research article optimal trajectory planning and linear

Research ArticleOptimal Trajectory Planning and Linear VelocityFeedback Control of a Flexible Piezoelectric Manipulator forVibration Suppression

Junqiang Lou1 Yanding Wei2 Guoping Li1 Yiling Yang2 and Fengran Xie2

1Zhejiang Provincial Key Lab of Part Rolling Technology College of Mechanical Engineering and Mechanics Ningbo UniversityNingbo 315211 China2China Key Laboratory of Advanced Manufacturing Technology of Zhejiang Province School of Mechanical EngineeringZhejiang University Hangzhou 310027 China

Correspondence should be addressed to Junqiang Lou loujunqiangnbueducn

Received 26 June 2015 Revised 4 August 2015 Accepted 5 August 2015

Academic Editor Mario Terzo

Copyright copy 2015 Junqiang Lou 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

Trajectory planning is an effective feed-forward control technology for vibration suppression of flexible manipulators Howeverthe inherent drawback makes this strategy inefficient when dealing with modeling errors and disturbances An optimal trajectoryplanning approach is proposed and applied to a flexible piezoelectric manipulator system in this paper which is a combinationof feed-forward trajectory planning method and feedback control of piezoelectric actuators Specifically the joint controller isresponsible for the trajectory tracking and gross vibration suppression of the link during motion while the active controller ofactuators is expected to deal with the link vibrations after jointmotion In the procedure of trajectory planning the joint angle of thelink is expressed as a quintic polynomial function And the sumof the link vibration energy is chosen as the objective functionThengenetic algorithm is used to determine the optimal trajectory The effectiveness of the proposed method is validated by simulationand experiments Both the settling time and peak value of the link vibrations along the optimal trajectory reduce significantly withthe active control of the piezoelectric actuators

1 Introduction

Flexible manipulators have been used in various applicationsincluding field inspection space exploration medical appli-cation and hazardous environment Compared with theirrigid counterparts flexiblemanipulators exhibitmany advan-tages such as higher manipulation speed greater payload toweight ratio and lower energy consumption Moreover theycan be quickly adapted to changing situations and productdesign variations [1] However the conflicting requirementsbetween high speed and high positioning accuracy havemade the control of flexible manipulators a real challengingproblem Structural flexibility leads to the appearance ofundesirable vibrations during motion So flexible manipula-tor systemexhibits combined behaviors of rigid bodymotions

and vibrations [2] These oscillations lead to deterioration ofpositioning accuracy and efficiency making the control ofsuch system extremely difficult

To address these problems quite a lot of researches havebeen conducted on the vibration control of flexiblemanipula-torsThe existing vibration control techniques can be broadlyclassified into two categories namely feedback control andfeed-forward control The advantage of feed-forward controlis that it does not require any additional sensors or actuatorsHence the control system can be manufactured at lowcosts thus being more economical One special feed-forwardcontrol strategy known as input command shaping hasbeen studied widely since its first appearance [3] Mohamedet al [4] presented investigations into vibration control ofa flexible manipulator using input shaping techniques with

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 952708 11 pageshttpdxdoiorg1011552015952708

2 Shock and Vibration

positive and negative input shapers Alam and Tokhi [5]developed the design of a command shaping controller forvibration suppression of a flexible manipulator using multi-objective genetic optimization process More recently Coleand Wongratanaphisan [6] described an adaptive FIR inputcommand shaping methodology to achieve zero residualvibration control of a rigid-flexible manipulator system Theinput command shaping method is effective to ensure thatthe input does not excite vibrations of the manipulator Butit usually costs a large amount of computation when dealingwith multiple vibration modes And it cannot suppress theinduced and existing vibrations

An alternative feed-forward control strategy to achievevibration control of flexible manipulators is the trajectoryplanning approach Wu et al [7] proposed an optimaltrajectory planning method for vibration reduction of adual arm space robot with front flexible links in whichthe trajectory was described using a fourth-order quasi-uniform B-spline Heidari et al [8] considered the problemon the rest-to-rest motion of a flexible manipulator andformulated the trajectory planning problem as a Pontrya-gin optimal problem Springer et al [9] focused on time-optimal trajectory planning for robots with flexible linksand determined the minimum time trajectory for avoidingelastic vibration of the link Choi et al [10] described anew trajectory planning method for output tracking of linearflexible system using exact equilibrium manifolds Howeverthe trajectory planning technique is a feed-forward controlstrategy vibration modes of the flexible manipulator are stillcontrolled in open loop [11] When dealing with variousdisturbances and parameters variations there would still belarge residual vibrations that need to be controlled [12]

To overcome the drawbacks of feed-forward controlschemesmany researchers have tried to solve the problems oflink vibration suppression using feedback control techniquesAmong them some researchers concentrated on reducingthe link vibrations using joint motor [13] However theexistence of nonminimum phase dynamics between the tipposition and the input torque applied at the joint motormakes the system difficult to stabilize [14] A rather differentapproach for feedback control of link vibrations has beendeveloped in recent years relying on the use of smartmaterials [15] Among those smart materials piezoelectric(PZT) materials have been found extensive applications instructure vibration control due to their lightweight fastresponse large bandwidth and so on [16] Sun et al [17]realized the rigid motion control and vibration damping ofa flexible manipulator using a combined scheme of a PDcontroller for the joint motor and a linear velocity feedback(L-type) controller for the piezoelectric actuators Gurses etal [18] extended the previous work studied and introduceda serial array of fiber optic sensors to the control of flexiblemanipulators which allowed simultaneous linear (L-type)and angular (A-type) velocity feedback Li et al [19] presenteda study on vibration suppression of a flexible piezoelectricmanipulator using adaptive fuzzy sliding mode control Choiet al [20] studied the position control of a single-link flexiblemanipulator and designed a hysteresis compensator for thepiezoelectric actuators

However in most studies on active control of flexiblemanipulator attention seems to be focused on how toimprove the performance of the piezoelectric controller Thecontributions of the feed-forward torque or the requiredtrajectory planning are rarely considered A more reasonableand efficient approach seems to emerge if a combined strategyis considered Specifically the joint motor is responsiblefor the gross vibrationless motion of the structure throughtrajectory planning approach while the PZT actuators areused to deal with vibrations that may arise from modelingerrors and external disturbances after joint motion by meansof active control The remainder of this paper is organized asfollows In Section 2 the flexible piezoelectric manipulatorsystem is introduced and the governing equations of thesystem are derived Section 3 proposes the optimal trajectoryplanning approach The joint angle of the link is expressed asa quintic polynomial functionThe constraint conditions theinitial and final conditions of the joint angle angular velocityand angular acceleration are applied The sum of the linkvibration energy is chosen as the objective function Thenthe optimal trajectory planning algorithm is constructed bythe use of genetic algorithm In Section 4 numerical analysisand simulations are conducted The optimal trajectory ofthe flexible piezoelectric manipulator in a point-to-pointmotion is obtained Section 5 describes an experimentalapparatus of the flexible manipulator system together withthe experimental results before concluding in Section 6

2 Dynamic Modeling

The schematic diagram of the flexible piezoelectric manipu-lator system is shown in Figure 1 It is composed of a rotaryjoint a single-link flexible manipulator with a rectangularcross section and a pair of collocated piezoelectric (PZT)actuators The manipulator is attached to the hub in such away that it rotates in the horizontal plane only The effect ofgravity can be ignored so the link can bemodeled as anEuler-Bernoulli beamThe axis 119900119909

119900is the inertial reference lineThe

axis 119900119909 is the tangential line to the neutral axis of the beam atthe hub For the sake of simplicity the same assumptions aremade as [17]

It is assumed that the dynamic behavior of the flexiblemanipulator during a point-to-point motion is dominated bythe first several vibration modes so the contributions of thehigher vibration modes are negligible Using assumed modemethod the deflection 119908(119909 119905) of the link can be representedas

119908 (119909 119905) =

119898

sum

119894=1

120593119894(119909) 119902119894(119905) = Φ (119909) q (119905) (1)

where 120593119894(119909) and 119902

119894(119905) devote the 119894thmode shape function and

corresponding generalized coordinate respectively Φ(119909) =

[1205931sdot sdot sdot 120593119894sdot sdot sdot 120593119898] and q(119905) = [119902

1sdot sdot sdot 119902119894sdot sdot sdot 119902119898] represent the

mode shape function and generalized coordinate indexrespectively119898 is the mode number

Shock and Vibration 3

Flexible manipulator

Piezoelectric actuatorso xooo

yo

y

120579(t)

w(x t)

120591(t)

P

x

Figure 1 Schematic diagram of the flexible manipulator system

Extended Hamiltonrsquos principle is used to derive the sys-tem equation The derivation process and parameter expres-sions are explicitly presented in Appendix The governingequations of the system can be derived as

119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905) (2)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904) (3)

It should be noticed that (2) represents the rigid motionof the piezoelectric manipulator while (3) describes thedynamic behavior of the flexible link with the control of PZTactuators If the desired trajectory can be realized by meansof tracking control for example trajectory control can berealized when the servo motor works in the speed controlmode In the case that tracking control is possible only (3)is used for planning a trajectory where the elastic vibrationof the link is eliminated or minimum Augmenting theproportional damping the dynamic equation of the flexiblemanipulator system can be expressed as

m119902119902q +m119879

120579119902

120579 + C119902q + K

119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(4)

It is known that it is feasible to use linear velocityfeedback in the controller formulation Thus the linearvelocity feedback (L-type) approach is readily implementableFurthermore the stability of the close-loop system includingthe L-type controller has been confirmed by Sun et al in[17] as well as the effectiveness of the controller Thereforethe linear velocity feedback (L-type) approach is employedto design the applied voltage of the PZT actuators in thisstudy In the L-type control strategy the control voltage of theactuators is defined as

119881 (119905) = minus119896119871[ (119909

119904+ 119897119901) minus (119909

119904)] = minus119896

119871Φ (119909119904) q (5)

where 119896119871is a positive control gainThen substituting (5) into

(4) leads to

m119902119902q +m119879

120579119902

120579 + C119902q + 119888119896

119871[Φ1015840

]

119879

(119909119904) Φ (119909

119904) q + K

119902q

minusm119902119902q 1205792

= 0

(6)

It can be observed from (6) that the rigid motion ofthe joint elastic vibration of the link and active control of

the PZT actuators are all considered in the procedure of tra-jectory planning However it is known that the piezoelectricparameters are very small numbers which means that thecontribution assured by the PZT control is relatively lessthan that of the feed-forward trajectory planning during jointmotionTherefore the essence of the optimal trajectory plan-ning approach is that the feed-forward trajectory planning isresponsible for the gross vibrationlessmotion of the structureduring joint motion while the PZT actuators are used to dealwith vibrations after joint motion bymeans of L-type controlstrategy

3 Optimal Trajectory Planning

This study focuses on the problem related to a point-to-point(PTP) motion of the flexible manipulator system rotatingfrom the initial joint angle 120579

0to the final joint angle 120579

119891over

a fixed traveling time 119905119891[21] In practice elastic vibrations

of the flexible link are unavoidable due to the flexible-rigidcoupled dynamics Considering the presence of parameteruncertainties and external disturbances the residual vibra-tions of the link are still exciting at the end of PTP motionin spite of active control of PZT actuators For a given PTPmaneuver in joint space the objective is to find an optimaljoint trajectory that minimizes link vibrations during andafter joint motion The total vibration energy of a trajectoryis adopted as the performance index which is defined as

PI (119909) = 1205821int

119905119891

1199050

q119879 (119905) q (119905) 119889119905 + 1205822int

119905119889

119905119891

q119879 (119905) q (119905) 119889119905 (7)

where 119905119889is the settling time of the link vibrations with

piezoelectric controlThe first part in (7) represents vibrationenergy of the link during joint motion while the second partrepresents residual vibration energy after joint motion 120582

1

and1205822are theweighting indexes of the twoparts respectively

To realize a smooth motion smooth and continuousfunctions are typically adopted to plan the joint trajectoryMoreover the first and second derivatives of the trajectoryare also expected to be smooth and continuous Among thosecommon trajectory functions the quintic polynomial func-tion has the advantages of being accurate precise and easyto design and control Moreover it has the extra advantage ofcontrolling the acceleration at the initial point and goal [22]So the quintic polynomial function is selected to interpolatethe optimal trajectory curves which can be written as

120579 (119905) = 1198620+ 1198621119905 + 11986221199052

+ 11986231199053

+ 11986241199054

+ 11986251199055

(8)

where 1198620 1198621 1198622 1198623 1198624 and 119862

5are the coefficients to be

determined from the initial and final boundary conditionsTo get a smooth motion the initial angle angular velocityand angular acceleration are set to be zero Meanwhile thefinal angular velocity and angular acceleration are also set tobe zero

To get the optimal trajectory that minimizes the per-formance index in (7) the travelling time 119905

119891is divided

into 119899 divisions each of which has an equal time interval


Alternative trajectory

t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 tf

Quintic polynomialΨ(t1)

Ψ(t2)

Ψ(t3)

Ψ(t4)

Ψ(t5)

Ψ(t6)

Ψ(t7)

Ψ(t8)Ψ(t9)

1

2

3 4

56

7

8

9

ΔF6

ΔF7

120579f

1205790

Figure 2 Discrete joint angle of the flexible manipulator system 120579119894

(119899 = 10)

If the times at divided knots can be denoted by 1199050(= 0)

1199051 119905

119894 119905

119899minus1 119905119891 the joint angle at 119905

119894is denoted by

120579119894

To construct the optimal trajectory the quintic polynomialfunction is employed to interpolate the discrete angles Theangular velocity and angular acceleration are obtained bydifferentiating the quintic polynomial curves with respect totime Finally the discrete angels of the link are defined byintroducing parameters Δ120579

119865119894as

120579119894= Ψ (119905

119894) + Δ

120579119865119894 (9)

where Ψ(119905) represents the quintic polynomial motion withthe above boundary conditions

Consequently the parameter Δ120579119865119894denotes the deviation

from the trajectory for the quintic polynomial motion Thediscrete joint angles of the flexible link are shown in Figure 2

Genetic algorithm (GA) approach is gaining popularityfor solving complex problems in robotics It is a population-based stochastic and global searchmethod Its performance issuperior to that revealed by classical optimization techniquesand has been successfully used in robot trajectory planningtechnique [23] As the procedure discussed above it ispossible to express the optimal trajectory

120579119894by using the

parameterΔ120579119865119894 whose number is 119899minus1 for the 119899divisions of 119905

119891

The trajectory planning procedures of the link using GA arecarried out as followsThe parametersΔ120579

119865119894(119894 = 1 2 119899minus1)

are considered to be optimized First the angular velocity120579(119905) and angular acceleration

120579(119905) are obtained from the

quintic polynomial interpolation of the discrete joint angles120579119894 Then the dynamic response of the flexible piezoelectric

manipulator is calculated through numerical integration of(6) Finally the optimal trajectory can be derived throughminimizing the performance index in (7) using geneticalgorithmThe main steps are summarized as below

Step 1 Division number 119899 is determined the discrete anglesof the quintic polynomial motion [120595(119905

1) 120595(1199052) 120595(119905

119894)

120595(119905119899minus1

)] are calculated Meanwhile the parameters of theGA are also determined

Step 2 The initial population for the first generation is gener-ated through a random generator Every individual includes119899minus1 genes corresponding to the interior points to be selectedwhich is defined as x

119896= [1199091198961 1199091198962 119909

119896119894 119909

119896(119899minus1)] The

gene of the individual 119909119896119894represents the optimized parameter

Δ120579119865119894 and the dimension of the search space is 119899 minus 1

Step 3 The trajectory of the link is generated through thequintic polynomial interpolation and the vibration responseof the link is derived through numerical simulations Thefitness values PI(119909) of all the individuals in (7) are evaluated

Step 4 A set of genetic operators which are selectioncrossover and mutation are used in succession to create anew population of chromosomes for the next generation

Step 5 The process of evaluation and creation of new succes-sive generations is repeated until the termination condition issatisfied And the optimal trajectory for vibration suppressionis finally decided where the fitness value PI(119909) is minimum

4 Simulation Studies

Numerical simulations are carried out to evaluate the per-formance of the proposed trajectory planning approachusing MATLAB The parameters of the proposed flexiblemanipulator system are shown in Table 1 In simulation theactuator pairs are chosen to lie near the root of the linkwhere the strain energy of the structure is the highest [24] Forsimplicity only the first vibrationmode is taken into accountwhich is found to be at 120596

1= 245 rads (39Hz) with modal

damping 1205771= 002 And the parameters of theGAare given in

Table 2Theweighting indexes of the vibration energy duringand after joint motion are set to 120582

1= 025 120582

2= 075

A fast motion case is considered here The traveling time119905119891is set to be 05 seconds The initial and final angles of the

joint are set to be 0 rad and 1205872 rad respectively And thenumber of trajectory partitions 119899 is set to be 10 Meanwhileto speed up the convergence of GA search the search spaceof the optimized parameters Δ120579

119865119894is taken to be

Δ120579119865119894isin [minus05

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

1003816100381610038161003816100381605

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

10038161003816100381610038161003816] (10)

For a real flexible manipulator system the joint accelera-tion the torque of the motor and control voltage of the PZTactuator should have constraint values [25] The constraintconditions of the flexible manipulator system are defined as

0 lt120579119894lt 120579119891

(119894 = 1 2 119899 minus 1)

120579 le

120579max

119881 le 119881max

(11)

where the parameter119881max is set to be 150V and themaximumof the joint acceleration

120579max is assumed to be 80 rads2


Table 1 Parameters of the flexible manipulator system

Parameter Flexible link PZT actuatorMaterial Epoxy PZT-5Length (mm) 6200 400Width (mm) 300 100Thickness (mm) 30 08Youngrsquos modulus (Gpa) 346 1170Density (Kgm3) 18400 75000PZT coefficient 119889

31(CN) NA 187 times 10

minus12

Table 2 Parameters of genetic algorithm

Parameter ValuePopulation type Binary stringNumber of genes 48 bitsPopulation size 25Crossover ratio 06Mutation ratio 005Generations 100

The evolution history of the GA search is shown inFigure 3 It can be seen that the fitness reduces with anincreasing value of the generation number And the optimaltrajectory can be approximately obtained at the 40th genera-tionThe results of numerical calculations and simulations arepresented in Figure 4The angular displacement velocity andacceleration of the obtained optimal trajectory are depicted inFigures 4(a)ndash4(c) respectively The tip vibration response ofthe flexible link along the optimal trajectory is illustrated inFigure 4(d) For comparison the simulation results along thequintic polynomial trajectory are also presented correspond-ingly

As seen in Figure 4(d) dynamic response of the link alongthe optimal trajectory shows nearly no trace of oscillationsafter joint motion That is to say there is nearly no residualvibration in the flexible link (ie the modal displacementafter 05 s) while the dynamic response of the link alongthe quintic polynomial trajectory owns large oscillations allthrough the simulation interval The presence of residualvibrations after joint motion is clearly visible even after aperiod of 3-fold traveling time 119905

119891

5 Experimental Results

A photograph of the experimental setup is shown in Figure 5The flexible piezoelectric manipulator is driven by an ACservo motor (Yaskawa SGMAH01A1A2S) with a reductiongear (Harmonic Drive CSF-17-50-2A) 501 A built-in 16-bitincremental encoder is installed within the motor to measurethe rotation angle The link is clamped at the shaft of thereduction gear through a coupling and is limited to rotateonly in the horizontal plane Meanwhile a pair of straingauges assembled in half-bridge configuration (resistance120Ω sensitivity 208) are placed at the base of the link to

0 20 40 60 80 1000

50

100

150

200

250

Generation

Fitn

ess v

alue

Best fitnessMean fitness

Figure 3 Evolution history of the genetic algorithm

measure the vibrations A pair of piezoelectric patches (PZT-5A) are attached to the link for the vibration suppressionlocated close to the strain sensor to get the best control effect

The control system is realized with an industrial com-puter A DAQ device (Advantech PCI-1742U) is used pro-vidingmultichannel DA AD and counter modules for dataacquisition and control output For joint motion control theAC servomotor is operated in the speed control mode Theoutput signal of the encoder is fed back to the industrialcomputer through the counter module of the DAQ And theinput torque determined from the PD controller is appliedto the motor through the DA converter and servo driver(Yaskawa SGDH01AE) Meanwhile the vibration signal ofthe flexible manipulator is low-pass filtered and amplifiedthrough a strain amplifier which can amplify the signalto a voltage range of minus10V to 10V and then fed into theindustrial computer through an AD converter To suppressthe link vibrations actively the input voltage determined fromthe controller amplified 15 times with a power amplifier issupplied to the PZT actuators through a DA converter Theschematic diagram of the experimental setup is shown inFigure 6

The first experiment was conducted to rotate the linkfrom 0 rad to 1205872 rad in 05 seconds along the quintic poly-nomial trajectory But the active control of PZT actuators wasnot applied Experimental results are presented in Figure 7The angular displacement velocity and acceleration of thequintic polynomial trajectory are depicted in Figures 7(a)ndash7(c) respectivelyThemeasured vibration response of the linkby the strain gauges is shown in Figure 7(d)

As seen in Figures 7(a) and 7(b) the experimental resultsof joint angle and angular velocity are in perfect agreementwith the reference curves Thus it can be inferred thatthe joint reference trajectory of the quintic polynomial was


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)

Optimal trajectoryQuintic polynomial

(a) Angular displacement

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)

(b) Angular velocity

0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

(c) Angular acceleration

Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5

(d) Modal displacement

Figure 4 Comparison of simulation results between the optimal trajectory and quintic polynomial trajectory

closely followed However unwanted oscillations with largeamplitude both during and after the joint motion were clearlyobserved As shown in Figure 7(d) the peak value of themeasured vibration response exceeds the range of the ADconverter even after the joint motor has stopped for a periodof 1 s Furthermore the settling time of the uncontrolled link

vibrations which is defined as the damping period when thepeak value of the sensor voltage does not exceed 01 V isfound to be nearly 20 times longer than that of the motormotion (05 s) reaching 95 s Thus vibration suppression ofthe link is necessary to attain a high positioning accuracy andefficiency


Base

Servo AC motor

Strain gauge

Flexible linkPZT actuators

Reduction gear (50 1)

Figure 5 Photograph of the experimental setup

The following experiment was the implementation ofthe optimal trajectory approach But vibration control ofthe PZT actuators was still not applied Figures 8(a)ndash8(d)show the angular displacement velocity acceleration andmeasured vibration response of the link along the optimaltrajectory respectively The link vibrations obtained fromthe quintic polynomial trajectory are compared with thoseobtained from the optimal trajectory strategy as depicted inFigure 8(d) It can be observed from Figures 8(a) and 8(b)that experimental results of the angular displacement andvelocity fairly coincide with the desired curves Therefore itcan be confirmed that the trajectory tracking control of theflexible manipulator system is realized Although the peakvalue of the link vibrations still exceeds the range of theAD converter during the joint motion residual vibrationscorresponding to the optimal trajectory are significantly lessthan those caused by the quintic polynomial trajectory Thesettling time of the link vibrations reduces to 43 s as shownin Figure 8(d) However residual vibrations of the link arenot perfectly suppressed as the simulation results plotted inFigure 4(d) Unwanted residual vibrations occur after jointmotion This is because the joint motor is not capable ofabsolutely tracking the given reference trajectory as assumedand tracking error is unavoidable In additionmodeling erroralso exists in simulation

The last experiment was the combination of optimaltrajectory and feedback control of the PZT actuators Thoseexperimental results are presented in Figure 9 As the jointtrajectory velocity and acceleration were not significantlychanged these results were not givenThe vibration responseof the link is presented in Figure 9(a) It can be seen thatthe peak value of the link vibrations still exceeds the rangeof the AD converter during joint motion but the residualvibrations which can be explained by the tracking error andunmodelled high frequency characteristics of the link aresignificantly suppressed by the combined effect of the optimaltrajectory and active control of PZT actuators Moreoverthe results obtained from the last experiment are comparedwith those obtained from the above two experiments Boththe peak value and settling time of the measured vibrationresponse after joint motion are given in Table 3

Table 3 Comparison of the experimental results

Peak value afterjoint motion (V)

Settling time afterjoint motion (s)

Quintic polynomialtrajectory gt10 9

Optimal trajectory withoutPZT control 128 38

Optimal trajectory withPZT control 072 23

As shown in Table 3 it can be observed that residualvibration of the link obtained from the proposed approachis only 072V less than 10 percent of those obtained byassuming that the trajectory corresponds to the quinticpolynomial motion Meanwhile compared with the resultsalong the quintic polynomial trajectory the settling time ofthe residual vibrations along the optimal trajectory descendsapproximately 75 percent with the active control of thePZT actuators reducing to 23 s from 9 s As a conclusionthe proposed optimal trajectory planning approach is effec-tive for vibration suppression of the flexible manipulatorsystem

6 Conclusions

The above sections have presented an optimal trajectoryplanning approach for vibration suppression of a flexiblepiezoelectric manipulator system with bonded PZT actua-tors The proposed approach is a combination of the feed-forward trajectory planning method and L-type velocityfeedback control of piezoelectric actuators based on theLyapunov approach The dynamic equations of the linkare derived using Hamiltonrsquos principle and assumed modemethod In the procedure of trajectory planning the flexiblepiezoelectric manipulator system is taken as a whole rigidmotion of the joint elastic vibration of the link and activecontrol of the PZT actuators are all considered Specificallythe joint controller is responsible for trajectory trackingand gross vibration suppression of the link during motionwhile the active controller of actuators is expected to dealwith residual vibrations after joint motion The joint angleof the link is expressed as a quintic polynomial functionAnd the sum of the link vibration energy is chosen as theobjective function Then the optimal trajectory planningapproach is constructed using genetic algorithm Simulationand experimental results validate the effectiveness of theproposed method Both the settling time and peak valueof the link vibrations along the optimal trajectory descendsignificantly with the active control of the PZT actuatorsAs a conclusion the proposed optimal trajectory planningapproach is effective for vibration suppression of the flexiblepiezoelectric manipulator system


Industrial computerPCI-1742UDAQ board

DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier

Servo driverServomotor

Reductiongear

PZTactuators

Strain gauge

Figure 6 Schematic diagram of the experimental setup

0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Desired trajectoryActual trajectory

Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6

Desired velocityActual velocity

Ang

ular

velo

city

(radmiddotsminus

1)


Desired accelerationActual acceleration

Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01

(d) Vibration response of the link

Figure 7 Experimental results of the link along the quintic polynomial trajectory


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10

Quintic polynomialOptimal trajectory

0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1


Figure 8 Experimental results of the link along the optimal trajectory

Appendix

The position vector p of the link relative to the inertial frameat an arbitrary location 119909 can be expressed as

p = [

119901119909

119901119910

] = [

(119903 + 119909) cos 120579 (119905) minus 119908 (119909 119905) sin 120579 (119905)(119903 + 119909) sin 120579 (119905) + 119908 (119909 119905) cos 120579 (119905)

] (A1)

where 119903 is the radius of the rigid hub 120579(119905) is the joint angleand 119908(119909 119905) is the elastic deflection of the link

The kinetic energy and the potential energy of the flexiblemanipulator system can be written as

119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)

Optimal trajectoryOptimal trajectory with PZT

1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1

(a) Vibration response of the link

0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50

(b) Control voltage of the PZT actuators

Figure 9 Experimental results of the link along the optimal trajectory with PZT control

Here 120588119860 is the effective mass per unit length of the structure120588119887 119860119887 119897119887and 120588

119901 119860119901 119897119901devote the density cross-sectional

area and length of the link and PZT actuators respectively119909119904is the start coordinate of the PZT actuators to the clamped

side And 119864119868 is the equivalent flexural rigidity of the linkwith PZT actuators 119864

119887 119868119887and 119864

119901 119868119901devote the modulus

of elasticity the moment of inertia of the link and PZTactuators respectively

The virtual work carried out by the moment119872(119905) of PZTactuators and the input torque acting on the hub 120591(119905) can beevaluated as

120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)

As each PZT actuator is boned to each side of the link thebending moment119872(119905) resulting from the voltage applied tothe PZT actuators is given by

119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)

where11988931is the piezoelectric constant 119887

119901andℎ119901are thewidth

and thickness of the PZT actuators respectively and 119881(119905) is auniform control voltage applied to the PZT actuators

Extended Hamiltonrsquos principle is used to derive the sys-tem equation Hamiltonrsquos principle is given by the followingvariational statement

int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)

Substituting (A1)sim(A3) incorporating (1) into (A5) thegoverning equations of the system can be derived as

119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)

where the corresponding coefficients are defined as

119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)

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 Sci-ence Foundation of China (no 51375433) and ZhejiangProvincial Natural Science Foundation of China (Grant nosLY13E050008 andLQ15E050002)The authorswould also liketo thank the K C Wong Magna Fund in Ningbo Universityfor its support

References

[1] C T Kiang A Spowage and C K Yoong ldquoReview of controland sensor system of flexible manipulatorrdquo Journal of Intelligentand Robotic Systems Theory and Applications vol 77 no 1 pp187ndash213 2014

[2] M Chu G Chen Q-X Jia X Gao and H-X Sun ldquoSimul-taneous positioning and non-minimum phase vibration sup-pression of slewing flexible-link manipulator using only jointactuatorrdquo Journal of Vibration and Control vol 20 no 10 pp1488ndash1497 2013

[3] W Singhose ldquoCommand shaping for flexible systems a reviewof the first 50 yearsrdquo International Journal of Precision Engineer-ing and Manufacturing vol 10 no 4 pp 153ndash168 2009

[4] ZMohamedA K Chee AW IMohdHashimMO Tokhi SH M Amin and R Mamat ldquoTechniques for vibration controlof a flexible robotmanipulatorrdquoRobotica vol 24 no 4 pp 499ndash511 2006

[5] M S Alam andMO Tokhi ldquoDesigning feedforward commandshapers with multi-objective genetic optimisation for vibrationcontrol of a single-link flexible manipulatorrdquo Engineering Appli-cations of Artificial Intelligence vol 21 no 2 pp 229ndash246 2008

[6] M O T Cole and T Wongratanaphisan ldquoA direct methodof adaptive FIR input shaping for motion control with zeroresidual vibrationrdquo IEEEASME Transactions on Mechatronicsvol 18 no 1 pp 316ndash327 2013

[7] H Wu F C Sun Z Q Sun and L C Wu ldquoOptimal trajectoryplanning of a flexible dual-arm space robot with vibrationreductionrdquo Journal of Intelligent and Robotic Systems vol 40 no2 pp 147ndash163 2004

[8] H R Heidari M H Korayem M Haghpanahi and V F BatlleldquoOptimal trajectory planning for flexible linkmanipulators withlarge deflection using a new displacements approachrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 72no 3-4 pp 287ndash300 2013

[9] K Springer H Gattringer and P Staufer ldquoOn time-optimaltrajectory planning for a flexible link robotrdquo Proceedings of theInstitution of Mechanical Engineers Part I Journal of Systemsand Control Engineering vol 227 no 10 pp 752ndash763 2013

[10] Y Choi J Cheong and H Moon ldquoA trajectory planningmethod for output tracking of linear flexible systems using exactequilibrium manifoldsrdquo IEEEASME Transactions on Mecha-tronics vol 15 no 5 pp 819ndash826 2010

[11] J Lessard P Bigras Z Liu and B Hazel ldquoCharacterizationmodeling and vibration control of a flexible joint for a roboticsystemrdquo Journal of Vibration and Control vol 20 no 6 pp 943ndash960 2014

[12] M Moallem M R Kermani R V Patel and M OstojicldquoFlexure control of a positioning system using piezoelectrictransducersrdquo IEEE Transactions on Control Systems Technologyvol 12 no 5 pp 757ndash762 2004

[13] V B Nguyen and A S Morris ldquoGenetic algorithm tuned fuzzylogic controller for a robot arm with two-link flexibility and

two-joint elasticityrdquo Journal of Intelligent and Robotic SystemsTheory and Applications vol 49 no 1 pp 3ndash18 2007

[14] J Das and N Sarkar ldquoPassivity-based target manipulationinside a deformable object by a robotic system with non-collocated feedbackrdquo Advanced Robotics vol 27 no 11 pp 861ndash875 2013

[15] C-Y Lin and C-M Chang ldquoHybrid proportional deriva-tiverepetitive control for active vibration control of smartpiezoelectric structuresrdquo Journal of Vibration and Control vol19 no 7 pp 992ndash1003 2013

[16] J-J Wei Z-C Qiu J-D Han and Y-C Wang ldquoExperimentalcomparison research on active vibration control for flexiblepiezoelectric manipulator using fuzzy controllerrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 59no 1 pp 31ndash56 2010

[17] D Sun J K Mills J J Shan and S K Tso ldquoA PZT actuatorcontrol of a single-link flexible manipulator based on linearvelocity feedback and actuator placementrdquoMechatronics vol 14no 4 pp 381ndash401 2004

[18] K Gurses B J Buckham and E J Park ldquoVibration control ofa single-link flexible manipulator using an array of fiber opticcurvature sensors and PZT actuatorsrdquoMechatronics vol 19 no2 pp 167ndash177 2009

[19] L Y Li G B Song and J P Ou ldquoAdaptive fuzzy slidingmode based active vibration control of a smart beam with massuncertaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[20] S-B Choi M-S Seong and S H Ha ldquoAccurate positioncontrol of a flexible arm using a piezoactuator associated witha hysteresis compensatorrdquo Smart Materials and Structures vol22 no 4 Article ID 045009 2013

[21] Y S Bian and Z H Gao ldquoImpact vibration attenuation for aflexible robotic manipulator through transfer and dissipation ofenergyrdquo Shock and Vibration vol 20 no 4 pp 665ndash680 2013

[22] A A Ata ldquoInverse dynamic analysis and trajectory planning forflexible manipulatorrdquo Inverse Problems in Science and Engineer-ing vol 18 no 4 pp 549ndash566 2010

[23] M D G Marcos J A Tenreiro Machado and T-P Azevedo-Perdicoulis ldquoA multi-objective approach for the motion plan-ning of redundant manipulatorsrdquo Applied Soft Computing Jour-nal vol 12 no 2 pp 589ndash599 2012

[24] T Nestorovic and M Trajkov ldquoOptimal actuator and sensorplacement based on balanced reduced modelsrdquo MechanicalSystems and Signal Processing vol 36 no 2 pp 271ndash289 2013

[25] W C Sun H J Gao and O Kaynak ldquoAdaptive backsteppingcontrol for active suspension systems with hard constraintsrdquoIEEEASME Transactions on Mechatronics vol 18 no 3 pp1072ndash1079 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



positive and negative input shapers Alam and Tokhi [5]developed the design of a command shaping controller forvibration suppression of a flexible manipulator using multi-objective genetic optimization process More recently Coleand Wongratanaphisan [6] described an adaptive FIR inputcommand shaping methodology to achieve zero residualvibration control of a rigid-flexible manipulator system Theinput command shaping method is effective to ensure thatthe input does not excite vibrations of the manipulator Butit usually costs a large amount of computation when dealingwith multiple vibration modes And it cannot suppress theinduced and existing vibrations

An alternative feed-forward control strategy to achievevibration control of flexible manipulators is the trajectoryplanning approach Wu et al [7] proposed an optimaltrajectory planning method for vibration reduction of adual arm space robot with front flexible links in whichthe trajectory was described using a fourth-order quasi-uniform B-spline Heidari et al [8] considered the problemon the rest-to-rest motion of a flexible manipulator andformulated the trajectory planning problem as a Pontrya-gin optimal problem Springer et al [9] focused on time-optimal trajectory planning for robots with flexible linksand determined the minimum time trajectory for avoidingelastic vibration of the link Choi et al [10] described anew trajectory planning method for output tracking of linearflexible system using exact equilibrium manifolds Howeverthe trajectory planning technique is a feed-forward controlstrategy vibration modes of the flexible manipulator are stillcontrolled in open loop [11] When dealing with variousdisturbances and parameters variations there would still belarge residual vibrations that need to be controlled [12]

To overcome the drawbacks of feed-forward controlschemesmany researchers have tried to solve the problems oflink vibration suppression using feedback control techniquesAmong them some researchers concentrated on reducingthe link vibrations using joint motor [13] However theexistence of nonminimum phase dynamics between the tipposition and the input torque applied at the joint motormakes the system difficult to stabilize [14] A rather differentapproach for feedback control of link vibrations has beendeveloped in recent years relying on the use of smartmaterials [15] Among those smart materials piezoelectric(PZT) materials have been found extensive applications instructure vibration control due to their lightweight fastresponse large bandwidth and so on [16] Sun et al [17]realized the rigid motion control and vibration damping ofa flexible manipulator using a combined scheme of a PDcontroller for the joint motor and a linear velocity feedback(L-type) controller for the piezoelectric actuators Gurses etal [18] extended the previous work studied and introduceda serial array of fiber optic sensors to the control of flexiblemanipulators which allowed simultaneous linear (L-type)and angular (A-type) velocity feedback Li et al [19] presenteda study on vibration suppression of a flexible piezoelectricmanipulator using adaptive fuzzy sliding mode control Choiet al [20] studied the position control of a single-link flexiblemanipulator and designed a hysteresis compensator for thepiezoelectric actuators

However in most studies on active control of flexiblemanipulator attention seems to be focused on how toimprove the performance of the piezoelectric controller Thecontributions of the feed-forward torque or the requiredtrajectory planning are rarely considered A more reasonableand efficient approach seems to emerge if a combined strategyis considered Specifically the joint motor is responsiblefor the gross vibrationless motion of the structure throughtrajectory planning approach while the PZT actuators areused to deal with vibrations that may arise from modelingerrors and external disturbances after joint motion by meansof active control The remainder of this paper is organized asfollows In Section 2 the flexible piezoelectric manipulatorsystem is introduced and the governing equations of thesystem are derived Section 3 proposes the optimal trajectoryplanning approach The joint angle of the link is expressed asa quintic polynomial functionThe constraint conditions theinitial and final conditions of the joint angle angular velocityand angular acceleration are applied The sum of the linkvibration energy is chosen as the objective function Thenthe optimal trajectory planning algorithm is constructed bythe use of genetic algorithm In Section 4 numerical analysisand simulations are conducted The optimal trajectory ofthe flexible piezoelectric manipulator in a point-to-pointmotion is obtained Section 5 describes an experimentalapparatus of the flexible manipulator system together withthe experimental results before concluding in Section 6

2 Dynamic Modeling

The schematic diagram of the flexible piezoelectric manipu-lator system is shown in Figure 1 It is composed of a rotaryjoint a single-link flexible manipulator with a rectangularcross section and a pair of collocated piezoelectric (PZT)actuators The manipulator is attached to the hub in such away that it rotates in the horizontal plane only The effect ofgravity can be ignored so the link can bemodeled as anEuler-Bernoulli beamThe axis 119900119909

119900is the inertial reference lineThe

axis 119900119909 is the tangential line to the neutral axis of the beam atthe hub For the sake of simplicity the same assumptions aremade as [17]

It is assumed that the dynamic behavior of the flexiblemanipulator during a point-to-point motion is dominated bythe first several vibration modes so the contributions of thehigher vibration modes are negligible Using assumed modemethod the deflection 119908(119909 119905) of the link can be representedas

119908 (119909 119905) =

119898

sum

119894=1

120593119894(119909) 119902119894(119905) = Φ (119909) q (119905) (1)

where 120593119894(119909) and 119902

119894(119905) devote the 119894thmode shape function and

corresponding generalized coordinate respectively Φ(119909) =

[1205931sdot sdot sdot 120593119894sdot sdot sdot 120593119898] and q(119905) = [119902

1sdot sdot sdot 119902119894sdot sdot sdot 119902119898] represent the

mode shape function and generalized coordinate indexrespectively119898 is the mode number




yo

y

120579(t)

w(x t)

120591(t)

P

x



119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905) (2)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904) (3)


m119902119902q +m119879

120579119902

120579 + C119902q + K

119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(4)


119881 (119905) = minus119896119871[ (119909

119904+ 119897119901) minus (119909

119904)] = minus119896

119871Φ (119909119904) q (5)


(4) leads to

m119902119902q +m119879

120579119902

120579 + C119902q + 119888119896

119871[Φ1015840

]

119879

(119909119904) Φ (119909

119904) q + K

119902q

minusm119902119902q 1205792

= 0

(6)






119891over



PI (119909) = 1205821int

119905119891

1199050

q119879 (119905) q (119905) 119889119905 + 1205822int

119905119889

119905119891

q119879 (119905) q (119905) 119889119905 (7)



1



120579 (119905) = 1198620+ 1198621119905 + 11986221199052

+ 11986231199053

+ 11986241199054

+ 11986251199055

(8)

where 1198620 1198621 1198622 1198623 1198624 and 119862




119891is divided




t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 tf


Ψ(t2)

Ψ(t3)

Ψ(t4)

Ψ(t5)

Ψ(t6)

Ψ(t7)

Ψ(t8)Ψ(t9)

1

2

3 4

56

7

8

9

ΔF6

ΔF7

120579f

1205790


(119899 = 10)


1199051 119905

119894 119905


119894is denoted by

120579119894


119865119894as

120579119894= Ψ (119905

119894) + Δ

120579119865119894 (9)







119891


119865119894(119894 = 1 2 119899minus1)






1) 120595(1199052) 120595(119905

119894)

120595(119905119899minus1



119896= [1199091198961 1199091198962 119909

119896119894 119909

119896(119899minus1)] The











1= 025 120582

2= 075




Δ120579119865119894isin [minus05

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

1003816100381610038161003816100381605

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

10038161003816100381610038161003816] (10)


0 lt120579119894lt 120579119891

(119894 = 1 2 119899 minus 1)

120579 le

120579max

119881 le 119881max

(11)







minus12





119891



0 20 40 60 80 1000

50

100

150

200

250

Generation

Fitn

ess v

alue








0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)



0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)


0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)


Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5






Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












yo

y

120579(t)

w(x t)

120591(t)

P

x



119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905) (2)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904) (3)


m119902119902q +m119879

120579119902

120579 + C119902q + K

119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(4)


119881 (119905) = minus119896119871[ (119909

119904+ 119897119901) minus (119909

119904)] = minus119896

119871Φ (119909119904) q (5)


(4) leads to

m119902119902q +m119879

120579119902

120579 + C119902q + 119888119896

119871[Φ1015840

]

119879

(119909119904) Φ (119909

119904) q + K

119902q

minusm119902119902q 1205792

= 0

(6)






119891over



PI (119909) = 1205821int

119905119891

1199050

q119879 (119905) q (119905) 119889119905 + 1205822int

119905119889

119905119891

q119879 (119905) q (119905) 119889119905 (7)



1



120579 (119905) = 1198620+ 1198621119905 + 11986221199052

+ 11986231199053

+ 11986241199054

+ 11986251199055

(8)

where 1198620 1198621 1198622 1198623 1198624 and 119862




119891is divided




t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 tf


Ψ(t2)

Ψ(t3)

Ψ(t4)

Ψ(t5)

Ψ(t6)

Ψ(t7)

Ψ(t8)Ψ(t9)

1

2

3 4

56

7

8

9

ΔF6

ΔF7

120579f

1205790


(119899 = 10)


1199051 119905

119894 119905


119894is denoted by

120579119894


119865119894as

120579119894= Ψ (119905

119894) + Δ

120579119865119894 (9)







119891


119865119894(119894 = 1 2 119899minus1)






1) 120595(1199052) 120595(119905

119894)

120595(119905119899minus1



119896= [1199091198961 1199091198962 119909

119896119894 119909

119896(119899minus1)] The











1= 025 120582

2= 075




Δ120579119865119894isin [minus05

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

1003816100381610038161003816100381605

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

10038161003816100381610038161003816] (10)


0 lt120579119894lt 120579119891

(119894 = 1 2 119899 minus 1)

120579 le

120579max

119881 le 119881max

(11)







minus12





119891



0 20 40 60 80 1000

50

100

150

200

250

Generation

Fitn

ess v

alue








0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)



0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)


0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)


Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5






Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 tf


Ψ(t2)

Ψ(t3)

Ψ(t4)

Ψ(t5)

Ψ(t6)

Ψ(t7)

Ψ(t8)Ψ(t9)

1

2

3 4

56

7

8

9

ΔF6

ΔF7

120579f

1205790


(119899 = 10)


1199051 119905

119894 119905


119894is denoted by

120579119894


119865119894as

120579119894= Ψ (119905

119894) + Δ

120579119865119894 (9)







119891


119865119894(119894 = 1 2 119899minus1)






1) 120595(1199052) 120595(119905

119894)

120595(119905119899minus1



119896= [1199091198961 1199091198962 119909

119896119894 119909

119896(119899minus1)] The











1= 025 120582

2= 075




Δ120579119865119894isin [minus05

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

1003816100381610038161003816100381605

10038161003816100381610038161003816120595 (119905119894) minus 120579119891

10038161003816100381610038161003816] (10)


0 lt120579119894lt 120579119891

(119894 = 1 2 119899 minus 1)

120579 le

120579max

119881 le 119881max

(11)







minus12





119891



0 20 40 60 80 1000

50

100

150

200

250

Generation

Fitn

ess v

alue








0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)



0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)


0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)


Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5






Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













minus12





119891



0 20 40 60 80 1000

50

100

150

200

250

Generation

Fitn

ess v

alue








0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)



0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)


0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)


Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5






Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)

Ang

ular

disp

lace

men

t (ra

d)



0 01 02 03 04 050

1

2

3

4

5

6

Time (s)


Ang

ular

velo

city

(radmiddotsminus

1)


0 01 02 03 04 05

0

20

40

60

Time (s)


minus60

minus40

minus20

Ang

ular

acce

lera

tion

(radmiddotsminus

2)


Time (s)0 05 1 15 2

0

5

10

15

Tip

vibr

atio

n re

spon

se (m

m)


minus15

minus10

minus5






Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Base

Servo AC motor

Strain gauge













6 Conclusions




DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











DAconverter

Counter

ADconverter

ADconverter

Encoder

Flexible link

Analogfilter

Strainamplifier

Poweramplifier


Reductiongear

PZTactuators

Strain gauge


0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5

6


Ang

ular

velo

city

(radmiddotsminus

1)



Time (s)0 01 02 03 04 05

0

10

20

30

40

Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus40

minus30

minus20

minus10


Time (s)0 2 4 6 8 10

0

5

10

Sens

or v

olta

ge (V

)

85 9 95 10

0

01

minus10

minus5

minus01




0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 050

02

04

06

08

1

12

14

16

Time (s)


Ang

ular

disp

lace

men

t (ra

d)


Time (s)0 01 02 03 04 05

0

1

2

3

4

5


Ang

ular

velo

city

(radmiddotsminus

1)


Time (s)0 01 02 03 04 05

0

20

40

60


Ang

ular

acce

lera

tion

(radmiddotsminus

2)

minus60

minus40

minus20


Time (s)0 2 4 6 8 10

0

5

10


0 1 2 3

0

1

Sens

or v

olta

ge (V

)

minus10

minus5

minus1



Appendix


p = [

119901119909

119901119910

] = [


] (A1)



119879 =

1

2

int

119897119887

0

120588119887119860119887(119909) (

2

119909+ 2

119910) 119889119909

+

1

2

int

119909119904+119897119901

119909119904

2120588119901119860119901(2

119909+ 2

119910) 119889119909

119881119901=

1

2

int

119897119887

0

11986411988711986811988711990810158401015840

(119909 119905)

2

119889119909

+

1

2

int

119909119904+119897119901

119909119904

11986411990111986811990111990810158401015840

(119909 119905)

2

119889119909

(A2)


0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5

0

2

4

6

8

10

Time (s)

Sens

or v

olta

ge (V

)


1 2 3 4

0

1

minus10

minus8

minus6

minus4

minus2

minus1


0 1 2 3 4 5

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus150

minus100

minus50







119887 119868119887and 119864




120575119882 = 120591 (119905) 120575120579 (119905) + 119872 (119905) 1205751199081015840

(119909119904 119905) (A3)


119872(119905) = minus211986411990111988931119887119901ℎ119901119881 (119905) = 119888119881 (119905) (A4)





int

1199052

1199051

(120575119879 minus 120575119881 + 120575119882) 119889119905 = 0 (A5)


119868120579120579

120579 + q119879m

119902119902q 120579 + 2q119879m

119902119902q 120579 +m

120579119902q = 120591 (119905)

m119902119902

119902 +m119879120579119902

120579 + K119902q minusm

119902119902q 1205792

= 119888119881 (119905) [Φ1015840

]

119879

(119909119904)

(A6)


119868120579120579=

1

3

120588119887119860119887[(119903 + 119897

119887)3

minus 1199033

]

+

1

3

2120588119901119860119901[(119903 + 119909

119904+ 119897119901)

3

minus (119903 + 119909119904)3

]

m119902119902= int

119897119887

0

120588119887119860119887Φ119879

(119909)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901Φ119879

(119909)Φ (119909) 119889119909

m120579119902= int

119897119887

0

120588119887119860119887(119909 + 119903)Φ (119909) 119889119909

+ int

119909119904+119897119901

119909119904

120588119901119860119901(119909 + 119903)Φ (119909) 119889119909

K119902= int

119897119887

0

119864119887119868119887[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

+ int

119909119904+119897119901

119909119904

119864119901119868119901[Φ10158401015840

]

119879

(119909)Φ10158401015840

(119909) 119889119909

Φ (119909119904) = Φ (119909

119904+ 119897119901) minusΦ (119909

119904)

(A7)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article optimal trajectory planning and linear

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