Performance Measure of Residual
Vibration Control
Chul-Goo Kang
Professor
Department of Mechanical Engineering,
Konkuk University,
Hwayang-dong, Gwangjin-gu,
Seoul 143-701, Korea
e-mail: cgkang@konkuk.ac.kr
The robustness of residual vibration control, such as input shap-
ing, has conventionally been evaluated from the ratio of residual
vibration amplitude with input shaping to that without input shap-
ing at the time of the final impulse. However, in that robustness
evaluation, vibration-suppressing speed due to each residual vi-
bration control has not been considered, which is also an impor-
tant aspect of residual vibration control. In this paper, a perfor-
mance measure including robustness to modeling errors and the
effect of vibration-suppressing speed is defined, and the validity of
the performance measure is demonstrated by simulations and ex-
perimental works. DOI: 10.1115/1.4003377
Keywords: residual vibration, performance measure, robustness
function, input shaping
1 Introduction
Input shaping, a method of residual vibration control, is a feed-
forward control technique for reducing vibrations in undamped or
underdamped systems, which is implemented by convolving a se-
quence of impulses with a desired command. The amplitudes and
time locations of the impulses for input shaping are determined
from the system’s natural frequencies and damping ratios by solv-
ing a set of constraint equations.
The early form of the input shaping called posicast control de-
veloped by Smith 1 in the late 1950s was motivated by a simple
wave cancellation concept for the elimination of the oscillatory
motion of the underdamped system and was unfortunately sensi-
tive to modeling errors of natural frequency and damping ratio.
Because of this sensitivity problem to modeling errors and lack of
microprocessor technology at that time, the posicast control did
not come into widespread use for real systems. However, an input
shaping paper published in 1990 by Singer and Seering 2 re-
newed interest in prefiltering reference inputs for residual vibra-
tion reduction, which is a method that improved robustness to
modeling errors by adding additional constraints on the derivative
of residual vibration magnitudes. Given its robustness, input shap-
ing has been implemented on a variety of systems including
cranes 3, disk drives 4, flexible spacecrafts 5,6, and industrial
robots 7,8 using microprocessor technology. Since Singer and
Seering’s work 2 for linear second-order systems, the input
shaping technique has progressed in such a way that is effective
for multimode systems 9, for nonlinear systems 5,10, for time-
varying systems 11, and for nonlinearities such as Coulomb fric-
tion 12, backlash 13, and on-off thrusters 14.
Robustness to modeling error is an important issue for practical
usefulness of residual vibration control. The robustness of residual
vibration control such as input shaping has conventionally been
evaluated using sensitivity curves plotted from the ratio of re-
sidual vibration amplitude with input shaping to that without input
shaping at the time of the final impulse 4,9,11,15–18. However,
this sensitivity curve does not represent the effect of vibration-
suppressing speed due to each residual vibration control, which is
also an important aspect of the residual vibration control.
In this paper, the robustness function and the performance mea-
sure including the effect of vibration-suppressing speed are de-
fined. The validity of the aforementioned parameters is verified
through simulation studies using SIMULINK models and experi-
mental studies that use an up-down motion control apparatus with
a flexible horizontal beam.
Section 2 briefly describes the idea of input shaping control.
Section 3 defines the robustness function and the performance
measure for residual vibration control and utilizes simulation stud-
ies to verify the definitions. Section 4 describes the design of the
experimental apparatus and demonstrates the validity of the ro-
bustness function and performance measure by experiments. Sec-
tion 5 concludes the paper.
2 Input Shaping Control
There are several ways to suppress unwanted residual vibra-
tions. The most well-known technique is input shaping control.
Figure 1 shows a block diagram for the input shaping control in
which an input shaper convolves command input rt with a se-
quence of impulses. For a single mode vibration system with an
underdamped second-order linear dynamics, the output of the so-
called zero vibration ZV input shaper, r
t, is represented by
r
t = rt
A
1
t - t
1
+ A
2
t - t
2
=
0
t
r · A
1
t - - t
1
+ A
2
t - - t
2
d 1
where
indicates the convolution integral of two functions and
t indicates the Dirac delta function. If rt is given by a step
function c · ht, then
r
t = cA
1
ht - t
1
+ A
2
ht - t
2
2
where ht indicates the Heaviside step function and c is a con-
stant value. The parameters t
1
, t
2
, A
1
, and A
2
in Eq. 2 are ob-
tained from constraint equations given by the ZV shaper 4–6,17.
t
1
= 0, t
2
=
n
1-
2
3
A
1
=
e
/
1-
2
1+ e
/
1-
2
, A
2
=
1
1+ e
/
1-
2
4
where t
1
and t
2
are the time locations of impulses, A
1
and A
2
are
the magnitudes of the impulses, and
n
and are the natural
frequency and damping ratio of the flexible structure, respectively.
Note that t
1
=0 and A
1
+ A
2
=1 without loss of generality. The du-
ration of t
2
is one-half period of the damped vibration.
A zero vibration and derivative ZVD input shaper composed
of three impulses is more robust to modeling errors than the ZV
input shaper. This is achieved by adding additional constraints on
the derivative of residual vibration magnitudes and is given by
4–6,17
t
1
= 0, t
2
=
n
1-
2
, t
3
=
2
n
1-
2
5
A
1
=
e
2/
1-
2
1+ e
/
1-
2
2
, A
2
=
2e
/
1-
2
1+ e
/
1-
2
2
Contributed by the Dynamic Systems Measurement and Control Division of
ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CON-
TROL. Manuscript received June 26, 2010; final manuscript received November 28,
2010; published online March 31, 2011. Assoc. Editor: YangQuan Chen.
Journal of Dynamic Systems, Measurement, and Control JULY 2011, Vol. 133 / 044501-1
Copyright © 2011 by ASME
Downloaded 01 Apr 2011 to 203.252.154.29. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm