KINEMATICS - Biomechanics and Motor Control of Human Movement - page 74

3.4.5 Comparison of Some Smoothing Techniques
It is valuable to see the effect of several different curve-fitting techniques on
the same set of noisy data. The following summary of a validation experiment,
which was conducted to compare (Pezzack et al., 1977) three commonly used
techniques, illustrates the wide differences in the calculated accelerations.
Data obtained from the horizontal movement of a lever arm about a vertical
axis were recorded three different ways. A goniometer on the axis recorded
angular position, an accelerometer mounted at the end of the arm gave tan-
gential acceleration and thus angular acceleration, and cinefilm data gave
image information that could be compared with the angular and acceleration
records. The comparisons are given in Figure 3.22. Figure 3.22 a compares
the angular position of the lever arm as it was manually moved from rest
through about 130 ◦ and back to the original position. The goniometer signal
and the lever angle as analyzed from the film data are plotted and compare
closely. The only difference is that the goniometer record is somewhat noisy
compared with the film data.
Figure 3.22 b compares the directly recorded angular acceleration, which
can be calculated by dividing the tangential acceleration by the radius of
the accelerometer from the center of rotation, with the angular acceleration
as calculated via the second derivative of the digitally filtered coordinate
data (Winter et al., 1974). The two curves match extremely well, and the
finite-difference acceleration exhibits less noise than the directly recorded
acceleration. Figure 3.22 c compares the directly recorded acceleration with
the calculated angular acceleration, using a polynomial fit on the raw angular
data. A ninth-order polynomial was fitted to the angular displacement curve
to yield the following fit:

35 t 2 210 t 3 430 t 4 400 t 5 θ (t ) = 0 . 064 + 2 . 0 t − + − +

170 t 6 25 t 7 2 . 2 t 8 0 . 41 t 9 − + + − rad (3.11)

Note that θ is in radians and t in seconds. To get the curve for angular
acceleration, all we need to do is take the second time derivative to yield:

α (t ) =− 70 + 1260 t − 5160 t 2 + 8000 t 3 − 5100 t 4

+ 1050 t 5 + 123 t 6 − 29 . 5 t 7 rad/s 2 (3.12)

This acceleration curve, compared with the accelerometer signal, shows
considerable discrepancy, enough to cast doubt on the value of the polynomial
fit technique. The polynomial is fitted to the displacement data in order to get
an analytic curve, which can be differentiated to yield another smooth curve.
Unfortunately, it appears that a considerably higher-order polynomial would
be required to achieve even a crude fit, and the computer time might become
too prohibitive.