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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

0y(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

2πr3EI

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