Wavelet series based iterative learning controller design for industrial robot manipulators

Int. J. Computer Applications in Technology, Vol. 35, No. 1, 2009 61

Wavelet series based iterative learning controller

design for industrial robot manipulators

Selvaraj Gopinath*

Control and Optimization Group,Research Department,Corporate Research,ABB Global Industries and Services Ltd.,White field Road, Bangalore 560048, IndiaE-mail: [email protected]*Corresponding author

Indra Narayan Kar and R.K.P. Bhatt

Digital Control Laboratories,Department of Electrical Engineering,Indian Institute of Technology Delhi,Hauz Khas, New Delhi 110016, IndiaE-mail: [email protected]: [email protected]

Abstract: In this paper, a wavelet series based learning controller has been proposed for thetracking control of robot manipulator. Wavelet series approximation is used to approximatethe desired and actual trajectories of the system into finite number of wavelet coefficients.A learning controller is designed in wavelet domain which forces the wavelet coefficients ofthe actual output approach to the corresponding wavelet coefficients of the desired trajectory,which are known constants, such that the tracking of robot manipulator is achieved. Numericalexperimentation studies show the ability of the learning controller designed in wavelet domainin the presence of uncertainties in robot manipulator system.

Keywords: wavelet series; integral wavelet transforms; learning control; robot control.

Reference to this paper should be made as follows: Gopinath, S., Kar, I.N. and Bhatt, R.K.P.(2009) ‘Wavelet series based iterative learning controller design for industrial robotmanipulators’, Int. J. Computer Applications in Technology, Vol. 35, No. 1, pp.61–72.

Biographical notes: Selvaraj Gopinath received his BEng (2000) in Electrical and ElectronicsEngineering at Madurai Kamaraj University, Madurai, India and his MTech (2002) andPhD (2007) in Control Engineering and Automation at the Indian Institute of Technology,Delhi, India. He is currently working as a Associate Scientist at ABB Global Industries andServices Limited, Corporate Research Center, Bangalore, India. His research interests includemachine learning, iterative learning control, signal processing and intelligent control.

Indra NarayanKar received his BEng (1988) from Bengal Engineering College, and hisMTech(1991) and PhD (1997) from the Indian Institute of Technology, Kanpur, India. He wasa research student at Nihon University, Tokyo under the Japanese government Monbushoscholarship programme from 1996 to 1998. He joined as a Lecturer in the Department ofElectrical Engineering, Indian Institute of Technology, Delhi, where he became AssistantProfessor (1999), Associate Professor (2005) and is presently working as Professor. His researchinterests include robust, intelligent control, embedded systems and non-linear control. He haspublished more than 45 papers in international journals and conference proceedings.

R.K.P. Bhatt received his BEng (1969) from Ravishankar University, India, his MTech (1971)from the Indian Institute of Technology, Kanpur, and his PhD (1981) from the Indian Instituteof Technology, Delhi, India. Currently, he is working as a Professor with the Departmentof Electrical Engineering at the Indian Institute of Technology, Delhi. His research interestsinclude image processing and adaptive systems.

Copyright © 2009 Inderscience Enterprises Ltd.

62 S. Gopinath et al.

1 Introduction

ILC has been an active research area for the pastfew decades and great efforts have been put into thedevelopment of different learning controllers. ILC canbe considered for improving the transient responseand tracking performance of systems that execute thesame operation in a repetitive manner (Arimoto, 1996).Industrial robotic operations, chemical processes aresuch situations, where ILC can be used to improve theperformance. Basic block diagram of ILC scheme is shownin Figure 1, where, uk(t) is the input signal at kth iterationapplied to the system results the output trajectory yk(t)and yd(t) denotes the desired trajectory, defined on a finiteinterval t ∈ [0, T ]. These signals are stored in memoryuntil the current (kth) iteration is over, at which time theyare processed offline by the ILC algorithm. The learningcontroller compares yd(t) and yk(t) and adds an updatetermwithuk(t) to produceuk+1(t), the refined input signalwill be given to the system for the (k + 1)th iteration. InILC, the updation occurs sequentially in time upto therequired error goal is reached.

Figure 1 Basic block diagram of ILC (see online versionfor colours)

Arimoto et al. (1984) introduced a D-type ILC algorithmuk+1(t) = uk(t) + Γek(t), which is considered to be thefirst work on ILC. The above ILC law uses the derivativeof the tracking error, which is defined as ek(t) = yd(t) −yk(t) along with a suitable learning gain (Γ). Bondi et al.(1988) set the learning controller with a sufficient linearfeedback, is shown to ensure the tracking performance.Further research led to the development of new algorithms(Atkeson and McIntyre, 1986; Kuc et al., 1991; Janget al., 1995; Moore, 1998; Gopinath and Kar, 2004) andimplementation issues of ILC on time varying, non-linearsystems with conventional feedback controllers. Someof the nonlinear adaptive and robust ILC algorithmsdesigned in time domain are also addressed (Xu andTan, 2003; Tayebi, 2004). During the learning processof ILC, certain higher frequency components (possiblynon-repetitive) enter into the learning feed-forwardcontrolled process at appropriate time instants. These highfrequency components lead to poor learning behaviourin conventional time domain based ILC approaches.To handle the poor learning transients and long-termstability problems, zero-phase low pass filtering techniqueis employed in time domain based ILCs (Tan et al., 2001;

Elci et al., 2002). In time-domain based ILC analysisthe issues of good transients and long-term stability arenot addressed, which motivates the design of learningcontrollers in frequency domain.

Frequency domain based learning controller designand analyses are reported in Kavli (1992) and Lucaet al. (1992). Learning controller scheme using basisfunctions like Fourier and B-splines as input shapingfunctions to approximate the feed-forward control signal,with convergence and stability analyses under repeatabledisturbances are reported in Lee et al. (1993), Gorinevskyet al. (1997), Velthuis et al. (2000), Chen andMoore (2001)and Chen et al. (2004). In Tang et al. (2000) proposedthe Fourier transformation method in order to reducethe tracking error due to high frequency and repeatabledisturbance, by individually controlling each harmoniccomponents of the system output.

In time-frequency based analysis techniques, thetime and frequency domain information of signal canbe preserved simultaneously. Wigner distribution basedtime-frequency analysis is used to design adaptive Q-filtersfor ILC as discussed in Rotariu et al. (2003) andBristow and Alleyne (2005). In time-frequency analysis,wavelet transform plays a vital role, which maps thesignal information into a two-dimensional array ofcoefficients, thus preserves the time and frequency domaininformation. Zhang et al. (2005) used discrete waveletpacket decomposition technique for cutoff frequencytuning in ILC scheme. Tzeng et al. (2005) proposedDiscreteWavelet Transform (DWT) based decompositiontechnique for the learning control design along withcontraction mapping theory based convergence issues.

In this paper we consider the wavelet series expansionapproach for ILC design by which the control signalcan be decomposed into basis functions in approximationand detailed subspaces. Here, we use a system functionapproximation methodology by utilising the universalapproximation (Zhang and Benveniste, 1992; Zhanget al., 1995) along with decompositions using wavelets.Properties such as Multiresolution Analysis (MRA) andorthogonality properties (Burrus et al., 1998) associatedwith wavelets are also exploited. In the proposed waveletseries based ILC algorithm, orthonormal wavelet basisfunctions are utilised to decompose the desired, actualtrajectories and the actual control input into respectiveapproximate wavelet coefficients. Then the learning ruleis implemented in updating these approximate controlinput coefficients of the learning controller. Convergenceand stability analysis of the proposed method ensuresthe convergence of approximate wavelet coefficients ofactual trajectory to the respective coefficients of desiredtrajectory. Thus achieves the tracking control in timedomain. Numerical experimentation studies show theability of the learning controller designed in waveletdomain in the presence of uncertainties.

Organisation of the paper is as follows. Section 2presents the problem statement and Section 3 briefs thedynamics of the robot manipulator including actuatordynamics. In Section 4, basics of wavelet series expansion

Wavelet series based iterative learning controller design for industrial robot manipulators 63

and its orthonormal properties are briefly discussed.Detailed design procedure of wavelet series based iterativelearning controller with stability and convergence issuesare also discussed. Efficacy of the learning algorithmhas been numerically experimented on single link robotmanipulator system are discussed in Section 5.

2 Problem statement

Consider a class of nonlinear time varying systemsdescribed in general as,

xk(t) = f(xk(t), t) + B(t)uk(t)(1)

yk(t) = C(t)xk(t)

where the index k corresponds to the kth iteration ofthe system; xk(t) ∈ n×1, uk(t) ∈ m×1 and yk(t) ∈ r×1

are the states, control input and output of the systemrespectively for t ∈ [0, T ] and f(x(t), t) : n × [0, T ] → n

is a non-linear continuous function of x(t) and t. Also thefunction f(.) satisfies the Lipschitz condition,

‖f(x1(t), t) − f(x2(t), t)‖ ≤ Lf‖x1(t) − x2(t)‖,(2)∀t ∈ [0, T ], ∀x1, x2 ⊆ n

where 0 < Lf < ∞ is the Lipschitz constant. Let thedesired trajectory yd(t) be continuously differentiable onthe finite time interval t ∈ [0, T ], such that for boundeddesired trajectory y(t), there exists a bounded input ud(t)for t ∈ [0, T ], for which system has unique boundedstates xd(t) and yd(t) = C(t)xd(t) for t ∈ [0, T ]. Thetrajectory-tracking problem can be stated as follows:For a desired trajectory yd(t), find a sequence of piecewisecontinuous control inputu(t) such that the system’s outputy(t) converges toyd(t). Forpractical systemsonlyboundedconvergence is expected. In the context of ILC, the trackingproblem is solved by modifying the control input uk(t)in each iteration (k) such that the control input uk(t)converges to the desired control input ud(t). In this paper,following norm definitions are used for a function f(t)stated as,

‖f(t)‖s = supt∈[0,T ]

e−st‖f(t)‖ (3)

‖f(t)‖∞ = supt∈[0,T ]

‖f(t)‖. (4)

3 Robot manipulator dynamics including theactuator model

This section presents a brief description about the statespace model of the manipulator system along with itsactuator dynamics used in tracking control problem. Thisstate space model is used to validate the proposed waveletseries based learning controller, later used in simulationstudies. The nonlinear dynamics of a serial link robot

manipulator having r-degrees of freedom can be describedby standard form as,

M(q(t))q(t) + C(q(t), q(t))q(t) + F (q(t)) + G(q(t))= uτ (t) (5)

where, M(q) is a (r × r) inertia matrix (bounded,symmetric and positive definite),C(q, q) is a (r × r)matrixcontaining the centripetal and coriolis terms,F (q) is a r × 1vector containing friction terms, G(q) is a (r × 1) vectorcontaining gravity terms, q(t) is a (r × 1) joint variablevector. The torque input vector is denoted by τ(t). Definethe desired trajectory vector qd(t) for the manipulatorjoints as, qd(t) = [q1d(t), q2d(t), . . . , qrd(t)]T for t ∈ [0, T ].The joints of the robot manipulator are driven by dcmotor and we seek to develop a dynamic model ofthe manipulator including dc motors, the actuators forthe robot manipulator. The combined model for all thearmature controlled dc motors is given by,

Ri(t) + Ldi(t)dt

+ eb = uv(t) (6)

where, uv(t) = [v1(t), v2(t), . . . , vr(t)]T , i(t) = [i1(t),i2(t), . . . , ir(t)]T are (r × 1) vectors corresponding toarmature input voltages and currents of the actuatorsfor the manipulator joints respectively; R and L are thediagonal (r × r) resistance and inductance matrices ofthe armature circuit. The relationship between actuatorsposition vector qm(t) and robot joint position vector q(t) isgiven as, qm(t) = Ntq(t). Back emf (eb) vector interms ofactuators velocity vector qm(t) is givenby, eb = Kbqm(t) =NtKbq(t) = Kbnq(t). The useful relation between theactuators torque vector uτ (t) and the armature currentvector is uτ (t) = NtKti(t) = Ktn(t), where, Nt being(r × r) turns ratio matrix of the robot joints (Nt > 0), Kb

and Kt are the (r × r) matrices corresponding to the backemf and torque constants of the actuators. It is known thatthis model captures the high frequency effects due to theelectrical factors like actuator resistance and inductancein case of high-speed operations. The dynamics of themanipulator including the actuator can be written as,

x1 = x2

x2 = M(x1)−1[−B(x1, x2) + Ktnx3] (7)

x3 = L−1[−Kbnx2 − Rx3 + uv]

with B(x1, x2) = C(x1, x2)x2 + F (x2) + G(x1). Herex1(t), x2(t) and x3(t) represents the state variablesrepresenting the angular position, velocity of thejoints and armature currents of actuators, i.e., x1(t) =[q1(t), q2(t), . . . , qr(t)]T ; x2(t) = [q1(t), q2(t), . . . , qr(t)]T ;x3(t) = [i1(t), i2(t), . . . , ir(t)]T . The control voltageinput v(t) to the system given in equation (7) can berepresented as,

v(t) = fv(q(t), q(t), i(t)) (8)

where, fv(q(t), q(t), i(t)) represents the relationshipbetween the voltage input to the actuator and the states of


the robot model. The objective of the proposed learningcontroller is to find the appropriate voltage v(t) to track thedesired trajectory qd(t). The voltage input function givenby equation (8), is in general an unknown nonlinear andcomplicated function. As iterations progress, the proposedwavelet series based learning controller searches for theappropriate voltage input sequence to attain the desiredvoltage input v∗(t), such that the tracking control of robotis achieved.

4 Wavelet series based learning controller design

This section begins with a brief description ofmultiresolution analysis and approximation of trajectoriesusing wavelet series. Then the learning controller designin wavelet domain using wavelet basis functions, stabilityand convergence issues are investigated.

4.1 MultiResolution Analysis (MRA)

Multiresolution formulation is designed to representsignals where a single event is decomposed into finerand finer detail. But it turns out also to be valuable inrepresenting signals where a time-frequency or time-scaledescription is desired even if no concept or resolutionis needed. Use of Fourier Transform (FT) techniquesmay lead to the loss of time domain information infrequency domain. But in wavelet transform the time andfrequency domain information of the signal are preservedsimultaneously. Consider the wavelet function ψ(t) calledas a mother wavelet function, whose binary dilations(1/2m) and dyadic translations (n/2m) generate the familyof functions ψm,n(t), which represents orthonormalwavelet basis of L2(R). The basis functions are defined as,

ψm,n(t) = 2m/2ψ(2mt − n). (9)

This wavelet basis equation (9) induces an orthogonaldecomposition of L2(R) as,

L2(R) = ⊕m

Wm (10)

where, Wm is a subspace spanned by 2m/2ψ(2mt −n)n=+∞

n=−∞. The wavelet ψ(t) is often generated from acompanion ϕ(t), known as the scaling function, throughthe following dilation equations:

ϕ(t) =√

2∑

k

hkϕ(2t − k) (11)

ψ(t) =√

2∑

k

gkϕ(2t − k) (12)

where, the coefficient sequences hk and gk represent apair of discrete quadrature-mirror filters (low-pass andhigh-pass) (Burrus et al., 1998), that are related through,

gk = (−1)k−1h−(k−1). (13)

The dilation and translations of the scaling function induceaMultiResolution Analysis (MRA) ofL2(R), i.e., a nestedchain of subspaces,

· · · ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 · · · ⊂ L2 (14)

such that,⋂m

Vm = 0, closure ⋃

m

Vm

= L2(R) (15)

where, Vm is the subspace spanned by 2m/2ϕ(2mt − n)n=+∞

n=−∞ or ϕm,n(t)n=+∞n=−∞. Thus the space

that contains higher resolution signals will contain thoseof lower resolution also. In addition, Vm and Wm arerelated by,

Vm+1 = Vm ⊕ Wm (16)

which provides a stable orthogonal decompositionof approximation subspace Vm+1 into approximationsubspace Vm and detail subspace Wm.

4.2 Wavelet series approximation

Definition 1: A function ψ ∈ L2(R) is called anorthonormalwavelet (or o.n. wavelet) if the family ψm,n,as defined in equation (9), is an orthonormal basis ofL2(R); that is 〈ψm,n, ψp,q〉 = δm,p · δn,q ,m, n, p, q ∈ Z andevery f(t) ∈ L2(R) can be written as,

f(t) =∞∑

m,n=−∞dm,nψm,n(t). (17)

The above wavelet series decomposition in L2 can be alsorewritten by using the set of scaling basis function and thefamily of wavelet basis function as,

f(t) =∞∑

n=−∞cM,nϕM,n(t) +

∞∑m=M

∞∑n=−∞

dm,nψm,n(t).

(18)

The coefficients cM,n and dm,n can be computed by thefollowing inner products,

cM,n = 〈f(t), ϕM,n〉 =∫ ∞

−∞f(t)ϕM,n(t)dt (19)

dM,n = 〈f(t), ψM,n〉 =∫ ∞

−∞f(t)ψM,n(t)dt. (20)

Applying equations (11) and (12) in equations (19) and (20)we get the following equations,

cM,n =∑m

h(m − 2n)cM+1,m (21)

dM,n =∑m

g(m − 2n)cM+1,m. (22)

(Appendix gives a brief explanation on equations (21)and (22) and in Burrus et al., 1998).


Thus any f(t) ∈ L2(R) canbe approximated arbitrarilyclosely in VM for some integer M . i.e., for any ε > 0, thereexists an M sufficiently large such that,∥∥∥∥∥f(t) −

∞∑n=−∞

cM,nϕM,n(t)

∥∥∥∥∥2

< ε. (23)

In equation (23), the approximation error (ε)consisting of signal components in detail subspacesWM , WM+1, . . . , W∞. It is negligible when M is large andis captured by ε.

4.3 Learning controller design

The signal f(t) is represented in approximationsubspace VM using coefficients cM,n being computed byequations (19) and (21). This low resolution representation(based on cM,n) satisfies the signal approximationcondition of equation (23). The following assumptions arefeasible.

• the system properties like internal dynamics arestable

• the desired trajectory for which the system has totrack is of finite time interval.

Subject to the above assumptions and by waveletseries analysis, the desired trajectory qd(t) for t ∈ [0, T ]can be decomposed into low resolution or low-frequency(approximations) components for a given set of Mand N as,

qd(t) N∑

n=0

cdM,nϕM,n(t) = CT

d Wϕ(t) (24)

where, the orthogonal basis vector Wϕ(t) = [ϕM,0(t),ϕM,1(t), . . . , ϕM,N (t)]. The corresponding coefficientsreferred as approximation coefficients for desiredtrajectory qd(t) are represented by, CT

d = [cdM,0, c

dM,1, . . . ,

cdM,N ]T . Similarly, the actual trajectory at kth iteration

qk(t) can be approximated as,

qk(t) N∑

n=0

ckM,nϕM,n(t) = CT

k Wϕ(t) (25)

where,CTk = [ck

M,0, ckM,1, . . . , c

kM,N ]T represents the vector

of approximation coefficients for actual trajectory qk(t).The approximation coefficients for desired and actualtrajectories are computed as per the inner productdefinition as,

cdM,n =

∫ T

0qd(t)ϕM,ndt,

(26)

ckM,n =

∫ T

0qk(t)ϕM,ndt, 0 ≤ t ≤ T.

Also note that,

cdM,n =

∑m

h(m − 2n)cdM+1,m;

(27)ckM,n =

∑m

h(m − 2n)ckM+1,m.

We consider the desired and actual trajectories at the finestresolution level (M + 1), i.e., qd(t) ∈ VM+1 and qk(t) ∈VM+1, where level (M + 1) is chosen such that entire signalenergy is contained in VM+1. Thus the approximationcoefficients corresponding to (M + 1)th level are thediscrete form of desired and actual trajectories. Thus wewrite,

cdM+1,m = qd(m∆T ) = qd[m];

(28)ckM+1,m = qk(m∆T ) = qk[m]

for m = 1, 2, . . . , K with K = (T/∆T ) where, ∆T is thesampling period.

At the kth iteration, tracking error is defined as,

ek(t) = qd(t) − qk(t). (29)

Apply the wavelet series expansion to error equation (29)we get,

ek(t) N∑

n=0

ekM,nϕM,n(t) = ET

k Wϕ(t) (30)

where ETk = [ek

M,0, ekM,1, . . . , e

kM,N ]T , denotes the vector

of approximation coefficients of error signal at kthiteration.

As per definition (1) the basis functions form anorthonormal set, and the approximation coefficients ofthe desired trajectory qd(t) are constants and have norelation with iteration number k. If all the coefficientsof ek(t) in equation (30) converge to zero, then theoutput tracking error in time domain will tend tozero, assuming that M is large enough so that allfunctions of interest can be represented exactly in VM .In order to increase the robustness of the closed loopsystem, a conventional PD-feedback controller is includedalong with the learning controller to deal with randomdisturbances. The PD controller is updated at every timeinstant but the learning controller will be updated aftereach iteration. At kth iteration the controller has thefollowing structure, which includes both the feedback andthe learning controller as,

vk(t) = vkfb(t) + vk

ff (t) (31)

where, vkfb = Kp(qd − qk) + Kd(qd − qk) denotes the

conventional proportional Plus Derivative (PD)controller with the feedback gains Kp and Kd forposition and velocity error components and the termvk

ff (t) is the estimation of desired feed-forward learningcontrol input v∗(t) at kth iteration. Applying the waveletseries approximation to the controller structure as inequation (31) we get,

N∑n=0

xkM,nϕM,n(t) =

N∑n=0

ykM,nϕM,n(t) +

N∑n=0

xkM,nϕM,n(t).

(32)


Here, xkM+1,n are the approximation coefficients of vk(t)

of equation (32). Further xkM+1,n can also be represented

as (using equations (8) and (19)),

xkM+1,n =

∫ T

0vk(t)ϕM+1,n(t)dt

=∫ T

0fvk

(qk(t), qk(t), ik(t))ϕM+1,n(t)dt

× 0 ≤ t ≤ T. (33)

Similarly the approximation coefficients ykM+1,n and

xkM+1,n in equation (32), represent the control inputs to the

system from the PD and learning controllers respectively.These are determined (similar to equation (28)) as,

ykM+1,n =

∫ T

0vk

fbϕM+1,n(t)dt = vkfb(n∆T ) (34a)

xkM+1,n =

∫ T

0vk

ffϕM+1,n(t)dt = vkff (n∆T ). (34b)

Using equation (27),

ykM,n =

∑m

h(m − 2n)ykM+1,m

=∑m

h(m−2n)vkfb[m], for m=1, 2, . . . , K (35a)

xkM,n =

∑m

h(m − 2n)xkM+1,m

=∑m

h(m−2n)vkff [m], for m=1, 2, . . . , K. (35b)

Equation (32) can be written in compact form using onlythe approximation coefficients as,

xkM,n = yk

M,n + xkM,n for n = 0, 1, 2, . . . , N. (36)

As iterations progress, in time domain the learningcontrol input vk

ff (t) will approach the desired controlinput v∗(t) and the feedback control input vk

fb(t) willapproach to zero. Equivalently, as the iterations progress,the approximation coefficients corresponding to learningcontrol input vk

ff (t) approach the same values as theapproximation coefficients xk

M,n to the coefficients ofdesired values implies the asymptotic convergence of thetracking error ek(t) to zero as k → ∞. The learningcontroller approximation coefficients are updated by usingthe gradient descent type learning law (Gopinath andKar,2004) in wavelet domain as,

xk+1M,n = xk

M,n + ΓykM,n. (37)

In equation (37), the learning gain value (Γ) can be selectedaccording to the sufficient conditions discussed in theSub-section 4.4.

Remarks: Since the ILC scheme is designed to learn thesystem dynamics by the iterative sense, the system must

satisfy the property of repeatable. In general, the systemdynamics must possess invariant dynamics and the systemdisturbances must be repeatable during the each iteration.In practice the assumptions over such repetitiveness aredifficult to satisfy, in such situations the tracking errorsignals obtained in iterations may not be repeatable, i.e.,certain non-repetitive, higher frequency components maybe present in it. Thus, the wavelet series approximation oftracking error as in equation (30) is valid, which containsonly low frequency approximation components and freefrom high frequency detailed components correspond tonon-repeatable components.

4.4 Convergence and stability issues

Let us define the difference operator as,

∆xkM,n = xk+1

M,n − xkM,n (38)

which represents the change in control signal betweentwo successive iterations in the form of approximationcoefficients. Similarly the tracking error change betweeniterations can be defined as,

∆ykM,n = yk+1

M,n − ykM,n. (39)

In time domain based ILC algorithms the followingconvergence condition has been stated and used by manyresearchers (Arimoto et al., 1984; Bondi et al., 1988;Atkeson and McIntyre, 1986; Kuc et al., 1991; Jang et al.,1995;Moore, 1998; Gopinath andKar, 2004; Xu and Tan,2003; Tayebi, 2004):

‖ek(t)‖s < Γ‖ek−1(t)‖s for 0 < Γ < 1 (40)

where the tracking error at kth iteration is defined inequation (29). Equation (40) implies that the error atkth iteration depends on the learning gain (Γ), whichcontrols the convergence rate of the learning algorithm.The following lemma can be stated for the purposeof convergence and stability issues of the learning lawdesigned using in wavelet domain.

Lemma: When the controller defined in equation (31) isapplied to the manipulator system in equation (7) and thelearning controller coefficients (xk

M,n) updated with thelearning law in equation (37), the sufficient conditions forthe convergence of the tracking error are (0 < Γ < 1) and∣∣∣∆xk

M,n

∆ykM,n

∣∣∣ < (1 − Γ) for all k.

Proof: Applying difference operator to equation (36)yields,

∆xkM,n = ∆yk

M,n + ∆xkM,n. (41)

The second term in RHS of equation (41) can be expressedusing the learning law equation (37),

∆xkM,n = xk+1

M,n − xkM,n = Γyk

M,n. (42)

Hence equation (41) becomes,

∆xkM,n = ∆yk

M,n + ΓykM,n (43)


(or)

∆xkM,n

∆ykM,n

= 1 + Γyk

M,n

∆ykM,n

. (44)

Using equation (39),

∆xkM,n

∆ykM,n

− 1 =Γyk

M,n(yk+1

M,n

ykM,n

− 1

) (45)

(or)

yk+1M,n

ykM,n

=Γ(

∆xkM,n

∆ykM,n

− 1

) + 1. (46)

Equation (46) implies that if the learning gain values

are selected as (0 < Γ < 1) and if∣∣∣∆xk

M,n

∆ykM,n

∣∣∣ < (1 − Γ)

then we get,∣∣∣yk+1

M,n

ykM,n

∣∣∣ < c < 1. This means the sequence of

approximation coefficients of the feedback control vkfb(t)

with iterations (k) can be written as,

|ykM,n| < c|yk−1

M,n| < c2|yk−2M,n| < · · · < ck|y0

M,n|. (47)

In the above sequence, for the bounded value of |y0M,n|,

the feedback controller coefficients will asymptoticallyconverges to zero as iterations (k) progress. Thus,theoretically in time domain we can define the final valueof feedback controller input to the system as,

vkfb = Kp(ek(t)) + Kd(ek(t)) = 0, for t ∈ [0, Ti]. (48)

The values of the feedback controller gains Kp and Kd arechosen in such a way that the system dynamics as definedin equation (8) will be stable. Hence, as the iterations (k)progress the tracking error will asymptotically converge tozero, i.e., limk→∞ ek(t) = 0.

4.5 Implementation procedure

The ILC scheme developed in Subsection 4.3 is applicableto the tracking control of general nonlinear system (1).However, here a step-by-step procedure has been describedto implement the proposed wavelet series based ILCalgorithm for the tracking control of robot manipulatorsystem.

Step (a): The desired trajectory of ith robot joint qdi (t),

t ∈ [0, Ti], has been sampled into Ki samples such that,qdi [m], m = 1, 2, . . . , Ki where Ki = (Ti/∆Ti) and ∆Ti isthe sampling period. Choose the learning gain (Γ) value as(0 < Γ < 1) and set the error index limiting value ε to asuitable small value.

Step (b): Choose the proper orthogonal wavelet basisfunctions say, Daubechies’ or Lemarie-Meyer’s wavelets.A proper level of decomposition (J) has to be chosen

such that the frequency band for approximation subspacecontains all useful and learnable information. Largerlevels of decompositions, may lead to the loss of usefulinformation now available in the detailed subspaces andthus it should be avoided. Detailed analysis of selection ofproper decomposition level (J) is presented in Section 5.

Step(c): Set the feedbackcontroller gainsKp andKd to getstable system. Set the learning iteration counter k = 1 andinitial approximation coefficients for learning controller tozero, i.e., xk

M,n = 0 for all n.

Step (d): By setting the index of sampling points m = 1,and the feedback controller wavelet coefficients yk

M,n = 0.

Step (e): Calculate the learning controller input vkff [m] =∑N

n=0 xkM,nϕM,n[m]. The feedback controller input at the

sampling instant m can be calculated as,

vkfb[m] = Kp(qd

i [m] − qki [m]) + Kd(qd

i [m] − qki [m]).

Total control input voltage to the actuator can becalculated as, vk[m] = vk

fb[m] + vkff [m].

Step (f): Increase the index m by 1 i.e., m → m + 1 andupdate the feedback controller wavelet coefficients as,yk

M,n → ykM,n + vk

ifb[m]h[m − 2n].

Step (g): Check ifm = Ki, then go to Step (d), else proceedto Step (h).

Step (h): Checkwhether thePerformance Index (I) definedin (2.49) at the kth iteration is less than ε. If yes, then goto Step (i). Else increase the iteration count k by 1 i.e.,k → k + 1 and calculate the learning controller waveletcoefficients for the next iteration by using the learning rule,xk+1

M,n = xkM,n + Γyk

M,n. Then proceed to Step (d).

Step (i): Learning iterations stop.

5 Simulation results

For a class of time varying nonlinear systems awavelet series based ILC approach is proposed inSection 4. In this section a single link robot manipulatorsystem is considered to validate the proposed approach.However during simulation, the manipulator systemis considered along with the time varying tip-mass(M), which represents a special case of time varyingsystem. Issues related to the implementation of proposedapproach and error convergence with iterations fortrajectory-tracking problem of manipulator system arediscussed. Performance of proposed ILC scheme iscompared with time domain PD-type ILC and Fourierseries ILC approaches.

5.1 Single-link manipulator model

In this study, we present an example of trajectory-trackingproblem of a single link manipulator system with a DC


motor actuator is shown in Figure 2. The dynamics of themanipulator system is described as,

Iq + vq +( 1

2m + M)gl sin q = u (49)

where, q is the angular displacement of the manipulator,u is the driving torque and I is moment of inertiadefined as I = Ml2 + 1

3ml2. The model parameters ofrobot joint are the mass (m) = 1 kg, damping coefficient(v) = 3 kgm2/s, length (l) = 0.5mand the tipmass (M) =2 kg. The parameters of actuator are, back emf constant =(Kb) = 1V/(rad/s), torque constant (Kt) = 1NmA−1,turns ratio (Nt) = 4 with armature resistance (R) = 1Ωand inductance (L) = 0.01H. These parameter valuesare chosen for the simulation purpose only. Howeverknowledge of these parameters is not needed for thecomputation of control. The learning controller has beenimplemented for the robot system with a conventionalPD-feedback controller having the gains Kp = 5 andKd = 10.

Figure 2 Single link manipulator model

Simulations are performed with the sampling time∆T = 0.01 s and the learning gain Γ is selected as 0.9.The performance of the learning controller was evaluatedusing an performance index I is defined as,

I =

√1T

∫(ei(t))2dt ≈

√√√√ 1K

K∑j=1

(ei[j])2

for K = T/∆T, (50)

where, the tracking error ei(t) is defined as the differencebetween desired and actual angular displacements of theith robot joint, i.e., ei(t) = qdi(t) − qi(t). DiscreteWaveletTransform (DWT) technique is used in the proposedlearning method. Simulation works are performed inMatlab-7.0 platform with the help of wavelet toolbox 3.0(Wavelet Toolbox – User’s Guide, 2008). In this paper,we choose ‘SymN’ family wavelets with the features ofcompact support, orthogonal, Finite Impulse Response(FIR) and near symmetrical, where N is the order.For simulation studies ‘Sym3’ is selected as the motherwavelet with the proper value for the decompositionlevel (J). The frequency band of basis functions ofapproximation subspace at Jth level of decompositionis, ∆fa = [0 2−(J+1)fs] Hz, where fs = 1/∆T being thesampling frequency. Similarly, for the basis functions

of detailed subspace at Jth level of decomposition is∆fd = [2−(J+1)fs 2−(J)fs] Hz (Vaidyanathan, 1993).

In this study three different trajectories are consideredto evaluate the performance of the proposed learningcontroller. As a first example, we consider the followingtrajectory:

qd(t) = sin(t) + cos(3t) for 0 ≤ t ≤ 10 s. (51)

The initial conditions on the joint variables and theirderivatives are as same as desired represented by,qk(0) = qd(0); qk(0) = qd(0) ∀k ≥ 1. The tip mass (M)of the robot system is varied to 3.5kg between 2.5 ≤t ≤ 3.5 s and varied to 3kg between 5.5 ≤ t ≤ 7.5 s.The decomposition level (J) is to be selected fordecomposing the feedback control signal, such that thehigh frequency non-repeatable components can be filteredout. Numerical simulation is performed to explain theeffect of decomposition level (J) on the error convergencewith the iterations as shown in Figure 3.

Figure 3 Effect of decomposition level (J) on errorconvergence (see online version for colours)

Figure 3 shows that the value of J , between 2 to 4leads to the lower convergence error as the iterationsprogress. For the level J > 4, the error convergence is notsatisfactory and even results in divergence, due to lose ofthe signal information in the form of detailed coefficientslying in high frequency bands. In this case, we chooseJ = 2, so that the approximationwavelet coefficients lies inthe sub-band [0 fs/8]Hz. Figure 4 shows the improvementin tracking performance using the proposed learningcontroller as iterations progress. In this study, we comparethe performance of proposed learning controller with thetime domain based PD-type ILC with a zero phase lowpass filter (Chen et al., 2004) and with Fourier seriesbased ILC (Chen and Moore, 2001). In PD-type ILC,the cut off frequency of the low pass filter is chosenas fs/8Hz.

From Figure 5, it can be observed that the learningcontroller based on wavelet series approach performs bestand converges to lowest tracking error in comparisonwith the time domain PD-type and Fourier domain ILC


approaches.To show the effect of learninggain (Γ)over theerror convergence, a simulation study was performed fordifferent values of learning gain (Γ) as shown in Figure 6.The performance of the learning and the error convergencewith the iterations gets improved with the increasing valueof learning gain.

Figure 4 Tracking performance of SDOF system forTrajectory-1 (see online version for colours)

Figure 5 Error convergence with iterations for Trajectory-1(see online version for colours)

As a second testing, the wavelet series based learningcontroller performancewas evaluated for a non-stationarytrajectory-tracking problem. The desired non-stationarytrajectory is defined as,

qd(t) =

π

3cos(4πt/3) for 0 < t ≤ 3 s

π

3cos(8πt/3) for 3 ≤ t ≤ 6 s

π

3cos(4πt/3) for 6 ≤ t ≤ 10 s

. (52)

The number of decomposition level (J) is chosen as 2 inthis case so that the signal information in approximationwavelet coefficients lies in the frequency sub-band

[0 fs/8]Hz. Figure 7 shows the tracking performance ofthe system by implementing the proposed wavelet seriesbased learning controller. The tracking error comparisonfor PD type ILC and Fourier and the proposed ILCapproach is shown in Figure 8. In this study, we choosetwo different values of decomposition levels (J = 2and 3), where the choice J = 2, gives comparatively betterperformance than other ILC approaches. In the proposedapproach, if we choose higher value of decomposition levelsay J = 3, the performance of time domain PD-type ILCwith low pass filter is better than the proposed approach.This may be due to the loss of useful information inapproximation coefficients (whichmay be now transferredto detailed coefficients) as decomposition level increases.

Figure 6 Effect of learning gain (Γ) over error convergence(see online version for colours)


As a final example, we consider the trajectory-3 asshown in Figure 9, for testing the performance of theproposed controller. It shows the actual position andvelocity tracking performance of the manipulator systemafter 50 learning iterations. Figure 10 shows the trackingerror convergence with iterations. In this case, we choosethe decomposition levels J = 2, which gives performancesimilar to that for PD-type ILC with low pass filter.


The cutoff frequency of the low pass filter is chosen sameas the cutoff frequency of approximation subspace.




5.2 Analysis and discussion

This section discusses the advantages and the limitationsof the proposed method. Issues related to the choice ofdecomposition levels (J) and dependency with the systembandwidth is also analysed.

5.2.1 Advantages of the proposed method

The proposed method uses only the approximatewavelet coefficients for learning process, thus avoidsthe non-repeatable system dynamics (high frequencycomponents). The proposed learning controller is appliedto the robot manipulator model by taking care of theactuator dynamics effects (electrical effects), which occurin very high-speed operations. This shows the efficacyof proposed method in practical applications. With theproposed method, the tracking error converges to lowervalue in comparison with the Fourier series and timedomain based ILCapproaches as shown inFigures 5 and 8.

5.2.2 Choice of decomposition level (J)

The choice of a proper level of decomposition level(J) is very important in the proposed wavelet seriesbased ILC approach. This approach uses only theapproximate wavelet coefficients for learning process.Larger decomposition levels, this may lead to the loss ofuseful signal information in thedetailed coefficients (higherfrequency bands). In such situation, proposed controllermay not provide performance better than time domainbased ILC methods as shown in Figure 10. The choice ofJ should avoid this kind information loss in the form ofdetailed coefficients.

5.2.3 Dependency on system bandwidth

Selection of a suitable decomposition level (J) alsodepends on system bandwidth (fb). Hence, the waveletseries based learning controller must preserve the originalsignal information in sub-band [0, fb], for maintainingthe system performance. The selection of desireddecomposition level (Jdesired) should be in such away that,the frequency band of the approximation coefficients mustaccommodate the systembandwidth (fb) information. Theproposed learning controller may not achieve the desiredoutput performance, if the decomposition level (J) exceedssuch desired value (Jdesired).

6 Conclusion

In this paper a learning controller design using waveletseries approach is proposed. Convergence analysis andimplementation issues of the proposed learning algorithmare presented.Theproposedmethod effectively reduces thetracking error as the iterations progress. The efficacy ofthe learning algorithm has been verified through detailedsimulation studies on a single link robot manipulatormodel. Through numerical simulations, for a wide class


of desired trajectories, the proposed wavelet series basedlearning controller shows its better performance overtime domain PD-type and Fourier series based ILCapproaches. The proposedmethod could be limited to suchapplications, where the useful information of trajectoriesnot lies in high frequency band.

Acknowledgements

The authors would like to express their thanks to theanonymous reviewers, and in particular, Editor-in-Chief,Dr. M.A. Dorgham, for their support and valuablecomments.

References

Arimoto, S. (1996) Control Theory of Nonlinear MechanicalSystems – A Passivity-based and Circuit-theoreticApproach, Clarendon Press, Oxford.

Arimoto, S., Kawamura, S. and Miyazaki, F. (1984) ‘Betteringoperation of robots by learning’, Journal of RoboticSystems, Vol. 1, No. 2, pp.123–140.

Atkeson, C.G. andMcIntyre, J. (1986) ‘Robot trajectory learningthrough practice’, Proceedings of the IEEE InternationalConference on Robotics and Automation, San Francisco,CA, pp.1737–1742.

Bondi, P., Casalino, G. and Gambaradella, L. (1988) ‘On theiterative learning control theory for robotic manipulator’,IEEE Transactions on Robotics and Automation, Vol. 4,No. 1, pp.14–22.

Bristow, D.A. and Alleyne, A.G. (2005) ‘BIBO stability ofan adaptive time-frequency iterative learning control withapplication to microscale robotic deposition’, Proceedingsof the American Control Conference, pp.4518–4524.

Burrus, C.S., Gopinath, R.A. andGuo, H. (1998) Introduction toWavelets and Wavelet Transforms-A Primer, Prentice-Hall,New Jersey.

Chen, Y-Q. and Moore, K.L. (2001) ‘Frequency domainadaptive learning feed-forward control’, IEEE Symposiumon Computer Intelligence in Robotics and Automation,pp.396–401.

Chen, Y-Q., Moore, K.L. and Bahl, V. (2004) ‘Learningfeedforward control using dilated B-spline network:frequency domain analysis and design’, IEEE Transactionon Neural Networks, Vol. 15, No. 2, pp.355–366.

Elci, H., Longman, R.W., Phan, M.Q., Juang, J.N. andUgoletti, R. (2002) ‘Simple learning control made practicalby zero phase filtering: applications to robotics’, IEEETransactions on Circuits and System-I: FundamentalTheory and Applications, Vol. 49, No. 6, pp.753–767.

Gopinath, S. and Kar, I.N. (2004) ‘Iterative learning controlscheme for manipulators including actuator dynamics’,Mechanism and Machine Theory, Vol. 39, pp.1367–1384.

Gorinevsky, D., Torfs, D.E. and Goldenberg, A.a. (1997)‘Learning approximation of feedforward controldependence on the task parameters with application todirect-drive manipulator tracking’, IEEE Transactions onRobotics and Automation, Vol. 13, pp.567–581.

Jang, T.J., Choi, C.H. and Ahn, H.S. (1995) ‘Iterative learningcontrol in feedback systems’, Automatica, Vol. 31, No. 2,pp.243–248.

Kavli, T. (1992) ‘Frequency domain synthesis of trajectorylearning controllers for robot manipulators’, Journal ofRobotic Systems, Vol. 9, pp.663–680.

Kuc, T.Y., Nam, K. and Lee, J.S. (1991) ‘An iterative learningcontrol of robotic manipulators’, IEEE Transactions onRobotics and Automation, Vol. 7, pp.835–841.

Lee, J.W., Lee,H-S. andBien, Z. (1993) ‘Iterative learning controlwith feedback using Fourier series with application to robottrajectory tracking’, Robotica, Vol. 11, pp.291–298.

Luca, A.D., Paesano, G. andUlivi, G. (1992) ‘Frequency domainapproach to learning controller design – implementationfor robot manipulator’, IEEE Transactions on IndustrialElectronics, Vol. 39, No. 1, pp.1–10.

Moore, K.L. (1998) ‘Iterative learning control – an expositoryoverview’, Applied and Computational Controls, SignalProcessing and Circuits, Vol. 1, pp.425–488.

Rotariu, I., Ellenbroek, R. and Steinbuch, M. (2003)‘Time-frequency analysis of a motion system with learningcontrol’, Proceedings of the American Control Conference,pp.3650–3654.

Tan, K.K., Dou, H., Chen, Y., Lee, T.H. (2001) ‘Highprecision linear motor control via relay-tuning and iterativelearning based on zero-phase filtering’, IEEE Transactionson Circuits and System-I: Fundamental Theory andApplications, Vol. 9, No. 2, pp.244–253.

Tang, X., Cai, L. and Huang, W. (2000) ‘A learningcontroller for robot manipulators using Fourier series’,IEEE Transaction on Robotics and Automation, Vol. 16,No. 1, pp.36–45.

Tayebi, A. (2004) ‘Adaptive iterative learning control for robotmanipulators’, Automatica, Vol. 40, pp.1195–1203.

Tzeng, K-S., Tzeng, D.C. and Chen, J.S. (2005) ‘An enhancediterative learning control scheme using wavelet transform’,IEEETransactions on Industrial Electronics, Vol. 52, No. 3,pp.922–924.

Velthuis, W.J.R., de Vries, T.J.A., Schaak, P. and Gaal, E.W.(2000) ‘Stability analysis of learning feedforward control’,Automatica, Vol. 36, pp.1889–1895.

Zhang, B.,Wang,D. andYe,Y. (2005) ‘Wavelet transform-basedfrequency tuning ILC’, IEEE Transactions on Systems,Man, and Cybernetics-Part B: Cybernetics, Vol. 35, No. 1,pp.107–114.

Zhang, Q. and Benveniste, A. (1992) ‘Wavelet networks’,IEEE Transactions on Neural Networks, Vol. 3, No. 6,pp.889–898.

Zhang, J. Walter, G.G., Miao, Y. and Lee, W.N.W. (1995)‘Wavelet neural networks for function learning’, IEEETransactions on Signal Processing, Vol. 43, No. 6,pp.1485–1497.

Wavelet Toolbox – User’s Guide (2008) The MathWorks Inc.

Vaidyanathan, P.P. (1993) Multirate Systems and Filter Banks,Prentice-Hall, New York.

Xu, J.X. and Tan, Y. (2003) Linear and Nonlinear IterativeLearning Control, Series of Lecture Notes in Controland Information Sciences, Springer-Verlag, 291, Berlin,Germany.


Appendix

From equation (11),

ϕ(2M t − n) =√

2∑

k

h(k)ϕ(2M+1t − 2n − k). (A-1)

After changing the variable m = 2n + k, equation (A-1)becomes,

ϕ(2M t − n) =√

2∑m

h(m − 2n)ϕ(2M+1t − m). (A-2)

If we denote approximation subspace VM as,

VM = Spann

2M/2ϕ(2M t − n) = ϕM,n(t)n=+∞n=−∞ (A-3)

then

f(t) ∈ VM+1 ⇒ f(t) =∑

n

cM+1,nϕM+1,n(t) (A-4)

is expressible at a scale of (M + 1) with scaling functionsonly. At one scale lower resolution, wavelets are necessaryfor the detail not available at a scale of M . We have,

f(t) =∑

n

cM,nϕM,n(t) +∑

n

dM,nψM,n(t). (A-5)

The M level approximation coefficients are found bytaking the inner product,

cM,n = 〈f(t), ϕM,n(t)〉 =∫ ∞

−∞f(t)ϕM,n(t)dt

=∫ ∞

−∞f(t)2M/2ϕ(2M t − n)dt. (A-6)

By using equation (A-2) and interchanging the sum andintegral, equation (A-6) can be written as,

cM,n =∑m

h(m − 2n)

×∫ ∞

−∞f(t)2(M+1)/2ϕ(2(M+1)t − m)dt. (A-7)

In equation (A-8), the integral is the inner product with thescaling function at a scale of (M + 1) gives,

cM,n =∑m

h(m − 2n)cM+1,m. (A-8)

The corresponding relationship for the detailedcoefficients is,

dM,n =∑m

g(m − 2n)cM+1,m. (A-9)

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Wavelet series based iterative learning controller design for industrial robot manipulators