A Performance Analysis System for the Sport of Bowling

Since the fundamental frequency and the minimum and maximum frequencies of the
waveform have already been found, and the manner in which the frequency changes over
time (monotonically increasing, with a possible inflection point, if roll out occurred) is
known, it is easy to develop a method to locate the local maxima and minima of the
filtered waveform.
This is accomplished by starting at the beginning of the filtered sample array (which
includes the pre-samples), and working through the array, looking for inflection points
(those points where the waveform changes direction). Each time an inflection point (a
local minimum or local maximum) is found, the index of the inflection point is recorded
in a separate array. Due to the monotonically increasing nature of the frequency of the
waveform, once the distance (the number of samples) between the first two inflection
points is found, that distance can be used to help estimate the location of the next
inflection point.
There are still two fractional revolutions that must be dealt with: from release until the
first peak, and from the last peak (or valley) to pin impact. Both of these fractional
values can be extrapolated from the angular velocity of the revolution immediate adjacent
to each fractional value.
Again let f be the sampling frequency (120 Hz), and j R be the index of the release sample,
j 1,0 be the index of the sample that begins the first full revolution, j F be the index of the
sample that ends the last full revolution, and j P be the index of the pin impact sample, and
ω 1 and ω F be the angular velocities during the first and last full revolutions of the ball,
respectively, then
R R = ((j 1,0 - j R ) / f) · ω 1 ,
and
R P = ((j P - j F ) / f) · ω F ,
where R R and R P are the fractional revolutions of the ball immediately after release and
immediately before pin impact, respectively.
The discussion in Section II established that there would be at least 15-20 samples per
revolution, allowing R R and R P to be resolved to at least 30° of rotation. Thus, it should
be possible to count the total revolutions of the ball to within 0.1 revolutions, yielding a
resolution of better than 1% for counts greater than 10 revolutions.
Even though a resolution of 1% has been obtained in counting the total revolutions, the
angular velocity can only be resolved to about 4% for a rotation rate of 300 rpms, using
the same technique. This discrepancy arises from the 120 Hz sampling frequency, which
yields a time resolution of 8.333 msecs. Although the waveform does not necessarily
capture the exact peaks and valleys of the waveform, it does at least capture a pair of
consecutive samples that form an 8.333 msec "window" around each true peak or valley.
It is possible to get around the sampling-induced resolution issue by interpolating
between each of those pairs of values to obtain very good approximations of the true
peaks and valleys of the waveform.
Figure 3-6 illustrates a simple interpolation technique for finding the "true" location of
the local maxima and minima of the filtered waveform. By finding the intersection point
of the slope lines of both sides of a peak or valley, the actual peaks and valleys can be
accurately located. Two sample values are used to find each slope line. These samples
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