Fuzzy-logic control of dynamic systems: from modeling to...

Fuzzy-logic control of dynamic systems: from modeling to design

M. Reza Emami*, Andrew A. Goldenberg, I. Burhan TuÈ rksen

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ont., Canada M5S 3G8

Received 1 September 1998; accepted 1 June 1999

Abstract

A systematic methodology for the synthesis and analysis of fuzzy±logic controllers for multi-input multi-output nonlineardynamic systems is proposed in this paper. A robust model-based control structure is suggested that includes the fuzzy±logic

dynamics model of the system and several robust fuzzy control rules. The fuzzy±logic model is systematically constructed fromthe input-output data, and the robust control rules are designed using the sliding-mode control theory. The stability andcompleteness of the control structure is guaranteed, based on a generalized formulation of the sliding-mode control developed in

this paper. The proposed fuzzy±logic control scheme has been applied to trajectory control of a four-degree-of-freedom robotmanipulator, and was compared with high-gain PID controllers. Superior tracking performance was achieved. # 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Variable structure systems; Sliding-mode control; Fuzzy-logic control; Robust control; Fuzzy systems; Robot manipulators

1. Introduction

The major part of the research on fuzzy±logic con-trol (FLC) has focused on practical implementations,and successful results have been reported in a widerange applications. Despite the diversity of theapproaches used in the development of fuzzy control-lers, most of them are designed based on `trial anderror'. Although this could be e�ective in some cases,it limits the rise of systematic approaches fuzzy±logiccontrol.

One route to the systematic synthesis and analysis ofthe fuzzy±logic systems is to consider the FLC as aparticular class of nonlinear systems, and to applytools taken from the classical nonlinear control sys-tems theory. A promising approach in this direction isbased on the fact that the FLC is a variable structuresystem (VSS). Variable structure control systems con-stitute a class of nonlinear feedback control systemswhose structure varies, depending on the state of the

system. Recently, new e�orts have been made to inves-

tigate the connection between fuzzy±logic and variable

structure control (Kawaji and Matsunaga, 1991;

Ghalia and Alouani, 1995; Wu and Liu, 1996). Based

on the analysis of these two control approaches, it was

concluded that, due to the partitioning of the input-

output space, the FLC is a qualitative extension of the

sliding-mode control. Some guidelines were speci®ed

for deriving the fuzzy IF-THEN control rules, and for

analyzing the stability and robustness of the fuzzy±

logic control, based on the variable structure system

theory (Palm, 1992), an approach that can be referred

to as `fuzzy sliding-mode control'. However, in the

above-mentioned e�orts, only single-input single-out-

put systems are considered. For multi-input multi-out-

put (MIMO) nonlinear systems, due to the state

interactions, more information from the system is

required, leading to a model-based fuzzy±logic control

approach that is the focus of this paper.

A few researchers have attempted to apply the fuzzy

sliding mode control approach to robot manipulators

(Chen et al., 1994; Tsay and Huang, 1994; Begon et

al., 1995). Despite successful results, the lack of a sys-

tematic approach to the design and analysis of FLC,

Engineering Applications of Arti®cial Intelligence 13 (2000) 47±69

0952-1976/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PII: S0952-1976(99 )00031 -7

www.elsevier.com/locate/engappai

* Corresponding author. Tel.: +1-416-946-3357; fax: +1-416-978-

7753.

E-mail address: [email protected] (M.R. Emami).

based on the sliding-mode control theory, can beobserved.

This paper ®rst introduces a structure for fuzzy±logic control of MIMO nonlinear systems in Section 2.The core of this structure is the knowledge of the sys-tem dynamics that is encapsulated in the form thefuzzy IF±THEN rules. Generally, the dynamics modelis expected to properly predict in real-time the systembehavior, in response to di�erent input trajectories andpayload conditions. Obtaining such a model throughanalytical methods involves two major burdens: (i) it isgenerally di�cult to accurately model some complexphenomena such as backlash, ¯exibility, friction, etc;and (ii) even if these e�ects are formulated, many in-ternal parameters must be accurately identi®ed inadvance, but this requires an exhaustive amount of ex-perimentation. Further, the resulting model would becomplicated and di�cult to use in real time. A fuzzy±logic `black-box' approach could be an alternative tothe analytical methods, provided that all the elementsof the fuzzy model are identi®ed from the input-outputdata. A systematic methodology of fuzzy±logic model-ing using the system input-output data has been intro-duced in (Emami, 1997; Emami et al., 1998a). Section3 brie¯y reviews this methodology, and Section 4 illus-trates how this methodology is applied to obtain thefuzzy±logic dynamics model of a 4-df robot manipula-tor. For the sake of comparison, the results of thefuzzy model are compared with those of a completeanalytical model. The simplicity in terms of systempresentation and model computation e�ort, and thecapability of capturing the complicated system beha-vior, is signi®cant.

By having a good fuzzy model of the system, themajor task of the proposed FLC is accomplished.However, in order to guarantee the stability androbustness of the system performance, and to compen-sate for system uncertainty and knowledge incomplete-ness, additional robust fuzzy rules are supplemented tothe FLC. A systematic approach to design of therobust fuzzy control rules and an analysis of the stab-ility and completeness of the control structure are the

main subjects of this paper. The key idea of the pro-posed approach is to consider the FLC (with crispinput and output) as a multi-dimensional nonlinearoperator with upper and lower limits. The FLC non-linear characteristics are due to its computationalstructure, i.e., fuzzi®cation, inference, and defuzzi®ca-tion. This requires the development of a formulationof the sliding-mode control for a class of nonlinearMIMO systems, that is suitable to the fuzzy±logicapproach. This formulation is crucial for two reasons:®rst, the approach to fuzzy±logic modeling and controltaken here is to consider the dynamic system as a`black box' without considering its interval parametersand structure. Hence, the system model and the robustcontrol rules must be obtained from the input-outputdata. The proposed generalized formulation of the slid-ing-mode control satis®es the above requirement.Second, for simplicity, it is desired to design the con-trol rules for each system state independently, despitethe state interactions, while the stability and robustnessof the entire system is guaranteed. This is another dis-tinct feature of the proposed formulation. In Section5, a generalized formulation of the model-based slidingmode control is developed for a class of nonlinearMIMO systems. In Section 6, the generalized formu-lation is used for generating the robust fuzzy±logiccontrol rules as a speci®c case of nonlinear control. InSection 7, the theoretical results are applied to the 4 dfrobot manipulator, and the practical steps are detailed.An experimental comparison study is made betweenthe proposed FLC and high-gain PID controllers.Concluding remarks are given in Section 8.

2. The proposed fuzzy±logic control structure

Figure 1 illustrates the proposed structure of theFLC for a nonlinear MIMO multi-variable second-order dynamic system. The controller consists of twoparts. In the ®rst part, a set of fuzzy IF-THEN rulesexpresses the dynamic behavior of the system. This`knowledge base' can be regarded as the (fuzzy±logic)

Fig. 1. The structure of the proposed fuzzy±logic control system.

M.R. Emami et al. / Engineering Applications of Arti®cial Intelligence 13 (2000) 47±6948

inverse dynamics model that represents the interactionbetween the system states as well as the other complexphenomena in the system. Unlike analytical models,the fuzzy±logic model is simple, and hence computa-tionally e�cient, and at the same time, as will be illus-trated for the robotic application, the fuzzy±logicmodel can represent complex phenomena of the systembehavior more precisely. Moreover, since the model isobtained directly from the input-output data, there isno need to identify the internal system parameters inorder to construct the model.

The second part of the FLC consists of `decoupled'robust fuzzy IF±THEN rules for each state indepen-dently, that guarantee system stability, and ensure thatthe desired performance is achieved. As shown in Fig.1, two (error) preprocessing units are also required toprovide suitable inputs to the FLC.

The proposed control structure is intuitive: based onprior knowledge, which may be incomplete or inaccur-ate, an attempt is made to control the system towardsthe desired performance; at the same time, by usingsome extra rules (fuzzy robusti®ers in Fig. 1), steps aretaken to ensure that the system remains stable anddoes not deviate from the desired behavior. In fact, fora simple system, these extra rules might be su�cientby themselves to control the system without anyfurther knowledge, as in the traditional fuzzy±logiccontrollers. However, as the complexity (such as largenumber of input variables, interaction between states,and wide variation of parameters) increases, more in-formation is required. This, it is proposed, will be

acquired as a fuzzy±logic knowledge base of the sys-tem. In essence, the proposed FLC is a robust model-

based control structure in which fuzzy IF±THEN rules

are used instead of the analytical formulation in order

to guarantee the desired system stability and perform-ance.

Based on the proposed structure, the systematic

methodology for design and analysis is proposed,based on the following steps:

1. Development of a fuzzy±logic model. The mainknowledge of the system is encapsulated in fuzzy

IF±THEN rules. The development of an objective

algorithm to extract this knowledge from the system

behavior (input±output data) is the heart of the

FLC. This task is brie¯y discussed in the modelingpart of the paper, Sections 3 and 4. The reader isreferred to (Emami, 1997; Emami et al., 1998a) formore details.

2. Design of the robust fuzzy IF-THEN rules for eachsystem state.

3. Proof of stability and completeness of the structure.In the proposed structure, for each system state, therobust fuzzy control IF-THEN rules are designedindependently. This `decoupling' characteristic pro-vides a simple approach to the design of the robustfuzzy rules. It should be proved that this is su�cientto guarantee the stability and robust performance ofthe entire system. Steps 2 and 3 are discussed in thecontrol part of this paper, Sections 5±8.

Part 1: modeling

3. A review of the systematic fuzzy±logic modeling

The central characteristic of fuzzy systems is thatthey are based on the concept of fuzzy partitioning ofthe information, and the decision-making ability of thefuzzy model depends on the existence of a set of rulesand a fuzzy reasoning mechanism. In the most generalform, the encoded knowledge of a multi-input multi-output (MIMO) system can be represented by fuzzymodels consisting of IF±THEN rules with multi-ante-cedent and multi-consequent variables (with r antece-dents, s consequents and n rules):

IF U1 is B11 AND U2 is B12 AND . . . AND Ur is B1r THEN V1 isD11 AND V2 is D12 AND . . . AND Vs is D1s

ALSO. . .ALSOIF U1 is Bn1 AND U2 is Bn2 AND . . . AND Ur is Bnr THEN V1 is Dn1 AND V2 is Dn2 AND . . . AND Vs is Dns

, �1�

where U1, U2, . . . , Ur are input variables, and V1, V2,. . . , Vs are output variables, Bij (i= 1, . . . , n, j = 1,. . . , r ) and Dik (i= 1, . . . , n, k = 1, . . . , s ) are fuzzysets of the universes of discourse X1, X2, . . . , Xr andY1, Y2, . . . , Ys, respectively. The set of rules operatingwith linguistic values of input-output variables appearsto be analogous to the system of equations used forthe presentation of linear and nonlinear systems. Thefuzzy sets Bij and Dik are parameters of the fuzzymodel, and the number of the rules determines itsstructure.

Conceptually, a system with multiple independentoutput variables can be considered as a set of single-output systems. Consequently, the general structure ofa MIMO fuzzy system can also be considered as a col-

M.R. Emami et al. / Engineering Applications of Arti®cial Intelligence 13 (2000) 47±69 49

lection of multi-input single-output (MISO) fuzzy sys-tems, such that for a system with s outputs, eachmulti-consequent rule is broken into s single-conse-quent rules. Although the number of rules in the newfuzzy system will be increased, modeling and inferencewould be more straightforward for MISO fuzzy sys-tems. That is the reason why the literature concen-trates on multi-input single-output rules as a genericpresentation of fuzzy systems. This research alsofocuses on the MISO fuzzy systems. The goal of thesystematic approach is to improve the objectivity offuzzy modeling by developing appropriate formu-lations and criteria to specify those features of themodel that are usually assigned heuristically in knownfuzzy modeling approaches. As a result, unlike ad hocfuzzy modeling techniques that are mainly based onexpert knowledge, the proposed methodology merelyexploits input±output data to extract some informationabout system characteristics. Fig. 2 illustrates the ¯ow-chart of the modeling methodology. The steps areexplained below.

3.1. Reasoning mechanism

The proposed methodology considers the inferencemechanism as an ìdenti®able' object of fuzzy systems.A uni®ed parameterized formulation was developedfor the reasoning process as follows. The reader isreferred to (Emami et al., 1999) for more details.

For the crisp input x�=(x1�, x2

�, . . . , xr�), the fuzzy

output of the system (1) (with single output) isobtained as (the index s is removed for convenience):

E�y� � b�1ÿ Sp�Tp�t1�x��, �D1� y��, . . . , Tp�tn�x��,�Dn� y�� 1ÿ b�Sp�Tp�t1�x � �, D1� y��, . . . ,

Tp�tn�x � �,Dn� y��,

�2�

where ti, called the `rule degree of ®ring' is computedas

ti�x�� Tq�Bi1�x�1�, Bi2�x�2�, . . . , Bir�x�r ��, �3�and

�Di� y� � 1ÿDi� y�: �4�

In Eq. (2), Sp is the n-ary t-conorm operator com-puted as follows

Sp�a1, a2, . . . , an�

� �a p1 � �1ÿ a

p1 ��a p

2 � �1ÿ ap2 ��. . . �a p

nÿ2

� �1ÿ apnÿ2��a p

nÿ1 � �1ÿ apnÿ1�a p

n �� . . .��1=p;

p > 0:

�5�

The operator Tw (w=p, q ) is the n-ary t-norm oper-ator that is calculated as

Tw�a1, a2, . . . , an�

� 1ÿ Sw��1ÿ a1�, �1ÿ a2�, . . . , �1ÿ an��:�6�

Equation (2) is a linear combination (with parameterb ) of two extreme reasoning approaches, Mamdani'sand logical, with adjustable parameters. The crisp out-put is then obtained by using the basic defuzzi®cationdistribution method as follows (Filev and Yager, 1991):

y� �

� y1

y0

y�E� y��ady� y1

y0

�E� y��ady0Ra <1: �7�

In the above reasoning formulation, i.e., Eqs. (2)and (7), four reasoning parameters p, q, a and b areintroduced whose variation will cause a continuousrange of variation of the reasoning mechanism.Consequently, unlike the traditional approach ofselecting the inference mechanism a priori, the optimalreasoning mechanism will be identi®ed by adjustingthe above parameters on the basis of the input±outputdata.

3.2. Identi®cation of the fuzzy structure

Fuzzy structure identi®cation involves assigning theoptimum number of rules, signi®cant input variables,input and output membership functions, and theamount of overlap between the membership functionsrequired for the fuzzy model. These are brie¯y dis-cussed in the following two sections.

3.2.1. Rule generation, output membership functionsAn intuitive approach to objective rule generation is

based upon the clustering of input±output data.However, in the proposed methodology, at ®rst onlythe output space is clustered, and then the input spacefuzzy partition is derived by projecting the outputspace partition onto each input space, separately.Simplicity and applicability, particularly for systems

Fig. 2. The ¯ow chart of fuzzy system modeling.


with a large number of input variables, are the mainadvantages of this approach.

The output fuzzy clusters are carried out by thefuzzy C-means (FCM) algorithm (Bezdek, 1981). Theidea of fuzzy clustering is to divide the output datainto fuzzy partitions that overlap with each other.Therefore, the containment of each datum yk to eachcluster i with a center vi is de®ned by a membershipgrade uik in the range [0, 1]. The membership gradesand cluster centers are obtained through an iterativeprocedure as follows:

uik,t �24Xc

j�1

��yk ÿ vi,tÿ1p��yk ÿ vj,tÿ1p

!2=�mÿ1�35ÿ1, �8�

vi,t �

XNk�1�uik,t�myk

XNk�1�uik,t�m

, �9�

where N is the number of data items, uik,t is the mem-bership grade of the output yk in the cluster i, and vi,tis the center of cluster i, at the tth iteration. Three cru-cial pieces of information are required prior to con-structing the suitable partitioning from the data: (i) anadequate number of rules for expressing the systembehavior; this, in most cases, equals to the number ofoutput clusters c; (ii) the order of fuzziness of the sys-tem model that represents the overlap of the fuzzyclusters, and it is adjusted by the parameter m calledthe `weighting exponent'; and (iii) a suitable initial lo-cation of the cluster centers, which a�ects the modelformation.

3.2.1.1. Speci®cation of the number of rules. The fol-lowing cluster validity index is developed for assigningthe optimum number of output clusters, and hence thenumber of rules, in the fuzzy model (Emami et al.,1998b):

scs �XNk�1

Xci�1�uik�m�� yk ÿ vi �2 ÿ �vi ÿ �v�2�, �10�

where �v is a weighted mean of data, considering theirmembership of each of the clusters de®ned as:

�v � 1Xci�1

XNk�1�uik�m

XNk�1

Xci�1�uik�myk: �11�

Minimization of scs will simultaneously increase thecompactness of the clusters and the separation betweenthem. Hence, the optimum number of clusters c corre-

sponds to the minimum scs. In most cases, c is equal tothe number of rules n of the fuzzy model (Emami etal., 1998b).

3.2.1.2. Speci®cation of the order of fuzziness. Theweighting exponent m controls the extent of member-ship sharing between the output fuzzy clusters in thedata set. In the range of (1, 1), the larger m is, the`fuzzier' are the membership assignments to each datapoint. For selecting m, the following index was devel-oped (Emami et al., 1998b):

ST �XNk�1

Xci�1�uik�m

!� yk ÿ �v�2: �12�

An appropriate value for m is what makes ST equalto a constant parameter K/2, where K is de®ned as:

K �XNk�1

2640@yk ÿ 1

N

XNj�1

y

1A2375: �13�

3.2.1.3. The initial cluster centers. The initial clustercenters for the FCM algorithm are assigned throughthe hard clustering techniques. This approach providesa more e�cient strategy compared to the previousapproach of randomly selecting the initial values(Emami et al., 1998b).

3.2.2. Input selection and input membership functionassignment

In order to identify the signi®cant input variablesamong a ®nite number of candidates, the output clus-ters are ®rst projected onto the space of each of theinput candidates. As a result, for each input candidatexj, the membership functions BÃ ij (i= 1, 2, . . . , n ) areformed. Then, the following index is calculated(Emami et al., 1998a):

pj � prodn

i�1

Gij

Gjj � 1, 2, . . . , r, �14�

where Gij is the range in which the membership func-tion BÃij is equal to one, Gi is the entire range of xj, n isthe number of rules, and rÃ is the number of input can-didates. A smaller pj illustrates a more dominant vari-able xj, and hence, signi®cant variables are selectedamong those that produce less p.

The convex membership functions Bij for the signi®-cant inputs xj ( j= 1, 2, . . . , r ) are then formed byusing the range Gij and by performing `fuzzy line clus-tering' as described in detail in (Emami et al., 1998a).


3.3. Identi®cation of the fuzzy parameters

The optimum values of the inference parameters ( p,q, a and b ) are identi®ed through a nonlinear con-strained optimization problem, by minimizing (Emamiet al., 1998a):

PI� p, q, a, b� �XNk�1� yk ÿ yk�2

�N, �15�

subject to the following constraints:

0 < p, q <1 and 0 < a <1 and 0 < b < 1,

where yk is the kth actual output and yÃk is the kthmodel output.

The input±output membership functions that havealready been identi®ed in the structure-identi®cationphase are approximated by trapezoidal functions, and

then an incremental tuning procedure is applied toadjust the membership function parameters, based onthe tuning data set and the performance index de®nedby Eq. (15).

4. Fuzzy modeling of a 4 df robot manipulator

4.1. The experimental setup

The systematic fuzzy±logic modeling approach wasapplied to a 4 df robot manipulator which is a part ofthe IRIS facility. This facility is a versatile, recon®gur-able and expandable setup, composed of several robotarms that can be easily disassembled and reassembledto provide a multitude of con®gurations. The basic el-ement of the system is the joint module, which has itsown input and output link. Each module is equippedwith a brushless DC motor coupled with harmonicdrive gear and instrumented with an optical encoder tomeasure the rotor angular displacement, and a ten-sion-compression load cell torque sensor to measurethe applied torque on the joint. The setup is controlledby a distributed computer system based on an AMD29050 RISC processor, tightly coupled with the hostcomputer, which is based on a 50 MHz Intel 80486processor accelerated with a large cache memory. Afast parallel I/O system allows up to a 5 kHz samplingrate. The modularity of the joints enables the user toarrange various con®gurations. Fig. 3 shows the speci-®ed con®guration of the IRIS arm used here for mod-eling purposes.

4.2. Test plan and data acquisition

A crucial phase of any system-identi®cationapproach is the planning of appropriate experiments,as the model's accuracy and robustness depend criti-cally on the data obtained through the experiments.This situation is even more critical in `black-box'approaches such as this one. In this application, themanipulator system has 12 input candidates, i.e., jointdisplacements, velocities, and accelerations, and 4 out-

Fig. 3. The desired con®guration of the IRIS arm.

Fig. 4. The block diagram of the system closed loop for identi®cation experiments.


put variables such as joint torque. The fuzzy±logicmodel of each joint is built up separately by consider-ing the e�ect of the dynamics of other joints. The planis to drive the manipulator joints along di�erent trajec-tories while the end-e�ector carries various amounts ofload. A simple control feedback was designed toensure stable performance in all cases, as shown inFig. 4. A user shell was also programmed on the hostcomputer to calculate joint velocities and accelerationson-line by using backward and central di�erence for-mulations, respectively.

Although the design of the experiments is problem-dependent, some general rules must be followed toguarantee the validity of the experimental information,as follows.

Causality: The input and output variables must bechosen so that the former causes the latter. In aclosed-loop experiment, there are obviously causaldependencies in both directions. However, if thecontrol signal is generated by a computer, as in thiscase, then due to the physical system delay, the con-sequence of `system state' and `control signal gener-ation' is closer in time than the consequence of the`control signal' and the new `system state'. Hence, itis appropriate to consider the system state (joint dis-placements, velocities and accelerations) as theinput, and the control signal (joint control torque)as the output. This is already de®ned as the ìnversedynamics model' of the system. The causality con-dition also implies that the measured joint torquecan not be considered as an identi®cation par-ameter, since it is neither a cause nor an indepen-dent e�ect.Su�cient excitation: Experiments should excite atleast all modes of the system that may be excitedwhen the model is used, in the same range of vari-ation. For robot manipulators, random joint trajec-tories that cover the desired range of input/outputparameters are suggested as `proper' input signals(Gautier and Khalil, 1992), provided that the jointvelocity and acceleration do not exceed the physicallimitations. In this application, these trajectories aregenerated by an interactive program prepared inMATLAB2 environment. Moreover, in order toenrich the obtained information, the manipulatorwas also tested for several step joint trajectories andsinusoidal trajectories with di�erent frequencies(due to the system bandwidth) and maximumdesired amplitude. Experiments were performedunder three di�erent loading conditions, i.e., light,moderate and heavy loads. The load e�ect is con-sidered as a disturbance to the system, and thefuzzy±logic model is expected to be robust enough,due to various loading conditions.Repeatability: Another condition is required to

ensure that the same system output (within anacceptable range of variation) is obtained fromrepeated experiments. In this study, reproducibilitytests were performed, and as a result, joint displace-ments, velocities and control torque signals werefound to be quite repeatable. However, as expected,the noise spectrum of the calculated joint accelera-tions was high, and hence some post-processing wasrequired for ®ltering the signal.Separability: The input signal to the system shouldbe independently generated, or at most, be in¯u-enced by the past and present system output only.A closed-loop experiment does satisfy this con-dition. It should be veri®ed, however, that the chan-nels transmitting input and output data do not haveany cross-interference. This condition was alsoexamined closely in this application.

4.3. Data processing and data selection

One of the advantages of the proposed methodologyis that, unlike classical approaches, there is no assump-tion of a zero-mean signal; hence there is no need toremove the trend and outliers in the data. Further, noscaling is required for the algorithm, either. However,the following data processing steps are useful for themodeling tasks.

Sampling frequency: A lower bound on the samplingfrequency is obtained from Shannon's sampling the-orem, which states that in order to recover a con-tinuous-time signal exactly, the sampling frequencyos should be chosen such that the signal does notcontain any useful frequencies above the Nyquistfrequency os/2. By considering the power spectraldensity of the measured joint torque, the highestuseful bandwidth can be obtained. As a result ofthis analysis, a Nyquist frequency around os/2=[20±30] Hz determines the width of the meaning-ful bandwidth in the experiments.Low-pass ®ltering: The second derivative of the jointdisplacement signal contains wide-band noise. Usinginput data with high bandwidth for system identi®-cation makes the model too sensitive, with a highererror variance. Therefore, it is desirable to ®lter theacceleration signal within an appropriate range offrequency. The measured joint torque signal can bea useful hint to specify the maximum meaningfulfrequency of the acceleration of each joint. Whatthe torque sensor measures is the torque loadapplied on each joint as a result of the dynamics ofall joints. Thus, this signal contains the e�ect ofjoint accelerations, as well. As a result, spectralanalysis of the measured torque signal provides asuitable cut-o� frequency for ®ltering the calculatedjoint accelerations. A low-pass digital Butterworth


Fig. 5. Fuzzy-logic model of the 4 df IRIS arm.


®lter was used to eliminate the e�ect of higher fre-quencies. This ®lter is characterized by a magnituderesponse that is maximally ¯at in the pass-band,and monotonic overall. Monotonicity and smooth

behavior are the advantages of this type of ®lter(Little and Shure, 1992). However, care must betaken to preserve the wave shape of signals in thepassband, as a result of the phase distortion. This

Fig. 6. Comparison of the fuzzy model and simulation joint torque with the experimental data.


can be examined by checking the ®ltered signalagainst the original one.Data selection and categorization: The sampling fre-quency for system identi®cation is not necessarily ashigh as the sampling frequency for data collection.In fact, as discussed before, in order to reduce thesensitivity of the model to noise, it is recommendedto choose a sampling frequency less than os for theidenti®cation. Also, it is desirable to reduce theamount of data used for the identi®cation. For re-duction of the sampling frequency, ®rst, aButterworth low-pass ®lter with a cut-o� frequencyaround Nyquist frequency os/2 was implemented, inorder to prevent aliasing e�ects. Then the data wascarefully reduced such that the relevant informationcontained in the signal was not lost. This proceduredepends on the signal and the applied trajectory.The resulting data from all the experiments were®nally combined and categorized into three di�erentsets:

* (i) Training set: for constructing the fuzzy modelstructure, i.e., clustering the input-output dataand forming the membership functions (processof Section 3.2)

* (ii) Tuning set: for adjusting the inference andmembership function parameters (process ofSection 3.3).

* (iii) Testing set: for examining the validity of themodel (to be discussed in Section 4.4).

4.4. Comparison with the analytical model

The fuzzy±logic model of the IRIS arm, made fromthe input-output data, is presented in Fig. 5. In paral-lel, an analytical simulation of the manipulator wasalso developed, that contains the e�ects of the inter-action of the nonlinear dynamics, motor and harmonicdrive dynamics, friction, and joint ¯exibility. Thereader is referred to (Emami, 1997) for a complete dis-cussion of the modeling process. Fig. 6 shows theresults of the fuzzy and analytical models for a testingsinusoidal trajectory under a moderate loading con-dition, compared to the experimental data. A betterperformance of the fuzzy model is clearly observablefor all joints. The same performance is also observedfor di�erent trajectories and loading conditions. Whilethe fuzzy±logic model performs almost uniformly forall joints in di�erent situations, the outcome of theanalytical model depends critically on how the variousphysical e�ects in the system are modeled. Forinstance, the behavior of joints 2 and 3 is more com-plicated, due to the weight e�ect and undesired back-lash and friction. These phenomena are hardlycaptured by the analytical formulation, and, therefore,

the simulation performance for these joints isdegraded.

Part 2: control

5. A generalized formulation of sliding mode control

Without loss of generality and for the sake ofclarity, second-order dynamic systems are consideredhere. The dynamic model of such systems with n sys-tem states and n input variables is represented as fol-lows:

�q � f�q, _q; t� � B�q, _q; t�u�t� � G�q, _q, u; t�, �16�where q=[q1, q2, . . . , qn]

T is the vector of systemstates, and q

.=[q

.1, q

.2, . . . , q

.n]T and qÈ=[qÈ1, qÈ2, . . . , qÈn]

T

are the state velocities and accelerations, respectively.Function f $ Rn is a nonlinear vector-valued functionthat represents the system dynamics, and B $ Rn�n isalso a nonlinear matrix function that acts as a non-linear control gain; vector u $ Rn is the control inputvector. The acceleration vector of the entire system canbe considered as a nonlinear and time-varying vectorfunction, G $ Rn.

The following assumptions are made with respect tothe system represented in Eq. (16).

1. By suitable transformations, it is possible to transferthe system dynamics into the form presented in Eq.(16) such that the system dynamics is linear in termsof the control input u (although nonlinear in thestates).

2. The control gain matrix B is nonsingular andbounded positive de®nite over the entire state space,i.e.,

8tr0; 8q, _q 2 Rn: bIRB�q, _q; t�Rb�q, _q; t�I, �17�where b is a positive constant and b

-(q, q

.; t ) is a

positive de®nite real function.

Based on the above assumptions, the inversedynamics model of the system can be presented as

u �M�q, _q; t� �q� h�q, _q; t� � F�q, _q, �q; t�, �18�where M=Bÿ1, and h=ÿBÿ1f.

Since B is nonsingular, from (17) it follows that:

8tr0; 8q, _q 2 Rn: mIRM�q, _q; t�Rm�q, _q; t�I, �19�where

m � 1

band m � 1

b:

In this analysis, the goal is to derive a formulationby using the nonlinear function F instead of its com-


ponents M and h. In this direction, considering thefact that:

M �q � F�q, _q, �q; t� ÿ F�q, _q, 0; t�, �20�the boundary matrix inequality (19) can be rewrittenin the following form of scalar inequalities:

8tr0; 8q, _q, �q 2 Rn:

mk �qk2R �qT�F�q, _q, �q; t� ÿ F�q, _q, 0; t��

Rm�q, _q; t�k �qk2:

�21�

The control task is to follow a desired qd and q.d in

the presence of system parameter variation and uncer-tainties. The tracking error e=qÿqd and the rate oferror e

.=q

.ÿq.d are to be observed. A generalized errorvector is de®ned as follows:

s � _e� Pe�Q

�t

0

e dt, �22�

where P > 0 and Q> 0 are n � n diagonal matrices.The integral of error is included in the generalizederror to ensure zero o�set error. Based on the theoryof sliding-mode control (Itkis, 1976), the tracking con-trol problem can be formulated as keeping the errorvector e on the sliding surface de®ned as follows:

_s � �e� P _e�Qe � 0: �23�An optimum response for each error state is

obtained if the system (23) is critically damped, where:

P � 2L and Q � L2, �24�and matrix L is an n � n positive de®nite diagonalmatrix.

At this point, one should consider the conditionsthat guarantee that for each state qi, the system trajec-tory will approach the sliding surface from any non-zero initial error, within a desired period of time. Inorder to avoid the chattering e�ect, i.e., undesired rela-tively high-frequency oscillations of the control signal,prevalent in sliding mode control (Hung et al., 1993),the condition is relaxed to asymptotic convergence ofthe system states to a small neighborhood of their cor-responding switching surfaces. For each state, thisneighborhood is de®ned as:

Bi�si,Fi� � fqi 2 R:jsijRFi g; Fi > 0: �25�The above task can be achieved if the control law u is

designed such that for each state a Lyapunov-like con-dition for system stability holds (Slotine and Li, 1990)

1

2

d

dt�s2i �Rÿ Zi�jsij ÿ Fi �; Zi,Fi > 0; i � 1, 2, . . . , n,

�26�

or in sum

1

2

d

dt�sTs�Rÿ

Xni�1

Zi�jsij ÿ Fi �; Zi, Fi > 0: �27�

The parameter Fi is the thickness of the boundarylayer and Zi is a design parameter that sets the timethe system trajectory requires to reach the boundarylayer from an outside of the boundary initial con-dition.

From Eqs. (23) and (24), the dynamics of the gener-alized error vector s is determined as:

_s � � �e� 2L _e� L2e� � � �qÿ � �qd ÿ 2L _eÿ L2e��: �28�The acceleration vector qÈ can be obtained from the

system dynamics. Eq. (16) can be rewritten as

�q �Mÿ1�uÿ h�: �29�Inserting (29) into (28) results in

_s �Mÿ1�uÿ �M� �qd ÿ 2L _eÿ L2e� � h��: �30�In Eq. (30), the term (M(qÈdÿ 2Le.ÿL 2e )+h ) is the

system inverse dynamics with the input accelerationcalled the `reference' acceleration and de®ned as fol-lows:

�qr � �qd ÿ 2L _eÿ L2e: �31�On this basis, the desired control input is de®ned as

ud �M�q, _q; t� �qr � h�q, _q; t� � F�q, _q, �qr; t�: �32�Because of the system uncertainty and variation, the

inverse dynamics model of the system (in this case afuzzy±logic model) is an approximation of the real sys-tem. Hence:

ud � M�q, _q; t� �qr � h�q, _q; t� � F�q, _q, �qr; t�: �33�Therefore, the control input u is de®ned as

u � ud � uc, �34�where the control term uc is the compensating inputdue to model uncertainty. It should be speci®ed suchthat the sliding condition (27) is satis®ed. By replacingu (Eq. (34)) in Eq. (30),

_s �Mÿ1�uc � �F�q, _q, �qr; t� ÿ F�q, _q, �qr; t��

�Mÿ1uc �Mÿ1DF,�35�

where DF is the uncertainty vector of the inversedynamics model. Consequently, the left-hand side ofthe sliding condition (27) becomes

sT _s � sTMÿ1uc � sTMÿ1DF: �36�


It is assumed that the uncertainty DF is bounded as

kDF�q, _q, �qr; t�kRr�q, _q, �qr; t� <1: �37�The following theorem is required before proceeding

further.

Theorem 1 (Emami, 1997). Suppose that for a positivede®nite n � n matrix B $ Rn�n there exists a positivereal scalar b > 0 such that bIrB where I is the n � nunity matrix. Suppose that for a vector y $ Rn there isan upper bound6y6R r. Then, for any arbitrary vectorx $ Rn, the following inequality holds:

xTByRbrkxk: �38�

By using Theorem 1, for the positive de®nite matrixB and bounded vector DF,

sTBDFRbrksk, �39�or

sTMÿ1DFR 1

mrksk, �40�

and therefore, Eq. (36) changes to the followinginequality:

sT _sR 1

mrksk � sTMÿ1uc: �41�

Considering the fact that

kskRXni�1jsij, �42�

the following inequality can be inferred from (41):

sT _sR 1

mrXni�1jsij � sTMÿ1uc: �43�

In order to satisfy the sliding condition (27), a con-tinuous uc is chosen such that

1

mrXni�1jsij � sTMÿ1ucRÿ

Xni�1�Zijsij ÿ ZiFi �, �44�

or in another form

sTMÿ1ucRÿXni�1

��rm� Zi

�jsij ÿ ZiFi

�: �45�

Assume that for each state i, uci is chosen as a func-tion of si, uci =Gi(si), such that it satis®es the follow-ing properties:

Gi�si � is continuous; �46a�

Gi�si � is monotonically decreasing for 0 < jsij < Fi;

�46b�

Gi�0� � 0: �46c�Gi(si) is expressed as follows:

uci � ÿgi�si �dsgn�si �, �47�where gi(si ) is a positive function for all si, anddsgn(si) is a function de®ned on the entire R as

dsgn�si � �8<:ÿ1; si < 00; si � 0�1; si > 0

�48�

From Eq. (47), the vector uc can be represented as

uc � ÿGOs, �49�where G is an n � n positive de®nite diagonal matrixwith gi(si) as its diagonals, and O is an n � n positivede®nite diagonal matrix with 1/|si| as its entries ifsi$0; otherwise the diagonal element corresponding tosi=0 is zero.

By inserting Eq. (49) into inequality (45) and multi-plying both sides by ÿm,

msTMÿ1GOsrmXni�1

��rm� Zi

�jsij ÿ ZiFi

�: �50�

At this point, another theorem is required as fol-lows, relating the left-hand-side of inequality (50) to aquantity without the inertia matrix M.

Theorem 2 (Emami, 1997). Consider M $ Rn�n as ann � n positive de®nite matrix, and K $ Rn�n as an n � ndiagonal positive de®nite matrix. If there exists a posi-tive real number m > 0 such that mIrM, then forevery arbitrary vector x $ Rn,

mxTMÿ1KxrxTKx, �51�

Since GO is positive de®nite (or at least positivesemide®nite), according to Theorem 2,

msTMÿ1GOsrsTGOs �Xni�1

gisi � dsgn�si �: �52�

Therefore, inequality (50) will be satis®ed if the fol-lowing inequality holds:

Xni�1

gisi � dsgn�si �rXni�1

�m

�rm� Zi

�jsij ÿmZiFi

�: �53�

It is su�cient that condition (53) holds for eachterm of the summation. Hence


gi � dsgn�si �rm

�rm� Zi

� jsijsiÿ mZiFi

si; i � 1, 2, . . . , n,

�54�or

girm

�rm� Zi

�ÿ mZiFi

jsij ; i � 1, 2, . . . , n: �55�

From inequality (54), the general condition for uc toguarantee the compensation for system uncertainty isobtained as8>>>>>><>>>>>>:

if si > 0�)uci < ÿ"m

�rm� Zi

�ÿm

ZiFi

jsij

#

if si < 0�)uci >

"m

�rm� Zi

�ÿm

ZiFi

jsij

# �56�

Figure 7 shows the domain in which each uci cancompensate for system uncertainties, here referred toas the `robustness region'. This region and the proper-ties speci®ed in (46) help to assign the robust controlterm uci for each state, independently.

In the methodology, the control terms uci are pro-duced by suitable fuzzy IF±THEN rules. The pro-cedure is as follows. Consider the nonlinear MIMOsystem (16) with the assumptions (i), (ii) and (iii), andwith the following inverse dynamics:

u � F�q, _q, �q; t�: �57�First, the fuzzy±logic inverse dynamics model of the

system is generated as

ud � Ffuzz�q, _q, �qr; t�, �58�based on a known bounded error DF. Then the controlinput is of the form

u � Ffuzz�q, _q, �qr; t� � uc, �59�where qÈr is de®ned by Eq. (31). The general conditionsof uc are: for each state i, uci should satisfy properties(46), and be located in the domain de®ned by (56) andillustrated in Fig. 7. The robustness region depends onthe design parameters li, Zi and Fi and parameters m,m and r, which are de®ned as follows:

r�q, _q, �qr; t�rkDF�q, _q, �qr; t�k, �60�

mR 1

k �qk2 � �qT�Ffuzz�q, _q, �q; t� ÿ Ffuzz�q, _q, 0; t��

Rm�q, _q, �q; t�:�61�

It is noted that for those trajectory points that haveclose to zero acceleration vector (qÈ=0), the term ininequality (61) becomes ambiguous, as both the de-nominator and numerator approach zero in the sameorder. This illustrates the fact that trajectory pointswith zero acceleration can not provide any informationabout the system inertia, as is obvious from Eq. (18).In practice, points with zero acceleration, if any, areremoved from the data set before calculating the re-lation (61).

In conclusion, this section has shown that it is poss-ible to design the `decoupled' robust fuzzy controlterms uci (i = 1, 2, . . . , n ) as presented in the controlstructure, Fig. 1. Furthermore, the general conditionswere developed for uci to ensure the stability androbustness of the entire system. These conditionsdepend on the bounds of the model error and systemparameters which, in this formulation, can be achievedfrom the inverse dynamics model (Eqs (60) and (61)).Therefore, in the proposed formulation, the systemdynamics is considered as a `black box', without thenecessity to specify its internal parameters explicitly.

6. Design of the robust fuzzy control rules

Section 3 discussed an approach to the control of aclass of nonlinear MIMO systems, which consists ofimplementing an inverse dynamics fuzzy model anddesigning a robust control term uci for each systemstate independently. Each function uci (si) should satisfyconditions (46) and (56). In this section, the fuzzy IF±THEN rules are designed so that they satisfy theseconditions.

From Fig. 7, the characteristic relationship betweenuci and si can be qualitatively expressed as: ùci is inver-

Fig. 7. The speci®ed domain for robust control term uci.


Fig. 8. Membership functions of the generalized error si for robust

control rules.

Fig. 9. The robust fuzzy control characteristics.

sely as large as si within certain limits'. The abovecharacteristic is interpreted by the following seven IF±THEN rules:

8>>>>>>>><>>>>>>>>:

IF s is Positive Big �PB�, THEN uc is Negative Large,IF s is Positive Medium �PM �, THEN uc is Negative Medium,IF s is Positive Small �PS �, THEN uc is Negative Small,IF s is Almost zero �AZ �, THEN uc is Almost Zero,IF s is Negative Small �NS �, THEN uc is Positive Small,IF s is Negative Medium �NM �, THEN uc is Positive Medium,IF s is Negative Large �NL�, THEN uc is Positive Large:

�62�

The above set of rules would express the robust con-trol commands for each system state, if the appropri-ate inference mechanism and input-output membershipfunctions are assigned. For the inference mechanism, itis feasible to apply the comprehensive parameterizedreasoning formulation developed for the system model-ing phase (Eqs. (2) and (7)). However, by having acomplete fuzzy±logic model of the system in the con-trol loop, the robust fuzzy control rules could have asimple structure. The modi®ed Sugeno's reasoning for-mulation (Sugeno and Yasukawa, 1993) is used, whichprovides a simpler and faster inference mechanism forthe robust fuzzy rules. Accordingly, given the input si,the crisp output uci is derived as:

uci �

X7k�1

Aik�si �bikX7k�1

Aik�si �, �63�

where Aik (si) is the membership function of si in theantecedent fuzzy set of the kth rule, and bik is the cen-troid of the consequent fuzzy set of the kth rule.

The problem is to assign suitable membership func-

tions for the robust control rules such that conditions(46) and (56) are satis®ed. It should be noted thataccording to the reasoning formulation (63), for theconsequent fuzzy sets, only their centroids arerequired. Considering the input membership functionsshown in Fig. 8, and seven consequent fuzzy-set cen-troids bi

1 for àlmost zero', bi2, bi

3, bi4 for `positive small,

medium, large' and bi2, bi

3, bi4 for `negative small, med-

ium, large', the siÿuci relation can be represented asshown in Fig. 9. For the sake of simplicity, and with-out loss of generality, the input membership functionsare arranged such that they always overlap at thedegree of membership equal to 0.5. Therefore, for eachinput si, at most two rules are ®red. Furthermore, asymmetric behavior for uci (si) is assumed. Hence

a2i � ÿa2i ; a3i � ÿa3i ; a4i � ÿa4i ; and b2i � ÿa2i ;

b3i � ÿb3i ; b4i � ÿb4i :�64�


From Fig. 9, some of the membership parameterscan be assigned immediately. First, conditions (46)require that:

a1i � b1i � 0: �65�By using inference formulation (63) and the mem-

bership functions shown in Fig. 8, a piece-wise linearcharacteristic is produced for the robust fuzzy controlfunction of each system state i (i = 1, 2, . . . , n ), whichcan be formulated as follows:

for a ji Rjsij < a j�1

i �)uci � ÿKji

j ji

si � u jcisgn�si �;

j � 1, 2, 3,

�66�

where

K ji � b j�1

i ÿ b ji ; j j

i � a j�1i ÿ a j

i �67�and

�u jci �

8>><>>:ÿXjÿ1s�1

Ksi �

K ji

j ji

Xjÿ1s�1

jsi j � 2, 3

0 j � 1:

: �68�

In order to assign the membership parameters, con-sider the dynamic behavior of the generalized errorvector s (Eq. (35)), which can be rewritten as:

_s � Buc � BDF, �69�or in component form:

_si � Biiuci �Xnk�1k 6�i

Bikuck �Xnk�1

Bik�DF �k;

i � 1,2, . . . ,n:

�70�

Substituting (67) into (70) results in:

_si � BiiK j

i

j ji

si � Biiujcisgn�si � �

Xnk�1k 6�i

Bikuck

�Xnk�1

Bik�DF �k;

i � 1, 2, . . . , n, j � 1, 2, 3:

�71�

Equation (71) represents the behavior of a state-dependent ®rst-order ®lter with corner frequency equalto Bii (Kij/fij) and system uncertainties and state inter-action dynamics as input to the ®lter. Based on thetheory of sliding-mode control (Itkis, 1976), a suitableselection of the corner frequency for such a ®lter to

remove system uncertainties and unmodeled frequen-cies is:

BiiK j

i

j ji

Rli, �72�

where li is the lower band of the unmodeled frequen-cies. Since b=1/m is an upper value of the gain matrixB, a reliable break-away frequency for ®lter (71) canbe assigned such that

K ji

j ji

Rmli: �73�

However, within the boundary layer, the unmodeledfrequencies and uncertainties can a�ect the system per-formance only when si is close to zero, i.e., for the ®rstsegment where |si|Yai

2. Therefore, for each state i,parameters ai

2 and bi2 should be selected such that:

b2ia2i

Rmli: �74�

For a larger distance between the state and theswitching line, since the unmodeled frequencies andstate interactions can not change the sign of the con-trol input, one could assign higher break-away fre-quencies that would provide better control andconsequently, faster response without any performancedegradation. Therefore, parameters ai

3 and bi3 are

designed such that:

b3i ÿ b2ia3i ÿ a2i

rb2ia2i: �75�

The capability of choosing di�erent slopes for therobust control signal within the boundary layer pro-vides more ¯exibility in the design of the robust con-trol than the sliding-mode control approach (Emami,1997). Moreover, as illustrated in Fig. 9, designingpiece-wise linear robust control function ensures thatthe state trajectory remains inside the `robustnessregion'.

Similar to the sliding-mode control, the trackingquality is guaranteed by choosing (Slotine and Li,1990):

K3i

j3i

� b4ia4i� mli: �76�

Outside the boundary layer, the maximum value ofthe robust control is assigned to be the lower bound ofthe `robustness region'; therefore, from (56):

b4i � m

�rm� Zi

�, �77�

and from (76) and (77), the boundary layer thickness


Fi=ai4 is speci®ed as:

Fi � a4i �m

m

�rm� Zi

�1

li: �78�

Equations (64), (65), (74), (75), (77) and (78) assignthe membership parameters of the robust fuzzy IF±THEN rules (62). Obviously, these parameters dependon the design parameters li and Zi. The natural fre-quency li speci®es the rate of convergence to the slid-ing surface, and this should be less than the minimumfrequency associated with the largest unmodeled timedelay tm and the frequency associated with thesampling rate ts. A suggested criterion for selecting liis (Slotine and Li, 1990):

lRmin

�1

3tm,

1

5ts

�: �79�

The above two criteria for assigning li are inverselyrelated to the accuracy of the model developed. Themore accurate the model is, the more computationaltime is required, and a higher sampling rate should beused. An ideal solution is to implement modeling para-digms that provide simpler interpretations with highaccuracy. This is the approach taken in the fuzzy±logicmodeling approach.

The design parameter Zi re¯ects the time requiredreaching the boundary layer from an outside initialcondition. A higher value of Zi results in a faster tran-sient response. However, since this parameter controlsthe maximum value of the robust control term (Eq.(77)), its magnitude is physically limited.

7. Application to robot manipulators

In this section, the proposed robust fuzzy controlrules are generated for the 4 df IRIS arm. A fuzzy±

logic dynamics model of the system was constructedfrom the input-output data given in the modeling partof the paper. Although it was illustrated that themodel performs quite well for the testing data, robustcontrol rules will be required to ensure that stabilityand satisfactory performance are maintained underdi�erent trajectory and load conditions.

7.1. Design and analysis

Referring to conditions (56), three system par-ameters should be speci®ed in advance namely r, mand m. Eqs (60) and (61) indicate the necessary re-lationship between the values of these parameters. Inorder to assign these values, the experimental data ofthe training, tuning and testing sets were used. First,the fuzzy model error vector DF (Eq. (34)) is obtainedand its norm is calculated. Fig. 10 shows the errornorm for some of the testing data. According to Eq.(60), the value of r should be the upper bound of theerror domain. Theoretically, it is possible to identify ras a function of the system states, and then apply theidenti®ed IF±THEN rules for calculating r at each sys-tem state. However, in order to avoid complexity, aconstant value was selected as an upper bound of mostof the experimental data. From Fig. 10, a value of 0.5was assigned to this parameter. Next, the lower andupper bounds of the inertia matrix M (Eq. (18)) arespeci®ed from the experimental data using Eq. (60).For each set of {q, q

., qÈ} obtained from the experiment,

the fuzzy model output F(q, q., qÈ) and the output F(q,

q., 0) for the same state but zero acceleration were

obtained, and then the following inertia value was cal-culated:

M � �qT ~Mq

k �qk2 ��qT�F�q, _q, �q� ÿ F�q, _q, 0��

k �qk2 : �80�

Fig. 10. Error norm of DF for some of the testing data.Fig. 11. The value of the inertia M for some of the experimental

data.


Fig. 12. Robust fuzzy control characteristics and membership functions for the IRIS arm.


Figure 11 shows M for the experimental data fromwhich a proper estimation of m and m is 0.05 and0.25, respectively.

The next step is to assign suitable values for designparameters li, Zi and Fi (i=1, . . . , 4). An increase inbandwidth li will reduce the e�ect of uncertainties inthe system. However, it should be far enough fromunmodeled frequencies, such as these due to the timedelay of the actuators, and from the real-time control-loop sampling frequency. A suitable value of li isobtained from Eq. (64). The sampling frequency of thecontrol loop is set to be 500 Hz, which is fast enoughsuch that it is not an actual limitation for l. Theunmodeled time delay is di�cult to estimate from theblack-box modeling approach. However, since the ex-perimental data was ®ltered by a cut-o� frequency of50 Hz, it is reliable to start with li=50/3 for all joints.During the experiments, li is gradually increased, foreach joint at a time, to reach the maximum possiblevalue. The design parameter Zi, as discussed in Section6, controls the duration of the transient response ofthe system. For fast response, a high value of Zi is rec-ommended. However, by increasing Zi the maximumvalue of uci is also increased; this is limited by theampli®er's current range. In this application, the totalcontrol signal is limited to 20 N m, and 20% of thisamount is considered to be allocated to the robustcontrol signal uci.

The parameter Fi is another design parameter thata�ects the system stability. A critically damped per-formance in the domain of [ÿFi, +Fi] assigns thevalues of Fi to be equal to (uci)max/mli, as discussed inSection 5.

Membership functions of the sliding variable s andthe robust control signal uc were obtained, based onthe selected values of design parameters, and based ondiscussions in Section 5. Fig. 12 shows the stabilityregion of the IRIS joints, the designed piece-wiserobust control signals and the associated membershipfunctions of the fuzzy control rules for each of thefour joints.

7.2. Experimental results

Similar to the modeling phase, several typical trajec-tories such as random, sinusoidal, and step were usedunder di�erent loading conditions to produce di�erentaccelerating, uniform speed, and decelerating motionsegments. In order to be able to assign the displace-ment, velocity and acceleration of each path segment,®fth-order polynomials were used for path segments ofrandom and step trajectories. Hence, each segment isplanned as follows:

q�t� � a0 � a1t� a2t2 � a3t

3 � a4t4 � a5t

5, �81�

where

a0 � q0; a1 � _q0; a2 � �q02;

a3 � 20�qf ÿ q0� ÿ �8 _qf � 12 _q0�tf ÿ �3 �q0 ÿ �qf�t2f2t3f

;

a4 � 30�qf ÿ q0� � �14 _qf � 16 _q0�tf � �3 �q0 ÿ 2 �qf�t2f2t4f

;

a5 � 12�qf ÿ q0� ÿ �6 _qf � 6 _q0�tf ÿ � �q0 ÿ �qf�t2f2t5f

: �82�

q0, q.0, qÈ0 and qf , q

.f , qÈf are joint states at the beginning

and end of each segment, respectively, and tf is thedesired time to pass the segment.

For each trajectory, two control schemes (the pro-posed fuzzy±logic control and a high-gain PID con-trol) were implemented, and the results werecompared. Figs. 13±15 illustrate the tracking perform-ance of these controllers for typical random, sinusoi-dal, and step trajectories, respectively, under a mediumloading condition. For each trajectory and each joint,the displacement error and velocity error are shownwith the corresponding applied input control torque.The PID gains were designed for each trajectory separ-ately, using MATLAB Control Toolbox and the ana-lytic model of the system, and then experimentallytuned for di�erent trajectories in order to provide thebest performance. On the other hand, the design par-ameters of the proposed fuzzy controller were ®xed forall trajectories and loading conditions.

7.3. Comparison of the results

Generally, tracking errors for a speci®c con®gur-ation depend on the payload and the trajectory. Ascan be observed from the results, the proposed fuzzy±logic controller outperforms the servo controller for allthe di�erent trajectories. For fast-changing trajectoriessuch as the step input shown in Fig. 15 and the high-frequency sinusoidal trajectory shown in Fig. 14, non-linear dynamic e�ects are dominant; hence, the betterperformance of the fuzzy control is more signi®cant asa result of the embedded knowledge of the systemdynamics. For the random trajectory shown in Fig. 13,fuzzy control still provides better tracking perform-ance, while the PID servo can also perform satisfac-torily in the absence of tight dynamic interactions.Similarly, the control input torque of the fuzzy andservo control schemes are closer to each other whennonlinear dynamic e�ects are less signi®cant, as isobserved for the random trajectory in Fig. 13. On the


Fig.13.Comparisonoftheproposedfuzzy±logic

controlandPID

controlsoftheIR

ISarm

forrandom

trajectory.



controlandPID

controlsoftheIR

ISarm

forsinusoidaltrajectory.



controlandPID

controlsoftheIR

ISarm

forstep

trajectory.


other hand, for step and (fast) sinusoidal trajectories,the fuzzy control input delivers more compensation foruncertainty signals, and is therefore signi®cantly di�er-ent from the servo control input. Small oscillations areobserved for the fuzzy control torque signal; these aredue to robust sliding-mode characteristics to compen-sate for friction, ¯exibility, backlash and other systemuncertainties. The amplitude of these high-frequencysignals is quite small so as not to cause chattering.Therefore, the overall system behavior remainssmooth. The best performance was achieved for thesinusoidal trajectory shown in Fig. 14. This is mainlydue to the complete knowledge of the controller aboutsystem sinusoidal behavior, as this can be inferredfrom the outcome of the identi®ed fuzzy model.Displacement and velocity errors of the fuzzy control-ler response to the step trajectory in Fig. 15 illustrate afaster and still more accurate performance than thePID control. However, more torque should be appliedin this case, which leads to more likely control signalsaturation. This basically implies that the desired steptrajectory is somewhat beyond the physical ability ofthe system. The ®rst overshoot of the joint error ateach `jump' illustrates the above fact.

It must be emphasized at this point that, unlike theparameters of the proposed fuzzy control, the PIDgains were adjusted for each trajectory separately. Asan example, the proportional gain of the servo control-ler of joint 2 for a step trajectory is three times as highas the one for a random trajectory. Trajectory-depen-dent gain adjustment was performed to assure the bestachievable performance for the servo control.

In summary, a better performance was achieved, interms of both joint displacement and velocity tracking,by the proposed fuzzy±logic control compared withhigh-gain servo control for all the typical trajectoriesimplemented in the experiment. The major part of thissuperior performance is attributed to the embeddedknowledge of the system behavior. The simplicity ofthe proposed controller makes it as easily applicable asservo control while the performance resembles perfectrobust model-based control schemes.

8. Conclusions

This research is an attempt to construct a systematicframework for the new paradigm for the modeling andcontrol of complex systems. By exploiting the conceptof `fuzziness' in the de®nition of real-world phenom-ena, and by applying the method of àpproximatereasoning' to the deduction of results from obser-vations, the new paradigm provides a strong potentialfor representing and manipulating ill-de®ned systems,i.e., those that are too complicated to be modeled byanalytical methods. Nevertheless, without a concise

methodology, this potential can not be fully exploited,and remains as the heuristic and ad hoc technique ithas been so far.

This research provides an opening to a newapproach to fuzzy modeling and control: a systematicand algorithmic approach. The result is signi®cant: itwas hypothesized and demonstrated that, althoughàpproximation' is inherent in fuzzy modeling and con-trol, based on a ®rm theoretical ground, one canachieve more àccuracy' and better `performance' thanwith analytical approaches, without sacri®cing the sim-plicity and applicability. This proposal was illustratedwith a typical complex system, a 4 df robot manipula-tor. The expectations from traditional approaches(analytical methods) should be limited by the degree ofcomplexity of the system. New paradigms such as thefuzzy±logic approach must be employed if simple andrelevant interpretations are required, together withhigh accuracy and satisfactory performance, at thesame time. The main conclusion to be drawn here isthat this is possible only with the help of a systematicframework.

The performance of the proposed FLC must still becompared to other applicable model-based controlschemes for di�erent applications, a task that shouldbe accomplished in the future.

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