Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
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extremely simplified representation of a recorded sports movement, with a coordinate
(r) expressed by the equation:
r = 2 sin 4πt + 0.02 sin 40πt
The first term on the right-hand side of this equation (2 sin 4πt) represents the motion
being observed (known, in this context, as the ‘signal’). The amplitude of this signal
is 2 – in arbitrary units – and its frequency is 4π radians/s (or 2 Hz) – indicated by
sin 4πt. The second term is the noise; this has an amplitude of only 1% of the signal
(this would be a low value for many sports biomechanics studies) and a frequency
of 40π rad/s (or 20 Hz), 10 times that of the signal. The difference in frequencies is
because human movement generally has a low-frequency content and noise is at a
higher frequency. Figure 4.8(a) shows the signal with the noise superimposed; note
that there is little difference between the noise-free and noisy displacements. The
above equation can now be differentiated to give velocity (v), which in turn can be
differentiated to give acceleration (a). Then:
v = 8π cos 4πt + 0.8π cos 40πt
a = −32π 2 sin 4πt − 32π 2 sin 40πt
The noise amplitude in the velocity is now 10% (0.8π/8π × 100) of the signal amplitude
(Figure 4.8(b)). The noise in the acceleration data has the same amplitude,
32π 2 , as the signal, which is an intolerable error (Figure 4.8(c)). Unless the errors in the
displacement data are reduced by smoothing or filtering, they will lead to considerable
inaccuracies in velocities and accelerations and any other derived data. This will be
compounded by any errors in body segment data (see pages 137–9).
Data smoothing, filtering and differentiation
Much attention has been paid to the problem of removal of noise from discretely
sampled data in sports biomechanics. Solutions are not always (or entirely) satisfactory,
particularly when transient signals, such as those caused by foot strike or other impacts,
are present. Noise removal is normally performed after reconstruction of the movement
coordinates from the image coordinates because, for three-dimensional studies, each
set of image coordinates does not contain full information about the movement coordinates.
However, the noise removal should be performed before calculating other
data, such as segment orientations and joint forces and moments. The reason for this
is that the calculations are highly non-linear, leading to non-linear combinations of
random noise, which can adversely affect the separation of signal and noise by low-pass
filtering.
The three most commonly used techniques to remove high-frequency noise from the
low-frequency movement coordinates use digital low-pass filters, usually Butterworth
filters or Fourier series truncation, or spline smoothing; the last of these is normally