ORIGINAL ARTICLE

Miguel Atencia Æ Gonzalo Joya Æ Francisco Sandoval

Parametric identification of robotic systemswith stable time-varying Hopfield networks

Received: 3 May 2004 / Accepted: 7 May 2004 / Published online: 8 October 2004� Springer-Verlag London Limited 2004

Abstract In this work, a novel method for on-line iden-tification of non-linear systems is proposed based uponthe optimisation methodology with Hopfield neuralnetworks. The original Hopfield model is adapted so thatthe weights of the resulting network are time-varying. Arigorous analytical study proves that, under mildassumptions, the estimations provided by the methodconverge to the actual parameter values in the case ofconstant parameters, or to a bounded neighbourhood ofthe parameters when these are time-varying. Time-vary-ing parameters, often appearing in mechanical systems,are dealt with by the neural estimator in a more naturalway than by least squares techniques. Both sudden andslow continuous variations are considered. Besides, incontrast to the gradient method, the neural estimatordoes not critically depend on the adjustment of the gain.The proposed method is applied to the identificationof a robotic system with a flexible link. A reducedoutput prediction error and an accurate estimation ofparameters are observed in simulation results.

Keywords Dynamical systems identification ÆParameter estimation Æ Hopfield neural networks ÆStability Æ Optimisation Æ Robotic systems

Abbreviations ODE: Ordinary differential equation ÆLIP: Linear in the parameters (system)

1 Introduction

In this paper, the problem of parametric identification ofnon-linear systems is approached by an adaptation ofthe optimisation methodology with Hopfield neuralnetworks. System identification, which is a crucial taskwithin the framework of systems engineering, consists inmodelling the internal model of a dynamical systemfrom observations of its external behaviour. Identifica-tion techniques can be broadly classified according towhether a model of the plant or system has been previ-ously formulated, either by application of physical lawsor by intuition of experts. When such a model is notavailable, linear models are usually adopted, due to theirsimplicity. Furthermore, this approach takes advantageof existing methods for the identification of linear sys-tems, which are, nowadays, well established [1]. When,on the other hand, a model of the system is known, up toa reasonable degree of accuracy and simplicity, thisknowledge should not be wasted. Methods that incor-porate previous knowledge about the system are notonly appealing from a methodological point of view, butthey also attain a more accurate identification. In fact,when the overall structure of a model of the system is athand, identification reduces to determining the numericvalue of the various parameters intervening in themodel. Hence, in this case, identification is usually re-ferred to as parametric identification or, simply, param-eter estimation [2]. In the context of robotics, precisemodels stem from physical principles, as long as somesimplifying assumptions are adopted. Furthermore,parameters of robotic systems usually have some phys-

This is a considerably extended version of a paper presented at theconference on Engineering Applications of Neural Networks(EANN), held in September 2003 at Malaga, Spain.

M. Atencia (&)Departamento de Matematica Aplicada,E.T.S.I.Telecomunicacion, Complejo Politecnico,Universidad de Malaga,Campus de Teatinos, s/n,29071 Malaga, SpainE-mail: matencia@ctima.uma.esTel.: +34-95-2137170Fax: +34-95-2132766

G. Joya Æ F. SandovalDepartamento de Tecnologıa Electronica,E.T.S.I.Telecomunicacion, Complejo Politecnico,Universidad de Malaga,Campus de Teatinos, s/n, 29071 Malaga, Spain

Neural Comput & Applic (2004) 13: 270–280DOI 10.1007/s00521-004-0421-4

ical meaning—masses, lengths, etc.—and obtaining theirnumeric values is important per se. Yet, there is a priceto pay for incorporating the knowledge of a physicalmodel: this model is usually non-linear, hence, thedevelopment of methods for parametric identification ofnon-linear systems has become the subject of intenseresearch in the last decade [3]. In order to fix notation, itis worth mentioning that this contribution sticks to timedomain models, which are formulated by means of or-dinary differential equations (ODEs), in contrast tofrequency domain models, which require a higher degreeof mathematical sophistication. In doing so, we join thecurrent tendency of favouring time domain models inthe context of robotics [2].

In this paper, we examine the problem of estimatingthe parameters in a model that is formulated by a non-linear ODE. In general, the equation of such a modelhas the form _x ¼ f x; u; hð Þ; where x(t) is the vector ofsystem states at instant t, u(t) is the vector of inputs orcontrol signals and h is the vector of parameters. Notethat time derivatives are indicated with the dot notation,the usual in robotics, and we adhere to this usage in thispaper. In particular, models of robotic systems [4] areusually linear in the parameters (LIP), despite the factthat they are non-linear with respect to the states. Hence,these models can be rewritten as:

y ¼ A x; uð Þh ð1Þ

where the vector y collects all the terms that do notdepend on the parameters and A is a matrix that, ingeneral, is not square. In the rest of this paper, we re-strict ourselves to LIP systems. With this notation, aparameter estimation method is an algorithm that pro-vides an estimation h; intended to minimise the estima-tion error:

~h ¼ h� h ð2Þ

In practice though, the estimation error is not a fea-sible way to measure the goodness of an estimator, sincethe actual values of parameters are unknown. Thus,most parametric identification techniques are built uponthe notion of prediction error, i.e. the difference betweenthe measured output and the predicted one [2, 5]:

e ¼ y� A x; uð Þh ð3Þ

Two families of methods for parameter estimation havebeen proposed: on one hand, algorithms based on leastsquares, on the other hand, gradient methods. Therationale behind the standard least squares estimator isthe minimisation of the squared prediction error, sum-med along the whole temporal evolution of the system.Thus, it is natural to think of least squares as a batchalgorithm, which is only performed after the systemobservation has finished. In contrast, on-line estimationis often a requirement, e.g. in adaptive control systems[6], that continuously use the current estimation in orderto calculate the appropriate instantaneous control

signal. Although some on-line variants of least squarestechniques have been proposed, it is at the cost of anincreased computational complexity. Furthermore,when the parameters vary with time, additional correc-tions must be added to least squares estimators, andthere is no general rule to adjust their characteristics sothat they work in a wide range of applications. On thecontrary, the gradient method is naturally formulated asan on-line estimator, thanks to the exceedingly simpleidea that supports it: the direction of maximal decreaseof the prediction error is its—negative—gradient, whenconsidered as a function of the estimation. In morerigorous terms, the formulation of this idea, also wellknown in optimisation, leads to the following dynamicalequation of the estimator:

_h ¼ �gr eTe

� �ð4Þ

where e is regarded as a function of the estimation h: Thechoice of the estimation gain g is a key aspect of theestimator design. A small value of g leads to a slowconvergence of the estimation towards the actualparameter, causing a sustained large estimation error.Although an increase in the value of g produces, at first,better results, beyond some limit, the estimation be-comes violently oscillating, and the estimation erroragain degrades. This problematic trade-off usually re-sults in a conservative adjustment of the gain g thatcauses the gradient method to be regarded as sufferingfrom slow convergence. The neural on-line estimator,proposed in this paper, is inspired by the gradientmethod, but the need for heuristic adjustment of the gainis eliminated.

In order to overcome the limitations of classicalparameter estimation methods, several techniquesbased upon computational intelligence have been pro-posed. In particular, since the trend-setting paper byNarendra and Parthasarathy [7], artificial neural net-works have become the basis for a variety of neuralidentification techniques (see [8] and references therein).Most of these methods are based upon the extension ofback-propagation learning to recurrent networks,leading to structures such as dynamic back-propaga-tion, Elman networks and so on. These learning algo-rithms suffer from an extremely slow convergence andare computationally intensive, hence, they are not wellsuited for on-line estimation. Furthermore, neuralestimators provide a replica of the original system,from which it is impossible to obtain an explicit model,due to the complex internal structure of neural net-works. Therefore, to the best of our knowledge, neuralnetworks have not been applied to parametric identi-fication, which is the most useful framework in roboticsystems. This contribution aims at filling this gap byusing Hopfield networks, which find a natural appli-cation to on-line parameter estimation due to theircontinuous definition. The proposed method has al-ready been applied to a system without inputs, whichmodels an epidemiological process [9]. Here, we further

271

explore this method applied to a more complex system:a robotic manipulator with one flexible link. A rigorousstability proof is provided and the ability of themethod to deal with time-varying parameters is em-phasised. Interestingly, within the field of control andidentification, neural networks have been considered asa mature technology, ready for its implementation inindustrial environments [10]. Conversely, neural net-works are mistrusted by many applied engineers, whoclaim that the stochastic response of neural networks isunreliable in real-world problems. This criticism, al-though unfair, is partly supported by the lack of rig-orous proofs, trial-and-error implementation designsand irreproducible simulation results that many pub-lished contributions contain. We firmly believe that, inorder to compete with traditional identification tech-niques, neural-network-based methods require solidfoundations. This belief has led us to dedicate consid-erable effort to providing a stability proof of the pro-posed neural estimator, even in the realistic setting oftime-varying parameters.

This paper is structured as follows: in Sect. 2, theproposed estimation method is formulated as an adap-tation of the optimisation methodology with Hopfieldnetworks, which is briefly reviewed. The main features ofthe method are emphasised, revealing that it is wellsuited to robotic applications, where linearity in theparameters hold, a model is available, and states andtheir derivatives can be accurately measured with highsampling rate. Section 3 undertakes a thorough analyt-ical study that, under the usual assumption of persistentexcitation, establishes the favourable stability propertiesof the estimator. The estimations asymptotically con-verge to the correct parameter values, as long as theparameters are constant. In the case of time-varyingparameters, convergence to a bounded neighbourhoodof the parameters is proved. The theorems also providerules to design the estimator so that this boundedneighbourhood can be made as small as necessary. InSect. 4, the proposed method is applied to the identifi-cation of a robotic arm with a flexible link, which ismodelled by a system of ordinary non-linear differentialequations, after some simplifications. The performanceof the neural estimator is assessed by means of graphicalrepresentations, which are in agreement with the theo-retical results. The main conclusions and directions forfurther research end the paper in Sect. 5.

2 Hopfield networks for parameter estimation

In this section, we present the parametric identificationmethod, starting from the optimisation methodologywith Hopfield neural networks. Soon after the formu-lation of the continuous Hopfield model [11], their use-fulness for combinatorial optimisation problems wasdescribed [12] and widely used (see [13] for a recent re-view). A closely related model was proposed by Abe [14],defined by the following dynamical equation:

_pi ¼X

j

wijsj � Ii si tð Þ ¼ tanhpi tð Þb

� �ð5Þ

where pi is the input potential to neuron i, si is the stateof neuron i and b is a parameter to eventually controlthe slope of the hyperbolic tangent. The network definedin Eq. 5 possesses a Lyapunov function with the simpleform:

V sð Þ ¼ � 1

2

X

i

X

j

wijsisj þX

i

Iisj ¼ �1

2sTWsþ sTI

ð6Þ

The Abe formulation has been considered advantageousfor optimisation [15], in contrast to the original Hopfieldformulation, whose Lyapunov function contained anadditional integral term. The procedure for the appli-cation of Hopfield networks to optimisation problemsconsist in matching the Lyapunov function to the targetfunction of the optimisation, since the point where thenetwork settles must be a—possibly local—minimum ofthe Lyapunov function. Noticeably, the presence of thehyperbolic tangent in Eq. 5 forces neuron states to re-main inside the unit hypercube s 2 [�1, 1]n.

Simply stated, the proposed estimator applies theabove methodology to minimising the prediction error,as defined in Eq. 3. Firstly, a remark on the notation isin order: let us define a vector hn of nominal values ofparameters, with the best approximation that can beguessed a priori. Then, the estimator response h providesthe estimated deviation h from the nominal values,rather the parameter itself, which is now hn þ h: Hence,it is reasonable to initialise the estimation at the originh t ¼ 0ð Þ ¼ 0; so that the estimated parameter is the apriori approximation. With this notation, the predictionerror is defined by:

e ¼ y� A x; uð Þ hn þ h� �

ð7Þ

It is natural that the definition of the target function isthe—squared—norm of the prediction error V ¼ 1

2 eTe;

which, after some algebra, results in:

V ¼ 1

2eTe

¼ 1

2hTATAhþ h

TATAhn � ATy� �

þ 1

2y� Ahnk k2 ð8Þ

where the last summand does not depend on the esti-mation h; so it is neglected. Finally, matching Eq. 8 tothe standard Lyapunov function in Eq. 6 results in theweights and biases:

W ¼ �ATA I ¼ ATAhn � ATy ð9Þ

of the neural network, whose states s represent, at eachinstant, the provided estimation h:

Several peculiarities of the method should be men-tioned before it is applied to a particular system.Firstly, the weights and biases are time-varying, hence,

272

the usual proofs of stability of Hopfield networks arenot valid in this case. The next section is dedicated toaddressing this issue. Secondly, due to the presence ofthe hyperbolic tangent function, neuron states remainwithin the unit hypercube. Thus, the real values ofparameters must also belong to the interval [�1, 1],which is the range of possible values of the estimation.If the feasible range of parameters is not coincidentwith this interval, the adequate linear transformationcan be applied, together with the definition of nominalvalues, but anyway, the parameter range must beknown in advance. This is not an overly restrictiverequirement in robotic systems, where the physicalinterpretation of parameters provides reasonablebounds. Conversely, the estimation is guaranteed to bebounded, in contrast to the gradient method that mustoften be complemented with normalisation or projec-tion algorithms [5], which result in higher computa-tional cost and hinder the stability analysis. Finally, inorder to calculate the weights of the network, both thevector y and the matrix A must be available, hence,continuous measurements of the inputs, the states andtheir derivatives need to be acquired. In roboticapplications, this is usually possible, since there existsensors that provide accurate measures of positions,velocities or accelerations. If, on the other hand, thefull state is not available, some filtering technique canbe applied [4] and derivatives can be obtained bynumerical differentiation. To summarise, the charac-teristics of the method suggest that it is well suited toapplications in mechanical systems.

3 Stability analysis of the neural estimator

In this section, we undertake a rigorous stabilityanalysis of the proposed estimation method. Due to thepresence of time-varying weights, the neural network isa non-autonomous system, hence, its analysis is morecomplicated than in the autonomous case (see [16] forbackground and reference). The analysis comprises oftwo different parts: in the first, constant parameters areconsidered and the convergence of the estimationstowards the actual values of the parameters is proved;in the second part, parameters are allowed to contin-uously vary and the exact convergence of the estima-tion cannot be proved, but the analysis guarantees thatthe estimation error reaches a bounded region of theorigin. Furthermore, the theoretical result provideshints to further reduce the estimation error and thatalso suggest which systems are more difficult to dealwith: those with many parameters, fast parametervariation and lack of persistent excitation. DifferentLyapunov functions and techniques are used in eachpart. Observe that systems whose parameters suddenlychange at discrete instants can be regarded as havingconstant parameters, as long as the changes are not toofrequent so that the estimation settles before theparameters change again.

First of all, the analysis below requires rewriting thedynamics of the neural network, given in Eq. 5, as asingle ODE:

_si ¼dsi

dpi_pi ¼

1

b1� s2i� � X

j

wijsj � Ii

!

ð10Þ

which can be put in vectorial form, recalling that thestates s represent the estimations h at each instant:

_h ¼ 1

bD Wh� I� �

ð11Þ

where the diagonal matrix D=(dij) is defined by:

dij ¼ 1� h2i if i ¼ j0 if i 6¼ j

�ð12Þ

Recall from Eq. 2 that the estimation error ~h is definedby ~h ¼ h� h: Then, the dynamics that rule the estimationerror is:

_~h ¼ _h� _h ¼ _h� 1

bD Wh� I� �

¼ _h� 1

bDATA~h ð13Þ

where the definition of the weights and biases—Eq.9—has been substituted. Now the main theoretical re-sults of the paper can be stated and proved.

3.1 Constant parameters

We prove the convergence of the neural estimator to theexact values, in the case of constant parameters _h ¼ 0;by defining an appropriate Lyapunov function andapplying a standard result in the analysis of non-autonomous systems (Lemma 4.3 in [2]), which is closelyrelated to the celebrated Barbalat’s Lemma:

Theorem 1 Let y=A(x, u)(hn+h) be the ODE thatdefines a dynamical system, in the LIP form, where theparameters h are constant. Assume all the componentsof matrix A are differentiable functions of the states s

and the inputs u, with all the partial derivatives contin-uous. Assume further that inputs u and their temporalderivatives, as well as the states and the output vector y,are bounded. Define the matrix B(t)=A

TA and let

kmin>0 be a lower bound of its eigenvalues, i.e. for everyeigenvalue k of B, the following holds:

k tð Þ>kmin > 0 8t ð14Þ

Then, the neural network defined by the weights andbiases in Eq. 9 provides estimations that asymptoticallyconverge to the exact values of the parameters.

Proof Consider the following candidate to Lyapunovfunction:

V ¼�12

Xn

i¼1ln 1� hi�~hi

� �� �1�hi

1þ hi�~hi

� �� �1þhi�

ð15Þ

273

Straightforward calculus shows that V is lower boun-ded and its partial derivatives are @V

@~hi¼ ~hi

1�h2i; so the

application of the chain rule results in the temporalderivative:

_V ¼Xn

i¼1

@V

@~hi

_~hi ¼ rVð ÞT _~h ¼ D�1~h� �T

� 1

bDATA~h

� �

¼ � 1

b~hTATA~h ¼ � 1

b~hTB~h ð16Þ

Now, since B is symmetric, the theorem of Rayleigh–Ritz [17] is applicable, hence:

_V ¼ � 1

b~hTB~h6� kmin

b~hT~h ¼ � kmin

b~h2

\0 ð17Þ

whenever ~h 6¼ 0: Finally, we have assumed that all theterms appearing when deriving Eq. 16 are bounded,thus, the second temporal derivative €V is bounded,which proves that _V is uniformly continuous. Since wehave proved all conditions of Lemma 4.3 in [2], weconclude that the estimation error converges to the set_V ¼ 0; which reduces to ~h ¼ 0:

Remark The conditions of the above theorem are alittle more restrictive than strictly necessary. In partic-ular, the requirement kmin>0, which is a sort of persis-tent excitation condition, is stronger than similarconditions found in the literature. Nevertheless, inpractice, simulation results show that accurate estima-tions are attained in mechanical systems with no needfor testing the condition. The relaxation of the condi-tions of the theorem is left for further research.

3.2 Time-varying parameters

In this section, we consider the case of parameters thatvary with time, but the rate of variation is bounded:

_h 6C: Due to the effect of parameter variation, thenegativity of the Lyapunov function can no longer beguaranteed. Hence, we are limited to proving that theestimation error converges to a bounded region of theorigin. The basic technique for this proof is theorem 4.18in [16]. As a result, some hints for enhancing the esti-mator performance stem from the proof of the theorem.Firstly, a remark on notation is in order. Consider the

function V ¼ 12~hT~h and, by Eq. 13, its temporal deriva-

tive results in:

_V ¼ rVð ÞT _~h ¼ ~hT _~h

¼ ~hT _h� 1

b~hTDATA~h6C ~h

� 1

b~hTDATA~h ð18Þ

where the diagonal matrix D is defined in Eq. 12. Be-cause of continuity, if h is near 0, say hi

������\hmax 8i with

hmax > 0; then D is a small perturbation from the iden-tity matrix. Let us call this perturbation e, then, as in Eq.17, we have:

� 1

b~hTDATA~h ¼ e � 1

b~hTATA~h

� �\� ekmin

b~h2

ð19Þ

Notice that these defined variables ~hmax; kmin and e donot have an obvious physical meaning but, rather, theyare mathematical auxiliaries in the following theorem. Inparticular, calculating the values of these variables is notpossible in practical applications. Hence, the exact sizeof the region of convergence cannot be determined.Instead, the theorem provides qualitative hints forimproving the design of the estimator. This issue isfurther addressed in the remark after the theorem proof.

Now the main theorem of this section can be statedand proved:

Theorem 2 Let y=A(x, u)(hn+h) be the equation of adynamical system, where the parameter variation is

bounded: _h 6C: Assume that all the conditions in

theorem 1 hold. Let hmax > 0 be the maximum of the

estimation, such that the expression � 1b~hTDATA~h re-

mains negative, i.e. for some small perturbation e>0,the following holds:

hi

������\hmax 8i) � 1

b~hTDATA~h\� ekmin

b~h2

ð20Þ

Assume the parameters remain within some boundedinterval, which is a subset of the allowable estimationvalues:

hi 2 hmin; hmax½ � � �hmax; hmax

h i8i ð21Þ

Let r ¼ min �hmax � hmin

������; hmax � hmax

������

� �and assume:

max hminj j; hmaxj jð Þ\r ð22Þ

Assume further l<r, where l ¼ bCekmin

:. Then, if the ori-gin is the initial value of the estimation, the estimation

error reaches the neighbourhood ~h\l and stays

within this bounded neighbourhood.

Proof The proof proceeds by proving that all theconditions of theorem 4.18 in [16] hold. Firstly, suppose

the parameter error belongs to a ball of radius r : ~h\r:

Then, from the definition of r, together with the

parameter bound in Eq. 21, the condition hi

������\hmax 8i

follows. Now recall the expression for the derivative of Vin Eq. 18 and the bound given in Eq. 20, to yield:

_V 6C~hT � 1

b~hTDATA~h6C ~h

� ekmin

b~h2

¼ ~h C � ekmin

b~h

� �ð23Þ

Thus, if the estimation error increases beyond ~h > l;

the derivative becomes negative. Finally, observe that

the condition in Eq. 22 guarantees ~h t ¼ 0ð Þ

¼

274

h t ¼ 0ð Þk k\r; since the initial value of the estimation is

the origin: ~h t ¼ 0ð Þ

¼ 0: All the conditions of theorem

4.18 in [16] have been proved, so we conclude the ulti-

mate bound ~h\l:

Remark The compatibility of the hypotheses of theabove theorem is illustrated in Fig. 1. From bottom totop: the two black horizontal bars represent the range ofparameter values and attainable estimations, where thelatter must include the former if Eq. 21 holds; the seg-ment marked r, which is the distance from �hmax to hmin,is projected by symmetry to the right of hmin, and mustpass through the origin in order to satisfy Eq. 22; the topsegment represents the length l, which must be shorterthan r so that the condition l<r holds. An importantaspect of theorem 2 is that it provides hints to improvethe estimator performance by realising which correctiveactions can be accomplished when the conditions are notsatisfied. Firstly, if the condition in Eq. 21 does not hold,the parameter range is too wide. In this case, parameterscan be rescaled as h0 ¼ kh; with k<1. The condition inEq. 22 may fail due to a bad choice of the nominalvalues, thus, suggesting the need for a previous analysisin order to improve the a priori guess. Finally, twodifferent reasons may lead to l>r. On one hand, if theparameter variation is too fast, a large value of C occurs,suggesting that the parameter variation should bemodelled in the dynamics. On the other hand, a smallvalue of the product ekmin suggests the lack of sufficientpersistent excitation. In either case, an increase in theestimation gain by means of a decrease of the value of bcan contribute to alleviating the problem. Furthermore,observe that a reduction in the value of l is alwaysdesirable, since it is the radius of the bounded regionthat the estimation error converges to. It is noticeablethat the results in this section, despite their theoreticalnature, provide practical hints for improving the esti-mator performance. A more detailed discussion can befound in [18].

4 Identification of a robotic arm with a flexible link

In this section, we apply the proposed estimator withHopfield neural networks to the identification of arobotic model: a manipulator with a single flexible link.In contrast to rigid manipulators, flexible structures are

far more complicated. Nonetheless, their study isnecessary in order to obtain realistic models ofmechanical systems, hence, robots with flexible links andflexible joints are receiving increased attention from thescientific community (see, e.g. [19] and other papers inthe same special issue). Since we aim at highlighting thecharacteristics of the estimation method rather thandeveloping an extremely exact model, we consider asimplified model that can be formulated with ODEs,avoiding the complications of partial differential equa-tions. Even with this simplification, the model presents acomplex, non-linear dynamical system, which is a usefulbenchmark for identification techniques. On the otherhand, it retains most typical properties of robotic sys-tems. In particular, it can be cast into the LIP form,which allows for the application of the proposed iden-tification techniques.

A single flexible link can be regarded as a couple ofrigid links joined by a spring, as shown in Fig. 2, whereu is the torque applied by the motor, m is the massplaced at the end of the link, k is the constant of thespring, l1 is the length from the fixed motor to the jointand l2 is the length from the joint to the spinning mass.This simplification leads to the following model, whichwas included in the Web page of the course onnon-linear control by J. Ostrowski (http://www.cis.upenn.edu/�jpo, 2003):

aþ c cos x2 aþ 12 c cos x2

aþ 12 c cos x2 a

!€x1€x2

� �

þ�c sin x2 _x1 _x2 þ 1

2_x22

� �

12 c sin x2 _x21

!

þ0

�kx2

� �¼

u

0

� �

ð24Þ

where a=ml22 and c=ml1l2. In order to apply a

parameter estimation method, this model must be con-verted into the general LIP form y=A(x, u)(hn+h) bymeans of an algebraic transformation and the properdefinition of the parameters hn þ h ¼ a b kð ÞT; thematrix A and the vector y, where the latter collects allthe terms that do not depend on the parameters:

A¼€x1þ€x2 €x1 cosx2þ1

2cosx2€x2�sinx2 _x1 _x2þ12_x22

� �0

€x1þ€x2 12

€x1 cosx2þsinx2 _x21� �

�x2

0

@

1

A

y¼ u0

� �ð25Þ

Fig. 1 Schematic representation of the hypotheses of theorem 2:Eqs. 21 and 22 and condition l<r Fig. 2 a A flexible link. b A simplified model with a spring

275

Two experimental settings are simulated to assess theperformance of the estimator in the presence of suddenparameter changes and continuous variation. In theformer case, the values of parameters are first constantlyassigned as m=3, l1=1, l2=0.5, k=2, then the mass hasbeen changed to m=5 at t=4p. The mass change sim-ulates the robot picking up a new load. The completesimulation extends from t=0 to t=8p. Secondly, a dif-ferent experimental setting has been adopted with theaim of testing the proposed method in the presence ofslow, continuous parameter variation. The parameter l1is defined as l1=1.2+0.05sin t/10, thus, simulating astructural change in the elastic properties of the link, dueto temperature, vibration or some other external influ-ence. In both experiments, the initial velocities are nulland the positions are defined as x1=0, x2=p/8, so thatthe initial state is not an equilibrium point. With regardto the control signal u, it has been chosen as a sum ofsinusoids with the Schroeder phase [1], so as to obtainpersistent excitation of the system:

u ¼ 1

2sin 2tð Þ þ sin 2t � 2p

3

� �þ sin 5t � 2pð Þ

� �ð26Þ

Finally, the nominal values are set to hn ¼ 1 2 2:4ð ÞT;so that the absolute deviation of the actual values ofparameters from the nominal values is always kept lessthan unity. This guarantees that neuron states, whichfulfil si 2 [�1, 1], are able to reach the correct solution.Since the main aim of the present paper is proving theability of the method to track time-varying parameters,

no comparison with other methods is included in theexperiment. A critical analysis of the proposed algo-rithm when compared to the classical gradient methodcan be found in [20]. Besides, a different application isaddressed in [9].

In Figs. 3–8, the simulation results for both experi-ments are shown. The prediction error e has not beenrepresented, since it tends to zero very fast in bothsimulations, which is a logical consequence of its defi-nition as the target function of the optimisation. InFig. 3, the first graph plots the squared estimation error

~h2

¼ h� h

2

; which vanishes after a short transient.

Also, when a sudden change of the parameters occurs,the neural estimator is able to quickly recover, in spite ofthe fact that the instantaneous error is even larger thanthe initial one. Figures 4 and 5 show how the estimatorcorrectly tracks the varying parameters. Next, the resultswith time-varying parameters are presented, startingfrom Fig. 6, which again shows that the estimation errorsettles near zero. Figure 7 clearly presents a good fittingof the estimation to the continuously time-varyingparameter c. For comparison purposes, in Fig. 8, theparameter definition as been changed tol1=1.2+0.05sin t, i.e. the variation is ten times faster,but it has the same amplitude. Now the estimator isnot able to accurately track the parameter, since theparameter frequency is in the same range as the sys-tem dynamics. This could be alleviated by a reductionof the gain b, according to the theoretical results

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (π seconds)

Squ

ared

nor

m o

f the

est

imat

ion

erro

r

|| e||2

Fig. 3 Estimation error along asudden parameter change

276

above presented. To summarise, the simulation resultsshow that the performance of the neural estimator issatisfactory with respect to both prediction and esti-

mation. The study of multiple-link manipulators,where computational cost may be an issue, is left forfurther research.

0 1 2 3 4 5 6 7 81.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Time (π seconds)

Par

amet

er γ

(kg

m2 )

Actual parameter γ

Estimation for γ

Fig. 5 Estimation of c with asudden parameter change

0 1 2 3 4 5 6 7 80.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Time (π seconds)

Par

amet

er α

(kg

m2 )

Actual parameter α

Estimation for α

Fig. 4 Estimation of a with asudden parameter change

277

0 1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (π seconds)

Squ

ared

nor

m o

f the

est

imat

ion

erro

r

|| e||2

Fig. 6 Estimation error forslowly varying parameters

0 1 2 3 4 5 6 7 81.75

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

Time (π seconds)

Par

amet

er γ

(kg

m2 )

Real value of γ

Estimation for γ

Fig. 7 Estimation of slowlyvarying c

278

5 Conclusions

A novel method for parametric identification ofdynamical systems is presented in this paper. The basisfor the proposed estimation method is the optimisationmethodology with Hopfield neural networks, whereasthe target function is defined as the output predictionerror. Since the prediction error evolves together withthe dynamical evolution of the system, the resultingnetwork has time-varying weights and biases, hence,the classical stability proofs based on a semi-definiteLyapunov function are not enough to guarantee theconvergence of the estimation. Consequently, a thor-ough stability analysis is accomplished, which has twodifferent parts. On one hand, the case of constantparameters is dealt with, and the convergence of theestimations to the exact parameter values can beproved. On the other hand, when parameters are time-varying, it is proved that the estimation error asymp-totically approaches a bounded region of the origin.Despite the theoretical nature of these results, somepractical hints that guide the implementation of theestimator can be obtained in order to enhance theperformance of the estimator if it is not satisfactory:the parameters can be rescaled; the guess of the nom-inal values should be as accurate as possible; persistentexcitation must be provided to the system; and lastly,the value of b can be reduced in order to increase theestimation gain, which partly compensates for the otherdifficulties. In contrast to existing least squares meth-ods, the neural estimator is able to deal with time-

varying parameters, whereas it does not require thecritical adjustment of the gain that hinders the per-formance of gradient methods. The estimator withHopfield networks happens to be particularly wellsuited to robotic applications, due to the typical char-acteristics of these systems: an analytic model is avail-able, the full state and its derivatives are measurable,measures can be obtained at high sampling rates andthe physical meaning of parameters suggest approxi-mate nominal values. Simulation results show that, inthe presence of persistent excitation, both the predic-tion and the estimation errors of the neural estimatortend to zero, as long as the parameters can be con-sidered as piece-wise constant. When parameters aresubject to continuous variation, the estimation errorremains within a small neighbourhood of zero. The sizeof this neighbourhood depends not only on theamplitude of the parameter oscillations but also on itsfrequency, as forecast by the theoretical analysis. Theestimation method with Hopfield networks can beconsidered a promising technique, yet it must be ap-plied to multiple-link robotic systems that impose morestringent requirements on the estimator. Currently, weare involved in the design of the estimator, e.g. thechoice of b, in order to deal with these more complexsystems, as well as the analysis and reduction of thecomputational costs involved.

Acknowledgements This work has been partially supported byproject no. TIC2001-1758, funded by the Spanish Ministerio deCiencia y Tecnologıa (MCYT) and FEDER funds. The commentsof the anonymous reviewers are gratefully acknowledged.

0 1 2 3 4 5 6 7 81.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1

Time (π seconds)

Par

amet

er γ

(kg

m2 )

Real value of γ

Estimation for γ

Fig. 8 Estimation of c with fastvariation

279

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