Research Article Active Vibration Suppression of a 3-DOF Flexible...
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Research Article Active Vibration Suppression of a 3-DOF Flexible Parallel Manipulator Using Efficient Modal Control Quan Zhang, Jiamei Jin, Jianhui Zhang, and Chunsheng Zhao State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Correspondence should be addressed to Quan Zhang; [email protected] Received 1 May 2014; Accepted 20 June 2014; Published 20 October 2014 Academic Editor: Jeong-Hoi Koo Copyright © 2014 Quan Zhang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper addresses the dynamic modeling and efficient modal control of a planar parallel manipulator (PPM) with three flexible linkages actuated by linear ultrasonic motors (LUSM). To achieve active vibration control, multiple lead zirconate titanate (PZT) transducers are mounted on the flexible links as vibration sensors and actuators. Based on Lagrange’s equations, the dynamic model of the flexible links is derived with the dynamics of PZT actuators incorporated. Using the assumed mode method (AMM), the elastic motion of the flexible links are discretized under the assumptions of pinned-free boundary conditions, and the assumed mode shapes are validated through experimental modal test. Efficient modal control (EMC), in which the feedback forces in different modes are determined according to the vibration amplitude or energy of their own, is employed to control the PZT actuators to realize active vibration suppression. Modal filters are developed to extract the modal displacements and velocities from the vibration sensors. Numerical simulation and vibration control experiments are conducted to verify the proposed dynamic model and controller. e results show that the EMC method has the capability of suppressing multimode vibration simultaneously, and both the structural and residual vibrations of the flexible links are effectively suppressed using EMC approach. 1. Introduction With increasing demands of high speed and lightweight manipulators, robots with flexible links are designed and applied in many industrial fields, such as semiconductor manufacturing, astronautics, and automatic microassembly. Compared with traditional rigid manipulator, such robots have the advantages of high speed and acceleration, lower energy consumptions, and greater payload-to-arm weight ratio [1]. However, the vibrations will be introduced in the system due to the flexibility, leading to longer settling time and lower motion tracking accuracy of the end-effector. Hence, vibration control technologies play a critical role in the field of flexible robots. Compared with conventional passive vibration control approaches, active vibration control can suppress multiple vibration modes simultaneously, has the ability to adapt the system and environmental variations, and usually is more effective than passive method [2]. Dur- ing active vibration control process, the vibration actuators bonded on the flexible parts generate forces according to the feedback signals of the displacement or velocity measured by the vibration sensors. If the output voltages of the vibration sensors are amplified properly through the active vibration controller, the actuator forces will increase the stiffness and damping of the flexible parts, thus attenuating the amplitude of the unwanted vibrations. With the development of the PZT materials, more and more flexible structures are mounted with multiple PZT transducers to achieve self-sensing and active vibration control [3, 4]. e dynamic modeling and control of these smart manipulators have been investigated by many researchers. Early investigations have focused primarily on modeling of serial flexible space arms or single flexible beams [5–7], and a detailed review was provided by Dwivedy and Eberhard [8]. In contrast, little research literature on vibration suppression of parallel robot with smart flexible links has been published. Wang and Mills [9] presented a finite element model (FEM) of a PPM with elastic linkages for active vibration analysis using substructure methodology. Piras et al. [10] analyzed the dynamics of a planar fully parallel robot with flexible links using FEM. Zhang et al. [11] Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 953694, 10 pages http://dx.doi.org/10.1155/2014/953694
Transcript of Research Article Active Vibration Suppression of a 3-DOF Flexible...
Research ArticleActive Vibration Suppression of a 3-DOF Flexible ParallelManipulator Using Efficient Modal Control
Quan Zhang Jiamei Jin Jianhui Zhang and Chunsheng Zhao
State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and AstronauticsNanjing 210016 China
Correspondence should be addressed to Quan Zhang quanzhangnuaaeducn
Received 1 May 2014 Accepted 20 June 2014 Published 20 October 2014
Academic Editor Jeong-Hoi Koo
Copyright copy 2014 Quan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper addresses the dynamic modeling and efficient modal control of a planar parallel manipulator (PPM) with three flexiblelinkages actuated by linear ultrasonic motors (LUSM) To achieve active vibration control multiple lead zirconate titanate (PZT)transducers aremounted on the flexible links as vibration sensors and actuators Based on Lagrangersquos equations the dynamicmodelof the flexible links is derived with the dynamics of PZT actuators incorporated Using the assumed mode method (AMM) theelastic motion of the flexible links are discretized under the assumptions of pinned-free boundary conditions and the assumedmode shapes are validated through experimental modal test Efficient modal control (EMC) in which the feedback forces indifferent modes are determined according to the vibration amplitude or energy of their own is employed to control the PZTactuators to realize active vibration suppression Modal filters are developed to extract the modal displacements and velocitiesfrom the vibration sensors Numerical simulation and vibration control experiments are conducted to verify the proposed dynamicmodel and controllerThe results show that the EMCmethod has the capability of suppressingmultimode vibration simultaneouslyand both the structural and residual vibrations of the flexible links are effectively suppressed using EMC approach
1 Introduction
With increasing demands of high speed and lightweightmanipulators robots with flexible links are designed andapplied in many industrial fields such as semiconductormanufacturing astronautics and automatic microassemblyCompared with traditional rigid manipulator such robotshave the advantages of high speed and acceleration lowerenergy consumptions and greater payload-to-arm weightratio [1] However the vibrations will be introduced in thesystem due to the flexibility leading to longer settling timeand lower motion tracking accuracy of the end-effectorHence vibration control technologies play a critical rolein the field of flexible robots Compared with conventionalpassive vibration control approaches active vibration controlcan suppress multiple vibration modes simultaneously hasthe ability to adapt the system and environmental variationsand usually is more effective than passive method [2] Dur-ing active vibration control process the vibration actuatorsbonded on the flexible parts generate forces according to the
feedback signals of the displacement or velocity measured bythe vibration sensors If the output voltages of the vibrationsensors are amplified properly through the active vibrationcontroller the actuator forces will increase the stiffness anddamping of the flexible parts thus attenuating the amplitudeof the unwanted vibrationsWith the development of the PZTmaterials more and more flexible structures are mountedwith multiple PZT transducers to achieve self-sensing andactive vibration control [3 4] The dynamic modeling andcontrol of these smartmanipulators have been investigated bymany researchers Early investigations have focused primarilyon modeling of serial flexible space arms or single flexiblebeams [5ndash7] and a detailed review was provided by Dwivedyand Eberhard [8] In contrast little research literature onvibration suppression of parallel robot with smart flexiblelinks has been published Wang and Mills [9] presented afinite element model (FEM) of a PPM with elastic linkagesfor active vibration analysis using substructure methodologyPiras et al [10] analyzed the dynamics of a planar fullyparallel robot with flexible links using FEM Zhang et al [11]
Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 953694 10 pageshttpdxdoiorg1011552014953694
2 Shock and Vibration
formulated a dynamic model of a PPM based on pinned-pinned boundary conditions based on the assumed modemethod Yu et al [12] conducted both theoretical and exper-imental studies on the dynamic analysis of a 3-RRR flexibleparallel robot However the dynamic modeling of flexibleparallel manipulator is a challenging work due to its complexcoupling characteristics between the rigid motion and elasticdeformation
Based on the dynamic model and vibration transducersa variety of active vibration control approaches have beenstudied since the 1970s In [13] two different vibrationcontrol laws namely velocity feedback control and positiveposition feedback control (PPF) are studied based on a singleflexible beam mounted with single pair of PZT transducerZhang et al [14] conducted strain rate feedback controlto suppress unwanted oscillation of a 3-PRR flexible par-allel manipulator with smart flexible links Combined withinput shaper and multimode PPF controller Chu et al [15]suppressed the residual vibration of a high speed flexiblemanipulator Considering the exact boundary conditionsZhang et al [16] achieved active vibration control of a flexibleparallel robot using input shaping control approach Toachieve high accuracy trajectory tracking while attenuatingthe structural vibration simultaneously Zhang et al [17]implemented variable structural control (VSC) and directoutput feedback control (strain and strain rate feedback) ona flexible parallel manipulator driven by LUSM Howeverusually only a limited small number of modes are selectedto be controlled in practice and hence the uncontrolledmodesmay lead to spillover a phenomenon inwhich controlsenergy flows to the uncontrolled modes of the system [18]To prevent the recoupling of modal equations throughfeedback and spillover Meirovitch and Baruh [19] proposedthe independent modal space control (IMSC) approach Inthis method the control problem of continuous structure canbe understood as controlling several single degree of freedomsystems simultaneously and each mode of the continuousstructure is controlled by one independent controller relatedto its own modal displacement and velocity However thefeedback gains of IMSC are calculated off-line and the controlvoltage for high modes is relatively high When some vibra-tion modes are excited to a greater extent by the disturbancesduring the operation neither priority nor adjustment of feed-back gains is given to these excited modes Based on IMSCBaz and Poh [20] developed modified independent modalspace control (MIMSC) to overcome these disadvantagesthrough controlling the different vibration modes dependingupon its own energy The vibration energy of each modeis identified at every intervals and the mode with highestvibration energy is controlled first Although the differentmodes are prioritized and the applied voltages are decreasedwhen using MIMSC it will put computational efforts onthe digital controller since the MIMSC needs to weigh andcompare the vibration energies of allmodes at every intervalsSingh et al [21] proposed an efficient modal control (EMC)method to suppress multiple vibration modes of a cantileverbeam Compared with IMSC and MIMSC the EMCmethodhas a simpler feedback gains and lower amplitude of controlvoltages which can be readily implemented on a controller
Hence the EMC method is adopted to suppress the vibrationof the flexible link in this study
This paper addresses the dynamic modeling and activevibration control of a PPM with three smart flexible linksFirstly the dynamic model of the flexible links mountedwithmultiple PZT transducers is formulated using Lagrangersquosequations and AMM and the experimental modal tests ofthe flexible links are implemented to verify the assumedmode shapes Then based on the presented dynamic modelthe efficient modal control strategies are adopted to realizethe active vibration control of the flexible links throughmultiple PZT transducers Finally MATLAB simulations andvibration control experiments are provided to validate theeffectiveness of the proposed control approach
2 Dynamic Model of the Flexible LinksIncorporated with PZT Transducers
21 Introduction of the Flexible Parallel Manipulator Con-sidering the beneficial characteristics of parallel structuremanipulator with flexible links a 3-PRR-type PPMwith lightlinks driving by LUSMs is developed to achieve planarmotiontracking and positioning task that is with coordinates of119909-119910-120593 The prototype and coordinates system of the PPMare shown in Figure 1 Variables 120572
119894and 120573
119894(119894 = 1 2 3)
represent the layout angle of the three linear guides and theangle between the 119909-axis of the static frame and 119894th linksrespectively 119871
119894is the length of the linkage The prismatic
motion of the LUSM is defined as 120588119894 The coordinates of
the moving platform are represented as a vector of 119883119901
=
(119909119901 119910119901 120593119901)119879 in the static coordinate system and 119908
119894(119909119894) is
the elastic deformation at the 119909119894(0 le 119909
119894le 119871119894) of the 119894th
linkage The detailed parameter definitions are shown in ourpreliminary study [17]
22 Discretization of the Elastic Motion The elastic motions119908119894in Figure 1 need to be discretized first for further dynamic
analysis Since the length of the flexible link is muchlonger than its thickness the Euler-Bernoulli beam theoryis adopted to model the flexible link and hence only thetransverse vibration of the link is considered According toAMM the elastic motion of the 119894th flexible links can bepresented as
119908119894 (119909 119905) =
119903
sum
119895=1
119884119894119895 (119909) 119902119894119895 (
119905) 119894 = 1 2 3 (1)
where 119902119894119895(119905) represents the unknown generalized coordinates
of the 119895th mode of the 119894th link and 119884119894119895(119909) are the spatial shape
functionsLiteratures [11 16] show that the moving platform will
vibrate fiercely due to the deformation of the flexible link insuch rigid-flexible PPM especially under the operation ofhigh speed and high acceleration Hence we may considerthe beginning of the flexible link is pinned at point 119861
119894while
the end of the link is free but modeled with constraint forcesapplied by joint 119862
119894 Hence the normalized shape functions
Shock and Vibration 3
XO
Y
P
Flexible link
Moving platform
CCDStatic platform
LUSM
Flexible link 1
Flexible link 2
Flexible link 3
Desired motion
LUSM 1 LUSM 2
LUSM 3
1205723
1205721
1205722
1205883
1205881
1205882
A3 A1
A2
B3
B1
B2
1205733
1205732
1205731
120593pC2
C3
C1
w2
w3
w1
X998400Y998400
Figure 1 Prototype and schematic model of the PPM
matching pinned-free boundary conditions are adopted tomodel the flexible links as follows [22]
119884119894119895 (119909)
=
1
radic119898119894119871119894sin 120579119895
[sin(120579119895119909119894
119871119894
) +
sin 120579119895
sinh 120579119895
sinh(120579119895119909119894
119871119894
)]
(2)
where 120579119895= (119895 + 025)120587 119895 = 1 2 119903 and 0 le 119909
119894le 119871119894
23 Modal Testing Experiment To validate the assumptionsof pinned-free mode shapes used in the dynamic modelexperimental modal tests are carried out As shown inFigure 2 the vibration properties of the flexible linkage thatis mode shapes and natural frequencies are identified byusing an impact hammer (PCB 086B02) an accelerometer(PCB 333B32) and a dynamic response analyzer (Agilent35670A) Based on the dynamic response analyzer the firsttwo mode shapes are validated as illustrated in Figure 3 Thefirst natural frequency is 925Hz with a damping ratio of0041 and the second frequency is 2413Hz with a dampingratio of 0013 Figure 3 shows that the estimated mode shapesmatch well with the pinned-free mode shapes but therestill exists some difference at the end of the linkage Thereason is that the elastic motions of the three flexible linksare coupled together through the moving platform Whenone flexible link suffers an impact and starts to vibratethe other two flexible links are also forced to vibrate dueto the coupled dynamic characteristic and the closed-loopnature of the parallel structure This clearly explains why themoving platform generates vibration during the operation
Accelerometer
Impact hammer
Dynamic responseanalyzer
LUSM
Flexible link
(1) Moving platform(2) Static base
Figure 2 Photography of the modal test experiment setup
The detailed analysis of the modal test experiment is shownin [17 23]
24 Dynamic Equations of the Flexible Links Since we focuson the vibration control of the flexible link in this study onlythe dynamic equations of the elastic motion of the flexiblelinks are formulated For the rigid body motions the Kineto-Elasto dynamics (KED) assumptions are employed to providea prescribed rigid motion
4 Shock and Vibration
0 20 40 60 80 100 120 140 160 180 200minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
x coordinate along beam (mm)
Am
plitu
de
e first assumed mode shapee first mode shape measurede second assumed mode shapee second mode shape measured
Figure 3 First two mode shapes of the flexible link identified
The total kinetic energy of the PPM is presented as
119879 =
3
sum
119894=1
1
2
int
119871 119894
0
120588119860119894
119903
2
119894119889119909 +
3
sum
119894=1
1
2
119898119878119894
1205882
119894
+
1
2
119898119901(2
119901+ 1199102
119901) +
1
2
1198681199012
119901
(3)
119903119894= [119909119886119894+ 120588119894cos120572119894+ 119909 cos120573
119894minus 119908119894 (119909 119905) sin120573119894] 119894
+ [119910119886119894+ 120588119894sin120572119894+ 119909 sin120573
119894+ 119908119894 (119909 119905) cos120573119894] 119895
(4)
where the first and the second items of (3) represent thekinetic energies of the flexible links and the sliders respec-tively and the last two items are the kinetic energy of themoving platform Equation (4) is defined as the positionvector of 119909
119894on the 119894th flexible links Variable 119898
119878119894is the mass
of the sliders119898119901is the mass of the moving platform 119868
119901is the
mass moment of the platform 120588119860119894is the mass per unit length
of the 119894th sliders and (119909119886119894 119910119886119894) are the coordinates of point119860
119894
in the static frameSince the manipulator is moving in the 119909-119910 plane the
potential energy changes because gravity is ignored andhence only the potential energy caused by the elastic motionof the flexible links is consideredThe total potential energiesof the PPM are presented as
119881 =
3
sum
119894=1
1
2
int
119871119894
0
119864119894119868119894(11990810158401015840
119894)
2
119889119909 (5)
where 119864119894and 119868119894represent the Young modulus of elasticity
and the second moment of area of the 119894th flexible linkrespectively
To implement the efficient modal control for vibrationcontrol of the PPM multiple PZT transducers are mounted
on the flexible links as vibration sensors and actuators andthe control strategies are shown in Figure 4 According to[24 25] the generalized modal control forces applied byPZT actuators corresponding to themodal coordinates 119902
119894119895are
given as
119865119894119895
119906=
119898
sum
119896=1
120582119886[1198841015840
119894119895(119897119896
2) minus 1198841015840
119894119895(119897119896
1)]119881119894
119886119896(119905) (6)
where 120582119886is the constant of PZT actuator 119897119896
2and 1198971198961are the left
and right end positions of the 119896th PZT actuator respectivelyand119881119894
119886119896(119905) represents the control voltages imposed on the 119896th
PZT actuator of the 119894th flexible linkThe general form of Lagrangersquos equations for elastic
generalized coordinates is given as
119889
119889119905
(
120597 (119879 minus 119881)
120597 119902119894119895
) minus
120597 (119879 minus 119881)
120597119902119894119895
= 119865119894119895
119906 (7)
where 119894 = 1 2 3 and 119895 = 1 2 119903Then substituting (3)ndash(6) into (7) and writing the results
in matrix form with considering 119894 = 1 2 3 yields thefollowing equations
119872119902 + 119870119902 = 119865
119903+ 119865119906 (8)
where 119902 = [11990211sdot sdot sdot 1199021119903
11990221
sdot sdot sdot 1199022119903
11990231
sdot sdot sdot 1199023119903]119879 119865119906=
[11986511
119906sdot sdot sdot 1198651119903
11990211986521
119906sdot sdot sdot 1198652119903
11990611986531
119906sdot sdot sdot 1198653119903
119906]
119879
119865119903is the excita-
tion modal forces arising from the rigid body motion andcoupling effect between elastic and rigid motion and119872 and119870 are the positive mass matrix and stiffness matrix of thethree flexible links respectively The detailed expressions of119872119870 and 119865
119903are given in the appendix
3 Efficient Modal Control
31 Independent Modal Space Controller Since the EMC isbased on the IMSC the IMSC is analyzed in this chapter firstIndependent modal space control has the ability to controleach independentmode instead of controlling the continuousstructures with no coupling among the targeted modesAccording to the dynamic equation (8) the independentmodal equation for the 119895th mode of the 119894th flexible link isgiven as
119902119894119895+ 1205962
119894119895119902119894119895=
(119865119894119895
119903+ 119865119894119895
119906)
119898119894119895
= 119891119894119895
119903+ 119891119894119895
119906 (9)
where120596119894119895reflects themodal frequency for the 119895thmode of the
119894th flexible link and119898119894119895= int
119871 119894
01205881198601198941198842
119894119895119889119909 is a positive constant
With IMSC the modal control force 119891119894119895119906is designed to
only depend on the modal displacement and modal velocityas
119891119894119895
119906= minus119896119894119895
119901119902119894119895minus 119896119894119895
119889119902119894119895 (10)
Shock and Vibration 5
x
l 11 l12 lk1 lk2 lm1 lm2 Ci
Bi
kth1st mth
Vis Vi
a
Efficient modal controlModal
controllerModal filter
kijp kijd (j ne 1) are weighed according to modal
displacement or energy in that mode
Modalsynthesizer
(qi1
qir
)= Wis (Vi
s1
Vism
)
(fi1u
firu
) =(minuski1p qi1 minus ki1d qi1
minuskirp qir minus kird qir
) (Via1
Viam
) = Wia (fi1
u
firu
)
PZT actuator
PZT sensor
Figure 4 Schematic of efficient modal control
Substituting (10) to (9) yields the closed-form equationsfor the 119895th mode of the 119894th flexible link as
119902119894119895+ 119896119894119895
119889119902119894119895+ (1205962
119894119895+ 119896119894119895
119901) 119902119894119895= 119891119894119895
119903 (11)
Equation (11) clearly indicates that the active dampingforces and the positive stiffness forces are imported tothe modal equation through the vibration actuator thusincreasing the damping and stiffness features of the flexiblelinks The initial values of feedback gains of 119896119894119895119901 and 119896
119894119895
119889can be
determined using either pole assignment or optimal controlmethod According to the optimal control approach [26] thecost function which is related to the potential energy 1205962
1198941198951199022
119894119895
kinetic energy 1199022
119894119895 and the required control input (119891119894119895
119906)2 is
selected as
119869 = int
infin
0
[(1205962
1198941198951199022
119894119895+ 1199022
119894119895) + 120582(119891
119894119895
119906)
2
] 119889119905 (12)
where weighting factor 120582 represent a compromise betweenthe required control input and vibration control efficiency
Based on [27] the solution of (12) is expressed as
119896119894119895
119901= (1205962
119894119895+ 120582minus1)
12
minus 120596119894119895
119896119894119895
119889= (2120596
119894119895((1205962
119894119895+ 120582minus1)
12
minus 120596119894119895) + 120582minus1)
12
(13)
32 Efficient Modal Control Since the control gains forhigher vibration modes are much larger than the lowermodes in the IMSC method the applied control voltageswill attain high values if the feedback gains are used directlywithout modifying The goal of the proposed controller isto attenuate the multimode vibration simultaneously withrelatively small control force In EMC method [21] thefeedback gains for each mode are modified according to itsown modal displacement or energy and hence the highermodeswith lower vibration amplitude can be suppressedwith
low damping applied According to EMC only the feedbackgains for the first mode are obtained through optimal controlmethod but the others are optimized using energy weightingmethod as follows
1198961198941
119901 119896119894119895
119901 119896119894119903
119901= 1
119864119894119895
1198641198941
times
1205961198941
120596119894119895
119864119894119903
1198641198941
times
1205961198941
120596119894119903
(14)
where 119864119894119895= 1205962
1198941198951199022
119894119895+ 1199022
119894119895reflects the modal vibration energy for
the 119895th mode of the 119894th link and 119903 represents the number ofmodes to be controlled
33 Modal Filter and Synthesizer Real-time monitoring ofmodal coordinates plays a significant role in design ofmodal feedback controller As anymodal feedback controllerthe modal displacements and velocities are required to beextracted from the vibration sensors in the proposedmethodThe most common methods used to measure the modalcoordinates include state observers temporal filters andmodal filters Compared with state observers and temporalfilters modal filters extract modal coordinates from vibrationsensors independent of the control work and hence it canbe directly applied to any modal feedback controller Fur-thermore modal filters also have the advantage of preventingobservation spillover from the residual modes Thereforemodal filterswith discrete PZT sensors are adopted to providemodal coordinates separation
According to [17 25] the resultant voltage generated inthe PZT sensor is expressed with respect to the charge as
119881119896
119904(119905) =
11988711986411990111988931119905119890119902
119862119897
119901
(
120597119908 (119897119896
2 119905)
120597119909
minus
120597119908 (119897119896
1 119905)
120597119909
) (15)
where 119864119901is the Youngmodulus of PZT sensor 119889
31represents
the constant of PZT materials 119887 is the width of the PZTsensor 119862119897
119901is the equivalent constant of piezoelectric capac-
itance 119905119890119902is defined as the distance between neutral axis of
beamsensor and that of midplane of beam and sensor and
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
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VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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Advances inOptoElectronics
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
2 Shock and Vibration
formulated a dynamic model of a PPM based on pinned-pinned boundary conditions based on the assumed modemethod Yu et al [12] conducted both theoretical and exper-imental studies on the dynamic analysis of a 3-RRR flexibleparallel robot However the dynamic modeling of flexibleparallel manipulator is a challenging work due to its complexcoupling characteristics between the rigid motion and elasticdeformation
Based on the dynamic model and vibration transducersa variety of active vibration control approaches have beenstudied since the 1970s In [13] two different vibrationcontrol laws namely velocity feedback control and positiveposition feedback control (PPF) are studied based on a singleflexible beam mounted with single pair of PZT transducerZhang et al [14] conducted strain rate feedback controlto suppress unwanted oscillation of a 3-PRR flexible par-allel manipulator with smart flexible links Combined withinput shaper and multimode PPF controller Chu et al [15]suppressed the residual vibration of a high speed flexiblemanipulator Considering the exact boundary conditionsZhang et al [16] achieved active vibration control of a flexibleparallel robot using input shaping control approach Toachieve high accuracy trajectory tracking while attenuatingthe structural vibration simultaneously Zhang et al [17]implemented variable structural control (VSC) and directoutput feedback control (strain and strain rate feedback) ona flexible parallel manipulator driven by LUSM Howeverusually only a limited small number of modes are selectedto be controlled in practice and hence the uncontrolledmodesmay lead to spillover a phenomenon inwhich controlsenergy flows to the uncontrolled modes of the system [18]To prevent the recoupling of modal equations throughfeedback and spillover Meirovitch and Baruh [19] proposedthe independent modal space control (IMSC) approach Inthis method the control problem of continuous structure canbe understood as controlling several single degree of freedomsystems simultaneously and each mode of the continuousstructure is controlled by one independent controller relatedto its own modal displacement and velocity However thefeedback gains of IMSC are calculated off-line and the controlvoltage for high modes is relatively high When some vibra-tion modes are excited to a greater extent by the disturbancesduring the operation neither priority nor adjustment of feed-back gains is given to these excited modes Based on IMSCBaz and Poh [20] developed modified independent modalspace control (MIMSC) to overcome these disadvantagesthrough controlling the different vibration modes dependingupon its own energy The vibration energy of each modeis identified at every intervals and the mode with highestvibration energy is controlled first Although the differentmodes are prioritized and the applied voltages are decreasedwhen using MIMSC it will put computational efforts onthe digital controller since the MIMSC needs to weigh andcompare the vibration energies of allmodes at every intervalsSingh et al [21] proposed an efficient modal control (EMC)method to suppress multiple vibration modes of a cantileverbeam Compared with IMSC and MIMSC the EMCmethodhas a simpler feedback gains and lower amplitude of controlvoltages which can be readily implemented on a controller
Hence the EMC method is adopted to suppress the vibrationof the flexible link in this study
This paper addresses the dynamic modeling and activevibration control of a PPM with three smart flexible linksFirstly the dynamic model of the flexible links mountedwithmultiple PZT transducers is formulated using Lagrangersquosequations and AMM and the experimental modal tests ofthe flexible links are implemented to verify the assumedmode shapes Then based on the presented dynamic modelthe efficient modal control strategies are adopted to realizethe active vibration control of the flexible links throughmultiple PZT transducers Finally MATLAB simulations andvibration control experiments are provided to validate theeffectiveness of the proposed control approach
2 Dynamic Model of the Flexible LinksIncorporated with PZT Transducers
21 Introduction of the Flexible Parallel Manipulator Con-sidering the beneficial characteristics of parallel structuremanipulator with flexible links a 3-PRR-type PPMwith lightlinks driving by LUSMs is developed to achieve planarmotiontracking and positioning task that is with coordinates of119909-119910-120593 The prototype and coordinates system of the PPMare shown in Figure 1 Variables 120572
119894and 120573
119894(119894 = 1 2 3)
represent the layout angle of the three linear guides and theangle between the 119909-axis of the static frame and 119894th linksrespectively 119871
119894is the length of the linkage The prismatic
motion of the LUSM is defined as 120588119894 The coordinates of
the moving platform are represented as a vector of 119883119901
=
(119909119901 119910119901 120593119901)119879 in the static coordinate system and 119908
119894(119909119894) is
the elastic deformation at the 119909119894(0 le 119909
119894le 119871119894) of the 119894th
linkage The detailed parameter definitions are shown in ourpreliminary study [17]
22 Discretization of the Elastic Motion The elastic motions119908119894in Figure 1 need to be discretized first for further dynamic
analysis Since the length of the flexible link is muchlonger than its thickness the Euler-Bernoulli beam theoryis adopted to model the flexible link and hence only thetransverse vibration of the link is considered According toAMM the elastic motion of the 119894th flexible links can bepresented as
119908119894 (119909 119905) =
119903
sum
119895=1
119884119894119895 (119909) 119902119894119895 (
119905) 119894 = 1 2 3 (1)
where 119902119894119895(119905) represents the unknown generalized coordinates
of the 119895th mode of the 119894th link and 119884119894119895(119909) are the spatial shape
functionsLiteratures [11 16] show that the moving platform will
vibrate fiercely due to the deformation of the flexible link insuch rigid-flexible PPM especially under the operation ofhigh speed and high acceleration Hence we may considerthe beginning of the flexible link is pinned at point 119861
119894while
the end of the link is free but modeled with constraint forcesapplied by joint 119862
119894 Hence the normalized shape functions
Shock and Vibration 3
XO
Y
P
Flexible link
Moving platform
CCDStatic platform
LUSM
Flexible link 1
Flexible link 2
Flexible link 3
Desired motion
LUSM 1 LUSM 2
LUSM 3
1205723
1205721
1205722
1205883
1205881
1205882
A3 A1
A2
B3
B1
B2
1205733
1205732
1205731
120593pC2
C3
C1
w2
w3
w1
X998400Y998400
Figure 1 Prototype and schematic model of the PPM
matching pinned-free boundary conditions are adopted tomodel the flexible links as follows [22]
119884119894119895 (119909)
=
1
radic119898119894119871119894sin 120579119895
[sin(120579119895119909119894
119871119894
) +
sin 120579119895
sinh 120579119895
sinh(120579119895119909119894
119871119894
)]
(2)
where 120579119895= (119895 + 025)120587 119895 = 1 2 119903 and 0 le 119909
119894le 119871119894
23 Modal Testing Experiment To validate the assumptionsof pinned-free mode shapes used in the dynamic modelexperimental modal tests are carried out As shown inFigure 2 the vibration properties of the flexible linkage thatis mode shapes and natural frequencies are identified byusing an impact hammer (PCB 086B02) an accelerometer(PCB 333B32) and a dynamic response analyzer (Agilent35670A) Based on the dynamic response analyzer the firsttwo mode shapes are validated as illustrated in Figure 3 Thefirst natural frequency is 925Hz with a damping ratio of0041 and the second frequency is 2413Hz with a dampingratio of 0013 Figure 3 shows that the estimated mode shapesmatch well with the pinned-free mode shapes but therestill exists some difference at the end of the linkage Thereason is that the elastic motions of the three flexible linksare coupled together through the moving platform Whenone flexible link suffers an impact and starts to vibratethe other two flexible links are also forced to vibrate dueto the coupled dynamic characteristic and the closed-loopnature of the parallel structure This clearly explains why themoving platform generates vibration during the operation
Accelerometer
Impact hammer
Dynamic responseanalyzer
LUSM
Flexible link
(1) Moving platform(2) Static base
Figure 2 Photography of the modal test experiment setup
The detailed analysis of the modal test experiment is shownin [17 23]
24 Dynamic Equations of the Flexible Links Since we focuson the vibration control of the flexible link in this study onlythe dynamic equations of the elastic motion of the flexiblelinks are formulated For the rigid body motions the Kineto-Elasto dynamics (KED) assumptions are employed to providea prescribed rigid motion
4 Shock and Vibration
0 20 40 60 80 100 120 140 160 180 200minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
x coordinate along beam (mm)
Am
plitu
de
e first assumed mode shapee first mode shape measurede second assumed mode shapee second mode shape measured
Figure 3 First two mode shapes of the flexible link identified
The total kinetic energy of the PPM is presented as
119879 =
3
sum
119894=1
1
2
int
119871 119894
0
120588119860119894
119903
2
119894119889119909 +
3
sum
119894=1
1
2
119898119878119894
1205882
119894
+
1
2
119898119901(2
119901+ 1199102
119901) +
1
2
1198681199012
119901
(3)
119903119894= [119909119886119894+ 120588119894cos120572119894+ 119909 cos120573
119894minus 119908119894 (119909 119905) sin120573119894] 119894
+ [119910119886119894+ 120588119894sin120572119894+ 119909 sin120573
119894+ 119908119894 (119909 119905) cos120573119894] 119895
(4)
where the first and the second items of (3) represent thekinetic energies of the flexible links and the sliders respec-tively and the last two items are the kinetic energy of themoving platform Equation (4) is defined as the positionvector of 119909
119894on the 119894th flexible links Variable 119898
119878119894is the mass
of the sliders119898119901is the mass of the moving platform 119868
119901is the
mass moment of the platform 120588119860119894is the mass per unit length
of the 119894th sliders and (119909119886119894 119910119886119894) are the coordinates of point119860
119894
in the static frameSince the manipulator is moving in the 119909-119910 plane the
potential energy changes because gravity is ignored andhence only the potential energy caused by the elastic motionof the flexible links is consideredThe total potential energiesof the PPM are presented as
119881 =
3
sum
119894=1
1
2
int
119871119894
0
119864119894119868119894(11990810158401015840
119894)
2
119889119909 (5)
where 119864119894and 119868119894represent the Young modulus of elasticity
and the second moment of area of the 119894th flexible linkrespectively
To implement the efficient modal control for vibrationcontrol of the PPM multiple PZT transducers are mounted
on the flexible links as vibration sensors and actuators andthe control strategies are shown in Figure 4 According to[24 25] the generalized modal control forces applied byPZT actuators corresponding to themodal coordinates 119902
119894119895are
given as
119865119894119895
119906=
119898
sum
119896=1
120582119886[1198841015840
119894119895(119897119896
2) minus 1198841015840
119894119895(119897119896
1)]119881119894
119886119896(119905) (6)
where 120582119886is the constant of PZT actuator 119897119896
2and 1198971198961are the left
and right end positions of the 119896th PZT actuator respectivelyand119881119894
119886119896(119905) represents the control voltages imposed on the 119896th
PZT actuator of the 119894th flexible linkThe general form of Lagrangersquos equations for elastic
generalized coordinates is given as
119889
119889119905
(
120597 (119879 minus 119881)
120597 119902119894119895
) minus
120597 (119879 minus 119881)
120597119902119894119895
= 119865119894119895
119906 (7)
where 119894 = 1 2 3 and 119895 = 1 2 119903Then substituting (3)ndash(6) into (7) and writing the results
in matrix form with considering 119894 = 1 2 3 yields thefollowing equations
119872119902 + 119870119902 = 119865
119903+ 119865119906 (8)
where 119902 = [11990211sdot sdot sdot 1199021119903
11990221
sdot sdot sdot 1199022119903
11990231
sdot sdot sdot 1199023119903]119879 119865119906=
[11986511
119906sdot sdot sdot 1198651119903
11990211986521
119906sdot sdot sdot 1198652119903
11990611986531
119906sdot sdot sdot 1198653119903
119906]
119879
119865119903is the excita-
tion modal forces arising from the rigid body motion andcoupling effect between elastic and rigid motion and119872 and119870 are the positive mass matrix and stiffness matrix of thethree flexible links respectively The detailed expressions of119872119870 and 119865
119903are given in the appendix
3 Efficient Modal Control
31 Independent Modal Space Controller Since the EMC isbased on the IMSC the IMSC is analyzed in this chapter firstIndependent modal space control has the ability to controleach independentmode instead of controlling the continuousstructures with no coupling among the targeted modesAccording to the dynamic equation (8) the independentmodal equation for the 119895th mode of the 119894th flexible link isgiven as
119902119894119895+ 1205962
119894119895119902119894119895=
(119865119894119895
119903+ 119865119894119895
119906)
119898119894119895
= 119891119894119895
119903+ 119891119894119895
119906 (9)
where120596119894119895reflects themodal frequency for the 119895thmode of the
119894th flexible link and119898119894119895= int
119871 119894
01205881198601198941198842
119894119895119889119909 is a positive constant
With IMSC the modal control force 119891119894119895119906is designed to
only depend on the modal displacement and modal velocityas
119891119894119895
119906= minus119896119894119895
119901119902119894119895minus 119896119894119895
119889119902119894119895 (10)
Shock and Vibration 5
x
l 11 l12 lk1 lk2 lm1 lm2 Ci
Bi
kth1st mth
Vis Vi
a
Efficient modal controlModal
controllerModal filter
kijp kijd (j ne 1) are weighed according to modal
displacement or energy in that mode
Modalsynthesizer
(qi1
qir
)= Wis (Vi
s1
Vism
)
(fi1u
firu
) =(minuski1p qi1 minus ki1d qi1
minuskirp qir minus kird qir
) (Via1
Viam
) = Wia (fi1
u
firu
)
PZT actuator
PZT sensor
Figure 4 Schematic of efficient modal control
Substituting (10) to (9) yields the closed-form equationsfor the 119895th mode of the 119894th flexible link as
119902119894119895+ 119896119894119895
119889119902119894119895+ (1205962
119894119895+ 119896119894119895
119901) 119902119894119895= 119891119894119895
119903 (11)
Equation (11) clearly indicates that the active dampingforces and the positive stiffness forces are imported tothe modal equation through the vibration actuator thusincreasing the damping and stiffness features of the flexiblelinks The initial values of feedback gains of 119896119894119895119901 and 119896
119894119895
119889can be
determined using either pole assignment or optimal controlmethod According to the optimal control approach [26] thecost function which is related to the potential energy 1205962
1198941198951199022
119894119895
kinetic energy 1199022
119894119895 and the required control input (119891119894119895
119906)2 is
selected as
119869 = int
infin
0
[(1205962
1198941198951199022
119894119895+ 1199022
119894119895) + 120582(119891
119894119895
119906)
2
] 119889119905 (12)
where weighting factor 120582 represent a compromise betweenthe required control input and vibration control efficiency
Based on [27] the solution of (12) is expressed as
119896119894119895
119901= (1205962
119894119895+ 120582minus1)
12
minus 120596119894119895
119896119894119895
119889= (2120596
119894119895((1205962
119894119895+ 120582minus1)
12
minus 120596119894119895) + 120582minus1)
12
(13)
32 Efficient Modal Control Since the control gains forhigher vibration modes are much larger than the lowermodes in the IMSC method the applied control voltageswill attain high values if the feedback gains are used directlywithout modifying The goal of the proposed controller isto attenuate the multimode vibration simultaneously withrelatively small control force In EMC method [21] thefeedback gains for each mode are modified according to itsown modal displacement or energy and hence the highermodeswith lower vibration amplitude can be suppressedwith
low damping applied According to EMC only the feedbackgains for the first mode are obtained through optimal controlmethod but the others are optimized using energy weightingmethod as follows
1198961198941
119901 119896119894119895
119901 119896119894119903
119901= 1
119864119894119895
1198641198941
times
1205961198941
120596119894119895
119864119894119903
1198641198941
times
1205961198941
120596119894119903
(14)
where 119864119894119895= 1205962
1198941198951199022
119894119895+ 1199022
119894119895reflects the modal vibration energy for
the 119895th mode of the 119894th link and 119903 represents the number ofmodes to be controlled
33 Modal Filter and Synthesizer Real-time monitoring ofmodal coordinates plays a significant role in design ofmodal feedback controller As anymodal feedback controllerthe modal displacements and velocities are required to beextracted from the vibration sensors in the proposedmethodThe most common methods used to measure the modalcoordinates include state observers temporal filters andmodal filters Compared with state observers and temporalfilters modal filters extract modal coordinates from vibrationsensors independent of the control work and hence it canbe directly applied to any modal feedback controller Fur-thermore modal filters also have the advantage of preventingobservation spillover from the residual modes Thereforemodal filterswith discrete PZT sensors are adopted to providemodal coordinates separation
According to [17 25] the resultant voltage generated inthe PZT sensor is expressed with respect to the charge as
119881119896
119904(119905) =
11988711986411990111988931119905119890119902
119862119897
119901
(
120597119908 (119897119896
2 119905)
120597119909
minus
120597119908 (119897119896
1 119905)
120597119909
) (15)
where 119864119901is the Youngmodulus of PZT sensor 119889
31represents
the constant of PZT materials 119887 is the width of the PZTsensor 119862119897
119901is the equivalent constant of piezoelectric capac-
itance 119905119890119902is defined as the distance between neutral axis of
beamsensor and that of midplane of beam and sensor and
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
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Active and Passive Electronic Components
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Shock and Vibration 3
XO
Y
P
Flexible link
Moving platform
CCDStatic platform
LUSM
Flexible link 1
Flexible link 2
Flexible link 3
Desired motion
LUSM 1 LUSM 2
LUSM 3
1205723
1205721
1205722
1205883
1205881
1205882
A3 A1
A2
B3
B1
B2
1205733
1205732
1205731
120593pC2
C3
C1
w2
w3
w1
X998400Y998400
Figure 1 Prototype and schematic model of the PPM
matching pinned-free boundary conditions are adopted tomodel the flexible links as follows [22]
119884119894119895 (119909)
=
1
radic119898119894119871119894sin 120579119895
[sin(120579119895119909119894
119871119894
) +
sin 120579119895
sinh 120579119895
sinh(120579119895119909119894
119871119894
)]
(2)
where 120579119895= (119895 + 025)120587 119895 = 1 2 119903 and 0 le 119909
119894le 119871119894
23 Modal Testing Experiment To validate the assumptionsof pinned-free mode shapes used in the dynamic modelexperimental modal tests are carried out As shown inFigure 2 the vibration properties of the flexible linkage thatis mode shapes and natural frequencies are identified byusing an impact hammer (PCB 086B02) an accelerometer(PCB 333B32) and a dynamic response analyzer (Agilent35670A) Based on the dynamic response analyzer the firsttwo mode shapes are validated as illustrated in Figure 3 Thefirst natural frequency is 925Hz with a damping ratio of0041 and the second frequency is 2413Hz with a dampingratio of 0013 Figure 3 shows that the estimated mode shapesmatch well with the pinned-free mode shapes but therestill exists some difference at the end of the linkage Thereason is that the elastic motions of the three flexible linksare coupled together through the moving platform Whenone flexible link suffers an impact and starts to vibratethe other two flexible links are also forced to vibrate dueto the coupled dynamic characteristic and the closed-loopnature of the parallel structure This clearly explains why themoving platform generates vibration during the operation
Accelerometer
Impact hammer
Dynamic responseanalyzer
LUSM
Flexible link
(1) Moving platform(2) Static base
Figure 2 Photography of the modal test experiment setup
The detailed analysis of the modal test experiment is shownin [17 23]
24 Dynamic Equations of the Flexible Links Since we focuson the vibration control of the flexible link in this study onlythe dynamic equations of the elastic motion of the flexiblelinks are formulated For the rigid body motions the Kineto-Elasto dynamics (KED) assumptions are employed to providea prescribed rigid motion
4 Shock and Vibration
0 20 40 60 80 100 120 140 160 180 200minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
x coordinate along beam (mm)
Am
plitu
de
e first assumed mode shapee first mode shape measurede second assumed mode shapee second mode shape measured
Figure 3 First two mode shapes of the flexible link identified
The total kinetic energy of the PPM is presented as
119879 =
3
sum
119894=1
1
2
int
119871 119894
0
120588119860119894
119903
2
119894119889119909 +
3
sum
119894=1
1
2
119898119878119894
1205882
119894
+
1
2
119898119901(2
119901+ 1199102
119901) +
1
2
1198681199012
119901
(3)
119903119894= [119909119886119894+ 120588119894cos120572119894+ 119909 cos120573
119894minus 119908119894 (119909 119905) sin120573119894] 119894
+ [119910119886119894+ 120588119894sin120572119894+ 119909 sin120573
119894+ 119908119894 (119909 119905) cos120573119894] 119895
(4)
where the first and the second items of (3) represent thekinetic energies of the flexible links and the sliders respec-tively and the last two items are the kinetic energy of themoving platform Equation (4) is defined as the positionvector of 119909
119894on the 119894th flexible links Variable 119898
119878119894is the mass
of the sliders119898119901is the mass of the moving platform 119868
119901is the
mass moment of the platform 120588119860119894is the mass per unit length
of the 119894th sliders and (119909119886119894 119910119886119894) are the coordinates of point119860
119894
in the static frameSince the manipulator is moving in the 119909-119910 plane the
potential energy changes because gravity is ignored andhence only the potential energy caused by the elastic motionof the flexible links is consideredThe total potential energiesof the PPM are presented as
119881 =
3
sum
119894=1
1
2
int
119871119894
0
119864119894119868119894(11990810158401015840
119894)
2
119889119909 (5)
where 119864119894and 119868119894represent the Young modulus of elasticity
and the second moment of area of the 119894th flexible linkrespectively
To implement the efficient modal control for vibrationcontrol of the PPM multiple PZT transducers are mounted
on the flexible links as vibration sensors and actuators andthe control strategies are shown in Figure 4 According to[24 25] the generalized modal control forces applied byPZT actuators corresponding to themodal coordinates 119902
119894119895are
given as
119865119894119895
119906=
119898
sum
119896=1
120582119886[1198841015840
119894119895(119897119896
2) minus 1198841015840
119894119895(119897119896
1)]119881119894
119886119896(119905) (6)
where 120582119886is the constant of PZT actuator 119897119896
2and 1198971198961are the left
and right end positions of the 119896th PZT actuator respectivelyand119881119894
119886119896(119905) represents the control voltages imposed on the 119896th
PZT actuator of the 119894th flexible linkThe general form of Lagrangersquos equations for elastic
generalized coordinates is given as
119889
119889119905
(
120597 (119879 minus 119881)
120597 119902119894119895
) minus
120597 (119879 minus 119881)
120597119902119894119895
= 119865119894119895
119906 (7)
where 119894 = 1 2 3 and 119895 = 1 2 119903Then substituting (3)ndash(6) into (7) and writing the results
in matrix form with considering 119894 = 1 2 3 yields thefollowing equations
119872119902 + 119870119902 = 119865
119903+ 119865119906 (8)
where 119902 = [11990211sdot sdot sdot 1199021119903
11990221
sdot sdot sdot 1199022119903
11990231
sdot sdot sdot 1199023119903]119879 119865119906=
[11986511
119906sdot sdot sdot 1198651119903
11990211986521
119906sdot sdot sdot 1198652119903
11990611986531
119906sdot sdot sdot 1198653119903
119906]
119879
119865119903is the excita-
tion modal forces arising from the rigid body motion andcoupling effect between elastic and rigid motion and119872 and119870 are the positive mass matrix and stiffness matrix of thethree flexible links respectively The detailed expressions of119872119870 and 119865
119903are given in the appendix
3 Efficient Modal Control
31 Independent Modal Space Controller Since the EMC isbased on the IMSC the IMSC is analyzed in this chapter firstIndependent modal space control has the ability to controleach independentmode instead of controlling the continuousstructures with no coupling among the targeted modesAccording to the dynamic equation (8) the independentmodal equation for the 119895th mode of the 119894th flexible link isgiven as
119902119894119895+ 1205962
119894119895119902119894119895=
(119865119894119895
119903+ 119865119894119895
119906)
119898119894119895
= 119891119894119895
119903+ 119891119894119895
119906 (9)
where120596119894119895reflects themodal frequency for the 119895thmode of the
119894th flexible link and119898119894119895= int
119871 119894
01205881198601198941198842
119894119895119889119909 is a positive constant
With IMSC the modal control force 119891119894119895119906is designed to
only depend on the modal displacement and modal velocityas
119891119894119895
119906= minus119896119894119895
119901119902119894119895minus 119896119894119895
119889119902119894119895 (10)
Shock and Vibration 5
x
l 11 l12 lk1 lk2 lm1 lm2 Ci
Bi
kth1st mth
Vis Vi
a
Efficient modal controlModal
controllerModal filter
kijp kijd (j ne 1) are weighed according to modal
displacement or energy in that mode
Modalsynthesizer
(qi1
qir
)= Wis (Vi
s1
Vism
)
(fi1u
firu
) =(minuski1p qi1 minus ki1d qi1
minuskirp qir minus kird qir
) (Via1
Viam
) = Wia (fi1
u
firu
)
PZT actuator
PZT sensor
Figure 4 Schematic of efficient modal control
Substituting (10) to (9) yields the closed-form equationsfor the 119895th mode of the 119894th flexible link as
119902119894119895+ 119896119894119895
119889119902119894119895+ (1205962
119894119895+ 119896119894119895
119901) 119902119894119895= 119891119894119895
119903 (11)
Equation (11) clearly indicates that the active dampingforces and the positive stiffness forces are imported tothe modal equation through the vibration actuator thusincreasing the damping and stiffness features of the flexiblelinks The initial values of feedback gains of 119896119894119895119901 and 119896
119894119895
119889can be
determined using either pole assignment or optimal controlmethod According to the optimal control approach [26] thecost function which is related to the potential energy 1205962
1198941198951199022
119894119895
kinetic energy 1199022
119894119895 and the required control input (119891119894119895
119906)2 is
selected as
119869 = int
infin
0
[(1205962
1198941198951199022
119894119895+ 1199022
119894119895) + 120582(119891
119894119895
119906)
2
] 119889119905 (12)
where weighting factor 120582 represent a compromise betweenthe required control input and vibration control efficiency
Based on [27] the solution of (12) is expressed as
119896119894119895
119901= (1205962
119894119895+ 120582minus1)
12
minus 120596119894119895
119896119894119895
119889= (2120596
119894119895((1205962
119894119895+ 120582minus1)
12
minus 120596119894119895) + 120582minus1)
12
(13)
32 Efficient Modal Control Since the control gains forhigher vibration modes are much larger than the lowermodes in the IMSC method the applied control voltageswill attain high values if the feedback gains are used directlywithout modifying The goal of the proposed controller isto attenuate the multimode vibration simultaneously withrelatively small control force In EMC method [21] thefeedback gains for each mode are modified according to itsown modal displacement or energy and hence the highermodeswith lower vibration amplitude can be suppressedwith
low damping applied According to EMC only the feedbackgains for the first mode are obtained through optimal controlmethod but the others are optimized using energy weightingmethod as follows
1198961198941
119901 119896119894119895
119901 119896119894119903
119901= 1
119864119894119895
1198641198941
times
1205961198941
120596119894119895
119864119894119903
1198641198941
times
1205961198941
120596119894119903
(14)
where 119864119894119895= 1205962
1198941198951199022
119894119895+ 1199022
119894119895reflects the modal vibration energy for
the 119895th mode of the 119894th link and 119903 represents the number ofmodes to be controlled
33 Modal Filter and Synthesizer Real-time monitoring ofmodal coordinates plays a significant role in design ofmodal feedback controller As anymodal feedback controllerthe modal displacements and velocities are required to beextracted from the vibration sensors in the proposedmethodThe most common methods used to measure the modalcoordinates include state observers temporal filters andmodal filters Compared with state observers and temporalfilters modal filters extract modal coordinates from vibrationsensors independent of the control work and hence it canbe directly applied to any modal feedback controller Fur-thermore modal filters also have the advantage of preventingobservation spillover from the residual modes Thereforemodal filterswith discrete PZT sensors are adopted to providemodal coordinates separation
According to [17 25] the resultant voltage generated inthe PZT sensor is expressed with respect to the charge as
119881119896
119904(119905) =
11988711986411990111988931119905119890119902
119862119897
119901
(
120597119908 (119897119896
2 119905)
120597119909
minus
120597119908 (119897119896
1 119905)
120597119909
) (15)
where 119864119901is the Youngmodulus of PZT sensor 119889
31represents
the constant of PZT materials 119887 is the width of the PZTsensor 119862119897
119901is the equivalent constant of piezoelectric capac-
itance 119905119890119902is defined as the distance between neutral axis of
beamsensor and that of midplane of beam and sensor and
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
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Active and Passive Electronic Components
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
4 Shock and Vibration
0 20 40 60 80 100 120 140 160 180 200minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
x coordinate along beam (mm)
Am
plitu
de
e first assumed mode shapee first mode shape measurede second assumed mode shapee second mode shape measured
Figure 3 First two mode shapes of the flexible link identified
The total kinetic energy of the PPM is presented as
119879 =
3
sum
119894=1
1
2
int
119871 119894
0
120588119860119894
119903
2
119894119889119909 +
3
sum
119894=1
1
2
119898119878119894
1205882
119894
+
1
2
119898119901(2
119901+ 1199102
119901) +
1
2
1198681199012
119901
(3)
119903119894= [119909119886119894+ 120588119894cos120572119894+ 119909 cos120573
119894minus 119908119894 (119909 119905) sin120573119894] 119894
+ [119910119886119894+ 120588119894sin120572119894+ 119909 sin120573
119894+ 119908119894 (119909 119905) cos120573119894] 119895
(4)
where the first and the second items of (3) represent thekinetic energies of the flexible links and the sliders respec-tively and the last two items are the kinetic energy of themoving platform Equation (4) is defined as the positionvector of 119909
119894on the 119894th flexible links Variable 119898
119878119894is the mass
of the sliders119898119901is the mass of the moving platform 119868
119901is the
mass moment of the platform 120588119860119894is the mass per unit length
of the 119894th sliders and (119909119886119894 119910119886119894) are the coordinates of point119860
119894
in the static frameSince the manipulator is moving in the 119909-119910 plane the
potential energy changes because gravity is ignored andhence only the potential energy caused by the elastic motionof the flexible links is consideredThe total potential energiesof the PPM are presented as
119881 =
3
sum
119894=1
1
2
int
119871119894
0
119864119894119868119894(11990810158401015840
119894)
2
119889119909 (5)
where 119864119894and 119868119894represent the Young modulus of elasticity
and the second moment of area of the 119894th flexible linkrespectively
To implement the efficient modal control for vibrationcontrol of the PPM multiple PZT transducers are mounted
on the flexible links as vibration sensors and actuators andthe control strategies are shown in Figure 4 According to[24 25] the generalized modal control forces applied byPZT actuators corresponding to themodal coordinates 119902
119894119895are
given as
119865119894119895
119906=
119898
sum
119896=1
120582119886[1198841015840
119894119895(119897119896
2) minus 1198841015840
119894119895(119897119896
1)]119881119894
119886119896(119905) (6)
where 120582119886is the constant of PZT actuator 119897119896
2and 1198971198961are the left
and right end positions of the 119896th PZT actuator respectivelyand119881119894
119886119896(119905) represents the control voltages imposed on the 119896th
PZT actuator of the 119894th flexible linkThe general form of Lagrangersquos equations for elastic
generalized coordinates is given as
119889
119889119905
(
120597 (119879 minus 119881)
120597 119902119894119895
) minus
120597 (119879 minus 119881)
120597119902119894119895
= 119865119894119895
119906 (7)
where 119894 = 1 2 3 and 119895 = 1 2 119903Then substituting (3)ndash(6) into (7) and writing the results
in matrix form with considering 119894 = 1 2 3 yields thefollowing equations
119872119902 + 119870119902 = 119865
119903+ 119865119906 (8)
where 119902 = [11990211sdot sdot sdot 1199021119903
11990221
sdot sdot sdot 1199022119903
11990231
sdot sdot sdot 1199023119903]119879 119865119906=
[11986511
119906sdot sdot sdot 1198651119903
11990211986521
119906sdot sdot sdot 1198652119903
11990611986531
119906sdot sdot sdot 1198653119903
119906]
119879
119865119903is the excita-
tion modal forces arising from the rigid body motion andcoupling effect between elastic and rigid motion and119872 and119870 are the positive mass matrix and stiffness matrix of thethree flexible links respectively The detailed expressions of119872119870 and 119865
119903are given in the appendix
3 Efficient Modal Control
31 Independent Modal Space Controller Since the EMC isbased on the IMSC the IMSC is analyzed in this chapter firstIndependent modal space control has the ability to controleach independentmode instead of controlling the continuousstructures with no coupling among the targeted modesAccording to the dynamic equation (8) the independentmodal equation for the 119895th mode of the 119894th flexible link isgiven as
119902119894119895+ 1205962
119894119895119902119894119895=
(119865119894119895
119903+ 119865119894119895
119906)
119898119894119895
= 119891119894119895
119903+ 119891119894119895
119906 (9)
where120596119894119895reflects themodal frequency for the 119895thmode of the
119894th flexible link and119898119894119895= int
119871 119894
01205881198601198941198842
119894119895119889119909 is a positive constant
With IMSC the modal control force 119891119894119895119906is designed to
only depend on the modal displacement and modal velocityas
119891119894119895
119906= minus119896119894119895
119901119902119894119895minus 119896119894119895
119889119902119894119895 (10)
Shock and Vibration 5
x
l 11 l12 lk1 lk2 lm1 lm2 Ci
Bi
kth1st mth
Vis Vi
a
Efficient modal controlModal
controllerModal filter
kijp kijd (j ne 1) are weighed according to modal
displacement or energy in that mode
Modalsynthesizer
(qi1
qir
)= Wis (Vi
s1
Vism
)
(fi1u
firu
) =(minuski1p qi1 minus ki1d qi1
minuskirp qir minus kird qir
) (Via1
Viam
) = Wia (fi1
u
firu
)
PZT actuator
PZT sensor
Figure 4 Schematic of efficient modal control
Substituting (10) to (9) yields the closed-form equationsfor the 119895th mode of the 119894th flexible link as
119902119894119895+ 119896119894119895
119889119902119894119895+ (1205962
119894119895+ 119896119894119895
119901) 119902119894119895= 119891119894119895
119903 (11)
Equation (11) clearly indicates that the active dampingforces and the positive stiffness forces are imported tothe modal equation through the vibration actuator thusincreasing the damping and stiffness features of the flexiblelinks The initial values of feedback gains of 119896119894119895119901 and 119896
119894119895
119889can be
determined using either pole assignment or optimal controlmethod According to the optimal control approach [26] thecost function which is related to the potential energy 1205962
1198941198951199022
119894119895
kinetic energy 1199022
119894119895 and the required control input (119891119894119895
119906)2 is
selected as
119869 = int
infin
0
[(1205962
1198941198951199022
119894119895+ 1199022
119894119895) + 120582(119891
119894119895
119906)
2
] 119889119905 (12)
where weighting factor 120582 represent a compromise betweenthe required control input and vibration control efficiency
Based on [27] the solution of (12) is expressed as
119896119894119895
119901= (1205962
119894119895+ 120582minus1)
12
minus 120596119894119895
119896119894119895
119889= (2120596
119894119895((1205962
119894119895+ 120582minus1)
12
minus 120596119894119895) + 120582minus1)
12
(13)
32 Efficient Modal Control Since the control gains forhigher vibration modes are much larger than the lowermodes in the IMSC method the applied control voltageswill attain high values if the feedback gains are used directlywithout modifying The goal of the proposed controller isto attenuate the multimode vibration simultaneously withrelatively small control force In EMC method [21] thefeedback gains for each mode are modified according to itsown modal displacement or energy and hence the highermodeswith lower vibration amplitude can be suppressedwith
low damping applied According to EMC only the feedbackgains for the first mode are obtained through optimal controlmethod but the others are optimized using energy weightingmethod as follows
1198961198941
119901 119896119894119895
119901 119896119894119903
119901= 1
119864119894119895
1198641198941
times
1205961198941
120596119894119895
119864119894119903
1198641198941
times
1205961198941
120596119894119903
(14)
where 119864119894119895= 1205962
1198941198951199022
119894119895+ 1199022
119894119895reflects the modal vibration energy for
the 119895th mode of the 119894th link and 119903 represents the number ofmodes to be controlled
33 Modal Filter and Synthesizer Real-time monitoring ofmodal coordinates plays a significant role in design ofmodal feedback controller As anymodal feedback controllerthe modal displacements and velocities are required to beextracted from the vibration sensors in the proposedmethodThe most common methods used to measure the modalcoordinates include state observers temporal filters andmodal filters Compared with state observers and temporalfilters modal filters extract modal coordinates from vibrationsensors independent of the control work and hence it canbe directly applied to any modal feedback controller Fur-thermore modal filters also have the advantage of preventingobservation spillover from the residual modes Thereforemodal filterswith discrete PZT sensors are adopted to providemodal coordinates separation
According to [17 25] the resultant voltage generated inthe PZT sensor is expressed with respect to the charge as
119881119896
119904(119905) =
11988711986411990111988931119905119890119902
119862119897
119901
(
120597119908 (119897119896
2 119905)
120597119909
minus
120597119908 (119897119896
1 119905)
120597119909
) (15)
where 119864119901is the Youngmodulus of PZT sensor 119889
31represents
the constant of PZT materials 119887 is the width of the PZTsensor 119862119897
119901is the equivalent constant of piezoelectric capac-
itance 119905119890119902is defined as the distance between neutral axis of
beamsensor and that of midplane of beam and sensor and
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
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International Journal of
Shock and Vibration 5
x
l 11 l12 lk1 lk2 lm1 lm2 Ci
Bi
kth1st mth
Vis Vi
a
Efficient modal controlModal
controllerModal filter
kijp kijd (j ne 1) are weighed according to modal
displacement or energy in that mode
Modalsynthesizer
(qi1
qir
)= Wis (Vi
s1
Vism
)
(fi1u
firu
) =(minuski1p qi1 minus ki1d qi1
minuskirp qir minus kird qir
) (Via1
Viam
) = Wia (fi1
u
firu
)
PZT actuator
PZT sensor
Figure 4 Schematic of efficient modal control
Substituting (10) to (9) yields the closed-form equationsfor the 119895th mode of the 119894th flexible link as
119902119894119895+ 119896119894119895
119889119902119894119895+ (1205962
119894119895+ 119896119894119895
119901) 119902119894119895= 119891119894119895
119903 (11)
Equation (11) clearly indicates that the active dampingforces and the positive stiffness forces are imported tothe modal equation through the vibration actuator thusincreasing the damping and stiffness features of the flexiblelinks The initial values of feedback gains of 119896119894119895119901 and 119896
119894119895
119889can be
determined using either pole assignment or optimal controlmethod According to the optimal control approach [26] thecost function which is related to the potential energy 1205962
1198941198951199022
119894119895
kinetic energy 1199022
119894119895 and the required control input (119891119894119895
119906)2 is
selected as
119869 = int
infin
0
[(1205962
1198941198951199022
119894119895+ 1199022
119894119895) + 120582(119891
119894119895
119906)
2
] 119889119905 (12)
where weighting factor 120582 represent a compromise betweenthe required control input and vibration control efficiency
Based on [27] the solution of (12) is expressed as
119896119894119895
119901= (1205962
119894119895+ 120582minus1)
12
minus 120596119894119895
119896119894119895
119889= (2120596
119894119895((1205962
119894119895+ 120582minus1)
12
minus 120596119894119895) + 120582minus1)
12
(13)
32 Efficient Modal Control Since the control gains forhigher vibration modes are much larger than the lowermodes in the IMSC method the applied control voltageswill attain high values if the feedback gains are used directlywithout modifying The goal of the proposed controller isto attenuate the multimode vibration simultaneously withrelatively small control force In EMC method [21] thefeedback gains for each mode are modified according to itsown modal displacement or energy and hence the highermodeswith lower vibration amplitude can be suppressedwith
low damping applied According to EMC only the feedbackgains for the first mode are obtained through optimal controlmethod but the others are optimized using energy weightingmethod as follows
1198961198941
119901 119896119894119895
119901 119896119894119903
119901= 1
119864119894119895
1198641198941
times
1205961198941
120596119894119895
119864119894119903
1198641198941
times
1205961198941
120596119894119903
(14)
where 119864119894119895= 1205962
1198941198951199022
119894119895+ 1199022
119894119895reflects the modal vibration energy for
the 119895th mode of the 119894th link and 119903 represents the number ofmodes to be controlled
33 Modal Filter and Synthesizer Real-time monitoring ofmodal coordinates plays a significant role in design ofmodal feedback controller As anymodal feedback controllerthe modal displacements and velocities are required to beextracted from the vibration sensors in the proposedmethodThe most common methods used to measure the modalcoordinates include state observers temporal filters andmodal filters Compared with state observers and temporalfilters modal filters extract modal coordinates from vibrationsensors independent of the control work and hence it canbe directly applied to any modal feedback controller Fur-thermore modal filters also have the advantage of preventingobservation spillover from the residual modes Thereforemodal filterswith discrete PZT sensors are adopted to providemodal coordinates separation
According to [17 25] the resultant voltage generated inthe PZT sensor is expressed with respect to the charge as
119881119896
119904(119905) =
11988711986411990111988931119905119890119902
119862119897
119901
(
120597119908 (119897119896
2 119905)
120597119909
minus
120597119908 (119897119896
1 119905)
120597119909
) (15)
where 119864119901is the Youngmodulus of PZT sensor 119889
31represents
the constant of PZT materials 119887 is the width of the PZTsensor 119862119897
119901is the equivalent constant of piezoelectric capac-
itance 119905119890119902is defined as the distance between neutral axis of
beamsensor and that of midplane of beam and sensor and
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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RotatingMachinery
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Advances inOptoElectronics
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
6 Shock and Vibration
119897119896
2and 1198971198961are the left and right end positions of the 119896th PZT
sensor respectivelyTherefore the modal filter expression for 119894th smart link
with119898 PZT sensor is given as [28]
119902119894(119905) = 119882
119894
119904119881
119894
119904(119905) (16)
where 119902119894(119905) = [1199021198941
(119905) 1199021198942(119905) sdot sdot sdot 119902
119894119903(119905)]
119879 is the modaldisplacement vector of the 119894th smart link 119881
119894
119904(119905) =
[119881119894
1199041(119905) 119881
119894
1199042(119905) sdot sdot sdot 119881
119894
119904119898(119905)]
119879
is the output voltage vector ofthe 119898 PZT sensors attached on the 119894th smart link and 119882
119894
119904
is a 119903 times 119898 transformation matrix expressed as
119882119894
119904=
1
120582119904
119860119879(119860119860119879)
minus1
(17)
where 120582119904= 11988711986411990111988931119905119890119902119862119897
119901is PZT sensor constant and the
matrix 119860 related to the mode shape function of the 119894th smartlink is written as
119860
=
[
[
[
[
1198841015840
1198941(1198971
2) minus 1198841015840
1198941(1198971
1) 1198841015840
1198942(1198971
2) minus 1198841015840
1198942(1198971
1) sdot sdot sdot 119884
1015840
119894119903(1198971
2) minus 1198841015840
119894119903(1198971
1)
1198841015840
1198941(1198972
2) minus 1198841015840
1198941(1198972
1) 1198841015840
1198942(1198972
2) minus 1198841015840
1198942(1198972
1) sdot sdot sdot 119884
1015840
119894119903(1198972
2) minus 1198841015840
119894119903(1198972
1)
1198841015840
1198941(119897119898
2) minus 1198841015840
1198941(119897119898
1) 1198841015840
1198942(119897119898
2) minus 1198841015840
1198942(119897119898
1) sdot sdot sdot 119884
1015840
119894119903(119897119898
2) minus 1198841015840
119894119903(119897119898
1)
]
]
]
]
(18)
During the efficient modal control process the outputvoltages of the PZT sensors are first transferred from phys-ical coordinates to modal coordinates through modal filterand then the modal coordinates are provided to the EMCcontroller to calculate the active modal forces At last thecomputed modal forces are converted to the control voltagescorresponding to the PZT actuators in physical space usingmodal synthesizerThe overall control process is illustrated inFigure 4 and the modal synthesizer for 119894th link is expressedas
119881
119894
119886(119905) = 119882
119894
119886119891119906119894(119905) (19)
where 119891119906119894(119905) = [119891
1198941
119906(119905) 119891
1198942
119906(119905) sdot sdot sdot 119891
119894119903
119906(119905)]
119879
is the modalcontrol force vector calculated from EMC method for the119894th smart link 119881119894
119886(119905) = [119881
119894
1198861(119905) 119881
119894
1198862(119905) sdot sdot sdot 119881
119894
119886119898(119905)]
119879
is thecontrol voltage vector applied to the 119898 PZT actuators and119882119894
119886is a119898 times 119903 transformation matrix expressed as
119882119894
119886=
1
120582119886
119860(119860119879119860)
minus1
(20)
where 120582119886is the constant of PZT actuator
A problem that must be mentioned is about the trans-formation matrices 119882119894
119904and 119882
119894
119886 Taking the matrix 119882119894
119904used
in the modal filter for example if the number of vibrationsensors used to extract modal coordinate is as many as thevibration modes namely119898 = 119903 in (18) the matrix119882119894
119904would
be square and the extracted modal coordinates would beequal to the ideal modal coordinate However the vibration
Table 1 Parameters of PPM and PZT transducers
Symbol Terms Values119898119901
Mass of moving platform 025 kg119871119894
Length of the link 200mm119898119887
Mass of the link 0011 kg119898119904
Mass of the sliders 015 kg120572119894
Angle of linear guide 120∘ 240∘ 0∘
119864119901
PZT Youngrsquos modulus 66 times 1010 Nm2
11988931
PZT piezoelectric constant minus180 times 10minus12 CN119905119887
Beam thickness 2mm119905119901
PZT transducers thickness 02mm
modes of the flexible structure are infinite and only a limitednumber of sensors can be mounted on the flexible structureHence the modal coordinates extracted frommodal filter areonly an approximation of the ideal modal coordinates andthe accuracy of the approximation is related to the numberof the vibration sensors In practice increasing the numberof the vibration sensors will put computational burden andmake the system more complicated The same problem alsoexists in the matrix 119882119894
119886of the modal synthesizer Therefore
a compromise between real-time computing capability andthe number of vibration transducers must be made whendesigning the overall system
34 Simulation Results Numerical simulations in MATLABsoftware are performed first to verify the proposed controlmethod During the simulation a circular trajectory motionof the moving platform is selected as the rigid motion inputas illustrated in Figure 1 The circular motion is given as
119909119901= 10 cos (10120587119905) minus 10 (mm)
119910119901= 10 sin (10120587119905) (mm)
120593119901=
120587
6
(rad)
(21)
The distance among points 119860119894and 119862
119894is 600mm and
120mm respectively The effective stroke of each LUSM is250mm Three pairs of PZT transducers are mounted at thelocations of 50mm 100mm and 150mm from 119861
119894on the
flexible link as vibration sensors and actuators The otherparameters of the system are detailed inTable 1Thefirst threemodes of the flexible links are selected to be controlled inthe simulation work Using the energy weightingmethod themodified ratio for the second and third modes of the threeflexible links are derived from (14) as
11989611
119901 11989612
119901 11989613
119901= 11989611
119889 11989612
119889 11989613
119889= 1 0315 0041
11989621
119901 11989622
119901 11989623
119901= 11989621
119889 11989622
119889 11989623
119889= 1 0384 0036
11989631
119901 11989632
119901 11989633
119901= 11989631
119889 11989632
119889 11989633
119889= 1 0291 0044
(22)
Due to the different driving forces applied by LUSM thevibration responses of the three flexible links are different
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
minus5
0
5
minus4minus2
024
minus5
0
5
e
1st l
ink
(m)
e
2nd
link
(m)
e
3rd
link
(m)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus5
Figure 5 The vibration responses at the three quarters of the threeflexible links
Since the modified ratios are based on the uncontrolledsystem response it can be observed in (22) that the feedbackratios for the three flexible links are not identical Thevibration responses of the three flexible links are shown inFigure 5 which reveal that the oscillations of the three flexiblelinks are suppressed rapidly with the proposed EMC strategyFigure 6 shows that the first three modesrsquo vibrations of thefirst flexible link are all attenuated effectively which furthervalidate that the EMC method can suppress multimodevibration simultaneously
4 Experimental Results
The trajectory of the moving platform in the experimentis the same as that used in the numerical simulation Toguarantee the accuracy of the desired motion the kinematiccalibration work of the flexible PPM is carried out basedon the visual feedback and the particle swarm optimization(PSO) algorithmThemotion control card (DMC-1842 Galil)is used to control the three LUSMs through three LUSMdrivers (made by NUAA)The feedback signal is provided bythe linear grating sensor (LIA20 NUMERIK JENA) A DSPcontrol board (Seed DEC2812) is adopted to realize the activevibration control PZT power amplifiers (XE-501 XMT) areadopted to amplify the control voltage of the PZT actuatorsIn the experiment only the first flexible link is targeted to becontrolled due to the limitation of the hardware Three pairsof PZT sensors and actuators (PZT5 CSSC) are mounted
minus5
0
5
e
1st m
ode
(m)
minus1
0
1
e
2nd
mod
e (m
)minus2minus1
012
e
3rd
mod
e (m
)0 005 01 015 02 025 03 035 04
Time (s)
0 005 01 015 02 025 03 035 04Time (s)
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
times10minus5
times10minus5
times10minus6
Figure 6The first three mode vibrations at the three quarters of thefirst flexible link
Moving platformEnlarged view
(PZT actuators)Enlarged view
(PZT sensors)
Flexible link
Grating sensor
Linear guideLUSM
Figure 7 Arrangement of the PZT transducers
on the location of quarter midpoint and three quarters ofthe first flexible link as shown in Figure 7 The sizes of eachsensor are 10times 10times 02mm and the sizes of each actuator are30 times 10 times 02mm
Experiments with and without EMC method are carriedout to validate the effectiveness of the proposed approachBefore the experiment the uncontrolled systems are analyzedfirst to provide the required information such as modaldisplacements and velocities The first two modes of the firstflexible link are chosen to be suppressed in the experimentBased on the proposed EMC approach the feedback gains for
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 8 Vibration response at the quarter point of the first link
0 005 01 015 02 025 03 035 04Time (s)
Without controlWith EMC
minus8
minus6
minus4
minus2
0
2
4
6
10
8
Resp
onse
(V)
Figure 9 Vibration response at the midpoint of the first link
the firstmode of the first link are calculated as 11989611119901= 1187 and
11989611
119889= 243 and the ratio for the second mode is optimized as
11989611
119901 11989612
119901= 11989611
119889 11989612
119889= 1 0259 The response voltages
of the three PZT sensors mounted on the first flexible linkare illustrated in Figures 8 9 and 10 which clearly show thatboth the structural vibrations and the residual vibration ofthe first link are suppressed effectively with the EMCmethodFurthermore Figures 8ndash10 also demonstrate that the residualvibrations can be completely attenuated within 008 s aftermotion stops Based on the fast Fourier transform (FFT) thefirst two vibration modes extracted from the PZT sensors areanalyzed Figures 11-12 show that the vibration amplitudesfor the first two vibration modes at 92Hz and 244Hz arereduced by 5778 and 4496 respectivelyThese PSD plots
0 005 01 015 02 025 03 035 04minus8
minus6
minus4
minus2
0
2
4
6
8
Time (s)
Resp
onse
(V)
Without controlWith EMC
Figure 10 Vibration response at the three-quarter point of the firstlink
verify that the proposed EMC method has the capability ofsuppressing multimode vibration simultaneously
From Figures 11-12 we also find that there still exist otherfrequencies besides the dominant natural frequencies In factas mentioned in Section 33 if the vibration sensors are asmany as vibration modes then the modal coordinates canbe extracted completely from the other frequencies and theother vibration frequencies will not be observed in Figures11-12 However the vibration modes are infinite and usuallythe hardware is limited in practice (in our study only threePZT sensors are adopted) Hence the number of vibrationmodes which can be sensed are limited and the residualmodemay participate in the extracted vibrationmode for examplethe first natural frequency of 92Hz is observed in Figure 12and the second natural frequency of 244Hz is observed inFigure 11 Besides many other forced vibration componentsare also shown in Figures 11-12 such as 33Hz 64Hz and144Hz These forced vibration frequencies are mainly fromthe movement of the sliders and flexible links the dynamicsof the LUSM and the coupling effect between the rigidbody motion and elastic motion Since these forced vibrationfrequencies are usually away from the natural frequenciesthe active damping force has nearly no effect on theseforced vibration Hence it is shown in Figures 11-12 that theamplitudes of these forced vibrations are almost unchangedduring the control experiment For further suppressing theseforced vibrations the joint motion control methods such assingular perturbation approach or nonlinear control methodmay be adopted to optimize the driving forces of the threeLUSM
5 Conclusions
This paper addresses the dynamic modeling of a PPM withflexible linkage driven by LUSMs The Lagrange equations
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 3037
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 40005
101520253035
X 9277Y 1282
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 11 FFT of the first vibration mode of the first link
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 9676
Frequency (Hz)
PSD
(dB)
Without control
(a)
0 50 100 150 200 250 300 350 400 450 50002468
1012
X 2441Y 5326
Frequency (Hz)
PSD
(dB)
With EMC
(b)
Figure 12 FFT of the second vibration mode of the first link
and the AMM method are adopted to deduce the dynamicequations of the flexible link With multiple PZT transducersmounted on the flexible links as vibration sensors andactuators the EMC method is designed to suppress theunwanted vibration of the flexible link Computer simulationand experiments are both implemented and the results showthat both the structural and residual vibrations of the flexiblelinks are effectively suppressed when using EMC approachFurthermore the multimode vibration control capability ofthe proposed EMC method is also validated through FFTanalysis In the near future the vibration control throughjoint motion controller may be employed to combine withthe EMC approach to achieve further vibration suppressionof the flexible link
Appendix
Consider
119872 =[
[
1198721
0 0
0 1198722
0
0 0 1198723
]
]
isin 1198773119903times3119903
119872119894=
[
[
[
[
[
[
[
int
119871 119894
0
1205881198601198941198842
1198941119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot int
119871 119894
0
1205881198601198941198842
119894119903119889119909
]
]
]
]
]
]
]
isin 119877119903times119903
119865119903
=
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
12058811198781int
1198711
0
120588119860111988411119889119909 +
1205731int
1198711
0
120588119860111990911988411119889119909 minus
1205732
111990211int
1198711
0
12058811986011198842
11119889119909
12058811198781int
1198711
0
12058811986011198841119903119889119909 +
1205731int
1198711
0
12058811986011199091198841119903119889119909 minus
1205732
11199021119903int
119871119894
0
12058811986011198842
1119903119889119909
12058821198782int
1198712
0
120588119860211988421119889119909 +
1205732int
1198712
0
120588119860211990911988421119889119909 minus
1205732
211990221int
1198712
0
12058811986021198842
21119889119909
12058821198782int
1198712
0
12058811986021198842119903119889119909 +
1205732int
1198712
0
12058811986021199091198842119903119889119909 minus
1205732
21199022119903int
1198712
0
12058811986021198842
2119903119889119909
12058831198783int
1198713
0
120588119860311988431119889119909 +
1205733int
1198713
0
120588119860311990911988431119889119909 minus
1205732
311990231int
1198713
0
12058811986031198842
31119889119909
12058831198783int
1198713
0
12058811986031198843119903119889119909 +
1205733int
1198713
0
12058811986031199091198843119903119889119909 minus
1205732
31199023119903int
1198713
0
12058811986031198842
3119903119889119909
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
isin 1198773119903times1
119870 =
[
[
[
[
1198701
0 0
0 1198702
0
0 0 1198703
]
]
]
]
isin 1198773119903times3119903
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Shock and Vibration
119870119894=
[
[
[
[
[
[
[
[
119864119868int
119871 119894
0
11988410158401015840
1198941
2
119889119909 sdot sdot sdot 0
sdot sdot sdot
0 sdot sdot sdot 119864119868int
119871 119894
0
11988410158401015840
119894119903
2
119889119909
]
]
]
]
]
]
]
]
isin 119877119903times119903
(A1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grants nos 91223201 51375225 and51175264) the Natural Science Research Project of Jiangsu(Grant no BK2012797) the Open Fund of PostgraduateInnovative Base (Laboratory) for Nanjing University of Aero-nautics and Astronautics (kfjj120201) and the FundamentalResearch Funds for the Central Universities and a ProjectFunded by the Priority Academic Program Development ofJiangsu Higher Education Institutions
References
[1] I Payo V Feliu and O D Cortazar ldquoForce control of a verylightweight single-link flexible arm based on coupling torquefeedbackrdquoMechatronics vol 19 no 3 pp 334ndash347 2009
[2] S Zhou and J Shi ldquoActive balancing and vibration control ofrotating machinery a surveyrdquo Shock and Vibration Digest vol33 no 5 pp 361ndash371 2001
[3] Y Shen and A Homaifar ldquoVibration control of flexible struc-tures with PZT sensors and actuatorsrdquo JVCJournal of Vibrationand Control vol 7 no 3 pp 417ndash451 2001
[4] X Zhang J K Mills and W L Cleghorn ldquoMulti-mode vibra-tion control and position error analysis of parallel manipulatorwith multiple flexible linksrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 34 no 2 pp 197ndash2132010
[5] K Krishnamurthy and L Yang ldquoDynamic modeling and simu-lation of two cooperating structurally-flexible robotic manipu-latorsrdquo Robotica vol 13 no 4 pp 375ndash384 1995
[6] C M A Vasques and J Dias Rodrigues ldquoActive vibration con-trol of smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers and Structuresvol 84 no 22-23 pp 1402ndash1414 2006
[7] Q Hu and G Ma ldquoVibration suppression of flexible spacecraftduring attitude maneuversrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 377ndash380 2005
[8] S K Dwivedy and P Eberhard ldquoDynamic analysis of flexiblemanipulators a literature reviewrdquo Mechanism and MachineTheory vol 41 no 7 pp 749ndash777 2006
[9] X Wang and J K Mills ldquoDynamic modeling of a flexible-link planar parallel platform using a substructuring approachrdquoMechanism and Machine Theory vol 41 no 6 pp 671ndash6872006
[10] G Piras W L Cleghorn and J K Mills ldquoDynamic finite-element analysis of a planar high-speed high-precision parallel
manipulator with flexible linksrdquo Mechanism and Machine The-ory vol 40 no 7 pp 849ndash862 2005
[11] X Zhang J K Mills andW L Cleghorn ldquoVibration control ofelastodynamic response of a 3-PRRflexible parallelmanipulatorusing PZT transducersrdquo Robotica vol 26 no 5 pp 655ndash6652008
[12] Y Yu Z Du J Yang and Y Li ldquoAn experimental study on thedynamics of a 3-RRR flexible parallel robotrdquo IEEE Transactionson Robotics vol 27 no 5 pp 992ndash997 2011
[13] J Shan H T Liu and D Sun ldquoSlewing and vibration control ofa single-link flexible manipulator by positive position feedback(PPF)rdquoMechatronics vol 15 no 4 pp 487ndash503 2005
[14] X Zhang J K Mills and W L Cleghorn ldquoFlexible linkagestructural vibration control on a 3-PRR planar parallel manip-ulator experimental resultsrdquo Proceedings of the Institution ofMechanical Engineers I Journal of Systems andControl Engineer-ing vol 223 no 1 pp 71ndash84 2009
[15] Z Chu J Cui and F Sun ldquoVibration control of a high-speed manipulator using input shaper and positive positionfeedbackrdquo inKnowledge Engineering andManagement pp 599ndash609 Springer Berlin Germany 2014
[16] Q Zhang J Mills K L W Cleghorn J Jina and Z SunldquoDynamic model and input shaping control of a flexible linkparallel manipulator considering the exact boundary condi-tionsrdquo Robotica 2014
[17] Q Zhang K J Mills LW Cleghorn et al ldquoTrajectory trackingand vibration suppression of a 3-PRR parallel manipulator withflexible linksrdquoMultibody System Dynamics pp 1ndash34 2013
[18] X Zhang Dynamic Modeling and Active Vibration Control ofa Planar 3-PRR Parallel Manipulator with Three Flexible LinksUniversity of Toronto Toronto Canada 2009
[19] L Meirovitch and H Baruh ldquoOptimal control of dampedflexible gyroscopic systemsrdquo Journal of Guidance Control andDynamics vol 4 no 2 pp 157ndash163 1981
[20] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988
[21] S P Singh H S Pruthi and V P Agarwal ldquoEfficient modalcontrol strategies for active control of vibrationsrdquo Journal ofSound and Vibration vol 262 no 3 pp 563ndash575 2003
[22] L Meirovitch Fundamentals of Vibrations Waveland Press2010
[23] Q Zhang L Zhou J Zhang et al ldquoEfficient modal controlfor vibration suppression of a flexible parallel manipulatorrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 13ndash18 2013
[24] X Zhang X Wang J K Mills and W L Cleghorn ldquoDynamicmodeling and active vibration control of a 3-PRR flexibleparallel manipulator with PZT transducersrdquo in Proceedings ofthe 7th World Congress on Intelligent Control and Automation(WCICA rsquo08) pp 461ndash466 June 2008
[25] N Jalili Piezoelectric-Based Vibration Control From Macro toMicroNano Scale Systems Springer New York NY USA 2010
[26] F Lin Robust Control Design An Optimal Control ApproachJohn Wiley amp Sons 2007
[27] L MeirovitchDynamics And Control Of Structures JohnWileyamp Sons New York NY USA 1990
[28] H Sumali K Meissner and H H Cudney ldquoA piezoelectricarray for sensing vibration modal coordinatesrdquo Sensors andActuators A Physical vol 93 no 2 pp 123ndash131 2001
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
