IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005 537

Learning-Based Identification and Iterative LearningControl of Direct-Drive Robots

Björn Bukkems, Dragan Kostic, Member, IEEE, Bram de Jager, and Maarten Steinbuch, Senior Member, IEEE

Abstract—A combination of model-based and iterative learningcontrol (ILC) is proposed as a method to achieve high-quality mo-tion control of direct-drive robots in repetitive motion tasks. We in-clude both model-based and learning components in the total con-trol law, as their individual properties influence the performanceof motion control. The model-based part of the controller compen-sates much of the nonlinear and coupled robot dynamics. A newprocedure for estimating the parameters of the rigid body model,implemented in this part of the controller, is used. This procedureis based on a batch-adaptive control algorithm that estimates themodel parameters online. Information about the dynamics not cov-ered by the rigid body model, due to flexibilities, is acquired exper-imentally, by identification. The models of the flexibilities are usedin the design of the iterative learning controllers for the individualjoints. Use of the models facilitates quantitative prediction of per-formance improvement via ILC. The effectiveness of the combi-nation of the model-based and the iterative learning controllers isdemonstrated in experiments on a spatial serial direct-drive robotwith revolute joints.

Index Terms—Dynamics, identification, iterative learning con-trol (ILC), model-based control, robotics.

I. INTRODUCTION

I NCREASING demands on the performance of modernrobotic manipulators have led to the development of various

motion control approaches. These approaches can be dividedinto three main categories [1]. The first one is decentralizedcontrol, relying on independent feedback loops that implementconventional proportional-integral-derivative (PID) controllaws [1], [2]. Nonlinear couplings in the robot dynamics aretreated as disturbances. Because of simplified implementa-tion and tuning requirements, decentralized control is still adominant solution in practice. However, this solution is alsorecognized as being unable to provide high performance mo-tion control, which in this paper is defined as high accuracyin realizing the reference robot trajectory, without violation ofstability criteria, without excitation of parasitic robot dynamics(e.g., flexibilities) and without amplification of noise (distur-bances).

The second category is model-based control [1], [2], whichis realized by integrating the nonlinear robot dynamics into the

Manuscript received April 5, 2004; revised October 22, 2004. Manuscriptreceived in final form February 2, 2005. Recommended by Associate EditorM. de Mathelin.

B. Bukkems, B. de Jager, and M. Steinbuch are with the Dynamicsand Control Technology Group, Department of Mechanical Engineering,Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherlands(e-mail: B.H.M.Bukkems@tue.nl; A.G.de.Jager@wfw.wtb.tue.nl; M.Stein-buch@tue.nl).

D. Kostic is with the Embedded Systems Institute, 5600 MB Eindhoven, TheNetherlands (e-mail: Dragan.Kostic@esi.nl).

Digital Object Identifier 10.1109/TCST.2005.847335

control design. This control approach is computationally moreintensive than decentralized control. However, it is supposedto provide better tracking accuracy. By augmenting the non-linear model-based controller with feedback laws that includemodels of the robot flexibilities and disturbances, higher robust-ness against these undesirable effects can be achieved. Model-based control can be used for the design of both feedback andfeedforward control laws.

The third category derives control input signals througha learning process. Learning means refinement of the inputsignals, based on past performance in executing the referencemotion. The refined control input should provide performanceimprovement. A special approach within this category is iter-ative learning control (ILC). This approach is very attractivein robotics, because of its performance improving capabilitiesalong repetitive trajectories, often met in practice. A learning al-gorithm calculates an ILC input based on the tracking error andthe ILC input of the previous trial. The tracking performance isimproving through the repetition of trials, until the reproducibleparts of the tracking errors are reduced, and nonreproducibleerrors become dominant.

This paper investigates the combination of nonlinearmodel-based robot motion control and ILC. The particularobjective is to realize high performance motion control ofdirect-drive robotic manipulators. Because of direct-drive actu-ation, dynamics of these manipulators are highly nonlinear andcoupled, which impedes motion control of high quality. Theadditional requirement is to realize high motion performancewhile implementing feedback controllers that are weaklytuned. Weak tuning means less stiffness in the robot joints,which in turn implies that the structure of the manipulatorwill exert less force/torque during an unexpected contact withsome object (e.g., when hitting an obstacle). The model-basedcontroller, with a conventional feedback control law, is usedfor stabilization and compensation of much of the nonlinearand coupled robot dynamics. Imperfections of the nonlinearmodel-based controller at low frequencies are tackled via ILC.Both model-based and learning control components are im-portant for the performance of motion control, and, therefore,they will be equally emphasized in this paper. In particular,we will present experimental procedures to identify rigid-bodyand flexible dynamics of a robot. The identified rigid-bodydynamics are used in the model-based controller. The flexibledynamics are considered in the ILC design. The theories behindidentification and control designs will be presented in sufficientdetail. These theories will be practically demonstrated on a spa-tial serial direct-drive robotic manipulator with three revolutejoints. Experimental results will confirm performance improve-

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538 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

ment when the model-based and the learning controllers areapplied in combination.

The paper is organized as follows. In the next section, wesummarize the relevant literature. We also indicate what is inno-vative in our concept of joining model-based control and ILC.In Section III, we formulate a model of both the rigid body dy-namics and the flexible robot dynamics. Identification of therobot dynamics is considered in Section IV. The design of theiterative learning controller is explained in Section V. In Sec-tion VI, we present the robotic setup used in the experiments,and the experimental results are given in Section VII. The con-clusion and final remarks are given in Section VIII.

II. MODEL- AND LEARNING-BASED ROBOT MOTION CONTROL

Opportunities in model-based robot control are boosted bythe computational power of modern digital processors, by ad-vances in the theory on robot modeling and identification [1],[3]–[6], and by incorporation of control theory in general. Amodel may be used to compensate for nonlinear dynamic cou-plings between robot axes, enabling linear robust feedback con-trol designs that ensure accurate and disturbance-free robot op-eration in an outer loop [7]. A model improves the control per-formance to the extent it matches the real robot dynamics. Amodel is of sufficient quality, if its structure describes all rel-evant aspects of the physics (i.e., inertial, Coriolis/centripetal,gravity, and friction effects), and if its parameters fit the realvalues.

So far, a number of theoretical and experimental studies onestimating the model parameters have been reported. Each ex-ploits the well-known property that a model of the rigid bodyrobot dynamics can be represented linearly in a set of identifi-able parameters, called the base parameter set (BPS) [3]. Theelements of the BPS are nonlinear combinations of robot iner-tial parameters, such as masses and moments of inertia of therobot links, as well as the Cartesian coordinates of the link cen-ters of masses. Since friction can cause control problems (e.g.,static errors, limit-cycles) [8]–[10], the model should also ac-count for friction effects. The friction force can be counteractedusing model-based [8], [9] or nonmodel based techniques [10].If model-based techniques are used, then the relevant values ofthe friction parameters are needed in addition to the BPS.

In general, we can distinguish between least squares-like [4],[5], maximum-likelihood [4], [6], and adaptive techniques [11],[12] for estimation of the BPS elements and friction parameters.If the inertial and friction parameters are jointly estimated, thenone should consider a friction model that admits a linear repre-sentation in its parameters.

In this paper, we use a particular batch adaptive control tech-nique for the problem of estimating both the BPS and the fric-tion model parameters. The control technique was originallyproposed in [13], as a way to accurately realize a repetitivereference trajectory, when the correct parameters of the robotdynamic model with friction are unknown. A newer versionof this technique has appeared in [14], improving the robust-ness against disturbance and parasitic conditions, as well asspeeding-up the convergence characteristics. Being computa-tionally cheap, the batch adaptive control technique admits a

relatively simple online implementation. When it realizes a ref-erence trajectory which excites the robot dynamics persistently[12], the technique can deliver accurate estimates of the BPSelements and the friction parameters. Design rules for a persis-tently exciting trajectory, later on referred to as the excitationtrajectory, are well known in robotics [4], [5]. Among others,the design suggested in [4] results in a cyclic excitation trajec-tory of which the frequency resolution and the frequency con-tent can be directly assigned. Control of the frequency contentis an important precaution against excitation of dynamics thatis not covered by the model (parasitic dynamics) of which theBPS is estimated. The design suggested in [4] is promoted inthis paper, too. A preliminary presentation of the parameter es-timation using batch adaptive control can be found in [15].

The batch adaptive algorithm has several advantages with re-spect to other identification procedures, e.g., least-squares [4],maximum-likelihood [4], [6], and normal adaptive techniques[11]. First of all, the algorithm admits a relatively simple onlineimplementation, which contributes to the efficiency of identifi-cation. Consequently, variable robot loads can be easily takeninto account. While the maximum-likelihood technique of [6]searches for estimates using a nonconvex optimization strategy,the batch adaptive algorithm takes advantage of the model rep-resentation that is linear in the parameters, and which facilitatesthe convexity of the estimation. Consequently, the batch adap-tive algorithm is more efficient than the maximum-likelihoodtechnique. When compared with the standard least-squares esti-mation, the batch adaptive algorithm features higher robustnessagainst measurement noise and against the initial guess of theestimates, together with fast convergence of the estimates to thesteady-state values [14].

It is recognized that a model-based approach is capable ofimproving the performance of motion control [1], [7]. It is evi-dent that the control quality is sensitive to modeling errors, e.g.,parameter inaccuracy or neglected (unmodeled) dynamics. Sta-bilization and high performance motion control can be realizedby augmenting the model-based controller with feedback loops,that are shaped such to achieve high-gain feedback [7] in thefrequency range of the reference trajectory, and rejection of dis-turbances and parasitics beyond that range. Because of the highfeedback gain, the robot joints are usually very stiff, while thefeedback controllers are typically of high order. If the referencetrajectory is repetitive, which is common for industrial robots,then motion control of high quality can be achieved by means ofILC. With ILC, there is no need for high-gain feedback control.The combination of model-based control and ILC will result inhigh performance control for a given repetitive trajectory.

After being introduced in robotics [16], ILC has evolved intoa control discipline utilizing a number of algorithms [1], [13],[14], [17]–[19]. Coarsely speaking, variants of ILC proposedfor robotic problems have become almost as numerous as thenumber of practitioners applying ILC in robotics. Commonly,the ILC design requires very little knowledge about the robot dy-namics, either looking for just an approximate dynamic modelwhich parameters are only roughly known [1], [13], [14], [17],or even considering only linearized dynamics [18], [19]. Re-garding the domain in which the learning controller is designed,a distinction can be made between time-domain [1], [13], [14],

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[16], and frequency-domain ILC techniques [18], [19]. Com-monly, the time-domain techniques only make use of rigid robotdynamics. Their rules for tuning the learning controllers are usu-ally rather conservative, since they do not allow one to predictthe maximum effect of ILC on the tracking performance [20].On the other hand, frequency-domain techniques can take intoaccount additional knowledge about the robot dynamics, e.g.,about dynamics not covered by the model-based compensation.Hence, frequency-domain techniques enable a formalism for de-signing the learning controller. A limitation is that the controllerdesign is based on the linearized robot dynamics, so nonlineareffects cannot be directly integrated.

In this paper, we consider a frequency-domain ILC technique.The design of the learning controller is based on the linearizeddynamics that remain after decoupling the robot with a model-based controller. Although already suggested as a possible so-lution for robots whose dynamics feature high nonlinearities,like direct-drive robots [18], it seems that the combination ofnonlinear model-based and ILC has not been practically in-vestigated yet. To the best of our knowledge, this paper is thefirst study of ILC applied to direct-drive robots, where the fre-quency-domain design of the learning controllers is enabledby nonlinear model based compensation. Preliminary results ofthis study were presented in [21]. A notable difference betweenother frequency-domain techniques used in robotics and our ap-proach in this paper, is that we provide a formalism for tuningthe learning filters based on the dynamics that are experimen-tally measured on the robot.

A direct-drive robot with three revolute joints was used toexperimentally demonstrate the considered techniques for pa-rameter estimation and ILC. The robot joints are implementedas a waist, shoulder, and elbow, which is a kinematic structure,often met in industry. Therefore, the results to be presented inthis paper should be representative for industrial case studies.

III. ROBOT DYNAMICS

We represent the rigid-body dynamics of a general serial di-rect-drive robotic manipulator with actuated joints using theEuler–Lagrange formalism [22]

(1)

where , and are the vectors of joint motions, speeds, andaccelerations, respectively, is the inertia matrix, , ,and are the vectors of Coriolis/centripetal, gravity, and fric-tion effects, respectively, and is the vector of control inputs(joint forces/torques). We assume that the analytical expressionsof , , , and are readily available, while the parameters in-volved in these expressions have to be determined.

The objective of the motion control problem is steering thejoint motions along the reference trajectory . Thisproblem can be solved by using the model (1) in the followingnonlinear model-based compensator:

(2)

Fig. 1. Motion control using model-based and ILC.

where represents the vector of new control inputs thatshould ensure stable robot motions and accurate tracking of thereference. Assuming is known, can be postulated as

(3)

where is the reference acceleration, is the controlinput calculated by a suitable feedback controller, and isthe output of the iterative learning controller. The implementa-tion of the motion controller (2), (3) is illustrated in Fig. 1. Todesign the feedback and learning controllers, we need the robotdynamics that are not covered by the nonlinear model-basedcompensator (2). To design an iterative learning controller, aparametric representation of these dynamics is required.

Ideally, when the controller (2) is applied to the dynamics (1),linear plants remain

(4)

This holds only if , , and .In practice, this rarely happens, especially in the presence offlexible effects that are not included in (1). As discussed in [7]and [23], the dynamics in the robot joints are more complexthan the ideal one in (4). In practice, the model-based compen-sation strongly reduces the robot nonlinearities, but it hardlyeliminates the nonlinear effects completely. Although present,the remaining nonlinearities are usually less significant for mo-tion performance than the parasitic robot dynamics, i.e., the dy-namics not covered by the rigid-body model. Therefore, the re-maining nonlinear effects are not considered hereafter, while thelinear parasitic dynamics in joint can be represented by thetransfer function

(5)

with representing the Laplace variable. In (5), is not justa double integrator, as suggested by (4), but it has poles andzeros due to the appearance of resonances and anti-resonances.The frequencies and damping of these resonances vary as therobot configuration changes. Capturing all possible variationsin the plant dynamics with a single model could be a difficulttask. Instead, we choose a strategy of adopting a nominal modelto represent an average of the linearized dynamics. Differencesfrom the nominal model are interpreted as perturbations in thedynamics (model uncertainty) and have to be counteracted byILC. This strategy is discussed in Section IV-B.

540 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

IV. LEARNING-BASED PARAMETER IDENTIFICATION

In this section, we first suggest identification of the parame-ters of the robot rigid-body dynamic model with friction usinga batch adaptive control algorithm [13], [14]. As mentioned inSection II, the algorithm is applicable to the rigid-body dynamicmodel with friction when this is linear in the unknown param-eters. The unknown friction coefficients are concatenated withthe elements of the BPS, forming the complete parameter vectorto be estimated. The next subsection will briefly describe thebatch adaptive control algorithm. For a more detailed descrip-tion, the reader is referred to [14]. Second, the parameterizedmodeling of the flexible dynamics, obtained after applying themotion controller (2), (3), and needed for the design of the iter-ative learning controllers, is discussed in Section IV-B.

A. Parameter Estimation via Batch Adaptive Control

A representation of (1) linear in the parameters is

(6)

where is the regression matrix, is the vectorof BPS elements and friction parameters, with the number ofparameters to be estimated. The robot can be stabilized using aconventional PD feedback controller

(7)

where and are diagonal matrices of positive pro-portional and derivative gains, respectively, the vector withnew input signals from the batch adaptive controller, and thetracking error

(8)

If (7) is applied to (6), then the closed-loop system takes theform

(9)

When all motions, speeds, and accelerations in (9) are chosento be reference ones, the input signal is obtained

(10)

and the residual error dynamics, with input, is given by

(11)

For appropriate choices of and [13], the residual errordynamics can be made passive for the output

(12)

where is a positive constant scalar, and is an vector ofgradients of a potential function. For a suitable choice of thispotential function, the reader is referred to [13] and [14].

Before showing a batch adaptive update law, suitable for theestimation of the BPS elements and parameters of the frictionmodel, several definitions and assumptions are needed

(13)

(14)

Here, denotes the duration of one trial of the repetitive trajec-tory. The matrix is assumed to be nonsingular, which holds if

in (6) contains a minimum number of BPS elements and fric-tion parameters, and if the reference trajectory excites the robotdynamics persistently.

The batch adaptive update law is given as follows [14]:

(15)

where denotes the number of the trial ,is an positive definite matrix called the adaptation gain,and represents the inner product between and

, weighted by a positive definite matrix . For twomatrices of equal number of rows, this inner product is definedas

......

. . .

(16)

where and are the th column of and the th columnof , respectively. The weighting matrix is omitted if it isidentity. The inner product of two columns is defined as follows:

(17)

If the adaptation gain satisfies

(18)

then the convergence of the tracking error (8)

(19)

can be proven [14], with denoting a signal norm. The conver-gence of the tracking error implies convergence of the elementsin to steady-state values . Theoretically, thereis no restriction on the initial guess of , required in (15). Prac-tically, when control inputs are bounded, cannot be chosenarbitrarily, but should be constrained to a certain feasible regioncorresponding to limits on the control inputs.

In the algorithm presented, the tuning parameters are con-tained in the matrices , , and . The coefficient shouldbe tuned as well, and there is also a possibility to use various

by choosing different potential functions, as long as thesefunctions satisfy the properties specified in [13] and [14]. The

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PD gains should ensure passivity of the robot dynamics whenrealizing the reference trajectory. The matrix , the scalar ,and the vector function influence the rate of convergenceto the steady-state values of . Inequality conditions that mustbe fulfilled when tuning and can be found in [14].

1) Design of Excitation Trajectories: With the algorithmpresented in the previous subsection, all elements of the vector

converge after a sufficient repetition of trials. We make useof this property to obtain an estimate of that provides a closematch between the model (1) and the real robot dynamics. Ob-taining such an estimate is the goal of robot identification, andit can be obtained if the robot realizes an appropriate excitationtrajectory.

To design such a trajectory, one should optimize some prop-erty of the information matrix

...

(20)

This matrix is formed by vertically stacking the matrices thatcorrespond to different points of the excitation trajectory. Somechoices of the performance criteria to be optimized are the con-dition number of the information matrix, its extreme singularvalue, its Frobenius condition number, or the determinant of theweighted product between the information matrix and its trans-pose [5]. In our opinion, “a good trajectory” is one that providesus with estimates that are relevant for at least those motions thatthe robot is supposed to realize in practice.

An excitation trajectory that meets the given requirements canbe determined by postulating it in the form of a finite Fourierseries [4]

(21)

with the prescribed fundamental frequency , and the param-eters , , and to be computed via optimization. Con-straints on mechanical limits in the robot joints and permissiblelevels of joint speeds and accelerations, must be taken into ac-count. The fundamental frequency defines the resolution of thefrequency content of the excitation trajectory. Together with thenumber of harmonics , which affects persistency of excita-tion, the fundamental frequency determines the frequency con-tent of excitation. This content should be rich enough to coverall possible motions of the robot, but also well below the eigen-frequency of the first flexible mode of the robot structure.

No matter which property of is optimized via , ,and , the resulting optimization problem is essentially non-convex, which means that the global optimum for the excitationtrajectory cannot be obtained easily. Different initial conditionsmay lead to different excitation trajectories, all corresponding tolocal minima of the optimization problem. Our experience [23],though, shows that even suboptimal excitation trajectories pro-vide satisfactory estimates.

Fig. 2. Principle of ILC.

B. Parameterized Modeling of Flexible Dynamics

A nominal parametric model , needed for the ILC design,can be determined based on the frequency response functions(FRFs) directly measured on the robot. A procedure to mea-sure these functions is explained in [7]. In short, FRFs for eachjoint are measured separately in a single-input–single-output(SISO) framework and under closed-loop operating conditions.Each FRF is computed assuming as excitation andas response. The model is based on FRFs that correspondto different robot configurations. For instance, assume FRFs

, , have been determined. Then, there areat least two possibilities to calculate the nominal FRF

(22)

or

(23)

Possibility (22) minimizes the distance between the nominal andthe measured FRFs at each frequency , while (23) calculatestheir average. We choose (23), because it provides smoothermagnitude and phase characteristics. Smoother data will facili-tate fitting the parametric model onto the data . Perturba-tions from will be taken into account during the ILC design.The fitting itself can be done in the least-squares sense, usingan output-error model structure [24]. The model order is up tothe designer, whose objective is to achieve a sufficient match be-tween and the Bode plot of . The obtained model willbe used in the ILC design, which will be explained in the nextsection. As this design is performed offline, the order of istypically not critical for the design. In practice, arbitrarily highmodel orders might not be admissible due to numerical prob-lems. Consequently, the designer chooses the order such that,on one hand, a reasonable match between the measured dataand the model is achieved, and, on the other hand, no numericalproblems arise.

V. ILC

This section formulates the design of the iterative learningcontrollers in the frequency domain. Consider the block diagramin Fig. 2. The SISO linear plant is described by its transferfunction (5). The feedback controller generates the feedbackcontrol input , according to (3). Apart from stabilization,the feedback controller is used to generate position errors thatare necessary to initialize the ILC algorithm. The input

542 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

represents the output of the learning controller, and it is updatedafter each trial .

In the Laplace domain,1 the update rule of the learning algo-rithm is defined as follows [25]:

(24)

where is the number of the trial, is the learning filter, andis the robustness filter. According to Fig. 2, the errors in two

successive trials can be calculated from the closed-loop transferfunctions from the control input, , to the tracking error,

(25)

(26)

where denotes the sensitivity function:

(27)

and the product is the process sensitivity. Substitu-tion of (24) and (25) into (26), results in

(28)

If the second term on the right-hand side of (28) is negligible,then the condition for the amplitude of the error in trial tobe smaller than the amplitude of the error in trial , is given by

(29)

with denoting the infinity norm. The inequality (29) isknown as the condition for error convergence toward zero. Thecontribution of the second term on the right-hand side of (28)can be minimized by suitable design of the -filter, as explainedin the following. To maximize the speed of error convergence,the learning filter should be identical to the inverse of theprocess sensitivity

(30)

To compute the -filter, a parametric model of the processsensitivity has to be made. By virtue of (27) and (30), the orderof the -filter depends on the orders of the plant model and thecontroller. As discussed in Section IV-B, a high-order processsensitivity model is acceptable, as long as it does not cause nu-merical problems. In the next section, we will present an ex-ample of the plant model and its corresponding process sensi-tivity model. If the plant is nonminimum phase, the -filterwill be unstable if the process sensitivity is inverted directly. To

1Because we assume identical initial conditions in each trial, the analysis inthe Laplace domain is allowed, even though each trial has finite time-length[25].

Fig. 3. RRR-robotic arm. (a) Photo. (b) Kinematic parameterizationsaccording to the DH notation.

avoid this, the zero phase error tracking algorithm for digitalcontrol [26] can be used. This algorithm provides a stable ap-proximation of the inverted process sensitivity.

The role of the robustness filter is to ensure error conver-gence in the presence of modeling errors. Namely, the -filter isbased on a parametric model which, in practice, cannot cover alldynamics of the real system, but only some low-frequency part.The -filter should guarantee the validity of (29) for the fre-quency components at which the model is inaccurate. Since inpractice modeling errors arise in the high-frequency range,is typically chosen to be a low-pass filter. Below its cutoff fre-quency , the -filter has a passband equal to 1, while above

its magnitude is decreasing. Thus, within the passband of ,the second term on the right-hand side of (28) diminishes. Sincethis term depends on the reference joint trajectory, of which theharmonic content is typically within the passband of the -filter,it also has negligible contribution to the error outside the pass-band of the -filter.

The -filter should influence the frequency content ofonly by its magnitude characteristics and should not introduceany phase distortion. To explain this further, let us discuss theerror (28). The objective of learning is to achieve the error intrial to be smaller than the error in the previous trial, i.e.,

. The convergence criterion (29) ensures decreaseof the part of which directly depends on . As alreadymentioned, the second term on the right-hand side of (28) is can-celed if is a real scalar equal to 1. Therefore, the -filter mustnot introduce any phase shift, especially within its passband,since this would violate the condition . Consequently,the -filtering is a noncausal operation that is performed of-fline, as indicated in Fig. 2.

VI. EXPERIMENTAL SETUP

The robotic arm, shown in Fig. 3, is the subject of our casestudy. It is an experimental facility for the research in motioncontrol [27]. The photo and kinematic parameterizations ac-cording to the well-known Denavits–Hartenberg (DH) notation[28], reveal three revolute degrees of freedom (DOF), whichmakes such a kinematic structure referred to as RRR. Each DOFis actuated by a gearless brushless dc direct-drive motor andhas an infinite range of motions, thanks to the use of slipringsfor the transfer of power and sensor signals. The actuators are

BUKKEMS et al.: LEARNING-BASED IDENTIFICATION AND ITERATIVE LEARNING CONTROL 543

TABLE IDH PARAMETERS OF THE RRR-ROBOTIC ARM

Dynaserv motors with nominal torques of 60, 30, and 15 [Nm],respectively. The servos are driven by power amplifiers withbuilt in current controllers. Joint motions are measured usingincremental optical encoders, with a resolution of rad .Both amplifiers and encoders are connected to a PC-basedcontrol system. This systems consists of a MultiQ I/O boardfrom Quanser Consulting (8 13-b analog-to-digital converter(ADC), 8 12-b DAC, 8 digital I/O, 6 encoder inputs, and3 hardware timers), combined with a real-time controller forMatlab/Simulink (Wincon). This facilitates the design of con-trollers in Simulink and their real-time implementation. Thecontrol system features a time delayed joint angular responseto the given control input. For a sampling time of 1 ms ,there is a delay of [23].

Detailed closed-form models of the robot kinematics and dy-namics are available in [23]. From Fig. 3, one can determinethe DH parameters, whose numerical values are presented inTable I. The DH parameters are: twist angles , link lengths

, joint displacements , and link offsets .

VII. EXPERIMENTAL RESULTS

This section contains the results of the experiments conductedto show the effectiveness of the identification procedure, pre-sented in Section IV, as well as the performance improving ca-pabilities of the ILC algorithm discussed in Section V.

A. Identification Results

1) Parameter Estimation Results: The rigid body dynamicsof the RRR-robot, as presented in [23], feature 15 BPS elements.Due to the symmetry in distribution of masses in the second andthe third robot link, two BPS elements are identically equal tozero [23]. Consequently, only 13 BPS elements had to be iden-tified using the strategy presented in Section IV. The adoptedfriction model covers Coulomb and viscous effects

(31)

where is a 3 vector of friction torques, while and are 33 positive definite diagonal matrices of Coulomb and viscous

friction parameters, respectively. This friction model admits alinear parametrization, which facilitates joint identification ofthe BPS elements and friction parameters.

The excitation trajectories, needed in the identification exper-iments, were calculated using (21). For all three joints, the fun-damental frequency was chosen to be 0.1 [Hz], which is usuallytaken as sufficient [4]. The number of harmonics was 100, sothe frequency contents were constrained up to 10 [Hz], whichis sufficient to cover operations the RRR-robot performs and to

Fig. 4. Motions of the excitation trajectories: � —thick solid, � —thin solid,and � —dashed.

not excite flexible dynamics. The tuning parameters , ,and were computed by optimizing the condition number ofthe information matrix , and the constraints were

rad/s rad/s

rad/s rad/s (32)

After optimization, the motions of the excitation trajectories,depicted in Fig. 4, were obtained.

In order to demonstrate the full potential of the estimationalgorithm from Section IV-A, three different identification ex-periments were performed. In the first experiment, no load hasbeen attached to the third robot joint, in the second experiment,a steel rod of 1.22 [kg] (Fig. 3) has been attached, whereas inthe third one, an extra load of 0.45 [kg] has been attached ontothe tip of the steel rod. Before all three experiments, the initialvalue of the vector of parameters to be estimated, , was set tozero. Consequently, the extra input in (7) was also zero duringthe first trial, see (15).

For online implementation of the batch adaptive algorithm,the feedback controller gains and in (7) were weaklytuned for each load separately, such as to stabilize the systemand to avoid excitation of parasitic dynamics (flexibilities). Thescalar in (12) and the weighting matrix in (15), were tunedsuch as to achieve fast convergence of the estimates to steady-state values.

Fig. 5 shows the results of the identification experiments withdifferent loads. The parameters represent the BPSestimates, whereas the parameters represent the es-timates of the friction parameters. By inspection of the givenplots, it can be seen that the parameters are updated at the endof each trial, i.e., after every 10 [s]. Almost all parameters havereached their steady-state values after not more than eight trials.It can be also noticed that the steady-state estimates of , ,

, , , , and , are not the same when estimated withdifferent loads. This can be expected, as these BPS elementsdepend on the inertial properties of the third robot link [23].Obviously, the variations in the mass attached to the third joint,induces differences in the estimates.

544 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

Fig. 5. Convergence of the model parameters: without the last link (black,thick), with the last link (black, thin), and with the extra load attached to thelast link (gray).

To evaluate if the estimates provide a close match between themodel (1) and the real robot dynamics for all three loads, vali-dation experiments have been conducted. In these experiments,the RRR-robot performed a writing task, suggested in [29], anddepicted in Fig. 6. This task is recognized as demanding for therobot dynamics, due to fast motions with discontinuous accel-erations. The task consists of writing the presented sequence of

Fig. 6. Writing task. (a) Reference path of the robot tip in x � z plane.(b) Corresponding joint motions.

letters in the vertical plane, as shown in Fig. 6(a). The corre-sponding joint motions are depicted in Fig. 6(b).

During the validation experiments, the robotic arm has beenstabilized using a conventional PD feedback controller

(33)

where and ane diagonal matrices of positiveproportional and derivative gains, respectively. To verify agree-ment of torques rendered by the model (1) and measured torquesunder PD control (33), the joint motions, speeds, and accelera-tions were reconstructed online using a Kalman observer de-scribed in [23]. In Fig. 7, outputs of the PD controllers of thesecond joint have been depicted for the three different valida-tion experiments, together with the torques reconstructed using(1). From this figure, it can be seen that in all cases, the torquesrendered by the model closely match the outputs of the PD con-trollers, which indicates that relevant estimates have been ob-tained.

Small deviations between the output of the PD feedbackcontroller and the reconstructed torques can be seen when thetorques peak, corresponding to the changes in the sign of thejoint speed. These indicate that the adopted friction model doesnot compensate properly for the real friction. In Fig. 7(c), itcan be seen that the output of the feedback controller is clippedat its maximum permissible value (30 [Nm]). The saturationoccurs as the extra load attached to the robot tip requires moreactuation power than permissible. Under normal operatingconditions, i.e., when no extra load has been attached to therobot tip [Figs. 7(a) and 7(b)], the saturation effect does notoccur.

2) Parameterized Models of Flexible Dynamics: For clarity,it is pointed out that the results discussed in this subsection havebeen obtained using the RRR-robotic arm with the third linkconnected to the third joint, but without the extra mass attachedto the robot tip. After applying the nonlinear model-based com-pensator (2) and stabilizing the robot using a PD feedback con-trol law

(34)

with and again representing diagonal matrices of posi-tive proportional and derivative gains, the remaining dynamicshave been identified in all three joints. We explain the identi-fication results of these remaining flexible dynamics in the firstjoint. However, the experimental results given in this subsection

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Fig. 7. Results of the validation experiments. (a) No load attached to the thirdjoint. (b) Steel rod attached to the third joint. (c) Steel rod and an extra massattached to the third joint.

cover the case of identification of the flexible dynamics in allthree joints. Fig. 8 shows the FRFs measured in the first joint,corresponding to 16 static postures, spanning the com-plete range of joints 2 and 3: ,

.

Fig. 8. Experimentally obtained FRFs G ; . . . ; G (gray) and the nominalFRF G (black).

The FRFs have been measured under closed-loop conditions,since the plant itself is not asymptotically stable. White noise isadded to the control input and the transfer function from thewhite noise signal to the total control input signal, the sensitivityfunction, is measured. Since the stabilizing feedback controller(34) is known, the remaining plant dynamics can be extractedfrom this sensitivity function. The results of this identificationprocedure are reliable outside the bandwidth of the closed-loopsystem, since in that range the coherence of the measurement isgood [7]. Fig. 8 shows that the theoretically expected dynamicsof a double integrator (4) holds at low frequencies only (below20 [Hz]), while the real dynamics are more involved at higherfrequencies. Also, up to 4 Hz, the slope of the magnitudes isless steep than 2. This frequency range is within the band-width of the closed-loop system established via (34). Becauseof the authority of the PD controller, we cannot achieve suf-ficient coherence of the measurements within the closed-loopbandwidth of 4 [Hz]. Consequently, the part of the identifiedFRF up to 4 [Hz] is not reliable. Note the resonance around 28[Hz], caused by vibrations at the robot base, and more profoundresonances at higher frequencies. The phase has a lag growingwith increasing frequency, which is caused by the time delay inthe control system [23]. The nominal FRF , calculated ac-cording to (23), is shown in Fig. 8 as well. It is also depicted inFig. 9, together with its parametric fit . As obvious from thisfigure, there is a discrepancy between the unreliable data below4 [Hz] and the fit. The order of the model is 8, which allows anaccurate fit between 4 and 70 [Hz]. Fitting the high frequent dy-namics is omitted, as we do not expect it would be possible tolearn above 70 [Hz].

B. ILC Results

In this subsection, the ILC design for the first joint is demon-strated, but the experimental results cover the case with learningcontrollers for all three joints.

A PD feedback controller , with 1000 and, is used to stabilize the first robot joint. Given and

546 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

Fig. 9. Nominal FRF G (thick line) and its parametric fit P (thin line).

Fig. 10. FRF of the L-filter for the first joint (black, thick), the processsensitivities calculated for 16 distinct robot postures (gray) and their products(black, thin).

, the parametric model of the process sensitivity can be calcu-lated. As explained in Section V, the process sensitivity model isused to calculate the -filter, which implies a fixed structure forthis filter. For the given plant model and the controller, the orderof the -filter is 9. This fairly high order is justified by a goodmatch between the -filter and the measurements in the lowerfrequency range. The good match at low frequencies is obviousfrom Fig. 10, which shows the -filter and the process sensitiv-ities calculated from all 16 measured FRFs . Thesame figure depicts the products between the -filter and theprocess sensitivities. A perfect -filter would result in 0 [dB]magnitude and 0 phase shift of these products for all fre-quencies. From Fig. 10, it can be seen that the actual -filtermatches the inverse of the process sensitivities until approxi-mately 20 [Hz] and in the range from approximately 32 [Hz]until 40 [Hz]. Between 20 and 32 [Hz], the varying location anddamping of the first resonance mode of the first robot joint causethe deviations from the 0 [dB] magnitude and 0 phase line.

Fig. 11. Reference path of the (a) robot tip and (b) its corresponding jointmotions.

In order to preserve the convergence criterion (29) in the fre-quency ranges where the -filter mismatches the inverse of theprocess sensitivities, the robustness filter is used, as explainedin Section V. This filter is chosen to be a fourth-order low-passdigital Butterworth filter with a passband magnitude equal toone. The cutoff frequency of the -filter can never exceedthe frequency up to which the -filter is a good representationof the inverse of the process sensitivity model. In determiningthe optimal cutoff frequency, the convergence criterion (29) hasto be evaluated for all process sensitivities calculated from the

measured FRFs, rather than for the nominal one only, to besure that the convergence criterion is satisfied for the completeset of robot postures. The cutoff frequency of the first -filteris chosen to be 18 [Hz]. The cutoff frequencies of the -filtersfor the second and third joint have been derived similarly andare 25 [Hz] and 27 [Hz], respectively. These frequencies canbe compared with the crossovers of the three SISO closed-loopsystems. The crossover frequency is defined as the first 0 [dB]crossing of the open-loop gain ( 1, 2, 3). Sincethe crossover frequencies of the three SISO systems are all ap-proximately equal to 9 [Hz], we can conclude that learning takesplace well above the crossovers.

To demonstrate the effectiveness of the proposed control de-sign, experiments have been conducted. In these experiments,the robotic arm had to perform the writing task, which hasalso been used in the validation experiments discussed in Sec-tion VII-A. The reference trajectory has been modified slightly,in order to obtain a repetitive trajectory. In this writing task,shown in three dimensional space in Fig. 11(a), the robot-tiphad to track the path starting from “A” in the direction ofthe arrows. The corresponding joint motions are depicted inFig. 11(b).

During the experiments, the robot motions have been stabi-lized by the PD feedback controllers that have also been used inthe derivation of the -filters. After each trial, the new ILC inputis calculated according to (24). This means filtering of the mea-sured tracking errors with the corresponding -filters, addingthe previous ILC input to this filtered signal, and, finally, fil-tering of the obtained signals with the corresponding -filters.The result is the ILC input for the next trial. No numerical prob-lems have been experienced during the filtering operations, de-spite the fairly high orders of the filters involved.

The performed experiment consisted of 18 trials. Since the er-rors had converged after 10 trials, and no further accumulationof error powers was observed during the other trials, we discuss

BUKKEMS et al.: LEARNING-BASED IDENTIFICATION AND ITERATIVE LEARNING CONTROL 547

Fig. 12. Tracking errors in the first (upper), second (middle) and third joint(bottom) during trial 1 (black, thick), trial 2 (gray) and trial 10 (black, thin).

Fig. 13. Root-mean-square values of the errors in all joints.

the results up to trial 10. In Fig. 12, the tracking errors during thefirst, second, and tenth trial in all three joints are depicted. Fromthis figure, it can be seen that the tracking errors have alreadyreduced considerably in the second trial, i.e., after just one iter-ation. Final tracking errors in the tenth trial are approximately10 times smaller than the errors in the first trial.

As depicted in Fig. 13, the root-mean-square values of the er-rors in all joints keep decreasing until they reach lower bounds,that cannot be further reduced. For the first two joints, seventrials are needed, while for the third joint it takes one trial more.The root-mean-square values of the errors after the learningprocess are over ten times smaller than before learning.

For the frequency analysis of the tracking errors in jointspace, auto power spectra of these errors have been calculated.Fig. 14 shows the auto power spectra of the tracking errorsin all joints, based on data measured during the first, second,and tenth trial. The spectra show a significant reduction offrequency components until approximately 10 [Hz] in the firsttwo joints and until 12 [Hz] in the third joint. This is very

Fig. 14. Auto power spectra of the tracking errors in the (a) first, (b) second,and (c) third joint for trial 1 (black, thick), trial 2 (gray) and trial 10 (black, thin).

important as the frequency content of the reference motionsis within these ranges. It can also be seen that the frequencycontent of the errors in the high frequency range in the tenthtrial does not differ from the frequency content in the firsttrial, which was expected, since no learning takes place in

548 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005

this range. From Fig. 14(a) and (b), it can be seen that at theend of the learning process, the errors in the first and in thesecond joint have slightly amplified power between 13 [Hz]and 20 [Hz]. The amount of amplification is not an issue for themotion performance. However, it is peculiar why amplificationoccurs in the frequency range within the bandwidth of thecorresponding robustness filters (18 [Hz] for joint 1 and 25[Hz] for joint 2). This phenomenon might be caused by nonre-producible effects, such as variable friction in the robot jointsand variable sampling times of the PC-based control system.It is well-known that nonreproducible errors can be slightlyamplified by ILC within the passband of the -filter [30].Both effects are evident in the experiments on the RRR-robot.Another reason for the amplification phenomenon could be theinfluence of residual nonlinear behaviors contained in the mea-sured data. The model-based compensation strongly reducesrobot nonlinearities but hardly eliminates these. The residualnonlinearities influence the outcome of ILC design, which doesnot directly take into account nonlinear effects. The design itselfassumes linear plant dynamics only. Therefore, an interestingtopic for future research could be analysis of the influenceof nonlinear behaviors on the considered ILC algorithm. Theamplification of the harmonic components might have beenavoided by reducing the passband magnitude of the robustnessfilter. However, this would slow down the convergence rateof learning, i.e., more trials would be needed for the errors toreach their lower bounds. There is no amplification of the errorharmonic content between 13 [Hz] and 20 [Hz] in the thirdjoint, as seen in Fig. 14(c).

VIII. CONCLUSION

A procedure to achieve high quality performance in direct-drive robot motion control is presented. It starts with the de-sign of a nonlinear model-based controller that compensates fornonlinear robot dynamics, leaving mostly decoupled and lineardynamics in the joints. The rigid body dynamic model with fric-tion, used for this compensation, is assumed to be known. Aprocedure for identifying the parameters of this model has beenpresented. The identification is based on a batch adaptive con-trol algorithm that admits online implementation, which is quiteappealing for use in practice.

The robot dynamics that remain after the model-based com-pensation, is identified using FRF measurements. Using thesemeasurements, parametric models of the remaining dynamicsare made. These models are used in constructing the filters ofthe iterative learning controllers. A procedure for filter tuning isgiven, taking into account learning capabilities and convergenceissues.

The identification procedure, as well as the design of thelearning controller, are experimentally demonstrated on a se-rial spatial direct-drive robotic manipulator. Experiments showa considerable improvement of motion control performance inall robot joints.

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Björn Bukkems received the M.Sc. degree (cumlaude) in mechanical engineering from the Tech-nische Universiteit Eindhoven, Eindhoven, TheNetherlands, in 2003.

Since 2003, he has been a Research Assistantin the Dynamics and Control Systems TechnologyGroup, Department of Mechanical Engineering,Technische Universiteit Eindhoven. His researchinterests include the multidisciplinary design andanalysis of dynamic embedded systems. Specialfocus is on using hybrid formalisms for modeling

industrial motion systems, and on supervisory controller synthesis for this kindof systems.

Dragan Kostic (S’98–M’05) received the GraduateEngineer and M.Sc. degrees in automatic control androbotics from the Faculty of Electronic Engineering,University of Nis, Serbia, in 1994 and 2000, respec-tively, and the Ph.D. degree from the Dynamics andControl Technology Group, Department of Mechan-ical Engineering, Technische Universiteit Eindhoven,Eindhoven, The Netherlands, in 2004.

He is a Research Fellow at the Embedded SystemsInstitute, Eindhoven, The Netherlands. From 1996 to1999, he was a Research and Teaching Assistant with

the Group for Automatic Control, Faculty of Electronic Engineering, Univer-sity of Nis, where he was investigating problems of biomechanical analogiesin robotics. His current research interests include dynamic modeling, systemidentification, and robust model-based control of robots, as well as data-basedcontrol of motion systems.

Bram de Jager received the M.Sc. degree in mechanical engineering from DelftUniversity of Technology, Delft, The Netherlands, and the Ph.D. degree fromthe Technische Universiteit Eindhoven, Eindhoven, The Netherlands.

He was with Delft University of Technology and Stork Boilers BV, Hengelo,The Netherlands. Currently, he is with the Technische Universiteit Eindhoven.His research interests include robust control of (nonlinear) mechanical systems,integrated control and structural design, control of fluidic systems, control struc-ture design, and application of symbolic computation in nonlinear control.

Maarten Steinbuch (S’83–M’89–SM’02) receivedthe M.Sc. degree (cum laude) in mechanical engi-neering and the Ph.D. degree in modeling and con-trol of wind energy conversion systems from DelftUniversity of Technology, Delft, The Netherlands, in1984 and 1989, respectively.

He is currently a Full Professor in the Departmentof Mechanical Engineering, Technische UniversiteitEindhoven, Eindhoven, The Netherlands. From 1984to 1987, he was a Research Assistant at Delft Uni-versity of Technology and KEMA (Power Industry

Research Institute), Arnhem, The Netherlands. From 1987 to 1998, he was withPhilips Research Laboratories, Eindhoven, as a Member of the Scientific Staff,working on modeling and control of mechatronic applications. From 1998 to1999, he was a Manager of the Dynamics and Control Group, Philips Centerfor Manufacturing Technology. His research interests are modeling and controlof motion systems. He was an Associate Editor of IFAC Control EngineeringPractice from 1994 to 1996 and is currently Editor-at-Large of the EuropeanJournal of Control.

Dr. Steinbuch was an Associate Editor of the IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY from 1993 to 1997 and of the IEEE ControlSystems Magazine from 1999 to 2002. In 2003, he received the Best-Teacher2002/2003 award from the Department of Mechanical Engineering, TechnischeUniversiteit Eindhoven.