INPUT SHAPING CONTROL TO REDUCE RESIDUAL VIBRATION OF A FLEXIBLE BEAM

Abstract

In this paper, three control algorithms based on input shaping method are developed to suppress the residual vibration of a flexible beam. The flexible beam is modeled as an under-damped system. Three input shapers, ZV, ZVD, and ZVDD, are used to control the flexible beam. The three control algorithms are implemented by using the Mechatrolink-III motion system. The experiments are performed to verify the effectiveness of the three control algorithms.

Full text

Journal of Computer Science and Cybernetics, V.32, N.1 (2016), 73–88
DOI: 10.15625/1813-9663/32/1/6765
INPUT SHAPING CONTROL TO REDUCE RESIDUAL
VIBRATION OF A FLEXIBLE BEAM
QUOC CHI NGUYEN1, HA QUANG THINH NGO2
1,2Head of Control and Automation Laboratory, Hochiminh City University of Technology;
1nqchi@hcmut.edu.vn; 2thinhpfiev2003@yahoo.com
Abstract. In this paper, three control algorithms based on input shaping method are developed to
suppress the residual vibration of a flexible beam. The flexible beam is modeled as an under-damped
system. Three input shapers, ZV, ZVD, and ZVDD, are used to control the flexible beam. The three
control algorithms are implemented by using the Mechatrolink-III motion system. The experiments
are performed to verify the effectiveness of the three control algorithms.
Keywords. Flexible beam, input shaping, industrial network, motion control, Mechatrolink-III,
residual vibration.
1. INTRODUCTION
For industrial motion system, the vibration suppression is one of key techniques which re-
searchers and engineers want to address the problem. Especially, with growth of accuracy
manufacturing system, a need for the development of residual vibration technique has there-
fore arisen. One of efforts to suppress residual vibrations has been tried by modifying the
motion profile. The proposed input motion design procedure in [1] was to define asymmetri-
cal S-curve velocity profiles with fast acceleration and slow deceleration of AC servo motors
which intermittently move the punching machine to desired positions with the reduced am-
plitude of residual vibrations. The authors in [2] developed an asymmetric S-curve profile
method with jerk bounded to obtain high precision and reduce the residual vibration. Ha et
al. [3] introduced a jerk ratio to scale down the jerks during the deceleration period. When
the jerk ratio increases, the residual vibration decreases in motion system. Particularly, the
high running speed can cause strong excitations in precision positioning machines. Thus,
in [4], a low-vibration motion profile generation method to lessen systematic excitations was
presented. The acceleration profile is designed by using a level-shifted sinusoidal waveform
to have an S-shape in order to control its change rate. In [5, 6] the constraints between
first natural frequency and deceleration time of motion profile was introduced. In these ex-
periments, the acceleration, constant velocity and deceleration time intervals of trapezoidal
velocity profile are selected by considering the lowest natural frequency at the stopping po-
sition. The root mean square (RMS) values are lowest if the inverse of the deceleration time
equals to the first natural frequency. It is highest if the inverse of the deceleration time
equals to the half of the first natural frequency.
Input shaping strategy is an attractive issue for reducing residual vibration in motion
control system due to the robustness and effectiveness. Since Singhose et al. in [7] described
a method for limiting vibration by adding constraints on the derivative of residual vibration
magnitudes, various approaches to suppress the residual vibrations has been achieved such
c
2015 Vietnam Academy of Science & Technology

74 QUOC CHI NGUYEN, HA QUANG THINH NGO
as hybrid input shaping [8], a three-impulse sequence input shaper [9] or lookup table con-
trol method [10]. However, Singhose’s work performs the robustness to modeling errors and
effectiveness to apply in the industrial motion system. In this method, engineer needs to
determine the natural frequency and damping ratio in order to build an input shaping con-
troller. Recently, researchers in [11] studied the effect of natural frequency error to residual
vibration in three input shaping controller. ZVD and ZVDD shaper are more robust than
ZV shaper if the error in natural frequency of flexible beam exists.
Mechatrolink-III protocol is a real-time protocol that is based on Ethernet technology
and was developed by Yaskawa [12]. Mechatrolink-III is chosen to investigate because of their
advantages such as fast communication, high reliability and rapid design. The performance
of Mechatrolink-III was analyzed in [13] and slave station was designed in [14]. The protocol
guarantees the short cycle time which is considered as a typical factor in motion control
fields.
In this paper, the input shaper ZV, ZVD, and ZVDD are applied to suppress the residual
vibration of a flexible beam when the beam moves to the desired position. The control
algorithm is implemented using Mechatrolink-III. The experiments are carried out to verify
the effectiveness of the control algorithm.
This paper is organized as follows. In Section 2, the model formulation of flexible beam is
presented. Section 3 proposes the implementation of input shaper as an engineer solution. In
Section 4, the hardware development and software development of Mechatrolink-III motion
controller are shown. In Section 5, the experimental verifications and results are carried out.
Finally, this paper ends with conclusions in Section 6.
2. FLEXIBLE BEAM MODEL
Consider a mass mounted on the end of flexible beam as Fig.1 (a). Both of the beam and the
mass of the tip end are made from aluminum alloy. The material parameters of the beam
are as follows. The beam has length Land mass per unit length ρ, as shown in Fig.1 (a).
The modulus of elasticity is E, and the internal bending moment of the beam is I.
Assuming that the end-mass is much greater than the mass of the beam. Define mtas
the total mass at the tip end of the beam. It should be noted that mtcan be computed as
follows.
mt=mbeam +m, (1)
where mbeam is the effective mass of the beam, and mis the mass of the tip end. Define the
effective mass of the beam mbeam as
mbeam =ρZL
0y(x)
ymax(L)2
dx. (2)
Using the first mode, the shape function of the beam (without the mass of the tip end
m) can be obtained as follows.

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 75
Figure 1. (a) Cantilever beam model; (b) free-body diagram; (c) a section of beam
y(x) = ρLg
EI −x3
6+Lx2
2,(3)
where y(x) is the deflection of the static beam.
Substituting Eq. (3) to Eq. (2) yields.
mbeam = 0.23357ρL. (4)
The free-body diagram of beam is described in Fig.1 (b). Let Rbe the reaction force,
and MRbe the reaction bending moment. Applying Newton’s law for static equilibrium
point, the following equations can be obtained.
R−mtg= 0,(5)
MR−mtgL = 0.(6)
In Fig.1 (c), at the left boundary of the flexible beam, the sum of bending moments M
at the right side can be as follows.
M=MR−Rx. (7)
Based on the differential equations for the elastic curve of a beam [13, Eq. (9.12), p.
297], the relation of the bending moment Mand the deflection y(x) of the flexible beam can
be obtained as follows.
M=−EI d2y(x)
dx2.(8)
It should be noted that the curvature is negative. Substituting Eq. (8) into Eq. (7),
results in the following equation at the right boundary.

76 QUOC CHI NGUYEN, HA QUANG THINH NGO
d2y(L)
dx2=−mg
EI (L−x).(9)
Integrating Eq. (9) twice with the zero boundary conditions
dy(L, 0)
dx = 0,(10)
y(L, 0),(11)
The following equation that describes the deflection of the tip of the beam (at right
boundary) is obtained.
y(L) = mtgL3
3EI .(12)
In this paper, the flexible beam with total mass mtat the end is assumed to be a linear
spring with the stiffness.
k=3EI
L3.(13)
The natural frequency ωnof the cantilever beam at the tip end can be computed as.
ωn=1
2πr3EI
mtL3.(14)
Theoretically, the damping ratio ζof an under-damped system in the time domain can
be identified by using logarithmic decrement method in Fig. 2. The amount of logarithmic
decrement δis the natural logarithm of the ratio between the two amplitudes consecutive
peaks xiand xi+1.
δ=1
nln xi
xi+n
.(15)
Then, the damping ratio is defined as follows.
ζ=δ
q(2π)2+δ2
.(16)
3. INPUT SHAPER DESIGN
In this paper, the control objective is to suppress the residual vibration while the beam is
driven to the desired angle. It is assumed that the flexible beam system can be considered
as an under-damped second order system, which can be expressed as the transfer function
G(s).
G(s) = Y(s)
U(s)=ω2
n
s2+ 2ζωns+ω2
n
.(17)
In the time domain, the impulse response of the system is given as

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 77
y(t) = Aωn
p1−ζ2e−ζωn(t−t0)sin ωnp1−ζ2(t−t0),(18)
where Aand t0are the amplitude and the time of impulse, respectively. Basic technical point
of input shaping is described in Fig. 3. When first impulse is applied, it results in vibration
to a system (solid line). Then, if we continue to apply second impulse with appropriate
amplitude and time location, this
Figure 2. Illustration of logarithmic decrement method to calculate the damping ratio
Figure 3. Basic concept of input shaping [7]

78 QUOC CHI NGUYEN, HA QUANG THINH NGO
impulse generates vibration (dash line), which can suppress the vibration caused by first
impulse. This effect results in zero vibration of the system. A sequence of impulses that
caused no vibration when applying to system is called input shaper.
The percentage residual vibration between the single unity-magnitude sequences is given
as [7]
V(ωn, ξ) = e−ξωntnpC(ωn, ξ)2+S(ωn, ξ)2,(19)
where
C(ωn, ξ) =
n
X
i=1
Aieξωnticos(ωdti),(20)
C(ωn, ξ) =
n
X
i=1
Aieξωnticos(ωdti).(21)
In the Eqs. (20) and (21) Aiandtiis the amplitude and time location of the i-th impulse,
nis the number of impulses in the impulse sequence, ωnis natural frequency, ζis damping
ratio, and ωd=ωnp1−ζ2is defined as damped natural frequency.
ZV is the simplest and earliest input shaper, it can be obtained by solving following
constraints [7]:
1) V(ω, ζ ) = 0;
2) PAi= 1;
3) Ai>0;
4) t1= 0;
5) min(tn).
(22)
ZV input shaper is given as the following matrix.
Aj
tj=

1
1 + K
K
1 + K
0 0.5Td

,(23)
where Td=2π/ωdis the damped period of vibration, andK= exp(−ξπ/p1−ξ2).
In this paper, the robust input shaper ZV D [15] is also used to improve the control
performance of the flexible beam system.
Aj
tj=

1
JZV D
0
2K
JZV D
0.5Td
K2
JZV D
Td

,(24)
where
JZV D = 1 + 2K+K2.(25)
To achieve more robustness than ZVD, the input shaper ZVDD [15] is introduced as
follows.
Aj
tj=

1
J
3K
J
3K2
J
K3
J
0 0.5TdTd1.5Td

,(26)

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 79
where
J=K3+ 3K2+ 3K+ 1.(27)
Figure 4. Block diagram of hardware design
4. CONTROL IMPLEMENTATION
In this paper, the controller of the flexible beam system is built by using the Mechatronlink-
III network components, which includes one C1 master and various slaves which can be
servo drive, stepping motor and I/O module. They can be connected in cascade, star or mix
topology if using hub. In the network, C1 master receives command profiles from slaves and
determines the kind of devices. After master sends commands, slaves receive and execute
commands, and later slaves reply their monitoring information. The speed of network to
transmit data is at 100 Mbps. The Mechatrolink-III protocol supports both synchronous
and asynchronous communication modes. In synchronous mode, master station sends the
command data at any required timing and the slave station responds to the sent command
data. Asynchronous communication can be used in a system where synchronous operation
is not needed, i.e., collecting necessary information for synchronous communication from
slaves.
4.1. Hardware development
In this section, the design of Mechatrolink-III master device is introduced. Figure 4 presents
the block diagram of hardware design. The controller communicates with host PC by PCI
interface. Then, FPGA chipset is used for timing synchronization of controller. Based
on the powerful and fast computation, motion generator and other calculations of signals
are embedded into DSP. In this diagram, ARM CPU plays an important role. It handles
data between internal controller and Mechatrolink-III network. Finally, ASIC JL-100A chip
control Mechatrolink-III sending/receiving frame inside network. In Table 1, specifications
of Mechatrolink-III controller are listed.

80 QUOC CHI NGUYEN, HA QUANG THINH NGO
4.2. Software development
In Windows platform, the static library 32-bit is programmed to input data from end-user.
The essential parameters to apply input shaping are shown.
AxmAdvISTSetParameter(lAxis,dFreq,dDampingRatio,dImpulseCount)
where lAxis is the number of axis to control; dFreq and dDampingRatio are natural frequency
and damping ratio of motion system. dImpulseCount displays the number of impulse to
convolve.
dImpulseCount = 2: ZV control scheme
dImpulseCount = 3: ZVD control scheme
dImpulseCount = 4: ZVDD control scheme
AxmAdvISTEnable(lAxis,dEnable)
The value TRUE of dEnable is to activate the input shaping technology, otherwise normal
motion is execute. Once, natural frequency and damping ratio are input, amplitude and time
cycle of impulse are known. A number of impulses depend on input shaping modes. When
input shaping control is activated, a reference data from profile generator is driven into ZV,
ZVD or ZVDD shaper scheme.
In low level, in Fig. 5, a loop which is responsible for the real-time control of data
exchange is defined. The firmware includes the main program and interrupts service routine.
In the first stage, PLL clock, timer, system control register, interrupt service routine and
motion parameters are predetermined. Later, ASIC JL-100A is initialized in three times.
If error still occurs, LED displays to notify. Otherwise, infinite loop that switches between
asynchronous and synchronous mode is used to update data. As shown in Fig. 6, the input
shaping strategies are implemented in ISR. After confirming the error system is in a range,
motion profile is generated. Value of time is computed and compared with sampling time
in system to determine the period of impulse. Then, a mathematical operation is convolved
between two signals as an output of input shaper. The result is transmitted to slave station
in Mechatrolink-III network in order to execute command profile.
Table 1. Specifications of Mechatrolink-III controller
Form Factor PCI (32 bit/33 MHz)
Protocol Communication Mechatrolink-III
Speed 100 Mbps
Transmission Cycle 250 us
Data Frame 48 byte
Support Communication Cyclic, Event-driven
Mechatrolink-III Chipset JL100A
The input shaping for S-curve motion profile is illustrated in Fig. 7. In fact, the con-
ventional motion controller generating S-curve trajectory to drive AC Servo motor yield
vibrations of the beam. In the case of the input shaping applied, the convolution product
between S-curve profile and two-pulse command yield the better results, i.e., the dynamic

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 81
Figure 5. Flowchart of main program
response is smoother, and the residual vibration is suppressed when the motion is completed.
The difference between the conventional and the proposed methods is the suppression of the
vibration at the end of the motion. In the case of the conventional method, the vibration
suppression is based on the viscous damping of the system. This can be considered as a
passive vibration suppression method. In the case of the proposed control method, the vi-
bration eliminated at the time the motion is completed. Therefore, the proposed control
method may yields two advantages: (i) The vibration is suppressed quickly; (ii) The time
for vibration suppression can be predicted.
5. EXPERIMENTAL RESULTS
Experiments were conducted with proposed Mechatrolink-III controller and slave servo drive
as shown in Fig. 8 and its corresponding parameters can be seen in Table 2. An aluminum
thin beam is used as flexible beam. Two bolts are hanged in each side of beam as mass or
load. Then the beam is directly mounted with motor shaft which is vertically suspended on
frame.
In order to identify exact natural frequency of beam, a laser sensor is used to measure
the vibration the flexible beam. The flexible beam rotates around vertical direction and
then, laser sensor detects residual vibration of beam. In Figs. 9 and 10, spectrum frequency
and system vibration that measured by laser sensor are displayed. The measured natural
frequency is 33.3 Hz. Hence, the calculated damping ratio based on logarithmic decrement

82 QUOC CHI NGUYEN, HA QUANG THINH NGO
Figure 6. Flowchart of interrupt service routine
method is 0.008. These parameters are used in all tests. However, depending on each kind
of input shapers, impulse counts are selected as 2, 3 or 4
Table 2. Parameters of experimental motion system
Young’s modulus of beam 70 GPa
Density 2.7 g/cm3
Length of beam 85 mm
Width of beam 15 mm
Height of beam 1.8 mm
Mass of two bolts 1.2 g
The experimental results of the case of ZV, ZVD and ZVDD shapers and of the conven-

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 83
Figure 7. Input shaping for S-curve profile
Figure 8. Experiment system is set-up
tional method are described in Figs. 11, 12, 13 and 14 correspondingly. To illustrate visually,
speed and position of servo motor are plotted on left and right side of vertical axis. Their
units are revolutions per minute (rpm) and the number of rotations (rev) around motor
shaft. In these experiments, angle of the beam is obtained by using the 17-bit incremental

84 QUOC CHI NGUYEN, HA QUANG THINH NGO
Figure 9. Spectrum frequency measured by laser sensor
encoder in the servo motor. Based on a motion planning trajectory, controller generates the
reference (command) speed to drive AC servo motor and obtains the actual speed. Fig. 11
shows the results in the case of the conventional method applied. There is the oscillation in
the actual speed, especially in period of constant velocity and deceleration. When the ZV
shaper is applied, the oscillation appears in the speed signal is decreased in comparison to
the case of the conventional method, as shown in Fig. 12. The ZVD and ZVDD shapers
yield the better results than ZV, as shown in the Figs. 13 and 14, where the oscillations
are suppressed completely. Fig. 15 shows experimental data of torque control in the case
of ZV, ZVD and ZVDD input shapers and of the conventional method. It is obvious that
there is the torque of the conventional method oscillates with a big amplitude There are still
small residual vibrations with the ZV and ZVD shapers, but residual vibrations are zero in
the case of the ZVDD shaper applied The experiments show that input shapers yield good
performance in Mechatrolink-III motion control system
Table 3. Comparison of RMS tracking errors and reduction ratios in test cases
Control Scheme RMS Reduction %
Without Input Shaper 1.4721e-3 -
With ZV Input Shaper 1.0572e-3 28.18
With ZVD Input Shaper 0.7233e-3 50.86
With ZVDD Input Shaper 0.2615e-3 82.23
The comparison of the control performance of the three proposed control methods, which
has been illustrated, is shown in Table 3. It should be noted that the tracking error is an
important factor to evaluate motion control systems. In the case of the conventional control,
the experimental result shows large RMS tracking error. The input shaper ZV, ZVD, and
ZVDD, yield the reduction of the RMS tracking error, where ZVDD generates the best

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 85
Figure 10. System vibration measured by laser sensor
Figure 11. Experimental speed profile without input shaper
control performance.
6. CONCLUSIONS
The three input shaping algorithms for a flexible beam have been presented. A mathematical
model of beam has been used to analyze natural frequency and damping ratio. According

86 QUOC CHI NGUYEN, HA QUANG THINH NGO
Figure 12. Experimental speed profile with ZV shaper
Figure 13. Experimental speed profile with ZVD shaper
to the experimental results, the motion profiles are smooth and reduced residual vibration.
The implementation of the three input shapers is performed by using the Mechatrolink-III
network. It is indicated that the input shapers have provided good control performance.
With large buffer enough in Mechatrolink-III, this successful implementation provides an
opportunity to apply the input shaper for multi-axes in industrial network motion system.

INPUT SHAPING CONTROL TO REDUCE RESIDUAL... 87
Figure 14. Experimental speed profile with ZVDD shaper
Figure 15. Experimental torque control with ZV, ZVD shaper, ZVDD shaper and without
shaper
ACKNOWLEDGMENTS
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.04-2012.37 and by Vietnam National
University Hochiminh City (VNU-HCM) under grant number C2013-20-01.

88 QUOC CHI NGUYEN, HA QUANG THINH NGO
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Received on August 24 - 2015
Revised on May 13 - 2016