Is it necessary to save the algorithm of Collins and De Luca?
Pierre - Marie GAGEY

     Although the paper ' Open-loop and closed-loop control of posture… ' Of Colliins and De Luca ( 1993 ) was accepted by the review Experimental Brain Research and widely quoted in the international literature, it is filled with inaccuracies, errors and so striking ignorances that I have never considered useful taking the feather to criticize it. But now Michel Lacour puts forward this paper to use it in his program for stabilometric signal analysis, PosturoPro. Would there be anything to save in this paper which can be of a some utility? If it is the case, as Michel Lacour thinks apparently, then it is necessary to take time to criticize this paper and separate the scorias from the gold nugget it would contain.

The absence of metrological data
     Proudly, in conclusion of his work, Collins announces us: " The act of maintaining an erect posture could be viewed, in part, as a stochastic process " but some lines further, he is not bothered saying that his stabilometric signal was sampled at 100Hz! When one analyzes a partially aleatory signal, it is a little bit ridiculous congratulating himself for the discovering, thanks to signal analysis, that it is partially aleatory!
     For those who do not understand, I explain. Collins in the statement of its results shows us that the quadratique distance gone through by the centre of pressure in one second, about, is of the order of 20mm2, that's to say a real distance of the order of 4,5 mm, and that in the part of the the signal where it moves most quickly (Figure 2b and 3 of the article). The sampling at100 Hz cuts this average distance in 100 portions that will measure on average 0,045 mm. If one considers that the distribution of these real elementary lengthes is gaussian, then one should admit that half of the measures which Collins analyzes are lower than four hundredth of millimeter. Considering the available technology in the 90s, the reader can not admit a priori that the platform used by Collins was endowed with such a power of resolution. Not to discuss this aspect of things is manifestly to state a regrettable indistinctness. It is very likely that half of the measures analyzed by Collins corresponded to nothing in reality.

The absence of mathematical exactness
     The absence of mathematical exactness in this paper was widely stigmatized by Delignières and coll. (2003) who reminded that using the diffusion property of the Brownian fractional movements has no sense in the case of limited signals.

The ignorance of the previous works
     Collins congratulates with having discovered thanks to his algorithm, that the stabilometric signal could be decomposed into two parts the limit of which it places in the neighborhood of 1 Hz. But it was known for 20 years!!! Gurfinkel (1973) had shown by a theoretical study of the differential equation of an inverted pendulum that in the frequency band 0/0,6 Hz, the stabilometric signal could be considered as representing the movements of the centre of gravity to an error not exceeding 50 %, but that beyond 0,6 Hz the stabilometric signal represent nothing more than the movements of the centre of pressure. Some years later by an experimental study, we showed that in fact this distinction centre of pressure / centre of gravity appeared very sharply only at a little higher frequency, around 1 Hz (Gagey and al ., 1985).

The absence of biomechanical reflection
     At no time in his article, Collins envisages that movements of the centre of gravity could be controlled by movements of the centre of pressure. What would have nevertheless avoided him this stupidity to declare that above one Hertz the movements of the centre of pressure evolve there in open loop, as if they did not intervene in the feedback loop which characterizes the upright postural system and which has the effect of stabilizing the position of the centre of gravity. We all made this exercise which consists in holding one brooms knocked down in balance on the end of one of the fingers, and one knows well that the movements of the centre of pressure - our finger - have for object the preservation of the position of the centre of gravity of brooms in limits compatible with an absence of fall. The mechanical model is childish, what does not prevent it from being very real.
     This absence of distinction in the thought of Collins between controlled variable (Position of the centre of gravity) and controlling variable (movements of the centre of pressure) brings Collins to utter another stupidity: the postural system would react when the movements of the centre of pressure exceed a certain value threshold " These fluctuations left unchecked by the postural control system until they exceed some systematic threshold, after which corrective feedback mechanisms are called into play. " Each knows that what moves with a frequency superior to one Hertz, it is not the centre of gravity, because the physical mass acts as a low-pass filter, and the resonance frequency of the human pendulum is situated around 0,3 Hz.! So according to Collins it would be excursions of the controlling variable that would activate control mechanisms. A snake which bites itself the tail. Nevertheless I listened still, I do not know where, somebody to express himself as if the position of the critical point was an evaluation of the threshold of reaction of the postural system!.

Conclusion of the negative criticism
     It is evident, and necessary to say, that this paper of Collins and Luca does not hold the road, neither from the metrological point of view, nor from the mathematical point of view, nor from the biomechanical point of view for lack of a sufficient documentation.
     Then? Where is the gold nugget?

The algorithm of Collins and Luca
     To study the so called Brownian fractional movements, Collins and De Luca used a simplified algorithm which shows itself golden! And until them, to my knowledge, nobody had thought of it.
I explain:
The basic algorithm is given by the formula (Figure 2a in the paper)


    The mean quadratic length gone through by the centre of pressure in the whole of the intervals of time that separate all the sampled positions distant from the same number, m, of elementary intervals

     The squared length that the centre of pressure goes through in the ith interval of this series of m elementary intervals.


     The total number of the positions sampled by the centre of pressure


FIG. 1 - Illustration of the basic algorithm.
      The algorithm estimates all the distances, l, squared, which separate a series of sampled positions of the centre of pressure separated by m sampling intervals [3 sampling intervals on this figure] then it makes the average of them, L.
      This basic algorithm is repeated for all the values of m, from 1 to (N - 1).
      The result for each of N-1 studied intervals of sampling is put back in a Cartesian space having for abscissas the time and for ordinate the distances.
     One obtains a representation of the quadratic distances according to time, so an expression approached of the speed, taking the square root of this quadratic distance let us having exactly the mean speed the signal moves in each of these N-1 intervals of sampling.

FIG. 2 - Representation of the results of N-1 applications of the basic algorithm. Every point has for abscissa the duration of the studied interval and for ordinate the mean quadratic length that the signal goes through during this interval of time.

     This algorithm allows us so to obtain at low cost an approached image of the speed of the movements of the centre of pressure (before the critical point) and of the speed of the movements of the centre of gravity (after the critical point). [I say 'at low cost' because that needs only a double loop in data processing].
     It is evident that the speed of movement of the centre of pressure is a new parameter and very credibly a good marker of the stability of the subject: the weaker this speed the less fast and\or important the movements of the centre of gravity, so the more stable the subject. Also the rapport of these two speeds is obviously a marker of the efficiency of the postural control (central or peripheral? This distinction remains to be studied).

     Patrice Rougier, who worked hard inside Collins theories, finally came to work on the difference CdP-CdG (see 2005, for instance), his instinct drove him to find where is the truth in the Boston's researchers paper. Finally, the errors, the inaccuracies, the ignorances of these authors don't matter because they brought us an interesting algorithm that has been in the fashion wind ! I think that Michel Lacour is completely right integrating it into his new kit of signal analysis.


Collins J.J., De Luca C.J. (1993) Open-loop and closed-loop control of posture: a random-walk analysis of center-of-pressure trajectories. Exp. Brain Res., 95: 308-318.

Delignières D., Deschamps T., Legros A., Caillou N. (2003) A methodological note on non-linear time series analysis: Is Collins and De Luca (1993)'s open- and closed-loop model a statistical artifact? Journal of Motor Behavior, 35, 86-96.

Gagey P.M., Bizzo G., Debruille O., Lacroix D. (1985) The one Hertz phenomenon. In Vestibular and visual control on posture and locomotor equilibrium. Igarashi M., Black F.O., Karger, Basel, 89-92,.

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

Parré F. Qualification d’une plate-forme de Stabilométrie. Rapport DESS de Physique.

Rougier P. (2005) Compatibility of postural behavior induced by two aspects of visual feedback: time delay and scale display. Exp Brain Res 165: 193–202

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