of the extreme positions of the centre of pressure. Figure 5: Mean and 90-Decile of the amplitude spectrum of the forward-backward oscillations of 91 normal subjects. (According to Gagey, 1986)

The statistical studies on the amplitude spectrum of normal subjects show that the fundamental is at 0.04 Hz, that the attenuation at 0.4 Hz is amply superior to 10% (Figure 5)(Gagey, 1986).

We can therefore consider that we make a minor error by assimilating the displacements of the centre of pressure to the displacements of the centre of gravity to measure the mean stability of the normal man, on condition that the hypothesis of the inverted pendulum that was at the basis of the calculation is respected.

References :

Bizzo G (1993) tude dynamique de la plate-forme. in Gagey P M (Eds) Huit Leons de Posturologie.: 33-38. AFP (Paris).

Gurfinkel V S - Physical foundations of stabilography. Agressologie, 14, C, 9-14, 1973,a.