Intelligent Control and Automation, 2011, 2, 77-85
doi:10.4236/ica.2011.22009 Published Online May 2011 (http://www.SciRP.org/journal/ica)
Probabilistic Fuzzy Control of Mobile Robots for Range
Sensor Based Reactive Navigation
Chunlin Chen, Tiaojun Xiao*
School of Management and Engineering, Nanjing University, Nanjing, China
Received December 4, 2011; revised March 29, 2011; accepted April 2, 2011
Abstract
In this paper, a probabilistic fuzzy approach is proposed for mobile-robot reactive navigation using range
sensors. The primary motivation is an integrated reactive navigation control system with good real-time per-
formance under uncertainty. To accomplish this aim, a probabilistic fuzzy logic system (PFLS) is introduced
to range measurement and reactive navigation in local environments. PFLS is first adopted to handle the
fuzzy and stochastic uncertainties in range sensors and to provide more precise distance information in un-
known environments. Consequently these sensor data are sent to a probabilistic fuzzy rule-based inference
system with reactive behaviors for local navigation. The feasibility and effectiveness of the proposed ap-
proach are verified by simulation and the experiments on a real mobile robot.
Keywords: Mobile Robots, Probabilistic Fuzzy System, Range Measurement, Reactive Navigation
1. Introduction
Reactive navigation control of autonomous mobile ro-
bots in the unknown environment is a fundamental and
crucial issue in robotics, cybernetics and artificial intel-
ligence [1-4]. In a local environment it is necessary for a
robot to possess the capabilities of obstacle avoidance,
moving toward sub-goal, escaping from local traps, etc.
[5, 6]. Real-time and precise reactive navigation has been
a challenging task for mobile robots because the envi-
ronment may be unpredictable and the sensory informa-
tion may be incomplete or not accurate enough for deci-
sion-making. Several reactive control approaches have
been proposed to implement local navigation. For exam-
ple, occupancy maps have been built using sonar sensors
to model the environment and have led to a series of lo-
calization and path planning methods [7]. The potential
field and virtual force field have been used for navigation
in a local environment, respectively, [8]. These methods
are mainly employed to deal with stationary obstacles. In
the middle of 1990s, some researchers presented behav-
ior-based control methods such as the subsumption ar-
chitecture [9], where the stimulus-action pairs are de-
fined for the decision and control rules of a robot after a
proper coordination if it is necessary. Various soft com-
puting and machine learning methods have been applied
to reactive navigation to improve the control perform-
ance [2,3,5,6,10,11]. To accomplish the better navigation
performance, two problems need to be solved with suit-
able approaches. One problem is that the data of onboard
sensors are always uncertain, which confines the detect-
ing ability of mobile robots in the unknown environ-
ments. The other problem is how to make decision using
the sensor data to navigate in the unknown and changing
environments.
The navigation using range sensors, namely ultrasonic
sensors, infrared sensors, laser scanner, etc., has been
recognized as one of the most fundamental and important
problems in the increasing applications of autonomous
mobile robots [2,12-14]. It is well known that all meas-
urement processes are always accompanied by uncer-
tainty [15] and various uncertainties can be classified
into nonstochastic uncertainty and stochastic uncertainty
[16]. The uncertainties in the range data measured by
range sensors with random noises and unpredictable
conditions may include incomplete information, vague-
ness and stochastic uncertainty [17]. In order to handle
these uncertainties, probabilistic approaches have been
adopted to various robot systems [13,14,18,19]; but most
of the existing results focus on sensor fusion, localization
and behavior selection. Generally, fuzzy logic systems
(FLS) [20] have the capability to deal with multiple un-
certainties without a precise mathematical formula [6,10].
The type-2 FLS [21-23] has also been recently studied in

C. L. CHEN ET AL. 78
detail to better modeling the fuzziness of a fuzzy relation,
which could improve the ability to handle inexact infor-
mation. However, ordinary FLS could not catch the sto-
chastic uncertainty and the traditional fuzzy technique is
not the best choice to process the stochastic uncertainty
[25,26]. Stochastic uncertainty arises with random prob-
ability that cannot be predicted in advance or recognized
accurately during the process of measurement. There are
various circumstances that are full of stochastic uncer-
tainties. For example, it is possible that random noises,
time-varying stochastic uncertainty, stochastic mutative
temperature or weather, and the like occur as stochastic
uncertainty. It is equally possible that the range sensor
data measured by different sensors provide stochastic
uncertainty. In addition, random disturbance involving
the probability that someone walking around the range
sensor will help make serious difference between the
measured distance and the real distance. Thus, without
ruling out all the possibilities above, the range sensor
could not measure the precise distance under various
circumstances.
The probabilistic fuzzy logic system (PFLS) [25-27] is
different from the ordinary FLS and it uses probabilistic
fuzzy sets instead of ordinary fuzzy sets to capture the
information with stochastic uncertainties. The main dif-
ferences between the proposed PFLS approach and most
ordinary FLS or probabilistic methods are two-fold. 1) In
the proposed PFLS approach, both of the nonstochastic
and stochastic uncertainties are processed for sensor fu-
sion and decision-making, while most of the existing
results only focus on one of them. So the PFLS method
can dramatically improves the performance of the meas-
urement and reasoning for mobile robots, especially in
unknown dynamic environments. 2) PFLS bridges the
gap of the numerical sensory fusion and fuzzy reasoning,
which provides an alternative way for the robot to ac-
quire data-driven human-like intelligence. Hence PFLS
will be more valuable for the processing of various un-
certainties and the reactive navigation control of mobile
robots.
In this paper, PFLS for range sensor is first proposed
to process stochastic uncertainty and can effectively re-
duce the disturbance caused by stochastic uncertainty, so
that the distance information can approximate to the ac-
tual measured values more accurately. Then these sensor
data are sent to a probabilistic fuzzy rule-based inference
system with reactive behaviors for the local navigation to
attain better performance under uncertainty. The rest of
the paper is organized as follows. The range measure-
ment problem is formulated in Section 2 for reactive
navigation of mobile robots. In Section 3, a general
probabilistic fuzzy system is designed with range sensor
based reactive behaviors for robot navigation. Both the
simulative and experimental results on a real mobile ro-
bot are shown and analyzed to test the presented ap-
proach in Section 4. Conclusions are given in Section 5.
2. Problem Formulation
The robot employed in this study is a MT-R mobile robot.
Its main sensors and configurations are shown in Figure
1. It is a two-wheel driven robot with 6 pairs of range
sensors and each pair of sensors consists of an ultrasonic
sensor and an infrared sensor. The detailed specifications
of robot MT-R are listed as shown in Table 1 and the
configuration of the 6 pairs of range sensors is shown as
in Figure 1.
To navigate the robot in a clustered environment, the
mobile robot detects the surrounding environment and
then decides the motion commands. The robot has six
range sensor pairs and can detect the obstacles from six
directions (Figure 1). So the inputs of the reactive navi-
gation control system are the six obstacle distances df, dlf,
drf, dl, dr, db obtained from the front, left-front, right-front,
left, right and back sensor pairs. The outputs are the mo-
tion commands to the two wheels with encoders. For
MT-R, the sensitive range of the ultrasonic sensor is 0.2
m ~ 7 m and that of the infrared sensor is 0.1 m ~ 0.8 m.
These two kinds of range sensors are always combined to
detect the obstacles in front of them. As addressed in
Section 1, the sensory inputs are always full uncertain.
Moreover, the robot control system has to judge the input
distance is “far” or “close” and to decide the output mo-
tion command is “forward”, “turn left” or “back” and so
on. To handle these vague and stochastic uncertainties,
Table 1. Specifications of mobile robot MT-R.
Specifications of Robot MT-R
Dimensions
Base: d = 0.490 m
Height: 0.495 m
Weight: 30 kg
Motion Control
DSP + CPLD
2 DC motors (MAXON 24 V, 70 W) with 2
shaft encoders
Max speed: 2.5 m/s
Exteroceptive
Sensors
6 sonar sensors:
Resonance frequency (20 KHz)
Sensitivity Range (0.2 m ~ 7 m)
6 infrared sensors:
Measurement frequency (100 HZ)
Sensitivity Range (0.1 m ~ 0.8 m)
CCD camera
1.3 M pixel
30 frames/s
USB interface
Other
equipments
Wireless communication: 54 M,
etc.

C. L. CHEN ET AL. 79
(a) (b)
Figure 1. Mobile robot MT-R and its configurations. (a) MT-R mobile robot with rang sensors; (b) Configuration of Range
sensors.
the probabilistic fuzzy approach is adopted as an inte-
grated control scheme for the reactive navigation control
of mobile robots in unknown environments.
Figure 2 shows three distance data measured by range
sensor under different stochastic circumstances, respec-
tively. For instance, Situation 1 represents the distance
data under normal condition without any disturbance.
Situation 2 expresses the data under man-made distur-
bances such as the movement of someone around the
range sensor during measurement. Situation 3 presents
the distance information with the disturbance of random
noises. Under each of these stochastic conditions, the
error (1,2,3)
ii

 between actual measured distance
and precise distance can be used for fuzzification with
diverse membership function. In addition, each of the
stochastic situations possesses a probability with a cer-
tain probabilistic distribution function in continuous case
or discrete case. Due to the specific probability held by
each of the stochastic uncertainties, PFLS for range
measurement can be implemented to effectively reduce
the measurement error caused by stochastic uncertainty.
3. Reactive Navigation with PFLS
Range sensors are most widely used for mobile robots to
move autonomously with obstacle-avoidance. In this
section, PFLS method is first introduced for processing
the sensory inputs, and then an integral control scheme is
designed and implemented to achieve robust and precise
reactive navigation for mobile robots.
3.1. Probabilistic Fuzzy Logic System
Ordinary fuzzy logic is based on the theory of fuzzy set
which is composed of discrete or continuous elements
possessing degree of membership. An ordinary fuzzy set
can be represented as a set , where an input
variable

,SIU

x
I

and

 

0, 1Uux is its fuzzy
membership grade. If


12
, ,,
n
I
xxx and n is the
number of the elements in fuzzy set, then the fuzzy set
(,SI)U

can be expressed as








12
1
12
,
N
ni
i
ni
ux ux ux ux
SIU
x
xx



x
.
In comparison with the ordinary fuzzy logic system
(FLS), a probabilistic fuzzy logic system (PFLS) simi-
larly includes fuzzification, fuzzy rules, fuzzy inference
and defuzzification. Nevertheless, the distinct difference
of PFLS to FLS is that the fuzzification and defuzzifica-
tion procedure are based on probabilistic fuzzy sets in-
stead of ordinary fuzzy sets [25-27].
3.1.1. Fuzzification in PFLS
Definition (Probabilistic Fuzzy Set) The probabilistic
fuzzy set can be donated as a probability space of
S



P,SS
, where {}{(, )
j
j
SS xu

 is
the set of all possible events and 1, 2,,}jm
x
I is the input
variable. For all element event
j
SS





0, ΣΣ ,1
jjj
PSP SPSPS . 
The probabilistic fuzzy set can be formulated as the
union of the finite space as follows:




,,
xI
SIU


P
In PFLS, the fuzzy membership grade is a ran-
dom variable with a certain probabilistic distribution
function (PDF)

ux




,Pxux . For example, Figure 3 pre-
sents an instance of a discrete probabilistic fuzzy set
in a three-dimension ordinary fuzzy space.
S




1,2,3, 4
,,
ii
i
SIu


P

C. L. CHEN ET AL. 80
Figure 2. Demonstration of three measured distance data under different stochastic circumstances.
Figure 3. An instance of discrete probabilistic fuzzy set in three-dimension coordinate.
where
123
{,,}{1,2,3}Ixxx
{0.6, 0.2, 0.1}uP
10.4P
0.2P
, ; ,
1
{0.1, 0.3, 0.7}u
20.1
0.3P
2{0,0.5,0.3}u
{0.4,0.8,0.5}
3, ; , . 34 4
u
3.1.2. Inference in PFLS
The jth rule of a PFLS is usually expressed as follows:
Rule j: IF x1 is 1,
j
A
 and x2 is 2,
j
A
 and  and n
x

C. L. CHEN ET AL.81
is ,nj
A
; Then is q
j
B

,
.
where ,)
ij
(1,2,)( 1,2,,
A
inj
m and
j
B
 are pro-
babilistic fuzzy sets. For more details, please refer to
[25-27].
3.1.3. Defuzzification in PFLS
The method of probabilistic defuzzification is deduced
from ordinary fuzzy sets. In ordinary FLS, it is usually
used for defuzzification with a centroid calculation.
However, the operation of defuzzification in PFLS is
realized by computing centroid calculation with the as-
sociation of the mathematical expectation.
For each possible input event, the output has a fuzzy
set which has L elements that each element is assigned a
value k and every number is corre-
sponding to a membership grade
where the input variable 12
(1,2,vk,L)
(,)(1,2,, )
vk
uxv kL
{,,, }
n
x
Ixx x

. Thus,
with the centroid calculation, the centroid output is ob-
tained


1
1
,
,
M
kv k
k
d
,M
,2, 
X
M
vk
k
vu xv
v
uxv




)
)
d

There are M possible events that each of them has an
output d through the defuzzification of
centroid calculation, and accordingly the probability dis-
tribution is . In the stochastic circum-
stance, the mathematical expectation from is
used as the terminative output of PFLS
(1
d
Pd
,2,vd
(1,Mvd
v
 
1
M
dd
d
vEvvP



3.2. PFLS for Reactive Navigation
Generally, reactive navigation refers to the navigation
control without map-building and global planning. In this
paper, the reactive navigation system is designed to help
the robot moves in a local area freely and safely. The
robot used in our study is called MT-R as shown in Fig-
ure 1. It is a two-wheel driven mobile robot and takes
range sensory data as inputs to detect the surrounding
environment and try to seek free regions after processing
these range data. So to achieve effective reactive naviga-
tion, firstly we have to adopt a range sensor data proc-
essing method to get more accurate range information;
secondly suitable reactive behavior and decision-making
approach should be designed to navigate the robot. In
[28], the probabilistic fuzzy system has been shown to be
effective to deal with both the nonstochastic and stochas-
tic uncertainties in range measurement. In addition,
PFLS is also a good candidate for the behavior control of
mobile robots. Hence we present an integral reactive
navigation approach based on range sensors using PFLS.
The overall reactive navigation system is described as
Figure 4. For each control step, the range sensory date
{,,,,, }
f
lfrflr b
dddddd are first sent to a PFLS, after
being processed, the outputs of range data are used for
the reactive behavior control. For the details of the PFLS
based method for range measurement, please refer to
[28]. As for the reactive behavior, the decision-making is
based on the range information {,,,,, }
f
lfrfl r b
dddddd.
Here three kinds of primitive behaviors are designed for
the local navigation control: emergency behavior, obsta-
cle-avoidance and goal-seeking. The reactive navigation
task is decomposed in terms of all these primitive be-
havior that respond to the immediate sensory inputs.
Emergency behavior is directly added to the control
system without probabilistic fuzzy inference to avoid
collision with a dynamic obstacle or a stationary wall
when a possible sensing failure. The emergency behavior
has the priority to guarantee very safe navigation. When
an obstacle is closer than a threshold and the robot has no
time to avoid it while moving, the robot has to stop im-
mediately and retreat from the imminent danger. The
threshold distances are selected online according to the
robot velocity.
Obstacle-avoidance has always been the basic ability
to navigate in a local area. The distance information for
obstacle-avoidance is gained by detecting the environ-
ment with range sensors {,,,,, }
f
lfrfl r b
dddddd. If the
threshold value δ = 0.15 m is set as the safe distance and
the maximum sensing range L = 7 m, then the obsta-
cle-avoidance behavior is controlled by using a probabil-
istic fuzzy inference engine within the range δ ~ L.
As for goal-seeking behavior, an autonomous robot
has the ability of recognizing or knowing where the goal
is. For the behavior of turning to goal or subgoal, the key
is to recognize the goal and measure the distance be-
tween the robot and the goal. In this paper, the location
of the goal is assumed to be known to the robot.
As shown in Figure 4, all of the behavior are imple-
mented using the probabilistic fuzzy controller and the
output data 1 and 2 refer to the control output after
defuzzification for the two DC motors with encoders.
m m
4. Experimental Results
To demonstrate the performance of the presented reac-
tive navigation system based on PFLS, several groups of
experiments are carried out using a simulated navigation
platform and the real mobile robot MT-R, respectively.
Figure 5 shows the range measurement results for the

C. L. CHEN ET AL. 82
Figure 4. The PFLS controller for reactive navigation.
Figure 5. Experimental results of range measurement for mobile robots.

C. L. CHEN ET AL. 83
reactive navigation in a local environment. The per-
formance of range measurement using PFLS is compared
with the ordinary FLS method. It is clear that FLS can
reduce the distance errors, but it can not process the sto-
chastic errors and is difficult to further improve the range
measurement performance. On the contrary, PFLS based
range data processing method works better.
Figure 6 demonstrates the navigation results in the
simulation environment, which is built up using Visual
C++ with the setting of 600 × 400 (Grid representation).
In each case, the environment is assumed to be com-
pletely unknown for the mobile robot except the start and
goal states. The robot has to explore the environment
using onboard range sensors. It is shown in Figure 6 that
the PFLS method navigates the robot more safely and
effectively. Then we further test the reactive navigation
performance on the real robot MT-R in our office build-
ing. As shown in Figure 7, the robot walks through a
corridor with clustered obstacles quickly and safely. All
these results also demonstrate the success and practica-
bility of the proposed reactive navigation control ap-
proach.
5. Conclusions
PFLS has been demonstrated to be an effective approach
to map the typical non-linear relation of input-output
model with stochastic and fuzzy uncertainties. In this
paper, PFLS is further extended to a general method for
range measurement based reactive navigation. First,
PFLS for range sensor is proposed to process stochastic
uncertainty and can effectively reduce the disturbance
caused by stochastic uncertainty, so that the distance
information can approximate to the actual measured val-
ues more accurately. Then these sensor data are sent to a
probabilistic fuzzy rule-based inference system with de-
signed reactive behaviors for the local navigation. Both
of the simulated experiments and the experiments on a
real mobile robot MT-R show that the presented prob-
abilistic fuzzy approach can help obtain more precise
sensory information robustly and improves the perform-
ance of the reactive navigation in uncertain environ-
ments.
Our future work will focus on the application of PFLS
to more sensor systems and the combination of robot
learning systems. The probabilistc fuzzy rules of behav-
ior-selection are mostly configured by experiences. It
will be more useful and practical for the robot to adjust
the existing control rules through learning.
6. Acknowledgements
This work was supported by the National Natural Science
Figure 6. Simulated results of reactive navigation.
Figure 7. Performance demonstration of reactive navigation in a local environment using MT-R.

C. L. CHEN ET AL. 84
Foundation of China (No. 60805029, 70971060, and
70731002).
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