KINEMATICS - Biomechanics and Motor Control of Human Movement - page 67

where: ω o = fundamental frequency

n = harmonic number

X n = amplitude of n th harmonic

θ n = phase of n th harmonic

To get the velocity in the x direction V x , we differentiate with respect to
time:

N

dx
dt =

V x = n ω 0 X n cos (n ω 0 t + θ n ) (3.4)

n = 1

Similarly, the acceleration A x is:

N

A x dV x
dt

(n ω 0 ) 2 X n sin (n ω 0 t =− + θ n ) (3.5)

n = 1

Thus, the amplitude of each of the harmonics increases with its harmonic
number; for velocities they increase linearly, and for accelerations the increase
is proportional to the square of the harmonic number. This phenomenon
is demonstrated in Figure 3.17, where the fundamental, second, and third
harmonics are shown, along with their first and second time derivatives.
Assuming that the amplitude x of all three components is the same, we can
see that the first derivative (velocity) of harmonics increases linearly with
increasing frequency. The first derivative of the third harmonic is now three
times that of the fundamental. For the second time derivative, the increase
repeats itself, and the third harmonic acceleration is now nine times that of
the fundamental.
In the trajectory data for gait, x 1 might be 5 cm and x 20 = 0 . 5 mm. The 20th
harmonic noise is hardly perceptible in the displacement plot. In the velocity
calculation, the 20th harmonic increases 20-fold so that it is now one-fifth
that of the fundamental. In the acceleration calculation, the 20th harmonic
increases by another factor of 20 and now is four times the magnitude of the
fundamental. This effect is shown if you look ahead to Figure 3.19, which
plots the acceleration of the toe during walking. The random-looking signal is
the raw data differentiated twice. The smooth signal is the acceleration calcu-
lated after most of the higher-frequency noise has been removed. Techniques
to remove this higher-frequency noise are now discussed.

3.4.4 Smoothing and Curve Fitting of Data
The removal of noise can be accomplished in several ways. The aims of each
technique are basically the same. However, the results differ somewhat.