Journal of Intelligent Learning Systems and Applications, 2012, 4, 291-302 Published Online November 2012 (
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.
Received May 9th, 2012; revised July 31st, 2012; accepted August 7th, 2012
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-
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-
Copyright © 2012 SciRes. JILSA
Interval Type-2 Fuzzy Logic Control of Mobile Robots
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
1, 0,1
denotes the union of all admissible
. 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
FOU represents a complete description of an IT2
FS. It uses an
UMF Upper Membership Function,
LMF Lower Membership Function; The
F UM and the
LMF of
are two T1 MFs that
bound the
UMF represents the upper
bound of
and is denoted
, and the
LMF represent the lower bound of
and is
For an IT2 FS,
 
X . In
the rest of this paper, we will use only the IT2 fuzzy sets
in the design of our work.
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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-
2.2.1. Mamdani Type-2 Fuzzy Logic Controller 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. 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
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.
iii ii
xx x
 
designed respectively
upper and lower membership grades of
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. 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
Copyright © 2012 SciRes. JILSA
Interval Type-2 Fuzzy Logic Control of Mobile Robots
in the following equation:
 
 
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
one output
Y, and with M rules. The th rule can
be expressed as:
:IFisand andis
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,
are type-2 fuzzy sets and
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 :
  
 
Wx wxwx
wx xx
wx x
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
IFisand and
lll l
yccxLcx 
T2 Model I 11
IFand andis
xisFx F
YCCx Cx 
T2 Model II 11
IFisand andis
yccx cx 
T2 Model III 11
IFisand andis
xF xF
YCCx Cx 
,,, ,,,
where i
, and , thus for
each rule we will obtain l and . Since all sets are
crisp, the Equation (9) results to:
ii ii
ll rl
wy wy
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
Copyright © 2012 SciRes. JILSA
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
lllr a
Figure 3. Representation of robot sensor positions.
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:
:IFis andis andis
andis andis andis
ll rl a
lr lr
dF dF dF
 
 
 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%.
Copyright © 2012 SciRes. JILSA
Interval Type-2 Fuzzy Logic Control of Mobile Robots
, 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
Wab ba
ll ll
ll ll
 
 
 
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
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.
Copyright © 2012 SciRes. JILSA
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
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)
(c) (d)
(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
Copyright © 2012 SciRes. JILSA
Interval Type-2 Fuzzy Logic Control of Mobile Robots
Copyright © 2012 SciRes. JILSA
(a) (b)
(c) (d)
(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:
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
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
Copyright © 2012 SciRes. JILSA
Interval Type-2 Fuzzy Logic Control of Mobile Robots
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-
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.
Copyright © 2012 SciRes. JILSA
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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