Measurement of Acceleration by Differentiation of Displacement

Supplied by Assistant Professor Margaret Mak, The Hong Kong Polytechnic University

Technical Details

Normal male subject performing a sit-to-stand activity at two different speeds (movie). The upper-limbs were held clasped against the chest, and the subject was sitting on a chair on a force platform, with his feet on another platform. The total vertical force (Fz) was recorded for both platforms.

Data was also collected by a two-dimensional Peak Performance Technologies video-based motion analysis system, and was used to determine the body centre of gravity using anthropometry from Dempster, and assuming body symmetry. Since F = ma, the vertical force should equal the vertical acceleration of the centre of gravity, plus gravity (9.81 m/s2), multiplied by body mass: F = m.(a + g).

In the curves shown below, however, there is a discrepancy in the vertical force (in Newtons) as derived by differentiation from the kinematics (red), and that measured by the force platform (blue)...

Note that for the purposes of this inverstigation, the data was not filtered, so as to exclude filtering as a cause of this discrepancy, but when filtered at 5Hz the difference between the peak forces in each case is around 50 N. When the subject was asked to stand up at a faster speed, the difference between the two curves was accentuated, with the peak force derived from kinematics some 150 N lower than that registered by the platform...

Note, however, that in both cases the baseline of the two curves are equivalent, and that the negative-going force is faithfully recorded. The experiments have been repeated many times with the same results.


  • Why is the up-going vertical force derived by differentiation of displacement data of a smaller peak amplitude than that derived from the force platform signal?
  • Is the discrepancy caused by inaccuracies in the data acquisition process or the model used to determine body centre of mass?
  • Why is this discrepancy not seen in the down-going peak (deceleration phase)?

  • What we said on the CGA list


    The discrepancy was caused by neglecting the head segment in the model. An abrupt downwards deceleration of the head is translated into an upwards acceleration, which is of a surprising magnitude, and nicely fills in the missing force when input to the Body CoM calculation:

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    Last modified on 2-August-97.