An Adaptive Fuzzy Controller for Robot Manipulators: Theory andExperimentation

MIGUEL A. LLAMA, RAFAEL KELLY, VICTOR SANTIBANEZ, and HUGO CENTENO Instituto Tecnologico de la Laguna, Apdo. Postal 49, Adm. 1

Torreon Coahuila, 27001, MEXICOFax: +52 871 7 05 13 26

email: santibanez@ieee.org, vsantiba@itlalaguna.edu.mx,mllama@itlalaguna.edu.mx, hugocnt@faraday.itlalaguna.edu.mx

Division de Fsica Aplicada, CICESE, Carretera TijuanaEnsenada km 107Ensenada, B. C., 22800, MEXICO

email: rkelly@cicese.mx

Abstract: this paper addresses the tracking con-trol of robot manipulators for the case when thedynamics parameters are unknown. To this end,we resort to the important property of fuzzysystems as universal approximators. Via Lya-punov theory we prove, under the fact that SISOfunctions approximation error can be achievedwith any prespecified degree of accuracy, thatthe closedloop tracking errors are ultimatelybounded. We extend the single-input-single-output indirect adaptive fuzzy approach previ-ously introduced in the literature to the moregeneral case of multi-input-multi-output fuzzycontrol for nonlinear robot manipulators. Theexposed theory is validated experimentally on atwo degrees of freedom robot arm.

KeyWords: fuzzy control, adaptive control,robot control, stability analysis, experimentaltest.

1 IntroductionThe fuzzy control of robot manipulators has beenwidely discussed in recent years [1][6] and it has beenimproved by incorporating other control techniques tothe classical ones such as adaptive fuzzy techniques,and fuzzy neural networks schemes [7][19]. The highlynon linearity, the complexity in the dynamics modelingof robotic systems and the presence of unknown andvariant parameters, turn difficult the task of designingeffective tracking controllers.

A control technique that has proved a very nice perfor-mance in the tracking control problem of robot manipu-lators is the computed torque control (inverse dynamicsapproach) [20]; this control scheme linearizes the sys-tem and applies a linear PD control. However, the mostaccuracy knowledge of the model of the plant is requiredin order to operate properly. If the dynamics of the ro-bot are well known, the computer torque control canbe implemented and even improved with the additionof fuzzy systems that tune PD gains or compute the PDcontrol contribution as it is shown in [1], [3] and [7].Nevertheless, not always the robot dynamics are wellknown or they may change during a task execution.Therefore adaptive fuzzy techniques have been intro-duced to the control of robot manipulators so as to es-timate the parameters that describe the system dynam-ics and hereby compute the control law. Adaptive fuzzycontrols can be classified in: direct, when the fuzzy con-troller is a single fuzzy system constructed (initially)from the control knowledge; and indirect, when thefuzzy controller comprises a number of fuzzy systemsconstructed (initially) from the plant knowledge [8]. Avery comprehensive study is shown in [9], where adap-tive fuzzy control was both simulated and implementedin simple non linear systems.Some direct adaptive fuzzy approaches have been ap-plied to the control of robot manipulators in [8],[10],[11], and with cubic feedback in [12]. Indirect adaptivefuzzy approaches are developed in [8], [13] and with ge-netic algorithms in [14]. However, none of these workswere experimentally validated.The indirect adaptive fuzzy control proposed in this pa-per is characterized by a simplified design based on clas-

sical computed torque control, identifying the robot dy-namics through adaptive fuzzy systems. The proposedstability analysis proves and guarantees via Lyapunovtheory that position errors are uniformly bounded andultimately bounded.The practical feasibility of the proposed controller isshown in this paper by means of experiments on a twodegrees of freedom directdrive vertical robot arm. Theresults show that the experimental response of the clas-sical computed torque control is similar to that of theadaptive fuzzy control, standing out the latter becauseit does not require a knowledge of the robot dynam-ics. However, if the knowledge of the initial conditionsis available, this would help to have a faster positiontracking convergence.

2 Dynamics of robotmanipulators and control

problem formulationThe dynamics of a serial n-link robot can be written as[20]:

M(q)q + C(q, q)q + g(q) + fv(q) = (2.1)

where q is the n 1 vector of joint displacements, q isthe n 1 vector of joint velocities, is the n 1 vectorof applied torque inputs, M(q) is the n n symmetricpositive definite manipulator inertia matrix, C(q, q)qis the n 1 vector of centripetal and Coriolis torques,g(q) is the n 1 vector of gravitational torques and fv(q) stands for the n 1 vector of friction viscousand Coulomb torques.The motion control problem addressed in this papercan be stated in the following terms. Assume that theparameters of the robot dynamics are constant but un-known, and the joint position q and joint velocity qare available for measurement. Let the n 1 desiredjoint position vector qd be a twice differentiable vectorfunction. The control problem is to find a controller todetermine the actuator torques in such a way thatthe joint positions q track a desired joint position tra-jectory qd as close as possible: limt q(t) = qd(t).One conventional solution to this control problem forthe case of known parameters is provided by the com-puted torque control given by [20]

=M(q)[qd+Kv q+Kpq]+C(q, q)q+ fv(q)+g(q)

where the joint position and velocity error vectors aredenoted respectively by the n 1 vectors q = qd qand q = qd q, with qd being the n 1 vector of de-sired velocity, and Kp and Kv are symmetric positivedefinite matrices proportional and derivative (PD)

gains, respectively. It is well known [20] that thiscontroller yields global exponential stability at least forconstant PD gainsKp and Kv. However, the latter factmay limit seriously the control system performance inmany applications. A stable computed-torque controlvia fuzzy self-tunning was introduced in [1] to improvethe performance of the response. Nevertheless, in bothcases the parameters of the robot model are rigorouslyrequired and they are even more sensitive in the con-troller response than the PD gains, which it means thatthose parameters must be well known for an optimumoperation of the closed loop system. If the behavior ofthe plant is properly described, the classical computedtorque control (CTC) provides an appropriate trackingcontrol for robotic systems. Therefore, a complete pa-rameter identifier system based on fuzzy systems andan adaption law will provide the functions that describethe model of the robot regardless the unknown parame-ters.

3 Adaptive fuzzy controlFrom the dynamics of the robot we can write the jointacceleration vector as follows:

q = M(q)1[C(q, q)q + g(q) + fv(q)] +M(q)1(3.1)

Now consider the general form for a single-input single-output non linear system

xn = f(x) + g(x)u (3.2)y = x y, u IR, x IRn

with g(x) 6= 0 so as to ( 3.2) be controllable.According to the latter, ( 3.1) can be rewritten in amulti-input multi-output scheme as

q = F (q, q) +G(q, q) (3.3)

with F (q, q) = M(q)1[C(q, q)q+g(q)+ fv(q)] andG(q, q) =M(q)1. In this case F (q, q) and G(q, q) aren 1 and nn matrices respectively. As the functionsF (q, q) andG(q, q) will be estimated by a fuzzy system,the computed torque control law takes the form

= G(q, q)1[qd +Kv q +Kpq F (q, q)] (3.4)

Where F (q, q) =

f1(q, q)...

fn(q, q)

and G(q, q) =

g11(q, q) g1n(q, q)...

. . ....

gn1(q, q) gnn(q, q)

denote the estimates of

the respective function matrices with fi(q, q) andgij(q, q) being their respective elements.

This controller is known as indirect adaptive fuzzycontrol, because the control law is computed from theestimates of the dynamic functions of the plant [8].

3.1 Fuzzy systemA fuzzy rule base system, for the case of two inputx1, x2, and one output y, consists of a set of fuzzyIFTHEN rules comprising the following rules (namedRl1l2)

IF x1 is Al11 AND x2 is Al22 THEN y is Bl1l2 (3.5)

where for each input fuzzy set Alii and output fuzzy setBl1l2 exists an input membership function Alii (xi) andoutput membership function Bl1 l2 (y), respectively,with li = Ni12 , ,1, 0, 1, ,

Ni12 , and i = 1, 2

for the case of two inputs. N1 is the number of fuzzysets for input 1, and N2 is the number of fuzzy sets forinput 2.In order to achieve the control law we recall the fuzzysystems property as universal approximators. Fuzzysystems can perform a non linear mapping with themore accuracy as the design involves; this is discussedin the Theorem 1 [8].

Theorem 1. Suppose that the input universe of dis-course U is a compact set in IRn. Then, for any givenreal continuous function g(x) on U and an arbitrary > 0, there exists a fuzzy system f(x) in the form of

f(x) =

N1i1=1

N2i2=1 y

i1i2(Ai11 (x1)Ai22 (x2)

)

N1i1=1

N2i2=1

(Ai11 (x1)Ai22 (x2)

)

(3.6)

such that

supxU

|f(x) g(x)| < (3.7)

where yi1i2 is the center of the (i1, i2)th output mem-bership function. It means that fuzzy systems are uni-versal approximators; that is, they can approximate anyfunction on a compact set to arbitrary accuracy (de-noted by ). The accuracy of the approach dependsupon the information of the system to approxime andthe complexity of the fuzzy systems. The approxima-tion accuracy can be found as

||g(x) f(x)|| g(x)x1

h1 +

g(x)x2

h2

+ . . .+g(x)xn

hn (3.8)

where || || = supxU |d(x)| is the infinite norm, hi =max1jNi1 |c

j+1i c

ji |, with cki as the center of the

membership function k = j or k = j + 1 (according tothe case) of the input i, where i=1,2. . . n, and n is thenumber of inputs of the fuzzy system [8].4In the proposed control law, we can approach the un-known functions using fuzzy systems with singletonfuzzifier, Ni (odd) triangular membership functions foreach input, with i = 1, 2 (see Fig. 3.1), n =M adapta-tion parameters as singleton membership function forthe output (see Fig. 3.2), rule base defined by ( 3.5)for two inputs, max-prod inference and center averagedefuzzifier as follows:

fj(x|f ) =

p1l1=1

pnln=1 y

l1lnf

(2i=1 Alii (xi)

)

p1l1=1

pnln=1

(2i=1 Alii (xi)

)

for j = 1, . . .n (3.9)

gjk(x|g) =

q1l1=1

qnln=1 y

l1lng

(2i=1 Blii (xi)

)

q1l1=1

qnln=1

(2i=1 Blii (xi)

)

for j = 1, . . .n, k = 1, . . . n (3.10)

Where yl1lnf are the free parameters in f

IRn

i=1 pix1, that is f =

y1...1f...

yl1...lnf

with pi = l1 ln,

and yl1lng are the free parameters in g IRn

i=1 rix1,

that is g =

y1...1g...

yl1...lng

with ri = l1 ln, so we can

rewrite ( 3.9) and ( 3.10) as

fj(x|f ) = Tf (x) (3.11)

gjk(x|g) = Tg (x) (3.12)

with (x) a vector of dimensionsn

i=1 pi 1, whoseelements are

l1ln(x) =

2i=1 Alii (xi)p1

l1=1 pn

ln=1

(2i=1 Alii (xi)

) (3.13)

and (x) a vector of dimensionsn

i=1 qi 1, whose el-ements are

l1ln(x) =

2i=1 Blii (xi)q1

l1=1 qn

ln=1

(2i=1 Blii (xi)

) (3.14)

Fig. 3.1: Input membership functions

Fig. 3.2: Output membership functions

Then, introducing the robot variables, the fuzzy ap-proximators can be written in a compact form as

F (q, q|f ) = Tf (q, q) (3.15)

G(q, q|g)1 = Tg (q, q), (3.16)

while (q, q) and (q, q) are n 1 adaptive para-

meters vectors and f =

f1 0 0

0. . . 0

0 0 f2

IRnpinpi , (q, q) =

1...n

IRnpi ,

g =

g11 g1n...

. . ....

gn1 gnn

IRnrinri ,

(q, q) =

1...n

IRnri .

The designed fuzzy rule base is summarized in the lookup Table 3.1.

Fig. 3.3: Closed loop system

3.2 Stability analysis

The closed loop system (see Fig. 3.3) is obtained bycombining the robot dynamic model ( 3.3) with thecontrol law ( 3.4). Introducing the acceleration of thedesired trajectory (qd) we get the acceleration error

q = Kv q Kpq+F (q, q|f ) + G(q, q|g) F (q, q)G(q, q)

(3.17)

Define =[1 2 n

]T = [F (q, q|f ) F (q, q)] + [G(q, q|g) G(q, q)] as the minimumapproximation error where F (q, q|f ) and G(q, q|

g)

are the best (max-prod) approximators of F (q, q) andG(q, q) respectively between all the fuzzy approxima-tors of the form ( 3.9) y ( 3.10); then ( 3.17) can beequivalently rewritten as

ddt

[qq

]:= e = Ae+B (3.18)

where

A =[

0 IKp Kv

] IR2n2n (3.19)

B =[

0F (q, q|f ) F (q, q|f ) + [G(q, q|g) G(q, q|g)] +

]

IR2n

In order to carry out the stability analysis we propose

Table 3.1: Lookup table for the fuzzy rule base.

l1 = 2 l1 = 1 l1 = 0 l1 = 1 l1 = 2x2\ x1 NB NS ZE PS PB

l2 = 2 NB 1 2 3 4 5l2 = 1 NS 6 7 8 9 10l2 = 0 ZE 11 12 13 14 15l2 = 1 PS 16 17 18 19 20l2 = 2 PB 21 22 23 24 25

the following Lyapunov function candidate:

V (q, q, f , g) =12[

q

T q

T ]P[

q

q

]

+n

i=1

12i

[fi fi ]

T [fi fi ]

+n

i=1

n

j=1

12ij

[gij gij ]

T [gij gij ]

with P = P T > 0 satisfying the Lyapunov equationATP PA = Q where Q = QT > 0. The first termof V (q, q,f ,g) is a positive definite function with re-spect to q and q because the positive definiteness ofP . The remain terms are also positive because theyare quadratic terms. So that, V (q, q,f ,g) is globallypositive-definite and radially unbounded function.The time derivative of the Lyapunov function candidatealong of the trajectories from ( 3.18) is given by

V (q, q,f ,g) =12[ qT qT

]P[qq

]

+12[qT qT

]P[ qq

]

+n

i=1

1i[fi

fi ]

T fi

+n

i=1

n

j=1

1ij

[gij gij ]

T gij

(3.20)

Inspired from [21] we take P = P T > 0 as

P =[Kp + Kv I

I I

](3.21)

where the smallest eigenvalue of Kv (min{Kv}) satis-

fies min{Kv} > > 0. We can reduce ( 3.20) asV (q, q, f , g) =

12[

q

T q

T ]Q[

q

q

]

+n

i=1

(qi + qi)i

+n

i=1

1i[

fi fi]T[

fi + i(qi + qi)i(q, q)]

+n

i=1

n

j=1

1ij

[

gij gij

]T

[

gij + ij(qi + qi)ij(q, q)j]

(3.22)

From ( 3.22) we can obtain the following update lawsfor the parameters such that their respective termsare cancelled

fi = i(qi + qi)i(q, q) (3.23)gij = ij(qi + qi)ij(q, q)j (3.24)

with i = 1, 2 . . . n and j = 1, 2 . . . n.When accomplishing the previous update laws, the timederivative of the Lyapunov function is reduced to

V (q, q,f ,g) = 12[qT qT

]Q[qq

]

+n

i=1(qi + qi)i (3.25)

Let min{Q} be the smallest eigenvalue of Q, then from( 3.25) we have

V 12[min{Q} 1]

qq

2

12

[qq

2

2[qT qT

] [

]+

2]

+12

[

] 2

12[min{Q} 1]

qq

2

+12

2

(3.26)

Since the fuzzy systems are universal approximators,we can make the minimum approximation error arbi-trarily small by using more rules to construct the fuzzy

system (

< ). Hence, by stability theory of

perturbed systems is possible to conclude that[qq

]

and[fg

]are uniformly bounded, and

[qq

]is uni-

form ultimately bounded.Furthermore, integrating both sides of ( 3.26) and se-lecting [min{Q}] > 1 we have t

0

q

q

2

d

2min{Q} 1

[

V (0) + t

0

2

d]

= 2min{Q} 1V (0) + 2min{Q} 1

t

0

2

d

if is square integrable, then[qq

]is also square

integrable and from ( 3.18),[ qq

]is also bounded;

then, from Barbalat Lemma we can conclude that

limt[qq

]=[00

] IR2n.

So we have proven the following

Proposition 1. Consider the indirect adaptive fuzzycontrol system given by ( 3.4), ( 3.15), ( 3.16), (3.23) and ( 3.24) in closed loop with the robot dy-

namics ( 2.1). The states[qq

]and

[figij

]are

uniformly bounded, and[qq

]is uniformly ultimately

bounded. Furthermore, if is square integrable, then

limt[qq

]=[00

] IR2n.

4

4 Experimental evaluation4.1 Experimental setup

In order to verify the effectiveness of the proposed adap-tive fuzzy control, an extensive number of realtimecontrol experiments on a well identified directdrive ro-bot arm were carried out. The experimental robot is atwo axis directdrive robot arm built at CICESE Re-search Center [22], [23]. The arm moves in the verticalplane as shown in Figure 4.1. The actuators are directdrive brushless motors operated in torque mode, so theyact as torque source and accept an analog voltage as a

reference of torque signal. The control algorithm is ex-ecuted at 2.5 msec sampling period in a control board(based on a DS1103 32/64bit floating point DSP boardrunning with a 400Mhz microprocessor) mounted on aPC host computer.

Fig. 4.1: Robot arm

The entries of the dynamics ( 2.1) of this two degreesoffreedom directdrive robotic armC(q, q) has beenchosen using Christoffel symbols, are given by [22]and [23]

M(q) =[

2.351 + 0.168 cos(q2) 0.102 + 0.084 cos(q2)0.102 + 0.084 cos(q2) 0.102

]

,

C(q, q) =[

0.084 sin(q2)q2 0.084 sin(q2)(q1 + q2)0.084 sin(q2)q1 0

]

,

g(q) = 9.81[

3.921 sin(q1) + 0.186 sin(q1 + q2)0.186 sin(q1 + q2)

]

,

fv(q) =[

2.288q1 + 8.049 sgn(q1)0.186q2 + 1.734 sgn(q2)

]

According to the actuators manufacturer, the directdrive motors are able to supply torques within the fol-lowing bounds

|1| max1 = 150 [Nm],|2| max2 = 15 [Nm].

4.2 Design parameters

The proposed controller have to estimate the F (q, q|f )and G(q, q|g)1 functions in order to calculate thecontrol law. These functions are computed with six

different fuzzy systems

F (q, q|f ) =[f1(q, q)f2(q, q)

]=[Tf11(q, q)Tf22(q, q)

]

G(q, q|g)1 =[g11(q, q)1 g12(q, q)1g21(q, q)1 g22(q, q)1

]

=[Tg1111(q, q)

Tg1212(q, q)

Tg2121(q, q) Tg2222(q, q)

]

From the knowledge of the structure of the 2 degreesoffreedom robot model we know that Tg1212(q, q) =Tg2121(q, q), thus we have not six but five differentfuzzy systems to be constructed. Also we can expectTg2222(q, q) to be a constant value. Every fuzzy sys-tem receives two inputs whose universe of discoursewere partitioned in five fuzzy sets: A2j = NB (Nega-tive Big), A1j = NS (Negative Small), A0j = ZE (Zero),A1j = PS (Positive Small), and A2j = PB (Positive Big),with j = 1, 2. We selected triangular membership func-tions

Aljj(xj) for both inputs.

In all cases the input membership functions are sym-metrical with respect to zero. This is shown in Figures3.1 and 3.2, where pxj = {p2j ,p1j , p0j , p1j , p2j} isthe set of the bounds of fuzzy set support (also calledfuzzy partition of the universe of discourse) which de-fines the input membership functions. The final parti-tions of the universes of discourse were (in agreementwith the used notation pA = {p2,p1, p0, p1, p2}):For q:

pq1 = {60,30, 0, 30, 60} [degrees]

pq2 = {60,30, 0, 30, 60} [degrees].

For q:

pq1 = {180,60, 0, 60, 180} [degrees/s]

pq2 = {180,60, 0, 60, 180} [degrees/s].

The F (q, q|f ) fuzzy sets have q1 and q2 as inputswhereas q2 and q2 are the inputs for the G(q, q|g)1fuzzy sets.It can be seen that, for any input, the sum of the mem-bership values of two adjacent fuzzy sets is one, and themembership value for any xj > p2j , xj < p2j is equalto one. Also, as we can see from Figure 3.1, only fourrules are fired in a given instant. Since both inputs have5 fuzzy sets (N1 = 5 and N2 = 5), then the number ofrules is M = 25 and it involves n = 25 product terms;every of these terms have its corresponding singletonmembership function whose center is given by an adap-tation parameter . This fact converts the fuzzy systeminto adaptive fuzzy system.

4.3 Desired taskThe desired position trajectory, is given by

qd(t) =

[

b1[1 ed1t3] + c1[1 ed1t

3] sin (1t)

b2[1 ed2t3] + c2[1 ed2t

3] sin (2t)

]

[rad]

(4.1)and was inspired from the structure of desired trajecto-ries used by other authors for experimental evaluationof control algorithms [24, 25].In our application, the first term of ( 4.1) was chosento exhibit a motion profile without abrupt changes inposition, velocity and acceleration but at the same timeto exploit the arm in its fastest motion without invadingthe actuators saturating zone.In expression ( 4.1), 1 and 2 represent the frequencyof desired trajectory for the shoulder and elbow jointsrespectively. In our experimental tests, we use 1 = 7.5rad/s and 2 = 1.75 rad/s.To quantify the control performance, we use the rootmean square average of tracking error (based on the L2norm of the tracking errors q) given by

L2[q] =

1

T t0

T

t0qT q dt (4.2)

where T represents the total experimentation time andt0 is the initial time of interest.Besides the adaptive fuzzy control (AFC), a classicalcomputed torque control (CTC) was tested so as tocompare their closed loop response and make conclu-sions.

4.4 Experimental resultsThe experimental results are depicted in the Figs. 4.2to 4.4. Fig. 4.2 shows position joints, position errorsand applied torques for the classical computed torquecontrol. For the desired trajectory the computed torquecontrol achieves a good tracking in both joints. It can-not reduce near to zero the position errors due to nonmodeled friction in the joints. Also, notice that the ap-plied torques are far of the manufacturer torque limits.Now, Fig. 4.3 refers to the position joints, position er-rors and applied torques for the indirect adaptive fuzzycontrol. The robot response are quite well for bothjoints. We recall that this controller does not resort inknown parameters of the robot, but only in the estima-tion of descriptive functions of the model via adaptivefuzzy systems. Comparing to the computed torque con-trol, the main difference lies in the position error of thesecond joint, which is a little bigger with this proposedcontroller.Alternatively, Fig. 4.4 shows the estimated functionsfrom the adaptive fuzzy systems and the real value of

0 2 4 6 8 1020

0

20

40

60

t [seg]

1[N

m]

Applied torques

0 2 4 6 8 101

0.5

0

0.5

1

t [seg]

q 1[deg]

Position errors

0 2 4 6 8 101

0.5

0

0.5

1

t [seg]q 2

[deg]

0 2 4 6 8 10

4

2

0

2

4

6

t [seg]

2[N

m]

0 2 4 6 8 100

20

40

60

t [seg]

q 1[deg]

Desired and actual positions

0 2 4 6 8 10

50

0

50

100

150

200

t [seg]

q 2[deg]

qd1

q2qd2

q1

Fig. 4.2: Graphic results for the CTC

0 2 4 6 8 1020

0

20

40

60

t [seg] 1

[Nm]

Applied torques

0 2 4 6 8 101

0.5

0

0.5

1

t [seg]

q 1[deg]

Position errors

0 2 4 6 8 102

1

0

1

2

t [seg]

q 2[deg]

0 2 4 6 8 10

4

2

0

2

4

6

t [seg]

2[N

m]

0 2 4 6 8 100

20

40

60

t [seg]

q 1[deg]

Desired and actual positions

0 2 4 6 8 10

50

0

50

100

150

200

t [seg]

q 2[deg]

q1qd1

q2qd2

Fig. 4.3: Graphic results for the AFC

0 2 4 6 8 1020

0

20

40

60

t [seg]

f 2,f

2

[1

seg2

]

0 2 4 6 8 100

0.05

0.1

0.15

0.2

t [seg]

g1

12,g1

12

[1

kgm

2

]

0 2 4 6 8 1050

0

50

t [seg]

f 1,f

1

[1

seg2

]

0 2 4 6 8 102.1

2.2

2.3

2.4

2.5

2.6

t [seg]

g1

11,g1

11

[1

kgm

2

]

0 2 4 6 8 100.1014

0.1016

0.1018

0.102

0.1022

0.1024

t [seg]

g1

22,g1

22

[1

kgm

2

]

f1

f1

f2

f2

g111

g112

g122

g122

g112

g111

Real and approximated functions

Fig. 4.4: Real and estimated functions

Table 5.1: Performance indexes.

Controller L2[q]CTC 0.3421 AFC 0.4272

those functions computed through the known model ofthe robot. It is not guaranteed that the estimators fol-lows the real values of the functions, but it does notmean that the control objective is not accomplished.

4.5 Concluding remarks ofexperiments

According to the desired trajectory which was used totest the controllers, the classical computed torque con-trol and the indirect adaptive fuzzy control have similarresponses, but the indirect adaptive fuzzy control hasthe advantage that does not require the model of therobot to compute the control law. Performance indexesare the L2 norm of the tracking position errors, whichare shown in Table 5.1 and Fig. 5.1. Fig. 5.2 shows theL2 norm of the tracking position errors for each joint.

5 ConclusionsIn this paper an indirect adaptive fuzzy control for ro-bot manipulators was proposed. To this end we ex-tended the single input single output indirect adap-tive fuzzy approach to the general case of multiinputmultioutput systems. Via Lyapunov theory and sta-bility theory of perturbed systems, we proved that thestates of the closed loop system, that is, the velocity andposition errors and the parameters of the fuzzy systemare bounded and furthermore the velocity and positionerrors are uniformly and ultimately bounded.The controller was experimentally validated on a 2 dofrobot manipulator achieving excellent results. A per-formance comparison was made between the proposedadaptive fuzzy control and the classic computed torquecontrol, resulting with a similar response but our pro-posed adaptive fuzzy control has the advantage of notrequiring the knowledge of any parameter of the robotmodel in contrast with the computed torque controlwhich require the overall knowledge of the dynamicsrobot.

ACKNOWLEDGEMENT

Work partially supported by DGEST and CONACyT(grant 45826).

REFERENCES

0

0.1

0.2

0.3

0.4

0.5

Norm

orerrors

[deg]

1

0.3421

1

0.4272

1

CTC

1

AFC

1

Fig. 5.1: L2 norm for both control techniques

0

0.2

0.4

0.6

0.8

Norm

orerrors

[deg]

1

0.4361

1

0.2046

1

0.6497

1

0.2480

1

joint 1

1

joint 2

1

CTC

1

AFC

1

Fig. 5.2: L2 norm of each joint for both control techniques

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