Applied Soft Computing 2 (2002) 24–38

Fuzzy logic control for flexible link robot arm by singularperturbation approach

K.Y. Kuo a, J. Linb,∗a National Central Police University 56, Shu Jen Rd., Takang Chun, Kwei Shan, Taoyuan, Taiwan 333, ROC

b Department of Mechanical Engineering, Ching Yun Institute of Technology, 229 Chien-Hsin Road, Jung-Li City 320, Taiwan, ROC

Received 21 September 2001; received in revised form 20 March 2002; accepted 31 May 2002

Abstract

This work addresses flexible-link robot arm control using fuzzy logic, and a singular perturbation approach. A singularperturbation approach is introduced to derive the slow and fast subsystems and thus reduce the effect of spillover. Consequently,a two-time scale fuzzy logic controller will be applied for such systems. The fast-subsystem controller damps out the vibrationof the flexible structure by the optimal control method. Thus, the slow-subsystem fuzzy controller dominates the tracking ofthe trajectory. The stability of the internal dynamics was guaranteed by adding a boundary-layer correction based on singularperturbations. The first investigation of hybrid LQ-fuzzy controller yielded a favorable output response for the flexible linkrobot arm than does the traditional outer-loop PD control with the boundary layer stabilizer. The fuzzy control method isreliable and robust.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Flexible-link; Singular perturbation; Fuzzy logic control; Composite control

1. Introduction

A flexible link arm is a distributed parameter sys-tem of infinite order, but must be approximated bya lower-order model and controlled by a finite-ordercontroller due to onboard computer limitations, sensorinaccuracy, and system noise. The so-called “controlspillover” and “observation spillover” effects thenoccur, which under certain conditions can lead toinstability [3,12,13].

Literature on the flexible arm is surveyed in[2].This work shows that flexibility imposes serious limi-tations on the effectiveness of standard rigid-arm con-trol schemes. Several control schemes have recentlybeen proposed for flexible robot arms[4,10,11,19].

∗ Tel.: +886-3-4581196x3300; fax:+886-3-4595684.E-mail address:jlin@cyit.edu.tw (J. Lin).

A controller that is based upon a reduced-ordermodel is proposed in[6,9] to maintain reasonablecomputational loading. In recent years, singular per-turbation theory has been demonstrated to providea convenient means “reduced-order modeling”. Thedynamics of singularly perturbed systems can be ap-proximated by the dynamics of the correspondingreduced-order and boundary layer subsystems forsufficiently small values of the singular perturba-tion parameter. The aim is to simplify the softwareand hardware implementation of control algorithms,while improving their robustness. A composite con-trol approach, based on a two-time scale model of theflexible-link arm has been derived in[15,19], and al-lows a definition of a slow subsystem that correspondsto a rigid body and a fast subsystem that describesthe flexible motion. The approach also supports a fastsubsystem that describes the flexible motion.

1568-4946/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S1568-4946(02)00026-1

J. Lin / Applied Soft Computing 2 (2002) 24–38 25

For many years, classical control engineers begantheir work with a mathematical model and did not ac-quire further knowledge of the system. Today, controlengineers use all of the above sources of information.Aside from a mathematical model whose use is clear,numerical (input/output) data can be considered to de-velop an approximate model and a controller, basedon fuzzy IF-THEN rules[24,26].

The application of the concepts of fuzzy set theoryin structural control has recently attracted increasinginterest. Fuzzy controllers afford a simple and ro-bust framework for specifying nonlinear control lawsthat accommodate uncertainty and imprecision. Suchcontrollers may be implemented using a fuzzy math-ematical model of the plant and controller. Fuzzycontrollers may be determined without a mathemati-cal model when linguistic descriptions of the controlare available or can be formulated. Implementationsthat depend on a linguistic synthesis approach havebeen proposed[7,8,17,23–25,27]and shown to betheoretically and practically applicable.

This work examines the feasibility of applyingcontrol algorithms based on fuzzy logic to a classof hybrid structural control systems shown in[1,10,16,18]. The investigation included both analyticaland experimental verification of the fuzzy controlalgorithm. The guidelines for implementing a fuzzyactive control strategy for civil engineering struc-tures are considered discussed in[5,20,21]. Thosestudies focus on the gap between a successful nu-merical example and the technical design of thedevice.

This work concentrates mainly on fuzzy logic,and the use of the singular perturbation approach inflexible-link robot arm control. A singular perturba-tion approach is also introduced to derive the slow andfast subsystems and thus reduce the effect of spillover.Therefore, a two-time scale fuzzy logic controllerwill be applied for such systems. The fast-subsystemcontroller will dampen the vibration of the flexiblestructure by the optimal control method. Hence, theslow-subsystem fuzzy controller dominates the track-ing of the trajectory. The stability of the internaldynamics is guaranteed by adding a boundary-layercorrection, based on singular perturbations. This fuzzylogic/singular perturbation approach brings togetherand employs the concepts elucidated in several of thepapers mentioned above.

2. Robot dynamics and sensor system design

2.1. Robot dynamics

A convenient and approximate form of the dynam-ics of a flexible robot arm can be derived by the as-sumed mode shape method. The deflection of elasticbeamw(η, t) can be expressed as a summation of theinfinite series terms by assuming a Bernoulli–Eulerbeam model[14]:

w(η, t) =∞∑i=1

qi(t)φi(η), (1)

where qi(t) are generalized modal coordinates andφi(η)are mode shape functions that depend onthe boundary-value problem (i.e. pinned-pinned,clamped-free, clamped-loaded,. . . ), where η repre-sents displacement along the neutral axis of the link.

The components of the dynamic model should beexplicitly separated into matrix form to specify theinertia (M), centrifugal/Coriolis/damping (D), stiffness(K), friction F(x, x), and gravitational constantG.

M(x)x +D(x, x)x +K(x)x+F(x, x)+G(x) = Bτ,

(2)

where B is the input matrix that depends on theclamped link assumptions), andτ represents the con-trol input torque. Here,x = [q0, q1, . . . , qn]T isdefined, whereq0 is the rigid mode,q1, . . . , qn arethe flexible modes, andn is the number of retainedmodes inEq. (1).

A pseudo-linear state-space representation for theflexible arm is given by

X =[

0

−M−1K

I

−M−1D

]X

+[

0

M−1B

]τ +

[0

−M−1

]τn, (3)

with X =[x x

]T, and the nonlinear forcing terms,

τn = [F(x, x) + G(x)].Given the robot dynamics (3), the output is defined

as

y = Cq, (4)

26 J. Lin / Applied Soft Computing 2 (2002) 24–38

whereC is generally a function ofq. The aim hereis to determine the control torque inputsτ(t) thatguarantee suitable performance with respect to thebehavior ofy(t).

2.2. Sensor system design

2.2.1. Rigid mode measurementEach axis of rotation uses an optical increment en-

coder that is mounted on the motor to measure posi-tion. The measurement of position is extracted fromthe quadrature signals by incrementing or decrement-ing a counter, as appropriate with every pulse edge.Eq. (5)specifies the measurement of the joint angle

[ϕ] = [ 1 φ′1(0) φ′

2(0) · · · ]

q1

q2...

qn

. (5)

2.3. Flexible mode measurement

Several sensors have been used to measure thevibration of a flexible beam. The most popular hasbeen the strain gauge, which measures the deflectionof the beam. The advantages of strain gauges are iso-lation of beam variables from rigid rotations, lack ofrestrictions on work positioning, high compatibilitywith harsh industrial environments, and low cost. Therelationship between the strain,δ, and the generalizedcoordinatesqi(t) is

δ = c∂2w(η, t)

∂η2= c

∞∑i=1

qi(t)d2φi(η)

dη2, (6)

wherec is the curvature of the beam.Eq. (6)can be expanded to relate each flexible mode

to the measured strain at each location,a, b, . . . , m.This relationship can be presented in matrix form as

δ(a, t)

δ(b, t)

...

δ(m, t)

= c

d2φ1(a)

dx2

d2φ2(a)

dx2· · · d2φn(a)

dx2

d2φ1(b)

dx2

d2φ2(b)

dx2· · · d2φn(b)

dx2...

......

...

d2φ1(m)

dx2

d2φ2(m)

dx2· · · d2φn(m)

dx2

q1

q2...

qn

(7)

Eq. (7) is rewritten as

δ = c[�]

q1

q2...

qn

. (8)

The flexible modes and strain gauge numbers are in-dicated as below.

(a) If n = m, then [�] is a square matrix, and[q1 q2 . . . qn

]T = 1c[�]−1δ.

(b) If n = m, then [�] is not a square matrix. From the

pseudo inverse matrix,[q1 q2 · · · qn

]T =1c([�]T[�])−1[�]Tδ.

3. Link-tip position control using I/O feedbacklinearization

This section applies input/output (I/O) feedback lin-earization to design a controller for the link-tip po-sitions,y ∈ Rnr . Moreover,Section 4completes thedesign using singular perturbation theory to design acontroller that gives a boundary-layer correction tostabilize the flexible modes.

3.1. Input/output feedback linearization

In Eq. (4)y(t) is twice differentiated using standardI/O feedback linearization techniques, and is then sub-stituted forq, usingEq. (2)to obtain the reduced-ordersystem,

y = u. (9)

The auxiliary control input,u(t), is defined by thereduced-order computed torque (ROCT) control

J. Lin / Applied Soft Computing 2 (2002) 24–38 27

law,

τ = (CM−1B)−1[CM−1(Dq + Kq + F + G) + u].

(10)

System (9) consists ofnr subsystems, each with twointegrators in series, and is said to be in Brunovskycanonical form.

When a finite number of flexible modes are retainedin the dynamic model, the matrixCM−1B is nonsingu-lar. This matrix has been shown to become singular asthe number of modes approaches infinity as in the ex-act model. In fact, retaining a finite number of modescorresponds to “approximate feedback linearization”of systems with an ill-defined relative degree, and issufficient in designing controllers with adequate per-formance.

The tracking error,e(t) = yd(t) − y(t), is definedwith yd(t) as the desired trajectory, andKd > 0, Kp >

0 is chosen. Then, an outer-loop PD control law for atrajectory is

u = yd + Kde + Kpe. (11)

Unfortunately, this obvious selection foru(t) doomsthe control scheme. Even though the ROCT controlconsists of (11), and (10) decouples thee(t) subsystemfrom the remainder of the plant and stabilizes it, theROCT almost certainly fails to stabilize the remainderof the plant because the zero dynamics are unstable.This issue was explored in[19,22].

3.2. Inverse system dynamics

Partitioned matrices were used as follows to eluci-date the detailed structure of the closed-loop systemafter ROCT.Eq. (2) is rewritten, neglecting frictionfor simplicity as[Mrr Mrf

Mfr Mff

]qr

qf+

[Drr Drf

Dfr Dff

] [qr

qf

]

+[

0 0

0 Kff

] [qr

qf

]+

[Gr

0

]=

[Br

Bf

]τ. (12)

The inverse of the mass matrix can be defined as[Hrr Hrf

Hfr Hff

]=

[Mrr Mrf

Mfr Mff

]−1

, (13)

andEq. (12)can be multiplied byEq. (13)from theleft, and the terms rearranged:

qr = −Darrqr − Da

rf qf − Karf qf − Ga

r + Bar τ (14a)

qf = −Dafr qr − Da

ff qf − Kaff qf − Ga

f + Baf τ (14b)

with

Darr = HrrDrr + HrfDfr , Da

rf = HrrDrf + HrfDff ,

Dafr = HfrDrr + Hff Dfr , Da

ff = HfrDrf + Hff Dff ,

Karf = HrfKff , Ka

ff = Hff Kff ,

Gar = HrrGr, Ga

f = HfrGr,

Bar = HrrBr + HrfBf , Ba

f = HfrBr + Hff Bf .

In the case of the general flexible-link robot arm,Bf is smaller thanBr. A sufficient condition thatguarantees the existence of(Ba

r )−1 in terms of the

matrix-induced norm (for example, the maximumsingular value[11]) is easily shown to be

||Bf || < (||B−1r H−1

rr Hrf ||)−1. (15)

Performing feedback linearization to (14) is equiv-alent to selecting

τ = (Bar )

−1(Darrqr + Da

rf qf + Karf qf + Ga

r + u) (16)

to obtain the dynamics

qr = u (17)

qf = −Dbfr qr − Db

ff qf − Kaff qf − Gb

f + Bbf u, (18)

whereu(t) is an auxiliary input and the Schur comple-ments are defined as

Dbfr = Da

fr − Baf (B

ar )

−1Darr,

Dbff = Da

ff − Baf (B

ar )

−1Darf ,

Kbff = Ka

ff − Baf (B

ar )

−1Karf ,

Gbf = Ga

f − Baf (B

ar )

−1Gar ,

Bbf = Ba

f (Bar )

−1.

28 J. Lin / Applied Soft Computing 2 (2002) 24–38

A state-space representation of the dynamics is

d

dt

qrqr

qf

qf

=

0 I 0 0

0 0 0 0

0 0 0 I

0 −Dbfr −Kb

ff −Dbff

qrqr

qf

qf

+

0I

0Bb

f

u +

00

0−I

Gb

f . (19)

Control law (16) is a refined expression of theROCT control (10), and system (19) is the completeclosed-loop system after ROCT. The given rigid-modetrajectory,yd(t), in (17) can be obtained by appro-priate selection ofu(t). Consequently, the function ofthe inverse system is to construct the unique flexiblemode trajectory that is associated with the corre-sponding rigid-mode trajectory. Selectingu(t) basedonly on the desired rigid subsystem performancedoes not guarantee a stable inverse system, becausethe zero dynamics are not stable for the flexible-linkrobot arm. Hence, the following section selection ofu(t) to achieve the desired rigid-mode performanceand stabilize the inverse dynamics, using a time scaleseparation in the next section.

4. Stabilizing internal dynamics usingboundary-layer correction

This section applies singular perturbation theory tostabilize the zero dynamics after I/O feedback lin-earization. As will be seen, this application affordsgreater accuracy of the time-scale separation assump-tion. A typical procedure for singular perturbation the-ory is reviewed in[9]. The feedback-linearized systemis singularly perturbed by rewriting (17) and (18) as

qr = u, (20)

qf + Dbfr qr + Db

ff qf + Kbff qf + Gb

f = Bbf u. (21)

The singular perturbation analysis depends onputting (20) and (21) in standard form and solving forsome fast variables. An appropriate parameter mustfirst be extracted from the fast subsystem because theflexible subsystem is faster than the rigid one.

By defining

U =Hff − [HfrBr + Hff Bf ]

× [HrrBr + HrfBf ]−1Hrf (22)

Kbff = Kcl = UKff (23)

whereKcl indicates “closed-loop stiffness” andKff de-notes the open-loop stiffness that appears inEq. (16).M−1

ff is generally large, such thatKcl = UKff is of alarger scale thanKff . Therefore, introducing a singularperturbation based onKcl rather than onKff is moreaccurate. Therefore, a positive scaling factor,κ, andfactor,Kcl, are introduced as

Kcl = κKcl, (24)

where Kcl is an invertible constant matrix and isindependent of the closed-loop stiffness factor,κ.

Variablesε2 = (1/κ) andε2ξ(t) = qf (t) are thendefined. This work addresses a case in which the stiff-nessκ is sufficiently large. The primary aim of thecontrol design is to enable the rigid motion,qr(t), tracka desired trajectory,yd(t). The output,y(t), can be gen-erally written asy = qr +f (qf ). In the pinned-pinnedmode case,f ( ·) ≡ 0, while in the clamped-free modecase,f is not constantly zero. Accordingly, the outputof the system is set as,y = qr(t); restated a modifiedoutput is used in the clamped-free case. Therefore,rewriting (21), (22) in terms ofξ(t) andε yields,

y(t) = u, (25)

ε2ξ = −Dfr (qr, qr, ε2ξ, ε2ξ )y

−Dff (qr, qr, ε2ξ, ε2ξ )ε2ξ

− Kcl(qr, ε2ξ)ξ − Gf (qr, ε

2ξ)

+Bf (qr, ε2ξ)u, (26)

where for simplicity, the superscript “b” has beendropped. A control,u, is designed to allow(y, y) track(yd, yd) closely and to stabilize the systems (25) and(26).

The control input,

u = u + uf (27)

is defined withu(t) as the slow control componentanduf (t) as a fast control component. Settingε = 0yields the slow dynamics,

¨y = u, (28)

J. Lin / Applied Soft Computing 2 (2002) 24–38 29

and an algebraic relationship iny, y, and ξ .

0 = −Dfr ˙y − Kclξ − Gf + Bf u. (29)

Notably, an overbar denotes the evaluation of nonlin-ear functions, withε = 0.

Settingε = 0 and solving forξ in Eq. (29)yieldsthe slow manifold equation,

ξ = (Kcl)−1(−Dfr ˙y − Gf + Bf u) (30)

The dynamics (28) are valid on this manifold.Statesς1 ≡ ξ − ξ and ς2 ≡ εξ are selected and

Eq. (29)written as,

ες2 = −Dfr y−Dff ες2−Kcl(ς1 + ξ ) − Gf + Bfu,

(31)

where a time-scale change ofτ = t/ε yields,

dς1

dτ= ς2

dς2

dτ= −Dfr y − Dff ες2 − Kcl(ς1 + ξ )

−Gf + Bf (u + uf ) (32)

Setting nowε = 0 and substituting forξ fromEq. (30)gives the fast dynamics part of the followingslow subsystem description of the feedback-linearizedarm:

¨y = u (33)

Moreover, the fast subsystem can be found to be

dς1

dτ= ς2 (34)

dς2

dτ= −Kclς1 + Bfuf , (35)

which describes a linear system parameterized in theslow variables.

4.1. Tracking requirement

According to (33)–(35),uf (t) can be selected toyield stable zero dynamics, provided that(Kcl, Bf )

is stabilizable. If(Kcl, Bf ) is controllable, no zero

dynamics exist, anduf (t) can be selected to control[ ςT

1 ςT2 ]T.

A composite control strategy can be pursued, asdemonstrated by the two reduced-order subsystems(33)–(35). The design of a feedback control for the fullsystems (25) and (26), can be split into two separatedesigns of feedback controls,u(t) anduf (t), for thetwo reduced-order systems. Formally,

u = u + uf . (36)

Moreover, a modified tracking outputy is consideredin [22]. Using Tikhonov’s Theorem,

y = y + O(ε), (37)

qf = ε2(ξ + ς1) + O(ε), (38)

whereξ (t) is given byEq. (30)and O(ε) denotes termsof orderε.u(t) is thus selected for suitable tracking behavior

in the slow subsystem (33). Then,uf (t) is selectedfor the stability of the fast part (34) and (35). Thefast control law is used to stabilize the inverse systemdynamics since the fast subsystem is nothing but alinearization of the flexible (internal system) dynamicsinduced by the slow manifoldξ(t). In this work,uf (t)

can be designed as an optimal control for the boundarylayer. The performance index will be a function ofthe slow state variables. The feedback gain matricescan also be designed on the basis of the final jointconfiguration, if and only if for that particular choiceof ς1, ς2 will not be unstable on the slow trajectory,since the main purpose of a flexible link robot armcontrol is to dampen the steady state deflections asrapidly as possible. Accordingly, the need to solve aRiccati equation for each joint configuration can beavoided[19].

Substituting the composite control (36) into the con-trol law (16) gives the entire control, applied to theflexible-link robot arm.

In this work, Eq. (38) shows that the flexiblemode trajectory has two parts. The slow part,ξ (t) in(30), depends on the rigid mode trajectories and thecontrol u(t) used to construct them. The fast part,ς1, is a boundary-layer correction added for stabili-zation.

The composite control requires full state feedbackof the rigid and flexible modes and their derivatives.

30 J. Lin / Applied Soft Computing 2 (2002) 24–38

In practice, an observer might be required to recon-struct the velocities from position and strain gaugemeasurements. Such techniques are commonly de-scribed in the literature[6,15] and straightforwardlyapplicable.

5. Fuzzy control

The field of fuzzy system and control has seengreat progress, motivated by practical success in con-trolling industrial processes. Fuzzy systems can beused as closed-loop controllers. In such cases, a fuzzysystem measures the outputs of the process and con-tinuously controls the actions involved in the process.The fuzzy controller uses a form of quantification ofimprecise information (input fuzzy sets) to generateby an inference scheme, which is based on a knowl-edge base of control force to be applied to the system[20].

The benefit of this quantification is that the fuzzysets can be represented by a unique linguistic expres-sion, such as small, medium, or large. The linguisticrepresentation of a fuzzy set is known as aterm, anda collection of such terms defines a term-set, or a li-brary of fuzzy sets. Fuzzy control converts a linguisticcontrol strategy, typically based on expert knowledge,into an automatic control strategy. The logical con-troller is comprised of four primary components. (1)Fuzzifier (the values of the state variables monitoredduring the process are fuzzified into fuzzy linguis-tic terms); (2) a knowledge base that contains fuzzyIF-THEN rules and membership functions; (3) fuzzyreasoning (the result of which is a fuzzy output foreach rule), and (4) a defuzzification interface (whichproduces a crisp control output)[5].

Fuzzy processing enables the quantification of anirregular input by sets that can be treated according totheir significance. A rule in the knowledge base canbe defined, such that the control force will be zero,without detrimentally affecting the system. Similarly,any set can be treated differently from any other set.The fuzzy controller offers several advantages. (1) Itcan discriminate among groups of input values of highor low importance, and treat them accordingly, witha small number of operations; (2) it can easily elim-inate actions that may be detrimental to the system;(3) using trapezoidal fuzzy sets implicitly filters input

signals, and (4) output aggregation and defuzzifica-tion schemes reduce the influence of input noise andcomputer imprecision.

5.1. Fuzzy control structure of the system

5.1.1. Fast subsystem stabilizerufThe control scheme of the flexible link robot arm

must be able to control the motion in the rigid bodymode and suppress the vibration modes of the arm. Inthis work, the fast subsystem controller is designed tostabilize all the vibration modes of the arm. Therefore,an LQ design was applied to the fast subsystem toselect appropriate PD gains.

5.1.2. Slow subsystem fuzzy controlleru

A slow subsystem fuzzy controller is designed, us-ing the tracking error,e, and error rate,e, to implementinput/output feedback linearization/singular perturba-tion approach control. Hence,u is selected for suitabletracking in the slow subsystem.Fig. 6 shows blockdiagrams of two-time scale fuzzy logic controller forthe flexible link robot arm.

5.2. Membership function and fuzzy rule base

Two difficulties in designing any fuzzy control sys-tem are the shape of the membership functions and thechoice of the fuzzy rules. The decision-making logicis the way in which the controller output is generated.The decision-making uses input fuzzy sets, and thedecision is governed by the values of the inputs. Fur-thermore, the knowledge base consists of knowledgeof the application domain and the attendant controlgoals. It includes a database and a fuzzy controlrule base.

A fuzzy logic controller is designed using thetracking error, e, and error rate,e, to implementinput–output feedback linearization/singular pertur-bation approach control. Very many researchers havecommended the use of equally spaced triangularmembership functions, “common sense” rules, andthen finding a way to “modify”, “tune”, or “adapt”them to the plant’s variations, unmodelled dynamics,or external disturbances. A control system is saidto be an adaptive fuzzy control system if a set offuzzy rules is used either to modify or to changean existing fuzzy controller’s architecture (including

J. Lin / Applied Soft Computing 2 (2002) 24–38 31

Fig. 1. Configuration of flexible-link robot arm.

membership functions and/or rules).Figs. 1–3showfuzzy membership functions ofe, and its derivative,e. Seven fuzzy membership functions correspond tolarge negative(LN), medium negative(MN), smallnegative(SN),zero(ZE), small positive(SP),mediumpositive (MP), large positive(LP) values of the er-ror, e, and the error rate,e. The support vectors,e,and e are between−0.1 and 0.1. However, neitherenor e are restricted to belong to this support, since

Fig. 2. Membership function ofe.

the fuzzy membership function can be extended to±∞. Therefore, anye or e that exceeds 0.1 willbe involved the large positive (LP) membershipfunction.

Fig. 4 presents the fuzzy set of the slow subsystemcontroller output,u. Seven fuzzy membership func-tions are the same ase and e.

After the input and output values are assigned todefine fuzzy sets, each possible input condition must

32 J. Lin / Applied Soft Computing 2 (2002) 24–38

Fig. 3. Membership function ofe.

be mapped onto an output condition. Such mapping iscommonly expressed as follows.

If (condition or antecedent) then (action or con-sequence): the fuzzy rule base can be illustrated as

Fig. 4. Membership function ofu(t).

a look-up table.Table 1 presents a look-up tablethat corresponds to the fuzzy sets depicted inFigs. 5and 6. This look-up table is a matrix of seven rows (thenumber of membership functions of thee fuzzy-set)

J. Lin / Applied Soft Computing 2 (2002) 24–38 33

Fig. 5. Viewer surface of the system.

and seven columns (the number of membership func-tions of thee fuzzy-set).

5.3. Defuzzification

Defuzzification is the conversion of a fuzzy quan-tity represented by a membership function, into a pre-cise or crisp value. Commonly used strategies may bedescribed asmaximum criterion, mean of maximum,and centroid methods. This study uses the centroidand the singleton methods to combine and defuzzifyoutputs into crisp values. The centroid method is firstused to determine each fired output value. A singletonor average weight method is used to combine those

Table 1Consequent table for fuzzy logic controller design

u e

LN MN SN ZE SP MP LP

LN LN MN LN LN ZE MP SPMN MN MN MN MN ZE MP MPSN LN MN SN SN ZE SP SPZE SN ZE ZE ZE ZE ZE SPSP SN SN ZE SP SP MP LPMP MN MN ZE MP MP MP MPLP SN SN ZE LP LP LP LP

output values into an executable single value. Theadvantage of the center of gravity defuzzifer is itsintuitive plausibility, but its disadvantage is its com-putational intensity.

6. Results and discussion

(1) ε = 0.05 Fig. 7 presents the tip position trackingerror of the flexible link robot arm with outer-loopPD control and boundary-layer.Fig. 8 shows thetip position tracking error of the flexible link robotarm, with fuzzy logic control and a boundary-layerstabilizer.y(t)|fuzzy clearly tracks the desired tra-jectory,yd(t), very closely. Moreover,Figs. 9 and10 reveal that the flexible mode deflections, usingthe fuzzy logic controller, are damped out morequickly than the outer-loop PD controller. In thecase of fuzzy logic control, the elastic vibrationsvirtually vanish after a short period and remain atzero.

(2) ε = 0.012 Fig. 11 shows the response of thetip position and the tip position tracking whenε = 0.012. Figs. 12–14depicts the response ofthe position-tracking error in using fuzzy logiccontroller. The figure shows that the fuzzy logic

34 J. Lin / Applied Soft Computing 2 (2002) 24–38

Fig. 6. Block diagram of fuzzy logic control with boundary-layer stabilizer.

control used here is better than that which usesouter-loop PD control in tip position tracking andin flexible modes.

(3) Varied ε. Next, the performance of this compos-ite controller is shown to improve when a moreappropriate value ofε is used. Comparing theperformance of the system usingε = 0.05 withthat usingε = 0.012, shows that the vibrationis damped out more quickly whenε is smaller.

Fig. 7. Tip position tracking error of flexible link robot arm with outer-loop PD control and boundary-layer stabilizer (ε = 0.05).

However, using a fuzzy logic controller to obtainvibration responses does not differ from. Thefuzzy control method appears to be quite reli-able and robust. Fuzzy logic controllers are veryrobust controllers in nonlinear systems becausethey act only on rules that are applied to mea-sured outputs, and can thus handle variations thata standard controller may not consider, for lackof explicit methods.

J. Lin / Applied Soft Computing 2 (2002) 24–38 35

Fig. 8. Tip position tracking error of flexible link robot arm with fuzzy logic control and boundary-layer stabilizer (ε = 0.05).

Fig. 9. Response of flexible modes of flexible link robot arm with outer-loop control and boundary-layer stabilizer (ε = 0.05).

Fig. 10. Response of flexible modes of flexible link robot arm with fuzzy logic control and boundary-layer stabilizer (ε = 0.05).

36 J. Lin / Applied Soft Computing 2 (2002) 24–38

Fig. 11. Tip position tracking error of flexible link robot arm with outer-loop PD control and boundary-layer stabilizer (ε = 0.012).

Fig. 12. Tip position tracking error of flexible link robot arm with fuzzy logic control and boundary-layer stabilizer (ε = 0.012).

Fig. 13. Response of flexible modes of flexible link robot arm with outer-loop PD control and boundary-layer stabilizer (ε = 0.012).

J. Lin / Applied Soft Computing 2 (2002) 24–38 37

Fig. 14. Response of flexible modes of flexible link robot arm with fuzzy logic control and boundary-layer stabilizer (ε = 0.012).

7. Conclusion

The primary contribution of this paper is in fuzzylogic, by providing a singular perturbation approachfor flexible-link robot arm control. A singular pertur-bation approach was introduced to reduce the effect ofspillover and thus derive slow and fast subsystems. Acomposite control design was adopted. Consequently,a two-time scale fuzzy logic controller can be appliedfor such systems. The fast-subsystem controller damp-ened out the vibration of the flexible structure by theoptimal control method. Hence, the slow-subsystemfuzzy controller dominated the trajectory tracking. Thestability of the internal dynamics was guaranteed byadding a boundary-layer correction based on singularperturbations. In conclusion, the first investigation ofhybrid LQ-fuzzy controller yielded a favorable out-put response for the flexible link robot arm than doesthe traditional outer-loop PD control with the bound-ary layer stabilizer. The fuzzy controllers are poten-tial candidates for deriving control in the presence ofthese structural nonlinearities. Future work will em-phasize a two-time scale hierarchical fuzzy logic con-troller of the multilink flexible arm and will be appearin a forthcoming issue.

Acknowledgements

The authors would like to thank the National Sci-ence Council of the Republic of China for financially

supporting this research under Contract no. NSC89-2218-E-231-001.

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