Robotica (2011) volume 29 pp 211ndash219 copy Cambridge University Press 2010doi101017S0263574710000056

Experimental backlash study in mechanical manipulatorsMiguel F M Limadaggerlowast J A Tenreiro MachadoDaggerand Manuel CrisostomosectdaggerDepartment of Electrical Engineering Superior School of Technology Polytechnic Institute of Viseu3504-510 Viseu PortugalDaggerDepartment of Electrical Engineering Institute of Engineering Polytechnic Institute of Porto 4200-072 Porto PortugalsectInstitute of Systems and Robotics Department of Electrical and Computer Engineering University of Coimbra Polo II3030-290 Coimbra Portugal

(Received in Final Form February 3 2010 First published online March 4 2010)

SUMMARYThe behavior of mechanical manipulators with backlashis analyzed In order to acquire and study the signals anexperimental setup is implemented The signal processingcapabilities of the wavelets are used for de-noising theexperimental signals and the energy of the obtainedcomponents is analyzed To evaluate the backlash effectupon the robotic system it is proposed an index basedon the pseudo phase plane representation Several tests aredeveloped that demonstrate the coherence of the results

1 IntroductionRobotic systems have nonlinearities in the actuators thatinclude deadzone backlash and saturation This problemis particularly important in robotic manipulation where highprecision is needed In fact the backlash is one of the mostimportant nonlinearities that limit the performance of themechanical manipulators This dynamic phenomenon hasbeen an area of active research but due to its complexitywell-established conclusions are still lacking

The backlash in robotic systems has two main aspects theidentification and the control Several authors have addressedthe problem of identification1ndash6 Dagalakis and Myers1

proposed a technique based on the coherence functionto detect backlash in robotic systems Another techniqueproposed by Stein and Wang2 was based on the analysis ofmomentum transfer for detecting the backlash in mechanicalsystems There have been also efforts toward the backlashdetection using artificial intelligence schemes and state spaceobservers45 Trendafilova and Brussel6 present several toolsto analyze the nonlinear dynamics of the robot joints

To mitigate the effects of the backlash several authorsstudied the control of the mechanical systems withthis nonlinearity7ndash9 Nordin and Gutman8 presented asurvey of the techniques for controlling mechanicalsystems with backlash The techniques included theuse of linear controllers such as proportionalndashintegralndashderivative controller (PID) state feedback and observer-based algorithms A control using the describing function

Corresponding author E-mail limamailestvipvpt

which is a common method for the analysis and synthesis ofnonlinear systems was also adopted7 Nonlinear10 neuro11

and fuzzy12 controllers have been also proposed Morerecently the fractional order controllers were also applied tosuppress the backlash in mechanical systems9 The problemof reducing the effects of backlash was also studied inother robotic applications including systems with joint13 andlink14 flexibility in grippers15 and anthropomorphic hands16

Modern robots use precision gears to reduce backlashHowever its elimination may be impracticable becausethere are several sources of the effect which are impossibleto remove completely1 Therefore in order to reduce thebacklash and its effects a good understanding of thisphenomenon is needed In this perspective we investigatethe behavior of a mechanical manipulator with backlash inthe joints

Bearing these ideas in mind this paper is organizedas follows Section 2 describes briefly the robotic systemused to capture the signals Sections 3 and 4 presentsome fundamental concepts and the experimental resultsrespectively Finally Section 5 outlines the main conclusionsand points out the plans for future work

2 Experimental PlatformIn order to analyze signals that occur in a robotic manipulatorwas developed an experimental platform The platform hastwo main parts the hardware and the software components17

The hardware architecture is shown in Fig 1 (left)Essentially it is made up of a mechanical manipulator acomputer and an interface electronic system The interfacebox is inserted between the arm and the robot controllerin order to acquire the internal robot signals The interfacesystem captures also external signals such as those arisingfrom accelerometers and forcetorque sensors The modulesare made up of electronic cards specifically designed for thiswork The function of the modules is to adapt the signals andto isolate galvanically the robotrsquos electronic equipment fromthe rest of the hardware required by the experiments

The software package runs in a Pentium 4 30 GHz PC andfrom the userrsquos point of view consists of two applicationsOne the acquisition application is a program made up of

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212 Experimental backlash study in mechanical manipulators

Fig 1 Block diagram of the hardware architecture (left) and accelerometer 2 mounted on the robot arm (right)

two parts the graphical user interface (GUI) module andthe real time module The other application is an AnalysisPackage that analyses the data obtained and recorded by theacquisition application The real time software running inthe Hyperkernel was developed in C based on a standardWindows NT2000 development tool (MS Visual Studio)and the robot controller software was implemented in theadvanced control language (ACL) proprietary languageThe Windows NT2000 Software is made up of the GUImodule of the acquisition system and Analysis Package Theacquisition system software was developed in C++ with MSVisual Studio The Analysis Package running off-line readsthe data recorded by the acquisition system and examinesthem The Analysis Package allows several signal processingalgorithms such as Fourier transform correlation timesynchronization etc With this software platform both theHyperkernel and the Analysis Package tasks can be executedon the same PC

The manipulator is a vertical articulated robot with fiverotational joints The third joint connects the upper armto the forearm of the robot This joint is driven througha servomotor coupled to the harmonic drive gear by atiming belt By adjusting the belt tension of the third jointtwo-stage transmission three distinct degrees of backlashwere introduced that are qualitatively classified as (i) lowbacklash (ii) medium backlash and (iii) high backlashThe vibration response is measured with the accelerometers1 and 2 that are mounted in the end of the upper armand forearm respectively (see Fig 1) The robot motion isprogrammed in a way such that only the third joint is drivenand consequently it oscillates over a predefined range fromthe vertical position The axis of the rotational joints 2 3 and

4 are parallel therefore the effects of the third axis rotationaffect directly the adjacent joints During the motion severalsignals are recorded with a sampling frequency of fs =750 Hz The tests developed show that this sampling fre-quency is adequate to capture the relevant dynamic behaviorof the system The signals come from different sensors suchas accelerometers wrist force and torque sensor positionencoders and joint actuator current sensors Figure 1 (right)depicts the accelerometer 2 mounted on the robot arm

3 Main ConceptsThis section presents briefly the fundamental conceptsinvolved in the experiments In order to deal with thenoisy signals captured by the accelerometers the de-noisecapabilities of wavelets18 are used Additionally the systembehavior with backlash is analyzed through the pseudo phasespace (PPS)6

31 The wavelet transformThe continuous wavelet transform (CWT) is a generalizationof the windowed Fourier transform (WFT) The concept ofthe WFT is very simple We multiply the signal to be analyzedx(t) by a moving window g(t minus τ ) and then we computethe Fourier transform of the windowed signal x(t) g(t minus τ )Each FT gives a frequency domain ldquoslicerdquo associated withthe time value at the window center Wavelet analysis isperformed in a similar way in the sense that the signal ismultiplied by a function called wavelet However in theCWT the width of the ldquowindowrdquo changes as the transformis computed Considering the wavelet function ψ centeredat time τ and scaled by s the CWT of the signal x(t) is

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Experimental backlash study in mechanical manipulators 213

Fig 2 Simplified diagram of a signal wavelet multiresolutionanalysis

represented analytically by

CWT(s τ ) = 1radic|s|int +infin

minusinfinx(t)ψ

(t minus τ

s

)dt (1)

The CWT of the signal x(t) is a function of twovariables translation τ which corresponds directly to timeand scaledilation s which indirectly relates to frequencyinformation The transforming function ψ(t) is called motherwavelet and consists in a prototype for generating the otherwindow functions The scale s in the wavelet transform issimilar to the scale used in maps High scales give a globalview of the signal corresponding in the frequency domainto the low frequencies Low scales give detailed informationof a signal corresponding in the frequency domain to thehigh frequencies

The digital version of the CWT is the discrete wavelettransform (DWT) that is considerably faster to implement ina computer A time-scale representation of a digital signalcan also be implemented using digital filtering techniquesAn efficient way to implement the DWT using filters wasdeveloped by Mallat19 Filters of different cutoff frequenciesare used to analyze the signal at different scales (see Fig 2)The signal is passed through a bank of high pass filtersto analyze the high frequencies giving detailed information(Dn) Additionally the signal is passed through a bank of lowpass filters to analyze the low frequencies giving a coarseapproximation (An) Then the decomposition of the signalinto different frequency bands is obtained by successive high-pass and low-pass filtering of the time domain signal

In what concerns to the wavelet function ψ(t) there isa wide variety of wavelet families proposed by differentresearchers that includes the Haar Daubechies MexicanHat and Morlet wavelets The function ψ(t) should reflectthe type of features present in the time series20 Therefore the

wavelet adopted in this paper is the Haar (Eq (2)) functiondue also to its simplicity and small computational time

In this work we use the signal processing capabilities ofthe DWT for de-noising the experimental signals

ψ(t) =⎧⎨⎩

1 0 le t lt 12

minus1 12 le t lt 1

0 otherwise(2)

32 The pseudo phase planeThe PPS is used to analyze signals with nonlinear behaviorFor the two-dimensional case it is called pseudo phaseplane (PPP)62122 The proper time lag Td for the delaymeasurements and the adequate dimension d isin N of thespace must be determined in order to achieve the phase spaceIn the PPP the measurement s(t) forms the pseudo vector y(t)according to

y(t) = [s(t) s(t + Td ) s(t + (d minus 1)Td )] (3)

The vector y(t) can be plotted in a d-dimensional spaceforming a curve in the PPS If d = 2 we have a two-dimensional time delay space

The procedure of choosing a sufficiently large d is formallyknown as embedding and any dimension that works is calledan embedding dimension dE The number of measurementsdE should provide a phase space dimension in which thegeometrical structure of the plotted PPS is completely unfoldand where there are no hidden points in the resulting plot

Among others21 the method of delays is the mostcommon method for reconstructing the phase space Severaltechniques have been proposed to choose an appropriate timedelay22 One line of thought is to choose Td based on thecorrelation of the time series with its delayed image Thedifficulty of correlation to deal with nonlinear relations leadsto the use of the mutual information This concept fromthe information theory23 recognizes the nonlinear propertiesof the series and measures their dependence The averagemutual information for the two series of variables t and t + Td

is given by

Iav(t t + Td ) =int

t

intt+Td

F1s(t) s(t + Td )) log2

times F1s(t) s(t + Td )F2s(t)F3s(t + Td ) dtd(t + Td) (4)

where F1s(t) s(t + Td ) is a bidimensional probabilitydensity function and F2s(t) and F3s(t + Td) are themarginal probability distributions of the two series s(t) ands(t + Td ) respectively

The index Iav allows us to obtain the time lag requiredto construct the PPS For finding the best value Td of thedelay Iav is computed for a range of delays and the firstminimum is chosen Usually Iav is referred62122 as thepreferred alternative to select the proper time delay Td

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214 Experimental backlash study in mechanical manipulators

Fig 3 Robot axis positions for joints 2ndash4 for the case (iii) (left) and electrical currents of robot axis motors 2ndash4 for the case (iii) (right)

Fig 4 Signals of accelerometer 1 (bottom) and accelerometer 2(middle and top)

4 ResultsAccording to the platform described in Section 2 we adopt anexperiment in which the robot arm moves as a consequenceof the joint 3 rotation By other words the robot motion isprogrammed in a way such that only the joint 3 is driven andconsequently it oscillates over a predefined range from thevertical position

Figures 3 and 4 present a typical time evolution of severalvariables All the captured robot signals were studied butdue to space limitations only the most relevant are depictedin this paper

Figure 3 (left) shows the axis positions of joints 2ndash4 for thecase of high backlash It is actuated only the joint 3 but dueto the dynamic coupling the adjacent joints 2 and 4 oscillateslightly Figure 3 (right) depicts the electrical currents ofrobot axis motors 2ndash4 for the high backlash (case (iii)) Thecurrents of the motors reveal the action of the robot controlsystem ensuring the position control

Figure 4 (middle and bottom) shows the robot accelera-tions for the case (iii) As referred previously theaccelerometers 1 and 2 are mounted in the end of the upperarm and forearm respectively (see Fig 1) The signals from

the accelerometers present a considerable noise and it isdifficult to extract information about the backlash effectSeveral tests demonstrated that the effects of the adjustedbacklash are captured only by the accelerometer 2 Thesignals from the others sensors present significant less noisebut the backlash effect is not observed by them A priorithe accelerometers constitute sensors that are more proneto capture the backlash effect but the truth is that theaccelerometer 1 revealed to be useless due to the high levelof noise

Figure 4 (top and middle) depicts the signal from theaccelerometer 2 for the cases (i) and (iii) respectively Thesignal corresponding to the case (iii) exhibits larger peakscompared with the case (i) In spite of this detail the effect ofthe backlash becomes difficult to analyze due to the amount ofnoise of the acceleration signals Therefore the signals mustbe filtered in order to reduce the noise First it was tried alow-pass Butterworth filter with different cutoff frequenciesThe resulting filtered signal presented a noise reductionbut the backlash effect was also reduced As an alternativethe wavelets were adopted Preliminary tests with severalwavelet families were performed to verify their capabilitiesThe Haar wavelet revealed good results and was adopted dueto its simplicity and low computational time

Figure 5 depicts the wavelet decomposition tree withthe resulting frequency bands of the correspondingapproximation (An) and detail (Dn) components for eachlevel Since the sampling rate of the captured signal is750 Hz the frequency ranges are approximately the valuesshown in the diagram

Figure 6 (left) shows the five level components obtainedthrough the process of decomposition of the accelerometer 2signal in the case (i) The original signal capturedfrom the accelerometer 2 is shown at the top Theapproximation component reveals the low frequency partof the acceleration 2 The five detail components show thehigh-frequency parts for the different frequency bands Amethod of filtering based on the threshold level for eachcomponent is used For the distinct components the valuesadopted for the threshold level based on an experimentalprocedure are [374 442 426 409 392 374] msminus2

for the wavelets components [A5 D1 D2 D3 D4 D5]respectively Figure 6 (right) shows at the top the resulted

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms

Experimental backlash study in mechanical manipulators 215

Fig 5 Wavelet decomposition and the resulting frequency bands

Fig 6 Wavelet decomposition of the accelerometer 2 signal for the case (i) noisy (left) filtered (right)

filtered signal where it is evident the effect of the backlashFigure 6 (right) shows also the six components of the filteredsignal corresponding to the same frequency bands shownin the left Comparing the five detail components of thefiltered signal with those ones from the original signal itis clear the effect of filtering Additionally the energy valuesof each wavelet component are shown in Fig 6 Comparingthe filtered with the noisy signal reveals that the process offiltering removes about 21 of the signal energy

To compare the wavelet transforms energies of each levelthe normalized values are adopted Therefore the total energyresulted from the sum of all the signal components isthe unity Figure 7 depicts the normalized energy for thesix components resulted from the decomposition processThe distribution of the energy in the frequency domain isnot uniform The signal has the energy concentrate at lowfrequency namely at A1 component Nevertheless the signalhas also a significant amount of energy at D1 to D3 details

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms

216 Experimental backlash study in mechanical manipulators

Fig 7 Wavelet components energy of the accelerometer 2 originaland filtered signals for the three cases

In terms of energy the approximation component A1 isthe most important either for the noisy signal or for thefiltered signal However the A1 component is not sensitiveto the backlash Additionally the details D1 of the noisysignal for the cases (i) (ii) and (iii) have the same energyapproximately while for the filtered signal the componentenergy for the three cases varies significantly The detailcomponents D2 and D3 present a similar behavior to the onedescribed for the component D1 Therefore comparing theD1 D2 D3 details we can say that these components ofthe noisy signal are not sensitive to the backlash while thefiltered versions components changes with the backlash Forthe noisy signal the detail components D4 D5 presentidentical energy while for the filtered signal the correspon-ding components energy for the three cases is negligible

In conclusion we can say that the approximationcomponent A1 is the signal responsible for the excitationof the system while the D1 D2 D3 details of the filteredsignal are the components sensitive to the backlash Finallythe detail components D4 D5 are essentially noise

The filtered periodic signal can be expanded to a Fourierseries in the form

xFS(t) = a0 +infinsum

k=1

[ak cos(kωt) + bk sin(kωt)] (5)

An infinite number of terms (ie the fundamental termand the harmonics) are required to fit the filtered signalx(t) with xFS(t) However we can assume that the filteredsignal x(t) (see Fig 6 right at top) is composed by thefundamental harmonic perturbed by the backlash effectTherefore for the low backlash (case (i)) with k = 1 weobtain (a0 a1 b1) = (minus053 minus569 072) Figure 8 showsthe filtered accelerometer 2 signal for the case (i) and itsfundamental harmonic

Figure 9 depicts the PPPs of the noisy and filteredaccelerometer 2 signals for the experiment with the low andhigh backlash and the fundamental harmonic As referredpreviously usually the time lag adopted for the PPS is basedon the correlation or mutual information of the time series

Fig 8 Filtered accelerometer 2 signal and its fundamental harmonicfor the case (i)

In this work practice reveals that the best time lag is that onecorresponding to one quarter of the period of the fundamentalharmonic which was adopted for the PPPs shown in Fig 9Once again we observe the advantages of filtering the signalComparing the PPPs of the noisy signal for the cases (i)(Fig 9a) and (iii) (Fig 9b) we can see the backlash effecthowever due to noise it is difficult to measure it This taskis simplified using the corresponding PPPs for the filteredsignal (Fig 9cndashd) In this line of thought was developed ametric based on the error between the filtered signal andits fundamental harmonic to analyze the influence of thedistinct levels of backlash in the robotic system For the twocomponents x(t) and x(t minus τ ) of the PPP the index is basedon the root mean square error in discrete time given by

RMSE2 = RMSE2t + RMSE2

tminusτ (6)

where

RMSEtx(t) =

1

N

Nsumk = 0

[x(kT ) minus x1(kT )]2

1

N

Nsumk = 0

x2(kT )

where x1(t) is the fundamental harmonic and T is thesampling period

The RMSE values computed for the three cases of backlashusing the filtered signal (RMSEi RMSEii RMSEiii) =(224 244 314) times 10minus7 show that the index magnitudeincreases with the level of backlash (see Fig 10) On theother hand the corresponding values for the unfilteredsignal are (RMSEi RMSEii RMSEiii) = (513 509 543) times10minus7 which reveals a different behavior due to the effect ofnoise Additionally a set of experiments were developed tostudy the influence of the amplitude and frequency of the joint3 motion upon the backlash dynamics Figure 11 depicts theRMSE of the accelerometer 2 signal for the nine experiments

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms

Experimental backlash study in mechanical manipulators 217

Fig 9 PPPs of the accelerometer 2 signal and its fundamental harmonic (a) noisy signal for the case (i) (b) noisy signal for the case (iii)(c) filtered signal for the case (i) (d) filtered signal for the case (iii)

Fig 10 RMSE for noisy and filtered versions of the accele-rometer 2 signal for the three cases

The results confirm in general the behavior of the RMSEthat increases with the level of the backlash However forsome experiments the index is not in complete accordance

Fig 11 RMSE of the accelerometer 2 signal for a set ofexperiments cases (i) (ii) and (iii)

with the expected results This fact can be caused by the noiseof the signal that is not filtered completely probably due tothe method adopted for the selection of the threshold level

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms

218 Experimental backlash study in mechanical manipulators

Fig 12 RMSE of the accelerometer 2 signal vs amplitude and frequency of joint 3 movements

The use of another method (eg an heuristic one) or the useof another type of wavelet could mitigate the problem of thenoise A deeper insight into the nature of this problem mustbe envisaged

Additionally Fig 12 (left) depicts the RMSE values ofthe accelerometer 2 signal vs the amplitude and frequencyof joint 3 movements for low backlash case The ninepoints shown in Fig 11 for each individual case producea smooth surface that relates the three variables There is amaximum of RMSE 106 times 10minus6 that occurs at (frequencyamplitude) = (06 65) Figure 12 (right) depicts the samevariables for the high backlash case and we verify that wehave a RMSE = 135 times 10minus6 maximum that occurs again at(frequency amplitude) = (06 65)

The tests developed in this article proved that the RMSEreveals to be a good index for the backlash analysis

5 ConclusionsThe tests demonstrate the usefulness of the wavelets on filter-ing the experimental signals reducing the noise while main-taining important features about the backlash The energyanalysis of the signals revealed the components responsiblefor the excitation of the system and for the backlash

An index based on the PPP was proposed to detect thebacklash effect on a robotic manipulator The tests proved thatthe RMSE is an appropriate index for the backlash analysis

In a future work we plan to pursue several researchesin order to further understand the backlash phenomenonThese include the study of the wavelet tuning parametersin the process of de-noising and the analysis of thebacklash with various static preloads applied on the robotend-effector

References1 N G Dagalakis and D R Myers ldquoAdjustment of robot joint

gear backlash using the robot joint test excitation techniquerdquoInt J Robot Res4(2) 65ndash79 (1985)

2 J L Stein and C-H Wang ldquoAutomatic detection of clearancein mechanical systems Theory and simulationrdquo Proc AmControl Conf 3 1737ndash1745 (1995)

3 N Sarkar R E Ellis and T N Moore ldquoBacklashdetection in geared mechanisms modeling simulation andexperimentationrdquo Mech Syst Signal Process 11(3) 391ndash408(1997)

4 G Hovland S Hanssen S Moberg T Brogardh SGunnarsson and M Isaksson ldquoNonlinear Identification ofBacklash in Robot Transmissionsrdquo Proceedings of the 33rdInternational Symposium on Robotics (ISR 2002) StockholmSweden (Oct 7ndash11 2002) Paper in CD-Rom proceedings

5 R Merzouki J A Davila J C Cadiou and L FridmanldquoBacklash Phenomenon Observation and IdentificationrdquoProceedings of American Control Conference MinneapolisMN (2006) pp 3322ndash3327

6 I Trendafilova and H Van Brussel ldquoNon-linear dynamicstools for the motion analysis and condition monitoring of robotjointsrdquo Mech Syst Signal Process 15(6) 1141ndash1164 (Nov2001)

7 A Azenha and J A T Machado ldquoVariable Structure Controlof Robots with Nonlinear Friction and Backlash at the JointsrdquoProceedings of the IEEE International Conference on Roboticsand Automation Minneapolis USA (1996) pp 366ndash371

8 M Nordin and P-O Gutman ldquoControlling mechanical systemswith backlash ndash A surveyrdquo Automatica 38 1633ndash1649(2002)

9 C Ma and Y Hori ldquoThe Application of Fractional OrderControl to Backlash Vibration Suppressionrdquo Proceedings ofAmerican Control Conference Boston MA (2004) pp 2901ndash2906

10 M Nordin and P-O Gutman ldquoNon-Linear Speed Controlof Elastic Systems with Backlashrdquo Proceedings 39th IEEEConf on Decision and Control Sydney Australia (Dec 2000)pp 4060ndash4065

11 David R Seidl Sui-Lun Lam Jerry A Putman and RobertD Lorenz ldquoNeural network compensation of gear backlashhysteresis in position-controlled mechanismsrdquo IEEE TransInd Appl 31 1475ndash1483 (6 NovDec 1995)

12 C-Y Su M Oya and H Hong ldquoStable adaptive fuzzy controlof nonlinear systems preceede by unknown backlash-likehysteresisrdquo IEEE Trans Fuzzy Syst 11(1) 1ndash8 (Feb 2003)

13 Z Shi Y Zhong W Xu and M Zhao ldquoDecentralized RobustControl of Uncertain Robots with Backlash and Flexibility atJointsrdquo IEEEInternational Conference on Intelligent Robotsand Systems Beijing China (2006) pp 4521ndash4526

14 K Xu and N Simaan ldquoActuation Compensation forFlexible Surgical Snake-like Robots with Redundant RemoteActuationrdquo IEEE Conf on Robotics and Automation OrlandoFL (2006) pp 4148ndash4154

15 E J Park and J K Mills ldquoStatic shape and vibration controlof flexible payloads with applications to robotic assemblyrdquoIEEEASME Trans Mechatronics 6 10 675ndash687 (2005)

16 T Mouri H Kawasaki and K Umebayashi ldquoDevelopments ofnew anthropomorphic robot hand and its master slave systemrdquoIEEE IROS 3225ndash3230 (2005)

17 M F M Lima J A T Machado and M CrisostomoldquoExperimental set-up for vibration and impact analysis inroboticsrdquo WSEAS Trans Syst 4(5) 569ndash576 (May 2005)

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms

Experimental backlash study in mechanical manipulators 219

18 S Mallat A Wavelet Tour of Signal Processing 2nd ed(Academic Press London UK 1999)

19 S Mallat ldquoA theory for multiresolution signal decompositionThe wavelet representationrdquo IEEE Trans Pattern Anal MachIntell 11(7) 674ndash693 (1989)

20 C Torrence and G P Compo ldquoA practical guide towavelet analysisrdquo Bull Am Meteorol Soc 79(1) 61ndash78(1998)

21 B F Feeny and G Lin ldquoFractional derivatives appliedto phase-space reconstructions special issue on fractionalcalculusrdquo Nonlinear Dyn 38(1ndash4) 85ndash99 (2004)

22 Henry D I Abarbanel Reggie Brown John J Sidorowich andLev Sh Tsimring ldquoThe analysis of observed chaotic data inphysical systemsrdquo Rev Mod Phys 65(4) 1331ndash1392 (1993)

23 C E Shannon ldquoA mathematical theory of communicationrdquoBell Syst Tech J 27 379ndash423 623ndash656 (July Oct 1948)

httpdxdoiorg101017S0263574710000056Downloaded from httpwwwcambridgeorgcore Open University Libraryy on 22 Dec 2016 at 173652 subject to the Cambridge Core terms of use available at httpwwwcambridgeorgcoreterms