Interval Type-2 Fuzzy Logic Control of Mobile Robots

doi:10.4236/jilsa.2012.44031

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Journal of Intelligent Learning Systems and Applications, 2012, 4, 291-302

http://dx.doi.org/10.4236/jilsa.2012.44031 Published Online November 2012 (http://www.SciRP.org/journal/jilsa)

291

Interval Type-2 Fuzzy Logic Control of Mobile Robots

Nesrine Baklouti1, Robert John2, Adel M. Alimi1

1REGIM: Research Group on Intelligent Machines, National Engineering School of Sfax (ENIS), Sfax, Tunisia; 2Centre for Compu-

tational Intelligence, De Montfort University, Leicester, UK.

Email: nesrine.baklouti@gmail.com, rij@dmu.ac.uk, adel.alimi@enis.rnu.tn

Received May 9th, 2012; revised July 31st, 2012; accepted August 7th, 2012

ABSTRACT

Navigation of autonomous mobile robots in dynamic and unknown environments needs to take into account different

kinds of uncertainties. Type-1 fuzzy logic research has been largely used in the control of mobile robots. However,

type-1 fuzzy control presents limitations in handling those uncertainties as it uses precise fuzzy sets. Indeed type-1

fuzzy sets cannot deal with linguistic and numerical uncertainties associated with either the mechanical aspect of robots,

or with dynamic changing environment or with knowledge used in the phase of conception of a fuzzy system. Recently

many researchers have applied type-2 fuzzy logic to improve performance. As control using type-2 fuzzy sets represents

a new generation of fuzzy controllers in mobile robotic issue, it is interesting to present the performances that can offer

type-2 fuzzy sets by regards to type-1 fuzzy sets. The paper presented deep and new comparisons between the two sides

of fuzzy logic and demonstrated the great interest in controlling mobile robot using type-2 fuzzy logic. We deal with the

design of new controllers for mobile robots using type-2 fuzzy logic in the navigation process in unknown and dy-

namic environments. The dynamicity of the environment is depicted by the presence of other dynamic robots. The per-

formances of the proposed controllers are represented by both simulations and experimental results, and discussed over

graphical paths and numerical analysis.

Keywords: Type-2 Fuzzy Logic; Control; Motion Planning; Mobile Robots

1. Introduction

Mobile robotic navigation in a dynamic and unknown

environment is a local path planning problem based on

sensory information with no knowledge about the form

and location of obstacles. Precise end-effector and posi-

tioning accuracy are required in the navigation process.

Due to their mechanic conception, Robots, by their very

nature, have significant uncertainties in their readings

and movements. These errors can be summarized as fol-

lows:

 Sensor measurements are usually noisy due to the

instruments. Generally a robot is equipped with in-

ternal sensors like encoder sensors and coupling sen-

sors and external sensors as laser sensors, infrared

sensors and cameras. Readings extracted from those

sensors are generally noisy [1],

 There are assembly errors that include linear and an-

gular errors produced during the assembly of the va-

rious robot mechanical components,

 Indeed, there are uncertainties provoked from a varia-

tion in the coupling of actuators characteristics with

environmental conditions. Measurement and actuator

create end-effector positioning errors. The resolution

of encoders and stepper motors are examples of such

errors. Although these errors are in the most cases

very small, they can be amplified to cause many er-

rors that can affect the accuracy of the system [1],

 Real environments are not ideal, so they can produce

random errors that change unpredictably. Those er-

rors are called non-systematic errors and are generally

caused by irregularities or roughness of the floor.

In order to overcome those uncertainties and to de-

velop a robust, flexible and on-line planner, type-1 fuzzy

logic has been used [2-5]. However type-1 fuzzy logic

cannot fully handle the stated uncertainties because it

uses precise type-1 fuzzy sets which don’t necessarily

cope well with all sources of vagueness and uncertainty.

There are at least three sources of uncertainty in type-1

fuzzy logic control:

 According to Mendel [6] “words mean different things

to different people”, so while designing a fuzzy con-

troller experts are unlikely to agree on the member-

ship functions,

 Consequents in the rules can be uncertain,

 And the database used in the conception of a fuzzy

controller is generally not ideal and contains errors,

even if the database has been constructed by an ex-

pert.

Interval Type-2 Fuzzy Logic Control of Mobile Robots

292

Then in mobile robot control by fuzzy logic, all the

cited forms of uncertainties will be multiplied over fuzzi-

fication, inference and defuzzification. Those errors can

degrade the performance of the whole robot controller.

Type-2 fuzzy logic has been used by researchers to try

and overcome some of these difficulties [6-8]. And since

many researchers have explored the use of type-2 fuzzy

logic controllers in various applications [9-16]. In fact,

type-2 fuzzy sets were initially introduced by Zadeh [17].

Firstly a general type-2 fuzzy set was defined, where it

represents a 3D set in which each membership grade is a

type-1 fuzzy set bounded in [0,1]. Due to the complexity

of the join (OR) and meet (AND) operations performing

the inference part [18] and type-reduction in the defuzzi-

fication part [19], the application of general type-2 fuzzy

sets has been limited. Hence, a simplified version of

general type-2 set called interval type-2 fuzzy set is used

more widely [6,20]. This kind of set has membership

grades that are crisp interval sets bounded in [0,1]. The

uncertainty here is represented as a 2D bounded region

that is called the Footprint of Uncertainty. Various re-

searchers have explored the advantages of interval type-2

fuzzy sets [21,22]. Moreover, a geometric approach has

been introduced by Coupland and John [23] distinguish-

ing between fuzzy logic over discrete and continuous do-

main. But this approach is not fast enough in control ap-

plications [24]. In the rest of this paper we treat only in-

terval type-2 fuzzy sets over discrete domains.

In mobile robotics, some researchers have explored the

control of mobile robots using interval type-2 fuzzy logic

[21,24-29]. As Hagras states in [21] control using type-2

fuzzy sets represents a new generation of fuzzy control-

lers. In [25] Hagras presented an interval type-2 fuzzy

logic controller to command a robot in indoor and out-

door unstructured environment. A robot was tested under

different sources of non-systematic errors. The results

showed that type-2 fuzzy logic outperforms its type-1

counterpart. This was shown through robot paths and

control surfaces. In [27], an interval type-2 fuzzy logic

was proposed for the control of a robot tracking a mobile

object in the context of robot soccer games. In this game

the robot has to track a ball. To evaluate the performance

of the type-2 fuzzy logic against its type-1 counterpart,

graphical paths analysis were presented showing the way

the player reaches the position of the ball. Also, an addi-

tional test was made presenting the ability of type-2 con-

troller to track the ball with less standard deviation error

than its type-1 counterpart.

In this paper we propose specific aspects of control of

mobile robots in unknown and dynamic environments

using type-2 fuzzy logic. The dynamicity of the envi-

ronment is depicted by the presence of other dynamic

robots. The performances of the proposed controllers are

represented by both simulations and experimental results,

and discussed over graphical paths and numerical analy-

sis. This paper has essentially two parts: In the first part

we designed an IT2TSK fuzzy logic controller for avoi-

ding obstacles using simulations. In the second part we

designed a Mamdani interval type-2 (IT2) fuzzy logic

controller for wall following behavior using the robot

Khepera II from LAMI [30]. The remainder of the paper

is organized as follows: Section II introduces type-2

fuzzy logic. Sections III presents the proposed Interval

type-2 fuzzy logic controller for obstacle avoidance be-

havior. Next section presents the conception of an Inter-

val type-2 fuzzy logic for a wall following behavior.

Simulations and experimental results in both applications

are presented and are discussed. Finally, some conclu-

sions are pointed out in Section VI.

2. Type-2 Fuzzy Logic

2.1. Overview on Type-2 Fuzzy Sets

Type-1 fuzzy sets are certain and crisp, whereas type-2

fuzzy sets are themselves fuzzy. Type-2 fuzzy sets were

first introduced by Zadeh [17]. An Interval T2 fuzzy set

 is described as in the following definition, where in

(1) all





Axu





,



Axu





.









1, 0,1

xXuJ

AxuJ









(1)

where



 denotes the union of all admissible

and



. An IT2 FS is represented by a bounded region limit-

ed by two membership functions, where corresponding to

each primary MF (which is in [0,1]), a secondary MF is

used to the primary one. The Uncertainty in the primary

membership function consists of the union of all mem-

bership functions. This Uncertainty represents a bounded

region that we call the Footprint of Uncertainty





UFO ,

i.e.,





FOU X





 (2)

The





FOU represents a complete description of an IT2

FS. It uses an





UMF Upper Membership Function,

and





LMF Lower Membership Function; The





F UM and the





LMF of



are two T1 MFs that

bound the





FOU .





UMF represents the upper

bound of





FOU

 and is denoted











FOU

, and the

LMF represent the lower bound of



 and is

denoted







:



FOU AxX













 (3)

For an IT2 FS,

 

XAA















X . In

the rest of this paper, we will use only the IT2 fuzzy sets

in the design of our work.

Interval Type-2 Fuzzy Logic Control of Mobile Robots 293

2.2. Interval Type-2 Fuzzy Logic Controller

In fact, a type-2 fuzzy logic system or controller uses the

same familiar notions as used in a type-1 fuzzy logic

controller as membership functions, rules, t-norms opera-

tions, fuzzification, inference, defuzzification. A type-1

fuzzy logic system consists basically of three blocs; fuz-

zification, inference and defuzzification as presented in

Figure 1. A type-2 fuzzy logic system is very similar to

type-1, where it follows the same methodology, but the

only difference is in the third block where we no longer

speak of only defuzzification but we speak about a type

reducer and defuzzification parts that constitute both the

output processing block. This difference is mainly asso-

ciated with the nature of the membership functions,

where type-reducer is needed due to the added degree in

the kind of fuzzy sets. Figure 2 presents a type-2 fuzzy

logic system.

Today, the two most popular fuzzy logic systems used

by engineers in control are the Mamdani and TSK sys-

tems.

2.2.1. Mamdani Type-2 Fuzzy Logic Controller

2.2.1.1. Fuzzification

In this part, we must first define the fuzzy sets of all in-

puts’ system. Those memberships can contain one or se-

veral type-2 fuzzy sets. Second, the fuzzifier maps in-

puts into the associated fuzzy sets to determine the de-

gree of membership of each input variable. We consider

only singleton fuzzification for which the inputs are crisp

values.

2.2.1.2. Inference

This block expresses the relationship that exists between

the input variables (expressed as linguistic variables) and

the output variables (also expressed as linguistic vari-

ables). As in type-1 fuzzy logic, in the design of a type-2

fuzzy logic we generally have IF-THEN rules. The for-

Figure 1. Type-1 fuzzy logic.

Figure 2. Type-2 fuzzy logic.

mulation of rules is the same. The only distinction be-

tween type-1 and type-2 is associated with the nature of

the membership functions. The inference engine com-

bines rules and makes a combination between input

type-2 fuzzy sets and output type-2 fuzzy sets. This is

ensured by searching unions and intersections of type-2

sets, as well as compositions of type-2 relations.

For a type-2 fuzzy logic with inputs and

XxX



 and one output

Y, and with M

rules. The th rule has the following form:



:IFisand andis

THEN is1

RxF x

yGl M













The firing strength of the ith rule is as in (4). The re-

sult of the input and antecedent operations is an interval

type-1 set.















,[,]

iiii

iii ii

FFFF

Fxfxfxff

xx x



















 

















(4)

where











and







 designed respectively

upper and lower membership grades of





 and











Fxx















(5)

Since generally we use the meet operation under

product or minimum t-norm. So, at each value of x the

intersection and union operations are referred to as the

meet and join operations, respectively.

2.2.1.3. Type Reducer and Defuzzification

In a type-1 fuzzy logic system the output of the inference

engine corresponding to each fired rule is a type-1 set.

The defuzzifier combines those output sets to obtain a

single output set. Using one of the existing methods of

defuzzification, for example the centroid of sets, the de-

fuzzifier searches the centroid of the obtained set to ob-

tain finally a crisp output. In a type-2 fuzzy logic, since

this kind of system deal with type-2 sets, then it is nec-

essary to have a type reducer block to map a T2 FS into a

T1 FS, and then defuzzification, as usual, maps that T1

FS into a crisp number. We can consider that the de-

fuzzification block of a T1 fuzzy logic is replaced by the

output processing block in a T2 fuzzy logic. That block

consists of type-reducer followed by defuzzification. In

fact, type Reducer was proposed by Karnik and Mendel

[7]. For now, there are five different type-reduction me-

thods. Karnik and Mendel [19] defined the centroid of an

IT2 FS which is an IT1 FS that is ensured using the Ex-

tension Principle. This IT1 FS is characterized by its left

and right end points l

and r

, which can be written

Interval Type-2 Fuzzy Logic Control of Mobile Robots

294

in the following equation:





111

,,,

MMMM

Mii

vwwvwwfffi





 





  

 





 (6)

This equation is derived from the consequent centroid

sets .







2.2.2. Interval Type- 2 TSK Fuzzy Logic Contro ller

The differences between a type-1 TSK fuzzy logic con-

troller [31] and a Mamdani T1 fuzzy logic consist essen-

tially of the definition of outputs and then on the conse-

quent part of rules. Consider we have a first order type-1

TSK fuzzy logic with inputs 11

p,,

XxX,

one output

Y, and with M rules. The th rule can

be expressed as:



011

:IFisand andis

THEN

RxFxF

yccx cx

 











A type-2 TSK fuzzy logic controller or system (T2

TSK fuzzy logic) was firstly introduced by Liang and

Mendel [32]. Although TSK type-1 fuzzy systems have

received a lot attention, the literature on TSK type 2

fuzzy systems is few. Liang and Mendel applied type-2

TSK systems in channel equalization of channels [12].

Where, according to them, there are three models of T2

TSK fuzzy logics depending on the kind of the antece-

dent and consequent part of rules, to have: T2 TSK-

Model I, T2 TSK-Model II and T2 TSK-Model III. We

can see in Tables 1 and 2 the difference between those

models. where are the consequent parameters,



are the outputs of the th rule, 





jp







are type-2 fuzzy sets and



jp



 are type-1

fuzzy sets. The firing strength of the ith rule





with meet operation under product or minimum t-norm is

an interval type-1 set expressed as follows :

  

 



iip

Wx wxwx

wx xx

wx x















(7)

The final output is also an interval type-1 set and is

calculated as follows:

Table 1. Models of T2 TSK FLS.

Model Antecedents Consequents

Model I Type-2 fuzzy sets type-1 fuzzy sets

Model II Type-2 fuzzy sets crisp numbers

Model III Type-1 fuzzy sets type-1 fuzzy sets

Table 2. Rules of T2 TSK FLS.

TSK FLS Rules l

Type-1 11

011

IFisand and

THEN

lll l

Fxis

yccxLcx 

F

T2 Model I 11

011

IFand andis

THEN

xisFx F

YCCx Cx 









T2 Model II 11

011

IFisand andis

THEN

xFxF

yccx cx 









T2 Model III 11

011

IFisand andis

THEN

xF xF

YCCx Cx 















,,, ,,,

Mii

yywwi

YYYWWy y





 



(8)

where i



, and , thus for

each rule we will obtain l and . Since all sets are

crisp, the Equation (9) results to:



,,1

iii

YyyiM





 



;

ii ii

ll rl

wy wy







(9)

And the defuzzified output is:





yyy (10)

3. Interval Type-2 TSK Fuzzy Logic for an

Obstacle Avoidance Behavior

3.1. Conception of Type-1 Behavior

We designed a zero-order TSK type-1 fuzzy logic con-

troller for the navigation of a mobile robot in dynamic

and unknown environment for obstacle avoidance be-

havior. The purpose of the controller is to perform the

navigation in unknown and dynamic environments for

polygonal mobile robot. In this behavior we defined six

sensor measurements inputs,as

are shown in Figure 3 where those distances represent

respectively the zones: central, left, right, lateral left,

lateral right and back. Each input expressed the distance

to the nearest obstacle in the zone of vision of its corre-

spondent sensor and is defined by two MFs Near and Far,

which are represented by trapezoidal membership func-

tions as in Figure 4.

,,, ,and

clrllrla

dddddd

Interval Type-2 Fuzzy Logic Control of Mobile Robots 295

The outputs of the behavior are the left and right

wheels’ rotational velocities noted respectively l

V and

r. We defined 64 TSK rules deduced by authors’ hu-

man expertise. For obstacles near the robot, we have

widely used the rotation without translation to avoid ob-

stacles instantly. An example of rules is expressed in

Equation (11), where we defined the case of robot finding

obstacles in his whole left side, so to guarantee avoiding

static or dynamic obstacles the robot turn little to his right.



IFis Far andis Near andis Far

andisNear andisFar andis Far

THEN ,0.5,0

clr

lllr a

ddd

VV 

(11)

Figure 3. Representation of robot sensor positions.

(a)

(b)

(c)

Figure 4. Membership functions of typ e-1 fuzz y logic inputs .

3.2. Conception of Type-2 Behavior

In the conception of type-2 fuzzy logic for obstacle avoi-

dance behavior, we extended the proposed type-1 con-

troller to a type-2 one by adding uncertainties in both the

antecedent and the consequent parts of each rule. The

main idea here consists in spreading the membership

functions’ values of the antecedent part by ± A%, and the

consequent part by ± C%. Therefore by this, our type-2

fuzzy logic is an IT2 TSK fuzzy logic Model I. So, the

six inputs of the behavior were extended by ± A%. The

following figures, Figures 5-7, show those inputs for ±

A% = ± 5%.

The outputs of the controller are the same as in type-1

fuzzy logic r

V and l

V. We have chosen to fix the

spread of the consequent parameter “C” equal to 1 rad/s.

We have defined thus 64 TSK T2 rules that are repre-

sented in Table 3 and are as the following form:

123

:IFis andis andis

andis andis andis

THEN ,,

clr

ll rl a

lr lr

RdFdFd

dF dF dF

VV CC







 



 



 6





Figure 5. MFs of the frontal input with ± 5%.

Figure 6. MFs of the lateral input with ± 5%.

Figure 7. MFs of the back input with ± 5%.

Interval Type-2 Fuzzy Logic Control of Mobile Robots

296

where





, 6



are type-1 fuzzy sets, and are the

consequent parameters for and













1,, 6j



1, 4

In the inference engine, we use the meet operation un-

der the product t-norm.

3.3. Simulation Results

Several simulations were tested while varying the spread

of the antecedent parameters ± A%. The results were

implemented in Matlab 6.5 under the operating system

Windows XP. We used an environment with dimensions

of 189 × 190 containing arbitrary complex obstacles,

using the SIMROBOT toolbox software [33]. A wheeled

mobile robot is considered under the assumptions of

non-slipping and pure rolling. Its kinematics can be de-

rived using Figure 8.

This robot has two degrees of freedom: y-translation

and either x-translation or z-rotation. Its dimensions are

taken into account when it navigates in an arbitrary com-

plex environment. The actuated inverse velocity solution

of this robot is as the following equation



Wab ba

ll ll













 





 



 







(12)

Table 3. Rules of the T2 FLC.

Rules Inputs Outputs

d l

d r

d ll

d rl

d a





1 L L L L L L [1.85, 1.85] ± C%

2 P P P P P P [0.95, −0.95] ± C%

. . . . . . . .

64 P P P P P P [0.71, −0.71] ± C%

Figure 8. Kinematic of the Simrobot robot.

where: ,



: are translational velocities of the robot

body in ms,



: is the robot z-rotational velocity in

rad s, 12



: are wheel rotational velocities in

rads , : is actuated wheel radius in and :

are distances of wheels from robot's axes in . More

details about the robot used kinematics can be found in

[33].

Rm,

To prove the efficiency of T2 fuzzy sets, we firstly

choose to test the robot navigation in a complex place,

like a narrow passageway. And, as we have not made any

learning study on the adequate footprint of uncertainty

(FOU) to our T2 fuzzy logic, we make simulations as

follows: We tested T1 fuzzy logic and T2 fuzzy logic for

different spread or (FOU), from the same initial position

and for the same number of steps, which were re-

spectively chosen to “[105 179]” and “110 steps”. Re-

sults are presented in the following part; Figure 9 re-

presents the initial robot position. Figure 10 shows the

trajectory generated by the robot using T1 fuzzy logic

controller. From this figure, we notice that while passing

through the narrow passage, the robot makes several

oscillations to attempt finally, at the end of 110 steps, the

position [140 178]. Sure our T1 fuzzy logic controller is

not perfect and may contains some issues of errors. This

explains well the obtained oscillations in the resulted

trajectory where several rules give opposite and acute

outputs reflecting an unstable navigation.

Figure 9. Initial chosen robot position (position = [105 179]).

Figure 10. TSK T1 fuzzy logic path for 110 steps.

Interval Type-2 Fuzzy Logic Control of Mobile Robots 297

Figures 11(a)-(f) represent the simulation results of

the T2 TSK fuzzy logic for different deviations of the

antecedent part ± A%, which correspond respectively to 5,

10, 15, 20, 30 and 40. It is clear from those figures the

ability of T2 fuzzy logics in avoiding obstacles or walls

as T1 fuzzy logic, but the most important things to high-

light are that all the generated trajectories by T2 fuzzy

logics are clearly smoother and contain less oscillations

and deviations than the one generated by type-1 fuzzy

logic in Figure 10. Thus T2 fuzzy sets can reduce uncer-

tainties coming from not perfect tuning or noisy database.

Besides, we remark that for the same number of steps,

the robot in all T2 fuzzy logics browsed more land than

in T1 fuzzy logic. So T2 controllers are more rapid and

allow the robot to arrive faster to a given destination

point.

To have a quantitative comparison, and to demonstrate

well the smoothness of the obtained T2 trajectories, we

choose to concentrate only on the narrow passageway;

we propose to visualize the angular velocities generated

by all T1 and T2 controllers, all from the same initial

position [105 179] to almost the same final one [140 178]

which represents the end of the passage way as shown in

Figure 12, while the number of steps is varying depend-

ing on the adopted controller to arrive quickly or slowly

to the final position. Thus, we obtain those results; Fig-

ure 13 represents the generated angular velocity by T1

fuzzy logic, where the positive values represent the left

robot turnings and negative ones represent the right turn-

(a) (b)

(e) (f)

Figure 11. TSK T2 FL paths for 110 steps. (a) A = 5; (b) A =

10; (c) A = 15; (d) A = 20; (e) A = 30; (f) A = 40.

Figure 12. Initial and final chosen robot positions.

Figure 13. Simulation generated by TSK T1 FLC.

ings. We notice from the figure that the robot continu-

ously oscillates right and left along the whole passage,

reflecting by thus the lack of stability and smoothness of

T1 fuzzy logic. Whereas Figures 14(a)-(f) denote the ge-

nerated angular velocity by T2 fuzzy logics for different

deviations A, which correspond respectively to 5, 10, 15,

20, 30 and 40. We remark clearly from those Figures 13

and 14, that all T2 generated trajectories contain less

peaks and oscillations than the T1 generated trajectory.

From those figures we extracted in Table 4 some re-

sults; we calculated first number of steps taken by each

simulation. We can remark that navigating from the same

initial position to the same final one takes from T1 fuzzy

logic the biggest number of steps which is 110 steps than

all the other T2 fuzzy logics that takes 93 steps for A = 5

to only 19 steps for A = 40. So, T2 fuzzy logics are faster

than T1 fuzzy logic.

Besides, in the table we extracted a smoothness meas-

urement: the angular velocity smoothness Index (AVSI).

AVSI represents the average accumulative angular ve-

locities made by the robot simulation during k steps.

Moreover, we calculate the Mean Square Error (MSE)

expressing the error between the actual generated output

Interval Type-2 Fuzzy Logic Control of Mobile Robots

298

(a) (b)

(e) (f)

Figure 14. Simulations generated by TSK T2 FLCs for different A. (a) A = 5; (b) A = 10; (c) A = 15; (d) A = 20; (e) A = 30; (f)

A = 40.

Table 4. Simulation results given by T1 and T2 controllers.

TSK FLC T1 T2 A = 5 T2 A = 10 T2 A = 15 T2 A = 20 T2 A = 30 T2 A = 40

Steps 110 93 58 49 38 36 19

AVSI (˚/s) 2.6437 2.5582 2.5522 2.4100 2.1224 1.2763 0.3242

MSE 57.08 37.895 33.541 32.075 19.073 19.061 2.0678

Interval Type-2 Fuzzy Logic Control of Mobile Robots 299

from the controller and the expected desired output in the

passage that normally might be a straight line path. The

equations of these measurements are given as follows:



AVSI

MSE

idi









(13)

It can be seen also from Table 4 that: while the pa-

rameter “A” is increasing, the navigator becomes smoo-

ther and with less oscillations. This is expressed mainly

with a degradation in AVSI and MSE measurements (A

= 40, AVSI = 0.3242˚/s; MSE = 2.0678).Sure there is a

limit of this decrease, but this point is not discussed in

this paper. Nevertheless, this remark is very interesting

as an optimization point in future work. The most impor-

tant thing to highlight from the table is that all the T2

generated trajectories represent better results than the T1

one in terms of AVSI or MSE.

Secondly, to prove the efficiency of type-2 fuzzy sets

in the setting of local avoiding obstacles, we tested the

controllers towards moving obstacles in dynamic and

unknown environment. For T2 controller, we choose to

test only an example of a T2 controller with a spread

equal to A = 20. The dynamicity of the environment is

manifested by the presence of several robots. We can see

in Figures 15(a) and (b) the ability of robots in both T1

and T2 controllers in avoiding obstacles and also in

avoiding themselves. Although we tested in both figures

the robots from same initial positions and for the same

number of steps, we can see that we have not the same

robots trajectories and this since the controllers did not

give the same instant outputs. Also, it is clear from both

figures the smoothness, the rapidity of T2 controller in

relation to T1 one, where robots in T2 figure browse

more land with smooth turnings.

To enhance the found result, we presented another

simulation in a different restricted complex place, where

the robot is supposed to be front with a corner as in Fig-

ures 16(a) and (b). We can see from the different gener-

ated trajectories that the T2 fuzzy logic with A = 20 pre-

sents the smoothest path essentially in the corner part.

Where T2 robot turns slowly towards the corner in a way

it seems following the wall. Whereas T1 robot presents

sharp turns in a way it turns back in other direction.

4. Interval Type-2 Fuzzy Logic for a Wall

Following Behavior

In this section we presented a type-2 mamdani fuzzy

logic for a wall following behavior for the miniature mo-

bile robot kheperaII from K-Team [30]. In reality, this

robot presents a good example of existing uncertainties,

(a) (b)

Figure 15. TSK T1 T2 paths (A = 20) from the same initial

robots positions.

(a) (b)

Figure 16. Simulations generated by T1 and T2 FLCs

against a corner.

where it is faced to a large amount of vagueness. The

kheperaII robot has 8 infrared proximity sensors. Each of

them has a maximum range of measurement of about

5cm with accuracy and resolution depending on the mea-

sured distance. Those sensors are imprecise and present

several kinds of errors like ambient light, color, shape

and intensity of the detected obstacle. It was proved in

[34] that the total error in the distance estimation by an

infrared sensor depends on the uncertainty in the read-

ings and the uncertainty in the angle of incidence of the

sensor. In our type-2 fuzzy controller we have three in-

puts, two outputs and eight rules, that is deduced from its

corresponding type-1 fuzzy logic by spreading the ante-

cedent and consequence parts by a footprint of uncer-

tainty. The robot have to navigate forward in the envi-

ronment until it detects an unexpected obstacle or a wall,

follows its contour on the right side with little turns. To

detect the wall, the robot used its six frontal proximity

sensors which are grouped two by two to constitute the

three inputs. Each of them is defined by two fuzzy sets

that are presented in Figure 17. The outputs are the left

and right wheels’ rotational velocities of khepera Vr and

Vl. The both are defined with two fuzzy sets and are pre-

sented in Figure 18.

Whereas, in the inference part we conceived eight

rules which are deduced from human expertise. Concer-

ning the implementation part, we used the kMatlab rou-

tines [30] to interact with Khepera robot over a serial

connection.

The two controllers are compared in same conditions,

where robots are fixed approximately at the same initial

position and then tested during the same number of steps

Interval Type-2 Fuzzy Logic Control of Mobile Robots

300

chosen equal to 130 steps. Figure 19 shows the robot in

the worked environment. As presented in previous simu-

lations, we extracted the angular velocities generated by

T1 fuzzy logic and T2 fuzzy logic illustrated respectively

in Figures 20 and 21.

We remark from these trajectories that type-2 path

presents less deviation with a smoothness measurement

index AVSI equal to 1.0934 than type-1 path that pre-

sents an AVSI equal to 0.5554. Figure 22 presents paths

generated by both controllers. We noted that in a part of

T1 trajectory, the robot touched the wall and did not keep

enough distance to it. Whereas T2 robot has kept ap-

proximately 8 cm to the wall. Besides, the T2 controller

has traveled 22.8 cm more in land than its equivalent T1

controller. Thus, type-1 fuzzy sets may not be robust en-

ough to handle uncertainties caused by infrared meas-

ures.

5. Conclusion

In this paper we presented T2 controllers for mobile ro-

bot navigation. We can highlight from above results that

type-2 fuzzy sets are very interesting in control of mobile

Figure 17. Inputs MFs of the wall following behavior.

Figure 18. Outputs’ MF of the wall following behavior.

Figure 19. The robot in the worked environment.

Figure 20. Angular velocities generated by T1 FLC.

Figure 21. Angular velocities generated by IT2 FLC.

Figure 22. Paths generated by T1 and T2 FLCs.

Interval Type-2 Fuzzy Logic Control of Mobile Robots 301

robots. We have shown deeply that the proposed T2 con-

trollers are more efficient in terms of saving time, smoo-

th trajectories and optimal distance than their counterpart

T1s. This was demonstrated through several paths of

robots and smoothness and error measures. So, Interval

type-2 fuzzy sets help to overcome uncertainties that can

exist in real environments.

6. Acknowledgements

The authors would like to acknowledge the financial sup-

port of this work by grants from the General Direction of

Scientific Research and Technological Renovation (DG-

RSRT), Tunisia, under the ARUB program 01/UR/11-02.

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