probabilistic fuzzy system for uncertain localization and map building of mobile robots

1546 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 6, JUNE 2012

Probabilistic Fuzzy System for UncertainLocalization and Map Building of Mobile Robots

Shuo Chen and Chunlin Chen, Member, IEEE

Abstract—Precise localization and map building for mobile ro-bots in unknown environments are fundamental and crucial issuesin robotics. In this paper, to deal with unavoidable uncertainties inperception and actuation, a probabilistic fuzzy approach is appliedto dead-reckoning-based localization and range measurement,respectively. Then, they are adopted to constitute a systematicmap-building method. Dead reckoning in autonomous localizationallows a mobile robot to determine its present position from aknown past position. Unfortunately, pure dead reckoning methodsare prone to accumulated errors that grow without bound overtime. In addition, various unpredictable errors in distance dataare also found in range measurement during exploration andmap building. It is analyzed that all these kinds of errors causedby various disturbances can be classified into nonstochastic andstochastic uncertainties. A probabilistic fuzzy system is designedto reduce both of these uncertainties for more precise localizationand map building. The experimental results demonstrate the suc-cess and robustness of the proposed method for more precise andreliable mobile-robot localization and map building with variousunexpected disturbances.

Index Terms—Dead reckoning, localization and map build-ing, probabilistic fuzzy system (PFS), range measurement,uncertainties.

I. INTRODUCTION

I T IS A fundamental and crucial issue for an autonomousmobile robot to know its position and to explore the un-

known complex environments [1]–[3]. Dead reckoning is acommon and basic method for localization, which enables amobile robot to determine its present position by projecting itspast courses steered and speeds over ground from a known pastposition [4]–[6]. However, new positions in dead reckoning arecalculated only from previous positions, so the displacementand orientation errors will accumulate and grow rapidly withtime [6]. These accumulated errors, if not being prohibited,

Manuscript received December 1, 2010; revised July 20, 2011; acceptedAugust 23, 2011. Date of publication March 1, 2012; date of current versionMay 11, 2012. This work was supported in part by the National Natural ScienceFoundation of China under Grant 60805029, by the Fundamental ResearchFunds for the Central Universities under Grant 1095011802, and by the projectfrom the State Key Laboratory of Industrial Control Technology, ZhejiangUniversity. The Associate Editor coordinating the review process for this paperwas Dr. John Sheppard.

S. Chen is with the Department of Control and System Engineering, NanjingUniversity, Nanjing 210093, China, and also with the Department of Electricaland Computer Engineering, University of Delaware, Newark, DE 19716 USA.

C. Chen is with the Department of Control and System Engineering andthe State Key Laboratory for Novel Software Technology, Nanjing University,Nanjing 210093, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2012.2186652

will induce the robot to increasingly deviate from the expectedposition. Therefore, dead reckoning is not suitable for long-distance navigation and brings great inconvenience in practicalapplications for mobile robots. In addition, uncertain errors arestill found in the range-sensor-based measurement during robotexploration even if multisensory techniques facilitate mobilerobots to detect the obstacles for exploration and map building[7]–[10]. With errors in range measurement, the robot willmistake the passable area for obstacles or wrongly considerthat the blocked area ahead can be passed. This fatal problemmust be addressed since it obviously increases the dangerand difficulty in robot exploration. Hence, the robot has toeffectively handle all kinds of errors and recover from variousdisturbances to achieve good performances in complex andunknown environments.

In order to effectively reduce the accumulated errors of deadreckoning and the range measurement errors, recent researchstudies generally focus on two approaches, i.e., landmark-based methods and data processing methods [11]–[18]. Thelandmark-based method has been implemented for localizing amobile robot in an environment with landmarks and estimatingthe position and orientation of the robot [11], [12]. Multi-landmarks are established in the possibly deviated position forrobot recognition, so that the robot can make amendment andget back to the expected location. Unfortunately, the landmarkmethod needs sufficient sensory information and predefinedlandmarks. It only works well in known or partially known en-vironments and is difficult to be applied to the exploration andmap-building tasks in an unknown unstructured environment.Data processing methods are also widely used to reduce theaccumulated errors. For example, Kalman filter combines theinformation of different uncertain sources to get more preciseresults [14], [15]. Furthermore, multisensor fusion approach[7], [16]–[18] increases both reliability and precision of theenvironmental observations used for the self-localization ofmobile robots. However, the data processing method empha-sizes the data processing such as sensory data from multisen-sors. It seldom models the possible stochastic events in theenvironments when the robot works. Thus, few of these dataprocessing methods can deal with various kinds of uncertain-ties effectively in dead-reckoning-based localization and rangemeasurement.

This paper considers fuzzy logic analysis of the data process-ing to achieve more precise and robust localization and mapbuilding. It is investigated that both accumulated errors in dead-reckoning-based localization and the errors in range-sensor-based exploration are caused by various uncertain disturbances[19], [20]. Generally, fuzzy logic systems (FLSs) [10], [21],

0018-9456/$31.00 © 2012 IEEE

CHEN AND CHEN: PFS FOR UNCERTAIN LOCALIZATION AND MAP BUILDING OF MOBILE ROBOTS 1547

Fig. 1. Demonstration of dead reckoning scheme. (a) World coordinate system and robot-centered coordinate system. (b) Localization using dead reckoning.

[22] have the capability to deal with multiple uncertaintieswithout precise mathematical formula. Nevertheless, the un-certainties in mobile-robot localization can be classified intononstochastic uncertainties (can be modeled as fuzzy uncer-tainties) and stochastic uncertainties. Stochastic uncertainties,including wheel slippage, random noises, human interference,stochastic temperature or weather influence, and the like, can-not be processed using ordinary FLS. Therefore, in this paper, aprobabilistic fuzzy system (PFS) [23]–[26] for mobile-robot lo-calization and map building is proposed to reduce the accumu-lated dead reckoning errors and range measurement errors byhandling both of the nonstochastic and stochastic uncertainties.In addition, as grid map is an approximate solution in robot mapbuilding and is not sensitive to the parameters in a particularsensing system, the occupancy grid map is adopted in this paperfor accurate map building. The precise localization and mapbuilding are implemented on various experimental platformsto demonstrate the feasibility and superiority of the proposedmethod. The experiment results show that PFS is more robustand reliable than ordinary FLS for uncertain localization andmap building.

The rest of this paper is organized as follows. The next sec-tion discusses the issues of dead-reckoning-based localization,range measurement for robot exploration, and map buildingand how current approaches address these issues. Section IIIdescribes the PFS for mobile-robot localization, exploration,and map building to effectively reduce the errors in localizationand range measurement. In Section IV, we test the presentedmethods with several groups of simulated experiments and areal mobile robot. The results are also analyzed and discussedin detail. Conclusions are given in Section V.

II. PROBLEM FORMULATION

In this section, the self-localization and map building formobile robots using dead reckoning and range measurementare introduced. Then, current approaches for eliminating theunpredicted errors are presented, and the existing problemsare discussed. Finally, the motivation of applying PFS to solvethese problems for more precise localization and map buildingis analyzed in detail.

A. Localization Based on Dead Reckoning

Robot localization is a fundamental issue in mobile roboticsthat the location and heading of robots can be estimated throughdetecting the internal states or the external environments [5]. Atpresent, the application of satellite Global Positioning Systemfor outdoor robot localization has been successfully imple-mented [27]. Unfortunately, due to the limitation of environ-ments, tasks, or other factors, most indoor mobile robots mainlyachieve self-localization only with onboard sensors and wheelrotation encoders (odometer). Dead reckoning method gives anestimate of the robot position from a known initial state byintegrating the movement information such as rotation of thewheels or vehicle acceleration.

We define the coordinate system of the external environmentas a 2-D world coordinate system. As shown in Fig. 1(a),(x(W ), y(W )) is denoted as the robot position (displacementcomponent) regarding the world coordinate, and θ(W ) is thetraveling orientation (rotation component). We distinguish itfrom the 2-D robot-centered coordinate system spanned by axesx(R) and y(R). In dead reckoning scheme, the initial positionof the robot is supposed to be known. For example, the twoactivated wheels of the robot are equipped with encoders torecord the wheel turns. At each sampling time, the distancetraveled by each wheel is calculated with the reading of the en-coders. The reference point of the robot’s relative position is themidpoint between two axles of wheels [Fig. 1(b)]. The displace-ment Δd(W ) and the rotation Δθ(W ) between two samplingpoints in the world coordinate system are formulated as follows:

Δd(W ) =Δd

(R)R + Δd

(R)L

2(1)

Δθ(W ) =Δd

(R)R − Δd

(R)L

Laxis(2)

where Δd(R)R and Δd

(R)L are defined as the distances covered

by the right and the left wheels, respectively. Laxis is the lengthof the wheel axis.

At the sampling step n(n = 0, 1, . . . , N) of encoders in thelocalization process, the robot location can be described as atriple unit (x(W )

n , y(W )n , θ

(W )n ). Then, as shown in Fig. 1(b), the


updated position (x(W )n+1, y

(W )n+1, θ

(W )n+1) using dead reckoning can

be expressed as follows [11]:

x(W )n+1 = x(W )

n + Δd(W )n cos

(θ(W )

n +Δθ

(W )n

2

)(3)

y(W )n+1 = y(W )

n + Δd(W )n sin

(θ(W )

n +Δθ

(W )n

2

)(4)

θ(W )n+1 = θ(W )

n + Δθ(W )n . (5)

Although dead reckoning is a basic and important method forthe position estimation of mobile robots, it is also well knownthat the accumulated errors are unavoidable, which makes it un-suitable for long-distance localization. Generally, these errorscan be formalized into displacement errors and rotation errors.Both of these errors are caused by various factors, which canbe classified into two categories: systematic and nonsystematicerrors [28]. Systematic errors include unequal wheel diameter,misalignment of wheels, uncertainty about the effective wheel-base, limited encoder resolution, and limited encoder samplingrate. Nonsystematic errors include the following: 1) wheelslippage due to slippery floors, overacceleration, fast turning(skidding), external forces (interaction with external bodies),internal forces (castor wheels), or nonpoint wheel contact withthe floor; 2) travel over uneven or irregular floors (bumps,cracks, or debris); 3) travel over unexpected objects on thefloor; 4) changes in indoor airflows; and 5) changes in indoortemperature. If the environment is ideal with most smoothindoor surfaces, mild air current, and proper temperature, thesystematic errors accumulate constantly. However, for actualapplications of robot localization in complex unknown environ-ments with precise robot structure design, nonsystematic odom-etry errors have become dominant because they are caused bythe interaction of the robot with unpredictable features of theenvironments and they always stochastically occur and cannotbe predicted.

To prohibit the accumulated errors and realize precise local-ization, recent research studies in robotics focus on two ap-proaches, i.e., the landmark-based methods and data processingapproaches. Landmarks are widely used to reduce the posi-tioning errors, where triangulation is a well-known techniquefor estimating a robot’s position and orientation in its envi-ronment [7]. However, some analysis shows that the landmarkmethods have some critical limitations: 1) It needs a globalmap of the environment known to the robot, but in real robotlocalization, the located position, the detected routine, and thesurrounding environments are often unknown and dynamic;2) most landmarks need to be predefined and preplaced man-ually according to the given environment; 3) the landmarkmethod needs sufficient sensory information; and 4) the appli-cation of landmarks also needs to meet many specific conditionsrequired by the users or the environments. All these limitationsmake it difficult to be applied to the exploration and map-building tasks in practice. On the other hand, as has been arguedin Section I, the data processing methods, such as Kalman filter[14], neural networks [15], and Bayesian method [16], cannoteffectively handle a variety of stochastic and unpredictable

Fig. 2. Mobile robot with multiple kinds of range sensors.

uncertainties. Therefore, in this paper, the PFS is proposedto reduce the accumulated errors by handling both of thestochastic and nonstochastic uncertainties and helps the robotachieve a more accurate self-localization.

B. Range Measurement for Environment Exploration

Range sensors are installed on mobile robots to measure thedistance between itself and the obstacle around, which helpsthe robot recognize the environments and avoid obstacles [9],[26]. For example, a mobile robot called MT-R is used in thisstudy. The main range sensors, including ultrasonic sensors,infrared sensors, and laser scanner, are shown in Fig. 2. Thesensitive range of the ultrasonic sensor is 0.2–7 m, and thatof the infrared sensor is 0.1–0.8 m. These two kinds of rangesensors are always combined to detect the obstacles in frontof them.

Ordinary FLS helps process the distance data with vaguenessdue to the limited performance of the sensors. However, theprocess of measurement by range sensors is always influencedby man-made disturbances and random noises from stochasticuncertainties. These unpredictable events cannot be recognizedby the range sensor or people who conduct the sensory in-struments. Thus, if we can try to find a specific probabilisticdistribution held by each of the stochastic uncertainties andimplement a PFS for range measurement, the measurementerror caused by stochastic uncertainties can be effectivelyreduced.

C. Map Building

Map building is to model the environments by combining theinformation of robot localization and perception [2], [9]. Withthe calibration of PFS on dead-reckoning-based localizationand range measurement process, a more precise map approx-imating to real surrounding environment can be constructed.In the indoor environments, there are two kinds of widelyused maps: topological map [29], [30] and occupancy grid map[31]–[33]. Topological map denotes the indoor environment asa topology graph with nodes and related edges, where nodesare important environmental points (corners, doors, elevators,stairs, etc.) and edges represent the relationship between thenodes, such as corridors. Topological map has high abstrac-tion, which is suitable for describing structured environments.Occupancy grid map is the map that divides the environment(the work space of the robot) into a series of grids, where each


Fig. 3. Processing configuration of PFS.

grid is given a possible value indicating the probability that thegrid is occupied. Grid map is easy to construct and maintainbut has very high computation complexity if the map is huge.Thus, currently, topology map and grid map are combinedto construct a hybrid topology–grid map to solve the map-building problem more effectively [29]. Since the grid map isthe foundation to construct a topological map or a hybrid map,we focus on grid map in this paper. In addition, grid map is anapproximate solution and is applicable in noisy and uncertainsensor measurement process. The key to get a precise gridmap is to make the localization and range measurement processmore reliable and accurate.

III. MOBILE-ROBOT LOCALIZATION AND

MAP BUILDING BASED ON PFS

In this section, PFS for mobile-robot localization, explo-ration, and map building is presented by combining fuzzylogic with the probabilistic processing method under stochasticuncertain circumstances. With the help of PFS, the accumulatederrors in dead-reckoning-based localization and the range mea-surement errors can be effectively reduced, which helps to builda more precise map of the environment.

A. PFS

In this paper, probabilistic fuzzy logic is adopted for the non-stochastic and stochastic uncertainty processing. The process-ing configuration of PFS is shown in Fig. 3. In PFS, theprobabilistic fuzzy logic integrates the probabilistic processingmethod into the ordinary fuzzy reasoning mechanism. In theprobabilistic fuzzification, the result of ordinary fuzzificationcan be obtained by membership-grade mechanism. Stochasticinformation can be handled through probabilistic calculation.Furthermore, with the assist of fuzzy rules, a fuzzy inferenceengine is used to process the fuzzy and stochastic informa-tion in the inference stage. Then, the crisp output can begained through probabilistic defuzzification, including ordinarydefuzzification and probabilistic processing. In PFS, the de-fuzzification process is similar to that of FLS using specifieddefuzzification calculation. The difference lies in that a mathe-matical expectation in probabilistic processing is computed toget the final crisp output.

Compared with FLS, PFS is special in the probabilisticfuzzification set and the defuzzification process, which havebeen discussed in our previous work [25], [26]. An ordinaryfuzzy set can be described as a set

S = (I, U) =u(x1)

x1+

u(x2)x2

+ · · · + u(xn)xn

=N∑

i=1

u(xi)xi

(6)

where an input variable is xi ∈ I = {x1, x2, . . . , xn} and itsfuzzy membership grade is u(xi) ∈ U ⊆ [0, 1].

The probabilistic fuzzy set S can be donated as a probabilityspace of S = (S, P ), where fuzzy membership grade u(x)is a random variable with a certain probabilistic distributionfunction P (x, u(x)). For example, Fig. 4 shows an instanceof a discrete probabilistic fuzzy set S in a 3-D ordinary fuzzyspace

S = ∪i=1,2,3,4

((I, ui), Pi) (7)

where I = {x1, x2, x3} = {1, 2, 3}, P1 = 0.4, u1 = {0.6, 0.2,0.1}, P2 = 0.1, u2 = {0, 0.5, 0.3}, P3 = 0.2, u3 = {0.1, 0.3,0.7}, P4 = 0.3, and u4 = {0.4, 0.8, 0.5}. The operation ofdefuzzification in PFS can be realized by centroid calculation.

B. PFS for Mobile-Robot Localization With Dead Reckoning

In this section, we introduce the odometry errors and theinterference factors that cause the accumulated errors in deadreckoning. A PFS-based approach is proposed to eliminate theposition errors and orientation errors for precise mobile-robotlocalization.

Analysis of Accumulated Errors in Dead Reckoning: In deadreckoning, new positions are calculated from previous posi-tions, so the displacement errors and orientation errors willaccumulate and grow rapidly when the robot moves. Fig. 5shows a moving path (consisting of five steps) of a robot. Therobot starts from (x(W )

0 , y(W )0 , θ

(W )0 ) = (0, 0, π/6) and stops at

(x(W )4 , y

(W )4 , θ

(W )4 ), which are labeled as START POINT and

TERMINAL POINT, respectively. Suppose that the robot moves1 m for each step and the rotation error of 3◦ occurs at thefirst, third, and fifth steps. It can be seen that the orientationerrors in dead reckoning calculation accumulate rapidly and


Fig. 4. Instance of discrete probabilistic fuzzy set in 3-D coordinate.

Fig. 5. Dead-reckoning-based localization with and without errors.

make the robot’s position deviate from the ideal location atevery sampling step, which leads the terminal point to a muchdifferent position from the expected one. Similarly, if the deadreckoning errors occur in the displacement component, therewill also be the inaccurate position estimation in localization. InFig. 5, it is clear that the position and orientation errors may oc-cur and accumulate as described in (1)–(5). In dead reckoning,the errors in both coordinates (ex(W )

n , ey(W )n ) and the heading

orientation eθ(W )n at the sampling step n(n = 0, 1, . . . , 4) are

as follows:

ex(W )n =x

(W )ABSn − x

(W )DRn

ey(W )n = y

(W )ABSn − y

(W )DRn

eθ(W )n = θ

(W )ABSn − θ

(W )DRn (8)

where ex(W )n , ey(W )

n , and eθ(W )n are the position and orientation

errors. x(W )ABSn, y

(W )ABSn, and θ

(W )ABSn are the absolute position

and orientation. x(W )DRn, y

(W )DRn, and θ

(W )DRn are the position and

orientation calculated with the dead reckoning method. It is

clear that the errors will accumulate and grow rapidly as therobot moves and will lead to large deviation from the absoluteposition and direction.

From all the aforementioned analysis, it is clear that mostof the troublesome errors in the position and orientation arenonsystematic errors that dominate in uncertain environments.These errors are caused by various stochastic interferences,and their characteristics may be acquired through statisticalmethods. For example, we have tested the MT-R robot in ourrobot laboratory and have gotten the results for some maindisturbances. The value bounds and occurrence probabilities ofthe errors regarding different types of disturbances are shownin Table I.

Probabilistic Fuzzification Processing: For probabilisticfuzzification processing, we first need to know what forms ofvariables can be used as the inputs of fuzzy system and howto deal with the random uncertainties. Position and orienta-tion errors are the main sources of the accumulated errors indead-reckoning-based localization. At the sampling step n(n =0, 1, . . . , N), both position and orientation errors come from thecombined stochastic disturbances Ej (j = 1, 2, . . . ,M). For


TABLE IBOUNDS AND OCCURRENCE PROBABILITIES OF LOCALIZATION ERRORS FOR DIFFERENT DISTURBANCES

TABLE IIMFS AND PROBABILITIES OF THE EVENTS CAUSING POSITION AND ORIENTATION ERRORS

Fig. 6. MFs used for the probabilistic fuzzification processing.

example, we take N = 40 and M = 4 and define the stochasticevents as follows: E1 is wheel slippage, E2 is uneven or irreg-ular floors, E3 is changes in indoor airflows, and E4 is changesin indoor temperature.Each random disturbance event can con-tribute to errors in both displacement and rotation forms, whichare denoted as a triple unit (ex(W )

n,j , ey(W )n,j , eθ

(W )n,j )T (n =

0, 1, . . . 40 and j = 1, 2, . . . , 4). Thus, at sampling step n, eachunit of these errors can be used as the input of fuzzificationand will be assigned membership functions (MFs) fj(x) ⊆(0, 1] according to specific disturbance events. Each eventcorresponds to one specific MF and probability P (Ej) in theprobabilistic fuzzification process (as shown in Table II).

The MFs used in this paper are shown in Fig. 6. The linguis-tic variables are set as {NE,ND,NC,NB,NA,Z, PA, PB,PC, PD,PE}, where N denotes negative, P denotes pos-itive, Z denotes zero, and E, D, C, B, and A render theconsequent descending degree wherein E is the most and Ais the least.Given the different bounds of position errors andthat of orientation errors, we use the errors (ex(W )

n,j , ey(W )n,j )T

and (eθ(W )n,j )T as the inputs of the PFS. Each unit of the triple

errors caused by every event Ej has its probabilistic fuzzy set

({u(j)};P (Ej)) with its ordinary membership grade set by thecalculation of MFs

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

{uex(j);P (Ej)}= {uj (aex(1)) , uj (aex(2)) , . . . , uj (aex(11)) ;P (Ej)}

{uey(j);P (Ej)}= {uj (aey(1)) , uj (aey(2)) , . . . , uj (aey(11)) ;P (Ej)}

{ueθ(j);P (Ej)}= {uj (aeθ(1)) , uj (aeθ(2)) , . . . , uj (aeθ(11)) ;P (Ej)} .

(9)

Probabilistic Inference Processing: The kth rule of the PFSis expressed as

IF e is Ak Then eu is Bk (10)

where Ak (k = 1, 2, . . . , P ) and Bk (k = 1, 2, . . . , P ) areprobabilistic fuzzy sets and are called the probabilistic fuzzysets of the antecedent and consequent parts, respectively. At thenth step of localization, (ex(W ), ey(W ))T and (eθ(W ))T canbe used as an input e in the inference process, respectively. Theinference output eu is the calibration variable to eliminate either


TABLE IIIINFERENCE RULES

position errors or rotation errors. The fuzzy control rules areshown in Table III.

The probabilistic fuzzy relation set is denoted as

RAk→Bk(x, y) =

⋃e∈X,eu∈Y

(∪Ak∩Bk, P ) (11)

and its fuzzy membership grade is expressed as a randomvariable

uRk= uAk

◦ uBk(12)

where ◦ denotes the t-norm operation and the minimum isadopted for most cases in this paper. ∪Ak∩Bk

is designed asa set of {uRk

∈ [0, 1]}, while uRk, uAk

, and uBkare defined

as the fuzzy membership grades of the probabilistic fuzzyrelation set RAk→Bk

(x, y), the antecedent fuzzy set Ak, and

the consequent fuzzy set Bk, respectively. The probabilities ofstochastic uncertainties can be processed as

P (EAk∩Bk) = P (EAk

) · P (EBk) (13)

where P (EAk∩Bk) is the probability of the stochastic circum-

stance EAk∩Bk. P (EAk

) and P (EBk) are equally the prob-

abilities possessed by uAkand uBk

, respectively. Then, theinference output eu can be computed as

eu = e ◦ RAk→Bk. (14)

Probabilistic Defuzzification Processing: With an outputfield sequence b ∈ H = {b1, b2, . . . , b11}, the inference outputeu in PFS can also be regarded to own its defuzzification set ofex(W ), ey(W ), eθ(W ) by the operation of the same MFs as thefuzzification processing⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

{vex(j);P (Ej)}= {vj (bex(1)) , vj (bex(2)) , . . . , vj (bex(11)) ;P (Ej)}

{vey(j);P (Ej)}= {vj (bey(1)) , vj (bey(2)) , . . . , vj (bey(11)) ;P (Ej)}

{veθ(j);P (Ej)}= {vj (beθ(1)) , vj (beθ(2)) , . . . , vj (beθ(11)) ;P (Ej)} .

(15)

For the defuzzification processing, the centroid of the outputfuzzy set can be computed as follows:

vj =

11∑w=1

bex,ey,eθ(w)v(b(w))

11∑w=1

v(b(w))

. (16)

It is presumed that each position or orientation error input instep n part of localization under the jth stochastic event has anoutput calibration vj (j = 1, 2, . . . , 4) and possesses a proba-

bility P (Ej). Thus, the final output calibration to displacementor rotation errors of PFS in the nth part of localization can beobtained by mathematical expectation v(n) from vj

vx,y,θ(n) = EX(vj) =M∑

j=1

vjPj . (17)

With each calibration output in PFS to a displacement or anorientation error caused by every probable random disturbance,the position of the robot in each localization step calculatedfrom dead reckoning is approximate to the precise value. Theaccurate position and orientation of terminal destination can beattained from the encoder and the dead reckoning computation.The calibration process is shown as follows:

x(W )ADJ0 =x

(W )DR0 + ex

(W )0 − vx(0) ≈ x

(W )ABS0

y(W )ADJ0 = y

(W )DR0 + ey

(W )0 − vy(0) ≈ y

(W )ABS0

θ(W )ADJ0 = θ

(W )DR0 + eθ

(W )0 − vθ(0) ≈ θ

(W )ABS0

x(W )ADJ1 =x

(W )ADJ0 + Δd

(W )0 cos

(θ(W )ADJ0 +

Δθ(W )0

2

)

+ ex(W )1 − vx(1) ≈ x

(W )ABS1

y(W )ADJ1 = y

(W )ADJ0 + Δd

(W )0 cos

(θ(W )ADJ0 +

Δθ(W )0

2

)

+ ey(W )1 − vy(1) ≈ y

(W )ABS1

θ(W )ADJ1 = θ

(W )ADJ0 + Δθ

(W )0 + eθ

(W )1 − vθ(1) ≈ θ

(W )ABS1

. . .

. . .

. . .

x(W )ADJn+1 =x

(W )ADJn + Δd(W )

n cos

(θ(W )ADJn +

Δθ(W )n

2

)

+ ex(W )n+1 − vx(n + 1) ≈ x

(W )ABSn+1

y(W )ADJn+1 = y

(W )ADJn + Δd(W )

n cos

(θ(W )ADJn +

Δθ(W )n

2

)

+ ey(W )n+1 − vy(n + 1) ≈ y

(W )ABSn+1

θ(W )ADJn+1 = θ

(W )ADJn + Δθ(W )

n + eθ(W )n+1

− vθ(n + 1) ≈ θ(W )ABSn+1 (18)

where vx(n), vy(n), andvθ(n), n = 0, 1, . . . , 40, are denoted asthe calibration components based on PFS to the position errors


Fig. 7. PFS for range measurement under different stochastic circumstances.

(ex(W )n , ey

(W )n ) and orientation errors eθ

(W )n . x

(W )ADJn, y

(W )ADJn,

and θ(W )ADJn, n = 0, 1, . . . , 40, are the results of the PFS-based

method which can be regarded as the results approximatingto the absolute values according to the calibration processpreviously mentioned.

C. Range-Measurement-Based Map Building

The environment around the robot can be modeled as a seriesof maps built in the robot to help with such tasks as explorationand navigation. The successful map building of mobile robotsis always based on the localization, perception, informationfusion, and various robust control methods. Due to the meritsof occupancy grid map, such as robustness and easy building,grid map is adopted in this paper to demonstrate the applicationof PFS for range measurement and accurate map building withuncertainties.

PFS for Range Measurement: It is necessary for mobilerobots to detect the unknown environments and make fullexploration of all uncertainties before building a complete map.Mobile robots achieve recognition on the surrounding environ-ments and avoid obstacles with onboard range sensors. Rangesensors can be used to measure distance information and exportrange data. Ordinary FLS helps process the distance data withvagueness due to the limited performance of the sensors. How-ever, the process of measurement by range sensors in unknownand complex circumstances is always influenced by randomnoises, vibration when the robot is moving, and changes in in-door flows or temperature from stochastic uncertainties. Theseunpredictable events cannot be recognized by range sensorsor people who conduct the sensor instruments. Consequently,these stochastic noises or interferences will render the sensoryoutput to be less accurate.

Fig. 7 shows three distance data measured by range sen-sors under different stochastic circumstances, respectively. Theprecise distance is set as 4 m. For instance, situation 1 rep-resents the distance data under stochastic condition with mu-tative indoor airflows or temperature. Situation 2 presents the

distance information with the disturbance of random noises.Situation 3 expresses the data under vibration when the robotis moving.

Under each stochastic condition Ei (i = 1, 2, 3), the distanceerror ei = hMEAi − hABSi, i = 1, 2, 3 between the precise dis-tance hABSi and the measured distance hMEAi can be used forthe fuzzification with diverse MFs according to specific inter-ference events. The types of MF and allocation of parametersare shown in Fig. 6. In addition, each of the stochastic situationspossesses a probability with a certain probabilistic distributionfunction in continuous case or discrete case.

In this paper, PFS processing for range measurement is thesame as the PFS control method on reducing the position andorientation errors in robot localization. Stochastic conditionsEi (i = 1, 2, 3) are defined as E1 being the mutative indoorflows or temperature, E2 being the random noises, and E3 beingthe vibration of the robot (as shown in Table IV).

According to the ranges of specific errors in Table IV, thebound of fuzzy field can be determined to have the same range.Each error ei (i = 1, 2, 3) has its probabilistic fuzzy set by thecomputation on the MFs. The elements in fuzzy filed can berepresented as {a(ei)

1 , a(ei)2 , . . . , a

(ei)11 } by equally dividing error

range into 11 parts, with its ordinary membership grade set bythe calculation of MFs {ue

i} = {ui(a1), ui(a2), . . . , ui(a11)}.The result of the probabilistic fuzzification processing containsthe fuzzy membership grade and its probability, which aredefined as fuzzy probability pairs ({ui}, P (Ei)).

Fuzzy inference rule is presented in Table III, withthe same probabilistic fuzzy relation set RAk→Bk

(x, y) =⋃e∈X,eu∈Y (∪Ak∩Bk

, P ) and the inference output eu = e ◦RAk→Bk

. The centroid calculation is used to implementthe defuzzification, and the result of the defuzzificationcan be attained as v1, v2, v3for eliminating each errorei (i = 1, 2, 3). Mathematical expectation v = EX(vi) =∑3

i=1 P (i)vi is computed to give the final output of PFS.Due to the specific probability held by each of the stochas-

tic uncertainties, PFS for range measurement can be imple-mented to effectively reduce the measurement errors caused by


TABLE IVMFS AND PROBABILITIES OF THE EVENTS CAUSING MEASURED ERRORS FROM RANGE SENSORS

Fig. 8. Exploration extension of range sensors (a) in practical cases and (b) in grid map simulation.

stochastic uncertainties. On the contrary, ordinary FLScan only handle the certain errors either under E1, E2,or E3.

Exploration and Map Building: The experiments on model-ing range sensors and exploration extension are based on themobile robot MT-AR that has two activated wheels and eightrange sensors for detecting the obstacles around, as shown inFig. 8(a). Under practical circumstances, sensor detection rangeis described as a circular structure. Hence, in the process ofbuilding a grid map, the detection range of sensors in simulationis discrete, with three units of grid distance as the widest rangefor the distance sensors to detect and with the assumption thatthe surrounding environment in all directions can be detected.It is also assumed that the robot is regarded as an ideal pointwhich is coincident with the geometry center of the eight rangesensors. The exploration extension of range sensors in grid mapbuilding is shown in Fig. 8(b).

Grid map is the map that divides the environment into aseries of grids, where each grid is given a value indicatingthe probability that the grid is occupied. In this paper, wesimplify the algorithm of grid possession using only threestatuses for each grid: occupied (denoted as OCC, for obstaclearea), void (defined as EMP, for passable area), and undetected(represented as UNEXPLORED, for unknown area). Beforeexploration, the grid map is initialized as UNEXPLORED, andthe changes on the state of the grid will occur only when therobot has explored this area. The robot moves one grid at each

Fig. 9. Robot movement (a) in one step and (b) in three steps.

step [see Fig. 9(a)], and the extension of the robot movingduring the first three steps is shown in Fig. 9(b).

The detection extension of range sensors is adopted for robotexploration of the environment, as shown in Fig. 8(b). If theobstacles are detected for the next step of robot movement,i.e., once there are obstacles on or inside the farthest edgeof the sensor exploration extension, the front grid will not bedetected further. Therefore, the area of the grid is representedas an unknown area in the map. Fig. 10 shows the indicationof available information in the next step exploration from thecurrent robot location. The movements at the next step in theright, down, left, and up are shown in Fig. 10. It is primaryfor robots to avoid obstacles in the next step moving, so ifthe status of the grid in the next step is OCC, the amountof information in the step is set to be negative infinity so


Fig. 10. Exploration method for two-step movement.

Fig. 11. Comparison of errors and calibration along the robot trajectory between PFS and FLS. (a) Distance errors and calibration. (b) Orientation errors andcalibration.

that the robot will not walk into the obstacles. The source ofnew information on occupancy of the next grid is to identifywhether the white grid has been explored. If not being detected,the amount of information in this direction contributed by thewhite grid is one which means that the area in this movingorientation can be explored. Otherwise, the contribution is zero.To ensure consistency with the sensor model, the robot is notallowed to cross the barrier to detect the grid behind the robot.The status of the adjacent grid shown in Fig. 10 is used todetermine whether the unexplored grid is detected. The choiceof robot rotation in a specific direction will encounter morethan one situation with the same information. Here, we usetwo main priority principles: 1) Primarily follow the originaldirection, similar to depth-first exploration algorithm to get agood depth of information, and 2) finally return to the originaldirection. The main purpose is to make the robot not to go backfrequently.

During the map-building process, the probabilistic fuzzyapproach is applied along with the robot moving at every stepto effectively reduce the range measurement errors and theaccumulated dead reckoning errors for better localization andperception, which helps in achieving a more precise and reliablemap building.

IV. EXPERIMENTAL RESULTS

To verify the effectiveness of the proposed approach, the ap-plications of PFS for localization, range measurement, and mapbuilding are tested with several groups of simulations and realexperiments. The performance of PFS is compared with thatof the ordinary FLS. PFS is constructed to effectively reduceboth of the range measurement errors and the localization errorsand to achieve an accurate map building for mobile robots inunknown environments with uncertainties. Then, the presented


Fig. 12. Comparison of performance between PFS and FLS for mobile-robot localization.

methods are further implemented on the real mobile robotMT-R with a simple exploration task.

A. PFS for Mobile-Robot Localization

The robot work space is defined in the x–y world coordinateswith located position variables (x(W ), y(W ), θ(W )). In theexperiments, the localization process consists of 41 samplingsteps (N = 40) with 1-m displacement per step. It is assumedthat the errors arise in the whole localization process. Fig. 11compares the positioning errors and calibration by FLSand PFS approaches along the robot trajectory. Fig. 11(a)shows the accumulated nonsystematic errors of displacementcomponent (ex(W )

n , ey(W )n ) at every step from odometers and

dead reckoning calculation and describes the accuracy fornonsystematic errors after ordinary FLS and PFS calibration.Fig. 11(b) shows the accumulated nonsystematic error ofrotation component eθ

(W )n in each sampling step from wheel

encoders and dead reckoning calculation and demonstrates theresult of accuracy for nonsystematic errors after ordinary FLSand PFS calibration.

From Fig. 11, it is clear that the PFS method greatly reducedboth of the displacement and orientation errors, while theerrors from pure dead reckoning computation and ordinary FLSmethod accumulate and grow rather rapidly as the robot movesand lead to large deviation from the absolute position andorientation. In addition, although ordinary FLS can reduce theaccumulative errors compared with the results without calibra-tion method, there still exist great position and rotation errorsusing ordinary FLS approach because FLS cannot effectivelyhandle the stochastic uncertainties.

The absolute robot trajectory and the path with differentcontrol methods to reduce the accumulated position errors androtation errors are shown in Fig. 12. It is assumed that therobot starts from a position (x(W )

0 , y(W )0 , θ

(W )0 ) = (0, 0, π/15)

and stops at (x(W )40 , y

(W )40 , θ

(W )40 ), which are labeled START

POINT and TERMINAL POINT, respectively. The termi-nal points of pure odometry, FLS path, PFS path, andthe absolute ideal path are (32.1219, 15.8731, −0.1704),(32.0149, 15.3484, −0.1868), (31.3467, 13.3777, −0.2554),

and (31.2768, 13.2501, −0.2618), respectively. From Fig. 12,it can be seen that, although FLS can reduce the accumulatederrors, the localization using FLS also possesses large positionand rotation errors. Ordinary FLS has the capacity to compen-sate the accumulated errors caused by one of the random cir-cumstances, but it cannot effectively process the uncertaintiesof all the stochastic events. On the contrary, the localizationbased on PFS shows more precise and robust performances,and the robot trajectory is approximate to the absolute tra-jectory. It is validated in this simulated experiment that PFScan handle both of the stochastic and vague uncertainties moreeffectively.

B. PFS for Range Measurement in Robot Exploration

The experimental results of distance errors in range mea-surement are shown in Fig. 13(a) with n1 = 50 samples incontinuous case and Fig. 13(b) with n2 = 1000 samples indiscrete case. The precise distance between the robot and thefront obstacle is set as 4 m. It is assumed that FLS can onlyprocess errors caused from stochastic event E2 but cannot pre-dict the occurrence of errors from stochastic events E1 and E3.Define the normalized mean square error (MSE) as

MSE =1n

n∑k=1

(y(k) − y(k))2

where y(k) is the desired output and y(k) is the estimated out-put of PFS. Normalized MSEs of PFS, FLS, and noncalibrationin continuous case and discrete case are shown in Tables Vand VI, respectively.

The experimental results demonstrate that the fluctuation ofdistance data errors in PFS is less than that in FLS understochastic circumstances and the distance measured with PFSprocessing is more approximate to the accurate distance thanthat measured with FLS.From the performance comparison,it is shown that PFS can handle both of the stochastic andvague uncertainties and its performance is better than that ofthe ordinary FLS.


Fig. 13. Comparison of performances between PFS, ordinary FLS, and no calibration for range measurement. (a) Fifty samples in continuous case.(b) On thousand samples in discrete case.

TABLE VCOMPARISON OF NORMALIZED MSE OF PFS, FLS, AND NONCALIBRATION IN CONTINUOUS CASE

TABLE VICOMPARISON OF NORMALIZED MSE OF PFS, FLS, AND NONCALIBRATION IN DISCRETE CASE

C. Map Building

In this group of map-building experiments, both of the struc-tured and unstructured environments are studied. A structuredenvironment (such as an office building) is shown in Fig. 14(a).The size of the terrain mesh is 65 × 38, and the initial positionof the robot is at (4, 36), marked as the little green circle.On the contrary, Fig. 15(a) shows an unstructured environment(a highly clustered room). The size of the grid map is 63 × 38,and the robot’s initial position is at (22, 26). In the simula-tion, white space represents passable region, and obstacles orunknown areas are represented as dark regions. It is also as-sumed that the errors in range measurement and dead reckoning

calculation arise every five grids when the robot moves. Theexploration path is marked as blue lines.

Figs. 14(b) and 15(b) show the results of map building withgreat measurement errors in range sensors and large positionand orientation errors calculated from dead reckoning. Due tolarge errors in range measurement, the robot will mistake thepassable area for obstacles or blockades that cannot be passed,or the obstacles will be wrongly recognized as the passablearea. On the other hand, wheel slippage and changes in indoorflows or temperature also lead the robot to deviate from itsexpected path; thus, some of the area that should be detectedby the range sensor cannot be explored. These mistakes in the


Fig. 14. Comparison of the real map and the map-building results of a structured environment. (a) Plan map of a structured environment. (b) Map building withgreat errors. (c) Map building with calibration of ordinary FLS. (d) Map building with calibration of PFS.

Fig. 15. Comparison of the real map and the map-building results of an unstructured environment. (a) Plan map of an unstructured environment. (b) Map buildingwith great errors. (c) Map building with calibration of ordinary FLS. (d) Map building with calibration of PFS.

recognition of environment also make the robot to not continuethe exploration process.Therefore, the robot exploration is notcompleted, and the map-building results are much differentfrom the real environment. The degree of the completion ofmapping in Fig. 14(b) is 43%, and that in Fig. 15(b) is 36%.

Figs. 14(c) and 15(c) show the result of map building withordinary FLS calibration for range measurement errors and

localization errors. Although the result of map creation is morecomplete than that with no calibration method, the stochasticerrors cannot be effectively reduced by FLS. Thus, there arestill many differences between the explored map and the realenvironment. The proportion of the area not being detected isstill relatively large. The degree of completion for map buildingin Fig. 14(c) is 68%, and that in Fig. 15(c) is 79%.


Fig. 16. Real experimental results of exploration and map building in a clustered office room.

Figs. 14(d) and 15(d) show the results of map building withPFS-based calibration for range measurement and localizationerrors. The exploration steps are 2500 and 3000, with thenumbers of explored grids of 15 149 and 17 753, respectively. Inspite of some areas not being explored by the robot, the detectedareas are greatly approximate to that of the real environment.Thus, the maps are well built, and the information of theenvironment around the localization path is relatively accurateand precise. The degree of completion of map building inFig. 14(d) is 90%, and that in Fig. 15(d) is 92%.

D. Real Experiments

The proposed methods are further applied to a real robot.The robot employed in this study is the mobile robot MT-Ras introduced in Section II-B. The robot MT-R has six rangesensor pairs (each range sensor pair consists of an ultrasonicsensor and an infrared sensor) and can detect the obstaclesfrom six directions. It also has two MAXON motors withtwo shaft encoders. To explore in a clustered office room,the mobile robot percepts, decides the motion commands, andmodels the surrounding environments with these range sensors.The parameter settings are the same as those in the simulatedexperiments. The difference lies in that the sensory inputs aresix range data attained through six range sensor pairs. Fig. 16shows the results of a complete exploration process.

As shown in Fig. 16, the robot trajectory is demonstratedwith a series of red circles, and its walking directions aremarked with arrows. Photographs (1)–(6) correspond to sixspecific positions 1–6 in this exploration task. The robot startsfrom a narrow corner around position 1 and then successfullyexplores this office room (door closed) by navigating throughpositions 2–6 in turn while avoiding obstacles. The degree ofcompletion of exploration and map building is approximatelyabove 90%, and some very narrow corners cannot be detecteddue to the physical limitations. More results also show that thePFS approach is practicable and robust to such problems asrange sensors and dead-reckoning-based localization and mapbuilding for mobile robots.

V. CONCLUSION

To navigate safely and effectively in unknown environmentswith various uncertainties, it is of great importance for a mobilerobot to determine its location and then to build a map ofthe environment [34], [35]. Dead-reckoning-based localizationmethod is prone to unbounded accumulated errors due tovarious disturbances. In addition, unavoidable errors are alsofound in range measurement for robot map building. It has beeninvestigated and analyzed that the accumulated localizationerrors and range measurement errors consist of two kinds ofuncertainties: nonstochastic and stochastic uncertainties. In thispaper, a PFS approach has been proposed to handle both ofthese kinds of uncertainties for more precise localization andperception. PFS is different from the ordinary FLS in thatit uses probabilistic fuzzy sets instead of ordinary fuzzy setsto represent and process the information with both stochasticand fuzzy uncertainties. Thus, the PFS approach presented inthis paper can help in reducing the errors in dead reckoningand range measurement, which is practical for a mobile robotto achieve more precise localization and map building. Inthe simulated experiments, the performances of the presentedPFS approach for dead-reckoning-based localization and range-measurement-based map building are comprehensively tested.Further experiments on a real mobile robot also show that thepresented method is practical and all the results demonstrate thesuccess of the proposed approach. In addition, the results alsoshow that PFS is more robust and reliable than ordinary FLS forprocessing uncertainties and can handle the various uncertain-ties in the localization, exploration, and map-building processesfor mobile robots. Although the PFS approach may cost a littlemore computation resources than the ordinary FLS approach,it will not affect the general performance of the robot becausethese tasks are not time consuming compared with other ro-bot tasks such as path planning in a grid-based map, large-scale map updating, and machine vision processing. Therefore,the PFS approach presented in this paper is a very goodcandidate for most uncertain localization and map-buildingapplications. Our future work will focus on the automatic


probability-distribution-obtaining method and the probabilisticfuzzy theories and algorithms for more applications.

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Shuo Chen received the B.E. degree in automaticcontrol from Nanjing University, Nanjing, China,in 2010. He is currently working toward the Ph.D.degree in the Department of Electrical and ComputerEngineering, University of Delaware, Newark.

He is also currently with the Department of Con-trol and System Engineering, Nanjing University.His research interests include intelligent control, mo-bile robotics, and high-performance computing.

Chunlin Chen (S’05–M’06) received the B.E. de-gree in automatic control and the Ph.D. degree inpattern recognition and intelligent systems from theUniversity of Science and Technology of China,Hefei, China, in 2001 and 2006, respectively.

He is currently an Associate Professor with theDepartment of Control and System Engineering,Nanjing University, Nanjing, China, where he is alsowith the State Key Laboratory for Novel SoftwareTechnology, Nanjing University. His research inter-ests include machine learning, intelligent control,

mobile robotics, and quantum algorithm.

probabilistic fuzzy system for uncertain localization and map building of mobile robots

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