CARTA REVISTA MEXICANA DE FfslCA 47 (5) 397-403 OCTUBRE 2001

Experimental analysis of chaos in undcractuatcd electromechanical systemsHugo G. González-Hernández

Laboratorio de Im'estigación y Desarrollo de Tecnología Avanzada, Coordinación General de InvestigaciónDirección de Posgrado e Investigación, Universidad La Salle

Benjamín Franklin 47, Col. Condesa, 06/40 México, D.F.. Mexicoe-mai!: hgoflz@ci.ulsa.l1Lt

Jaime Álvarez GallegosDepartamefllo de lngenier[a Eléctrica, Ce1ltro de Investigación)' de ESllldios Avanzados, Instituto Politécnico Nacional

Aparllldo postal/4-740, 07000 México, D.F., Mexicoe-mai!: jalvafez@enigma.red.á1l ••£.stav.nu

Joaquín Álvarez GallegosDepartamento de Electrónica y Telecomunicaciones, CeJltro de Im'estigaci6n Científica)' de Educación Superior de Ensenada

Km. 107 Carretera 7íjuQna-£llseflada, 22860 Ensenada, B.e.. Mexicoe-mail: jqa/l'tlr@cicese.lnr

Recibido el 30 de marzo de 2000; aceptado el 31 de mayo de 2001

An underactuntcd SYSlcmis a kind uf non-autonomous robotic systcm in which [herc are more links [han actuators. Thc complcxity of thedynamical behavior of [hese systems allows a wide variety of steady.state responses. Thc reconstruction of altractors based on time seriesobtained from measurements of one of the variables of a two-link, planar, underactuated robot called PendubOl, is developed. Time-delay,:oardinates. average mutual information, and pcrcentage of f31se nearest neighbars' mcthods are used to reconstruct the invariant sets. It isshown that. under the action of a pcriodic torque. the Pendubot can display a variety of steady-state dynamics, including strange aUractors.

Kl')'II'ords: Chaos; time series analysis; robotics; attractar recons[ruction

Un sistema electromecánico subactuado es un tipo de sistema robótico no autónomo que cuenta con más eslabones que actuadores. Lacomplejidad del comportamiento dinámico de estos sistemas permite una gran variedad de respuestas en estado estacionario. En este trabajose desarrolla la reconstrucción de atractares basada en series de tiempo obtenidas a partir de mediciones de una de las variables de un robotplanar de dos grados de libertad subactuado llamado Penduhot. A fin de reconstruir los conjuntos invariantes, se utilizan técnicas comoretraso de coordenadas, promedio de información mutua y porcentaje de falsos vecinos cercanos. Se muestra que bajo la acción de un IOrqueperiódico, el Pendubot puede desplegar una variedad de comportamientos dinámicos en estado estacionario incluyendo atractores extraños.

DescriplOres: Caos; análisis de series de tiempo; rohótica: reconstrucción de atraclOres

PACS: 05.45.Tp; 05.45.Ac; 45.40.l.n

1. Introduction

The dynamical analysis of certain systems is not always aneasy task, particularly when the system dynamics involvesterms that are difficult to know in a precise way. and the dy-namics is seriously changed with smal! disturbances of theseterms. These ones may inelude uncertain dynamics and pa-rameters. Therefore, it is not always possible to have a pre-cise mathematical model for the system we are dealing with.In this line of action there are lots of efforts which considerthe use ofmeasurements of. at least one ofthe variables ofthes)'stem in order to characterize the dynamics of such systems.these set of techniques is known as time series analysis.

One intcrcsting approach is the use of timc-delayed ver-sions of the time series from the measuremcnts to recons-truct the corresponding Iimit set in the state space [1]. AI-though this powerful technique is suitable for systems show-ing simpler dynamics, it is particularly useful when the sys-tem dynamics exhibits complex behavior. In this case, ob-taining fundamental invariants of [he s)'stelll, Iike the local

dimension of the steady state dynamics and the reconstruc-tion of the corresponding attractor is very important. but it isnot an easy problem.

Chaotic dynamics have been widely studied in severaldisciplines during the last decades. A chaotic signal is gene-rated by a deterministic dynamical systcm, but because of itssensitivity to initial conditions, it is long-term unpredictable.Sorne mcthods have been developed for situations where thesystem dynamics are known. However, when an accurate ma-thematical rnodel is not available, time series analysis hasshown to be a suitable alternative to this problem. this is thecase for underactuated robots.

The important distinction between a standard robotic sys-tem and an underactuated robotic system is the absence ofsorne aClUator in the overall configuration of the device: thismeans that the underactuated system has more links than ac-tuators and hence, sorne specific positions can not be reachedvia standard control strategies.

Undcractuated rnanipulators arise in a nurnber of impor-tant applications such as free-flying space robots, hyper-redundant manipulators and snake-like robots, manipulators

398 EXPERIMENTAL ANALYSIS OF CIlAOS IN UNDERACfUATED ELECfROMECIIANICAL SYSTEMS

2. Sorne conccpts on dynal11ical systcrns

2) Topologica/ rransitivity. For any two open sets U.ji e A, there exists tER sueh that 9(/, U) n ji i' 0.

They are not easy to analyze due to the absence of toolsallowing a good understanding of the phenomena. In reeentycars, many tcehniques haye been devcloped for the analysis

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3. Analysis of chaotic systcllls

whcre dimK is the capacity dimension. Then the set of pro-jections 11'": A -? E, such that 1r is injcetive, is dense amongall projeetions with respect to the norm operator topology.

Dcfinition 3. (Embedding Dirncnsinn). The dimensiondimE' = dE in Eq. (3) is called the embeddillg dimeflsiolland it is the dimcnsion for whieh the attraetor is fully un~foldcd; i.e., the dimcnsion in which lwo points far away theother in the original spaee are nat projected near each otherin the observatíon space.

Duc to this theorem, it is possiblc lo reconstruet the at-tractor in sorne prcyiously determined embedding dimensionoThe problem here is to find this dimension from a time series.In the next seetions sorne prescriptions fer finding this dimen-sion. and sorne other nceessary parameters for the atlractorreeonstruction will be givcn.

of the dynarnies of this kind of systems. Sorne of thcse tech-niques are described in the next seetion, but before we willgive sorne definitions and a usefulthcorem.

()cfinilioll 2. (Capacily Dimcnsion) [7J. Let A be abounded subset of Rn. Lel N, (A) be Ihe smallesl number ofsets of maximum diarneter 6"that cover A. Then, the eapaeitydimension is defined. if exists, by

There are no analytical solutions 10 equations describingchaotic phenomena. even an approximate solution is nat easyto find. Sorne analysis techniques for this kind of systemsinvolve perturbation rnethods [7] fer setting approxirnate so-lulions of Eq. (1). An importanl poinl here is thal usually, itis possiblc 10 measure at least one of the variables ¡nvolvedin the time evolution of the system. There are same mc-thods to analyzc the ehaotie phenomena by using time series.These methods are based on the embedding Iheorem for lhereconstruction of the attractor, and SOl1leprescriptions havebecn proposed to calculate sorne irnportant system parame.ters [0-11). Duc lO this thcorcm, it is possible lo rcconstruetthe attraetor if the embedding dimension is previously deter-mineu. Two problems arise hcre, the first one is how lo findthis dimension from a time series and the second one is howto determine the time del ay. In what follows sorne prescrip-tians for solYing this problems. and sorne other necessary pa-rametcrs for the attraetor reeonstruetion, will be given.

Typically, this quantity is not an integer number for a ehaotieattraetor A. When this situation oeeurs it is said that A is afractal sel.

Thcorcm. (Embcdding Thcorcm) [1]. Let .4 be a eom-paet and E a subspace of finite dimension such that

(1)i: = f(x, 1,/1),

Consider a systcm given by

1) Sensiril'e dependence on inirial condirimls. There existé > O sueh Ihat for any x E A and any neighbor-hood U of x, Ihere exist y E U and 1 > O sueh Ihal19(t,x) - 9(I,y)l > é.

whcre x E 3?'1 is the state, f: 3?" ---t ~'I is a smooth vectorfield. and JI denotes the system parameters. The solution ofEq. (1) is some vector funelion x = x(t) Ihal describes Ihetrajectories in the state space constructed with its eoordinates.Depending on the parameter values the system may displaydifferent steady states, ranging from equilibrium points tochaotic attraetors.

()cfinilion l. (Chaolic Atlractor) [6]. Consider an au-tonomous veelor field C' (r ~ 1) on Rn, defining a systemIike (1). Denole lhe flow generaled by (1) as ",(t,x), and as-sume that A e ~n is a compaet set, invariant under rjJ(t1 x).Then A is said lo be ehaoJie if show the following behavior:

with structural tlexibility, etc. Previous work on the modelingand control of such rnanipulators can be found in Refs. 2-4.Experimental analysis performed on different kinds of mani-pulators have shown the rich variety of steady states, rangingfrom equilibrium points lo strange atlractors.

In this papee we report the application of SOI1lC cecoos.tcuctian techniques in arder lo analyzc the Pcndubot's [5]stcady-state behavior when it is foreed by a periodic larque.The Pendubot is a two links planar underactuatcd robot andthe main objective of this papee is lo analyze its complex dy-namics in arder lo help in the further selection of a suítablecontrol strategy. This analysis ecHes only on the availabilityof a time series, obtained from a measured variable ofthe sys.temoThereforc, given a time series obtaincd from sorne mea-sured variables of Ihe Pendubol, possibly showing an irregu-lar behavior. the objective is to determine the stoehastic ordetenninistic nature of the system dynamics, as well as somefundamental parameters. In particular, we are interested incalculating the local dimension of the system dynamics, andto reeonstruet the eorresponding attraetor.

The paper is organized as follows. In Seco 2 sorne basiceoncepts on dynamical systems are revicwed. Sectian 3 givesa dcscription of same methods to analyze signals arising fromchaatic systems. In See. 4 an application of these methods tothe Pendubot, and sorne experimental results, are shown. Fi-nally, Seco 5 ineludes somc conelusions.

R,-v. Mex. Fís. 47 (5) (2001) 397-403

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HUGO G. GONZÁLEZ-HERNÁNDEZ, JAIME ÁLVAREZ GALLEGOS, AND JOAQuiN ÁLVAREZ GALLEGOS

tion between these coardinales would be numericalIy subjectlo be random-like wilh respeet to eaeh olher. In Ref. 10 it issuggested to base lhe seleetion of T in a fundamental aspeetof chaos: lhe information gencration. The average mutual in-formation coocept is based on lhe Shannon's idea foc infor-mation. Let us consider tWQ measurements ai aod bj fromseIs A = a, = x(n) and [J = bj = x(n + T) respeeti-vely. The mutual information betwecn measurernent ai aodmeasurement bj is lhe quantity learned by measurement aiabout measurernent bjo In bits, this is given as follows:

y(n) = {x(n), x(n + T), ... , xln + (d - I)TI}

There are several procedures to reconstruct a chaotic aUrae-tor from discrcte time mcasurcmcnts [1,7,111. In general,lhe solution relies 011 choosing a suitable sampling period foclhe signal such that topological characteristics of the attrac-lOr can be reproduced. The attractor reconstruction is thenaccornplished by using time delay vcrsions of a scalar quan-tity s(t). observcd fmm time to to sorne final timc, as coordi-nales for the state spaee. Let us define x(n) = s(to + n"'t),1l = 1,2, ... , for sorne ¡n¡tial timc to and a sampling inter-val .6.t. Prom the observations, el-dimensional veetors

3.1. Attractor rcconstruction

are used to trace out the orbit of the system. Thus, the pro-blem arising here is what values of time delay factor T andthe embedding dimcnsion el = elE to choose. The next t\\/osubscctions deal with those problems.

3.2. A",era~e mutual information (A1\II)

where PAB(a, f)) is the joint probabilistic density for mea-surements in A and [J, PA (a) and PB(b) are the individualprobability densities for lhe measurements in A and [J, res-pectivcly. The average of all these statistic information is cal-led average mutual information betwecn A and B and it maybe written as [2J

Before formally describing the idea of mutual information,we have to considcr sorne restrictions. First. if the period T istoo short, eoordinates x(n) and x(n + T) would nol be inde-pendent enough. And second. if T is lOo large, e\'ery connce-

I

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In lerms of x(n) and x(n + T):

{P[x(n),x(n + T)J }

I(T) = L, P[x(n),x(n + T)] log2 P[x(n)]P[x(n + T)] ,.r(ll),x(n+TJ

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The preseription for delermining if the values of x(n) andx(n + T) are independent enough such that we can use thcmto construct the vector y(n), is to take T whcre the first mi-nimum of lhe I(T) oeeurs.

3.3, Global false nearesl neighbors

Theorem tells us that if the attractor dimension definedby the orbits assoeialed to Eq. (1) is dimK(A), then theattractor will unfold in an integer cmbcdding dimcnsiondE > 2 dirnK(A) + 1 as a maximum value, In an embeddingdimcnsion that is too small to unfold the attractor, not all thepoints that Iie c10se to each other will be neighbors bccauseof lhe dynamies. some of lhem will aetually be far from eaehother and appear as neighbors, because the geornetric struc-tore of the attractor has becn projectcd down onto a smallerspace. In a el-dimensional space and denoting the r-th near-est neighbor of y(n) by Y,(n), the square of the euelideandistanee belween lhese lwo poinls is given by [9J

d-l

R~(n. r) =L [x(n + kT) - x,(n + kT)t (8)k=O

In a (d+ 1)-dimensional spaee we add x(n + dT) as a eoordi-nate to each ofthe vectors y(n). Again. the squared euclidean

Idistance in this dirnension bctween both points is (9]:

R~+ 1 (n, r) = R~(n, r) + [x(n+ kT) - x,(n +kT)t (9)

A criterion to find false neighbors may be the increase in dis-tanee between y(n) and y,(n) when going from dimension dto d + l. The inerease of distanee can be stated as [7J:

R~+I (n, r) - R~(n, r) Ix(n+dT)-x,(n+dT)1=~--~-~~-~R~(n, r) Rd(n, r)

> R-rH' (lO)where R-rH is sorne threshold nurnber. In our caseR-rll 2: 15, was founded experimentally,

4. App!icalion lo lhe Pcndubol

\Ve have chosen an underactuated rnechatronic system thatpresents a wide variety of behaviors. The system called Pen-dubol [5] eonsists oftwo rigid links, link I is direetly eoupledlo the shaft of a 90 V permanenl magnet OC molor mounledto the end of atable, this motor is the only one actuator ofthc systcm. Link 2 is couplcd to link 1 and is moved onlyby the motion of link 1. The angular position of both linksis rnonitored to a computer via optical encoders as shown inFig. 1.

/lev. Mex. Fí.<.47 (5) (2001) 397-403

400 EXPERIMENTAL ANALYSIS OF C/lAOS IN UNDERACTUATED ELECTROMECHANICAL SYSTEMS

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FIGURE1.The Pendubol.(a)

FIGURE 2. a) A1\lI for an amplitude A ::::0.5 V and b) pcrcentageof false nearesl neigbors for T == 13.

FIGURE 3. a) Measured time series for A == 0.5 V and b) recons-tructcd attractor for A = 0.5 V, T = 13 and rI¡.; = 3.

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computer with a OlA card and an encoder interface cardoUsing the standard software library routines supplied withthese interface cards we were able to program Qur algorithmsdirecIly in C-language. The vohage lo Ihe DC motor is sup-plied via a servo amplifier. Thcrc is a relationship betweenIhe supplied vohage and the applied torque to the DC motor,this relationship was experimentally determined lo be

T= (tl102+t"U!2)COS[~-(0.27fi3V+0.0335)], (11)

where T is the applied torque and V is the vohage suppliedlo the power amplifier. The pararnctcrs used here were mea-sured directly from the device and are: ti = 0.2fi987 111,

101= 5.1885 N, t, = 0.13494 m, ¡¡nd 102= 3.2824 N.,\Ve have applied several sinusoidal voltage inputs as

V = .hin <.Ji V to the Pendubot ¡¡nd measured the angularposition in the second link (the one without ao actuator). Thefrequency used was w == 4 rad/s and we have varied lhevohage amplitude from 0.1 V lo 1.7 V. In the following, weshO\\' some selected examples of attractor reconstruction ap~plying the average mutual information criterion for finding asuitable 7', and the false neighbors idea to find the embeddingdimensiono

The angular position of the second link of the Pendubothas becn sampled every 16 ms and we have taken the tran-sients off the time series.

E.wmple J. Paiadie Orbi!. Wc have consiJercd an am-plitude .4 = 0.5 V. Figure 2a shows the AMI for thissigna!. Using the criteria previously suggested we havetaken 7' = 13. Figure 2b shows the percentage of falsenearest neighbors for this value of T. Note that oncethe percentage of false nearcst neighbors has reachedlhe zero value, this percentage does not change any-more. The first value of d where the percenlagc is zerois the embcdding dimension d¡;;.

Figure 3 displays the reconstructed attractor usingT:::: 13 and dE = 3. It can be obscrvcd fmm this Figu-re that the reconstructed attractor is a periodic limit cy-ele, indeed this is a period-3 stable orbit. Thus we knowthat thefe should be sorne set of parameters 11, for whichthe system would exhibit chaotic behaviof [12J.

Reo'. Mex. Fís. ~7 (5) (2001) 397-403

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BUGO G GONZÁLEZ-BERNÁNDEZ, JAIME ÁLVAREZ GALLEGOS, AND JOAQulN ÁLVAREZ GALLEGOS

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FIGURE4, a) A~II for an amplilude of V = 0.9 V and b) pereen-tage oC falsc ncaresl ncigbors ror an amplitudc A = 0.9 V and alime deJay faclOr T = 8. 35

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Example 2. Clzaoric atIractor. We have analyzed thedynamics roc an input signal with and an amplitudeA=O.9 V. Figure 4a shows the average mutual infor-mation as a function of the time delay T, applying (heprcscription prcviously describcd. we find the embed-ding dimension to be de = 3. this is shown in Fig. 4b.In Fig. 5 it can be observed (he reconstructed attractor.In arder lO provc that this bchavior is chaotic, \••..c

have obtained lhe largest Lyapunov exponent (LE) [13]resulting.xl = 0,092. This positive .xl indieates thatthe time series is chaotic. although positivc LE's 31sodescribe noisy signals we can say that the bchaviorof (he system for A = 0.9 V is ehaotie beeause:a) the AMI of the signal has a striet local minimum,b) the pereentage of false nearest neighbors falls lozero in sorne finite dimension, e) the geometry of (hereeonstrueted attraetor, and d) the signal has a broadfrequeney speetrum. The frequeney speetrum of themeasurements was obtained using a standard FFT al-gorithm and is shown in Fig. 6.Emmple 3, Chaoclc AtlraClOr, The input signal con-sidered now has an amplitude A = 1.4 V. Figure 7ashows the average mutual ¡nformarion as a funetion oflhe lime delay T. In Fig. 7b it is shown the pereentageof false neighbors giving dE = 3. Figure 8 shows theattraetar reeonstruetion for these paramclers.

-os '~I ' .r""tI'-,_'5L_~ ~_~ __ ~_~

o 200 400 600 800 1000 1200r

FIGURE6. Frcquency speclrum in Hcnz for A = O.9V.

In Fig. 9 it is shown the speelrum PU) of the timeseries obtained using the standard fast Fourier trans-form (FFf) algorithm. It can be observed that lhis sig-nal has a braad speetrum as can be expected. Finally,we have obtained lhe largest Lyapunov exponent ofthetime series using the algorithm deseribed in [13] gi-ving.A) = 0.2.10 indicating also a chaotic behavior.

5. Conclusions

Thc main objective for using a manipulator rabol is to ac-complish tasks involving the exact tracking of sorne desiredtrajeetory. The exact tracking dcpends on the nature of the

lIer'. Mex. Fís. 47 (5) (2001) 397-403

402EXPERIMENTAL ANALYSIS OF CHAOS IN UNDERACTUATED ELECTROMECHANICAL SYSTEMS

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FIGURE 7. a) AMI for an amplitudc A. = 1.4 V and b) pcrcentagcof false ncaresl ncighors for A. = 1.4 V and a lime delay T = 8.

(b)

FIGURE 8. a) Mcasurcd time series for A = 1.4 V and b) rccon-true[ed auractor foe A = 1.4 V and lime delay T = 8.

mension ofthe space where the attractor can be observed; i.e.where the essential dynamics in stationary state is embedded;as the AMI, this method is usefu! for detcrmining the natureof the signal, because noisy signals are lypicaJly embeddedin an infinite-dirnensional space.

Finally, we can say that the attractor recanstruction dis-plays Ihe geomclry of an object topologicaJly equivalent tothe attractor constructed with the original coardinates. Be-sides, it offers a c1ear view of lhe different kinds of behaviordisplayed by Ihe system. In this sen se, it complements the in-formalion provided by Ihe spectrum and Ihe LE of Ihe signal.

FIGURE 9. Spcelrum in Hcrtz of the original signa\.

designed control algorithm, provided the proper analysis ofthe dynamics of lhe particular devicc. The absence of ao ae-tuator may Icad lo an improper performance of the systcm,bad tracking of lhe desired trajectory and 31solo a 1055of sta-bility. In additioo, as we have shown, underactuated robotslike the Pendubot may exhibit chaotic bchavior foc a certainset of parameters.

There is nol a general framework for chaos control andsuppressing undesircd complex dynamics, buI it is nceded,as a first stagc, lhe analysis of the dynamics of lhe systcmlo be controlled. In this paper we have shown a way to ana-Iyze the complex behavior of the Pendubol on the basis ofhaving measurements of only one of the variables of the sys-temo After performing severa! experiments we have noticedabout the existence of chaotic responses in this system. Thcsemelhods help us in analyzing the dynamics of any sYSlem re.quiring only a measurcment of one of its variables, particll-larly they are usefu! for extracting information from the sys-lem and besides, Ihey can be the first stage in the applicationof the method developed by 011, Grebogi, and Yorke [14] fmcontrolling chaotic behavior.

The average mutual information (AMI) is useful becauseit suggests a suitable sampling period for the signal and it of-fers a first sight about the stochastic or deterministic natureof the signa! because if we are dealing with a noisy signa!the AMI does not have a striet local minimum. Finding theglobal false nearest neighbors helps us in determining the di-

10

200 400 f 600 800 1000

Rev. Mex. Fís. 47 (5) (2001) 397-403

HUGO G GONZÁLEZ-HERNÁNDEZ. JAIME ÁLVAREZ GALLEGOS. AND JOAQUÍN ÁLVAREZ GALLEGOS 403

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Rel'. Mex. Fr,. 47 (5) (2001) 397-403