a nonlinear adaptive h//spl sup infinity//tracking control

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997 13

A Nonlinear Adaptive Tracking Control Designin Robotic Systems via Neural Networks

Yeong-Chan Chang and Bor-Sen Chen,Senior Member, IEEE

Abstract—An adaptive neural-network tracking control with aguaranteedH1 performance is proposed for robotic systems withplant uncertainties and external disturbances. A neural-networksystem is introduced to learn these unknown (or uncertain)dynamics by an adaptive algorithm. Moreover, the effects on thetracking error due to the approximation error via the adaptiveneural network must be attenuated to a prescribed level, i.e., anH1 tracking performance is achieved. Hence, in this study, both

the H1 tracking theory and adaptive neural-network controlscheme are combined together to achieve the nonlinear adaptiveH1 tracking control design for uncertain or unknown robotic

systems. The developed control scheme is smooth and semiglobalas well as very simple and computationally efficient, since it doesnot require a knowledge of either the mathematical model orthe parameterization of the robotic dynamics. Finally, extensivesimulations are given to illustrate the tracking performance of atwo-link robotic manipulator with the proposed adaptive neuralH1 control design.

Index Terms—Nonlinear adaptive control, H1 tracking con-trol, neural network, robotic control.

I. INTRODUCTION

T HE MOTION control of industrial manipulators is a cen-tral issue in the robotic area and has received a great deal

of attention in the past decade. Many approaches have beenintroduced to treat this control problem and various controlalgorithms have also been proposed in the literature. When themodel is known exactly, the control of robotic manipulatorsis a simple task. In this situation, the technique of feedbacklinearization of a nonlinear system has been developed to treatthe control of robots (it is a kind of “computed torque”). Thismethod uses a nonlinear state feedback to exactly cancel thenonlinear terms and factors, and then employs optimal controltechniques or VSS techniques to treat the equivalent lineardynamic problem (see, for instance, [11], [15], [19], and [30]).

Since uncertainties which cannot be knowna priori exactlyare inevitable in practical robotic systems, (for example, theload may vary while performing different tasks, the frictioncoefficients may change in different configurations, and someneglected nonlinearity, such as backlash, may appear as adisturbance at the control input), the robot arm may receiveunpredictable interference from the environment where itresides. Therefore, it is necessary to consider these effects due

Manuscript received October 21, 1994. Recommended by Associate Editor,S. Kumagai. This work was supported by National Science Council underContract NSC 84-2213-E-007-077.

The authors are with the Control and Signal Processing Laboratory, De-partment of Electrical Engineering, National Tsing Hua University, Hsinchu,Taiwan 30043, R.O.C.

Publisher Item Identifier S 1063-6536(97)00253-4.

to plant uncertainties which contain structured (or parametric)uncertainties and unstructured uncertainties (or unmodeleddynamics), and external disturbances. In order to compensatesuch uncertainties in the robot manipulator dynamic equa-tion, many control strategies have been proposed. There arebasically two underlying strategies to the control of suchuncertain systems: the robust control strategy and the adaptivecontrol strategy. In the recent development of robust controlalgorithms (see, for example, [2], [7], [30], and [32], etc.),if an a priori bound of uncertainty is known, then high-gainfeedback laws or saturating-type controllers can be proposedto treat the tracking of robot motion. Interested readers shouldconsult the survey paper by Abdallahet al. [2] and textbook bySpong and Vidyasagar [30] for the background of the roboticcontrol application of this technique.

On the other hand, in the recent development of adaptivecontrol algorithms (see, for example, [24], [28], and [29],etc.), the use of the regressor matrix has become ratherpopular in adaptive control of robotic manipulators. In thissituation, nonlinear dynamics of a rigid robot with unknown(or uncertain) system parameters are assumed to be expressedas a product of a regressor matrix and an unknown param-eter vector, that is, the property of linearity in the systemparameters is used in the derivation of adaptive control results.And then, a parameter update law has been used to estimatethe unknown parameters which are assumed to be constant orslowly-varying. However, there are some potential difficultiesassociated with this approach. For example, the unknownparameters may be quickly varying, the linear parameterizableproperty may not hold, computation of the regressor matrixis a time-consuming task, and implementation also requires aprecise knowledge of the structure of the entire robot dynamicmodel. Hence, the introduction of an alternative approach totreat the adaptive control of robotic systems with uncertaintiesis interesting.

Artificial neural networks offer the advantage of perfor-mance improvement through learning using parallel and dis-tributed processing. These networks are implemented by usingmassive connections among processing units and are attractivefor wide applications in identification, signal processing, andcontrol. Recently, adaptive neural-network algorithms havebeen used to solve highly nonlinear control problems [1],[12], [13], [18], [21]–[23], [25]. In general, the conventionaladaptive neural-network algorithm has not attacked the prob-lem of attenuation of the effects on the tracking error dueto the approximation error via neural-network approximation.However, the approximation error may cause the tracking

1063–6536/97$10.00 1997 IEEE

14 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997

performance of the closed-loop neural network-based controlsystem to be poor. Hence, it is still a challenge to introducean additional performance criterion such that the closed-looptracking performance is upgraded.

In the last decade, the approach of optimal controlhas been widely discussed for robustness and its capability ofdisturbance attenuation in linear control systems [9], [10] andin nonlinear time-invariant systems [3], [14], [33], [34]. In theconventional tracking control theory, the plant modelsmust be known, perhaps allowing a small perturbation. If theplant models have large uncertainties, the conventionaltracking control will meet additional difficulties. In this situ-ation, an alternative approach (for example, adaptive controltechnique, etc.) must be employed to treat this problem. To thebest of our knowledge, the solution of “nonlinear adaptivetracking control” for robotic systems under plant uncertaintiesand external disturbances is still an open problem.

Motivated by the above factors, an adaptive neural-networktracking control with a guaranteed performance is pro-posed for robotic systems under plant uncertainties and ex-ternal disturbances. A neural-network system is introduced tolearn these unknown (or uncertain) dynamics by an adaptivealgorithm, and then a neural network-based adaptive trackingcontroller is obtained so that the effects on the trackingerror due to the approximation error via the neural-networkapproximation must be less than or equal to a prescribedlevel, that is, the tracking performance is achieved. Bothsignular and nonsingular control cases are discussed. Atthe beginning, with the help of the technique of completingthe squares, a nonlinear Riccati equation provides a possiblesolution to our problem. In general, it is not easy to solve thisnonlinear Riccati equation. Fortunately, by virtue of the skew-symmetric property in robotic systems and via a proper choiceof Lyapunov function, the adaptive neural robotic trackingproblem only needs the solution of an algebraic equation. Itcan be shown that if a desired attenuation levelis prescribed,with an appropriate choice of the weighting matrix on the

control algorithm, an adaptive neural-network-basedcontrol law can be explicitly constructed for robotic trackingsystems. The proposed control law consists of two parts. Oneis an adaptive neural-network algorithm which can be viewedas rough tuning, and the other is an tracking controlalgorithm which can be viewed as fine tuning. The main meritsof the proposed adaptive neural network-based controllerare its smooth and semiglobal properties. Moreover, it doesnot require the linear parameterizable property of roboticsystems. The control scheme developed is very simple andcomputationally efficient since it does not require a knowledgeof either the mathematical model or the parameterization ofthe robotic dynamics.

The rest of this paper is organized as follows. In Section IIthe model description of an-degree-of-freedom rigid robotis given. The formulation of the adaptive neural track-ing control problem, which takes into account the effectsresulting from plant uncertainties and external disturbances,is presented. Section III presents the dynamic state feed-back controller with an adaptive neural parameter updatelaw. In Section IV, a simple design algorithm is proposed.

In Section V, extensive simulations are made for trackingcontrol of a two-link robotic manipulator with the proposed

tracking designs. Finally, the conclusions are given inSection VI.

In what follows, we shall use the following standard nota-tions. We denote by or the square norm of a vector

with the weighting matrix We saythat is in if and

is bounded (i.e., in iffor all

II. PRELIMINARY

A. Model Description for Robotic Systems

According to the Denavit–Hartenberg representation [36],the geometric configuration of a robotic manipulator withlinks can be determined by variables. Set these variablesas the entries of a column vector and from the context ofanalytical dynamics, a systematic way to derive the equationof motion is to apply the methods of the Lagrange theory [30].More precisely, for a given mechanical system havingrigidbodies with kinetic energy and potential energywe first recall that the Lagrangian of this system isdefined to be

(1)

and the Lagrange equation has the form

(2)

in which is the vector of the externally applied torques(or forces) along the directions of their corresponding gener-alized coordinates For rigid robotic systems, the kineticenergy is a quadratic function of the vector of theform

Hence, after substituting the Lagrangian into the La-grange equation (2), the equation of motion for-links rigidrobotic manipulator is of the following form:

(3)

where the matrix denotes the moment ofinertia. Other terms of (3) include coriolis, centripetal forces

and the gravitational forces

Although the equations of motion (3) are complex, theyhave two fundamental properties which can be exploited tofacilitate the control system design. We state these propertiesas follows [24].

P1) The inertia matrix is symmetric and positivedefinite for every

CHANG AND CHEN: NONLINEAR ADAPTIVE TRACKING CONTROL 15

P2) A suitable definition of makes matrixskew-symmetric, namely

In this paper, the desired reference trajectory to followis assumed to be available as bounded functions of time interms of generalized positions (the class of twicecontinuously differentiable functions) and their correspondingaccelerations and velocities All variables areassumed to be within the physical and kinematic limits of thecontrol object. The variables may be convenientlygenerated from a reference model of the type

(4)

with some bounded driving signal Of course, the dynamicsystem (4) with -matrices and needs to bestable.

In practical robotic systems, uncertainties which may effectthe tracking performance are inevitable. Hence, it is significantto consider the effects due to uncertainties. In this paper,plant uncertainties and external disturbances will be consideredsimultaneously. First, a neural-network system is introducedin the next section to learn these unknown (or uncertain)dynamics by an adaptive algorithm. Next, the effects on thetracking error due to the approximation error via the adaptiveneural network must be attenuated to a prescribed level, thatis, the nonlinear performance is achieved.

B. Problem Formulation with NonlinearTracking Performance

In order to describe the attenuation property from theapproximation error produced by the neural-network approxi-mation to the tracking error of states about the desired positionand velocity of the robotic system, it is necessary to derive asuitable state-space error dynamics.

The state tracking erroris defined as

(5)

Using (3) and (5), and taking into account the unmodeled dy-namics and external disturbances, denoted asthe dynamicequation for the state tracking erroris obtained as

(6)

where

For the convenience of design, take the following state-space transformation:

(7)

where are constant matrices to be adequatelydetermined later. is assumed to be a diagonal matrix,namely, for some Then, the dynamicequation for the error state can be obtained as

(8)

where and

(9)

Since the performance of the state tracking errorisconsidered in this paper, it is natural to derive the modifiederror dynamic equation of based on the dynamic equation(8) of Hence, we get

(10)

where

Remark 1: As the parameter matrices andof robotic systems are well known and available in the

design of control law, the term in (10) can be exactlycancelled by a suitable choice of control law However,in practical robotic systems, parametric uncertainties whichcannot bea priori known exactly are inevitable. Therefore,the system parameter matrices and are notavailable in the adaptive control design. Generally speaking, inthe previous arguments (see, for example, [24] and [28], etc.),the assumption of linear parameterization of in (10)is used in the derivation of adaptive control results. Hence,the term which depends on the uncertain parametermatrices and can be expressed as

where is an matrix of known functions, known asthe regressor, and the parameter is a constant -dimensionalvector with components depending on manipulator parameters(such as link masses, moments of inertia, etc.). The systemparameter , which may be unknown (or uncertain), is assumedto be constant or slowly-varying in analyzing the adaptivecontrol design. And then, an adaptive parameter update lawcan be proposed to learn the behavior of the unknown pa-rameter. However, in practice, the system parametermay


be quickly varying or the linear parameterizable property maynot hold. Moreover, computation of the regressor matrix is atime-consuming task. On the other hand, besides parametricuncertainties a robotic system may also be perturbed byunmodeled dynamics and external disturbances. Hence, theintroduction of an alternative and suitable method for learningthe behavior of the totally uncertain term isof interest. In this paper, a neural-network approximationapproach will be proposed to solve this problem.

A neural network is proposed here to approximatethe uncertain term in (10) where is a vectorcontaining the tunable network parameters.

Let

... and

... (11)

The neural networks for are com-posed of nonlinear neurons in every hidden layers and linearneurons in the input and output layers. For simplicity of design,the adjustable weightings for are put in theoutput layers of the following single-output neural networks:

(12)

with

... and ...

for , where according to the multilayer neural-network approximation theorem [13], must be a nonconstant,bounded, and monotonically increasing continuous function. Inthis work the following hyperbolic tangent function is used:

(13)

where is a function of the augmented stateRemark 2: 1) The neural networks for

in (12) are all of single-output network with onenonlinear layer containing hidden neurons forrespectively. 2) In general, the order of the neural net-work is given so that can approximate

as close as possible. 3) The weightings andbiases for and arespecified beforehand in this study.

Having the neural-network system be denoted by

......

......

......

......

(14)

let us define the optimal constant approximation parameter[12], [13]

(15)

where Here, the linearly parameterized net-work model [25], [27] is employed in the approximationprocedure of the uncertain dynamics. It should be noted thatexplicit expressions for computation of are not requiredsince this value can be learned by using an adaptive algorithmin this study.

By the optimal approximation in (15), if the applied torqueis chosen as the following form:

where stands for anew control variable which should bedesigned later, then the modified error equation in (10) canbe rewritten as

(16)

where

Since the approximation errorcan not be known exactly,it is impossible to find a good control strategy to cancel outthis effect completely. The philosophy of our tracking designis expected that the effect of the approximation erroron thetracking performance is as small as possible or at least withina prescribed attenuation level. That is, an adaptive neuralperformance criterion will be introduced. More explicitly, ourinvestigated problem is formulated as follows.

Adaptive Neural Tracking Problem (ANHTP):Consider the modified error dynamics equation (10). Itis said that the adaptive neural tracking problem can besolved if for a given desired reference trajectory anda prescribed attenuation level there exists a dynamicstate feedback controller

(17)

(18)


where stands for a new control variable, such that theclosed-loop system (10), (17), and (18) satisfies the followingrequirements [4].

1) Boundedness: If the approximation error has finite en-ergy, i.e., for all then all the variables

and are bounded.2) Attenuation: For any initial condition, the following

quadratic performance criterion is achieved:

(19)

for some matrices andwhere denotes the neural

parameter estimation error.

Remark 3: The tracking error of the closed-loop adaptivesystem is influenced by both the approximation error andthe initial choices of estimated states and neural parameters.Hence, without loss of generality, the nonzero initial conditioncan be viewed as an extra disturbance and has been includedinto the above performance criterion.

Remark 4 (Finite -gain characterization): While the er-ror dynamic (16) starts with and then the

tracking performance in (19) can be reduced to

(20)

Moreover, the performance (20) can be rewritten as [4]

(21)

where denotes the truncated norm at time Theabove expression means that the nonlinear (inducednorm from to must be less than or equal to

According to the above problem formulation, a two-stagecontrol design procedure will be proposed to solve the adaptiveneural tracking problem of perturbative robotic controlsystems. At the first stage, an adaptive neural-network al-gorithm with a parameter update law is introduced to learnthe behavior of uncertainties. At the second stage, a robustcontroller is introduced to guarantee the desired trackingperformance. In [13], they have shown that any nonlinearcontinuous function can be uniformly approximated as accu-rately as necessary by increasing the number of neurons inthe multilayer feedforward neural network. But, in practicalrobotic systems there may exist some discontinuous uncer-tainties. In this situation, neural-network systems may yield apoor approximation property. If only a pure neural network-based controller were designed to solve this problem, thenthe closed-loop robotic system might be unstable. Fortunately,in this paper, by virtue of the attenuation property theclosed-loop system can be proved to be stable without regardto the presence of apoor approximation error. In summary, ourproposed controller can be applied to treat the robotic systemunder a large class of uncertainties.

III. A DAPTIVE NEURAL NETWORK-BASED

CONTROL FOR ROBOTIC SYSTEMS

Based on the analysis in the above section, the adaptivemodel reference robotic control problem with a desired at-tenuation of the approximation error via the neural networkis formulated as a nonlinear adaptive neural (singular)tracking control problem. In this section, first, a nonlinearRiccati equation, which depends on the unknown (or uncertain)parameter matrices of robotic systems, is derived to providea possible solution to our problem. Next, by an adequatechoice of Lyapunov function and by virtue of skew symmetricproperty of robotic systems, this nonlinear Riccati equation canbe further simplified to an algebraic matrix equation which isindependent on the unknown parameter matrices.

Lemma 1: Consider the tracking error dynamic equation(10). If there exists a symmetric positive definite matrix

satisfying the following nonlinearRiccati equation:

(22)

for some matrix then the following adaptive neuralnetwork-based control law:

(23)

(24)

with

(25)

for any positive definite matrix guarantees themodel reference tracking performance (19).Proof: Let us choose an adequate Lyapunov function

(26)

Taking the time derivative of along the error dynamic (10)and taking (24) into account, we get

(27)

Substituting (25) into (27), after some rearrangements, the timederivative of can be rewritten as


(28)

Since

(29)

substituting (22) and (23) into (28), we get

(30)

Integrating the above inequality from to yields

(31)

Since the above inequality leads to

(32)

This completes the proof.Remark 5: In general, it is not easy to solve the above

nonlinear Riccati equation (22). Furthermore, andalso contain unknown parameter matrices and

of the robotic system. Hence, further discussion isneeded to treat the nonlinear Riccati equation (22).

However, in the robotic system, the nonlinear Riccati equa-tion (22) can be further simplified to an algebraic matrixequation which is independent on the unknown parametermatrices if an adequate matrix function is chosenand the property P2 of the skew symmetric matrix of roboticsystems is used. Because the state transformation (7) has beeninvolved in the modified error dynamic equation (10), wesuggest to be in a more explicit form [5], [15]

(33)

where is a positive definite symmetric constant matrixrepresenting the stiffness of a spring action around the givenreference position. The term containing representskinetic energy. We can view the value functionwith in (33) as a total energy storage at error stateestimation error variable and time

In the following paragraphs, we shall show that under someconditions this suggested matrix function in (33) is thesolution of nonlinear Riccati equation (22). Furthermore, theconstant matrices and can be solved from an algebraicRiccati equation which is independent on unknown parametermatrices.

Consider the second and third terms of the nonlinear Riccatiequation (22). By the skew–symmetric property P2 and (33),

and after some algebraic manipulations, we get

(34)

It can also be easy to check that

(35)

So, the fourth term of (22) becomes

Substituting (34) and (35) into (22), after some rearrange-ments, the nonlinear Riccati equation (22) becomes an alge-braic equation

(36)

Therefore, according to Lemma 1, if in (33) is thesolution of the nonlinear Riccati equation (22), the constantmatrices and must be the solutions of thealgebraic Riccati equation (33). Note that the algebraic Riccatiequation (33) contains no unknown parameter matrices of therobotic system and both and can be solved if and

are given. In order to guarantee the positivity ofin (36),the following inequality must hold:

(37)

Summarizing the previous derivation, we obatin the follow-ing main result:

Theorem 1: For the robotic systems described as (3) withplant uncertainties and external disturbances, if andthe adaptive neural network-based control law is chosen as

(38)

(39)

where is the solution of the following algebraic Riccati-likeequation:

(40)

then the ANHTP is solved.Proof: Choose the Lyapunov function as

(41)

where is defined in (33). From the analysis in the aboveparagraphs, the time derivative of is obtained as

(42)

By the same arguments in Lemma 1, this implies that theperformance (19) is achieved.


Next, we investigate the property of boundedness. Integrat-ing the two sides of (42) from zero to we get

(43)

Since for we can conclude that

(44)

This implies that and are all bounded forBy the definitions of and we can conclude

that and are all bounded. This completes theproof.

Remark 6: 1) It is worth noting that the initial states canbe arbitrarily preassigned, that is, a semiglobal convergenceresult is obtained in this study. For any bounded initial statesthe bounded compact sets and can also be explicitlyconstructed [20]. 2) If then byBabalat’s lemma we can conclude that and

[25]. Moreover, if, in addition, the regressormatrix is persistently exciting, then the learning parameter

converges to the optimal value i.e.,[27], [28], [29].

Remark 7: The proposed controller (39) consists of twoparts. The first part (i.e., is an adaptive neural networkand the second part (i.e., is the robustcontroller to achieve the desired tracking performance.Hence, in practice, this controller is a hybrid adaptive-robustcontroller. The adaptive neural network plays the role of roughtuning and the algorithm plays the role of fine tuning.Since the algorithm is used as fine tuning to efficientlyeliminate the approximation error in the proposed controlalgorithm, the number of the neurons in the neural-networkcontrol design can be decreased significantly. This result isvery useful from the practical control design point of view.

Remark 8: The problem formulated above falls into thecategory ofsingular nonlinear control since the desired

performance does not require any penalty on controlinput variables. However, it is also interesting to consider thenonsingularnonlinear control problem. In this situation,it is natural to question that what control effect should bepenalized for robotic systems. As pointed out in the work

of Chen [5], a selective applied torquewhich affects the kinetic energy only (see the

Appendix), is proposed to be penalized. An optimal controlhas been found in the sense of tracking designs.

If the control variable is also concerned in thequadratic performance criteria, then anonsingularnonlinear

control problem is formulated and the performance (19)should be changed as follows:

(45)

for some weighting matrix Similarly in Remark4, while the error dynamic (16) starts with and

the tracking performance in (45) can bereduced to

(46)

This means that the nonlinear norm from tomust be less than or equal to i.e.,

(47)

In the following, we shall show that the singular adap-tive neural network-based control law proposed in Theorem 1,with a little modification in the algebraic Riccati-like equationcan also be used to solve the nonsingular adaptive neuraltracking control problem. The procedure of proof is similar tothe arguments in Theorem 1.

Corollary 1: For the robotic systems described as (3) withplant uncertainties and external disturbances, if andthe adaptive neural network-based control law is chosen as

(48)

(49)

with

(50)

where is the solution of the following algebraic Riccati-likeequation

(51)

then the ANHTP with the performance (45) is solved.Proof: Substituting the applied torque (49) into (10),

we get the state trcking error dynamics (from to statetracking error which has the form

(52)

Select the Lyapunov function as

(53)

where is defined in (33). Taking the time derivative ofalong the error dynamics (52) and taking into account (34),

(35), and (48)–(51), we get


(54)

Moreover, by the similar procedure in Theorem 1, this proofis completed.

Remark 9: Observing inequality (45), if then weimmediately obtain that for every the resultinginput function and the error state are in

for arbitrary initial condition.Remark 10: The inequalities in Theorem 1 and

in Corollary 1 are to guarantee the symmetric positivedefinite solution of matrix They present the lower boundsfor in the quadratic performances (19) and (45), respectively.The lower bound for (45) is larger than the lower bound for(19). This is reasonable since the control inputmust berequired to satisfy the quadratic performance (45).

Remark 11: If then the quadratic performancecriteria (19) and (45) are always satisfied. In this situation, noattenuation is expected in tracking error. While the attenuationlevel by the lower-bound constraint on the desired level

the control variable i.e., the high-gain control isnecessary for an arbitrarily small attenuation of approximationerror.

Remark 12: If the constrained problem due to the optimalapproximation parameter that belongs to some preassignedcompact set is considered, then additional tools concerningabout projection algorithm [6], [17], [25], [26], [35] canbe used to analyze this bounded problem. Suppose

and let whereand Then the parameter update laws in (38) and

(48) must be modified as [17], [26]

if orand

otherwise

where It can be verified that ifthen and the performance

in (19) and (45) can still be guaranteed. Moreover, ifthen all the variables of the closed-loop system can

also be guaranteed to be bounded on a compact set [17].

IV. CONSTRUCTION OF ASIMPLE DESIGN ALGORITHM

In order to implement the adaptive neural network-based tracking control law of the robotic system, we need tofind and which solve the Riccati-like algebraic equations(40) or (51). Some techniques will be employed to simplifythe design procedure. First, let the positive definite symmetricmatrix be factorized by Cholesky factorization [5], [15], as

(55)

With this factorization and by the definition of and (40)can be split into the following four equations:

(56)

(57)

(58)

(59)

From (56) and (59), the matrix must bepositive definite since the matrices and arepositive definite. Perform the following factorization [5], [15]:

(60)

Substituting the above equation into (56) and (59), the subma-trices and can be obtained as

(61)

Substituting (61) into (57) and (58), we obtain the symmetricsolution as

(62)

Hence, in order to guarantee the positive definite ofin (62),the choice of in (55) must satisfy the following inequality:

(63)

Furthermore, in order to satisfy the requirement in ,which must be a diagonal matrix, some further constraintsmust be imposed on the weighting matricesand

Corollary 2: Given a desired attenuation level andweighting matrix let the weighting matrix be furthertaken as

(64)

with scale values and satisfying Then,if we choose

(65)

the ANHTP with the performance (19) is solved by theadaptive neural network-based control law

(66)

(67)

with

(68)


Proof: From (60), (61), and (64), we get

(69)

From the definition of and we get

(70)

Substituting (69) and (70) into (38) and (39), this proof iscompleted.

In the above arguments, a simplified adaptive control law forthe singular adaptive neural control problem is proposed.Similarly, with regard to the nonsingular case, we get thefollowing results. In this case, the matrixin (56)–(60) is replaced by

Corollary 3: Given a desired attenuation level andweighting matrices and let the weightingmatrices and be further taken as

(71)

with scale values and satisfying andThen the ANHTP with the performance (45) is solved

by the same adaptive neural network-based control law (66)and (67) in Corollary 2 with

(72)

In summary, from the results of Corollary 2 and Corollary3, it is important to note that the adaptive neural network-based controller for both singular and nonsingular controlproblems is of the same structure. This controller is drivenonly by the tracking error and can be easily implemented.More appealingly, from (68) and (72) we only need to tune theparameter for the singular and nonsingular casesas a controller (66) and (67) has been implemented. Finally,the adaptive neural (singular and nonsingular) trackingdesign for robotic systems can be outlined as a simple designalgorithm.

Design AlgorithmStep 1) Choose a desired attenuation level,Step 2) Choose the weighting matrix

with

and the weighting matrix such that

for singular casefor nonsingular case.

Step 3) Calculate the parameter i.e.,

for singular case

for nonsingular case.

Step 4) Select the neural-network functionsfor in (12).

Fig. 1. The two-link robotic manipulator.

Step 5) Obtain the corresponding adaptive neural network-based control law

The corresponding smooth projection update lawcan be obtained as in Remark 12.

V. A SIMULATION EXAMPLE

In this section, we test our proposed adaptive neuralcontrol design on the tracking control of a two-link robot byusing a computer. Consider a two-link manipulator describedin Fig. 1 with system parameters as link masses (kg),lengths (m), angular positions (rad), and appliedtorques (Nm). The parameters for the equation of motion(3) are [15]

where and the shorthand notationsare used.

Suppose that the trajectory planning problem for a weight-lifting operation is considered and the two-link manipulatorsuffers from the time-varying parametric uncertainties, unmod-eled friction forces, and exogenous disturbances.

The adaptive neural model reference tracking control withthe desired performance (19) is employed to treat thisrobotic trajectory planning problem. For the convenience of


simulation, the nominal parameters of the robotic system aregiven as kg, kg, m, m,

m/s and the initial conditionsand The desired reference trajectory

vector starting from is characterized by (4)with andwhich generates sinusoidal motions.

Suppose the parameters and are perturbed in thefollowing form:

and

respectively. The unmodeled friction forces and areexpressed as and respectively. Moreover, theexogenous disturbances and are square waves withperiod that is

Adopting the above parameters, the equation of motion (3)for two-links rigid robotic manipulators with parameter pertur-bations, unmodeled friction forces, and external disturbanceshas the form

where andNow, according to the proposed design algorithm, the

singular tracking control design is given by the followingsteps.

Step 1) In order to illustrate the capability of attenuationswith different choice of we deliberately designthe control laws to achieve the following fourdifferent levels of attenuation,

Step 2) Choose the weighting matricesand

forforforfor

Step 3) Calculate the parameter

forforforfor

Step 4) Let us denote

The neural-network functions in (12) are chosen tobe and

with components below

which are shown in Fig. 2. Moreover

where

For the convenience of simulation, chooseand Set and

Step 5) Obtain the adaptive neural network-basedcontrol law with a smooth projection update law

if orand

otherwise

where and isthe value obtained in Step 3) with respect to thedifferent cases, respectively.

Remark 13: The number of basis functions in the neural-network system heavily influences the complexity of a neural-network system. In general, the larger the number, the morecomplex is the neural-network system and the higher theexpected performance of the neural-network system. Hence,there is always a tradeoff between complexity and accuracy inthe choice of the number of basis functions. Their choice isusually quite subjective and based on some experiences. In theabove design, seven basis functions for both neural networks

and are chosen in which the biasesand for are selected asrespectively. Since one is interested in obtaining an accurateneural-network system to approximate the uncertain dynamicsin a (possibly large) neighborhood of the point it isintuitively evident that the biases should be clustered aroundzero in this neighborhood.

On the other hand, the weighting coeffieients andfor in the basis functions heavilyinfluence the smoothness of the input–output surface deter-mined by the neural-network system. In general, the sharper


Fig. 2. The neural-network basis functions.

Fig. 3. The angular positionq1(t):

the basis functions, the less smooth is the input–output surface.The choice of the coefficients and is also subjectiveand based on some experiences. Here, for convenience, theseweighting coefficients are all selected to be equal to one. Inthis situation, our proposed control law can also be guaranteedto be smooth.

The simulation results are shown in Figs. 3–11. The angularpositions and are represented in Figs. 3 and 4, respec-tively. The angular velocities and are depicted in Figs. 5

and 6, respectively. The applied torquesand are plottedin Figs. 7–10. From the simulation results of the above fourcases, we find that smaller may yield better performancein attenuating the effect of the approximation error, but thecontrol signal during the transient time which is very short(about 0.1 s) is larger than the case of largerHence, thereis a tradeoff between the attenuation level at which thetracking performance is achieved and the amplitude of the con-trol signal during the transient time. Moreover, the simulation



Fig. 5. The angular velocity_q1(t):

result in Fig. 11 indicates that the integrals of error signalsunder different prescribed attenuation levels are decreasedsequentially. Hence, the desired attenuation properties ofthe proposed designs have been achieved. They can be usedto diminish the effects due to the parametric uncertainties,unmodeled dynamics, and external disturbances in roboticsystems. More appealingly, the number of neurons used inthis example to achieve the tracking performance is small.This fact is of great interest from the engineering point of view.

Next, in order to illustrate the effectiveness of our proposedadaptive neural network-based control algorithm, we compare

our proposed design with two other approaches. One is withexactly known parametric matrices and in (10)and then the control law is designed as

The other is without employing the adaptiveneural-network control part in (39), but with a convenientPD control law. In this case, Here,the desired reference trajectory vector is assumed to be

and the initial conditions are chosen asThe attenuation level

weight matrices andthen the parameter The angular positions


Fig. 6. The angular velocity_q2(t):

Fig. 7. The applied torque�1(t) (transient response 0–0.5 s).

and are shown in Figs. 12 and 13, respectively. Itis obvious that the speed of convergence for the case ofknown models (dashed line) is faster than the others. It isreasonable since the parametric matrices have been assumedto be known exactly. However, in pratical applications, it maybe impossible and then the use of the adaptive neural-networkapproximator is of interest. The tracking performance for thecase with adaptive neural network-based control (dash-dottedline) is much better than the case without neural-network

control (dotted line). Therefore, incorporating the neural-network approximator with control clearly results insatisfactory tracking performance.

VI. CONCLUSIONS

An adaptive neural tracking control problem for roboticsystems under plant uncertainties and external disturbanceshas been proposed and solved from the viewpoint of the

tracking performance in this paper. The perfor-


Fig. 8. The applied torque�1(t) (steady-state response 0.5–20 s).

Fig. 9. The applied torque�2(t) (transient response 0–0.5 s).

mance consists of the desired attenuation property and theboundedness property of all the argumented variables. Bothsingular and nonsingular adaptive tracking control designproblems have been considered. We have shown that thesolution relies only on an algebraic matrix equation ratherthan a nonlinear time-varying partial differential equation. Theminimum achievable level of attenuation suffers a lower boundwhich merely relates to the magnitude of the weighting matrix

By the proposed adaptive neural network-based track-ing design, a hybrid adaptive-robust controller has been con-structed. First, a neural-network system has been used tolearn the behavior of the unknown (or uncertain) dynamicsin robotic systems. Next, an algorithm has been usedto attenuate the effect of the learning error on the trackingerror. Hence, a bridge between tracking theory andadaptive neural-network control scheme is built so that the

tracking control design can be applied to uncertain or


Fig. 10. The applied torque�2(t) (steady-state response 0.5–20 s).

Fig. 11. The response ofsT0

jj~x(t)jj2 dt: (� = 1: “—”; � = 0:5: “- - -”; � = 0:2: “� � � �”; � = 0:05: “ � � �”).

unknown robotic systems with the help of neural-networkapproximation. Furthermore, the performance of the neural-network control can be improved with the help oftechnique. The control signal is smooth and the solution issemiglobal in the sense that all variables are bounded forany bounded initial conditions. The number of the neuronsis decreased significantly because an algorithm is usedas fine tuning to efficiently eliminate the approximation errorin the proposed control algorithm. This result is very useful

from the practical control design point of view. Moreover, thedeveloped control scheme is very simple and computationallyefficient since it does not require a knowledge of either themathematical model or the parameterization of the roboticdynamics. More appealingly, the designed controller can beused to solve, simultaneously, both singular and nonsingularadaptive robotic tracking control problems.

Finally, from the simulation results with various prespecifiedattenuation levels, it is seen that our design has achieved the



Fig. 13. The angular positionq2(t): (Desired: “—”; known model: “- - -”; with N. N.: “� � � �”; without N. N.: “� � �”).

desired results. Further study may be necessary on the casewhen joint flexibility is presented or speed measurement isnot available.

APPENDIX

PHYSICAL MEANINGS OF THE VARIABLE

The control variable in Remark 8 has the followingphysical meanings which are proposed in Johansson [15].

Substituting the Lagrangian in (1) into the Lagrangeequation (2), we get

Because the changes in potential energy due to gravitationare inevitible and can be determined from the initial and endpoints only, there is no need to try to optimize the gravitation-dependent torques (or forces) Therefore, it is natural


to define the effective applied control torque of the roboticdynamics in the form

which affects the kinetic energy only. In view of (3), we have

(73)

Because the state transformation (7) has been involved inthe process of design, without loss of generality, the effectiveapplied control variable of the modified error dynamics (10)is chosen in a more general form as [5], [15]

(74)

It is obvious that if we let andthen the control variable in (74) can be reduced toin (73).Hence, the selective applied torqueaffects the kinetic energyonly.

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Yeong-Chan Changwas born in Tainan, Taiwan,R.O.C. He received the M.S. and Ph.D. degrees inelectrical engineering from the National Tsing HuaUniversity, Hsinchu, Taiwan, in 1991 and 1995,respectively.

He is now an Associate Professor at Kung ShanInstitute of Technology and Commerce, Tainan, Tai-wan. His interests include nonlinear control, adap-tive systems, robotics, and signal processing.

Bor-Sen Chen(M’82–SM’89) received the B.S. de-gree from Tatung Institute of Technology, in 1970,the M.S. degree from National Central University,Taiwan, R.O.C., in 1973, and the Ph.D. degree fromthe University of Southern California, Los Angeles,in 1982.

He was a Lecturer, Associate Professor, and Pro-fessor at Tatung Institute of Technology from 1973to 1987. He is now a Professor at National Tsing-Hua University, Hsinchu, Taiwan, R.O.C. His cur-rent research interests include robust control, adap-

tive control, and signal processing.Dr. Chen has received the distinguished research award from National

Science Council of Taiwan four times.

a nonlinear adaptive h//spl sup infinity//tracking control

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