Integral Resonant Control for Vibration Damping and Precise Tip-Positioning of a Single-Link...

232 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 2, APRIL 2011

Integral Resonant Control for Vibration Dampingand Precise Tip-Positioning of a Single-Link

Flexible ManipulatorEmiliano Pereira, Member, IEEE, Sumeet S. Aphale, Member, IEEE, Vicente Feliu, Senior Member, IEEE,

and S. O. Reza Moheimani, Senior Member, IEEE

Abstract—In this paper, we propose a control design method forsingle-link flexible manipulators. The proposed technique is basedon the integral resonant control (IRC) scheme. The controller con-sists of two nested feedback loops. The inner loop controls thejoint angle and makes the system robust to joint friction. The outerloop, which is based on the IRC technique, damps the vibration andmakes the system robust to the unmodeled dynamics (spill-over)and resonance frequency variations due to changes in the payload.The objectives of this work are: 1) to demonstrate the advantages ofIRC, which is a high-performance controller design methodologyfor flexible structures with collocated actuator–sensor pairs and2) to illustrate its capability of achieving precise end-point (tip)positioning with effective vibration suppression when applied toa typical flexible manipulator. The theoretical formulation of theproposed control scheme, a detailed stability analysis and experi-mental results obtained on a flexible manipulator are presented.

Index Terms—Flexible manipulators, integral resonant control(IRC), precise tip-positioning, vibration damping.

I. INTRODUCTION

THE PAST two decades have seen significant interest inthe control of flexible robotic manipulators. Novel robotic

applications demand lighter robots that can be driven usingsmall amounts of energy. An example of this is robotic boomsin the aerospace industry, where lightweight robot manipulatorswith high-performance requirements (high-speed operation andbetter accuracy) are demanded [1]. Unfortunately, the flexibilityof these robotic arms leads to oscillatory behavior at the tip of thelink, making precise-pointing or tip-positioning a daunting taskthat requires complex closed-loop control. In order to addresscontrol objectives, such as tip-position accuracy and suppressionof residual vibration, many control techniques have been appliedto flexible robots [2]–[4].

There are two main problems that complicate the control de-sign for flexible manipulators, viz.: 1) the high order of the sys-

Manuscript received July 21, 2009; revised October 27, 2009; acceptedNovember 20, 2009. Date of publication February 8, 2010; date of current ver-sion January 19, 2011. Recommended by Technical Editor J. Ueda. This workwas supported by the Spanish Government Research Program under ProjectDPI2006-13834 (Ministerio de Ciencia y Tecnologıa).

E. Pereira and V. Feliu are with the Universidad de Castilla-La Mancha,Ciudad Real 13071, Spain (e-mail: [email protected]; [email protected]).

S. S. Aphale is with the Center for Applied Dynamics Research, Schoolof Engineering, University of Aberdeen, Aberdeen, AB24 3UE, U.K. (e-mail:[email protected]).

S. O. R. Moheimani is with the School of Electrical Engineering and Com-puter Science, University of Newcastle, Newcastle, N.S.W. 2308, Australia(e-mail: [email protected]).

Digital Object Identifier 10.1109/TMECH.2009.2039713

tem and 2) the nonminimum phase dynamics that exist betweenthe tip position and the input torque applied at the joint. Theseproblems have motivated researchers to employ a wide rangeof control techniques such as linear quadratic Gaussian [5],linear quadratic regulator [6], pole placement [7], and inversedynamics-based control [8]. The drawback of these model-basedapproaches is that the complexity of the control algorithm in-creases significantly with the system order, and the stability ofthe closed-loop system is sensitive to: 1) changes in the robotpayload; 2) model parameter uncertainties; and 3) high-orderunmodeled dynamics, respectively, since a wide-control band-width may lead to spill-over effect [9]. In order to address theseproblems, solutions based on adaptive control [10], H∞ con-trol [11], μ-synthesis [12], sliding-mode control [13], neuralnetworks [14], and fuzzy-logic algorithms [15] have been in-vestigated. However, the mathematical complexity and difficul-ties in their implementation make their application to flexiblerobotics less attractive.

The use of alternative outputs has emerged as a potential so-lution to the problem of nonminimum phase dynamics. In [16],an alternative output, the so-called reflected tip position, wasproposed. It was demonstrated therein that the transfer functionfrom the motor torque to the reflected tip velocity is passive.Therefore, strictly passive controllers make the system stablein L2 sense. However, the passive relationship depends on thevalue of the hub inertia, which must be sufficiently small inrelation with beam inertia. In [17], an additional control loopto make the passive relationship independent of the system pa-rameters was proposed. However, the main limitation of thesepassivity-based control schemes is that they make the systemvery sensitive to joint friction. Other solutions based on twocontrol loops can be found in [18]–[20]. The inner loop basedon a collocated I/O arrangement, is used to damp the vibrations,whereas the outer loop is used for position control. However,in [18] and [19], the stability of the position control depends onthe link and motor parameters, which complicates the design.Although the vibration damping proposed in [20] was quite effi-cient when a motor without reduction gear was used, the residualvibration suppression is not effective when reduction gears areemployed.

Recently, a novel control-design scheme based on two con-trol loops was proposed in [21] to make the system stabilityindependent of the link and motor parameters. The difference inthis control scheme when compared with the philosophy of thecontrol schemes proposed in [18]–[20] is that the outer loop is

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PEREIRA et al.: IRC FOR VIBRATION DAMPING AND PRECISE TIP-POSITIONING OF A SINGLE-LINK FLEXIBLE MANIPULATOR 233

Fig. 1. Experimental platform.

used to damp the vibrations and the inner loop is used for posi-tion control. The position controller only depends on the motordynamics, and the outer loop always guarantees an effectivevibration suppression (with or without the reduction gear). Inaddition, the stability analysis is simple and can be made inde-pendent of the link dynamics. However, the proposed outer loopcontroller has some disadvantages like high sensitivity at lowfrequencies that complicates its implementation on a practicalexperimental platform (offset in the sensors and gravity effect).It also suffers from instability issues due to antialiasing filters.

This study builds on the control methodology of [21] andproposes a new approach based on the integral resonant control(IRC) scheme [22]. This methodology consists of: 1) designinga joint position controller robust to Coulomb friction and pay-load variations, thus allowing accurate positioning of the joint;and 2) designing an outer controller that imparts substantialdamping to link dynamics. This combined scheme results in aneasy-to-tune, low-order damping controller that damps multi-ple resonant modes, without instability issues due to unmodeledsystem dynamics. This work substitutes the outer controller usedin [21] by a modification of an IRC scheme, which retains theadvantages of [21] and alleviates the aforementioned problems.

This paper is organized as follows. Section II provides adescription of the experimental setup. The system model andthe associated parameters are briefly explained in Section III.Section IV explains the proposed control scheme and alsodemonstrates the closed-loop stability. A practical implemen-tation of the proposed control scheme on a single-link flexiblerobot and the obtained experimental results are presented inSection V. This section also gives a detailed comparison of po-sitioning performance achieved by various previously proposedand popular control schemes. Section VI concludes the paper.

II. EXPERIMENTAL SETUP

Fig. 1 shows a photograph of the single-link flexible ma-nipulator used as the experimental platform in this work. Thesetup consists of 1) a dc motor (Maxon Motor EC-60) with aharmonic-drive reduction gear 1:50 (HFUC-32-50-20H); 2) aflexible single-link comprised of a slender aluminium beamwith a rectangular cross section that is attached to the motorhub in such a way that it rotates only in the horizontal plane,

TABLE IPARAMETERS OF THE FLEXIBLE ARM

Fig. 2. Parametric representation of the flexible arm system.

so that the effect of gravity can be ignored; and 3) a mass withnegligible inertia at the end of the arm. In addition, two sensorsare used: 1) an encoder mounted at the joint of the manipula-tor to measure the motor angle; and 2) a strain-gauge bridgeplaced at the base of the beam to measure the coupling torque.The physical parameters of the system are given in Table I.The strain signal is amplified by the dynamic strain amplifier(Kyowa DPM600) and filtered by a second-order Butterworthfilter with its cutoff frequency set to 300 Hz. A National Instru-ments 6024E and a Measurement Computing CIO-DIO24 areused in a PC in conjunction with the real-time windows targetof MATLAB. The sampling time is set to 0.002 s. Finally, adual-channel Agilent-35670A spectrum analyzer was used fordetermining the frequency response functions (FRFs).

III. SYSTEM MODEL

Consider a flexible arm (see Fig. 2) composed of: 1) a motorand a reduction gear of 1:n at the base with total inertia (ro-tor and hub) J0 , dynamic friction coefficient ν, and Coulombfriction torque Γf ; 2) a flexible beam with uniform linear massdensity ρ, uniform bending stiffness EI , and length L; and 3)a payload of mass MP and rotational inertia JP . Furthermore,the applied torque is Γm , w(x, t) is the elastic deflection mea-sured from the undeformed beam, θm is the joint angle, and θt

is the tip angle. Note that, parametric representation of Fig. 2corresponds to the so-called pseudo-clamped configuration, in


which the noninertial frame (x, y) rotates with the motor and theoverall structure rotates in an inertial frame (X , Y ). Using thevariational calculus as [23], the next boundary value problemcan be defined

EIw′′′′(x, t) + ρ(xθm (t) + w(x, t)) = 0 (1)

Γm (t) = nKV (t) = J0 θm (t) + νθm (t) + Γcoup(t) + Γf (t)

(2)

in which the (˙) and (′) indicates derivation with respects to tand x, V is a voltage that controls the motor, K is a constantthat relates the motor torque (Γm ) and the control voltage (V ),and Γcoup is the coupling torque in the joint due to the link andthe payload and can be obtained using the formula

Γcoup(t) = −EIw′′(0, t) (3)

where w′′(0, t) is proportional to the strain measured at the baseof the link. The boundary conditions are

w(0, t) = 0, w′(0, t) = 0

EIw′′(L, t) = −JP (θm (t) + w′(L, t))

EIw′′′(L, t) = MP (Lθm (t) + w(L, t)). (4)

The boundary problem is solved by using separability of thevariables and modal expansion. Thus, the elastic deflection isassumed as follows:

w(x, t) =∞∑

i=1

φi(x)δi(t) (5)

where φi and δi are the spatial and time solution of (1), respec-tively, for the i vibration mode. Thus, proceeding in the sameway as described in [24], the model between w(x, t) [or anyfunction of w(x, t)] and Γm can be obtained, if Γf is not con-sidered in model deduction. If w′′(x, t) is calculated from (5)and it is substituted into (3), Γcoup is as following:

Γcoup(t) = −EI

∞∑i=1

φ′′i (0)δi(t). (6)

It can be proved that φ′′i (0) �= 0∀i, which guarantees that the link

vibration is zero if and only if Γcoup(t) = 0. In addition, Γcoupis also the result of the reaction torque in the link due to the jointacceleration and link vibrations, which makes Γcoup(s)/sθm (s)a passive system [21]. If Γf is neglected in (2), the transfer func-tion between θm (s) and Γcoup(s) can be expressed as follows:

θm (s)Γm (s)

=(nK/J0)/(s(s + ν/J0))

1 + ((nK/J0)/(s(s + ν/J0)))(Γcoup(s)/θm (s))(7)

which is used to obtain the expression of Γcoup(s)/θm (s). Itmust be noted that the zeros of θm (s)/Γm (s) are the polesof ((nK/J0)/(s(s + ν/J0)))(Γcoup(s)/θm (s)). Then, the ze-ros of θm (s)/Γcoup(s) are obtained by using the poles ofθm (s)/Γm (s) and the denominator of (7). Thus, the resultingexpression of the system Γcoup(s)/θm (s), which is deduced by

assuming a link with distributed mass, is as follows:

Γcoup(s)θm (s)

=(

Γm (0)θm (0)

) N∑i=1

s2/z2i + 1

s2/ω2i + 1

(8)

where N is the number of considered vibration modes, zi and ωi

are the zeros and poles of Γcoup(s)/θm (s), respectively. Notethat, the term θm in (1) involves two zeros in the origin in (8)(i.e., z1 = 0). Finally, the tip angle, which can be defined asfollows:

θt(t) = θm (t) + w(L, t) (9)

is accurately positioned to its desired position θ∗t , if θm ≡ θ∗tand w(L, t) = 0 for t ≥ ts , being ts the settling time. The con-dition w(L, t) = 0 is achieved in this work by actively dampingθm (s)/Γcoup(s).

IV. PROPOSED CONTROL SCHEME

The proposed control methodology accurately positions θm

and actively damps link vibrations to achieve a precise tip-positioning by an inner and an outer control loops, respectively.This section describes the utilized controllers and demonstratesthe stability of overall control scheme.

A. Inner Control Loop

The control law of the first loop is as follows:

V (t) =Γcoup(t)

nK + Kp(θ∗m (t) − θm (t)) − Kv θm (t)(10)

where Γcoup(t)/nK (denoted as decoupling term) makes thedesign of the proportional-derivative (PD) constants (Kp , Kv )independent of the link dynamics. The substitution of (10) into(2) yields

J0 θm (t) + (nKKv + ν)θm (t) + nKKpθm (t)

= nKKpθ∗m (t) + Γf (11)

which only depends on the motor dynamics. Thus, if Γf is con-sidered as a perturbation input, the motor angle can be expressedas follows:

θm (s) =θ∗m (s) + Γf (s)/(nKKp)

s2(J0/nKKp) + s ((nKKv + ν)/(nKKp)) + 1.

(12)It must be noted that, in addition to the independence be-tween θm (s) and the link dynamics, this inner loop is robust toCoulomb friction and to changes in the dynamic friction (withlarge values of Kp ) [25]. Thus, if the tuning of the parameters ofthe PD controller (Kp , Kv ) is carried out to achieve a criticallydamped second-order system, (12) can be approximated by

θm (s) ∼= Gm (s)θ∗m (s) =θ∗m (s)

(1 + αs)2 (13)

where α is the constant time of the transfer function betweenθm and θ∗m (Gm (s)). From (12) and (13), the values of Kp andKv are obtained.

Kp =J0

nKα2 Kv =(2J0 − υα)

nKα. (14)


Fig. 3. (a) Schematic of the second control loop formed by IRC. (b) ModifiedIRC proposed in [26].

Fig. 4. General control scheme.

B. Outer Control Loop

Let us denote G1(s) and G2(s) as the transfer functionsθt(s)/θm (s) and Γcoup(s)/θm (s), respectively. The outer con-trol loop imparts damping to link dynamics by using an IRCscheme for G2(s). As stated earlier, IRC is an easy-to-tune,low-order controller that imparts substantial damping to mul-tiple resonant modes of a flexible structure, without instabilityissues due to unmodeled system dynamics. If the IRC proposedin [22] is used [see Fig. 3(a)], the next two problems appear.The first one is that, the dc gain of θt(s)/θ∗t (s) is equal to zero.This can be proved by calculating the transfer function betweenθm (s) and R(s) [see Fig. 3(a)], which is

θm (s)R(s)

=s

s + Kc (G2(s) + Df ). (15)

Then, if θm = 0 rad in steady state and w(L, t) = 0, the tipangle θt is also zero [see (9)]. This problem is solved by usingthe alternative representation of IRC scheme, which is shown inFig. 3(b) [26]. The transfer function between θm (s) and R(s)is as follows:

θm (s)R(s)

=s + KcDf

s + Kc (G2(s) + Df )(16)

which has the same denominator as (15). In addition, the dc gainof (16) is equal to one because G2(0) = 0 due to the two zerosin the origin mentioned earlier. The second problem is derivedfrom the impossibility of actuating directly on θm . This is solvedby using θ∗m as error signal in the outer control loop. Thisrequires of a dynamic inversion of Gm (s), which is denoted byG−1

m (s) = (sα + 1)2 .1 Then, the term (sβ + 1) is included intothe outer controller to make it implementable with relative orderequal to zero. The general control scheme is shown in Fig. 4

1Ideally, α = α, but we consider the case of an inexact cancellation of closed-loop motor dynamics

and the transfer function of the overall system is as follows:

θt(s)θ∗t (s)

=Gm (s)G1(s)(s + KcDf )(sβ + 1)

βs(s + 1) + s + Kc(Gm (s)G−1m (s)G2(s) + Df )

.

(17)Note that, if β → 0 and α = α, the design of IRC gains (Df andKc ) is analogous to previous works (see [22] and [26]). Finally,it can be deduced that dc gain of θt(s)/θ∗t (s) is equal to one(G1(0) = 1 and G2(0) = 0). Thus, the precise tip-positioningis achieved if the vibrations are canceled.

C. Stability Analysis

Due to the fact that θt(s)/θ∗t (s) and Γcoup(s)/θ∗t (s) havethe same denominator, and Gm (s) is stable, the stability of(17) can be demonstrated by considering the negative feed-back interconnection of G2(s) and KcGm (s)G−1

m (s)/((sβ + 1)(s + KcDf )). At this point, the results proved in [27], whichare based on the feedback connection of systems with nega-tive imaginary frequency response (NIFR)2 are utilized. Moreprecisely, the Theorem 5 in [27] states that, the positive feed-back of two SISO systems M(s) and N(s), with j[M(jω) −M(jω)∗] ≥ 0 and j[N(jω) − N(jω)∗] > 0, respectively, for allω ∈ (0,∞), is internally stable if and only if M(0)N(0) < 1,M(∞)N(∞) = 0, and N(∞) ≥ 0.

As we have a negative feedback, we demonstrate that−G2(s)is NIFR, which is proved by using the passivity property ofG2(s)/s deduced in [21]. Note that, G2(s)/s is passive if andonly if it is positive real, i.e., Re[G2(jω)/jω] ≥ 0, where Re[�]is the real part of a complex number (see for example Section4.7.2 of [28]). Then, if Re[G2(jω)/jω] ≥ 0, it is easy to deducethe following:

j [−G2(jω) − (−G2(jω)∗)] = ω

[G2(jω)

jω− G2(jω)∗

−jω

]

= 2ωRe

[G2(jω)

jω

]≥ 0. (18)

Let us denote C(s) as the following transfer function:

C(s) =Kc(sα + 1)2

(s + KcDf )(sα + 1)2(sβ + 1). (19)

The NIFR definition j [C(jω) − C(jω)∗] > 0 ∀ω ∈ (0,∞), isanalogous to Im [C(jω)] < 0 ∀ω ∈ (0,∞), where Im[�] is theimaginary part of a complex number. If we substitute s = jωin (19) and multiply the numerator and denominator by thecomplex conjugate of the denominator, the result is as follows:

C(jω) =(1 − jβω)(1 − jγω)[(1 + ααω2) + jω(α − α)]2

Df (1 + β2ω2)(1 + γ2ω2)(1 + α2ω2)2

(20)

in which γ = 1/KcDf . Furthermore, the next restriction ob-tained from the imaginary part of the numerator of C(jω) can

2A system G(s) has NIFR when j[G(jω) − G(jω)∗] ≥ 0 (or > 0) for allω ∈ (0,∞)


be written as

1 − 2(α − α)β + γ

+ (4αα − α2 − α2 − 2(α − α)β + γ

(αα − βγ))ω2

+(

α2 α2 + 2(α + α)β + γ

ααβγ

)ω4 > 0. (21)

Note that, the left side of (21) is a polynomial of the variableω. Let us denote the variable z = ω2 and the polynomial coef-ficients of ω4 , ω2 , and ω0 as a2 , a1 , and a0 , respectively. Thus,(21) can be rewritten as follows:

φ(z) = a2z2 + a1z + a0 > 0. (22)

This condition is fulfilled, if a0 > 0 (φ(0) > 0) and a21 < 4a0a2

[there is no sign change in φ(z)]. After further simplification,these restrictions can be expressed as

1− 2(α−α)β + γ

> 0, 4(αα − β2)(γ2 − αα)− (α−α)2(β + γ)2

− 4(α − α)(β + γ)(αα − βγ) > 0. (23)

Although, this is not a explicit relation for obtaining the con-troller parameters, it is useful to analyze the affordable variationof α (due to motor parameters uncertainties) with respect to itsdesired value (α), as it will be shown later.

As mentioned earlier, the dc gain of G2(s) is zero (G2(0) =0), and subsequently, −G2(0)C(0) = 0 < 1. Moreover, it canbe deduced from (19) that C(∞) = 0, which implies thatG2(∞)C(∞) = 0 and C(∞) ≥ 0, respectively. Therefore, ac-cording to Theorem 5 of [27] the closed-loop system is internallystable. Note that, this stability property only depends on the con-troller parameters see (23)], i.e., the stability is independent ofthe link and payload parameters.

V. EXPERIMENTAL RESULTS

We set α = α = 0.01 and β = 0.001, respectively, which al-lows us to neglect Gm (s)G−1

m (s)/(sβ + 1) in the tuning of theIRC parameters and permits us to use the design strategy givenin [22]. In addition, this value of α makes the transfer functionGm (s) robust to Coulomb friction and does not saturate the dcmotor. From Table I and (14), the values of Kp and Kv were350.9 and 6.9, respectively. Identification for Gm (s)G2(s) withMP = 0 kg and JP = 0 kgm2 , respectively, which was usedfor IRC design, was carried out. The frequency response ofGm (s)G2(s) was obtained by applying a band-limited randomnoise signal (0.3–25 Hz) to the motor angle reference (θ∗m ). Boththe input (θ∗m ) and the output (Γcoup ) were recorded using thespectrum analyzer. Using a subspace-based system identifica-tion technique, an accurate model of the experimental systemwas obtained, [29]. Thus, the first three vibration modes areidentified with sufficient accuracy to design an IRC and testthe achievable damping in simulations. The identified transferfunction is as following:

Gm (s)G2(s) =s2

(0.01s + 1)2

(3.95

s2 + 0.07s + 43.5

+3.69

s2 + 0.25s + 1668+

2.68s2 + 0.41s + 1.32 · 104

)(24)

Fig. 5. Magnitude response of the measured (·–·) and modeled system (—) ofGm (s)G2 (s).

and the FRF of the measured and modeled system ofGm (s)G2(s) are plotted in Fig. 5.

IRC parameters were tuned using the strategy given in [22].First, a value Df > 0 was defined. Subsequently, Kc was tunedto impart maximum damping to (G2(s) + Df )/s. It must benoted that, while selecting the value of Df , there is a tradeoffbetween the time response of the system and the maximumdamping that can be imparted. Small values of Df correspondto a controlled system with a dominant pole close to the origin.In this case, though the damping that can be imparted is large,the time response of the system (i.e., trajectory tracking) is veryslow. If the value of Df is increased, this pole moves from theorigin and can impart good damping. Thus, the performanceof the system is improved to some extent by increasing Df .Any increase in the value of Df beyond this point results in apoor damping performance. Therefore, after careful root-locusanalysis, tuning of Df should be realized together with Kc .

Due to practical considerations mentioned earlier, the pro-posed outer controller parameters were set to: Df = 1 andKc = 0.7 [γ = 1/0.7 in (23)], respectively. This design con-sidered the transfer function Gm (s)G2(s) of (24) and the max-imum variation of tip mass (0.3 kg from Table I). In addition, itcan be deduced from (23) that the controller system is stable ifα ∈ (0.0012, 0.056), which makes the controlled system robustto moderate uncertainties in Gm (s). Fig. 6 shows the FRF ofthe open loop (Gm (s)G2(s)) and closed loop for MP = 0 kg[see Fig. 6(a)] and MP = 0.3 kg [see Fig. 6(b)], respectively.The first three vibration modes are attenuated by 20.4, 12.8, and0.8 dB, respectively, for MP = 0 kg, and by 13.95, 11.9, and7.2 dB, respectively, for MP = 0.3 kg. It can be seen that thedamping performance is quite robust to payload variations.

In order to illustrate the controller performance and showthe slowdown in the time response of the system when thepayload increases, experimental time responses were obtained.The used trajectory reference was a Bezier polynomial of fourth


Fig. 6. Magnitude response of Gm (s)G2 (s) (·–·) and the closed-loop system between coupling torque and tip-angle reference (—). (a) MP = 0.0 kg.(b) MP = 0.3 kg. Note that, in both the cases, there is substantial damping in the first two resonant modes.

Fig. 7. Time response of the joint angle. Reference (. . .), MP = 0.0 kg (—),and MP = 0.3 kg (·–·).

order with final value of 0.5 rad and trajectory time of 1 s. Thethree represented variables were the joint angle, the couplingtorque measured at the base of the arm, and the estimated tipangle, which was estimated using a full-order observer3 whoseinputs are the measured motor angle and the coupling torque,respectively. Fig. 7 shows the tip angle reference and the jointangle with IRC for MP = 0 kg and MP = 0.3 kg, respectively.Fig. 8(a) and (b) shows the coupling torque without and with IRCfor MP = 0 kg and MP = 0.3 kg, respectively. It can be notedfrom Figs. 7 and 8 that the joint angle is accurately positionedand the vibration is canceled for the two values of tip mass.Finally, Fig. 9(a) and (b) shows the reference and the estimatedtip angle of the open loop (Gm (s)G2(s)) and closed loop, forMP = 0 kg and MP = 0.3 kg, respectively. It can be observedthat the system reaches the final position without vibration forthe two values of tip mass. In addition, the settling time is biggerfor MP = 0.3 kg. Finally, the steady-state error in θm and θt

3This full-order observer was only used to estimate the system output forcomparison, and was not used for control purposes.

is approximately equal to 0.02% and is due to the Coulombfriction. This can be reduced even more by higher proportionalgain or by including an integral action in the inner loop.

Therefore, it can be concluded from these experimental re-sults that this novel control scheme is robust to big changes inthe payload (by almost 90% when compared to the mass of themanipulator) because the damping is virtually unchanged andthe stability is guaranteed. However, the time response is sloweddown because the aforementioned pole associated to Df movesto the origin. This is due to the dependence of Df on the naturalfrequency of the first vibration mode, which decreases as the tipmass increases.

A. Performance Comparison of the Most SignificantControl Schemes

The election of a control strategy depends on the plant andparticular application. This work has proposed a novel controlapproach that can be classified into the group of control strate-gies that use minimum-phase dynamics to damp the vibration,originated by link flexibility. These strategies are more conve-nient for making the controlled system robust to unmodeledhigh-frequency dynamics (spillover) than those strategies thatconsider nonminimum phase dynamics (e.g., dynamics betweentip position and motor torque). This is because the stability ofthe controlled system is simpler to obtain in practice.

The most significant references related to minimum-phasesystems, which have been cited in Section I, are compared in thissection. The precise tip-positioning can be achieved by using anoutput related to tip position [16], [17], [20] or by canceling thevibration and positioning the joint [18], [19], [21]. Although thedirect measurement of the tip presents the advantage of knowingthe tracking error, these sensors might not be suitable in manyindustrial applications due to the need of visual contact betweenthe tip and joint. This has motivated the use of other sensors suchas strain gauges, which have been successfully used in practice[18], [19], [21]. In addition to the precise positioning undernominal conditions, these control techniques exhibit one or more


Fig. 8. Time response of the coupling torque in open loop (·–·) and closed loop (—). (a) MP = 0.0 kg. (b) MP = 0.3 kg.

Fig. 9. Time response of the estimated tip angle obtained from experimental data. (a) MP = 0.0 kg. (b) MP = 0.3 kg. Reference (. . .) and estimated tipposition in open loop (·–·) and closed loop (—).

of the following characteristics: 1) robust to payload changes;2) robust to joint frictions; 3) robust to external disturbances;4) design simplicity; and 5) stability independent of payloadparameters. In the next paragraphs, a performance comparisonin terms of these characteristics is presented.

The reflected tip position and its generalization are usedin [16], [17], respectively, to accurately position the tip, whichis achieved by a PD controller. Their main advantages are therobustness to payload changes and design simplicity. In addi-tion, any external disturbance can be detected by the tip sensor.However, the passivity condition needed to guarantee the sta-bility does not permit the use of an integral action. In addition,the PD gains are designed to achieve a good time response (i.e.,imparted damping), which limits the value of these gains. There-fore, the controller is not robust to joint frictions. Moreover thispassivity depends on the relationship between hub and link-plus-payload inertia, making the system stability highly dependenton payload.

To solve this problem, other control methodologies basedon two control loops have been proposed. For example, in[18]–[20], an inner loop to actively damp the link vibrationsand an outer loop to achieve precise positioning are used. Sen-sors located in the link are used in [18] and [19] to impartdamping. According to the experimental results shown in theseworks, the feedback of strain measured at the base of the link(inner loop) together with a joint position control achieves pre-cise positioning robust to joint frictions. In addition, we havechecked in our experimental platform that this strategy is quiterobust to payload changes. Moreover, the strain measured at thebase can detect external disturbances, making these strategiesrobust to them. However, the design of the joint position controlis complicated because the dynamics of the controlled systemby the inner loop is considered. In addition, the stability demon-stration depends on system parameters if an integral action isused. The joint angle is used in [20] to impart damping to thesystem by a resonant controller. Then, an integral controller of


TABLE IICOMPARISON PERFORMANCE

the tip position is used to guarantee precise tip-positioning. Thiscontrol scheme is robust to changes in the payload and joint fric-tions. However, the stability of overall system depends on theposition control, and the order of the inner controller increaseswith the number of considered vibration modes. In addition, theresidual vibrations due to external disturbance are substantial ifa reduction gear is employed, thus delivering poor disturbancerejection performance.

The strategy proposed in this work uses an inner loop to ac-curately position the joint angle and an outer loop to impartdamping, which takes coupling torque as the output (propor-tional to the strain measured at the base of the link) and jointangle as the input. The joint control is designed without con-sidering the link and can be made robust to joint frictions. Theouter controller is based on the known inner loop dynamics andthe transfer function between coupling torque and joint angle,which allows the detection of external disturbances in the link.Based on the internal stability property of the positive feedbackof two systems with NIFR, the stability only depends on con-troller parameters. Thus, the stability of the overall system isindependent of payload (i.e., robust to large payload changes).Finally, a simple and systematic tuning procedure for inner andouter controller exists and has been provided.

Table II shows a summary of the performance comparisonexpounded above. It can be seen that the contribution of theproposed methodology is to formulate a control law, robust tojoint frictions and external disturbances (like [18], [19]) com-bined with a simple design methodology, which allows us tomake the stability of the overall system independent of payloadparameters.

VI. CONCLUSION

A new approach for the control of flexible manipulators wasformulated and experimentally verified. This work follows thedesign methodology proposed in [21] to formulate a novel con-troller, which is based on a modification of the IRC schemeoriginally used in [22] to damp collocated smart structures. Theresulting control scheme is internally stable if the given stabil-ity conditions are fulfilled, which are independent of payloadparameters.

The precise tip-positioning is guaranteed by accurately po-sitioning the joint angle and by damping the vibrations caused

by the reference trajectory. Experimental results have shownrobustness to joint frictions, large changes in payload param-eters, and spill-over effects, which have resulted in a precisetip-positioning. In addition, this new approach simplifies thepositioning and damping control task to the tuning of two pa-rameters (Kc and Df ) that can be adjusted intuitively withoutan exact knowledge of system dynamics. Therefore, the pre-cise positioning and damping achieved by this simple controlscheme makes it very suitable for many applications.

Future research endeavors will focus on multilink flexiblemanipulators as well as parameter optimization.

ACKNOWLEDGMENT

The authors would like to thank Dr. I. M. Dıaz for the help inthe development of the system model.

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Emiliano Pereira (M’07) received the Ph.D. (Hons.)degree in industrial engineering from the Universidadde Castilla-La Mancha, Ciudad Real, Spain, in 2009.

He is currently a Research Assistant at the Uni-versidad de Castilla-La Mancha, where he has beenengaged in monitoring and control of vibrations inflexible structures, a research project of the Span-ish Government. His research interests include activedamping of vibrations in flexible structures and dy-namic control of flexible robots.

Sumeet S. Aphale (M’09) received the B.E. de-gree in electrical engineering from the Universityof Pune, Pune, India, in 1999, and the M.S. andPh.D. degrees in electrical engineering from the Uni-versity of Wyoming, Laramie, in 2003 and 2005,respectively.

He was a Member of the Hexapod Research Lab-oratory involved in design and control of Gough–Stewart platform-type robotic manipulators. FromOctober 2005 to June 2008, he was a Member ofthe Australian Research Council’s Centre of Excel-

lence for Complex Dynamic Systems and Control, University of Newcastle,Newcastle, Australia. He was a Research Fellow in the Centre for Applied Dy-namics Research, University of Aberdeen, U.K, in 2008. He was appointed to aNew Blood Lectureship in the School of Engineering, University of Aberdeen,in 2009. His research interests include robot kinematics and control, vibrationcontrol applications, as well as design and control of nanopositioning systems.

Vicente Feliu (M’88–SM’08) received the M.S. de-gree (Hons.) in industrial engineering and the Ph.D.degree in automatic control from the PolytechnicalUniversity of Madrid, Madrid, Spain, in 1979 and1982, respectively.

From 1980 to 1994, he was in the Electrical En-gineering Department, Universidad Nacional de Ed-ucacion a Distancia, Madrid, where he was a FullProfessor during 1990, and the Head of the Depart-ment from 1991 to 1994. From 1987 to 1989, he was aFulbright Scholar at the Robotics Institute, Carnegie

Mellon University, Pittsburgh, PA. Since 1994, He has been the Dean of theSchool of Industrial Engineering, Universidad de Castilla-La Mancha, CiudadReal, Spain. His research interests include multivariable and digital control sys-tems, fractional dynamics and control, kinematic and dynamic control of rigidand flexible robots, mechatronics, automation of hydraulic canals, and computervision for robots. He his the author or coauthor of about 90 technical papersin international refereed journals and over 140 technical papers in internationalconference proceedings.

Dr. Feliu is a member of the International Federation of Automatic Control.He received the Best Paper Award for papers published in the Pattern Recogni-tion Journal in 2001.

S. O. Reza Moheimani (SM’00) received the Ph.D.degree in electrical engineering from the Universityof New South Wales at the Australian Defence ForceAcademy, Canberra, Australia, in 1996.

He is currently a Professor in the School of Electri-cal Engineering and Computer Science, University ofNewcastle, Newcastle, Australia, where he holds anAustralian Research Council Future Fellowship. Heis an Associate Director of the Australian ResearchCouncil Centre for Complex Dynamic Systems andControl, an Australian Government Centre of Excel-

lence. He has held several visiting appointments at the IBM Zurich ResearchLaboratory, Zurich, Switzerland. He is the author or coauthor of two books,several edited volumes, and more than 200 refereed articles in archival journalsand conference proceedings. His current research interests include applicationsof control and estimation in nanoscale positioning systems for scanning probemicroscopy, control of electrostatic microactuators in microelectromechanicalsystems, and data storage systems.

Prof. Moheimani is a Fellow of the Institute of Physics, U.K. He is a recipientof the 2007 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY Out-standing Paper Award and the 2009 IEEE Control Systems Technology Award.He has served on the Editorial Board of a number of journals including the IEEETRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, and chaired several in-ternational conferences and workshops.

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