Fuzzy moving sliding mode control with application to robotic manipulators

Automatica 35 (1999) 607—616

Brief Paper

Fuzzy moving sliding mode control with application torobotic manipulators1

Q.P. Ha*, D.C. Rye, H.F. Durrant-Whyte

Department of Mechanical and Mechatronic Engineering J07, The University of Sydney, 2006 NSW, Australia

Received 2 December 1997; revised 25 May 1998; received in final form 11 August 1998

Abstract

This paper presents a fuzzy tuning approach to sliding mode control for tracking-performance enhancement in a class of nonlinearsystems. The sliding surface can rotate or shift in the phase space in such a direction that the tracking behaviour can be improved. It isshown that with arbitrary initial conditions, the reaching time and tracking error in the approaching phase can be significantlyreduced. Chattering can also be reduced by fuzzy tuning of the controller parameters. A two-degree-of-freedom robotic manipulatorsubject to external disturbances is simulated to demonstrate the validity of the proposed method. ( 1999 Elsevier Science Ltd. Allrights reserved.

Keywords: Fuzzy tuning; Moving sliding mode; Fast tracking; Robustness

1. Introduction

In recent years, the sliding mode control methodologyhas been widely used for robust control of nonlinearsystems (Slotine and Li, 1991). Sliding mode control,based on the theory of variable structure systems, hasattracted a lot of research on control systems for the lasttwo decades. A comprehensive survey on variable struc-ture control was given in Hung et al. (1993). The salientadvantage of sliding mode control is robustness againststructured and unstructured uncertainties. In path track-ing systems, however, the system invariance propertiesare observed only during the sliding phase. In the reach-ing phase, tracking may be hindered by disturbances orparameter variations. The straightforward way to reducetracking error and reaching time is to increase the con-trol discontinuity gain. Nevertheless, this may cause se-vere chattering which is undesirable in some dynamic

—————

*Corresponding author. Tel.: #61 2 9351 3098; fax: #61 2 93517474; e-mail: [email protected].

1This work is supported by a grant from the Australian ResearchCouncil, by NS Komatsu Pty. Ltd., and by the Centre for MiningTechnology and Equipment. This paper was not presented at any IFACmeeting. This paper was recommended for publication in revised formby Associate Editor K.W. Lim under the direction of Editor C.C. Hang.

systems. Several chattering reduction methods based onthe use of a boundary layer in the sliding mode (Slotineand Sastry, 1983; Spurgeon, 1991) have been reported.These approaches are known to degrade robustness. Thecontinuous sliding mode control strategy proposed byZhou and Fisher (1992) can drive the system state toa chattering-free sliding mode but tends to produce con-servative designs. Another method for reducing chatter-ing without the sacrifice of robust performance is tocombine the attractive features of sliding mode controland fuzzy control (Lu and Chen, 1994; Choi and Kim,1997). In general, there is a direct trade-off betweenchattering and robustness. A combination of fuzzy logictuning schemes and sliding mode control (Ha, 1996b)is described for accelerating the reaching phase andreducing the influence of unmodelled uncertainties,thus improving system robustness. Whereas only linearor linearised systems are considered in Ha (1996b), thispaper applies fuzzy tuning to moving sliding surfaces forfast and robust tracking control for a class of nonlinearsystems. A moving sliding surface, proposed in Choi andPark (1994) and Choi et al. (1994) for fast tracking withrotating or shifting surfaces, is adaptable to arbitraryinitial conditions. In the proposed scheme, however, thesliding surface is shifted and/or rotated instantaneously,and after each movement the system representative pointis no longer on the surface. Consequently, the system is,

0005-1098/99/$—see front matter ( 1999 Elsevier Science Ltd. All rights reservedPII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 1 6 9 - 1

Fig. 1. Rotating sliding surface.

at least for some time, sensitive to disturbances andparameter uncertainties, as pointed out in Bartoszewicz(1995). In this work, the piecewise time-constant coeffi-cients of the sliding surface (Choi and Park, 1994; Choiet al., 1994) or their modified forms (Bartoszewicz, 1995)are substituted by the continuous-time functions ob-tained by fuzzy tuning. Using the fuzzy moving slidingsurface, it is shown that the tracking behaviour isimproved remarkably in terms of fast tracking androbustness. The path tracking control of a two-degree-of-freedom robotic manipulator subject to externaldisturbances and parameter variations is simulated todemonstrate some eminent advantages of the proposedmethod, and to verify its efficacy and superiority.

2. Problem formulation

Consider the dynamics of a rigid robotic manipulator(Choi and Kim, 1997; Choi and Park, 1994) described bya coupled second-order nonlinear system of the form

xR2k~1

(t)"x2k

(t),

x2k

(t)"fk(x,t)#b

k(x)u

k(t), k"1, 2,2 , m, (1)

x0"x(t

0),

where x"[x1(t) x

2(t) 2 x

2m]T3R2m is the state vector

with the initial conditions x0"x(t

0), u3Rm is the control

vector, and bk(x)'0 is the control gain. The nonlinear

dynamics fk, k"1, 2,2 , m are not known exactly, but

are estimated as fLk

with an error bounded by a knownfunction F

k(x):

D fk!fL

kD4F

k(x). (2)

Let x8 "[xJ1(t) xJ

2(t) 2 xJ

2m]T3R2m be the desired

trajectory with xJi3C1[t

0,R), i"1, 2,2 , 2m. The

control problem is to drive asymptotically the trackingerror

e"[e1(t) e

2(t) 2 e

2m]T

"[x1(t)!xJ

1(t), x

2(t)!xJ

2(t),2 , x

2m!xJ

2m(t)]T (3)

to zero for any initial states, x0"x(t

0), given at an initial

time t0. A derivation of the control law begins with the

choice of a typical sliding surface

Sk"e

2k#c

ke2k~1

, ck'0. (4)

To eliminate the reaching phase, the condition e0"0 is

imposed in Slotine and Li (1991). Then, given positiveconstants c

k, system (1) with the control law

uk"b~1

k[uL

k!(F

k#g

k) sgn(S

k)], (5)

where

uLk"!fL

k#x8R

2k!c

ke2k

(6)

satisfies the sliding condition

SkSQk4!g

kD S

kD, g

k'0. (7)

The initial errors may, however, be located arbitrarily inthe error state space and thus are generally not zero.Moreover, the system may be subject to parameter vari-ations and external disturbances. Consequently, the con-trol law (5) and (6) may be sensitive to uncertaintiesduring the reaching phase. The problem is how to adjustthe control law in order to achieve fast and robust track-ing of the state variables, x

2k~1, k"1, 2,2 , m, during

both the reaching phase and the sliding mode.

3. Fuzzy rotating and shifting sliding surface

Inspired by the works of Choi and Park (1994), Choiet al. (1994), and Bartoszewicz (1995), we propose fuzzytuning schemes to continuously move sliding surfacessuch that fast tracking is obtained. The sliding surfaceslopes, c

k, are bounded by maxima, c

k ,.!9, and minima,

ck,.*/

. Fig. 1 shows the region for possible slopes ofrotating sliding surfaces in the stable zone of the phaseplane (e

2k~1, e

2k) (the second and fourth quadrants).

System performance is sensitive to the sliding surfaceslope c

k. If large values of c

kare available the system will

be more stable but the tracking accuracy may be de-graded because of a longer reaching time of the represen-tative point to the surface S

k. In mechanical systems, the

value for ck,.!9

is typically limited by three factors: thefrequency of the lowest unmodelled structural mode,the largest unmodelled time delay, and the sampling rate(Stoline and Li, 1991). Conversely, if small values of c

kare

chosen, the convergence speed on the sliding surface itselfwill be slow, leading to longer tracking times. The valuefor c

k,.*/is therefore determined by the allowable track-

ing time.

608 Q.P. Ha et al. /Automatica 35 (1999) 607—616

Fig. 2. Shifting sliding surface.

Recalling that a larger value for ck

will result ina longer reaching time, we propose to decrease c

kwhen

De2k~1

D is large, and vice versa. The fuzzy rule for tuningckcan thus be formalised as

If error is large (EL) then slope is small (CS),If error is small (ES) then slope is large (CL).

By defining the following membership functions:

kES"2/[exp(!e

rk)#exp(e

rk)]"1/cosh(e

rk),

kEL"1!k

ES, (8)

kCL"G

1, ck"c

k,.!9,

0, ckOc

k,.!9,

kCS"G

1, ck"c

k,.*/,

0, ckOc

k,.*/,

and using the max—min inference for defuzzification (Ha,1996a) the value of c

kcan be found as

ck"c

k, .*/#*c

k/cosh(e

rk), (9)

where *ck"c

k, .!9!c

k, .*/, e

rk"e

2k~1/p

erkis some pos-

itive constant.

Theorem 1 (Rotating sliding mode). If the controller (5)is chosen as

uLk"!fL

k#x8R

2k!Cck#

*ckerk

sinh(erk)

cosh2(erk) D e

2k, (10)

with the value for ck

given by Eq. (9), the system (1)incorporating the sliding surface (4) then satisfies thesliding condition (7).

Proof. Let us define a Lyapunov function »k"1

2S2k

withSk"e

2k#c

ke2k~1

. The derivative of the Lyapunov func-tion can be expressed as

»Qk"S

kSQk"S

k[eR

2k#c

keR2k~1

#cRke2k~1

]

"Sk[xR

2k!x8R

2k#c

ke2k#cR

ke2k~1

]

"Sk[ f

k#uL

k!(F

k#g

k) sgn(S

k)!xJ R

2k#c

ke2k

#cRke2k~1

].

Using the fuzzy tuned expression (9) for the slope ck, we

have

cRk"2

*ck[exp(!e

rk)!exp(e

rk)]

perk

[exp(erk)#exp(e

rk)]2

e2k"

*cksinh(e

rk)

perk

cosh2(erk)e2k

.

Substituting Eq. (10) into the expression of »Qkthen yields

»Qk"S

k[ f

k!fL

k!(F

k#g

k) sgn(S

k)], g

k'0.

It can be shown from Eq. (2) that

»Qk4D S

kD[F

k!(F

k#g

k) sgn (S

k)],

so that the asymptotic stability condition (7)»Q

k"S

kSQk4!g

kD S

kD is satisfied by »Q

k. This concludes

the proof. h

It is clear that if initial conditions are located in thefirst and third quadrants, robustness derived froma rotating sliding surface may be absent during the reach-ing phase. Taking this into consideration, we propose thefollowing sliding surface which can shift towards therepresentative point in order to reduce the reaching time:

Sk"e

2k#c

ke2k~1

#ak. (11)

The intercept akis to be tuned by fuzzy logic according to

the tracking error e2k~1

. Fig. 2 illustrates the region ofpossible shifting sliding surfaces. Note that given a con-stant slope c

k, the intercept a

kplays the role of a bound-

ary layer (Slotine and Li, 1991). If ak'0 the sliding

surface is shifted increasing Skand if a

k(0, it is shifted

decreasing Sk. The value for a

kis bounded by a constant

ak,.!9

that may be determined by the guaranteed pre-cision for tracking. The larger the constant a

k, .!9is, the

more tolerant the tracking error can be expected. Hence,when D e

2k~1D is large we intend to increase D a

kD, and vice

versa. Also, if the tracking error is negative, the slidingsurface should shift upwards, i.e. a

kis positive, and vice

versa. These considerations result in the following fuzzyrules for tuning a

k:

If error is positive large (EPL) then intercept is nega-tive large (ANL),

If error is positive small (EPS) then intercept is nega-tive small (ANS)

or

If error is negative large (ENL) then intercept is posit-ive large (APL),

If error is negative small (EPS) then intercept is posit-ive small (APS).

Q.P. Ha et al. /Automatica 35 (1999) 607—616 609

By introducing the following membership functions:

kEPL

"tanh(esk), k

EPS"1!k

EPL,

kANL

"G1, a

k"!a

k,.!9,

0, akO!a

k,.!9,

kANS

"G1, a

k"0,

0, akO0,

or

kENL

"!tanh(esk), k

ENS"1!k

ENL,

kAPL

"G1, a

k"a

k,.!9,

0, akOa

k,.!9,

kAPS

"G1, a

k"0,

0, akO0,

(12)

the value for akcan be found as

ak"a

k, .!9tanh(e

sk) , (13)

where esk"e

2k~1/p

eskand p

eskis some positive constant.

Theorem 2 (Shifting sliding mode). If the controller (5) ischosen with a constant c

kand

uLk"!fL

k#x85

2k!Cck!

(ak,.!9

/pesk

)

cosh2(erk) D e

2k, (14)

then system (1) incorporating the sliding surface (11), withthe value of a

kgiven by Eq. (13) satisfies the sliding condi-

tion (7).

Proof. We define a Lyapunov function »k"1

2S2k

withSk"e

2k#c

ke2k~1

#ak. The derivative of the Lyapunov

function can be expressed as

»Qk"S

kSQk"S

k[eR

2k#c

keR2k~1

#aRk]

"Sk[xR

2k!x8R

2k#c

ke2k#aR

k]

"Sk[ f

k#uL

k!(F

k#g

k) sgn(S

k)!x8R

2k#c

ke2k#aR

k].

Using Eq. (13) for tuning the intercept ak, we have

aRk"!

(ak,.!9

/pesk

)

cosh2(erk)

e2k

.

Thus, substituting Eq. (14) into the expression for»Qkyields

»Qk"S

k[ f

k!fL

k!(F

k#g

k) sgn (S

k)], g

k'0.

From Eq. (2) it is evident that »Qksatisfies the asymptotic

stability condition (7):

»Qk"S

kSQk4!g

kD S

kD .

This concludes the proof. h

Remark 1 (Rotating and shifting sliding mode). Intuitively,we may combine the rotating and shifting sliding surfacesto get better results in terms of fast reaching time. It canbe shown that with the continuous control law uL

kcom-

bined from Eqs. (10) and (14) as

uLk"!fL

k#x8R

2k

!Cck#*c

kerk

sinh (erk)

cosh2 (erk)

!

(ak,.!9

/pesk

)

cosh2 (erk) D e

2k, (15)

system (1) incorporating the sliding surface (11), wherethe slope c

kand intercept a

kare, respectively, given in

Eqs. (9) and (13), satisfies the sliding condition (7).

4. Moving sliding surface

In practice, the control gain bk

in Eq. (1) is usuallyuncertain within known bounds

0(bk,.*/

4bk4b

k, .!9. (16)

Choosing the estimated gain to be the geometric mean ofits extrema

bLk"(b

k,.*/bk, .!9

)1@2, (17)

we have

b~1k

4bLk/b

k4b

kand b~1

k4b

k/bL

k4b

k, (18)

where bk"(b

k, .!9/b

k, .*/)1@251.

Control law (5) is rewritten as

uk"bL ~1

k[uL

k!K

ksgn (S

k)] , (19)

where the discontinuous control gain Kkwill be defined

in the following theorem.

Theorem 3 (Moving sliding mode). For the system (1)satisfying Eqs. (2) and (16), if the estimated continuouscontrol law uL

kis chosen as in Eq. (15) with the values of

ckand a

kgiven in Eqs. (9) and (13), and the controller (19) is

chosen with

Kk"b

k(F

k#g

k)#(b

k!1) D uL

kD , (20)

then the system (1) incorporating the sliding surface (11)satisfies the sliding condition (7).

Proof. We define a Lyapunov function »k"1

2S2k

withSk"e

2k#c

ke2k~1

#ak, where c

kand a

kare respectively

given in Eqs. (9) and (13). The derivative of the Lyapunovfunction can be expressed as

»Qk"S

kSQk"S

k[eR

2k#c

keR2k~1

#cRke2k~1

#aRk]

"Sk[xR

2k!x8R

2k#c

ke2k#cR

ke2k~1

#aRk]

"Sk[ f

k#b

kuk!x8R

2k#c

ke2k#cR

ke2k~1

#aRk].


Fig. 3. A two-degree-of-freedom ‘‘Stanford’’ manipulator.

Following the proofs of Theorems 1 and 2, substitutionof Eq. (15) into the expression for »Q

kyields

»Qk"S

kM f

k!b

kbL ~1k

[ fLk!x8R

2k!(c

ke2k#cR

ke2k~1

#aRk)

!Kksgn (S

k)]!x8R

2k#c

ke2k#cR

ke2k~1

#aRkN

"SkM f

k!b

kbL ~1k

fLk#(b

kbL ~1k

!1)[x8R2k!(c

ke2k

#cRke2k~1

#aRk)]!b

kbL ~1k

Kksgn (S

k)N.

»Qksatisfies the asymptotic stability condition (7) if

bkbL ~1k

Kk5D f

k!b

kbL ~1k

fLk#(b

kbL ~1k

!1)

][x8R2k!(c

ke2k#cR

ke2k~1

#aRk)] D#g

k

or

Kk5D bL

kb~1k

( fk!fL

k)#bL

kb~1k

fLk!fL

k#(1!bL

kb~1k

)

][x8R2k!(c

ke2k#cR

ke2k~1

#aRk)] D#bL

kb~1k

gk

"DbLkb~1k

( fk!fL

k)#(1!bL

kb~1k

)[!fLk#x8R

2k

!(cke2k#cR

ke2k~1

#aRk)] D#bL

kb~1k

gk

"D bLkb~1k

( fk!fL

k)#(1!bL

kb~1k

)uLkD#bL

kb~1k

gk.

Thus, taking Eq. (18) into consideration, if Eq. (20) satis-fies then we have S

kSQk4!g

kD S

kD . This concludes the

proof. h

Remark 2. While there is no increase in the discontin-uous control gain required in the cases of rotating andshifting sliding mode, the discontinuous control gain (20)has been increased in order to account for uncertaintyin the control gain. This may exacerbate chattering.A boundary layer around switching introduced in Slotineand Sastry (1983) may be used to remedy this situation.A compromise between sliding performance and chatter-ing can then be attained via an appropriate selection ofthe boundary thickness.

Remark 3. An alternative approach to reducing thechatter is to replace the signum function ½"sgn(S) bya saturating continuous approximation obtained byfuzzy inference. For this approach, the following fuzzyrules are introduced:

If S is positive (SP) then ½"1; If S is zero then ½"0;If S is negative (SN) then ½"!1;

Defining the following membership functions, forS50,

kSP"tanh(s

k), k

SZ"1!k

SP,

kY/1

"G1, ½"1,

0, ½O1,kANS

"G1, ½"0,

0, ½O0,

or for S40,

kSN"!tanh(s

k), k

ENS"1!k

ENL,

kY/~1

"G1, ½"!1,

0, ½O!1,kY/0

"G1, ½"0,

0, ½O0,(21)

we obtain a common expression for the two cases, S50and S40:

½"tanh(sk), (22)

where sk"S

k/p

Skand p

Skis some positive constant. Al-

though smoothing of the control signal can be obtained,tracking performance will then depend on properlytuning the parameter p

Sk.

5. Application to a two degree-of-freedom roboticmanipulator

A two-degree-of-freedom robotic manipulator, studiedby Fu and Liao (1990), Choi and Park (1994), and Choiand Kim (1997), is used here to illustrate the efficacy ofthe proposed method. The manipulator for hozizontaltracking is shown in Fig. 3. In this figure, k and M arethe mass of the motional link and the payload, respec-tively; J

1and J

2are the moments of inertia the motional

link with respect to the vertical axis through the centre ofmass C and the origin O, respectively; a is the distancefrom C to the centre of the payload; and the controlinputs u

1and u

2are a force and a torque required to

execute the desired motion.The state equation can be written (Choi and Park,

1994) as follows:

xR1"x

2,

xR2"[kx

1#M(x

1#a)]x2

4/(k#M)

#[u1(t)#d

1(t)]/(k#M),

xR3"x

4,

xR4"M!2[kx

1#M(x

1#a)]x

2x4#u

2(t)

#d2(t)N/[J

1#J

2#kx2

1#M(x

1#a)2],


Fig. 4. Trajectories with different CSSs.

where di(t) is the unknown but bounded external

disturbance, D di(t) D4d

i, i"1, 2. Assuming that the

payload mass is known, the system can be described byEq. (1) with

f1"M[kx

1#M(x

1#a)]x2

4#d

1(t)N/(k#M),

b1"1/(k#M),

f2"M!2[kx

1#M(x

1#a)]x

2x4#d

2(t)N/

[J1#J

2#kx2

1#M(x

1#a)2],

b2"1/[J

1#J

2#kx2

1#M(x

1#a)2].

Choosing fL1"[kx

1#M(x

1#a)]x2

4/(k#M) and

fL2"!2[kx

1#M(x

1#a )]x

2x4/[J

1#J

2#kx2

1#

M(x1#a)2], we have

D f1!fL

1D"D d

1/(k#M) D4d

1/(k#M)"F

1

and

D f2!fL

2D"D d

2/[J

1#J

2#kx2

1#M(x

1#a)2] D

4d1/(J

1#J

2)"F

2.

Suppose that a desired trajectory is chosen to be

x81(t)"G

!0.75 sin(nt/20) m, 04t410,

!0.75 m, t510,

x83(t)"G

2n sin(nt/20) rad, 04t410,

2n rad, t510,

so that the desired output motion (end-point trajectory)in the (x, y)-plane becomes

x(t)"(xJ1(t)#a) cos (x8

3(t)),

y(t)"(x81(t)#a) sin (x8

3(t)).

For the purpose of simulation the following numericalvalues are employed: k"M"1 kg, J

1"J

2"1 kgm2,

a"1 m, d1(t)"0.2 cos(5nt) N, d

2(t)"0.5 cos(5nt) Nm,

and the initial conditions are

[x1(0),x

2(0),x

3(0), x

4(0)]T"[!0.15,!0.2, 0.349,

0.987]T.

With d1"0.2 and d

2"0.5, we have F

1"0.1 and

F2"0.25. Choosing g

1"0.1 and g

2"0.25, we obtain

the same discontinuous control gains F1#g

1"0.2 and

F2#g

2"0.5 as are used in Choi and Park (1994).

Assume that the values for the slopes c1

and c2

liebetween c

i,.*/"1 and c

i, .!9"9, i"1,2. Using the con-

ventional sliding surface (CSS) with c1"c

2lying in the

interval [1,9] gives the tracking time ranging from 3.78 to7.16 s as shown in Fig. 4, where the tracking time ¹

5is determined on the basis that the tracking errorbetween the desired and actual trajectories becomes 1%.

Note that with the use of the optimal sliding surfaceand moving sliding surface (MSS) proposed by Choi andPark (1994), the tracking time is found to be 3.161 and1.953 s, respectively.

Using the proposed fuzzy rotating sliding surface(FRSS) with p

er1"p

er2"0.04, or c

i"1#8/

(cosh(ei/0.04)), i"1, 2, gives a tracking time of ¹

5"

1.894 s. Fig. 5 shows the control responses obtained byemploying FRSS in comparison with CSS (c

1"c

2"5).

It is observed that employing FRSS can provide a fastertracking response in comparison with the response ob-tained by employing CSS. The FRSS also provides slight-ly faster tracking than does the response using Choi andPark’s MSS technique. Generally, a faster response willrequire more control effort. It is inherent in the presentcontrollers that fast tracking response is associated withlarger peak values of the control force u

1.

A tracking time of ¹5"2.96 s is obtained by employ-

ing fuzzy shifting sliding surface (FSSS) with pesi"3.25,

and ai,.!9

"9.5, i"1,2. The control responses areshown in Fig. 6. Although it can be seen that the FSSSperformance clearly exceeds that obtained using a clas-sical (fixed) sliding surface, Choi and Park’s MSS tech-nique has a slightly shorter tracking time of 1.953 s. Thisis to be expected, as the MSS technique allows the slidingsurface to both rotate and shift, whereas the FSSS con-troller permits shifting only.

It is expected that a combination of fuzzy rotating andshifting sliding surface (FRSSS) will give better trackingperformance, as shown in Fig. 7. The tuning parametersused in the simulations are c

i,.*/"1 and c

i, .!9"9,

peri"0.04, p

esi"2.5 and a

i, .!9"2, i"1, 2. A tracking

time of ¹5"1.879 s is obtained by employing the result-

ing FRSSS, which is slightly faster that the tracking timereported by Choi and Park.

The simulations presented above demonstrate that theproposed technique can serve to enhance the trackingperformance of systems subjected to external distur-bances. It can also be applied to the case of parameter


Fig. 5. FRSS control responses: (a) trajectory, (b) error e1, and (c)

control u1.

Fig. 6. FSSS control responses: (a) trajectory, (b) error e1, and (c)

control u1.

variations. Indeed, assume that beside external distur-bances, the system is now suffering from a varyingpayload with the mass M within the range ofM

.*/"0 kg and M

.!9"1 kg. The control responses

obtained by employing FRSSS at M"0.5 kg are shownin Fig. 8.

Although the FRSSS responses are much more insensi-tive to parameter variations in comparison to the re-sponse obtained by CSS (Fig. 8) with an adequately fast

tracking time of ¹5"2.938 s, robustness of the system

can be further improved by using the fuzzy moving slid-ing surface (FMSS) described in Section 4. With theextremum values chosen as b

1,.!9"1/(k#M

.*/)"1,

b1,.*/

"1/(k#M.!9

)"0.5, b2,.!9

"1/(J1#J

2)"0.5,

and b2,.*/

"1/[J1#J

2#kX2

1m#M

.!9(X

1m#a)2]"

0.108 (X1m

"0.75), from Eq. (18) we obtain b1"1.41

and b2"2.15. The sigmoidal function (22) with

pS,i"0.03, i"1, 2 is used for smoothing the control


Fig. 7. FRSSS control responses: (a) trajectory, (b) error e1, and (c)

control u1

at M"1 kg.


control u1

at M"0.5 kg.

action due to increments in the discontinuous gains (20).The trajectory, tracking error e

1, and control force u

1at

M"0.5 and 1 kg that were obtained by employing theproposed FMSS are shown in Fig. 9. The tracking time isfound to be ¹

5"1.78 and 1.68 s for the values of payload

mass M"0.5 and 1 kg, respectively.It is apparent that the proposed fuzzy moving sliding

surface is able to provide remarkably fast and robusttracking with a smooth control action.

6. Conclusion

We have presented a new approach employing fuzzytuning schemes to adjust sliding surfaces during trackingcontrol of second-order nonlinear systems subject todisturbances and parameter variations. Starting fromany initial conditions of the system states the systemoutputs will asymptotically track the desired trajectorieswith a fast and robust response. The sliding surface



control u1

at M"0.5 and 1 kg.

incorporated with a conventional variable structure con-troller is designed to be continuously tuned with fuzzylogic via rotating and/or shifting according to the track-ing error. Employing the proposed moving sliding sur-face with additional fuzzy tuners significantly shortensthe tracking time and substantially improves the system

parameter and disturbance-insensitivity with smoothed-out control inputs. Simulation results for a two-degree-of-freedom robotic manipulator verifies the efficacy ofthe proposed technique and its superiority to conven-tional sliding mode control in the sense of trackingperformance enhancement in systems suffering fromsevere parameter variations and disturbances.

References

Bartoszewicz, A. (1995). A comment on ‘A time-varying sliding surfacefor fast and tracking control of second-order dynamic systems’.Automatica, 31, 1893—1895.

Choi, S. B., & Park, D. W. (1994). Moving sliding surfaces for fasttracking control of second-order dynamic systems. ASME J. Dyn.Systems Meas. Control, 116, 154—158.

Choi, S. B., Park, D. W., & Jayasuriya, S. (1994). A time-varying slidingsurface for fast and tracking control of second-order dynamic sys-tems. Automatica, 30, 899—904.

Choi, S. B., & Kim, J. S. (1997). A fuzzy-sliding mode controller forrobust tracking of robotic manipulators. Mechatronics, 7, 199—216.

Fu, L. C., & Liao, T. L. (1990). Globally stable robust tracking ofnonlinear systems using variable structure control and with applica-tion to a robotic manipulator. IEEE ¹rans. Automat. Control, 35,1345—1350.

Ha, Q. P. (1996a). PI controllers with fuzzy tuning. IEE Electron. ¸ett.,32, 1043—1044.

Ha, Q. P. (1996b). Robust sliding mode controller with fuzzy tuning.IEE Electron. ¸ett., 32, 1626—1628.

Hung, J. Y., Gao, W., & Hung, J. C. (1993). Variable structure control:A Survey. I EEE ¹rans. Ind. Electron., IE-40, 2—22.

Lu, Y. S., & Chen, J. S. (1994). A self-organising fuzzy sliding-modecontroller design for a class of nonlinear servo systems. IEEE ¹rans.Ind. Electron., IE-41, 492—502.

Slotine, J. J., & Sastry, S. S. (1983). Tracking control of non-linearsystems using sliding surfaces, with application to robot manipula-tors. Int. J. Control, 38, 465—492.

Slotine, J. J., & Li, W. (1991). Applied nonlinear control. EnglewoodCliffs, NJ: Prentice-Hall.

Spurgeon, S. K. (1991). Choice of discontinuous control component forrobust siding mode performance. Int. J. Control, 53, 161—179.

Zhou, F., & Fisher, D. G. (1992). Continuous sliding mode control. Int.J. Control, 55, 313—327.

Quang P. Ha received the BE degree inElectrical Engineering from Ho Chi MinhCity Polytechnic University, Vietnam, thePhD degree in Engineering Science fromMoscow Power Institute, Russia, and thePhD degree in Electrical Engineering fromthe University of Tasmania, Australia, in1983, 1992, and 1997, respectively. He wasa lecturer at the Faculty of Electrical En-gineering, Ho Chi Minh City PedagogicalUniversity of Technology, Vietnam, foreight years. Since 1997 he has been a senior

research associate at the Department of Mechanical and MechatronicEngineering, the University of Sydney, Australia. His research interestsinclude robust control systems, sliding mode control, and applicationsof artificial intelligence in engineering.


David C. Rye received a BE from the Uni-versity of Adelaide, Australia, and the PhDdegree from the University of Sydney, Aus-tralia, both in Mechanical Engineering, in1980 and 1986, respectively. From 1986 toDecember 1987, he served as a lecturer inmechanical engineering in the Departmentof Mechanical Engineering at the Univer-sity of Newcastle, Australia. Since 1988 hehas been with the Department of Mechan-ical and Mechatronic Engineering, theUniversity of Sydney, Australia, as a lec-

turer and then senior lecturer in mechanical engineering. His researchinterests include mechanics of tracked vehicles, autonomous excava-tion, crane dynamics and control, mechatronics and automation, andapplied and nonlinear control.

Hugh F. Durrant-Whyte received the BSc(Eng.) degree (145 class honours) in Mech-anical and Nuclear Engineering from theUniversity of London, UK, in 1983, theMSE and PhD degrees, both in SystemsEngineering, from the University of Pen-sylvania, USA in 1985 and 1986, respec-tively. From 1987 to 1995, he wasa University Lecturer in EngineeringScience, the University of Oxford, UK.Since July 1995 he has been Professor ofMechatronic Engineering at the Depart-

ment of Mechanical and Mechatronic Engineering, the University ofSydney, Australia, where he leads the Australian Centre for FieldRobotics. His research interests include sensor data fusion, sensorsystems, and mobile robotics.


Fuzzy moving sliding mode control with application to robotic manipulators

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