Closure to “Discussion of ‘Friction-
Compensating Command
Shaping for Vibration Reduction’ ”
(2006, ASME J. Vib. Acoust.,
128, p. 540)
Jason Lawrence
William Singhose
Woodruff School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA
Keith Hekman
Department of Mechanical Engineering,
The American University in Cairo,
Cairo, Egypt
DOI: 10.1115/1.2206714
Dr. Shahruz is correct in asserting that exact cancellation of
frictional terms is not possible. However, nowhere in our paper do
we claim exact cancellation, nor does our method require exact
cancellation of the friction effects. In fact, a large section of our
paper is devoted to experimentally examining the effects of inac-
curacies in the implementation of our technique. Therefore, we are
alarmed that he has come to such a conclusion and that this jour-
nal would publish such an erroneous statement.
Our paper shows that partially canceling the friction effects will
lead to an improvement in performance over standard input shap-
ing techniques. More specifically, consider the system model
given in our paper 1:
mx ¨ + b + K
D
x ˙ + K
P
x = K
D
u ˙ + K
P
u - signx ˙
k
N 1
Under the conditions stated in Sec. 3.3 of our paper, this equation
can be simplified:
mx ¨ + b + K
D
x ˙ + K
P
x = K
P
u - signx ˙
k
N 2
Equation 2 implies that friction can be viewed as a disturbance
force with a constant magnitude. Standard input shapers are not
designed to compensate for such disturbances. Therefore, if a
standard input-shaped step command was used as the input u, then
the system would not settle at the desired final setpoint and there
might be some residual vibration.
Now consider using the friction-compensated command
1
given
in our paper:
u =
K
N
K
P
signx ˙ + v 3
where
K
N are estimates of the coefficient of kinetic friction and
normal force. Substituting 3 into 2 yields
mx ¨ + b + K
D
x ˙ + K
P
x = K
P
v - signx ˙
K
N -
K
N 4
Comparing Eqs. 2 and 4, it is clear that a reasonable estimate
of
K
N will yield a reduction in the magnitude of the friction
“disturbance” force. As the magnitude of the term decreases, the
response to the shaped command will improve smaller steady
state error and less residual vibration. An exact cancellation of
the last term in 4 is not necessary for this improved perfor-
mance.
Contrary to the comments of Dr. Shahruz, this type of feed-
forward friction cancellation strategy is fairly common, although
applying it to input shaping is a new idea. Many motion control-
lers on the marker, such as Siemens, Adept, Newport, Panasonic,
etc., include analogous feed-forward friction compensation with
their PD and PID feedback control algorithms.
As for the issue of robustness, the great robustness properties of
input shaping are well known and have appeared in hundreds of
papers. We refer the reader to only a small fraction of these papers
here 2–11. This robustness is the primary reason input shaping is
installed on literally millions of machines worldwide ranging from
massive cranes to small piezo actuators. In fact, input shaping can
be made arbitrarily robust to modeling errors in natural frequency
and damping ratio 10. It is simply a question of trading off the
robustness with the rise time. Unfortunately, a method for making
the technique arbitrarily robust to errors in the kinetic coefficient
of friction has not been developed. This is an admitted limitation
of our approach. However, even with a poorly estimated coeffi-
cient, our technique will improve performance over standard input
shaping. Furthermore, our control system contains a feedback
controller that adds robustness in terms of final positioning accu-
racy and disturbance rejection.
The input shaping method proposed in our paper was not de-
signed to be extremely robust to modeling errors. Our main focus
was on compensating for the frictional effects, not generating
highly robust commands. However, it is possible to combine our
friction-compensation scheme with more robust input shaping ap-
proaches 11. Robustness information for the technique under
discussion was presented primarily in Figs. 14 and 15. These fig-
ures show the results from hundreds of experiments that were
performed while varying the impulse amplitudes and times in the
input shaper. This directly negates issue 2, item iii raised by
Dr. Shahruz. In addition, because the impulse times are derived
from the system frequency and the impulse amplitudes are derived
from the system damping ratio and kinetic friction, these results
also address issues 2, item i and 2, item ii.
In conclusion: 1 our method does not require exact cancella-
tion of the nonlinear friction term, and 2 the results from many
experiments well document the robustness properties of our ap-
proach.
References
1 Lawrence, J., Singhose, W., and Hekman, K., 2005, “Friction-Compensating
Command Shaping for Vibration Reduction,” ASME J. Vibr. Acoust., 127, pp.
307–314.
2 Singer, N. C., and Seering, W. P., 1990, “Preshaping Command Inputs to
Reduce System Vibration,” ASME J. Dyn. Syst., Meas., Control, 112, pp.
76–82.
3 Singh, T., and Vadali, S. R., 1993, “Robust Time-Delay Control,” ASME J.
Dyn. Syst., Meas., Control, 115, pp. 303–306.
4 Pao, L., and Lau, M., 2000, “Robust Input Shaper Control Design for Param-
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6 Singhose, W. E., Porter, L. J., Tuttle, T. D., and Singer, N. C., 1997, “Vibration
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Control, 119, pp. 320–326.
7 Park, U. H., Lee, J. W., Lim, B. D., and Sung, Y. G., 2001, “Design and
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8 Murphy, B. R., and Watanabe, I., 1992, “Digital Shaping Filters for Reducing
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10 Singhose, W., Seering, W., and Singer, N., 1996, “Input Shaping forVibration
Reduction With Specified Insensitivity to Modeling Errors,” presented at the
Japan-USA Symposium on Flexible Automation, Boston, MA.
11 Lawrence, J., Singhose, W., and Hekman, K., 2004, “Robust Friction-
Compensating Input Shapers,” presented at the 8th Cairo University Interna-
tional Conference on Mechanical Design and Production, Cairo, Egypt.
1
Note that Dr. Shahruz has confused our reference command signal with our
control law. Our system is under PD feedback control.
Journal of Vibration and Acoustics AUGUST 2006, Vol. 128 / 541 Copyright © 2006 by ASME
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