Mathematical Problems in Engineering

Volume 2015, Article ID 736720, 10 pages

http://dx.doi.org/10.1155/2015/736720

## Multivariable Fuzzy Control Based Mobile Robot Odor Source Localization via Semitensor Product

^{1}School of Electronic and Engineering, University of Jinan, Jinan, Shandong 250022, China^{2}School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Received 8 April 2015; Accepted 18 June 2015

Academic Editor: Xinguang Zhang

Copyright © 2015 Ping Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to take full advantage of the multisensor information, a MIMO fuzzy control system based on semitensor product (STP) is set up for mobile robot odor source localization (OSL). Multisensor information, such as vision, olfaction, laser, wind speed, and direction, is the input of the fuzzy control system and the relative searching strategies, such as random searching (RS), nearest distance-based vision searching (NDVS), and odor source declaration (OSD), are the outputs. Fuzzy control rules with algebraic equations are given according to the multisensor information via STP. Any output can be updated in the proposed fuzzy control system and has no influence on the other searching strategies. The proposed MIMO fuzzy control scheme based on STP can reach the theoretical system of the mobile robot OSL. Experimental results show the efficiency of the proposed method.

#### 1. Introduction

In natural world, many organisms such as drosophila, moth, and lobster use olfaction or/and vision cues to find the same species, avoid predators, exchange information, and search for food [1–3]. Inspired by those biological activities, in the early 1990s researchers started to build single or multiple mobile robots with onboard odor sensors or/and winds sensor to accomplish the odor source localization (OSL) task. Existing methods can be categorized along two lines. One is olfaction-based method, which mainly uses olfaction or/and wind information to search for gas sources without visual information. The other is vision-based method, which takes the visual information as an assistant of olfaction to accomplish the OSL task. Most work has been focused on the first field and it has become a mature and popular filed. However, the vision-based method is immature and needs to do deep study due to the late beginning. Russell [4], Meng and Li [5], Lilienthal et al. [6], Naeem et al. [7], Kowadlo and Russell [8], and Ishida et al. [9] have given relative reviews about mobile robot OSL from a different angle or application. The interested reader is referred to [4–9] for a comprehensive review of olfaction-based mobile robot OSL. Compared with organisms, robots can be deployed quickly, maintained at low cost, and work for a long time without fatigue. Moreover, they can enter the dangerous or harmful areas. Mobile robot OSL is a multidisciplinary research field with wide potential applications, such as judging toxic/harmful gas leakage location, checking contraband (e.g., heroin), locating unexploded mines and bombs, and fighting against terrorist attacks.

It is well known that human beings normally first look around to search for the most potential region or object and then identify whether the region or object is an odor source by olfaction. Vision contains abundant information, so visual sensor could be a good assistant of olfaction for mobile robot OSL. Meanwhile, large amount of leakage accidents indicate that some devices are more likely to leak, such as valves, bottles, and pipelines. In this paper, such devices are called potential gas sources and the areas which contain such devices are called plausible areas. It would improve the searching efficiency if these potential gas sources are recognized accurately and the plausible areas are determined rapidly in the searching process.

In recent years, a few researchers attempted to integrate vision and olfaction to localize the odor source. Kowadlo et al. [10] took crackles as the vision feature assisting olfaction to search for the odor source. Ishida et al. [11] proposed a color-based algorithm to deal with the vision information in the searching process. These methods were verified in the experiments, which indicate that vision as an assistant of olfaction for mobile robot OSL is efficient. Inspired by these researches, Jiang et al. [12] proposed a support vector machine based algorithm to localize an odor source and the author also presented a top-down visual attention mechanism-based algorithm [13] for mobile robot OSL. And then least square estimation was used to fuse the vision and olfaction information to accomplish the OSL task in stable airflow environment [14]. Meanwhile, Jiang and Zhang [15] attempted to integrate the vision and olfaction using subsumption architecture to accomplish the OSL task. However, how to fuse the uncertainty, ambiguity, vagueness, incompleteness, and granularity of the multisensor information from the mobile robot, especially vision and olfaction information, needs further study from the deep analysis to those few reports. It is noteworthy that multisensor data fusion is developed in recent years and new fusion algorithms and models are constantly emerging such as Dempster-Shafer evidence theory, probability theory, fuzzy theory, possibility theory, rough set theory, and the improved algorithms of these methods [16, 17]. Meanwhile, these methods have been successfully used in many fields, such as image processing, fault diagnosis, and target tracking. Inspired by these successful cases, we attempt to set up a multivariable fuzzy control system based on semitensor product for mobile robot OSL by fusing multisensor information and obtain some interesting results.

Fuzzy control as an intelligent control strategy needs no precise mathematical model for the objective system. They have found a great variety of applications ranging from control engineering, qualitative modeling, pattern recognition, signal processing, machine intelligence, and so on [18, 19]. In particular, fuzzy logic control (FLC), as one of the earliest applications of fuzzy sets and systems, has become one of the most successful applications. In fact, FLC has been proved to be a successful control approach to many complex nonlinear systems or even nonanalytic ones. The fuzzy control algorithm consists of a set of heuristic control rules, and fuzzy sets and fuzzy logic are used, respectively, to represent linguistic terms and to evaluate the rules. Since then, fuzzy logic control has attracted great attention from both academic and industrial communities and a lot of excellent books and tutorial articles on the topic have been published. However, it is difficult to infer the proper control input for a multivariable system since the dimension of its relation matrix is very large. The high dimensionality of the relation matrix might lead to not only computational difficulties but also memory overload. To solve this problem, a fuzzy control algorithm by which the multivariable fuzzy system is decomposed into a set of one-dimensional systems [18, 19]. The decomposition of control rules is preferable since it alleviates the complexity of the problem.

Recently, the semitensor product (STP) of matrices was proposed in [20]. And up to now, it has been widely applied in many fields, such as boolean network [21, 22] and coloring problems [23]. The logic expression can be expressed into an algebraic form by constructing its structure matrix. In [22], the observed data was expressed into a two-valued algebraic form. For the mobile robot odor source localization different sizes of the multisensor information play the different roles in the searching process. Therefore, the multisensor information for the mobile robot odor source localization cannot be divided into two-valued true and false cases simply. This multisensor information is expressed as multivalued algebraic form according to the actual demand. It is noted that the fuzzy logic also can be considered as an extended mix-valued logic in which the truth-values are the ones of memberships of all the elements in a fuzzy set, and the complex reasoning process can be converted into a problem of solving a set of algebraic equations via STP, which greatly simplifies the process of logical reasoning.

In this paper, we attempt to set up a multi-input multioutput (MIMO) fuzzy control framework based on STP for the mobile robot OSL. The multisensor information obtained by the mobile robot is the inputs and the relative searching strategies are the outputs. Several interesting results are obtained. The main contributions of this paper are as follows:(1)A MIMO fuzzy control system is set up for the mobile robot OSL.(2)Fuzzy control rules with algebraic equations are given according to the multisensor information.(3)Any output can be updated in this framework and has no influence to the others.(4)The proposed method based on MIMO fuzzy control scheme via STP for mobile robot OSL can reach the theory of this field.

The rest of this paper is organized as follows. Section 2 provides some necessary preliminaries on the semitensor product of matrices and the expression of logical function and logical variables. Section 3 presents the proposed algorithms for mobile robot OSL. Section 4 shows experimental results and analysis and the conclusion is given in Section 5.

#### 2. Matrices with Logical Variables

First, some notations are introduced, which will be used in this paper:(i): the th column of the identity matrix .(ii); especially, .(iii); to use matrix expression, “” and “” can be expressed with the following vectors, respectively: , .(iv), .(v)A matrix is called a logical matrix if the columns of , denoted by , are of the form ; that is, .(vi)Let denote the set of logical matrices; if , by definition, it can be expressed as ; for the sake of compactness, it is briefly denoted as .(vii)Each -valued logical value with a vector can be denoted as ; then,

In the following, we recall some definitions and basic properties about the STP [20].

*Definition 1. *Let and . Let denote the least common multiple of and . Then, the semitensor product of and is defined aswhere “” is the Kronecker product.

*Remark 2. *It is noted that when , the STP of and becomes the conventional matrix product. Hence, the STP is a generalization of the conventional matrix product. Because of this, we can omit the sign “” without confusion.

*Definition 3. *A swap matrix is an matrix. Its rows and columns are labeled by double index , the columns are arranged by the ordered multi-index , and the rows are arranged by the ordered multi-index . Then the elements at position are

*Remark 4. *Let and be column vectors; then . Let , , be column vectors; then

Let and . Assume that a logic mapping,can be expressed aswhere

Lemma 5. *Any logical function can be uniquely expressed into the multilinear form of**where is called the structural matrix of , , , and .*

Lemma 6. *Consider (5). For the sake of compactness, we denote . For any , we split into equal-size blocks as . If all the blocks are the same, then is a redundant variable. Thus, can be replaced by**where .*

#### 3. Multivariable FLC Based on STP for Mobile Robot OSL

Consider the linguistic control rules of the multivariable fuzzy system:where and are linguistic variables representing the process state and the control variable, respectively. denotes the th fuzzy inference rule, where , and is the number of fuzzy rules. , and , are the normalized fuzzy set of linguistic values on universes of discourses and , respectively. The control system is shown in Figure 1.