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Vibrational Error Extraction Method Based on
Wavelet Technique
Loay Alkafafi1, Carsten Hamm1, and Tomas Sauer2
1Siemens AG, Erlangen 91056, Germany,
loay.alkafafi@siemens.com,carsten.hamm@siemens.com,
2University of Passau, Passau 94032, Germany,
tomas.sauer@math.uni-giessen.de
Abstract. A key factor in developing and assessing any vibration atten-
uation technique for elastic systems is the measure that quantifies the oc-
curring vibrations. In this paper, we propose a general and instantaneous
vibration measure which allows for more subtle methods of localized vi-
bration attenuation techniques. This measure is based on extracting the
vibrational part from the conventional tracking error signal using wavelet
technique. The paper also provides a method for constructing a wavelet
function based on the system impulse response. This wavelet outperforms
the existing ones in representing the system behavior while guaranteeing
admissibility and providing sufficient smoothness and rate of decay in
both time and frequency domains.
Keywords: elastic system, vibration attenuation, vibrational error, im-
pulse response, wavelet transform, admissibility.
1 Introduction
Suppressing the vibrations in numerically controlled mechanical systems is a
challenge in many industrial applications. Vibrations typically occur when an
elastic system is driven by positioning commands that contain frequencies close
to the system’s critical frequency. There already exists a variety of techniques to
treat such a problem, the most popular of which is the input shaping technique
[10]. In general, all existing techniques share the same idea of filtering out the
system’s critical frequency from the driving commands.
A primary requirement for any such vibration suppression technique is to de-
rive a measure that quantifies the occurring vibrations. In literature, discussions
are often limited to performance measures for the techniques in use rather than
for the existing vibrations. A very good survey on the available key performance
measures is found in [3]. One of the limitations of the existing performance mea-
sures is that they “represent” the system vibrations merely after the command
completion and only for a specific type of input commands. On the other hand,
an instantaneous measure of vibration allows for more subtle methods for local
corrections of the input command. Providing such a measure will indeed be useful
2
in many practical applications, e.g., the attenuation of vibrations in CNC ma-
chine tools via local modification of their reference motion commands [5], where
a localized algorithm is needed to allocate the critical oscillation regions, and the
assessment of the influence of motion commands on the vibrations of industrial
CNC machine tools [1], where an instantaneous measure for the oscillations of
the machine is needed.
A general vibration measure that characterizes the system vibrations at any
point in time is presently not available. An exception to this can be found in
Barre [1]. In this paper, we propose a general vibration measure based on ex-
tracting the vibrational part from the conventional tracking error signal using
system adapted wavelet technique.
2 Elastic Systems and Vibrational Error
Elastic systems tend to vibrate whenever they are excited by fast motion. Their
oscillatory behavior is described by the vibrational modes which in turn are char-
acterized by an oscillation frequency ω0and a damping ratio ζthat defines how
fast the vibration will decrease in amplitude. The behavior of each vibrational
mode can be generally described by a second order underdamped system which
we will use, without loss of generality, as a representative model for elastic sys-
tems. As a mechanical system, a single-degree-of-freedom mass-spring-damper
system as shown in Fig. 1 will be used. The system is driven by a time vary-
y(t)
x(t) K
d
m
Fig. 1. Single-degree-of-freedom mass-spring-damper system.
ing displacement input function x(t). Using Newtons second law, the ordinary
differential equation (ODE) of the system is given by
m¨y(t) + d˙y(t) + Ky (t) = K x (t).(1)
The dynamical behavior of such a system is described by the quotient between
its output signal y(t) and input signal x(t) in the Laplace domain which is
known as the transfer function of the system. For the above system, the transfer
function is given by
G(s) = Y(s)
X(s)=ω2
0
s2+ 2ζω0s+ω2
0
,(2)
where ω0=qK
mand ζ=d
2√mK . Whenever an elastic system is driven with an
input that contains frequencies close to its natural frequency, oscillating behavior
3
is seen in its response. A conventional measure for the performance of such a
system is the tracking error signal (t) which measures the deviation of the
system response from the input command:
(t) := y(t)−x(t).(3)
By (2), the tracking error signal (3) can also be described in the Laplace domain
as the error model transfer function
E(s)
X(s)=−s(s+ 2ζω0)
s2+ 2ζω0s+ω2
0
.(4)
Fig. 2 shows an example of a tracking error signal for a second order under-
damped system in response to a jerk-limited step command. In technical ap-
plications, jerk is used for the third derivative with respect to time, i.e., the
variation of acceleration. In addition to the vibratory behavior of the system,
0 0.2 0.4 0.6 0.8 1 1.2
−6
−5
−4
−3
−2
−1
0
1
2
x 10−4
²(t) (m)
Time (sec)
Fig. 2. Tracking error signal for second order underdamped system in response to
jerk-limited step command.
the tracking error signal also shows various static deviations from the input com-
mand. In general, two types of error can be distinguished in the tracking error
signal [1]:
1. Aperiodical terms representing errors related to the tracking characteristics
of the system, denoted as ap (t),
4
2. Oscillatory periodical terms vib (t) related to the vibrational behavior of the
system,
where (t) = ap (t) + vib (t). In this paper, only the vibrational behavior of the
system, i.e. the oscillating terms of the tracking error signal, is of interest to us.
The practical reason is that in numerically controlled mechanical systems the
aperiodical terms result from the control loops of the system and can be neither
avoided nor compensated.
In [1], Barre derived a formula to describe the oscillatory terms in the track-
ing error signal as a function of the system parameters and the input command
signal, however, under the assumption that the input command signal can be
broken down into a sequence of steps with well-known amplitudes and time of
occurrence. In the case of jerk-limited commands, this will be a sequence of jerk
steps with amplitudes J= [J1, J2,···, Jn] and step times T= [T1, T2,···, Tn].
Using equation (4) and the assumption above, the tracking error signal is de-
scribed in the Laplace domain by:
E(s) =
n
X
k=1 −Jk
s3
s+ 2ζω0
s2+ 2ζω0s+ω2
0
e−
k
P
j=1
Tj·s
.(5)
From an inverse Laplace transform of (5) the oscillatory terms are then easily
extracted.
Barre’s approach has two main pitfalls. First, the existence of steps with
well-known amplitude and time locations can only be satisfied in theoretical
cases and for very simple input commands. In practical reality, the jerk signal
is often noisy and hard to describe. Thus, describing the jerk signal as needed
by this approach is practically impossible. Secondly, the approach assumes an
exact knowledge of the system transfer function which is also quite impossible
for real applications.
3 The Continuous Wavelet Transform
The continuous wavelet transform (CWT) is the correlation of a time signal with
a dictionary of translated and dilated versions of the analyzing (mother) wavelet
ψ. It decomposes a signal into a time-scale representation that elucidates the
transient characteristics of that signal. To fix notation, we recall that for a finite
energy signal f(t)∈L2(R) the continuous wavelet transform is defined by
W f (b, a) = hf, ψb,ai=
∞
Z
−∞
f(t)1
√aψt−b
adt, (6)
cf. [6] where aand bare the scaling and translation parameters, respectively.
5
Since the continuous wavelet transform is complete and maintains an energy
conservation, an inverse wavelet transform exists and given by
f(t) = 1
Cψ
∞
Z
0
∞
Z
−∞
W f (b, a)1
√aψt−b
adbda
a2.(7)
This inverse formula exists as long as ψsatisfies the admissibility condition
Cψ=
∞
Z
0
ˆ
ψ(ω)
2
ωdω < ∞,(8)
which in particular implies that ˆ
ψ(0) = 0, i.e., Rψ= 0.
4 Balanced Impulse Response Wavelet
One feature that makes wavelets attractive for practical use is the possibility to
select the wavelet function ψsuch that it fits the application at hand. Indeed, in
the literature a number of well-developed wavelet functions can be found which
cover a wide range of applications. For the extraction of vibrational error signals
we tried several wavelet candidates, e.g. Morlet wavelet, Mexican hat wavelet
and impulse response wavelet. All tested wavelets suffer from two main pit-
falls: first, an additional optimization method is always required to optimize the
wavelet shape parameters in order to achieve satisfactory results. Second, most of
the available wavelets are (relatively) symmetric and two-sided wavelets. Thus,
whenever such wavelets are used for reconstructing purposes, additional spurious
oscillating parts will show up in the shape of reconstruction error. Therefore, we
intend to design a new wavelet that overcomes such pitfalls.
Since the oscillatory behavior of an elastic system is characterized by its
impulse response, we construct the wavelet function ψfrom the system impulse
response as a template whose scaled and dilated occurrence we wish to detect
in the given signal. We call such a specific machine adapted wavelet balanced
impulse response wavelet. The impulse response wavelet itself is not new; different
forms of such a wavelet are available in the literature. The starting point for
building such a wavelet is the impulse response of an underdamped second order
system
h(t) = ω0
p1−ζ2e−ζω0tsin (ωdt),(9)
where ωd=ω0p1−ζ2is the damped natural frequency of the system. Since
the system impulse response usually does not satisfy the admissibility condition,
modifications have to be applied. Junsheng [2], for example, modified the impulse
response via direct mirroring to achieve the admissible wavelet
ψ(t) =
e−βωct
√1−β2sin (ωct), t ≥0,
e
βωct
√1−β2sin (ωct), t < 0,
(10)
6
where ωcis the wavelet center frequency and βis a damping or control parameter.
In practice these values are directly related to the damped natural frequency of
the system ωdand the system’s damping ratio ζwhich are normally estimated
from an experimental analysis of the dynamic system.
The Fourier transform of the impulse response wavelet is given by
ˆ
ψ(ξ) = βωc
ip1−β2"1
β2ω2
c
1−β2+i(2πξ −ωc)2−1
β2ω2
c
1−β2+i(2πξ +ωc)2#.(11)
The resulting impulse response wavelet in time and frequency domains is shown
in Fig. 3. By construction, ψis an odd function and thus has zero mean which is
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
ψ(t)
T ime (sec)
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
¯
¯
¯
ˆ
ψ(ξ)¯
¯
¯
F requ ency (Hz)
Fig. 3. Impulse response wavelet in time (left) and frequency (right) domain: with
ωc= 1 H z ,β= 0.2.
the essential part of the admissibility condition. On the other hand, the reflection
clearly results in a two-sided wavelet that will not reproduce the original system
response which, for example, had no symmetry in the beginning.
Our construction will complete a function with a damped oscillation behav-
ior in R+by adding a function with controllable support in R−such that the
resulting function satisfies not only the admissibility condition but also pro-
vides a certain amount of smoothness, so that ψand b
ψboth decay sufficiently
fast. Consequently, with a generalized form of the system impulse response h(t)
where the damping ratio ζis replaced with a general control parameter βas
g(t) = exp −βωc1−β2−1/2tsin (ωct), we define the wavelet function as
ψ(t) =
g(t), t ≥0,
f(t), τ ≤t≤0,
0, t < τ,
(12)
where τ < 0 is a freely chosen parameter that defines the support extension to
the negative axis and controls the time localization properties of the resulting
7
wavelet. Moreover, fis a function from a finite dimensional space that has to
satisfy the balancing condition
Z0
τ
f(t) dt=−Z∞
0
g(t)dt=−1
ωcβ2
1−β2+ 1,(13)
as well as for k= 0,...,n the smoothness conditions
f(k)(τ) = 0,(14)
and
f(k)(0) = g(k)(0) = ωk
cX
j≤(k−1)/2
(−1)k−j−1k
2j+ 1 β
p1−β2!k−2j−1
.(15)
The easiest way to build fis to use polynomial completions. The 2n+ 2 Hermite
conditions in (14) and (15) always have a unique solution in Π2n+1, thus the
complete problem defined by (13), (14) and (15) can be solved in Π2n+1 if the
solution pof (14) and (15) happens to satisfy the balancing condition of (13).
Otherwise, a solution in Π2n+2 is given by
f=p−q
R0
τq(t)dtZ0
τ
p(t)dt+Z∞
0
g(t)dt, q(t) = tn+1 (t−τ)n+1 ,
where q > 0 on (τ , 0), hence, R0
τq(t)dt > 0.
For example, for the case n= 1 the coefficients a0,...,a4of the polynomial
completion are the solutions of the system
1 0 0 0 0
0 1 0 0 0
1τ τ2τ3τ4
0 1 2τ3τ24τ3
−τ−τ2
2−τ3
3−τ4
4−τ5
5
a0
a1
a2
a3
a4
=
0
ωc
0
0
−1
ωcβ2
1−β2+1
,(16)
and the Fourier transform of the resulting wavelet is
b
ψ(ξ) = 1
2i
1
βωc
√1−β2+i(2πξ −ωc)−1
βωc
√1−β2+i(2πξ +ωc)
+a0
i1−e−i2πξτ
2πξ +a1
1−e−i2πξτ (1 −2iπξτ )
(2πξ)2
−a2
2i+e−i2πξτ 4πτ ξ +i4τ2π2ξ2−2
(2πξ)3
−a3
6 + e−i2πξτ 12τ2π2ξ2−6 + i8τ3π3ξ3−12τ πξ
(2πξ)4
+a4
24i−ei2πξτ 32τ3π3ξ3−48τπξ +i16τ4π4ξ4−48τ2π2ξ2+ 24
(2πξ)5,
8
which has a (removable) singularity at ξ= 0, since, due to the balancing property
(13) we have b
ψ(0) = Rψ= 0. Since the practical computation of the wavelet
transform requires a sampling of b
ψ, we recall that the Fourier transform in this
approach can be explicitly computed as
b
ψ(ξ) =
χ[τ,0]
2n+2
X
j=0
aj(·)j
∧
(ξ) + χ[0,∞]e−βωc(1−β2)−1/2·sin (ωc·)∧(ξ)
=
2n+2
X
j=0
aj −j!
(i2πξ)j+1 +e−i2π ξτ
(i2πξ)j+1
j
X
l=0
j! (i2πξτ )l
l!!+ (17)
1
2i
1
βωc
√1−β2+i(2πξ −ωc)−1
βωc
√1−β2+i(2πξ +ωc)
.
The first part of this expression is singular at ξ= 0 and therefore hard to sam-
ple in the neighborhood of the origin. The singularity is only removable due
to the choice of the coefficients ajwhich guarantees that b
ψis uniformly con-
tinuous. This dependency of the coefficients which requires that the numerator
is precisely zero in order to apply the l’Hˆopital rule cannot be maintained in
floating point computations, hence this formula is numerically very unstable in
the neighborhood of the origin. Fortunately, there is a series expansion of the
truncated polynomial which can be used close to the origin.
Lemma 1. For the truncated polynomial function
f=χ[τ,0]
2n+2
X
j=0
aj(·)j,
we define the convergent series representation as
b
f(ξ) = −∞
X
k=0
(iτξ)k
k!
2n+2
X
j=0
aj
τj+1
j+k+ 1.(18)
Proof. We first note that
b
f(k)(0) = ZR
(it)kf(t) dt=ikZ0
τ
2n+2
X
j=0
ajtj+kdt=−(iτ)k
2n+2
X
j=0
aj
τj+1
j+k+ 1.
Substituting this into the Taylor series
b
f(ξ) = ∞
X
k=0 b
f(k)(0)
k!ξk,
which exists since fis compactly supported, hence b
f∈C∞(R), gives (18). The
sum 2n+2
X
j=0 |aj||τ|j+1
j+k+ 1,
9
is bounded independently of kand the remainder of the series is the series
expansion of eiξτ , hence the series converges absolutely. ut
For small values of |τ ξ|, the series in (18) converges very fast and so (18) is
suitable and a very stable way for sampling b
ψclose to the origin, while for
large values of |τξ|, (17) is the more appropriate expression to evaluate. This
observation suggests the use of small values of |τ|which is in accordance with
our application of completing a single-sided wavelet without adding too much
support on the negative side. Of course, smaller values of |τ|will lead to larger
coefficients |aj|, and these numbers will diverge for τ→0.
In addition, Lemma 1 can be used to derive a convenient formula for the
Fourier transform of a spline function, which is easily obtained by shifting each
polynomial piece of the spline to a support interval of the form [τ, 0].
Corollary 1. Let t0<···< tmbe a knot sequence and fa piecewise polynomial
of the form
f=
m
X
`=1
χ[t`−1,t`)
n
X
j=0
a`j(·)k.
Then the Fourier transform of fis
b
f(ξ) =
m
X
`=1
et`ξ∞
X
k=0
(iτ`ξ)k
k!
2n+2
X
j=0
a`j
τj+1
`
j+k+ 1, τ`:= t`−1−t`.(19)
Again, (19) is an alternative to the formula in [7], in particular for frequencies
such that τ`ξis small.
Returning to our application, we first show the resulting balanced impulse
response wavelet for n= 1 in time and frequency domain in Fig. 4. The wavelet
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
ψ(t)
T ime (sec)
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
¯
¯
¯
ˆ
ψ(ξ)¯
¯
¯
F requ ency (Hz)
Fig. 4. Balanced impulse response wavelet in time (left) and frequency (right) domain:
with ωc= 1 H z ,β= 0.2, τ=−0.5.
is indeed a real single-sided one that satisfies the admissibility condition.
10
To compare this to the behavior of the conventional impulse response wavelet
given by (10), we use a test signal containing two impulse responses of a second
order underdamped system with damped natural frequency fd= 25 Hz and
damping ratio ζ= 0.1, defined as
g(t) = H(t−0.2) e−ζ2πfd
√1−ζ2(t−0.2) sin (2πfd(t−0.2)) (20)
+H(t−1) e−ζ2πfd
√1−ζ2(t−1) sin (2πfd(t−1)) ,
where H(·) is the Heaviside function. For this signal, we consider the two wavelet
transforms with an analyzing frequency of 25 Hz and identical center frequency
and damping parameter, ωc= 25 Hz and β= 0.1. For comparing the behavior
of the two wavelets in terms of their localization capabilities, it is sufficient to
consider the wavelet transforms with a single analyzing frequency and a noiseless
test signal so that we can highlight only the effects of the choice of wavelet. A
normalized version of the test signal and the modulus of the wavelet coefficients
is shown in Fig. 5. As the results demonstrate, the wavelet from our above
construction outperforms the conventional one in catching the impulse amplitude
envelope and their time locations. Thus, it provides a much better alternative for
applications where accurate detection of impulses amplitude and time location
are needed.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Amplitude
T ime (sec)
T est S ignal
I mp. W avelet M od ulus
P rop . W av elet M odu lus
Fig. 5. Comparison between conventional impulse response wavelet and the new pro-
posed wavelet: with ωc= 25 H z,β= 0.1, τ=−0.5.
11
5 Vibration Extraction Technique
To extract and remove the vibrational part of the tracking error signal we finally
use a method based on forward and inverse wavelet transforms, which makes
use of the redundant representation of the wavelet transform and its ability to
localize the signal information on the time-scale grid. The use of forward and
backward transforms requires a bit of care, cf. [8, 9], but can be performed in an
efficient and stable way. The method extracts the relevant error information by
performing forward and inverse continuous wavelet transforms on the tracking
error signal at a small number of selected scales only and consists of the following
steps:
1. Define the wavelet shape parameters (ωcand β) as the system damped
natural frequency ωdand damping ratio ζ, respectively.
2. Perform a forward wavelet transform on the tracking error signal (t). The
analyzing scales should cover a small band around the wavelet center fre-
quency, i.e. ω∈[ωc±υ] where ωis the analyzing frequencies and υis a
small percentage from the wavelet center frequency, typically in the range of
5 %.
3. If necessary, a simple soft thresholding can be applied to the resulting wavelet
coefficients for reducing the noise and highlighting interesting error features.
4. Perform inverse wavelet transform on the thresholded wavelet coefficients to
reconstruct the vibrational part of the error signal.
Fig. 6 shows the extracted vibrations from the tracking error signal shown in
Fig. 2 by means of our method. The results shows the capability of this method
to precisely extract the vibrational behavior from the tracking error signal. Us-
ing the balanced impulse wavelet eliminates the need to optimize any obscure
wavelet shape parameters as the parameters are now adapted exactly to the
problem. Furthermore, since the constructed wavelet is essentially single-sided,
the effect of extra side oscillations introduced by the reconstruction in the in-
verse wavelet transform is significantly reduced. This can be nicely seen from the
second oscillation in Fig. 6 where the extracted vibration only has one oscillation
prior to the peak of the error signal.
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12
0 0.2 0.4 0.6 0.8 1 1.2
−6
−5
−4
−3
−2
−1
0
1
2
x 10−4
²(t) (m)
T ime (sec)
Er ror S ignal
Extracted V ibrations
Fig. 6. Tracking error signal and extracted vibrational error using the proposed
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