Probabilistic Fuzzy System for Uncertain Localization ... - IEEE Xplore

1546

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

Probabilistic Fuzzy System for Uncertain Localization and Map Building of Mobile Robots Shuo Chen and Chunlin Chen, Member, IEEE

Abstract—Precise localization and map building for mobile robots in unknown environments are fundamental and crucial issues in robotics. In this paper, to deal with unavoidable uncertainties in perception and actuation, a probabilistic fuzzy approach is applied to dead-reckoning-based localization and range measurement, respectively. Then, they are adopted to constitute a systematic map-building method. Dead reckoning in autonomous localization allows a mobile robot to determine its present position from a known past position. Unfortunately, pure dead reckoning methods are prone to accumulated errors that grow without bound over time. In addition, various unpredictable errors in distance data are also found in range measurement during exploration and map building. It is analyzed that all these kinds of errors caused by various disturbances can be classified into nonstochastic and stochastic uncertainties. A probabilistic fuzzy system is designed to reduce both of these uncertainties for more precise localization and map building. The experimental results demonstrate the success and robustness of the proposed method for more precise and reliable mobile-robot localization and map building with various unexpected disturbances. Index Terms—Dead reckoning, localization and map building, probabilistic fuzzy system (PFS), range measurement, uncertainties.

I. I NTRODUCTION

I

T IS A fundamental and crucial issue for an autonomous mobile robot to know its position and to explore the unknown complex environments [1]–[3]. Dead reckoning is a common and basic method for localization, which enables a mobile robot to determine its present position by projecting its past courses steered and speeds over ground from a known past position [4]–[6]. However, new positions in dead reckoning are calculated only from previous positions, so the displacement and orientation errors will accumulate and grow rapidly with time [6]. These accumulated errors, if not being prohibited,

Manuscript received December 1, 2010; revised July 20, 2011; accepted August 23, 2011. Date of publication March 1, 2012; date of current version May 11, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 60805029, by the Fundamental Research Funds for the Central Universities under Grant 1095011802, and by the project from the State Key Laboratory of Industrial Control Technology, Zhejiang University. The Associate Editor coordinating the review process for this paper was Dr. John Sheppard. S. Chen is with the Department of Control and System Engineering, Nanjing University, Nanjing 210093, China, and also with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA. C. Chen is with the Department of Control and System Engineering and the 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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2012.2186652

will induce the robot to increasingly deviate from the expected position. Therefore, dead reckoning is not suitable for longdistance navigation and brings great inconvenience in practical applications for mobile robots. In addition, uncertain errors are still found in the range-sensor-based measurement during robot exploration even if multisensory techniques facilitate mobile robots to detect the obstacles for exploration and map building [7]–[10]. With errors in range measurement, the robot will mistake the passable area for obstacles or wrongly consider that the blocked area ahead can be passed. This fatal problem must be addressed since it obviously increases the danger and difficulty in robot exploration. Hence, the robot has to effectively handle all kinds of errors and recover from various disturbances to achieve good performances in complex and unknown environments. In order to effectively reduce the accumulated errors of dead reckoning and the range measurement errors, recent research studies generally focus on two approaches, i.e., landmarkbased methods and data processing methods [11]–[18]. The landmark-based method has been implemented for localizing a mobile robot in an environment with landmarks and estimating the position and orientation of the robot [11], [12]. Multilandmarks are established in the possibly deviated position for robot recognition, so that the robot can make amendment and get back to the expected location. Unfortunately, the landmark method needs sufficient sensory information and predefined landmarks. It only works well in known or partially known environments and is difficult to be applied to the exploration and map-building tasks in an unknown unstructured environment. Data processing methods are also widely used to reduce the accumulated errors. For example, Kalman filter combines the information of different uncertain sources to get more precise results [14], [15]. Furthermore, multisensor fusion approach [7], [16]–[18] increases both reliability and precision of the environmental observations used for the self-localization of mobile robots. However, the data processing method emphasizes the data processing such as sensory data from multisensors. It seldom models the possible stochastic events in the environments when the robot works. Thus, few of these data processing methods can deal with various kinds of uncertainties effectively in dead-reckoning-based localization and range measurement. This paper considers fuzzy logic analysis of the data processing to achieve more precise and robust localization and map building. It is investigated that both accumulated errors in deadreckoning-based localization and the errors in range-sensorbased 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

Fig. 1.

1547

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 uncertainties without precise mathematical formula. Nevertheless, the uncertainties in mobile-robot localization can be classified into nonstochastic uncertainties (can be modeled as fuzzy uncertainties) and stochastic uncertainties. Stochastic uncertainties, including wheel slippage, random noises, human interference, stochastic temperature or weather influence, and the like, cannot be processed using ordinary FLS. Therefore, in this paper, a probabilistic fuzzy system (PFS) [23]–[26] for mobile-robot localization and map building is proposed to reduce the accumulated dead reckoning errors and range measurement errors by handling both of the nonstochastic and stochastic uncertainties. In addition, as grid map is an approximate solution in robot map building and is not sensitive to the parameters in a particular sensing system, the occupancy grid map is adopted in this paper for accurate map building. The precise localization and map building are implemented on various experimental platforms to demonstrate the feasibility and superiority of the proposed method. The experiment results show that PFS is more robust and reliable than ordinary FLS for uncertain localization and map building. The rest of this paper is organized as follows. The next section discusses the issues of dead-reckoning-based localization, range measurement for robot exploration, and map building and how current approaches address these issues. Section III describes the PFS for mobile-robot localization, exploration, and map building to effectively reduce the errors in localization and range measurement. In Section IV, we test the presented methods with several groups of simulated experiments and a real mobile robot. The results are also analyzed and discussed in detail. Conclusions are given in Section V.

A. Localization Based on Dead Reckoning Robot localization is a fundamental issue in mobile robotics that the location and heading of robots can be estimated through detecting the internal states or the external environments [5]. At present, the application of satellite Global Positioning System for outdoor robot localization has been successfully implemented [27]. Unfortunately, due to the limitation of environments, tasks, or other factors, most indoor mobile robots mainly achieve self-localization only with onboard sensors and wheel rotation encoders (odometer). Dead reckoning method gives an estimate of the robot position from a known initial state by integrating the movement information such as rotation of the wheels or vehicle acceleration. We define the coordinate system of the external environment as a 2-D world coordinate system. As shown in Fig. 1(a), (x(W ) , y (W ) ) is denoted as the robot position (displacement component) regarding the world coordinate, and θ(W ) is the traveling orientation (rotation component). We distinguish it from the 2-D robot-centered coordinate system spanned by axes x(R) and y (R) . In dead reckoning scheme, the initial position of the robot is supposed to be known. For example, the two activated wheels of the robot are equipped with encoders to record the wheel turns. At each sampling time, the distance traveled by each wheel is calculated with the reading of the encoders. The reference point of the robot’s relative position is the midpoint between two axles of wheels [Fig. 1(b)]. The displacement Δd(W ) and the rotation Δθ(W ) between two sampling points in the world coordinate system are formulated as follows: (R)

ΔdR + ΔdL 2

Δθ(W ) =

ΔdR − ΔdL Laxis

(R)

II. P ROBLEM F ORMULATION In this section, the self-localization and map building for mobile robots using dead reckoning and range measurement are introduced. Then, current approaches for eliminating the unpredicted errors are presented, and the existing problems are discussed. Finally, the motivation of applying PFS to solve these problems for more precise localization and map building is analyzed in detail.

(R)

(R)

Δd(W ) =

(R)

(1)

(R)

(2)

where ΔdR and ΔdL are defined as the distances covered by the right and the left wheels, respectively. Laxis is the length of the wheel axis. At the sampling step n(n = 0, 1, . . . , N ) of encoders in the localization process, the robot location can be described as a (W ) (W ) (W ) triple unit (xn , yn , θn ). Then, as shown in Fig. 1(b), the

1548

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

(W )

(W )

(W )

updated position (xn+1 , yn+1 , θn+1 ) using dead reckoning can be expressed as follows [11]:   (W ) Δθn (W ) (W ) (W ) (W ) xn+1 = xn + Δdn cos θn + (3) 2   (W ) Δθn (W ) (W ) (W ) (W ) (4) yn+1 = yn + Δdn sin θn + 2 (W )

θn+1 = θn(W ) + Δθn(W ) .

(5)

Although dead reckoning is a basic and important method for the position estimation of mobile robots, it is also well known that the accumulated errors are unavoidable, which makes it unsuitable for long-distance localization. Generally, these errors can be formalized into displacement errors and rotation errors. Both of these errors are caused by various factors, which can be classified into two categories: systematic and nonsystematic errors [28]. Systematic errors include unequal wheel diameter, misalignment of wheels, uncertainty about the effective wheelbase, limited encoder resolution, and limited encoder sampling rate. Nonsystematic errors include the following: 1) wheel slippage due to slippery floors, overacceleration, fast turning (skidding), external forces (interaction with external bodies), internal forces (castor wheels), or nonpoint wheel contact with the floor; 2) travel over uneven or irregular floors (bumps, cracks, or debris); 3) travel over unexpected objects on the floor; 4) changes in indoor airflows; and 5) changes in indoor temperature. If the environment is ideal with most smooth indoor surfaces, mild air current, and proper temperature, the systematic errors accumulate constantly. However, for actual applications of robot localization in complex unknown environments with precise robot structure design, nonsystematic odometry errors have become dominant because they are caused by the interaction of the robot with unpredictable features of the environments and they always stochastically occur and cannot be predicted. To prohibit the accumulated errors and realize precise localization, recent research studies in robotics focus on two approaches, i.e., the landmark-based methods and data processing approaches. Landmarks are widely used to reduce the positioning errors, where triangulation is a well-known technique for estimating a robot’s position and orientation in its environment [7]. However, some analysis shows that the landmark methods have some critical limitations: 1) It needs a global map of the environment known to the robot, but in real robot localization, the located position, the detected routine, and the surrounding environments are often unknown and dynamic; 2) most landmarks need to be predefined and preplaced manually according to the given environment; 3) the landmark method needs sufficient sensory information; and 4) the application of landmarks also needs to meet many specific conditions required by the users or the environments. All these limitations make it difficult to be applied to the exploration and mapbuilding tasks in practice. On the other hand, as has been argued in Section I, the data processing methods, such as Kalman filter [14], neural networks [15], and Bayesian method [16], cannot effectively 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 proposed to reduce the accumulated errors by handling both of the stochastic and nonstochastic uncertainties and helps the robot achieve a more accurate self-localization. B. Range Measurement for Environment Exploration Range sensors are installed on mobile robots to measure the distance between itself and the obstacle around, which helps the robot recognize the environments and avoid obstacles [9], [26]. For example, a mobile robot called MT-R is used in this study. The main range sensors, including ultrasonic sensors, infrared sensors, and laser scanner, are shown in Fig. 2. The sensitive range of the ultrasonic sensor is 0.2–7 m, and that of the infrared sensor is 0.1–0.8 m. These two kinds of range sensors are always combined to detect the obstacles in front of them. Ordinary FLS helps process the distance data with vagueness due to the limited performance of the sensors. However, the process of measurement by range sensors is always influenced by man-made disturbances and random noises from stochastic uncertainties. These unpredictable events cannot be recognized by the range sensor or people who conduct the sensory instruments. Thus, if we can try to find a specific probabilistic distribution held by each of the stochastic uncertainties and implement a PFS for range measurement, the measurement error caused by stochastic uncertainties can be effectively reduced. C. Map Building Map building is to model the environments by combining the information of robot localization and perception [2], [9]. With the calibration of PFS on dead-reckoning-based localization and range measurement process, a more precise map approximating to real surrounding environment can be constructed. In the indoor environments, there are two kinds of widely used maps: topological map [29], [30] and occupancy grid map [31]–[33]. Topological map denotes the indoor environment as a topology graph with nodes and related edges, where nodes are important environmental points (corners, doors, elevators, stairs, etc.) and edges represent the relationship between the nodes, such as corridors. Topological map has high abstraction, 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

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

Fig. 3.

1549

Processing configuration of PFS.

grid is given a possible value indicating the probability that the grid is occupied. Grid map is easy to construct and maintain but has very high computation complexity if the map is huge. Thus, currently, topology map and grid map are combined to construct a hybrid topology–grid map to solve the mapbuilding problem more effectively [29]. Since the grid map is the foundation to construct a topological map or a hybrid map, we focus on grid map in this paper. In addition, grid map is an approximate solution and is applicable in noisy and uncertain sensor measurement process. The key to get a precise grid map is to make the localization and range measurement process more reliable and accurate. III. M OBILE -ROBOT L OCALIZATION AND M AP B UILDING BASED ON PFS In this section, PFS for mobile-robot localization, exploration, and map building is presented by combining fuzzy logic with the probabilistic processing method under stochastic uncertain circumstances. With the help of PFS, the accumulated errors in dead-reckoning-based localization and the range measurement errors can be effectively reduced, which helps to build a more precise map of the environment. A. PFS In this paper, probabilistic fuzzy logic is adopted for the nonstochastic and stochastic uncertainty processing. The processing configuration of PFS is shown in Fig. 3. In PFS, the probabilistic fuzzy logic integrates the probabilistic processing method into the ordinary fuzzy reasoning mechanism. In the probabilistic fuzzification, the result of ordinary fuzzification can be obtained by membership-grade mechanism. Stochastic information can be handled through probabilistic calculation. Furthermore, with the assist of fuzzy rules, a fuzzy inference engine is used to process the fuzzy and stochastic information in the inference stage. Then, the crisp output can be gained through probabilistic defuzzification, including ordinary defuzzification and probabilistic processing. In PFS, the defuzzification process is similar to that of FLS using specified defuzzification calculation. The difference lies in that a mathematical expectation in probabilistic processing is computed to get the final crisp output.

Compared with FLS, PFS is special in the probabilistic fuzzification set and the defuzzification process, which have been discussed in our previous work [25], [26]. An ordinary fuzzy set can be described as a set u(xn )  u(xi ) u(x1 ) u(x2 ) + + ··· + = x1 x2 xn xi i=1 (6) N

S = (I, U ) =

where an input variable is xi ∈ I = {x1 , x2 , . . . , xn } and its fuzzy membership grade is u(xi ) ∈ U ⊆ [0, 1]. The probabilistic fuzzy set S˜ can be donated as a probability space of S˜ = (S, P ), where fuzzy membership grade u(x) is a random variable with a certain probabilistic distribution function P (x, u(x)). For example, Fig. 4 shows an instance of a discrete probabilistic fuzzy set S˜ in a 3-D ordinary fuzzy space 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 of defuzzification 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 the interference factors that cause the accumulated errors in dead reckoning. A PFS-based approach is proposed to eliminate the position errors and orientation errors for precise mobile-robot localization. Analysis of Accumulated Errors in Dead Reckoning: In dead reckoning, new positions are calculated from previous positions, so the displacement errors and orientation errors will accumulate and grow rapidly when the robot moves. Fig. 5 shows a moving path (consisting of five steps) of a robot. The (W ) (W ) (W ) robot starts from (x0 , y0 , θ0 ) = (0, 0, π/6) and stops at (W ) (W ) (W ) (x4 , y4 , θ4 ), which are labeled as START POINT and TERMINAL POINT, respectively. Suppose that the robot moves 1 m for each step and the rotation error of 3◦ occurs at the first, third, and fifth steps. It can be seen that the orientation errors in dead reckoning calculation accumulate rapidly and

1550

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

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 at every sampling step, which leads the terminal point to a much different position from the expected one. Similarly, if the dead reckoning errors occur in the displacement component, there will also be the inaccurate position estimation in localization. In Fig. 5, it is clear that the position and orientation errors may occur and accumulate as described in (1)–(5). In dead reckoning, (W ) (W ) the errors in both coordinates (exn , eyn ) and the heading (W ) orientation eθn at the sampling step n(n = 0, 1, . . . , 4) are as follows: (W )

(W )

(W )

(W )

(W )

(W )

) ex(W = xABSn − xDRn n

eyn(W ) = yABSn − yDRn eθn(W ) = θABSn − θDRn (W )

(W )

(W )

(8)

where exn , eyn , and eθn are the position and orientation (W ) (W ) (W ) errors. xABSn , yABSn , and θABSn are the absolute position (W ) (W ) (W ) and orientation. xDRn , yDRn , and θDRn are the position and orientation calculated with the dead reckoning method. It is

clear that the errors will accumulate and grow rapidly as the robot moves and will lead to large deviation from the absolute position and direction. From all the aforementioned analysis, it is clear that most of the troublesome errors in the position and orientation are nonsystematic errors that dominate in uncertain environments. These errors are caused by various stochastic interferences, and their characteristics may be acquired through statistical methods. For example, we have tested the MT-R robot in our robot laboratory and have gotten the results for some main disturbances. The value bounds and occurrence probabilities of the errors regarding different types of disturbances are shown in Table I. Probabilistic Fuzzification Processing: For probabilistic fuzzification processing, we first need to know what forms of variables can be used as the inputs of fuzzy system and how to deal with the random uncertainties. Position and orientation errors are the main sources of the accumulated errors in dead-reckoning-based localization. At the sampling step n(n = 0, 1, . . . , N ), both position and orientation errors come from the combined stochastic disturbances Ej (j = 1, 2, . . . , M ). For

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

1551

TABLE I B OUNDS AND O CCURRENCE P ROBABILITIES OF L OCALIZATION E RRORS FOR D IFFERENT D ISTURBANCES

TABLE II MF S AND P ROBABILITIES OF THE E VENTS C AUSING P OSITION AND O RIENTATION E RRORS

Fig. 6.

MFs used for the probabilistic fuzzification processing.

example, we take N = 40 and M = 4 and define the stochastic events as follows: E1 is wheel slippage, E2 is uneven or irregular floors, E3 is changes in indoor airflows, and E4 is changes in indoor temperature.Each random disturbance event can contribute to errors in both displacement and rotation forms, which (W ) (W ) (W ) are denoted as a triple unit (exn,j , eyn,j , eθn,j )T (n = 0, 1, . . . 40 and j = 1, 2, . . . , 4). Thus, at sampling step n, each unit of these errors can be used as the input of fuzzification and will be assigned membership functions (MFs) fj (x) ⊆ (0, 1] according to specific disturbance events. Each event corresponds to one specific MF and probability P (Ej ) in the probabilistic fuzzification process (as shown in Table II). The MFs used in this paper are shown in Fig. 6. The linguistic variables are set as {N E, N D, N C, N B, N A, Z, P A, P B, P C, P D, P E}, where N denotes negative, P denotes positive, Z denotes zero, and E, D, C, B, and A render the consequent descending degree wherein E is the most and A is the least.Given the different bounds of position errors and (W ) (W ) that of orientation errors, we use the errors (exn,j , eyn,j )T (W )

and (eθn,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 the calculation 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 )} ⎪ ⎪ ⎪ ⎪ {u ⎪ ⎩ eθ (j); P (Ej )} = {uj (aeθ (1)) , uj (aeθ (2)) , . . . , uj (aeθ (11)) ; P (Ej )} . (9) Probabilistic Inference Processing: The kth rule of the PFS is expressed as ˜k IF e is A˜k Then eu is B

(10)

˜k (k = 1, 2, . . . , P ) are where A˜k (k = 1, 2, . . . , P ) and B probabilistic fuzzy sets and are called the probabilistic fuzzy sets of the antecedent and consequent parts, respectively. At the nth step of localization, (ex(W ) , ey (W ) )T and (eθ(W ) )T can be used as an input e in the inference process, respectively. The inference output eu is the calibration variable to eliminate either

1552

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

TABLE III I NFERENCE RULES

position errors or rotation errors. The fuzzy control rules are shown in Table III. The probabilistic fuzzy relation set is denoted as RA˜k →B˜k (x, y) = (∪A˜k ∩B˜k , P ) (11)

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

vx,y,θ (n) = EX(vj ) =

e∈X,eu∈Y

vj P j .

(17)

j=1

and its fuzzy membership grade is expressed as a random variable uRk = uAk ◦ uBk

(12)

where ◦ denotes the t-norm operation and the minimum is adopted for most cases in this paper. ∪A˜k ∩B˜k is designed as a set of {uRk ∈ [0, 1]}, while uRk , uAk , and uBk are defined as the fuzzy membership grades of the probabilistic fuzzy relation set RA˜k →B˜k (x, y), the antecedent fuzzy set A˜k , and ˜k , respectively. The probabilities of the consequent fuzzy set B stochastic uncertainties can be processed as P (EA˜k ∩B˜k ) = P (EA˜k ) · P (EB˜k )

(13)

With each calibration output in PFS to a displacement or an orientation error caused by every probable random disturbance, the position of the robot in each localization step calculated from dead reckoning is approximate to the precise value. The accurate position and orientation of terminal destination can be attained from the encoder and the dead reckoning computation. The calibration process is shown as follows: (W )

(W )

(W )

− vx (0) ≈ xABS0

(W )

(W )

(W )

− vy (0) ≈ yABS0

(W )

(W )

(W )

− vθ (0) ≈ θABS0 

xADJ0 = xDR0 + ex0 yADJ0 = yDR0 + ey0 θADJ0 = θDR0 + eθ0

where P (EA˜k ∩B˜k ) is the probability of the stochastic circumstance EA˜k ∩B˜k . P (EA˜k ) and P (EB˜k ) are equally the probabilities possessed by uAk and uBk , respectively. Then, the inference output eu can be computed as

(W ) xADJ1

eu = e ◦ RA˜k →B˜k .

(W ) yADJ1

(14)

Probabilistic Defuzzification Processing: With an output field sequence b ∈ H = {b1 , b2 , . . . , b11 }, the inference output eu in PFS can also be regarded to own its defuzzification set of ex(W ) , ey (W ) , eθ(W ) by the operation of the same MFs as the fuzzification 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 )} ⎪ ⎪ ⎪ ⎪ (j); P (Ej )} {v ⎪ eθ ⎩ = {vj (beθ (1)) , vj (beθ (2)) , . . . , vj (beθ (11)) ; P (Ej )} . (15) For the defuzzification processing, the centroid of the output fuzzy set can be computed as follows: 11

vj =

11

+

(W )

+ ex1 (W ) = yADJ0

+

(W )

+ ey1 (W )

(W )

(W )

(W ) Δd0

(W )

cos

(W )



(W )



Δθ0 + 2

(W ) θADJ0 (W )

− vx (1) ≈ xABS1  (W ) Δd0

cos

(W ) θADJ0

Δθ0 + 2

(W )

− vy (1) ≈ yABS1 (W )

θADJ1 = θADJ0 + Δθ0

(W )

+ eθ1

(W )

− vθ (1) ≈ θABS1

... ... ... (W ) xADJn+1

(W )

yADJn+1

 (W ) = xADJn

) + Δd(W cos n

(W )

(W ) θADJn

Δθn + 2

(W )

(W )

(W )

(W )



+ exn+1 − vx (n + 1) ≈ xABSn+1   (W ) Δθn (W ) (W ) (W ) = yADJn + Δdn cos θADJn + 2 + eyn+1 − vy (n + 1) ≈ yABSn+1

bex,ey,eθ (w)v(b(w))

w=1

(W ) = xADJ0

(W )

.

(16)

v(b(w))

w=1

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

(W )

(W )

(W )

θADJn+1 = θADJn + Δθn(W ) + eθn+1 (W )

− vθ (n + 1) ≈ θABSn+1

(18)

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

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

Fig. 7.

1553

PFS for range measurement under different stochastic circumstances.

(W )

(W )

(W )

(W )

(W )

(exn , eyn ) and orientation errors eθn . xADJn , yADJn , (W ) and θADJn , n = 0, 1, . . . , 40, are the results of the PFS-based method which can be regarded as the results approximating to the absolute values according to the calibration process previously mentioned. C. Range-Measurement-Based Map Building The environment around the robot can be modeled as a series of maps built in the robot to help with such tasks as exploration and navigation. The successful map building of mobile robots is always based on the localization, perception, information fusion, and various robust control methods. Due to the merits of occupancy grid map, such as robustness and easy building, grid map is adopted in this paper to demonstrate the application of PFS for range measurement and accurate map building with uncertainties. PFS for Range Measurement: It is necessary for mobile robots to detect the unknown environments and make full exploration of all uncertainties before building a complete map. Mobile robots achieve recognition on the surrounding environments and avoid obstacles with onboard range sensors. Range sensors can be used to measure distance information and export range data. Ordinary FLS helps process the distance data with vagueness due to the limited performance of the sensors. However, the process of measurement by range sensors in unknown and complex circumstances is always influenced by random noises, vibration when the robot is moving, and changes in indoor flows or temperature from stochastic uncertainties. These unpredictable events cannot be recognized by range sensors or people who conduct the sensor instruments. Consequently, these stochastic noises or interferences will render the sensory output to be less accurate. Fig. 7 shows three distance data measured by range sensors under different stochastic circumstances, respectively. The precise distance is set as 4 m. For instance, situation 1 represents the distance data under stochastic condition with mutative 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 robot is moving. Under each stochastic condition Ei (i = 1, 2, 3), the distance error ei = hMEAi − hABSi , i = 1, 2, 3 between the precise distance hABSi and the measured distance hMEAi can be used for the fuzzification with diverse MFs according to specific interference events. The types of MF and allocation of parameters are shown in Fig. 6. In addition, each of the stochastic situations possesses a probability with a certain probabilistic distribution function in continuous case or discrete case. In this paper, PFS processing for range measurement is the same as the PFS control method on reducing the position and orientation errors in robot localization. Stochastic conditions Ei (i = 1, 2, 3) are defined as E1 being the mutative indoor flows or temperature, E2 being the random noises, and E3 being the vibration of the robot (as shown in Table IV). According to the ranges of specific errors in Table IV, the bound of fuzzy field can be determined to have the same range. Each error ei (i = 1, 2, 3) has its probabilistic fuzzy set by the computation on the MFs. The elements in fuzzy filed can be (e ) (e ) (e ) represented as {a1 i , a2 i , . . . , a11i } by equally dividing error range into 11 parts, with its ordinary membership grade set by the calculation of MFs {uei } = {ui (a1 ), ui (a2 ), . . . , ui (a11 )}. The result of the probabilistic fuzzification processing contains the fuzzy membership grade and its probability, which are defined as fuzzy probability pairs ({ui }, P (Ei )). Fuzzy inference rule is presented in Table III, with the  same probabilistic fuzzy relation set RA˜k →B˜k (x, y) = e∈X,eu∈Y (∪A˜k ∩B˜k , P ) and the inference output eu = e ◦ RA˜k →B˜k . The centroid calculation is used to implement the defuzzification, and the result of the defuzzification can be attained as v1 , v2 , v3 for eliminating each error ei (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 stochastic uncertainties, PFS for range measurement can be implemented to effectively reduce the measurement errors caused by

1554

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

TABLE IV MFS AND P ROBABILITIES OF THE E VENTS C AUSING M EASURED E RRORS F ROM R ANGE S ENSORS

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

stochastic uncertainties. On the contrary, ordinary FLS can only handle the certain errors either under E1 , E2 , or E3 . Exploration and Map Building: The experiments on modeling range sensors and exploration extension are based on the mobile robot MT-AR that has two activated wheels and eight range sensors for detecting the obstacles around, as shown in Fig. 8(a). Under practical circumstances, sensor detection range is described as a circular structure. Hence, in the process of building a grid map, the detection range of sensors in simulation is discrete, with three units of grid distance as the widest range for the distance sensors to detect and with the assumption that the surrounding environment in all directions can be detected. It is also assumed that the robot is regarded as an ideal point which is coincident with the geometry center of the eight range sensors. The exploration extension of range sensors in grid map building is shown in Fig. 8(b). Grid map is the map that divides the environment into a series of grids, where each grid is given a value indicating the probability that the grid is occupied. In this paper, we simplify the algorithm of grid possession using only three statuses for each grid: occupied (denoted as OCC, for obstacle area), void (defined as EMP, for passable area), and undetected (represented as UNEXPLORED, for unknown area). Before exploration, the grid map is initialized as UNEXPLORED, and the changes on the state of the grid will occur only when the robot 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 moving during the first three steps is shown in Fig. 9(b). The detection extension of range sensors is adopted for robot exploration of the environment, as shown in Fig. 8(b). If the obstacles are detected for the next step of robot movement, i.e., once there are obstacles on or inside the farthest edge of the sensor exploration extension, the front grid will not be detected further. Therefore, the area of the grid is represented as an unknown area in the map. Fig. 10 shows the indication of available information in the next step exploration from the current robot location. The movements at the next step in the right, down, left, and up are shown in Fig. 10. It is primary for robots to avoid obstacles in the next step moving, so if the status of the grid in the next step is OCC, the amount of information in the step is set to be negative infinity so

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

1555

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 and calibration.

that the robot will not walk into the obstacles. The source of new information on occupancy of the next grid is to identify whether the white grid has been explored. If not being detected, the amount of information in this direction contributed by the white grid is one which means that the area in this moving orientation can be explored. Otherwise, the contribution is zero. To ensure consistency with the sensor model, the robot is not allowed to cross the barrier to detect the grid behind the robot. The status of the adjacent grid shown in Fig. 10 is used to determine whether the unexplored grid is detected. The choice of robot rotation in a specific direction will encounter more than one situation with the same information. Here, we use two main priority principles: 1) Primarily follow the original direction, similar to depth-first exploration algorithm to get a good depth of information, and 2) finally return to the original direction. The main purpose is to make the robot not to go back frequently.

During the map-building process, the probabilistic fuzzy approach is applied along with the robot moving at every step to effectively reduce the range measurement errors and the accumulated dead reckoning errors for better localization and perception, which helps in achieving a more precise and reliable map building. IV. E XPERIMENTAL R ESULTS To verify the effectiveness of the proposed approach, the applications of PFS for localization, range measurement, and map building are tested with several groups of simulations and real experiments. The performance of PFS is compared with that of the ordinary FLS. PFS is constructed to effectively reduce both of the range measurement errors and the localization errors and to achieve an accurate map building for mobile robots in unknown environments with uncertainties. Then, the presented

1556

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

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

methods are further implemented on the real mobile robot MT-R with a simple exploration task. A. PFS for Mobile-Robot Localization The robot work space is defined in the x–y world coordinates with located position variables (x(W ) , y (W ) , θ(W ) ). In the experiments, the localization process consists of 41 sampling steps (N = 40) with 1-m displacement per step. It is assumed that the errors arise in the whole localization process. Fig. 11 compares the positioning errors and calibration by FLS and PFS approaches along the robot trajectory. Fig. 11(a) shows the accumulated nonsystematic errors of displacement (W ) (W ) component (exn , eyn ) at every step from odometers and dead reckoning calculation and describes the accuracy for nonsystematic errors after ordinary FLS and PFS calibration. Fig. 11(b) shows the accumulated nonsystematic error of (W ) rotation component eθn in each sampling step from wheel encoders and dead reckoning calculation and demonstrates the result of accuracy for nonsystematic errors after ordinary FLS and PFS calibration. From Fig. 11, it is clear that the PFS method greatly reduced both of the displacement and orientation errors, while the errors from pure dead reckoning computation and ordinary FLS method accumulate and grow rather rapidly as the robot moves and lead to large deviation from the absolute position and orientation. In addition, although ordinary FLS can reduce the accumulative errors compared with the results without calibration method, there still exist great position and rotation errors using ordinary FLS approach because FLS cannot effectively handle the stochastic uncertainties. The absolute robot trajectory and the path with different control methods to reduce the accumulated position errors and rotation errors are shown in Fig. 12. It is assumed that the (W ) (W ) (W ) robot starts from a position (x0 , y0 , θ0 ) = (0, 0, π/15) (W ) (W ) (W ) and stops at (x40 , y40 , θ40 ), which are labeled START POINT and TERMINAL POINT, respectively. The terminal points of pure odometry, FLS path, PFS path, and the 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 accumulated errors, the localization using FLS also possesses large position and rotation errors. Ordinary FLS has the capacity to compensate the accumulated errors caused by one of the random circumstances, but it cannot effectively process the uncertainties of all the stochastic events. On the contrary, the localization based on PFS shows more precise and robust performances, and the robot trajectory is approximate to the absolute trajectory. It is validated in this simulated experiment that PFS can handle both of the stochastic and vague uncertainties more effectively.

B. PFS for Range Measurement in Robot Exploration The experimental results of distance errors in range measurement are shown in Fig. 13(a) with n1 = 50 samples in continuous case and Fig. 13(b) with n2 = 1000 samples in discrete case. The precise distance between the robot and the front obstacle is set as 4 m. It is assumed that FLS can only process errors caused from stochastic event E2 but cannot predict the occurrence of errors from stochastic events E1 and E3 . Define the normalized mean square error (MSE) as 1 (y(k) − y˜(k))2 n n

MSE =

k=1

where y(k) is the desired output and y˜(k) is the estimated output of PFS. Normalized MSEs of PFS, FLS, and noncalibration in continuous case and discrete case are shown in Tables V and VI, respectively. The experimental results demonstrate that the fluctuation of distance data errors in PFS is less than that in FLS under stochastic circumstances and the distance measured with PFS processing is more approximate to the accurate distance than that measured with FLS.From the performance comparison, it is shown that PFS can handle both of the stochastic and vague uncertainties and its performance is better than that of the ordinary FLS.

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

1557

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 V C OMPARISON OF N ORMALIZED MSE OF PFS, FLS, AND N ONCALIBRATION IN C ONTINUOUS C ASE

TABLE VI C OMPARISON OF N ORMALIZED MSE OF PFS, FLS, AND N ONCALIBRATION IN D ISCRETE C ASE

C. Map Building In this group of map-building experiments, both of the structured and unstructured environments are studied. A structured environment (such as an office building) is shown in Fig. 14(a). The size of the terrain mesh is 65 × 38, and the initial position of 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 simulation, white space represents passable region, and obstacles or unknown areas are represented as dark regions. It is also assumed that the errors in range measurement and dead reckoning

calculation arise every five grids when the robot moves. The exploration path is marked as blue lines. Figs. 14(b) and 15(b) show the results of map building with great measurement errors in range sensors and large position and orientation errors calculated from dead reckoning. Due to large errors in range measurement, the robot will mistake the passable area for obstacles or blockades that cannot be passed, or the obstacles will be wrongly recognized as the passable area. On the other hand, wheel slippage and changes in indoor flows or temperature also lead the robot to deviate from its expected path; thus, some of the area that should be detected by the range sensor cannot be explored. These mistakes in the

1558

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

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 with great 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 building with 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 continue the exploration process.Therefore, the robot exploration is not completed, and the map-building results are much different from the real environment. The degree of the completion of mapping 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 with ordinary FLS calibration for range measurement errors and

localization errors. Although the result of map creation is more complete than that with no calibration method, the stochastic errors cannot be effectively reduced by FLS. Thus, there are still many differences between the explored map and the real environment. The proportion of the area not being detected is still relatively large. The degree of completion for map building in Fig. 14(c) is 68%, and that in Fig. 15(c) is 79%.

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

1559

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 with PFS-based calibration for range measurement and localization errors. The exploration steps are 2500 and 3000, with the numbers of explored grids of 15 149 and 17 753, respectively. In spite of some areas not being explored by the robot, the detected areas are greatly approximate to that of the real environment. Thus, the maps are well built, and the information of the environment around the localization path is relatively accurate and precise. The degree of completion of map building in Fig. 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-R as introduced in Section II-B. The robot MT-R has six range sensor pairs (each range sensor pair consists of an ultrasonic sensor and an infrared sensor) and can detect the obstacles from six directions. It also has two MAXON motors with two shaft encoders. To explore in a clustered office room, the mobile robot percepts, decides the motion commands, and models the surrounding environments with these range sensors. The parameter settings are the same as those in the simulated experiments. The difference lies in that the sensory inputs are six range data attained through six range sensor pairs. Fig. 16 shows the results of a complete exploration process. As shown in Fig. 16, the robot trajectory is demonstrated with a series of red circles, and its walking directions are marked with arrows. Photographs (1)–(6) correspond to six specific positions 1–6 in this exploration task. The robot starts from a narrow corner around position 1 and then successfully explores this office room (door closed) by navigating through positions 2–6 in turn while avoiding obstacles. The degree of completion of exploration and map building is approximately above 90%, and some very narrow corners cannot be detected due to the physical limitations. More results also show that the PFS approach is practicable and robust to such problems as range sensors and dead-reckoning-based localization and map building for mobile robots.

V. C ONCLUSION To navigate safely and effectively in unknown environments with various uncertainties, it is of great importance for a mobile robot to determine its location and then to build a map of the environment [34], [35]. Dead-reckoning-based localization method is prone to unbounded accumulated errors due to various disturbances. In addition, unavoidable errors are also found in range measurement for robot map building. It has been investigated and analyzed that the accumulated localization errors and range measurement errors consist of two kinds of uncertainties: nonstochastic and stochastic uncertainties. In this paper, a PFS approach has been proposed to handle both of these kinds of uncertainties for more precise localization and perception. PFS is different from the ordinary FLS in that it uses probabilistic fuzzy sets instead of ordinary fuzzy sets to represent and process the information with both stochastic and fuzzy uncertainties. Thus, the PFS approach presented in this paper can help in reducing the errors in dead reckoning and range measurement, which is practical for a mobile robot to achieve more precise localization and map building. In the simulated experiments, the performances of the presented PFS approach for dead-reckoning-based localization and rangemeasurement-based map building are comprehensively tested. Further experiments on a real mobile robot also show that the presented method is practical and all the results demonstrate the success of the proposed approach. In addition, the results also show that PFS is more robust and reliable than ordinary FLS for processing uncertainties and can handle the various uncertainties in the localization, exploration, and map-building processes for mobile robots. Although the PFS approach may cost a little more computation resources than the ordinary FLS approach, it will not affect the general performance of the robot because these tasks are not time consuming compared with other robot tasks such as path planning in a grid-based map, largescale map updating, and machine vision processing. Therefore, the PFS approach presented in this paper is a very good candidate for most uncertain localization and map-building applications. Our future work will focus on the automatic

1560

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

probability-distribution-obtaining method and the probabilistic fuzzy theories and algorithms for more applications. R EFERENCES [1] M. A. Salichs and L. Moreno, “Navigation of mobile robot: Open questions,” Robotica, vol. 18, no. 3, pp. 227–234, May 2000. [2] E. Martín-Gorostiza, F. J. Meca, J. L. L. Galilea, E. Martos-Naya, F. B. Naranjo, and Ó. Esteban, “Coverage-mapping method based on a hardware model for mobile-robot positioning in intelligent spaces,” IEEE Trans. Instrum. Meas., vol. 59, no. 2, pp. 266–282, Feb. 2010. [3] T. W. Manikas, K. Ashenayi, and R. L. Wainwright, “Genetic algorithms for autonomous robot navigation,” IEEE Instrum. Meas. Mag., vol. 10, no. 6, pp. 26–31, Dec. 2007. [4] H. Chung, L. Ojeda, and J. Borenstein, “Accurate mobile robot deadreckoning with a precision-calibrated fiber-optic gyroscope,” IEEE Trans. Robot. Autom., vol. 17, no. 1, pp. 80–84, Feb. 2001. [5] C. C. Tsai, “A localization system of a mobile robot by fusing deadreckoning and ultrasonic measurements,” IEEE Trans. Instrum. Meas., vol. 47, no. 5, pp. 1399–1404, Oct. 1998. [6] M. Golfarelli, D. Maio, and S. Rizzi, “Correction of dead-reckoning errors in map building for mobile robots,” IEEE Trans. Robot. Autom., vol. 17, no. 1, pp. 37–47, Feb. 2001. [7] J. A. Castellanos, J. Neira, and J. D. Tardós, “Multisensor fusion for simultaneous localization and map building,” IEEE Trans. Robot. Autom., vol. 17, no. 6, pp. 908–914, Dec. 2001. [8] I. A. R. Ashokaraj, P. M. G. Silson, A. Tsourdos, and B. A. White, “Robust sensor-based navigation for mobile robots,” IEEE Trans. Instrum. Meas., vol. 58, no. 3, pp. 551–556, Mar. 2009. [9] T. Yang and V. Aitken, “Evidential mapping for mobile robots with range sensors,” IEEE Trans. Instrum. Meas., vol. 55, no. 4, pp. 1422–1429, Mar. 2006. [10] H. H. Lin, C. C. Tsai, and J. C. Hsu, “Ultrasonic localization and pose tracking of an autonomous mobile robot via fuzzy adaptive extended information filtering,” IEEE Trans. Instrum. Meas., vol. 57, no. 9, pp. 2024–2034, Sep. 2008. [11] M. Betke and L. Gurvits, “Mobile robot localization using landmarks,” IEEE Trans. Robot. Autom., vol. 13, no. 2, pp. 251–263, Apr. 1997. [12] D. Busquets, C. Sierra, and R. L. D. Mantaras, “A multi-agent approach to qualitative landmark-based navigation,” Autonom. Robots, vol. 15, no. 2, pp. 129–154, Sep. 2003. [13] P. Corke and D. Rus, “Localization and navigation assisted by networked cooperating sensors and robots,” Int. J. Robot. Res., vol. 24, no. 9, pp. 771–786, Sep. 2005. [14] N. Negenborn, “Robot localization and Kalman filters,” M.S. thesis, Utrecht Univ., Utrecht, The Netherlands, 2003. [15] K. S. Choi and S. G. Lee, “Enhanced SLAM for a mobile robot using extended Kalman filter and neural networks,” Int. J. Precision Eng. Manuf., vol. 11, no. 2, pp. 255–264, Apr. 2010. [16] P. U. Lima, “A Bayesian approach to sensor fusion in autonomous sensor and robot networks,” IEEE Instrum. Meas. Mag., vol. 10, no. 3, pp. 22–27, Jun. 2007. [17] D. Amarasinghe, G. K. I. Mann, and R. G. Gosine, “Landmark detection and localization for mobile robot applications: A multisensor approach,” Robotica, vol. 28, no. 5, pp. 663–673, Sep. 2010. [18] G. G. Rigatos, “Extended Kalman and particle filtering for sensor fusion in motion control of mobile robots,” Math. Comput. Simul., vol. 81, no. 3, pp. 590–607, Nov. 2010. [19] L. W. Finkelstein, “Strongly and weakly defined measurement,” Measurement, vol. 34, no. 1, pp. 39–48, Jul. 2003. [20] Z. Godec, “Standard uncertainty in each measurement result explicit or implicit,” Measurement, vol. 20, no. 2, pp. 97–101, Feb. 1997. [21] Y. Bai and H. Zhuang, “On the comparison of bilinear, cubic spline, and fuzzy interpolation techniques for robotic position measurements,” IEEE Trans. Instrum. Meas., vol. 54, no. 6, pp. 2281–2288, Dec. 2005. [22] P. Rusu, E. M. Petriu, T. E. Whalen, A. Cornell, and H. J. W. Spoelder, “Behavior-based neuro-fuzzy controller for mobile robot navigation,” IEEE Trans. Instrum. Meas., vol. 52, no. 4, pp. 1335–1340, Aug. 2003.

[23] Z. Liu and H. X. Li, “A probabilistic fuzzy logic system for modeling and control,” IEEE Trans. Fuzzy Syst., vol. 13, no. 6, pp. 848–859, Dec. 2005. [24] H. X. Li and Z. Liu, “A probabilistic neural–fuzzy learning system for stochastic modeling,” IEEE Trans. Fuzzy Syst., vol. 16, no. 4, pp. 898–908, Aug. 2008. [25] C. Chen, G. Rigatos, and D. Dong, “Partial feedback control of quantum systems using probabilistic fuzzy estimator,” in Proc. 48th IEEE Conf. Decision Control/28th Chin. Control Conf., Shanghai, China, Dec. 16–18, 2009, pp. 3805–3810. [26] S. Chen and C. Chen, “Probabilistic fuzzy logic system for range measurement,” Mediterranean J. Meas. Control, vol. 5, no. 3, pp. 119–125, 2009. [27] S. Panzieri, F. Pascucci, and G. Ulivi, “An outdoor navigation system using GPS and inertial platform,” IEEE/ASME Trans. Mechatron., vol. 7, no. 2, pp. 134–142, Jun. 2002. [28] J. Borenstein and L. Feng, “Measurement and correction of systematic odometry errors in mobile robots,” IEEE Trans. Robot. Autom., vol. 12, no. 6, pp. 869–880, Dec. 1996. [29] S. Thrun, “Learning metric-topological maps for indoor mobile robot navigation,” Arti. Intell., vol. 99, no. 1, pp. 21–71, Feb. 1998. [30] C. Chen, H. X. Li, and D. Dong, “Hybrid control for autonomous mobile robot navigation—A hierarchical Q-learning algorithm,” IEEE Robot. Autom. Mag., vol. 15, no. 2, pp. 37–47, Jun. 2008. [31] A. Elfes, “Sonar based real world mapping and navigation,” IEEE J. Robot. Autom., vol. RA-3, no. 3, pp. 249–265, Jun. 1987. [32] M. G. Plaza, T. Martínez-Marín, S. S. Prieto, and D. M. Luna, “Integration of cell-mapping and reinforcement-learning techniques for motion planning of car-like robots,” IEEE Trans. Instrum. Meas., vol. 58, no. 9, pp. 3094–3103, Sep. 2009. [33] B. A. Merhy, P. Payeur, and E. M. Petriu, “Application of segmented 2-D probabilistic occupancy maps for robot sensing and navigation,” IEEE Trans. Instrum. Meas., vol. 57, no. 12, pp. 2827–2837, Dec. 2008. [34] C. Chen and D. Dong, “Grey system based reactive navigation of mobile robots using reinforcement learning,” Int. J. Innovative Comput., Inform. Control, vol. 6, no. 2, pp. 789–800, 2010. [35] S. Garrido, L. Moreno, and D. Blanco, “Exploration and mapping using the VFM motion planner,” IEEE Trans. Instrum. Meas., vol. 58, no. 3, pp. 2880–2892, Aug. 2009.

Shuo Chen received the B.E. degree in automatic control from Nanjing University, Nanjing, China, in 2010. He is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, University of Delaware, Newark. He is also currently with the Department of Control and System Engineering, Nanjing University. His research interests include intelligent control, mobile robotics, and high-performance computing.

Chunlin Chen (S’05–M’06) received the B.E. degree in automatic control and the Ph.D. degree in pattern recognition and intelligent systems from the University of Science and Technology of China, Hefei, China, in 2001 and 2006, respectively. He is currently an Associate Professor with the Department of Control and System Engineering, Nanjing University, Nanjing, China, where he is also with the State Key Laboratory for Novel Software Technology, Nanjing University. His research interests include machine learning, intelligent control, mobile robotics, and quantum algorithm.