Sensor fusion by pseudo information measure: A mobile robot application

ISA TRANSACTIONS® ISA Transactions 41 共2002兲 283–301

Sensor fusion by pseudo information measure: application

A mobile robot

Mohammad Reza Asharif,a,b,* Behzad Moshiri,a,† Reza HoseinNezhada,‡ a

Department of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran b Department of Information Engineering, Faculty of Engineering, University of the Ryukyus, Okinawa, Japan,

共Received 19 February 2001; accepted 17 November 2001兲

Abstract In any autonomous mobile robot, one of the most important issues to be designed and implemented is environment perception. In this paper, a new approach is formulated in order to perform sensory data integration for generation of an occupancy grid map of the environment. This method is an extended version of the Bayesian fusion method for independent sources of information. The performance of the proposed method of fusion and its sensitivity are discussed. Map building simulation for a cylindrical robot with eight ultrasonic sensors and mapping implementation for a Khepera robot have been separately tried in simulation and experimental works. A new neural structure is introduced for conversion of proximity data that are given by Khepera IR sensors to occupancy probabilities. Path planning experiments have also been applied to the resulting maps. For each map, two factors are considered and calculated: the fitness and the augmented occupancy of the map with respect to the ideal map. The length and the least distance to obstacles were the other two factors that were calculated for the routes that are resulted by path planning experiments. Experimental and simulation results show that by using the new fusion formulas, more informative maps of the environment are obtained. By these maps more appropriate routes could be achieved. Actually, there is a tradeoff between the length of the resulting routes and their safety and by choosing the proper fusion function, this tradeoff is suitably tuned for different map building applications. © 2002 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Sensor/data fusion; Bayesian theory; Pseudo information measure; Occupancy grids; Path planning

1. Introduction Usually, mobile robots are equipped with several sensors. Sensor data fusion is one of the important issues in the research index of this application 关1兴. A general architecture of perception and planning for an autonomous mobile robot is given in Fig. 1. The sensors collect information about the environment. The collected data are trans*E-mail address: [email protected] †

E-mail address: [email protected] E-mail address: [email protected]

‡

formed to a common mathematical form that is called internal representation. This representation shows the selected model of the environment which is supposed to be generated in the robot’s mind 共memory兲. Occupancy grids is the usual model that is selected for grid-based 共cellular兲 mapping of the environment around the robot. Multiple sensory data are fused for generation of the map in the next step. Sensor data fusion can be applied both in mapping and in planning phases. In this research work we have concentrated on probabilistic sensor data fusion methods, applied for occupancy grids map building.

0019-0578/2002/$ - see front matter © 2002 ISA—The Instrumentation, Systems, and Automation Society.

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Fig. 1. General architecture of perception and planning for an autonomous mobile robot. Sensor data fusion methods, applied in the map building block have been concentrated in this research work.

One of the popular strategies for sensor fusion in mobile robotics is the Bayesian rule of probability combination. In this approach the occupancy grids map is the internal representation used for conversion of sensory data. In occupancy grids representation, the environment around the robot is divided into very small cells 共similar to the pixels of an image兲. Each cell in the map is associated with a probability value that indicates the state of the cell. This value introduces the probability of being occupied by obstacles. In Elfes formulation of map building by Bayesian fusion 关2,3兴, some conditional probability must be defined and calculated. Actually this conditional probability is a sensor model and the raw sensory information is transformed into occupancy probability values by this model. But there is no exact model for many sensing modalities and doing this transformation is usually difficult. Van Dam has tried to do this transformation for ultrasonic range finder sensors by using artificial neural networks 关4,5兴.

In many applications Bayesian fusion of sensory information is followed by the assumption that the sensors are measuring and sensing the environment in a conditionally independent manner. Of course the conditional independence assumption is not always a valid assumption and it is not close to reality in some cases. But the results show that a wide range of variety in Bayesian fusion formulation is obtained and it does not make any diverging error in the map building process and its drawbacks are compensated after enough instances of sensing. Refer to several recent works of Thrun in Refs. 关6 –9兴. A new concept is introduced in this paper. It is applied to extend Bayesian fusion of the information that is gathered from independent sources. More flexibility in selection of the fusion function is achieved by using this extended form of Bayesian fusion. Probabilistic approaches including Bayesian fusion are applied in Thrun and other similar works. In the Bayesian approach, there is no control on how the uncertainty existing in the sensory data appears in the global map of the environment. By using the flexibility attained by our new method the proper fusion function can be chosen. The appearance of the uncertainty in the environment map can be controlled in such a way that more occupied maps are obtained for more crowded environments and less occupied 共sharper兲 maps are obtained for less crowded environments, in which there are many alternatives for a path to be planned from a start to a goal point. The advantages of using the new idea in the map building process for mobile robots are shown in simulation and experimental results. This idea has been trained and developed during our research work. In the ICSPAT2000 conference 关10兴 the new concept of the pseudo information measure was introduced with some simple simulation results. Then experimental results on the Khepera robot for the mapping of a simple environment were presented at the AROB2001 conference 关11兴. The complete theory and analysis are presented in this paper with simulation and corresponding experimental results on mapping more complex and practical environments. The outline of the paper is as follows. In the second section, the new ideas and their application to data fusion are explained. Bayesian fusion of conditional independent sources of infor-

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mation is briefly reviewed in this section. Then, the definition of the new concept and its necessary properties are introduced. Fusion formulas and a general analysis of the fusion process in this framework are discussed at the end of the section. Bayesian and pseudo information fusion are used in a map building simulation in the third section. This section also includes an explanation of ultrasonic sensor models and path planning algorithm in two subsections and the results are presented in the third subsection. In the fourth section, map building experiments are presented. In these experiments, Bayesian and the new fusion formulas were applied to the infrared proximity sensors of a Khepera robot. Both in the simulation and in the implementation results sections, the created maps have been applied to a path planning algorithm. The resulting maps and routes show the advantages of the fusion methods. Finally in the last section, some conclusive remarks about the advantages and flexibilities of the new method are mentioned.

P 关 OCC共 C j 兲 兩 S 1 ,S 2 兴 P 关 S 1 兩 OCC共 C j 兲 ,S 2 兴 ⫻ P 关 OCC共 C j 兲 兩 S 2 兴

兺

P 关 OCC共 C j 兲 ,S 1 ,S 2 兴 P 关 S 1 ,S 2 兴 and the right-hand side is equal to

1 P关 S2兴 1 P 关 S 1 ,s 共 C j 兲 ,S 2 兴 ⫻ P关 S2兴

P 关 S 1 ,OCC共 C j 兲 ,S 2 兴 ⫻

兺

s共 C j 兲

⫽

P 关 OCC共 C j 兲 ,S 1 ,S 2 兴 P 关 S 1 ,S 2 兴

共2兲

where s ( C j ) means the ‘‘state’’ of the cell C j and it can be ‘‘occupied’’ or ‘‘free.’’ Also OCC stands for occupied. Here, conditional independence for the two sensors S 1 and S 2 is defined by the following relation:

P 关 S 1 兩 OCC共 C j 兲 ,S 2 兴 ⫽ P 关 S 1 兩 OCC共 C j 兲兴 . 共3兲

P 关 OCC共 C j 兲 兩 S 1 ,S 2 兴

Despite our discussion being formulated based on occupancy grids map building in robotics environment perception, it can be expressed for the fusion of any other type of information. Indeed, the proposition ‘‘the cell C j is occupied’’ in the following text can be replaced with any other proposition in other applications. For example, in the case of a military multiradar target detection and tracking, the proposition may be expressed as ‘‘detected object is enemy’’ or in the case of medical pattern recognition, the proposition may be expressed as ‘‘the tissue is infected.’’ Assume that two sensors S 1 and S 2 give two range measures r 1 and r 2 , and they give two occupancy probability values for some cells in the map. In order to integrate these two values, the fused state of some cell, e.g., C j , can be calculated by

s共 C j 兲

because the left-hand side is equal to

Hereby, by the assumption of conditional independence, Eq. 共1兲 results in the following equation:

2. Sensor fusion, using pseudo information measure

⫽

285

P 关 S 1 兩 s 共 C j 兲 ,S 2 兴 ⫻ P 关 s 共 C j 兲 兩 S 2 兴 共1兲

⫽

P 关 S 1 兩 OCC共 C j 兲兴 ⫻ P 关 OCC共 C j 兲 兩 S 2 兴

兺

s共 C j 兲

⫽

P 关 S 1 兩 s 共 C j 兲兴 ⫻ P 关 s 共 C j 兲 兩 S 2 兴

P 关 S 1 兩 OCC共 C j 兲兴 ⫻ P 关 OCC共 C j 兲 兩 S 2 兴 P 关 S 1 兩 OCC共 C j 兲兴 ⫻ 关 OCC共 C j 兲 兩 S 2 兴 ⫹ P 关 S 1 兩 EMP共 C j 兲兴 ⫻ P 关 EMP共 C j 兲 兩 S 2 兴

P关 S1兴 P 关 OCC共 C j 兲兴 ⫻ P 关 OCC共 C j 兲 兩 S 2 兴 ⫽ , P关 S1兴 P 关 OCC共 C j 兲 兩 S j 兴 ⫻ P 关 OCC共 C j 兲兴 ⫻ P 关 OCC共 C j 兲 兩 S 2 兴 ⫹ P 关 EMP共 C j 兲 兩 S 1 兴 P关 S1兴 ⫻ ⫻ P 关 EMP共 C j 兲 兩 S 2 兴 P 关 EMP共 C j 兲兴 P 关 OCC共 C j 兲 兩 S 1 兴 ⫻

where EMP stands for empty. Finally, P 1 , P 2 , and P are defined as the occupancy probability values for the cell C j , derived from the data attained by S 1 and S 2 and by Bayesian fusion of the information provided by the two sensors, respectively. They are expressed as follows:

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P 1 ⫽ P 关 OCC共 C j 兲 兩 S 1 兴 , P 2 ⫽ P 关 OCC共 C j 兲 兩 S 2 兴 , 共4兲

P⫽ P 关 OCC共 C j 兲 兩 S 1 ,S 2 兴 .

Hence the following simple fusion formula is achieved:

P⫽

P 1⫻ P 2 . P 关 OCC共 C j 兲兴 P 1⫻ P 2⫹ P 关 EMP共 C j 兲兴 ⫻ 共 1⫺ P 1 兲 ⫻ 共 1⫺ P 2 兲

共5兲

Since P 关 OCC( C j ) 兴 ⫽1⫺ P 关 EMP( C j ) 兴 the fusion formula 共5兲 can be rewritten by the following interesting form:

冉

冊冉

1 1 1 ⫺1⫽ ⫺1 ⫻ ⫺1 P P 关 EMP共 C j 兲兴 P1 ⫻

冉

冊

1 ⫺1 . P2

冊

Fig. 2. The mathematical shape of a pseudo information function in a general case.

共6兲

Usually the prior probability P 关 EMP( C j ) 兴 is considered to be equal to 21 with the maximum entropy assumption. So the fusion formula may be simply expressed by

冉

冊冉

冊

1 1 1 ⫺1⫽ ⫺1 ⫻ ⫺1 . P P1 P2

共7兲

or more explicitly

P⫽

P 1⫻ P 2 . P 1 ⫻ P 2 ⫹ 共 1⫺ P 1 兲 ⫻ 共 1⫺ P 2 兲

共8兲

We can extend Eq. 共7兲 by induction to the case where several sources of information S 1 ,S 2 ,...,S n exist. Assuming that these sources give the values P 1 , P 2 ,..., P n for a unique proposition 共e.g., occupancy of a cell C j in our map building discussion兲, the fusion formula can be expressed as follows: n

冉 冊

1 1 ⫺1 . ⫺1⫽ 兿 P i⫽1 P i

共9兲

It is desired that a quantity is discovered that is added up while the fusion takes place by Eqs. 共7兲 or 共9兲. Such a quantity may be interpreted as some quantitative description of the information existing in the proposition C j is occupied. If the function INFO( P ) is defined by

INFO共 P 兲 ⫽ln共 P 兲 ⫺ln共 1⫺ P 兲 .

共10兲

Subsequently, the combination of Eqs. 共7兲 and 共10兲 simply results the following desired property for INFO( P ) :

INFO共 P 兲 ⫽INF共 P 1 兲 ⫹INFO共 P 2 兲 .

共11兲

兩 INFO( P ) 兩 must increase as P approaches farther away from 21, because the proposition becomes more informative. But in order to distinguish between the propositions that are near to be false and the ones which are near to be correct, the function INFO( P ) is symmetrically negative in the former case and positive in the latter. Fig. 2 shows this behavior. We tried to extend this concept to a more general case. Actually there are some properties that if every other function satisfies them, then it can be accepted as a quantitative measurement for the information existing in a proposition. We nominate such functions as pseudo information measure functions and will use the symbol PINFO( P ) for them. The desired properties are as follows: 共i兲 It is defined on 关0,1兴. 共ii兲 It is symmetric around 21 and zero at 21. So, it can be defined as

PINFO共 P 兲 ⫽J 共 1⫺ P 兲 ⫺J 共 P 兲 .

共12兲

Clearly in the case of Eq. 共10兲 the function J ( P ) is ⫺ln(P).

Mohammad Reza Asharif et al. / ISA Transactions 41 (2002) 283–301

共iii兲 It must satisfy the following limits:

LimP→0 ⫹ PINFO共 P 兲 ⫽⫺⬁, LimP→1 ⫺ PINFO共 P 兲 ⫽⫹⬁,

共13兲

287

some proposition. They are fused in such a way that the pseudo information measure for the resulting probability is the algebraic sum of the values associated with the two probabilities. It is the same as Eq. 共11兲 rewritten as

and it is sufficient for the function J ( P ) to satisfy the following limits:

PINFO共 P 兲 ⫽PINFO共 P 1 兲 ⫹PINFO共 P 2 兲 .

LimP→0 ⫹ J 共 P 兲 ⫽⫹⬁,

Six examples of infinite possible definitions that satisfy the necessary conditions are as follows:

LimP→1 ⫺ 兩 J 共 P 兲 兩 ⬍⬁.

共14兲

共iv兲 It must be an increasing function of P 关equivalently J ( P ) must be a decreasing function of P兴. 共v兲 It must be a concave function for P⬍0.5 and convex for P⬎0.5. The first property is evident because the probabilities vary in the 关0,1兴 interval. The second property is essential because the PINFO function must behave in the same manner for P and 1⫺P. Actually if a proposition is declared to be true with a probability of P in a case, or with a probability of 1⫺ P in another case, the amount of information that exists in the two declarations are equivalent. A proposition with a probability of P ⫽ 12 does not contain any useful information, because there is no knowledge about whether the proposition is more likely to be true or false. Thus the condition PINFO( P ) ⫽ 21 is essential to be satisfied for any pseudo information measure function to be defined. The third property is based on the behavior of the fusion of two probabilities when one of them is 0 or 1. From a logical point of view, the fused probability must become 0 or 1 in these two cases, respectively. Since the pseudo information fusion is based on the summing of PINFO values 共it will be explained later兲, the third property guarantees this behavior. The fourth property guarantees that there is only one pseudo information measure value, associated with every probability value. Furthermore, this property guarantees the coincidence of the fusion results with the human logic 共it will be discussed later in this section兲. Finally, the convexity property is necessary so that the error sensitivity reduces during pseudo information fusion 共it will be discussed later兲. As soon as a pseudo information function is defined, a fusion formula can be attained. Suppose that two sources of information have been processed and expressed as two probability values for

J 1 共 P 兲 ⫽⫺ln共 P 兲 , J 4共 P 兲 ⫽

J 2共 P 兲 ⫽ 1 , P2

1 , P

J 5共 P 兲 ⫽

J 6 共 P 兲 ⫽ln

冉

J 3共 P 兲 ⫽

共15兲

1 , P 1.1

1 , e ⫺1 P

冊

1⫹ P 2 . P

These functions are just some samples. There was no basic reason for selection of these examples from the infinite possible alternatives. But they are a complete set of functions, having different possible behaviors while applied to map building applications. As we will see in the following text, some of them behave more softly and some behave more roughly with informative propositions while fusion takes place. Another reason for choosing J 2 , J 5 , and J 6 for pseudo information fusion is the fact that explicit or implicit formulas could be easily attained for fusion by these functions. Actually J 3 ( P ) and J 4 ( P ) have been brought just to declare that any function of the form 1/P q could be considered being a valid choice for J ( P ) , where q is any positive rational number. These two functions are not used for map building either in simulation or in experiments. A comparative representation for the mathematical shapes of the six pseudo information measure functions created by the functions J 1 ( • ) ⬃J 6 ( • ) , is given in Fig. 3. In this figure the behavior of the pseudo information values are compared while 兩 P⫺ 21 兩 increases. J 1 ( • ) is the function that gives the Bayesian fusion formula 共8兲. In J 2 ( • ) , J 3 ( • ) , and J 4 ( • ) fractional functions and in J 1 ( • ) , J 5 ( • ) , and J 6 ( • ) logarithmic or exponential functions are used. We can derive fusion formulas, associated with each definition of pseudo information by applying Eqs. 共12兲 and 共15兲. In some cases, an explicit formula can be directly calculated. For example, in

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The J ( • ) functions mainly differ in their behavior when 兩 P⫺ 21 兩 increases. Figure 3 shows this difference in behavior. Actually, if PINFOi ( P ) stands for the pseudo information measure function that is calculated by using J i ( P ) , it can be proved that

兩 PINFO6 共 P 兲 兩 ⭐ 兩 PINFO1 共 P 兲 兩 ⭐ 兩 PINFO2 共 P 兲 兩

⬵ 兩 PINFO5 共 P 兲 兩 ⭐ 兩 PINFO3 共 P 兲 兩 Ⰶ 兩 PINFO4 共 P 兲 兩 .

Fig. 3. The mathematical shapes of the six PINFO functions that are created by six functions J 1 (•) – J 6 (•). Since we had to sketch the function values in a logarithmic scale, they are plotted in the probability interval 共0.5,1兲.

the case of applying J 2 ( • ) for calculation of the pseudo information measure values, the following fusion explicit formula is attained:

P⫽

再

x⫺2⫹ 冑x 2 ⫹4 2x

if x⫽0

1 2

if x⫽0,

共16兲

where x⫽1/1⫺ P 1 ⫺1/P 1 ⫹1/1⫺ P 2 ⫺1/P 2 共refer to Ref. 关10兴 for more details兲. Also in the case of applying the exponential function J 5 ( • ) the following formula results:

P⫽ln

冉

冊

x 共 e⫹1 兲 ⫹ 冑x 2 共 e⫺1 兲 2 ⫹4e , 共17兲 关 3 共 x⫹1 兲兴

where x⫽J 5 ( 1⫺ P 1 ) ⫺J 5 ( P 1 ) ⫹J 5 ( 1⫺ P 2 ) ⫺J 5 ( P 2 ) 共refer to Ref. 关11兴 for more details兲. There is no direct formula for fusion in many cases and an implicit equation must be solved. For example, in the case of J 6 ( • ) , the following thirdorder algebraic equation must be solved: 共 1⫹x 兲 P 3 ⫺ 共 2⫹x 兲 P 2 ⫹ 共 2⫹x 兲 P⫺x⫽0,

共18兲

where

P 1 P 2 ⫹ 关 1⫹ 共 1⫺ P 1 兲 2 兴关 1⫹ 共 1⫺ P 2 兲 2 兴 . x⫽ 共 1⫺ P 1 兲共 1⫹ P 21 兲共 1⫺ P 2 兲共 1⫹ P 22 兲

共19兲

These inequalities show that the pseudo information measure, calculated by using J 6 ( • ) , behaves more softly with low and high probability values 共associated with more informative propositions兲 and the inverse second-order function J 4 ( • ) behaves roughly with informative propositions 共it has not been considered in the comparative simulation and implementation in this research work兲. When the occupancy probability P approaches 1, the Bayesian information function PINFO1 ( P ) grows much more rapidly than PINFO6 ( P ) , as shown in Fig. 3. Thus, fusion of two probability values that are close to be absolutely true leads to a higher probability value if the fusion is done by Bayesian fusion. In contrast, fusion of two probability values that are close to be absolutely false 共zero兲 leads to a lower probability value if the fusion is done by Bayesian fusion. In other words, during an environment mapping process, occupancy probabilities of the cells approach 1 or 0 more rapidly if Bayesian fusion is applied and more slowly if PINFO6 ( P ) fusion is applied. So the map resulting from PINFO6 ( P ) fusion will contain more gray areas instead of absolutely black or white 共occupied or free兲 areas. Hence it is expressed that PINFO6 ( P ) behaves more softly with low and high probability values. It is shown in the simulation and implementation sections that the behavior of the fusion is directly related to the associated pseudo information function. Suppose that two sources of information, S 1 and S 2 , provide two probability values, P 1 and P 2 , associated to a proposition. If P 1 ⬎0.5 and P 2 ⬎0.5, then human logic implies that the fused probability must be greater than both P 1 and P 2 共nearer to 1 to be absolutely true兲. Pseudo information fusion formula 共15兲 along with the necessary properties of a PINFO function show that the logical point, which was mentioned above, is sat-

Mohammad Reza Asharif et al. / ISA Transactions 41 (2002) 283–301

isfied while doing fusion by Eq. 共15兲. Three cases are considered here for more clarification: • If P 1 ⬎0.5 and P 2 ⬎0.5, then since PINFO is an increasing function and zero at P ⫽0.5, PINFO( P 1 ) and PINFO( P 2 ) will be positive and by Eq. 共15兲, PINFO( P ) will be greater than PINFO( P 1 ) and PINFO( P 2 ) and again since PINFO is an increasing function, P will be greater than both P 1 and P 2 , or in other words it will be nearer to 1. • In the case of P 1 ⬍0.5 and P 2 ⬍0.5, it can be similarly deduced that P will be less than both P 1 and P 2 , or in other words it will be closer to 0. • In the case of P 1 ⬎0.5 and P 2 ⬍0.5 and 兩 P 1 ⫺0.5兩 ⬎ 兩 P 2 ⫺0.5兩 then PINFO( P 1 ) ⬎0 and PINFO( P 2 ) ⬍0 and because of the symmetric behavior of the PINFO function, PINFO( P 1 ) ⬎⫺PINFO( P 2 ) and by Eq. 共15兲 PINFO( P ) will be positive and so P ⬎0.5. Also P⬍ P 1 because PINFO( P ) ⬍PINFO( P 1 ) . Similarly, in the case of 兩 P 1 ⫺0.5兩 ⬍ 兩 P 2 ⫺0.5兩 the resulting probability value will satisfy in P 2 ⬍ P⬍0.5. Finally, in the case of 兩 P 1 ⫺0.5兩 ⫽ 兩 P 2 ⫺0.5兩 , since the two probability values are located on equal distances from 0.5, by the symmetric property of PINFO function 共12兲 it can be simply shown that PINFO( P 1 ) ⫽⫺PINFO( P 2 ) and by Eq. 共15兲, PINFO( P ) ⫽0 and so P⫽0.5. The above results are clearly the same as expected from the common sense logic of information fusion. We are also interested in studying the error sensitivity of the fusion process. Basically, by the probability values, the uncertainty of the information provided by sensors are modeled. In many cases, besides the uncertainty, there are some errors existing in the process of conversion of the raw sensory data to probability values. For example, in the case of the experiments that we have done on the Khepera mobile robot, a neural network was used for conversion of infrared sensory data to probability values. Besides the uncertainty that exists in the raw sensory data, there is also some error in the conversion process which is performed by the neural network 共e.g., because of the limited set of the learning patterns which are applied to train the network兲. In the case of existing

289

errors 共or deviations兲 in P 1 and/or P 2 values, by Eq. 共15兲 we can write

⌬PINFO共 P 兲 ⫽⌬PINFO共 P 1 兲 ⫹⌬PINFO共 P 2 兲 , dPINFO dPINFO 共 P 兲 ⫻⌬ P⫽ 共 P 1 兲 ⫻⌬ P 1 dP dP ⫹

dPINFO 共 P 2 兲 ⫻⌬ P 2 , dP

dPINFO dPINFO 共 P1兲 共 P2兲 dP dP ⌬ P⫽ ⫻⌬ P 1 ⫹ dPINFO dPINFO 共P兲 共P兲 dP dP ⫻⌬ P 2 . Obviously 兩 ⌬ P/⌬ P 1 兩 defined as the output error relative sensitivity of the fusion process for P 1 共assuming ⌬ P 2 is zero兲 is equal to

冏

冏

dPINFO 共 P1兲 dP . dPINFO 共P兲 dP

If P 1 ⬎0.5 and P 2 ⬎0.5 then P⬎max(P1 ,P2) ⬎0.5 and because of the convexity of PINFO in this area, its derivative is an increasing function of P. Consequently,

冏

冏

dPINFO 共 P1兲 dPINFO dP dPINFO 共 P1兲⬍ 共 P 兲⇒ dP dP dPINFO 共P兲 dP ⬍1⇒⌬ P⬍⌬ P 1 .

Hence the output error relative sensitivity of the fusion process will be less than 1 in this case. Such a reasoning is true also in the other two cases. It means that any noise or error in the probability values are diminished during the fusion process. Usually in robotic map building applications, the environment around the robot is modeled with a map. This map is generated by using a sensor data fusion method. Then the robot applies this map to execute several tasks. We discuss the path planning task here. It is apparent that the resulting path from a start point to a destination point de-

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pends on the map which has been utilized by the path planning algorithm. Actually there is a tradeoff between the length and the safety of the generated paths, and by choosing the proper pseudo information function, we can tune it. This tradeoff can be explained as follows. Two different cases for the pseudo information functions are considered. A pseudo information function can be greater than the Bayesian 共pseudo兲 information function 关 J 1 ( 1⫺ P ) ⫺J 1 ( P ) 兴 or less than it. For example, the functions defined by J 2 ( 1⫺ P ) ⫺J 2 ( P ) and J 5 ( 1⫺ P ) ⫺J 5 ( P ) satisfy the former case and the pseudo information function defined by J 6 ( 1⫺ P ) ⫺J 6 ( P ) satisfy the latter 关refer to Eq. 共19兲兴. In the former case there is more exaggeration in the fusion process 共i.e., the probability values approach to their absolute values, 0 or 1, more quickly兲 and in the latter case, the probabilistic fusion takes place more softly with less exaggeration. That is why there is a little more gray-colored 共unsafe兲 area in the environment maps, generated by fusion formulas belonging to the latter case and thus the resulting paths are generated in such a way that obstacles are avoided more severely and routes are farther from them. Of course this phenomenon naturally leads to more lengthy routes. By the same discussion, in the former case, since there is more exaggeration in the fusion process, the resulting paths will be shorter but closer to the obstacle area.

Fig. 4. The two certainty functions f E 共top兲 and f O 共bottom兲 for a range reading r.

25° arc of circumference of radius r. Hence there is evidence that the cells, located in the proximity of this arc, are occupied. On the other hand, points well inside the circular sector of radius r are likely to be free. To model this knowledge, in Ref. 关12兴, two functions were introduced as follows:

f E 共 ␳ ,r 兲 ⫽

3. Simulation results The mobile robot that is simulated in this research work is similar to the Nomad 200™ vehicle. It has been used as a typical mobile robot by many researchers 关12–15兴. This cylindrical robot is equipped with 16 ultrasonic Polaroid range finders 关16兴. The robot has been simulated in this work, equipped merely with eight ultrasonic sensors. Before presenting the simulation environment and the resulting maps and path planning results, preprocessing of ultrasonic information and the A* path planning algorithm are briefly reviewed in two subsections. 3.1. Sensor information preprocessing Unless a multiple reflection occurred, a single reading r provides the information that one or more obstacles are located somewhere along the

再

0⭐ ␳ ⬍r⫺⌬r

k E, kE

冉 冊 r⫺ ␳ ⌬r

2

,

r⫺⌬r⭐ ␳ ⬍r

␳ ⬎r,

0,

共20兲

f O 共 ␳ ,r 兲

⫽

再

0,

冋 冉 冊册

k O 1⫺ 0,

r⫺ ␳ ⌬r

0⭐ ␳ ⬍r⫺⌬r,

2

,

r⫺⌬r⭐ ␳ ⬍r⫹⌬r

␳ ⬎r⫹⌬r. 共21兲

They describe, respectively, how the degree of certainty of the assertions empty and occupied vary with ␳ for a given range reading r. Here ␳ is the distance from the cell under consideration to the sensor, k E and k O are two constants, corresponding to the maximum values attained by the functions, and ⌬r is half of the width of the area considered proximal to the arc of radius r. The profiles of f E and f O are displayed in Fig. 4. The choice of

Mohammad Reza Asharif et al. / ISA Transactions 41 (2002) 283–301

Fig. 5. The angular modulation function m 1 ( ␪ ) 共top兲 and the radial modulation function m 2 ( ␳ ) 共bottom兲.

k E , k O , and ⌬r, as well as other parameters to be introduced, depends on the physical characteristics of the sensor and also the fusion method that is used for map building. This topic is discussed in detail in Ref. 关12兴 and in our simulations we have chosen the same values of Ref. 关12兴 with the exception of the case of fuzzy method using MAX operator where we have chosen k E ⫽1 and k O ⫽1. Since the intensity of the waves decreases to zero at the borders of the radiation cone, the degree of certainty of each assertion is assumed to be higher for the points close to the beam axis. This is realized by defining an angular modulation function: m 1共 ␪ 兲 ⫽

再

D共 ␪ 兲,

0⭐ 兩 ␪ 兩 ⭐ ␪ max

0,

兩 ␪ 兩 ⬎ ␪ max ,

291

Fig. 6. The polar coordinates 共␳,␪兲 of a cell with respect to a sensor.

where a smooth transition occurs from certainty to uncertainty. The motivation for introducing this function is twofold. First, since the possibility of multiple reflections increases as the beam makes a longer fly, the use of m 2 ( ␳ ) reduces the undesirable effects of multiple reflections. Besides, narrow passages and doors appear to be obstructed if seen from a large distance, due to the wide sensor radiation angle. For each range measurement r 共measured by one of the sensors in the ring in one of sensing time instances兲, the two measures m O and m E for each cell in the occupancy grids map can be calculated by the equations

共22兲

m E 共 ␳ , ␪ 兲 ⫽ f E 共 ␳ ,r 兲 ⫻m 1 共 ␪ 兲 ⫻m 2 共 ␳ 兲 , 共24兲

where D ( ␪ ) is the radiation directivity function and ␪ max is 12.5° in simulations. For ease of computation, the directivity function can be approximated inside the original lobe by a fourth-order polynomial 共instead of the Bessel function form in the exact formula 关12兴兲, shown in Fig. 5. Finally, a radial modulation function is defined as

m O 共 ␳ , ␪ 兲 ⫽ f O 共 ␳ ,r 兲 ⫻m 1 共 ␪ 兲 ⫻m 2 共 ␳ 兲 , 共25兲

m 2共 ␳ 兲 ⫽

1⫺tanh关 2 共 ␳ ⫺ ␳ v 兲兴 2

共23兲

to weaken the confidence of each assertion as the distance from the sensor increases 共Fig. 5兲. The parameter ␳ v plays the role of a visibility radius,

where ␳ and ␪ are the polar coordinates of the cell with respect to the sensor 共Fig. 6兲. Apparently, the calculations in Eqs. 共24兲 and 共25兲 are not necessary to do if the cell is located in the outside of the radiation cone of the sensor. 3.2. A * path planning algorithm In a path planning algorithm, where an appropriate path from a start point to a goal point is searched, an environment map is applied for the searching process. In the A* path planning algo-

292

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rithm, this map is an occupancy grids map and the uncertainty existing in a grid-based map of the environment affects the path that results from A* path planning 关12,18兴. In this subsection, the A* algorithm is explained in order to express this phenomenon. Assume that a map M of the environment has been constructed from the measurements of the sensors. A start cell S and a goal cell G are given. A path from S to G is a sequence of adjacent cells 兵 S,...,G 其 . Our path planner must explicitly take into account that the existing map has an uncertain nature and does not provide a separation between the free and the occupied space. A natural strategy is then to avoid the cells with large occupancy values. The algorithm uses an evaluation function e ( N ) associated with each cell N that estimates the cost of the optimal path from S to G constrained passing through N. It is obtained as follows:

e 共 N 兲 ⫽g 共 N 兲 ⫹h 共 N 兲 ,

共26兲

where g ( N ) is the cost of the current path from S to N. If this path is P, then g ( N ) is defined by

g共 N 兲⫽

兺

N i 苸P

␮共 Ni兲,

共27兲

where ␮ is the occupancy value measured for each cell in the map. It is either an occupancy probability or a combination of m O and m E such as Eq. 共33兲 that will be explained later. Actually, the uncertainty that is included inside the environment map affects the resulting path, while calculating the term of g ( N ) cost by Eq. 共27兲. h ( N ) is a heuristic estimate of the cost of the optimal path from N to G. The map M is searched from S toward G one cell by one cell. Any cell that has been visited but not expanded 共i.e., all of its successors have not been visited from that cell兲 is placed in a list, which is sorted by increasing values of the cost estimates of its current cells. Since it is open for new cells appended to it, we name it OPEN. At each iteration, the algorithm removes and expands the first cell N of OPEN. The following procedure is repeated for each successor N ⬘ : if N ⬘ has not been visited so far, it is inserted in OPEN; otherwise, it must be checked if the path leading to N ⬘ through N is more convenient. This is done using the test

g 共 N ⬘ 兲 ⬎g 共 N 兲 ⫹k 共 N,N ⬘ 兲 ,

共28兲

where K ( N,N ⬘ ) is the cost associated with the subpath joining N to N ⬘ . The algorithm assigns to each visited cell N ⬘ a pointer to its parent, namely its predecessor on the current path from S to N ⬘ . If Eq. 共28兲 is verified, then the value of e ( N ⬘ ) 共and hence the list OPEN兲 is updated and the pointer of N ⬘ is redirected to N. In this way, only a spanning tree T of the graph is memorized. The definition of k ( N,N ⬘ ) , which coincides with g ( N ) defined in Eq. 共27兲 is simply

k 共 N,N ⬘ 兲 ⫽ ␮ 共 N ⬘ 兲 .

共29兲

For the heuristic function, usually, the following equation is used:

h 共 N 兲 ⫽d 共 N 兲 • ␮ min

共30兲

where d ( N ) is the minimum number of the cells that compose a subpath from N to G and ␮ min is the smallest nonzero value of ␮ over M. In A* algorithm, in the case of one-adjacency, i.e., when each cell has four adjacent cells, d ( • ) is defined as follows:

d 共 N 兲 ⫽max兵 兩 x G ⫺x N 兩 , 兩 y G ⫺y N 兩 其

共31兲

and in the case of two-adjacency, i.e., when each cell has eight adjacent cells, d ( • ) is defined by

d 共 N 兲 ⫽ 兩 x G ⫺x N 兩 ⫹ 兩 y G ⫺y N 兩 .

共32兲

3.3. Resulting maps and paths A rectangular room with the dimensions of 15 ⫻20 m2 was selected to be simulated for exploration, map building, and path planning. It contains four walls on its perimeter and some obstacles inside. Actually this environment is a simplified model of a laboratory. Fig. 7 shows the map of the environment that has been divided into many small 10⫻10-cm2 cells. Here we explain the stepby-step process that has been used for map building. Initially, the mobile robot does not know anything about the location of the walls 共dimensions of the room兲 and the obstacles. There is an initially blank map in the robot’s memory with 150⫻200 cells. In an ideal case, this map converges to the map depicted in Fig. 7 gradually. The robot begins to navigate inside the room either by human aid or by reactive obstacle avoidance. We have presented a fuzzy method for reactive obstacle avoidance and navigation in Ref. 关17兴 and this method was used in the simulation. During this phase, each of

Mohammad Reza Asharif et al. / ISA Transactions 41 (2002) 283–301

Fig. 7. The real map of the simulated environment to be generated in the robot’s mind. The environment is a simulated version of a lab.

its ultrasonic sensors gives a distance measurement in every sensing instance. These values give m O ( C j ) 共occupancy measure兲 and m E ( C j ) 共emptiness measure兲 values for some cells in the map 共refer to Sec. 3.1 and Refs. 关12,13兴 for details about how several kinds of ultrasonic sensory data uncertainties are considered in this calculation of measure values兲. Either these measure values are combined through the fusion formulas 共in the case of fuzzy or Dempster-Shafer fusion methods兲 or they are converted to occupancy probabilities and then combined by Bayesian or pseudo information fusion formulas. Conversion of occupancy and emptiness measure values to probabilities can be derived as follows:

293

Fig. 8. Simulated map that is generated by fuzzy fusion with MAX union operator.

for each cell. These two maps were mixed by Eq. 共33兲 and only the mixed values 共which are interpreted as occupancy probabilities兲 have been plotted as the environment map. Besides, these are the values that were utilized as the input map for A* path planning algorithm. Although these methods lead to appropriate maps in some cases, as figures show the resulting maps for the case which is studied in our work are not appropriate enough. The map generated by Bayesian fusion is given in Fig. 11. Similar trials were examined by using pseudo information fusion formulas. Figures 12–14 show the maps generated by pseudo information fusion techniques when the PINFO function in Eq. 共15兲 was calculated by choosing

P 关 OCC共 C j 兲 兩 New Sensory Information兴 ⫽

1⫹m O 共 C j 兲 ⫺m E 共 C j 兲 . 2

共33兲

In the next step, a fusion method is applied to the map building process. Fuzzy fusion method 共by using MAX and Dombi operators for fusion兲 and Dempster-Shafer method 共by using Dempster’s rule of combination for fusion兲 were tried besides Bayesian and pseudo information fusion methods. The maps resulted by applying fuzzy methods 共MAX and Dombi’s union operators兲 and Dempster-Shafer methods for fusion purposes are shown in Figs. 8 –10, respectively. Actually there are two maps for each method. One map is the set of m O values and the other is the set of m E values

Fig. 9. Simulated map that is generated by fuzzy fusion with Dombi’s union operator.

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Fig. 10. Simulated map that is generated by Dempster’s rule of combination.

Fig. 12. Simulated map that is generated by PINFO2 ( P).

f 共 M ,M I 兲 J 2 ( P ) , J 5 ( P ) , and J 6 ( P ) , respectively. All of the maps were gradually generated during the activation of the sensor ring at 4500 different random points in the environment. In order to compare the maps, two factors are defined and calculated for each map. The first is a fitness measure to represent the similarity of the generated map to the ideal map 共depicted in Fig. 7兲. Generally, the existing fitness between an occupancy grids map M and an ideal map M I is defined by the following equation:

Fig. 11. Simulated map that is generated by Bayesian fusion.

⫽1⫺

兺

c苸B艛P

兩 M 共 c 兲 ⫺M I 共 c 兲 兩 2

Total number of the cells in B艛P 共34兲

where B is the set of the cells which fall on the blank areas in the ideal map, P is the set of the cells, falling on the perimeters of the obstacles in the ideal map, and M ( c ) is the occupancy probability of the cell c in the map M. In the case of perfect fitness, the above factor will be equal to 1. In the worst case of complete mismatch between

Fig. 13. Simulated map that is generated by PINFO5 ( P).

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295

Sec. 3.2 for more details on this algorithm and to Refs. 关12,18兴 for a complete discussion兲. Since we decided to achieve comparative results, 30 difficult paths have been chosen for path planning. Two quantities have been calculated for each path: the length of the path and the safety measure of the path. A suitable path must be not only as short as possible, but also as far as possible from the obstacles. For a path P its safety measure ␣ d ( P ) was defined and calculated by

␣ d共 P 兲 ⫽ Fig. 14. Simulated map that is generated by PINFO6 ( P).

M and M I , in the occupied area we have M I ( c ) ⫽1 and M ( c ) ⫽0 and vice versa. So, the numerator and denominator will be equal in this case and thus the fitness factor will be zero. It is evident that a higher value for the factor means that the maps are matched better. The cells that fall inside the obstacles in the real map have not been considered in this formulation because it is not important what the created maps judge about the state of such cells. Only the blank areas and the perimeter of the obstacles are significant to be identified as free and occupied cells. The other factor is a measure for the augmented occupancy existing in a generated map, compared to the ideal map 共the real case兲. It was simply calculated by

a O 共 M ,M I 兲 ⫽ 兺 M 共 c 兲 . c苸B

共35兲

Actually this augmented occupancy appears in the generated maps and it is mostly concentrated on the areas around the perimeter of the existing obstacles. From a safety point of view, this could be a useful characteristic, because it leads to safer routes 共i.e., farther from obstacles兲, generated by path planning algorithm. In order to gain more comparable results to analyze the performance of the fusion methods, we have also used these maps as inputs to the A* path planning algorithm. There are two inputs in this algorithm. The first is a cellular map of the environment and the second is the coordinates of some START and GOAL points in that map 共refer to

兺

C苸P

␥ ⫺d min 共 C 兲

共36兲

where d min ( C ) is the minimum of the distance between the cell C and the cells on the perimeter of an obstacle or a wall in the environment. It is calculated with respect to the true map of the environment. ␥ ⬎1 is a constant that controls the variations of ␣ d ( P ) , with d min ( C ) . In this simulation it was equal to 1.2. It is apparent that a path P is safer for lower values of ␣ d ( P ) . The sum of the above two quantities, associated with the 30 paths are displayed in two columns in Table 1. In each column, the results of summing over the paths which are generated by using different maps are presented in different rows. Also fitness and augmented occupancy calculation results are abstracted in this table. Since the fuzzy map 共generated by using Dombi’s union operator for fusion兲 is not clear or appropriate, it is not applied to path planning and only the map-related 共fitness and augmented occupancy兲 calculations are displayed for this map in the table. Table 1 shows that the resulting maps for fuzzy and Dempster-Shafer methods are not appropriate. Furthermore, the maps, generated by fusion of pseudo information measures, calculated by J 2 ( P ) and J 5 ( P ) contain less augmented occupancy compared to the Bayesian map. The inverse is true for the map, generated by fusion of pseudo information measures, calculated by J 6 ( P ) . This fact can be directly observed from the maps in the figures. According to the analysis that was given in the last part of the previous section, this result was expected because the pseudo information fusion by J 2 ( P ) and J 5 ( P ) are more exaggerating than Bayesian fusion and inversely the pseudo information fusion by J 6 ( P ) is less exaggerated than Bayesian fusion. Although the resulted augmented occupancy values seem to be large, they are rela-

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Table 1 Summary of the simulated mapping and path planning results. For each map, the sum of two factors for 30 paths are calculated as useful criteria for comparison. The factors are the length and their safety measure. Also it is shown that how close each map is to the ideal map and how much augmented occupancy exists in each map with respect to the ideal map. Fusion method

⌺ Len( P)

⌺ ␣ d ( P)

f (M fusion method ,M I )

ao(M fusion method)

Fuzzy 共MAX operator兲 Fuzzy 共Dombi operator兲 Dempster-Shafer Bayesian PINFO Fusion by J 2 ( P) PINFO Fusion by J 5 ( P) PINFO Fusion by J 6 ( P)

472共m兲 523共m兲 465共m兲 458共m兲 395.8共m兲 389.1共m兲 523.2共m兲

4253 3954 4385 4591 3846 3652 4792

0.8223 0.7349 0.8529 0.8753 0.8921 0.8952 0.8021

6255 6298 5811 5413 3381 3359 5750

tively small and acceptable. That is because they are sum of occupancy probability of more than 40 000 empty cells in the map. Obviously, path planning results in Table 1 show that the routes that are resulted from the maps provided by pseudo information fusion by J 2 ( P ) and J 5 ( P ) are shorter but more dangerous than the routes resulted from Bayesian map. On the other hand, the map provided by pseudo information fusion by J 6 ( P ) generates longer but safer routes. This tradeoff is the important advantage of our extension to Bayesian fusion 共considered to be represented in simulation and experiments兲 in map building for mobile robots. Thus using the large family of the new formulas that resulted from our new theory gives us the flexibility to choose the proper function based on the application.

proximity detectors, like the model for ultrasonic range finders that was introduced in Sec. 3.1 and applied in our simulation. We trained a feedforward multilayered perceptron to implement an inverse model for the sensors. The inputs of the network are the eight proximity values, provided by Khepera infrared sensors and the local coordinates of a cell in the occupancy grids map around the robot. The output of the network is the occupancy probability value of the cell. In order to train the neural network, N⫽78 points were selected in a simple environment with only one L-shaped obstacle. The location of these points were chosen in such a way that the neural net can distinguish many cases such as a straight wall, a concave corner, a convex corner, etc., in many different distances. About the directivity, in each of the N points the robot was rotated 5° step

4. Implementation on a real robot Similar to the case of the previous section, we have applied the Khepera miniature mobile robot to some real map building experiments. The Khepera robot is equipped with eight infrared sensors and two wheel encoders in its basic configuration 关19兴. Fig. 15 shows a photo of this robot and some of its infrared sensors mounted on it. In our experiments, these two groups of sensors were the only sources of information about the environment around the robot. Infrared data were applied to mapping directly. Wheel encoders data were utilized to estimate the robot’s pose by the deadreckoning technique. The values of the eight infrared sensors around Khepera were the only sources of information. There is no accurate inverse model for the infrared

Fig. 15. A photo of Khepera robot and its exploration environment.

Mohammad Reza Asharif et al. / ISA Transactions 41 (2002) 283–301

297

Fig. 17. Step-by-step diagram of map building process for Khepera mobile robot.

Fig. 16. Local maps for ten test positions. In each position, the true map 共right兲 and the map that is resulted by neural network 共left兲 are displayed.

by step. Thus the total number of patterns, applied for training of the network, is M ⫽ 360 5 ⫻N ⫽2808. The network was trained using on-line training, not batch training. But for performance measurement in the training phase, in each D steps, the following mean square error 共MSE兲 expression was used: D

MSE⫽

1 兺 共 T i ⫺O i 兲 2 . D i⫽1

共37兲

Here i counts the number of locations in a block of D samples taken at random. T and O are the values of the True occupancy probability 共0 or 1兲 and the actual Output of the network, respectively. We could reach the average measure of 0.03 for MSE expression and stopped training at this point. Fig. 16 shows local maps for ten test points. Each

couple contains the local map generated by the best 12-10-1 network trained on real robot data and the true world image with the walls and corners that generate the eight sensor values. The outer circle is the limit of the local map. In Figs. 16共e兲 and 共f兲 the space on the other side of the wall can be glimpsed to the left. The robot orientation is marked with a thin white line pointing outwards from the center of the robot. Fig. 17 shows a step-by-step demonstration of the mapping process. An occupancy grids global map of the environment is initialized with blank values. Then infrared sensory data and the robot wheels encoder data are acquired in the next step. Then they are fed to the neural network and a local occupancy grids map around the robot is obtained. Actually this local map is the set of occupancy probabilities for some cells around the current position of the robot. The local coordinates of each of these cells are given to the network inputs combining with the IR data, and each of the probabilities are calculated by the network separately. On the other hand, the position and orientation of the robot are estimated by the dead-reckoning technique 共i.e., by using encoder data兲. Having the

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Fig. 18. The Khepera robot and its eight infrared proximity sensors.

robot’s pose 共position and orientation兲 and a global map of the environment 共which is under construction兲, a local map is extracted from the global map. This local map is the set of occupancy probability values of the same local cells on the global map. In the next step, the two local maps are fused by integrating the probabilities of each couple of the associated cells. The resulting probabilities are substituted in the global map and thus it is updated. At this point, the algorithm proceeds to the next iteration and map upgrading and building process continues, leading to a complete map of the environment. While the robot is exploring the environment, it avoids the obstacles by the Braitenberg algorithm 关20兴. Also the additive position error existing in the dead-reckoning estimation is manually calibrated by putting the robot at one of the previously known points in the environment occasionally. We

have formulated a new approach for an automatic calibration that is also appropriate to be applied to Khepera infrared sensory data 关21兴. However, in the experiments which are brought in this paper, automatic calibration was ignored in order to concentrate on mapping and path planning processes. Fig. 18 shows the robot in the environment that its map is built by several approaches in our experiment. In Fig. 19 the ideal 共real兲 map of the exploration environment is depicted. Four map building experiments were tried, by using Bayesian fusion 共5兲 and the three pseudo information fusion formulas 共16兲–共18兲. In Figs. 20–23, the resulting maps 共only by 1000 sensing iterations兲 of the environment are shown, respectively.

Fig. 19. The real map of the environment, where the mapping experiments on the Khepera robot was performed.

Fig. 21. The map, created by fusion of pseudo information measures, defined by J 2 ( P).

Fig. 20. The map, created by Bayesian fusion.

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299

Fig. 22. The map, created by fusion of pseudo information measures, defined by J 5 ( P).

Fig. 23. The map, created by fusion of pseudo information measures, defined by J 6 ( P).

In order to achieve comparative results, fitness measure and augmented occupancy calculation and path planning experiments have been executed for the resulting maps. In this part, 30 different couples of start and goal points have been selected for route finding by the A* algorithm. Similar to the previous section, the total sum of the length and safety measure of the routes were calculated and listed in Table 2. This table also includes the fitness measure and augmented occupancy calculation for each map. Again the results in Table 2 express that the maps provided by pseudo information fusion by J 2 ( P ) and J 5 ( P ) contain less augmented occupancy around the obstacles and generate some routes that are shorter but more dangerous than the case of the Bayesian map. On the other side, the map provided by pseudo information fusion by J 6 ( P ) contains more augmented occupancy and generates longer but safer routes 关11兴. The results, abstracted in Table 2, agree with our discussion

about the achieved flexibility to choose the fusion function that generates maps and paths that are shorter or safer. 5. Conclusive remarks A new concept named ‘‘pseudo information measure’’ was introduced in this paper and applied for sensor data fusion purposes. Some new fusion formulas were derived. Fusion performance while using the new technique was examined by simulation and experimental results. Satisfying results show the capabilities of implementation of the novel approach for autonomous map building and intelligent path planning in mobile robot applications. Different parameters such as fitness and augmented occupancy of the maps and the length and safety of the paths were examined to evaluate the advantages and flexibility of the new approach, compared to the other fusion methods. Some of the new fusion formulas attained by our method

Table 2 Summary of the experimental mapping and path planning results. For each map, its fitness and augmented occupancy measures with respect to the true map and also the sum of the length and the sum of the safety measures for 30 paths are calculated. Fusion method Bayesian PINFO Fusion by J 2 ( P) PINFO Fusion by J 5 ( P) PINFO Fusion by J 6 ( P)

⌺ Len( P)

⌺ ␣ d ( P)

f (M fusion method ,M I )

ao(M fusion method)

21.2共m兲 18.4共m兲 18.8共m兲 23.5共m兲

146 154 168 135

0.6524 0.7432 0.7791 0.6133

14180 13905 13872 14646

300

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have more exaggeration and some have less. In the cases where a PINFO fusion formula with more exaggeration is used for map building, the resulting environment map is sharper. On the contrary, the maps created by less exaggerated fusion include some augmented occupancy around the obstacles. Path planning by using the former class of the generated maps provides shorter but more dangerous routes. Meanwhile, utilizing the latter case of maps for path planning leads to safer but more lengthy routes. As it is clearly shown by the simulation and experimental results, this tradeoff can be suitably controlled by using pseudo information fusion as an extension of the Bayesian fusion approach.

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Mohammad Reza Asharif was born in Tehran, Iran, on December 15, 1951. He received the B.Sc. and M.Sc. degree in electrical engineering from the University of Tehran, Tehran, in 1973 and 1974, respectively, and the Ph.D. degree in electrical engineering from the University of Tokyo, Tokyo in 1981. He was a senior researcher at Fujitsu Labs. Co., Kawasaki, Japan from 1985 to 1992. Then, he was an assistant professor in the Department of Electrical and Computer Engineering, University of Tehran, Tehran, Iran from 1992 to 1997. Dr. Asharif is now a full professor at the Department of Information Engineering, University of the Ryukyus, Okinawa, Japan since 1997. His research interests are in the field of digital signal processing. Professor Asharif has been a senior member of IEEE since 1998.

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Behzad Moshiri was born in Tehran, Iran in 1959. He received his B.Sc. in mechanical engineering from Iran University of Science and Technology 共IUST兲 in 1983. He pursued higher studies leading to the M.Sc. and Ph.D. degrees in control systems engineering from the University of Manchester, Institute of Science and Technology 共UMIST兲, U.K. in 1987 and 1991, respectively. He was a member of ISA in 1991-1992. Since 1992 he joined the department of electrical and computer engineering at the University of Tehran, Iran, where he is currently an associate professor of control engineering and the head of the Robotics & AI Division. Dr. Moshiri’s research interests include intelligent control, industrial process control, advanced instrumentation, and sensor and data fusion.

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Reza HoseinNezhad was born in Tehran, Iran, on June 1, 1973. He received his B.Sc. and M.Sc. degrees in electrical engineering from the University of Tehran, Tehran, Iran in 1994 and 1996, respectively. He was a research student in the Department of Information Engineering, University of the Ryukyus, Okinawa, Japan in 2000-2001. He received his Ph.D. degree in electrical engineering from the University of Tehran, Tehran, Iran, in March 2002. He is also doing research at the Institute of Studies on Physics and Math 共IPM兲. HoseinNezhad’s research interests include intelligent control, neural and fuzzy and neurofuzzy systems, mobile robotics, artificial life, and sensor data fusion.