Transcript No Slide Title

ACCURACY OF THE CRITICALLY DAMPED AND BUTTERWORTH FILTERS
USING THE ACCELERATION OF A FALLING OBJECT
Robyn Wharf, B.Sc. and D. Gordon E. Robertson, Ph.D., FCSB
School of Human Kinetics, University of Ottawa, Ottawa, Canada
Introduction
Results and Discussion
The Butterworth filter has been the gold standard in filtering
motion data since D.A. Winter et al. introduced it in 1974. It
was selected for its superior roll-off and its “flatness” in the
passband (Pezzack et al., 1977). Unfortunately, it has the
undesirable characteristic of being underdamped and
therefore overshoots the data during rapid transitions
(Robertson & Dowling, 2003). An alternate filter that solves
this problem is the critically damped filter (Robertson &
Dowling, 2003). This filter has poorer roll-off than the
Butterworth but by cascading the data through the filter a
similar roll-off can be achieved.
Figure 2 demonstrates that using 4th-order critically damped
or Butterworth filters (CD-4 & BW-4) do not yield acceptably
flat periods of constant acceleration during the flight phase
of the ball. Although the average values of the acceleration
(–10.00 & –10.05) were close to the correct value of –9.81
m/s2, their standard deviations were quite high (0.69 & 0.48).
Using 20th-order filters (CD-20 & BW-20) achieved better
results by reducing the variability (0.23 & 0.34), but only the
critically damped filter improved on predicting the correct
acceleration (–9.96 vs. –10.11).
Purpose
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The purpose of this study was to compare the effectiveness
of the critically damped and Butterworth filters using the
known acceleration of a falling body–a golf ball.
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CD-4
BW- 4
Gr av.
C D - 20
BW- 20
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- 11
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0.92
0.73
0.55
0.37
0.18
0.00
Figure 2. Accelerations of a ball in flight after various types of
filtering
Figure 1. Golf ball bounce digitized. Green line raw data,
teal line critically damped filter 12th order.
Methodology
To validate the filters, a golf ball was dropped and filmed by
a digital video camera in SP mode. The image was
calibrated by a 1.0 x 2.0 m grid of control points. The plane
was calibrated by a fractional linear transform that corrected
for any camera misalignment. The motion data were
captured and digitized using the Ariel Performance Analysis
System and then processed by the Biomech Motion
Analysis System (http://www.health.uottawa.ca/biomech/software). The latter software allows for filtering the data
with 4th, 8th, 12th, 16th and 20th-order Butterworth and the
critically damped filters.
Figure 3 show the flight phase results when the data file
included the bounces that occur before and after the fight. In
this case the 20th-order Butterworth caused significant
distortion of the signal after and particularly before the
bounces. Some of these distortions were removed by
clipping the flight phase so that the results did not exceed
the boundaries of the graph. The Butterworth filter caused
greater distortion as the order was increased whereas the
critically damped filter caused less distortion at the ends of
the flight phase as the order increased.
Figure 4 shows the 8th, 12th and 16th order filtering by both
the critically damped and the Butterworth filters. The
average accelerations of the critically damped filter ranged
from -9.79 m/s2 for the 8th order to -9.66 m/s2 and -9.44 m/s2
for the 12th and 16th orders, respectively. Nevertheless the
standard deviations were quite high, from 0.969 to 1.505.
Whereas the Butterworth filter was consistent throughout
the different orders with an average of -10.90 m/s2 at the
8thorder and -10.93 m/s2 at both the 12th and 16th orders.
0
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0
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CD-8
C D - 12
C D - 16
Gr av
BW- 8
BW- 12
BW- 16
68
60
52
44
36
28
20
Figure 4. Comparison of both critically damped and Butterworth
filters at 8th, 12th and 16th order.
Conclusion
The critically damped filter was shown to produce superior
acceleration results as compared to the Butterworth filter
when, as suggested, 20th-order was used to filter the
motion of a body accelerated due to gravity (Robertson &
Dowling, 2003). Higher order Butterworth filters caused
significant distortion in the accelerations of the ball when
rapid transitions precede or follow the flight phase. Caution
should therefore be used with the Butterworth filter when
higher order filtering is required.
References
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CD-4
BW- 4
Gr av.
C D - 20
BW- 20
-5
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1.07
0.87
0.67
0.47
0.00
Pezzack, J.C.; Winter, D.A. & Norman, R.W. (1977)
J Biomech, 10:377-82.
Robertson, D.G.E. & Dowling, J.J. (2003). J Electromyogr
Kines, 13:569-73.
Winter, D.A.; Sidwall, H.G. & Hobson, D.A. (1974)
J Biomech, 7:157-9.
Figure 3. Ball acceleration during flight between bounces after
various types of filtering
Biomechanics Laboratory