A new design of adaptive fuzzy hybrid force/position controller for robot manipulators

A

New Design of Adaptive Fuzzy Hybrid Force/Position Controller

for Robot Manipulators

Feng-Yih Hsul and Li-Chen Fu1t2

Dept. of Electrical Engineering'

Dept. of Computer Science

&

Information Engineering2

National Taiwan University, Taipei, Taiwan, R.O.C.

Abstract

T h e m a j o r problems of hybrid force/position con- trol arise f r o m uncertainty of t h e robot manipula- t o r s and u n k n o w n parameters of the task enviroment. In this paper, a n e w design method of t h e hybrid force/position control of t h e robot manipulators i s pro- posed t o solve these problems. T h e control objective is t o track t h e desired force and position trajectories si- multaneously regardless of t h e u n k n o w n parameters of t h e task e n v i r o m e n t and t h e existence of t h e manipu- lator dynamics, represented as a f u z z y rube-base. T h e algorithm embedded in t h e proposed architecture c a n automatically update t h e f u z z y rules and, consequently, guarantee t h e global stability and drive t h e tracking er- rors t o a neighborhood of zero. T h e present work is ap- plied t o t h e control of a five degree-of-freedom (DOF) articulated robot manipulator. S i m u l a t i o n results show t h a t the proposed control architecture is featured in f a s t convergence.

1

Introduction

Applying the force sensing t o the tasks of manip- ulating objects with robots in contact with surround- ing enviroment becomes more and more populatorly adopted. This is mainly due t o the technology im- provement and the cost-down stimulus of the sensor hardware as well as the rapid advance of the sig- nal processing technology. Typical examples of these tasks are assembly of mechanical parts, deburring, and contour-following operation [l]. These tasks usually require the end effector of robot manipulators t o con- tact the task environment with the desired force and t o follow the desired position trajectories. Because of the above-mentioned revolulutionary change, sophis- ticated methods of simultaneously achieving the two control objectives are thus more feasible.

During the past years, the force control schemes are often referred t o as compliant control or hybrid force/position control in the literature [7]-[8]. Gener- ally, the compliant control is trying t o simultaneously solve the force control problem and the position con- trol problem in the task frame, and provides recon- ciliation between the respective solutions of these two problem. Contrarily, the hybrid force/position control

is t o carefully formulate the task frame so that the associated task space can be easily decomposed into

force control subspace and position control subspace, arid then achieves the desired respective objectives. In this paper, we will address the problem of the hybrid force/position control.

Since the dynamics of robot manipulators are ex- tremely complex and their surrounding environment

is often changing t o perform versatile tasks, the non- linearity terms of the robot manipulators and the pa, ra.meters of the task environment, e.g., stiffness, can hardly be estimated exactly. Hence, some control which can automatically tune their control laws or up- date the parameters of the control schemes are neces- sa.ry and have been proposed such as adaptive control

14-

PI,

repetitive control [4] and learning control [5]-

6

.

In this paper, we propose an adaptive robust fuzzy control algorithm to solve the control problems un- der significant uncertainties in manipulator dynamics and its surrounding enviroment, in which the adaptive pitrt is t o on-line update the unknown parameters of the task environment. The overall control scheme not only guarantees the global stability of the closed-loop system but also drives the tracking errors to a neigh- borhood of zero.

2

Manipulator Dynamics in Con-

When the end effector of a robot manipulators moves in a force-control task along a constraint sur- face, the constraint coordinate frame attached t o the constraint surface are usually selected so as t o eas- ihy describe the desired force and position trajecto- ries. For simplicity, we assume that the position co- ordinates of end effector in constraint frame may be represented as a function of its position coordinates in the reference frame, i.e.,

straint Coordinate Frame

xc = H(x), (1)

where x, = [ ~ , ~ , . . . , z ~ ~ ] ~

(17:

5

6) is the position coordinates of the end effector in the constraint frame whereas x = [zl, a .

,

z,IT is its position coordinates

in, the reference frame [5]. Differentiating equation ( l ) , we can derive

IEEE international Conference on Robotics and Automation

- 863 -

where R ( x ) E Enx" is a transform matrix and is as-

sumed t o be nonsingular for x lying in a compact set in the task space, then there is a one-to-one mapping between

x,

and

x

in a properly defined compact set due t o the implicit function theorem [ll]. Besides, de-

note

J(x)

as a Jacobian transformation from the joint

coordinate frame t o the reference coordinate frame. Define a new transformation as J , (x) = R ( x )

J(x),

then we can derive the dynamics of robot manipula- tors in the constraint frame as follows. Consider a

class of articulated robot arms with n links whose end effectors move along some constraint surfaces. Then,

a general dynamic model can be described by

M(xc)xc

+

C(x,, X,)X,

+

G(xc)

+

D ( x , , X,)

+

f, = f,

where

Mc(xc)

E Rnxn is the inertia matrix,

C(x,,

xc)xc

is the vector representing the Coriolis forces, G(x,) is the vector of gravitational forces,

D(x,, X.) is the vector of friction forces,

f,

E

Rn

is the vector o contact forces, and f E

En

is the n x 1 vector of control input forces. Also, the torque T in joint co- ordinates can be derived as T = JCT(x) f . Generally, when an end effector contacts is in contact with the constraint surface, the desired force trajectories are usually designed for the force elements which are in the directions normal t o the surface, and the desired position trajectories are along the tangent directions of the surface, such as deburring tasks or contour- following tasks. Hence, normally the force elements along the direction of the desired trajectories are very small and are ignored. Therefore, the vector of the contact forces is defined as f, =

[f,:,

0,.

.

.,

0IT. On the other hand, owing t o the flexibility of applying robot manipulators, the manipulator payloads may not be necessarily known prior t o a variety of tasks. Because of the frequent change of payloads and of difficult measurements of the friction in the process uncertainties of modeling the manipulators dynamics become inevitable. As a robot, there are some un- negligible deviation between the nominal plant, i.e.,

M

=

M + A M ,

C = E + A C , G = G + A G ,

D

=

5

+

AD.

Our aim is to track both the desired trajectories of force and position, i.e., t o force the con- tact force fcl and the position vector of the end effec-

tor xc2 = [zcz,.

.

,

a C J T t o follow the desired trajec- tories f d l ( t ) and x d z ( t ) = [ x d 2 , - e . , Zd,lT(t). Hence,

the tracking force error ef = f d l - fcl and the po- sition error vectoreZ2 = [er2, - - e , eZ,JT are defined.

Assume that the robot manipulator is rigid and the de- formation of the task environment satisfies the Hook's where

IC,

denotes the stiffness, 3c, is the position of the constraint surface, and a d l is the desired position in the force direction. Because the end effector initially contacts the surface with the unknown environmen- tal deformation, the position

x,

is impossible to be known. Generally, if

k,

is exactly known, then ef =

k e ( x d l - zcl) so that i d l =

IC,-'

j d l will transfer the force control problem to the position control problem, i.e., t o force x, to approach xd = [ x d l , a d z , . - e ,

However, when

k,

is unknown and is estimated by

x,,

(3)

h

law, i.e., f c l = k e ( x c 1 as) and f d l = k e ( 2 d l - as),

the actual desired position x d l can hardly be obtained. Consequently, apart from the necessity of compensat- ing for the unknown nonlinearity of the robot manip- ulator, the stiffness of the environment, namely,

k,,

is also required t o be identified.

3

Virtual Adaptive Hy

sition

Control

-

A

Consider a virtual force signal f c l expressed as f c l =

k,

xC1 - k e a , , where

IC,

and k , x , are the estimate of

IC,

and k e a , , respectively. The difference between the actual c p t a c t for? and v&ual force is defined as

e p

=

f c l - f c l . With

k,

and k e a , given, a virtual de- sired position trajectory in the direction of the desired

h h

0.

-

contact force is expressed as

=

IC;'

(fdl+

k,~,),

where f d l is real desired force trajectory.

Our

aim IS

t o let

G1

approach the real desired position trajectory z d l and the position tracking errors e Z 2 ( t ) approach

zero. Therefore, a robust position controller have t o be given in order t o ensure the convergence of both the position tracking errors to zero. First, a nominal controller based on the computed-torque notion is designed as

where $ d = K s is defined as a

P D

controller, with s

=

[si,

- .

-

,

s,IT = e

+

Xe being defined as a sliding mode vector, e = [e^,l,e,2,...,ez,]T with e^,l = z d l - x c l ,

and

X

= diag(X1,.

.

-,

A,) as a diagonal positive defi- nite matrix; K = diag(k1,

- . - ,

k n ) is a positive definite gain matrix; x T = Xd

+

Xe is an augmented signal vec- tor; f d = [ f d l , 0 , .

. .

,

0lT is the desired force vector;

f,

=

[%

e^, 1, 0 .

.

.

,

0IT is an additional compensator. Moreover, the force error ef can be expressed as ef = f d i - f c i h A . h h

-

= k e ( G i -

~ c i ) - (he - k e ) a c l + (JceZs - k e a s )

k(G1

- . C l ) - ( f c i -

f c i )

= k,e^,l - ep (5) h A = A 4 . A

This equation induces

%ZZl

+

ef = k,s1 - e p . More-

over, assume that 0

<

up

<

IC,

<

uy and 1x,I

<

a y ,

then let the real controller be given as

f

= ? + A f (6)

Then, the dynamic equation involving the sliding mode vector s is given as

MS = - K , s - C S

+

ACS

+

h

+

AMX, - A f , _{( 7 )}

where K , = d i a g & / A ~

+

k l , k z , . . . , k J , h(q,q)

=

ACXc

+

AG

+

A D is a n uncertain nonlinear vector, ep =

[

e p , 0,

...,

0 I T . Now, consider the ideal case

where the uncertaity of the manipulator dynamics can

be exactly approximated, so that

A f

be designed as follows :

A f = h + A M X ,

+

A C s (8)

Then equation (7) becomes

M i

=

-K,s

- C S

+

A C s

+

h

+

AMX, - A f

= - K , s - C S

h h

Furthermore, let k , , k , z , be updated as follows:

-

h

ke =

-41

=

r l e p z c l i k e ( 0 )

>

0 ( 9 )

h

k e a ,

=

- 4 2 = -r2ep (10) h

where 41 and 4 2 are defined as 41

=

k ,

-

k , and

4 2

=

k e z s - k e z , , and and 7 9 are some positive

constants. Thus, the following proposition will sum- marize the resulting properties due t o the aforemen- tioned control.

Proposition 1 If

the control laws are given as

in

equations

(4),

(fi), (8) and the update law are given by equations (9),

[IO)

t h e n the force and position track- ing errors of the robot m a n i p d a t o r s s y s t e m (3) will converge t o zeros

Proof:

Define Lyapunov functions Vl(t) and V2(t) as Vl(t) =

s T M s and V2(t) =

&

41’

+

1

4z2,

and take their time derivatives, respectively, as

h 21.a 1

vi

-sT(Ak

-

2 C ) s

-

s T K , s 2

=

- s T K , s < O = -(41ac1)2

+

2 4 1 4 2 z c 1 - 4 2 2 = -(41%1 - 4 2 1 , - 2 1 _-((ke

-

_{L ) x c 1 - (kexs}

-

_{k e z s ) )}

=

5

o

Then, it is obvious that all signals inside the system are bounded. Howerever, by invoking the Barbalat’s lemma [ g ] , it can be easily verified that s + 0 and e p ---f 0, separately, as

t

--+ 00. This result implies

e --+ 0 as

t

-+ 00 [lo]. From equation (5), when

e z l . = 0 and e p =O are obtained, then e 0 is achieved. Hence, as

t

--$ 00, e,2 -+ 0 a n d :e --+ 0

are derived. This completes our proof. Q.E.D.

Remark:

To avoid the singularity in evaluating Z ; l , the update law

is

modified

as

follows.

If

x,

5

of,

A

otherwise as equations ( 9 ) , (10) will be assumed. This mlodification of the update law also guarantees the positivity of

2,

and, hence, the positive definiteness of

K,.

1, the problem of force control is reduced into the robust position control problem, which is how t o design A f properly. Furthermore, we let A f

=

f h

+

f,, where f h is designed t o approximate

the uncertainty of the manipulator dynamics, namely,

h, and

f ,

is regarded as a robust compensator for the remaining uncertainty.

A

proper

A f

is thus defined such that f h approximates

h

as closely as possible and

f, has a minimum gain. In the next section, we will build a robust controller using the scheme presented in the next section.

4

Adaptive Fuzzy Hybrid Force

/

Po-

Referring t o section 2, the robust control law is Based on Prop.

sition Control

given as follows:

f ( t )

=

?+

f ;

+

f ; , ₍₁₃₎

A

where

f ; =

h, which is supposed t o be a n optimal approximation of h in the terms of fuzzy-knowledge representation; f,’ is defined as follows:

f : k

=

( I % k l + UMIIXrlloo)Ssn(sk),

where k represents the k-th element of vector, E h =

h - h, e, = E h

+

A C s and

U M

=

I

A M I [ , are given. Here, we will approximate

f i

and

kz

by fuzzy knowl- edge represent

at

ion.

-

4,.1

Fuzzy

Knowledge Representation

of

and f :

via fuzzy knowledge representation, where a fuzzy rule-base generally consists of a collection of ifithen

rules. Now, take f: as a n example t o describe this representation. First, let Z h = [xCT x c

]

=

[ Z h l , . .

..,

Z h l n l T be denoted as an input linguistic

vector in the discourse universe

U,,

and

Lj

=

L.f

,

Lj”,

* * a ,

L s j ,

-.

.

,

Lrzhj as a family of fuzzy sets as-

sociated with the membership functions pLyjYs (see Fig. 1) with respect t o the variable Z h j , where

L.?

is a fuzzy set in L j . Besides, the cores of the family of fuzzy sets, L j , axe denoted as & j =

{ , ~ ~ j , Z ~ j , . . . , Z ~ ~ , . . . , Z ~ ~ j ~ , where 2:; is a core

point satisfying

<

z;:

<

- - -

<

2;;’

(allso see Fig. 1). Let

L

be defined as a product set of

L

=

n,=,

Li,

consisting of the families of fuzzy sets,

&,

i

= 1,

-

,

2n. Then, an example of the i-th fuzzy rule is represented as follows:

Uncertainties

Here our objective is t o approximate

f

* T T

3

<

zij

<

R[i]

: If z h is

La(;),

then y is Q B ( * ) ,

(14)

where

La(;)

E

L,

q i )

=

aq;)

x e-- x a2n(i) is a prod-

uct index associated with the i-th rule, y = f h =

[y1,.*.,ynlT is denoted as an output linguistic vec- another product index associated with the i-th rule, Q is another product set of Q

=

nk=

Q k , consist- ing of the families of fuzzy sets, Q k ,

f

= 1 , .

.

-,

n,

whereQk = { Q ~ , Q ~ , . . . , Q ~ ' , . . . , Q ~ * } i s d e n o t e d a s

a family of fuzzy sets associated with the membership functions p p h with respect t o the output variable yk with Qf' being a fuzzy set in the family Q k . And, let the cores of the family of fuzzy sets,

Q k ,

be denoted as

Yk

= CY,', Y f ,

- -

-,

ytk,

..

e , Y,""}, where

YE"

is a core

point satisfying Y:

<

Y$

<

. . .

<

yEb

<

. . .

<

yLY'.

Furthermore, every rule is fired with a weightingfunc- tion P i ( z h ) , which is determined by membership func- tions and a compositional operator. Hence, & ( Z h ) can be expressed as follows:

tor, and Qp(i) E

Q,

@(i)

=

_{x x} P n ( j ) is

(15) where p - J ( G l ( z h j ) is denoted as a membership func- tion, which is a positive function with p a,(;) ( z h , )

5

1

and reaches the maximum value when z h j = Z;:(') as shown Fig. 1. In expression (15), the compositional operator is selected to be either the sup-product or the sup-min operator. Finally, using defuzzification func- tion, the output variable yk is expressed as follows:

L J

-%

where D k ; ( f J i ( z h ) ) is denoted as a defuzzification func- tion with the core point

Y['(')

for the i-th rule com- monly expressed as follows:

{ p i ( z h ) , if center-average-defuzzifier;

if center-of-area defuzzifier;

(17)

where yk* and y; are solution values of yk satisfying & ( z h ) = p aho(yk) with y;

2

y k * given the values Zh with R h being the total number of rules generally equal to r h l x r h a

x

x

T ~ Then, we denote a ~ ~ ~ .

fuzzy basis function

&g.(zh)

expressed as follows:

Q k

so that equation (16) can be rewritten as: Rh

Yk

=

~ y ~ b " ) ~ h t i ( Z h )

=

@ h k T & k ( z h ) , ₍₁₉₎

i=l

where @ h k =

[Yk

@'(I)

,

y['(a)

,

. . .

,

y [ b ( R h ) ] T

is

re- garded as a parameter vector and &k = [&kl

,

[hk.

,

. . .

,

[ h k R h l T is regarded as a regressor vector. For

simple in notations, we define a product operator 8

for two matrices 0 E

RnxP,

( E W x n as

0 8

E

= [ @ I T & ,

. .

* ,

O k T E k , .

. .

,

OnTIJT (20)

where @I, is the k-th row vector of 0 and & is the k-

t h column vector of

<,

respectively. Hence, we denote the matrices @ h E R n x R h ,

&

E

RRhxn

with the k-th row vector denoted as Ohk and the k-th column vector denoted as E h k , respectively. Then the output vector

y is expressed as

y = Oh '8 <ha (21)

Apparently, when Rh is large, the computation time and memory-space usage must be considered in real implementation. As a result, we focus our attentions on computation time first, then the number of the fired rules and that of the total rules should be clearly distinguished. First, collect the indices of the fired rules as the following set:

I

= {i : & ( a h )

>

O}, (22)

where h ( z h ) is the weightingfunction of the i t h fired rules. Our aim is to figure out that the number of

I,

# n ( I ) , is less than the number of total rule R h durin firing the fuzzy rules along with the expression (167 satisfied.

At the beginning, define the domain set of the total rules with respect to z h as

E = { z h : Z i , < z h J i Z L : J , j = 1 , . . . , 2 n } Therefore, if z h falls into E , then the fuzzy rules are fired to compensate the unknown functions. Partition this domain set into the finite 2n - cells, which are defined as

E ,

= { z h : 2:; < Z h ,

5

~:;+l,j=

i,.-.,2n),

where a = a1 x . . x aJ x

. .

x azn is a product index and satisfies Zg E Z h ,

Z t :

<

Z ; r J . To union the all colection of E,, we can obtain E =

UzZEZh

E,.

A b - box is defined as an equivalent of E,,

n a ( z t , b ( z h ) ) = ( z h :

zt:

5

zhj

5

z::

+

( z h ) } ,

= E ,

where j = 1 , . - - , 2 n , 6 ( Z h ) E !J?2nX1 and the j-th el- ement of 6 ( 8 h ) is defined as 6 j ( Z h ) =

z,J+'

-

z:

;.

Define the set of all corner points of &box R, as P,

=

{ z c ( z h ) : c ( z h ) E

n:Z,{a,,a,

+

1)). Hence the number of

Pa,

#n(P,) is equal to 22n.

h,

Consider the membership function (Fig. 1): 1, as ZhJ

=

Z:;; x;', 3 0, as Zh,

2

z;:"

01 z h 3

5

z

;

:

-

'

;

z; ' E (0, l), otherwise, (23) -866-

for aj. = 1 , 2 , . -! l j and j

=

1 , 2 , . e - , 2n, then the

following proposition is given:

Proposition 2 If the membership f u n c t i o n s are given

as eq. (23) and Zh is the linguistic vector, Z h E

E ,

t h e n ezists a 6-boz, n , ( Z t , b ( z h ) ) such that z h E n a ( Z ; , 6 ( z h ) ) , and # n ( I )

5

#n(P,), where

#n(l>

and #n(P,) are the number of fired f u z z y rules and that of corner points of b-boz, respectively.

Hence, the computation time is drastically reduced, especially, as

Rh

is very large.

4.2

Adaptive Robust Fuzzy Hybrid Con-

trol

Based on the fuzzy knowledge representation in subsection 4.1, we will use this representation to ap- proximate the robust compensator

f

:.

Besides, the parameter matrices are also t o be determined by adap- tive scheme. Let z , = [ x C T i c T s T I T = [zS1,

.",

Z s k ,

.

- a , z s 3 J T be denoted as another input linguistic vec-

tor with respect t o

f,

in the discourse universe

Us,,

and the cores of the family of fuzzy sets with respect t o z , k are denoted as

z , k =

{ Z ~ ~ , Z ; ~ , . - . , Z ; ; ~ ] ,

where

z;

<

ZSfh

<

* - *

<

Z:;h and

k. = 1 , 2 , . ,3n.

Let

f,

is denoted as

f,

= [ f S l ,

. .

-,

f S k ,

. . .

,

f S n l T , where

f , k be expressed as follows:

f s k = @ a k T t s k

+

@ M k T h k l l i : , l l m , (24)

where @ , I , and @ M k are denoted as parameter vectors, ( s k , [M are denoted as regressor vectors as in equa- tion (197, and

R ,

denotes the total number of rules for

f,.

Hence the robust compensator vector

f,

can be expressed as follows:

fa = 0,

e9L

+ @ M

@EMIIX'rIlm,

₍₂₅₎

where the matrices

Os,

@M E X n X R s consit of the k-th row vectors @ M k , respectively, and &, [ M

E

R R s x n

consist of the k-th column vector

tSk,

&k for k = 1,

- -

.

,

n, respectively. Our goal is t o design the optimal parameter matrices such that the controlled system has minimal tracking errors and robust fea- tures. At the beginning, we must prove that bounds on approximation errors depend on our design. The the k-th column vectors of optimal parameter matrices

0;. 0: and 0;M are defined as follows:

Proposition 3

if

the linguistic vector z h f a h into

6

-hoz, n,(ZF, 6 ( z h ) ) , t h e n the k - t h element of optimal ajDproximation error vector c h ( z h ) will be bounded b y

I

Ehkl

-

<

g k ( z h ) T 6 ( z h ) , where c h ( z h ) = h - 0; @ [ h , g k

is the k - t h row vector of m a t r i z g ( z h ) , g ( z h ) E

E n x 2 n 2

and its element is defined as g i j = supZhEa,

[lvll

f o r i = l , . . * , n , j = 1 , . . . , 2 n

Now, let the control law be redesigned as

f

=

d(t)(fh

+

fs)

+

(1

-

d ( t ) ) G ( Z S )

+

T,

(29)

f i r s ( z s ) = ( h u ( z h )

+

MulbfFllm

+

C f I I S l l m ) S g n ( S ) (31)

Update laws are given as follows :

6 -

h - r t h ( z h ) d i a g ( s g n ( s a ) ) as Z h E E (32)

6*

= rtM(zs)diag(sgn(sA))llX,llm as z s E

E ,

(34)

6 ,

= r(,(z,)diag(sgn(sa)) as z , E

E,

(33)

T

fI3T Some T

>

0 and S A = [ s A l , . ' " s A k , ' . . , S a n ]

,

{

::

a k i as s k

<

ak;

S a g = s k

-

b k , sk

>

b k ; ₍₃₅₎

othewise. S A k = i k

where a,

b

E

?12"

are some constant vectors and their the k-th elements satisfying a k

5

z~~~~~

5

0

5

+ a , + h + l

L ~ ~ ~ , , + ~

Theorem 1 If the controllaw and the update laws are

griven as in eq. (29) and

in

eq. (32)-(34)) t h e n tracking errors will asymptotically converge to a neighborhood of zero.

5

b k for someone index C Y Z ~ + ~ .

5

Simulation and Experimental Re-

A five degree-of-freedom (DOF) articulated robot arm is set up in the Intelligent Robot Laboratory of CS&IE in NTU as shown Figure 2 . A simulation

sults

@+,,

=

arg min[supzhEE iOh:thk -

f l k [ ]

( 2 6 ) @ : k = arg m i n [ s z l p , ~ E E . @ s T t s k s g n ( s k )

2

/E,kl (27)

is run in assumption that inGtia1 matrix M , Cori- olis, centrifugal torques 6, and gravity vector G of the arm are unknown. To show the effectiveness of adaptive robust law,

f,

,we will neglect the compensa- tion function fh. the position trajectories is given to

@ b k = arg m i n [ s . l l p z ~ E E . @ M ~ t M k S g n ( s k )

2

U M k ] (28)

where

E,

is deined as follows: move a cirle with period, 4 secon2 and radius, 5 cm

in x, - y c plane, and in z, direction force trajectories

is given as f d l = 10

-

1 0 e z p ( - t )

Nt.

A simplified

is-given. compositional operators. The total rule number of

fuzzy controller is 5 x 21 x 21 and initial parameter ma- trices of fuzzy rules are set t o zeros. The real parame- ters are given as IC, = lOOOO(Nt/m) and x, = 5(mm),

and initial estimate values are given as

&(O)

= 5000 and

k

s

8

(

=

0

0. Simulation is run by 4-order Runge- the force error e f result are listed in Figure 3. At the beginning, sinice initial parameter matrices are aet to zeros, which is similar t o only use the PD controller t o compensate for the uncertainties in the first period of circle motion, after the first period the tracking error is quickly driven toward zero. On the other hand, the virtual force error ep is given in the Fig. 4, we can find the error is fast converging t o zero.

Kutta met

h

od with fixed sampling time 0.5ms and

6

Conel1rsions

We had proposed a novel fuzzy hybrid

force/position controller design method, which can up- date fuzzy rules t o compensate unknown uncertainty of robot manipulator along with adaptive estimation

of unknown parameters of task environment and guar-

antee the stability of the controlled robot manipulator systems. Besides, a dexterous use for fuzzy mathe- matics made the complicated scheme with computing efficiency. Present work is t o implement this control algorithm t o actual experiment.

eferences

[l] S. Chiaverini, L. Sciavicco “The Parallel Approach t o Force/Position Control of Robot Manipulators”

IEEE Transactions o n Robotics and A u t o m a t i o n vo. 9, n o .

4,

p p . 361-373, Aug. 1993

[2] R. Colbaugh, A. Engelmann “Adaptive Compliant

Motion Control of Manipulators: Theory and Ex-

periments ” IEEE Robotics and A u t o m a t i o n C o n f ,

~ ~ 2 7 1 9 - 2 7 2 6 , 1994

[3] G. Niemeyer, J-J

E.

Slotine, “ Performance in

Adaptive Manipulator Control

”,

T h e Interna- tional J o u r n a l of Robotics Research, vol. 10, n o . 2, p p . 149-161, 1991.

[4] K. Guglielmo, N . Sadegh “ Implementating a Hy-

brid Learning Force Control Scheme ” IEEE C o n -

t r o l S y s t e m s vo. 14, n o . I , pp. 72-79 Feb. I994

[5] D. Jeon, M. Tomizuka “Learning Hybrid Force and

Position Control of Robot Manipulators” IEEE

Transactions o n Robotics and A u t o m a t i o n vo. 9, n o .

4,

pp. 423-432, Aug. 1993

[6] Sheng Liu and H. Asada, “Teaching and Learn-

ing of Deburring Robots Using Neural Networks”

IEEE Conference o n Robotics and A u t o m a t i o n , pp. 339-345, 1993

[7] M. H. Raibert and J. J. Craig, “Hybrid Posi-

tion/Force Control of Manipulators”

ASME

Jor-

n a l of D y n a m i c S y s t e m s , M e a s u r e m e n t , and Con- trol vol. 102 p p . 126-133, I981

[8] Daniel E. Whitney, “Historical Perspective and State of the Art in Robot Force Control”, T h e In- ternational Journal of Robotics Research, vol. 6, n o . 1, p p . 3-14, 1987.

[9]

S. Sastry and

M.

Bodson, Adaptive Control: Sta- bility, Convergence, and Robustness, Prentice Hall, pp.19 1989.

[lo] J-J E. Slotine and W. Li, Applied Nonlinear C o n -

trol, Englewood Cliffs, NJ: Prentice Hall, pp. 278- 284, 1991.

[ll]

J.

Marsden, Elementary Classical Analysis. San Franciso: W. H. Freeman, 1974

5, !

zM ‘h, zhj z?

‘hJ-

Fig. 1 Tlie

fuzzy

enviroment of linguistic variable

Fig. 2. Tlie diagram of A-type robot

0 2 4 b 8 1 e l 4 ; 2 4 6 8 I O 21’ t i m e ( s e c )

Fig. 3. Force error (Nt)

V.S.

time(sec)

7 6

0 ‘ 2 4 6 8 I O P

Fig

4

Virtual force error (Nt) V.S time(sec)