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Journal of Automation, Mobile Robotics & Intelligent Systems

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VOLUME 11, N° 1

2017

pISSN 1897-8649 (PRINT) / eISSN 2080-2145 (ONLINE)

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JOURNAL OF AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 11, N° 1, 2017 DOI: 10.14313/JAMRIS_1-2017

CONTENTS 3

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A Binary Representation for Real-Valued, Local Feature Descriptors Mariusz Oszust DOI: 10.14313/JAMRIS_1-2017/1

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Fuzzy Modal Operators and Their Applications Anna Maria Radzikowska DOI: 10.14313/JAMRIS_1-2017/2 21

Fuzzy Relation-based Approximation Techniques in Supporting Medical Diagnosis Anna Maria Radzikowska DOI: 10.14313/JAMRIS_1-2017/3 30

Multi-Strategy Navigation for a Mobile Data Acquisition Platform Using Genetic Algorithms Fadi Halal, Marek B. Zaremba DOI: 10.14313/JAMRIS_1-2017/4 42

Application the GPS Observations in SPP Method for Aircraft Positioning in Flight Experiment in Dęblin, Poland (01.06.2010) Kamil Krasuski DOI: 10.14313/JAMRIS_1-2017/5

2

Articles

Particle Swarm Optimization for Tuning PSS-PID Controller of Synchronous Generator Amina Derrar, Abdelatif Naceri DOI: 10.14313/JAMRIS_1-2017/6 53

Fuzzy Switching for Multiple Model Adaptive Control in Manipulator Robot Behrouz Kharabian, Hosseyn Bolandi, Seyed Majid Smailzadeh, Seyed Kamaledin Mousavi Mashhadi DOI: 10.14313/JAMRIS_1-2017/7 57

An Efficiency No Adaptive Backstepping Speed Controller Based Direct Torque Control Abdelkader Ghezouani, Brahim Gasbaoui, Jamel Ghouili, Asma Amal Benayed DOI: 10.14313/JAMRIS_1-2017/8 64

Self-sensing Teleoperation System Based on 1-dof Pneumatic Manipulator Mateusz Saków, Karol Miądlicki, Arkadiusz Parus DOI: 10.14313/JAMRIS_1-2017/9 77

Implementation of Micro Airborne Radio Relay Karol Niewiadomski, Grzegorz Kasprowicz DOI: 10.14313/JAMRIS_1-2017/10

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Submi ed: 8th October 2016; accepted: 10th January 2017

Mariusz Oszust DOI: 10.14313/JAMRIS_1-2017/1 Abstract: The usage of real-valued, local descriptors in computer vision applica ons is o en constrained by their large memory requirements and long matching me. Typical approaches to the reduc on of their vectors map the descriptor space to the Hamming space in which the obtained binary strings can be efficiently stored and compared. In contrary to such techniques, the approach proposed in this paper does not require a data-driven binarisa on process, but can be seen as an extension of the floa ng-point descriptor computa on pipeline with a step that allows turning it into a binary descriptor. In this step, binary tests are performed on values determined for pixel blocks from the described image patch. In the paper, the proposed approach is described and applied to two popular real-valued descriptors, SIFT and SURF. The paper also contains a comparison of the approach with state-of-the-art binarisa on techniques and popular binary descriptors. The results demonstrate that the proposed representa on for real-valued descriptors outperforms other methods on four demanding benchmark image datasets.

Keywords: SIFT, SURF, LDAHash, binary tests, image matching, image recogni on

1. Introduc on Real-valued, local feature descriptors, such as Scale-Invariant Feature Transform (SIFT) [24] and Speeded Up Robust Features (SURF) [5] have already found their place in many computer vision applications, e.g., recognition [14], localisation [11, 12], tracking [8, 34], simultaneous localisation and mapping [17], or retrieval [6, 15]. However, there is a need for the development of more ef icient techniques, in terms of computation time, storage requirements, or robustness [6, 10, 12, 13, 25, 27]. Since SIFT and SURF are among the best performing loating-point descriptors [21, 26], there are many works which aim to preserve their distinctive properties while performing a dimensionality reduction of their long vectors [19], or storing them as binary strings after mapping the descriptor space into the Hamming space [35]. A transformation to binary strings allows for faster feature matching, since such strings can be ef iciently compared on modern CPUs. For example, in [35] LDAHash was introduced, in which a projection matrix is selected and computed minimising in-class covariance and maximising covariance across classes of SIFT features. Then, a threshold vector is used for bina-

risation of projections and providing binary strings which maximise recognition rate. In another datadriven work [31], a vector of medians was used for binarisation of SIFT keypoints. The same approach can be easily applied to SURF vectors. The recently introduced Bi-DCT descriptor for dense matching [22] converts loating-point vectors into binary strings, taking into account DCT coef icients. The method exploits frequency and orientation information based on 2D DCT. In [16], in turn, dimensionality reduction of SIFT descriptor was performed using matched descriptor pairs and a linear discriminant embedding.

PCA-SIFT technique [19] produces shorter real-valued descriptors based on principal component analysis (PCA) which is applied to SIFT keypoints. A more recent technique, shown in [18], uses a spectral embedding for transformation of the original feature space into an low-dimensional space, preserving the manifold structure and a relevance relationship among the images.

The approach introduced in this paper transforms high-dimensional information on the described image patch centred at the keypoint to a binary string. It is devoted to hand-crafted loating-point techniques for which the patch division into smaller pixel regions, or blocks, is easy accessible. In a typical binary descriptor, binary tests between intensities of pairs of points within the image patch are performed. This can be seen in Binary Robust Independent Elementary Features (BRIEF) [7], where point pairs are sampled from isotropic Gaussian distribution. Binary Robust Invariant Scalable Keypoints (BRISK) [23], in turn, uses a circular pattern with equally spaced points for this purpose, and Oriented FAST and Rotated BRIEF (ORB) [8, 10, 33] uses a learned sampling pattern and FAST [32] technique to generate keypoints. A retinal sampling pattern is used in Fast Retina Keypoint (FREAK) [2]. These approaches perform binary tests on pixel intensities, what can make them more sensitive to noise. Therefore, recently introduced binary descriptors perform tests on larger patch regions. For example, in Local Difference Binary (LDB) [39, 40] and Modi ied-LDB [3] (AKAZE), the described patch is divided into 4, 9, 16, and 25 disjunctive blocks of pixels (cells) and then binary tests are performed on their mean intensities and directional gradients. A more distinctive approach can be found in [30], where Binary Robust Fast Features (BRAF) descriptor employs four image patches with scale-dependent sizes are divided into 3 × 3 pixel blocks, or in Binary Descriptor with Shared Pixel Blocks (BDSB) [29], in which 3

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overlapping pixel regions are used. These pixel blockbased descriptors perform all-against-all tests separately for each patch and for each the type of information extracted from patches, i.e., intensities and gradients.

In the literature, apart from hand-crafted solutions, there are also data-driven binary approaches in which the image patch is divided into pixel regions and then values used for their description are compared optimising performance criteria. Tests on intensities and gradients of regional invariants can be seen in Ordinal and Spatial information of Regional Invariants (OSRI) [38]. In OSRI, a resulted binary string is long and requires further reduction. This is also present in LDB [40]. BinBoost descriptor [36] replaces binary tests with learned binary hash functions [36].

AdaBoost classi ier and gradient-based image features are used here. Receptive Fields Descriptor (RFD) [9] uses thresholded ields’ responses of rectangular or Gaussian pooling regions, and Binary Online Learned Descriptor (BOLD) [4] is independently optimised for each image patch. The state-of-the-art learning-based binary descriptors (e.g., BinBoost), as well as handcrafted block-based descriptors (e.g., LDB, AKAZE) have computation time close to loating-point techniques, what makes a binary descriptor build on the top of a loating-point technique still worthy of consideration. Furthermore, dimensionality reduction approaches with a projection matrix may also suffer from longer computation time due to multiplication of feature vectors by the matrix [35].

The technique introduced in this paper is related to pixel block-based binary descriptors, in which tests on values representing some image regions within the patch are performed. Since modi ications of SIFT and SURF are presented and evaluated in further parts of this paper, it can be said that they are novel binary representatives of these techniques.

The rest of this paper is organised as follows.

Section 2 covers the description of the proposed approach and its application. Experimental results with related discussions are presented in Section 3. Finally, Section 4 concludes the paper and indicates possible directions of future research.

2. Approach 2.1. Binary Tests for a Real-Valued Descriptor It can be assumed that a real-valued descriptor provides a description of the image patch centred at the detected keypoint. Let Bi denotes a pixel block within the patch, i = 1, 2 . . . , N , where N is the number of such blocks. Each i-th block is described with a vector Vji having M real-valued dimensions, i i i Vj=1 , Vj=2 , . . . , Vj=M . The size of the patch and its division depends on descriptor computation pipeline. In the proposed approach, the binary string is created using all-against-all binary tests on values representing blocks [3, 29, 30, 39, 40]. Thus, the computation of the binary string b can be written as: 4

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 2o−1 Tj  ,

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1≤o≤N C2

where o denotes a pair of blocks, Bl and Bk , l 6= k, {l, k} = 1, 2, . . . , N . In the equation, the sum sign denotes concatenation of strings with binary values.

There are N C2 = N ! \ (2!(N − 2)!) binary tests for each j-th dimension, the test Tj is calculated using eq.

(2):

1, if Vjl < Vjk Tj = (2) 0, otherwise.

2.2. Binary SIFT and SURF In SIFT [24], an image patch centred at keypoint location is extracted. Its size depends on the keypoint’s scale. Then, local gradients are used to provide a dominant orientation of the patch. The orientation is used for orienting local gradients determined for 4×4 grids placed within each of 16 pixel blocks. The gradients are quantised into eight angular bins, and then 128dimensional, weighted histogram is determined. The histogram is created using magnitudes and orientations. Magnitudes are weighted using Gaussian, and the resulting vector is normalised to unit vector to ensure robustness against illumination changes. Extension of SIFT descriptor pipeline using the approach presented in this paper is applied as follows. Each of 16 blocks (N = 16) is represented by eight-dimensional (M = 8), real-valued vector. Therefore, there are 120 binary tests (16 C2 ) for each dimension, and the inally obtained binary string has 960 bits. The string is long and that may require additional dimensionality reduction (as e.g., 21576-bit string in OSRI [38]). However, in some applications it would be more convenient to use a small number of longer descriptors with high discriminative properties than a large number of worse performing descriptors with short binary strings.

In SURF [5], similarly to SIFT, a square patch is divided into 16 pixel blocks (4 × 4), and the descriptor is created by a union of vectors resulted from sums of horizontal and vertical Haar wavelet responses and their absolute values. The Haar wavelet responses for each pixel block are computed at 5×5 regularly spaced sample points and then weighted with a Gaussian in order to introduce better robustness against geometrical transformations. Finally, each pixel block is described using four values, then the 64-dimensional vector is further transformed into unit vector to provide contrast invariance [5]. Since the patch is described using N = 16 blocks, and each block is represented by M = 4 values, the resulting binary string is composed of 480 bits (4 × 16 C2 ). It is worth noticing that recently introduced AKAZE has a similar length (486 bits).

3. Experimental Evalua on The binary versions of SIFT and SURF created with the feature representation introduced in this paper were evaluated and compared with their loatingpoint counterparts. The state-of-the-art techniques

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designed to reduce their high-dimensionality (LDAHash [35], binarised SIFT [31], binarised SURF [31]) were also used, as well as the state-of-the-art binary descriptors (AKAZE [3], BRISK [23], and ORB [33]).

AKAZE, being an example of hand-crafted pixel blockbased descriptor, is closer to the introduced technique, in terms of performed steps in descriptor’s computation pipeline, than binarisation techniques applied to the resulted loating-point vectors. All three compared binary descriptors are equipped in the full descriptor pipeline, i.e., they were designed to be able to detect and describe interest points.

In experiments, single-threaded applications with descriptors were run on a CPU with Intel Core i5-5200u 2.2 GHz, 8 GB RAM, and Microsoft Windows 7. Java implementations of SURF and SIFT from BoofCV library (http://boofcv.org/) were used [1], as well as Java interface to OpenCV implementations of AKAZE [3], BRISK [23], and ORB [33] descriptors (http://opencv.org, https://github.com/bytedeco/javacv).

The author of this work implemented LDAHash and approaches introduced by Peker [31] in Java using known projection matrices (LDAHash, http://cvlab.ep l.ch/research/detect/ldahash) or the published description [31].

3.1. Image Matching The compared techniques were evaluated in terms of the area under Recall vs. 1-Precision curve calculated for corresponding pairs of images that belong to image benchmarks designed to test matching performance of local feature descriptors [6,12,13,25,27]. For this purpose, Oxford [25] and Heinly et al. [13] datasets were used. The datasets contain base images and sequences of images that exhibit different amount of most popular transformations, e.g., rotation, scaling, viewpoint change, blur, illumination, exposure, and JPEG compression. Exemplary image sentences from these datasets can be seen on ig. 1. In order to calculate the area under Recall vs. 1-Precision curve, for the base image and its transformed equivalent, a predeined number of interest points on both images were detected and described. Then, for each keypoint from the base image its corresponding keypoint from the second image was determined, taking into account a distance ratio between the closest and the second closest keypoint from that image, localisation error of the found keypoint and an overlap size between described image patches. In this paper, the following values were used [13, 25]: 0.8 distance ratio, three pixel localisation error, and 40% overlap. Binary descriptors were compared using Hamming distance, while loatingpoint vectors were compared with Euclidean distance.

Precision of matching was calculated as the number of returned correct matches to the all matches, and Recall as the number of returned correct matches to all possible correct matches [13]. Finally, Recall vs. 1-Precision curves were obtained using threshold-based similarity matching [5] for 500 keypoints with the strongest response per image.

Obtained results are presented in Tab. 1 and Tab.

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Fig. 1. Exemplary images from Oxford and Heinly et al. banchmark datasets 2. On the basis of matching results it can be said that for Oxford dataset the introduced binary versions of SURF and SIFT performed slightly worse than their real-valued counterparts. However, for Heinly et al. dataset and the sequences from Oxford dataset with illumination and blur changes, they were better. Furthermore, SURFb and SIFTb clearly outperformed other feature binarisation techniques. Here, SIFTb was the leading binary technique for seven image sequences, and six times was the best descriptor in general.

SURFb , in turn, outperformed other techniques six times, and for two image sequences was the leading performer.

The comparison of description time of compared approaches, as well as their lengths, is given in tab.

4. In the table, the average timings measured per keypoint for Bikes sequence from Oxford dataset [25] are reported. It can be seen that the presented approach requires the addition of 2 to 6% of descriptor computation time in order to obtain binary representation of described image patch. This takes longer than simple thresholding [31], but is two times faster than more advanced LDAHash. Binary descriptors designed to perform fast detection and description of interest points are faster, but as it was shown in image matching experiments, as well as in recognition tests (Section 3.2), the usage of the introduced technique is justi ied by its superior performance. Such performance is often important in practical tasks, where unequivocal description of an image patch may lead to, e.g., near-duplicate content detection [20] The matching time depends on the length of the descriptor, and both descriptors are shorter than their loatingpoint counterparts. SIFTb ’s binary string allows for faster matching than 128-dimensional vector used by SIFT. However, a practical usage of both techniques may require an application of a salient bit-selection technique [29, 38, 40].

3.2. Recogni on The performance of descriptors was also compared in matching-based recognition tests on two demanding image collections, 5

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Tab. 1. Comparison of matching performance measured in terms of mean area under Recall vs. 1- Precision curves on Oxford dataset Image sequence and transformation Bark Bikes Boat Graf iti Leuven Ubc Wall Rotation Blur Rotation Viewpoint Illumination JPEG Viewpoint SURF [5] 0.278 0.578 0.386 0.180 0.587 0.746 0.438 SIFT [24] 0.144 0.534 0.212 0.177 0.642 0.604 0.454 LDAHash [35] 0.118 0.557 0.176 0.157 0.624 0.587 0.363 Bin. SURF [31] 0.045 0.400 0.113 0.047 0.408 0.496 0.105 Bin. SIFT [31] 0.103 0.341 0.110 0.074 0.485 0.496 0.322 AKAZE [3] 0.177 0.368 0.199 0.121 0.383 0.466 0.245 BRISK [23] 0.057 0.324 0.035 0.064 0.132 0.333 0.091 ORB [33] 0.033 0.060 0.043 0.025 0.133 0.181 0.090 SURFb 0.229 0.627 0.347 0.164 0.619 0.733 0.323 SIFTb 0.124 0.615 0.199 0.225 0.675 0.600 0.380 Note: The best value for each image sequence is written in boldface, the best result for the binary approach is underlined.

Method

Trees Blur 0.382 0.391 0.291 0.105 0.237 0.204 0.129 0.077 0.325 0.286

Tab. 2. Comparison of matching performance measured in terms of mean area under Recall vs. 1- Precision curves on Heinly et al. dataset Image sequence and transformation Ceiling Day and night Rome Semper Venice Rotation Illumination Rotation Rotation Scaling SURF [5] 0.458 0.061 0.563 0.318 0.652 SIFT [24] 0.616 0.158 0.658 0.62 0.151 LDAHash [35] 0.604 0.156 0.665 0.61 0.138 Bin. SURF [31] 0.257 0.037 0.358 0.239 0.326 Bin. SIFT [31] 0.542 0.057 0.407 0.549 0.120 AKAZE [3] 0.464 0.038 0.354 0.267 0.225 BRISK [23] 0.320 0.050 0.355 0.395 0.244 ORB [33] 0.083 0.019 0.125 0.119 0.135 SURFb 0.439 0.128 0.603 0.387 0.664 SIFTb 0.625 0.199 0.690 0.633 0.164 Note: The best value for each image sequence is written in boldface, the best result for the binary approach is underlined.

Method

Tab. 3. Comparison of recogni on performance on the BR and UKBench datasets Method

Test set in the BR dataset Autumn Winter Winter Spring night day night day Recognition rate (in %) SURF [5] 34.6 40.6 32.2 47.4 40.4 SIFT [24] 56.8 45.4 62.4 53.6 65.0 LDAHash [35] 42.6 37.8 51.2 47.2 54.8 Bin. SURF [31] 28.0 31.4 23.0 32.0 25.4 Bin. SIFT [31] 23.4 18.0 30.0 19.0 23.0 AKAZE [3] 32.2 40.8 28.8 44.0 40.6 BRISK [23] 2.60 2.40 4.40 2.40 4.00 ORB [33] 16.2 16.0 11.6 16.0 18.0 SURFb 52.2 68.4 47.2 65.2 54.0 SIFTb 58.0 54.6 72.0 60.6 65.8 Note: The best value for each test is written in boldface.

Autumn day

the UKBench [28] and the Beautiful Rzeszow (http://marosz.kia.prz.edu.pl/br.html) [30] datasets. Some images from these datasets are shown on ig. 2. The UKBench dataset contains images of 2550 objects. There are four images per object. According 6

UKBench Spring night 41.8 50.0 44.4 32.8 22.0 58.0 8.60 36.4 66.8 52.8

Score 2.444 3.000 2.859 2.262 2.342 2.757 1.347 2.199 2.791 3.071

to the proposed performance index [28], an average of top four results for the irst objectsâ&#x20AC;&#x2122; images is used.

The second benchmark used in recognition tests, the Beautiful Rzeszow (BR) dataset, contains 3000 images of 50 sites in Rzeszow, Poland. There are 10 images of

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Fig. 2. Exemplary images from object recogni on benchmarks: (a) UKBench, and (b) the BR Tab. 4. Comparison of descrip on me and length Method Description time (ms) Length SURF [5] 0.140 64 Floats SIFT [24] 0.752 128 Bytes LDAHash [35] SIFT + 38 us 128 Bits Bin. SURF [31] SURF + 2 us 64 Bits Bin. SIFT [31] SIFT + 3 us 128 Bits AKAZE [3] 0.141 486 Bits BRISK [23] 0.042 512 Bits ORB [33] 0.022 256 Bits SURFb SURF + 9 us 480 Bits SIFTb SIFT + 20 us 960 Bits Note: Description time is given per keypoint.

each site captured at a different time of the day (day and night) and season (spring, autumn, and winter), in 2015. The objects were photographed introducing viewpoint, scale, and rotation changes. The time the images were captured also resulted in challenging illumination conditions and many occlusions, which are not present in popular UKBench. The recognition performance on this dataset was tested using images taken at the given time of the day as queries and the top one returned results was assessed. Here, the recognition accuracy is presented as the percent of correctly recognised objects. In the object recognition tests, test images were recognised using k-nearest neighbour classi ier (k=1), which assigned learned labels taking into account the largest number of returned matched descriptor pairs between the test and the learning images. In matching, the threshold of 0.8 was used. For UKBench, due to long matching time for real-valued descriptors, the score was reported on the basis of the irst 3000 images.

Table 3 contains recognition results for these two datasets. It can be seen that both introduced binary descriptors, SURFb and SIFTb , were better than compared techniques. Due to high robustness of SIFT descriptor, tests with day images were easier for its derivative approaches (SIFTb and LDAHash). However, for

the night images, SURFb turned out to be better, since performed binary tests increased illumination invariance of the keypoint’s description. Interestingly, both techniques improved results obtained by SURF and SIFT, what with faster matching time and shorter descriptor length has a practical importance. This can be also observed for UKBench dataset, where SIFTb outperformed other methods. For this dataset, SIFT was better than SURF, what also indicates that the developed binary representation is limited by the distinctive properties of binarised real-valued descriptor.

4. Conclusion In this paper, an approach to an extension of loating-point descriptor’s computation pipeline with a step that allows turning them into binary descriptors was shown. The approach performed binary tests on values determined by loating-point descriptors for pixel blocks within the image patch in order to create a binary string. The resulted binary representation was evaluated and compared with popular feature binarisation techniques, as well as with state-ofthe-art binary descriptors equipped with full descriptor pipeline. Obtained results on four image benchmarks are promising and showed that two introduced binary representations built on top of SURF and SIFT techniques are highly competitive, outperforming other approaches in matching and recognition tasks.

Future works will consider dimensionality reduction of obtained binary strings, as well as experiments with other real-valued, local descriptors, or other descriptor types [37].

AUTHOR Mariusz Oszust – Department of Computer and Control Engineering, Rzeszow University of Technology, Wincentego Pola 2, 35-959 Rzeszow, Poland, e-mail: marosz@kia.prz.edu.pl, www: www.marosz.kia.prz.edu.pl.

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[2] A. Alahi, R. Ortiz, and P. Vandergheynst, “FREAK: Fast retina keypoint”. In: Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, 2012, 510–517.

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[4] V. Balntas, L. Tang, and K. Mikolajczyk, “BOLD - Binary online learned descriptor for ef icient image matching”. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015, 2367–2375.

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Submi ed: 13th January 2017; accepted: 16th February 2017

Anna Maria Radzikowska DOI: 10.14313/JAMRIS_1-2017/2 Abstract: In this paper we present some fuzzy modal operators and show their two possible applica ons. These operators are fuzzy generaliza ons of modal operators well-known in modal logics. We present an applica on of some composi ons of these operators in approxima ons of fuzzy sets. In par cular, it is shown how skills of candidates can be matched for selec ng research projects. The underlying idea is based on the observa on that fuzzy sets approxima ons can be viewed as intui onis c fuzzy sets introduced by Atanassov. Distances between intui onis c fuzzy sets, proposed by Szmidt and Kacprzyk, support the reasoning process. Also, we point out how modal operators are useful for represen ng linguis c hedges, that is terms like â&#x20AC;&#x153;veryâ&#x20AC;?, â&#x20AC;&#x153;deďŹ nitelyâ&#x20AC;?, â&#x20AC;&#x153;ratherâ&#x20AC;?, or â&#x20AC;&#x153;more or lessâ&#x20AC;?.

Keywords: Modal operators, Fuzzy sets, Approxima on operators, Intui onis c fuzzy sets, Linguis c hedges

1. Introduc on The term modal operators usually refers to logical connectives for modal logics which are characterized by expressing a modal attitude (necessity, possibility, belief, knowledge) about propositions they are applied to. Semantically, these operators are interpreted as mappings de ined on a universe of binary relations (in a nonempty domain in discourse, say đ?&#x2018;&#x2039;) and subsets of a domain đ?&#x2018;&#x2039;, which return another subset of đ?&#x2018;&#x2039;.

Such mappings are also called modal operators. The operators of possibility and necessity are typical examples of modal operators. Another pair of modal operators, usually referred to as suf iciency (or negative necessity) and dual suf iciency (or impossibility) were introduced in order to represent expressions like â&#x20AC;&#x153;necessary falseâ&#x20AC;? and â&#x20AC;&#x153;possibly falseâ&#x20AC;?, respectively (cf. Humberstone [12], Gargov [10], and Goranko [11]).

Modal operators found many interesting applications. Probably the most famous ones are rough sets introduced by Pawlak [16, 17] where necessity and possibility operators are used for set approximations.

In more general settings modal operators are mappings of the form đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) Ă&#x2014; â&#x201E;&#x2DC;(đ?&#x2018;&#x152;) â&#x2020;&#x2019; â&#x201E;&#x2DC;(đ?&#x2018;&#x2039;), where đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) stands for the family of all binary relations on two nonempty domains đ?&#x2018;&#x2039; and đ?&#x2018;&#x152;, and â&#x201E;&#x2DC;(đ?&#x2018;?) is the power set of đ?&#x2018;?. In formal concept analysis (FCA) the suf iciency operator is known as derivation operator (cf. Wille [31]). DuĚ&#x2C6; ntsch and Gediga [8] discussed compositions of modal operators in qualitative data analysis and in [9] they considered these operators in the context of knowledge and skills structures.

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Modal operators, as traditionally investigated and applied, are based on classical structures like sets and relations. From the standpoint of practical applications this approach is suf icient when we deal with precise data. However, when the available information is imprecise, or vague, more general structures are needed. Fuzzy set theory introduced by Zadeh [32] offers numerous tools and techniques for representing, processing, and analyzing information which is imprecise in its nature. For example, assume that we are to evaluate studentâ&#x20AC;&#x2122;s skills during some course. Clearly, it is essential to know to what extend studentâ&#x20AC;&#x2122;s knowledge and abilities match our requirements and the yes-no information is practically meaningless. Also, it is important to state to what extent some skill is required for realizing a particular research project and the useful/usefulness answers may highly restrict correctness of decision process concerning selection of proper candidates. Therefore, fuzzy sets and fuzzy relations are natural tools for representing this kind of data. Fuzzy generalizations of modal operators seem to be adequate for drawing conclusions from fuzzy information.

In this paper we present fuzzy generalization of modal operators and show two their applications. In the irst application it is shown how fuzzy modal operators can be applied for supporting process of selecting candidates for research projects. The underlying information is candidatesâ&#x20AC;&#x2122; skills and projectsâ&#x20AC;&#x2122; requirements. We point out that the compositions of fuzzy suf iciency and fuzzy dual suf iciency operators form fuzzy approximation operators. As observed, these approximations lead to Atanassovâ&#x20AC;&#x2122;s intuitionistic fuzzy sets. Distances between intuitionistic fuzzy sets determine decisions on selecting candidates for projects.

The second application is focused on the representation of linguistic hedges. Traditional, and still very popular representation, originally proposed by Zadeh [33], is a powering technique. Precisely, if đ??š is a fuzzy set representing some property đ?&#x2018;&#x192; (e.g., good, high, old), then for đ?&#x203A;ź > 1, đ??š stands for very P (or de initely P, extremely P), while â&#x2C6;&#x161;đ?&#x2018;&#x192; represents rather P (or more or less P, quite P). This approach is purely technical and, in our opinion, passes over the fact that linguistic hedges can be treated as speci ic modal expressions. Following this observation we present representations of linguistic hedges using fuzzy necessity and fuzzy possibility operators. In particular, given some property of objects (typically an adjective in natural language) represented by a fuzzy set đ?&#x2018;&#x192; in the set đ?&#x2018;&#x2039; of objects, we say that an object đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; is very P (e.g.,

Journal of Automation, Mobile Robotics & Intelligent Systems

very young) to the degree to which all objects from đ?&#x2018;&#x2039; resembling đ?&#x2018;Ľ posses the property đ?&#x2018;&#x192;. Similarly, đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; is rather P (e.g., rather young) to the degree to which some object from đ?&#x2018;&#x2039;, which resembles đ?&#x2018;Ľ, has the property đ?&#x2018;&#x192;. This way linguistic hedges provide characterizations of objects relatively to other objects.

The paper is organized as follows. In Section 2 we recall basic notions and terminology which will be used in the paper. Fuzzy modal operators are presented in Section 3. We de ine four basic operators and consider two their compositions. It is pointed out that, given a fuzzy set đ??š, these operators constitute fuzzy approximations of đ??š and lead to an intuitionistic fuzzy set. In Section 4 we propose how these operators may be useful for matching research projects for potential candidates taking into account requirements imposed on particular projects and skills shown by candidates.

The selection criterion is based on a distance between intuitionistic fuzzy sets. The next section is focused on modeling linguistic modi iers by means of fuzzy possibility and fuzzy necessity operators. Several schemes are presented and the corresponding representation is discussed. Concluding remarks complete the paper.

2. Preliminaries In this section we present basic notions and some of their properties which clarify our discussion in the present paper.

2.1. Fuzzy Sets Let đ?&#x2018;&#x2039; be a non-empty domain. A fuzzy set in đ?&#x2018;&#x2039; is any mapping đ??š â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; [0, 1]. For any đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ??š(đ?&#x2018;Ľ) is the degree to which đ?&#x2018;Ľ belongs to đ??š. Given two fuzzy sets in đ?&#x2018;&#x2039;, đ??´ and đ??ľ, we say that â&#x20AC;&#x201C; đ??´ in (totally) included in đ??ľ, written đ??´ â&#x160;&#x2020; đ??ľ, if đ??´(đ?&#x2018;Ľ) ⊽ đ??ľ(đ?&#x2018;Ľ) for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, â&#x20AC;&#x201C; đ??´ is (totally) equal to đ??ľ, written đ??´ = đ??ľ, if đ??´(đ?&#x2018;Ľ) = đ??ľ(đ?&#x2018;Ľ) for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;.

A kernel of a fuzzy set đ??´ in đ?&#x2018;&#x2039; is de ined as đ?&#x2018;&#x2DC;đ?&#x2018;&#x2019;đ?&#x2018;&#x;(đ??´) = {đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; â&#x2C6;ś đ??´(đ?&#x2018;Ľ) = 1} while the support of a fuzzy set đ??´ in đ?&#x2018;&#x2039; is the set đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;?(đ??´) = {đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; â&#x2C6;ś đ??´(đ?&#x2018;Ľ) > 0}.

The family of all fuzzy sets in đ?&#x2018;&#x2039; will be denoted by â&#x201E;ą(đ?&#x2018;&#x2039;).

A binary fuzzy relation in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; (or just a fuzzy relation in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152;) is a fuzzy set in đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x152;. For every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ľ is đ?&#x2018;&#x2026;-related with đ?&#x2018;Ś. We will write â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) to denote the family of all fuzzy relations in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152;. For đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), the converse relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x152;, đ?&#x2018;&#x2039;) is deined as đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;. If đ?&#x2018;&#x2039; = đ?&#x2018;&#x152;, then we have a binary fuzzy relation on đ?&#x2018;&#x2039; (fuzzy relation on đ?&#x2018;&#x2039;, for short). a fuzzy relation đ?&#x2018;&#x2026; on đ?&#x2018;&#x2039; is called â&#x20AC;&#x201C; re lexive if đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ) = 1 for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, â&#x20AC;&#x201C; symmetric if đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;,

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â&#x20AC;&#x201C; sup-min transitive if sup min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;§) ⊽ đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;§) â&#x2C6;&#x2C6;

for all đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;.

2.2. Intui onis c Fuzzy Sets Intuitionistic fuzzy sets, originally proposed by Atanassov [1, 2], is an interesting generalization of fuzzy sets where both degrees of membership and non-membership are involved. More speci ically, an intuitionistic fuzzy set in đ?&#x2018;&#x2039; is given by đ??´ = {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} where đ?&#x153;&#x2021; , đ?&#x153;&#x2C6; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) with đ?&#x153;&#x2021; (đ?&#x2018;Ľ)+đ?&#x153;&#x2C6; (đ?&#x2018;Ľ) â&#x2030;¤ 1 for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, are called a membership and a non-membership function, respectively. For đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x153;&#x2039; (đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; đ?&#x153;&#x2021; (đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x153;&#x2C6; (đ?&#x2018;Ľ) is a hesitation margin re lecting the lack of knowledge of whether đ?&#x2018;Ľ belongs to đ??´ or not. Clearly, every fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) is a speci ic intuitionistic fuzzy set with đ?&#x153;&#x2C6; (đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; đ?&#x153;&#x2021; (đ?&#x2018;Ľ), i.e., đ?&#x153;&#x2039; (đ?&#x2018;Ľ) = 0 for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;.

As argued by Szmidt and Kacprzyk [30], distance measures between intuitionistic fuzzy sets should involve all three parameters. Namely, for two intuitionistic fuzzy sets, đ??´ and đ??ľ, in a inite universe đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ }, the normalized Hamming distance is de ined as ( , ) |

( )

( )|

|

( )

( )|

|

( )

( )|.

2.3. Fuzzy Logical Connec ves Fuzzy logical connectives generalize logical connectives of classical logic. The most popular generalization of classical conjunction are triangular norms (t-norms, for short). Speci ically, a triangular norm is a function â&#x160;&#x2014; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] satisfying the following conditions (T1) đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś = đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;Ľ for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1] (commutativity), (T2) đ?&#x2018;Ľ â&#x160;&#x2014; (đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;§) = (đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś) â&#x160;&#x2014; đ?&#x2018;§ for all đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1] (associativity), (T3) increasing in both arguments, i.e., đ?&#x2018;Ľ ⊽ đ?&#x2018;§ implies đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś ⊽ đ?&#x2018;§ â&#x160;&#x2014; đ?&#x2018;Ś and đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;Ľ ⊽ đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;§ for all đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1], (T4) đ?&#x2018;Ľ â&#x160;&#x2014; 1 = đ?&#x2018;Ľ for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1] (boundary condition).

Typical examples of t-norms are: â&#x20AC;&#x201C; the standard t-norm (the largest t-norm) đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś = min(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x20AC;&#x201C; the product đ?&#x2018;Ľâ&#x160;&#x2014; đ?&#x2018;Ś = đ?&#x2018;Ľđ?&#x2018;Ś â&#x20AC;&#x201C; the Ĺ ukasiewicz t-norm đ?&#x2018;Ľâ&#x160;&#x2014; đ?&#x2018;Ś = max(0, đ?&#x2018;Ľ + đ?&#x2018;Ś â&#x2C6;&#x2019; 1).

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A t-norm is called left-continuous whenever it is leftcontinuous on both arguments. Note that all three tnorms mentioned above are left-continuous. For the extensive studies on t-norm we refer to Klement, Mesiar, and Pap [13].

A fuzzy implication â&#x2020;&#x2019; is a [0, 1] â&#x2C6;&#x2019; [0, 1] map with increasing irst and decreasing second partial mappings and satisfying 1 â&#x2020;&#x2019; 1 = 0 â&#x2020;&#x2019; 0 = 0 â&#x2020;&#x2019; 1 = 1, and 1 â&#x2020;&#x2019; 0 = 0. The well-known fuzzy implications are â&#x20AC;&#x201C; the Kleene-Dienes implication đ?&#x2018;Ľâ&#x2020;&#x2019;

đ?&#x2018;Ś = max(1 â&#x2C6;&#x2019; đ?&#x2018;Ľ, đ?&#x2018;Ś),

â&#x20AC;&#x201C; the Reichenbach implicator đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś = 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ + đ?&#x2018;Ľđ?&#x2018;Ś, â&#x20AC;&#x201C; the Ĺ ukasiewicz implication đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś = min(1, 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ + đ?&#x2018;Ś),

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set đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) may be interpreted as a description of an individual: for any property đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ??ľ(đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ś characterizes đ??ľ.

Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) be a fuzzy relation which represents characterizations of objects from đ?&#x2018;&#x2039;: for every object đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and for every property đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ľ posses đ?&#x2018;Ś. Given a graded information about objects represented by a relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), we can derive new information about two types of relationships, namely a relationship between objects from đ?&#x2018;&#x2039; determined by their properties and a relationship between properties from đ?&#x2018;&#x152; basing on objects having these properties. These relationships are represented by fuzzy information relations which are generalizations of information relations widely studied within the framework of the rough set-style data analysis (see, e.g., OrĹ&#x201A;owska [14], Demri and Orlowska [7]). Fuzzy information relations were investigated by Radzikowska and Kerre [23,27,29], logical systems capable to reason about such relations were considered

by Radzikowska [20].

For two objects đ?&#x2018;Ľ and đ?&#x2018;Ľ from đ?&#x2018;&#x2039;, we say that

â&#x20AC;&#x201C; the GoĚ&#x2C6; del implication đ?&#x2018;Ľâ&#x2020;&#x2019; đ?&#x2018;Ś=

VOLUME 11,

1 đ?&#x2018;Ś

for đ?&#x2018;Ľ ⊽ đ?&#x2018;Ś elsewhere.

A special class of fuzzy implications are residual implications: given a left-continuous t-norm â&#x160;&#x2014;, its residual implication is de ined for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1],

â&#x20AC;&#x201C; đ?&#x2018;Ľ is relevant to đ?&#x2018;Ľ to the degree to which all properties of đ?&#x2018;Ľ are also properties of đ?&#x2018;Ľ ; â&#x20AC;&#x201C; đ?&#x2018;Ľ and đ?&#x2018;Ľ are compatible to the degree to which they both share some common property; â&#x20AC;&#x201C; đ?&#x2018;Ľ and đ?&#x2018;Ľ are coherent to the degree to which they both do not have some property.

đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś = sup{đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1] â&#x2C6;ś đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;§ ⊽ đ?&#x2018;Ś}.

The Ĺ ukasiewicz the the Gaines implications are examples of residual implications based on â&#x160;&#x2014; and â&#x160;&#x2014; , respectively. For an extended survey on fuzzy implications we refer to BaczynĚ ski and Jayaram [3].

A fuzzy negation is a mapping ÂŹ â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1], non-increasing and satisfying ÂŹ0 = 1 and ÂŹ1 = 0. It is involutive iff ÂŹÂŹđ?&#x2018;Ľ = đ?&#x2018;Ľ for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1]. Residual implications induce fuzzy negations of the form ÂŹđ?&#x2018;Ľ = (đ?&#x2018;Ľ â&#x2020;&#x2019; 0). Since residual implications are de ined on the basis of left-continuous t-norms â&#x160;&#x2014;, these type of fuzzy negations will be referred to as fuzzy negations induced by â&#x160;&#x2014;. The standard fuzzy negation ÂŹđ?&#x2018;Ľ = 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ, đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1], is the involutive negation induced by the Ĺ ukasiewicz t-norm. It is well-known that it is the only involutive negation induced by leftcontinuous t-norms.

Using fuzzy logical connectives basic operations on fuzzy sets are de ined. In particular, for a t-norm â&#x160;&#x2014; and for a fuzzy negation ÂŹ, the â&#x160;&#x2014;-intersection of two fuzzy sets, đ??´, đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), is de ined for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, (đ??´ â&#x2C6;Šâ&#x160;&#x2014; đ??ľ)(đ?&#x2018;Ľ) = đ??´(đ?&#x2018;Ľ) â&#x160;&#x2014; đ??ľ(đ?&#x2018;Ľ), and a fuzzy ÂŹcomplementation of đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) is given by (ÂŹđ??´)(đ?&#x2018;Ľ) = ÂŹđ??´(đ?&#x2018;Ľ) for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;.

3. Fuzzy Modal Operators Let đ?&#x2018;&#x2039; be a set of objects and let đ?&#x2018;&#x152; be a set of properties. Any fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) may be viewed as a fuzzy attribute (property): for every object đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ??´(đ?&#x2018;Ľ) is the degree to which đ??´ characterizes đ?&#x2018;Ľ. Similarly, any fuzzy 12

In order to derive such relationships fuzzy modal operators are used. These operators, being generalizations of operators well-known in modal logics, were extensively studied by Radzikowska and Kerre [18, 19, 22, 24, 25, 25, 26, 28, 29]. Algebraic and logical aspects were presented by OrĹ&#x201A;owska, Radzikowska, and Rewitzky [15]. Recall that these operators are â&#x201E;ą(đ?&#x2018;&#x152;) â&#x2020;&#x2019; â&#x201E;ą(đ?&#x2018;&#x2039;) mappings de ined as follows. Let â&#x160;&#x2014; be a t-norm, let â&#x2020;&#x2019; be a fuzzy implication, and let ÂŹ be a fuzzy negation. Given a fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), for every fuzzy set đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??ľ(đ?&#x2018;Ľ) = inf (đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2020;&#x2019; đ??ľ(đ?&#x2018;Ś))

(1)

â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??ľ(đ?&#x2018;Ľ) = sup(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x160;&#x2014; đ??ľ(đ?&#x2018;Ś))

(2)

[[đ?&#x2018;&#x2026;]]â&#x2020;&#x2019; đ??ľ(đ?&#x2018;Ľ) = inf (đ??ľ(đ?&#x2018;Ś) â&#x2020;&#x2019; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś))

(3)

â&#x2C6;&#x2C6;

â&#x2C6;&#x2C6;

â&#x2C6;&#x2C6;

â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??ľ(đ?&#x2018;Ľ) = sup(ÂŹđ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x160;&#x2014; ÂŹđ??ľ(đ?&#x2018;Ś)).

(4)

â&#x2C6;&#x2C6;

The above operators are called fuzzy necessity, fuzzy possibility, fuzzy suf iciency, and fuzzy dual suf iciency, respectively. They have the following natural interpretation in data analysis: for any individual đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and for any đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, â&#x20AC;&#x201C; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??ľ(đ?&#x2018;Ľ) is the degree to which the object đ?&#x2018;Ľ is relevant to the individual đ??ľ; â&#x20AC;&#x201C; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??ľ(đ?&#x2018;Ľ) is the degree to which the individual đ??ľ and the object đ?&#x2018;Ľ are compatible;

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â&#x20AC;&#x201C; [[đ?&#x2018;&#x2026;]]â&#x2020;&#x2019; đ??ľ(đ?&#x2018;Ľ) is the degree to which the individual đ??ľ is relevant to the object đ?&#x2018;Ľ; â&#x20AC;&#x201C; â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??ľ(đ?&#x2018;Ľ) is the degree to which the individual đ??ľ and the object đ?&#x2018;Ľ are coherent.

Taking đ?&#x2018;&#x2026; and an attribute đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), in the similar way we obtain fuzzy relevance (resp. compatibility, coherence) between properties. E.g., for any property đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, [đ?&#x2018;&#x2026; ]đ??´(đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ś is relevant to đ??´.

Let us recall some basic properties of operators (1)â&#x20AC;&#x201C;(4).

Property 3.1 For all fuzzy sets đ??´, đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and for any fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), (a) đ??´ â&#x160;&#x2020; đ??ľ implies [đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??´ â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??ľ, â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´ â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??ľ, [[đ?&#x2018;&#x2026;]]â&#x160;&#x2014; đ??ľ â&#x160;&#x2020; [[đ?&#x2018;&#x2026;]]â&#x160;&#x2014; đ??´, and â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??ľ â&#x160;&#x2020; â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??´; (b) If â&#x2020;&#x2019; and â&#x2021;&#x2019; are two fuzzy implications such that â&#x2020;&#x2019; ⊽ â&#x2021;&#x2019; (i.e., đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś ⊽ đ?&#x2018;Ľ â&#x2021;&#x2019; đ?&#x2018;Ś for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1]), then [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; đ??´ and [[đ?&#x2018;&#x2026;]]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; [[đ?&#x2018;&#x2026;]]â&#x2021;&#x2019; đ??´; (c) If â&#x160;&#x2014; and â&#x160;&#x2122; are two t-norms such that â&#x160;&#x2014; ⊽ â&#x160;&#x2122;, then â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´ â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ??´ and â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??´ â&#x160;&#x2020; â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2122;,ÂŹ đ??´ for any fuzzy negation ÂŹ.

Now, take a left-continuous t-norm â&#x160;&#x2014;, the residual implication â&#x2020;&#x2019; based on â&#x160;&#x2014;, and the fuzzy negation ÂŹ induced by â&#x160;&#x2014;. Then all four operators (1)â&#x20AC;&#x201C;(4) can be indexed by â&#x160;&#x2014; only.

Now, let us de ine the following two â&#x201E;ą(đ?&#x2018;&#x152;) â&#x2C6;&#x2019; â&#x201E;ą(đ?&#x2018;&#x152;) operations for every đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;: â&#x2013;łâ&#x160;&#x2014; đ??ľ = â&#x;¨â&#x;¨đ?&#x2018;&#x2026;

â&#x;Šâ&#x;Šâ&#x160;&#x2014; â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014; đ??ľ

(5)

â&#x2C6;&#x2021;â&#x160;&#x2014; đ??ľ = [[đ?&#x2018;&#x2026;

]]â&#x160;&#x2014; [[đ?&#x2018;&#x2026;]]â&#x160;&#x2014; đ??ľ.

(6)

These operators have the following important approximation property: Property 3.2 Let â&#x160;&#x2014; be a left-continuous t-norm such that its residual implication induces an involutive fuzzy negation. Then for every đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and for every đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), â&#x2013;łâ&#x160;&#x2014; đ??ľ â&#x160;&#x2020; đ??ľ â&#x160;&#x2020; â&#x2C6;&#x2021;â&#x160;&#x2014; đ??ľ.

Note that the approximation property holds only for the Ĺ ukasiewicz t-norm â&#x160;&#x2014; since it is the only leftcontinuous t-norm which induces an involutive fuzzy negation. The operators (5) and (6) determined by â&#x160;&#x2014; will be written â&#x2013;ł and â&#x2C6;&#x2021; , respectively. For any fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and for any fuzzy set đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), â&#x2013;ł đ??ľ is a lower bound of đ??ľ whereas â&#x2C6;&#x2021; đ??ľ is an upper bound of đ??ľ with respect to đ?&#x2018;&#x2026;, respectively. The pair (â&#x2013;ł đ??ľ, â&#x2C6;&#x2021; đ??ľ) is called an (â&#x2013;ł , â&#x2C6;&#x2021; )â&#x20AC;&#x201C;approximation of đ??ľ with respect to đ?&#x2018;&#x2026;. For any đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, â&#x2013;ł đ??ľ(đ?&#x2018;Ś) may be viewed as the degree to which đ?&#x2018;Ś at least belongs to đ??ľ and â&#x2C6;&#x2021; đ??ľ(đ?&#x2018;Ś) can be read as the degree to which đ?&#x2018;Ś at most belongs to đ??ľ. Accordingly, for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, the value â&#x2C6;&#x2021; đ??ľ(đ?&#x2018;Ś) â&#x2C6;&#x2019; â&#x2013;ł đ??ľ(đ?&#x2018;Ś) is a hesitation region, that is, the degree to which it is unknown whether đ?&#x2018;Ś belongs to đ??ľ or not. This leads us to the following observation.

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Observation 3.1 Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and let đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;).

De ine the following two mappings đ?&#x153;&#x2021; , đ?&#x153;&#x2C6; â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; [0, 1] for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x153;&#x2021; (đ?&#x2018;Ľ) = â&#x2013;ł đ??´(đ?&#x2018;Ľ) đ?&#x153;&#x2C6; (đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; â&#x2C6;&#x2021; đ??´(đ?&#x2018;Ľ).

Then {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} is an intuitionistic fuzzy set.

An application of these operators will be shown in Section 4.

Now, let us consider a fuzzy relation đ?&#x2018;&#x2026; on đ?&#x2018;&#x2039;, i.e., đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;, đ?&#x2018;&#x2039;). Radzikowska and Kerre [26, 28, 29] showed the following property.

Property 3.3 Let â&#x160;&#x2014; be a left-continuous t-norm, let â&#x2020;&#x2019; be its residual implication, and let ÂŹ be the fuzzy negation induced by â&#x160;&#x2014;. Then for every đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), (a) đ?&#x2018;&#x2026; is re lexive iff â&#x2C6;&#x20AC;đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), đ??´ â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´ iff â&#x2C6;&#x20AC;đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; đ??´; (b) đ?&#x2018;&#x2026; is symmetric iff â&#x2C6;&#x20AC;đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; đ??´ iff â&#x2C6;&#x20AC;đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), đ??´ â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´. (c) ÂŹâ&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´ â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; (ÂŹđ??´) and â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; (ÂŹđ??´) â&#x160;&#x2020; ÂŹ[đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´; if ÂŹ is involutive, then both inclusions are equalities.

Then the following corollary follows.

Corollary 3.1 Let â&#x160;&#x2014; be a left-continuous t-norm and let â&#x2020;&#x2019; be its residual implication. Then for every re lexive and symmetric fuzzy relation đ?&#x2018;&#x2026; on đ?&#x2018;&#x2039; and for every đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ??´ â&#x160;&#x2020; đ??´ â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´ â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´.

4. Skills Assessment and Projects Matching In this section we show how fuzzy operators (5) and (6) can be applied to a proper selection of candidates to research projects that are to be carried out at some department.

Assume that a set đ?&#x2018;&#x192; of projects is given. Each one requires some skills guaranteed its accomplishment.

Let đ?&#x2018;&#x2020; be a set of these skills. Researchers responsible for projects present their requirements by determining to what extend particular skills are demanded for their projects. A natural way for representation of such descriptions is to use a fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x192;, đ?&#x2018;&#x2020;), where đ?&#x2018;&#x2026;(đ?&#x2018;?, đ?&#x2018; ) is the degree to which a skill đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020; is required for a project đ?&#x2018;?. Next, let a group đ??ś of candidates (students or researchers) apply for these projects.

They passed some tests which show their abilities in required skills: for each candidate đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś it was evaluated to what extent he/she posses particular skills from đ?&#x2018;&#x2020;. Again, fuzzy structures are useful for representation of candidatesâ&#x20AC;&#x2122; abilities. In consequence, we have another fuzzy relation đ?&#x2018;&#x201E; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ??ś, đ?&#x2018;&#x2020;) such that for every candidate đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś and for every skill đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;, đ?&#x2018;&#x2026;(đ?&#x2018;?, đ?&#x2018; ) is the degree to which đ?&#x2018;?â&#x20AC;&#x2122;s abilities coincide with the skill đ?&#x2018; .

13

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Tab. 1. Projectsâ&#x20AC;&#x2122; requirements đ?&#x2018;&#x2026; đ?&#x2018;? đ?&#x2018;? đ?&#x2018;?

Java 0.7 0.2 0.9

DBases 0.9 0.8 0.6

DMining 1.0 0.9 0.4

Statistics 0.6 0.8 1.0

Algorithmics 0.5 0.6 0.8

Tab. 2. Candidatesâ&#x20AC;&#x2122; skills đ?&#x2018;&#x201E; Tom Susan Jane Bill Mary Ted

Java 0.2 0.4 1.0 0.6 0.2 1.0

DBases 0.5 1.0 0.6 0.3 0.8 0.5

DMining 1.0 1.0 0.6 0.9 0.7 0.7

The task is to choose the most adequate candidates for each project.

First, observe that for any project đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;, đ?&#x2018;?đ?&#x2018;&#x2026; is its description in terms of skills it requires, and for every candidate đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś, đ?&#x2018;?đ?&#x2018;&#x201E; is his/her description in terms of his/her abilities. Then the simplest solution of our problem seems to take distances between fuzzy sets đ?&#x2018;?đ?&#x2018;&#x201E; and đ?&#x2018;?đ?&#x2018;&#x2026; â&#x20AC;&#x201C; the proper choice of a candidate for a project đ?&#x2018;? is pointed out by the shortest distance.

However, this method has a substantial drawback.

Observe that the relation đ?&#x2018;&#x2026; explicitly shows requirements for particular projects, but implicitly đ?&#x2018;&#x2026; gives information about relationships between projects and between skills. This implicit information should be taken into account when the candidate selection is to be made adequately.

Following the interpretation presented in Section 3, any fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2020;) can be viewed a problem and đ??´(đ?&#x2018; ) is the degree to which a skill đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020; is required to solve đ??´. Analogously, any fuzzy set đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x192;) may represent some feature and đ??ľ(đ?&#x2018;?) is the degree to which a project đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;, if carried out, requires đ??ľ. Taking a fuzzy implication â&#x2020;&#x2019;, a t-norm â&#x160;&#x2014;, and a fuzzy negation ÂŹ, for any problem đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2020;) and for any project đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;, - [[đ?&#x2018;&#x2026;]]â&#x2020;&#x2019; đ??´(đ?&#x2018;?) is the degree to which the problem đ??´ is relevant to the project đ?&#x2018;?; - â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??´(đ?&#x2018;?) is the degree to which the project đ?&#x2018;? and the problem đ??´ are coherent.

Similarly, for an attribute đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x192;) and for any skill đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;, [[đ?&#x2018;&#x2026; ]]â&#x2020;&#x2019; đ??ľ(đ?&#x2018; ) is the degree to which the attribute đ??ľ is relevant to the skill đ?&#x2018; , whereas â&#x;¨â&#x;¨đ?&#x2018;&#x2026; â&#x;Šâ&#x;Šâ&#x160;&#x2014;,ÂŹ đ??ľ(đ?&#x2018; ) is the degree to which the attribute đ??ľ and the skill đ?&#x2018; are coherent.

Now, take a left-continuous t-norm. For any problem đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2020;) and for any skill đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;, - â&#x2013;łâ&#x160;&#x2014; đ??´(đ?&#x2018; ) is the degree to which the skill đ?&#x2018; is coherent with the attribute â&#x;¨â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x;Šâ&#x160;&#x2014; đ??´, or equivalently, the degree to which some project incoherent with the problem đ??´ does not require the skill đ?&#x2018; ; - â&#x2C6;&#x2021;â&#x160;&#x2014; đ??´(đ?&#x2018; ) is the degree to which the attribute [đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??´ is relevant to đ?&#x2018; ; in other words, the degree to which all projects to which the problem đ??´ is relevant to, require the skill đ?&#x2018; .

14

Statistics 1.0 0.4 0.8 1.0 0.5 0.6

Algorithmics 0.4 0.6 0.7 0.0 0.1 0.9

By Property 3.2, any problem đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2020;) can be approximated using a relation đ?&#x2018;&#x2026; and any left-continuous t-norm â&#x160;&#x2014;. In particular, for a candidate đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś and a description đ?&#x2018;?đ?&#x2018;&#x201E; of his/her abilities, we have â&#x2013;łâ&#x160;&#x2014; đ?&#x2018;?đ?&#x2018;&#x201E; â&#x160;&#x2020; đ?&#x2018;?đ?&#x2018;&#x201E; â&#x160;&#x2020; â&#x2C6;&#x2021;â&#x160;&#x2014; đ?&#x2018;?đ?&#x2018;&#x201E;.

Hence, for all candidates đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś, we obtain lower and upper bounds of their abilities with respect to particular skills đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;. These approximated evaluations take into account both abilities of candidates resulting from tests they passed, and requirements for projects they applied for. Note that these requirements are twofold: on one hand they follow from researchersâ&#x20AC;&#x2122; needs (described directly in the relation đ?&#x2018;&#x2026;) and, in addition, those ones which result from relationships between both projects and skills (implicitly follow from the relation đ?&#x2018;&#x2026;). Consequently, the proposed approximation uses both explicit and implicit knowledge of all projectsâ&#x20AC;&#x2122; coordinators. Clearly, such an information is required for selecting proper candidates.

Example 4.1 Let đ?&#x2018;&#x192; be a set of three projects đ?&#x2018;? , đ?&#x2018;? , đ?&#x2018;? and let đ?&#x2018;&#x2020; be a set of ive skills required: Programming in Java (Java), Data Bases (DBases), Data Mining (DMining), Statistics (Statistics), and Algorithmics (Algorithmics). A fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x192;, đ?&#x2018;&#x2020;) given in Tab. 1 represents requirements for projects đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; in terms of skills đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;. Next, let đ??ś = {Tom, Alan, Jim, đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;Bill} be a group of four candidates for projects đ?&#x2018;&#x192;. A fuzzy relation đ?&#x2018;&#x201E; â&#x2C6;&#x2C6; â&#x201E;ą(đ??ś, đ?&#x2018;&#x2020;) presented in Tab. 2 represents candidatesâ&#x20AC;&#x2122; skills, Now, taking the Ĺ ukasiewicz logical connectives: t-norm â&#x160;&#x2014; , implications â&#x2020;&#x2019; , and the negation ÂŹ (in fact, the standard negation ÂŹ ), (â&#x2013;ł , â&#x2C6;&#x2021; )â&#x20AC;&#x201C; approximations of particular candidatesâ&#x20AC;&#x2122; abilities are presented in Tab. 3. â&#x2013;Ą However, there is still a question as to which candidate is the proper one for particular projects. To cope with this problem we adopt the methodology as in [21]. Namely, Step 1: Determine (â&#x2013;ł , â&#x2C6;&#x2021; )â&#x20AC;&#x201C;approximation of descriptions đ?&#x2018;?đ?&#x2018;&#x2026; of each project đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;;

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Tab. 3. Candidatesâ&#x20AC;&#x2122; assessments đ?&#x2018;&#x201E; Tom Susan Jane Bill Mary Ted

Java (0.2,0.4) (0.4,0.4) (0.8,1.0) (0.4,0.6) (0.2,0.2) (0.8,1.0.)

DBases (0.4,1.0) (0.4,1.0) (0.6,0.8) (0.2,1.0) (0.3,0.8) (0.5,0.9)

DMining (0.6,1.0) (0.6,1.0) (0.6.0.6) (0.4,0.9) (0.5,0.7) (0.6,0.7)

Statistics (0.3,1.0) (0.4,0.7) (0.6,0.9) (0.0,1.0) (0.1,0.6) (0.5,1.0)

Algorithmics (0.4,0.8) (0.5,0.6) (0.6,0.8) (0.0,0.9) (0.1,0.5) (0.5,0.9)

Tab. 4. Projectsâ&#x20AC;&#x2122; assessments đ?&#x2018;&#x2026; đ?&#x2018;? đ?&#x2018;? đ?&#x2018;?

Java (0.7,0.7) (0.2,0.2) (0.8,0.9)

DBases (0.4,0.9) (0.4,0.8) (0.2,0.6)

DMining (0.6,1.0) (0.6,0.9) (0.4,0.4)

Statistics (0.4,0.6) (0.3,0.8) (0.4,1.0)

Algorithmics (0.5,0.5) (0.4,0.6) (0.5,0.8)

Tab. 5. Intui onis c fuzzy rela on đ?&#x2018;&#x2026; (projects-skills) đ?&#x2018;&#x2026; đ?&#x2018;? đ?&#x2018;? đ?&#x2018;?

Java (0.7,0.3,0.0) (0.2,0.8,0.0) (0.8,0.1,0.1)

DBases (0.4,0.1,0.5) (0.4,0.2,0.4) (0.2,0.4,0.4)

DMining (0.6.0.0,0.4) (0.6,0.1,0.3) (0.4,0.6,0.0)

Statistics (0.4,0.4,0.2) (0.3,0.2,0.5) (0.4,0.0,0.6)

Algorithmics (0.5,0.5,0.0) (0.4,0.4,0.2) (0.5,0.2,0.3)

Tab. 6. Intui onis c fuzzy rela on đ?&#x2018;&#x201E; (candidates-skills) đ?&#x2018;&#x201E; Tom Susan Jane Bill Mary Ted

Java (0.2,0.6,0.2) (0.4,0.6,0.0) (0.8,0.0,0.2) (0.4,0.4,0.2) (0.2,0.8,0.0) (0.8,0.0,0.2)

DBases (0.4,0.0,0.6) (0.4,0.0,0.6) (0.6,0.2,0.2) (0.2,0.0,0.8) (0.3,0.2,0.5) (0.5,0.1,0.4)

Step 2: Calculate intuitionistic fuzzy relations, đ?&#x2018;&#x2026; and đ?&#x2018;&#x201E; determined by đ?&#x2018;&#x2026; and đ?&#x2018;&#x201E;, respectively, and the approximation operators (5) and (6): đ?&#x2018;&#x2026; = {(đ?&#x153;&#x2021; (đ?&#x2018;?, đ?&#x2018; ), đ?&#x153;&#x2C6; (đ?&#x2018;?, đ?&#x2018; ), đ?&#x153;&#x2019; (đ?&#x2018;?, đ?&#x2018; )) â&#x2C6;ś đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;, đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;}, đ?&#x2018;&#x201E; = {(đ?&#x153;&#x2021; (đ?&#x2018;?, đ?&#x2018; ), đ?&#x153;&#x2C6; (đ?&#x2018;?, đ?&#x2018; ), đ?&#x153;&#x2019; (đ?&#x2018;?, đ?&#x2018; )) â&#x2C6;ś đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś, đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;} are given by: đ?&#x153;&#x2021; (đ?&#x2018;?, đ?&#x2018; ) = â&#x2013;ł (đ?&#x2018;?đ?&#x2018;&#x2026;) đ?&#x153;&#x2C6; (đ?&#x2018;?, đ?&#x2018; ) = 1 â&#x2C6;&#x2019; â&#x2C6;&#x2021; (đ?&#x2018;?đ?&#x2018;&#x2026;)(đ?&#x2018; ) đ?&#x153;&#x2019; (đ?&#x2018;?, đ?&#x2018; ) = â&#x2C6;&#x2021; (đ?&#x2018;?đ?&#x2018;&#x2026;)(đ?&#x2018; ) â&#x2C6;&#x2019; â&#x2013;ł (đ?&#x2018;?đ?&#x2018;&#x2026;)(đ?&#x2018; ) and đ?&#x153;&#x2021; (đ?&#x2018;?, đ?&#x2018; ) = â&#x2013;ł (đ?&#x2018;?đ?&#x2018;&#x201E;) đ?&#x153;&#x2C6; (đ?&#x2018;?, đ?&#x2018; ) = 1 â&#x2C6;&#x2019; â&#x2C6;&#x2021; (đ?&#x2018;?đ?&#x2018;&#x201E;)(đ?&#x2018; ) đ?&#x153;&#x2019; (đ?&#x2018;?, đ?&#x2018; ) = â&#x2C6;&#x2021; (đ?&#x2018;?đ?&#x2018;&#x201E;)(đ?&#x2018; ) â&#x2C6;&#x2019; â&#x2013;ł (đ?&#x2018;?đ?&#x2018;&#x201E;)(đ?&#x2018; ),

DMining (0.6,0.0,0.4) (0.6,0.0,0.4) (0.6,0.4,0.0) (0.4,0.1,0.5) (0.5,0.3,0.2) (0.6,0.3,0.1)

Statistics (0.3,0.0,0.7) (0.4,0.3,0.3) (0.6,0.1,0.3) (0.0,0.0,1.0) (0.1,0.4,0.5) (0.5,0.0,0.5)

Algorithmics (0.4,0.2,0.4) (0.5,0.4,0.1) (0.6,0.2,0.2) (0.0,0.1,0.9) (0.1,0.5,0.4) (0.5,0.1,0.4)

approximations, are given in Tab. 5 and Tab. 6, respectively. Normalized Hamming distances between respective intuitionistic fuzzy sets are presented in Tab. 7. Therefore, Susan should be chosen for the project đ?&#x2018;? , she and Mary for đ?&#x2018;? , and Ted for đ?&#x2018;? . â&#x2013;Ą Tab. 7. Distances between candidatesâ&#x20AC;&#x2122; and projectsâ&#x20AC;&#x2122; descrip ons Tom Susan Jane Bill Mary Ted

đ?&#x2018;?

0.3 0.12 0.32 0.5 0.32 0.3

đ?&#x2018;?

0.18 0.16 0.36 0.44 0.16 0.32

đ?&#x2018;?

0.36 0.4 0.22 0.46 0.4 0.18

respectively.

Step 3: Distances between intuitionistic fuzzy sets point out the proper candidate selection: a candidate đ?&#x2018;? is chosen for a project đ?&#x2018;? whenever the distance đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;Ą(đ?&#x2018;?đ?&#x2018;&#x201E; , đ?&#x2018;?đ?&#x2018;&#x2026; ) is the shortest for all đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś.

Example 4.1 (cont.) Tab. 4 shows the results of (â&#x2013;ł , â&#x2013;ł )â&#x20AC;&#x201C;approximations of đ?&#x2018;?đ?&#x2018;&#x201E; for every đ?&#x2018;? â&#x2C6;&#x2C6; đ??ś. Intuitionistic fuzzy relations đ?&#x2018;&#x2026; and đ?&#x2018;&#x201E; , derived from these

The approximation operators (5) and (6) are not the only ones that enables to estimate information given by relations. Radzikowska [21] presented another pair of operators also constructed from modallike operators which allow us for similar approximations. Also, following rough set-style data analysis one can derive fuzzy information relations and on their basis approximate fuzzy sets using fuzzy necessity and fuzzy possibility.

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5. Modeling Linguis c Hedges In this section we show how fuzzy necessity and fuzzy possibility operators can be applied for modeling linguistic hedges. This approach was presented by De Cock, Radzikowska, and Kerre [5, 6] and then developed by De Cock and Kerre [4].

In natural language many properties of objects are normally expressed by adjectives, for example good, young, warm. Using fuzzy-set theoretical approach, they are represented by fuzzy sets. Linguistic modi iers (also referred to as linguistic hedges) are speci ic type of linguistic expressions like very, extremely, more or less, quite. While applied to adjectives, linguistic hedges allow us to express an emphasis we impose on the corresponding properties. In general, there are two types of linguistic hedges: intensifying and weakening. While the former strengthen the emphasis imposed on the term they are applied to (e.g., very good, extremely warm, de initely high), the latter weaken this emphasis (e.g., quite good, more or less warm, rather high).

In the literature two types of interpretation of linguistic hedges are use: inclusive and non-inclusive.

Roughly speaking, for a given property đ?&#x2018;&#x192;, in the inclusive interpretation any object quali ied as â&#x20AC;&#x153;very Pâ&#x20AC;? is also viewed as having the property đ?&#x2018;&#x192; and an object which posses the property đ?&#x2018;&#x192; is also referred to as quite P. For instance, if someone is called very tall, then he/she is also viewed as tall and quite tall. Therefore, when representing linguistic terms by fuzzy sets, the following semantic entailment holds: extremely P â&#x160;&#x2020; very P â&#x160;&#x2020; P â&#x160;&#x2020; more or less P â&#x160;&#x2020; rater P. On the other hand, in the non-inclusive interpretation objects quali ied as very P are not considered P (e.g., people of 90 years old or more, called very old, are not viewed as just old). In this section only the inclusive interpretation is considered and modi ied terms are represented by supersets or subsets of the original term.

In the literature there are many approaches for modeling linguistic hedges. Probably the most popular representation, proposed by Zadeh [33], is a powering technique: given a fuzzy predicate đ?&#x2018;&#x192; (stated a property of objects and represented by a fuzzy set), the modiied term is represented by đ?&#x2018;&#x192; with đ?&#x203A;ź > 1 for intensifying hedges and đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1) for weakening ones. One disadvantage of this approach is that both a kernel and a support of đ?&#x2018;&#x192; are preserved: đ?&#x2018;&#x2DC;đ?&#x2018;&#x2019;đ?&#x2018;&#x;(đ?&#x2018;&#x192;) = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2019;đ?&#x2018;&#x;(đ?&#x2018;&#x192; ) and đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x192;) = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x192; ). However, it seems counterintuitive: if John is 25 years old, he is obviously viewed as young to the degree 1, yet intuitively he is very young up to the lower degree, say 0.9. Moreover, this method is based on technical operations only and does not take into account any inherited meaning from modi ied terms. It is worth noting that linguistic hedges add a special emphasis to adjectives they are applied to. For example, while saying that â&#x20AC;&#x153;George is a very good doctorâ&#x20AC;? one wants to emphasise Georgeâ&#x20AC;&#x2122;s medical quali ications. This conviction may be viewed as an implicit reference to medical quali ications of other doctors. In this sense linguistic hedges have a rel16

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ative lavor and, as such, are in fact modal expressions, like certainly, sometimes, or presumably. Consequently, they are to be modeled in the similar way as modalities, that is using relational methods.

Following this idea we present another representation of linguistic hedges basing on the notion of resemblance. This approach was proposed by De Cock, Radzikowska, and Kerre [5, 6], and by De Cock and Kerre [4]. Intuitively, having a universe đ?&#x2018;&#x2039; of objects, any đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; is threated as resembling itself and, if đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; resembles đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, then also đ?&#x2018;Ś resembles đ?&#x2018;Ľ. This relationship is represented by re lexive and symmetric fuzzy relation (originally de ined on the basis of pseudo-metric spaces, yet in practice it is often assumed that the underlying pseudo-metric is the identity). Transitivity is not required since it may lead to counterintuitive results. For example, a temperature of 0â&#x2C6;&#x2DC; C resembles 1â&#x2C6;&#x2DC; C up to the degree 1, also 1â&#x2C6;&#x2DC; C totally resembles 2â&#x2C6;&#x2DC; C , the same with 10â&#x2C6;&#x2DC; C and 11â&#x2C6;&#x2DC; C, yet 0â&#x2C6;&#x2DC; C resembles 11â&#x2C6;&#x2DC; C to the degree de initely less than 1. Having established a resemblance relation đ?&#x2018;&#x2026; on a universe in discourse, linguistic hedges are modelled by fuzzy necessity (1) and fuzzy possibility (2) operators. Namely, given a fuzzy predicate đ?&#x2018;&#x192;, an intensifying modi ier đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018;, and a weakening modi ier đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018;, we use the following general schemas: đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018;(đ?&#x2018;&#x192;) = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018;(đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;.

For example, if đ?&#x2018;&#x192; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) stands for good, then very good is represented by [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192;, while quite good is represented by â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;. This representation re lects our underlying intuition: đ?&#x2018;Ľ is called very good up to the degree to which all objects resembling đ?&#x2018;Ľ are quali ied as good.

Various intensifying (resp. weakening) modi iers re lect different emphasis on terms they are applied to. Assume that two intensifying modi iers, đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; and đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; , are such that đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; re lects a stronger emphasis than đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; . Then, for any fuzzy predicate đ?&#x2018;&#x192;, đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) â&#x160;&#x2020; đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;).

(7)

The following three schemes are proposed for the representation of the resulting modi ied terms: Scheme 1.i: Take two fuzzy implications, â&#x2020;&#x2019; and â&#x2021;&#x2019;, such that â&#x2020;&#x2019; ⊽ â&#x2021;&#x2019;. Then đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; đ?&#x2018;&#x192;.

Here the inclusion (7) is guaranteed by Property 3.1(c).

Scheme 2.i: Take a t-norm â&#x160;&#x2014; and a fuzzy implication â&#x2020;&#x2019;. Then đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;?) = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192;.

Now, (7) holds due to Corollary 3.1.

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Scheme 3.i: Take two fuzzy implications â&#x2020;&#x2019; and â&#x2021;&#x2019;.

Then đ?&#x2018;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192;

đ?&#x2018;&#x2026;(10, 13 ) â&#x160;&#x2014; đ?&#x2018;&#x2026;(13 , 15 ) ⊽ min(đ?&#x2018;&#x2026;(10, 13 ), đ?&#x2018;&#x2026;(13 , 15 )) = min( , ) = ⊽̸ 0 = đ?&#x2018;&#x2026;(10, 15 ).

Using the above schemes we represent these terms as

Let a fuzzy set đ?&#x2018;&#x160; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), representing a term warm, be given by

extremely đ?&#x2018;&#x192; = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; [đ?&#x2018;&#x2026;] â&#x2021;&#x2019; đ?&#x2018;&#x2026; de initely đ?&#x2018;&#x192; = [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; đ?&#x2018;&#x192;

0

very đ?&#x2018;&#x192; = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; where fuzzy implications â&#x2020;&#x2019; and â&#x2021;&#x2019; satisfy â&#x2021;&#x2019; ⊽ â&#x2020;&#x2019;.

Similarly, for weakening modi iers we have three schemes. Assume that đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; and đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; are two weakening modi iers such that the former re lects weaker emphasis than the latter one. Consequently, for any fuzzy predicate đ?&#x2018;&#x192;,

đ?&#x2018;Ľâ&#x2C6;&#x2019;4

đ?&#x2018;&#x160;(đ?&#x2018;Ľ) = 1

0

(8)

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ?&#x2018;&#x192;

[đ?&#x2018;&#x2026;]

đ?&#x2018;Ľâ&#x2C6;&#x2019;

đ?&#x2018;&#x160;(đ?&#x2018;Ľ) = 1

0 [đ?&#x2018;&#x2026;]

[đ?&#x2018;&#x2026;]

đ?&#x2018;Ľâ&#x2C6;&#x2019;

đ?&#x2018;&#x160;(đ?&#x2018;Ľ) = 1

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;.

(8) is guaranteed by Property 3.1(d).

Scheme 2.w: Take a fuzzy implication â&#x2020;&#x2019; and a t-norm â&#x160;&#x2014;. The

0 đ?&#x2018;Ľâ&#x2C6;&#x2019;

â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160;(đ?&#x2018;Ľ) = 1

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;.

0

(8) holds by Corollary 3.1.

đ?&#x2018;Ľâ&#x2C6;&#x2019;

â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160;(đ?&#x2018;Ľ) = Scheme 3.w: Take two t-norms, â&#x160;&#x2014; and â&#x160;&#x2122;. Then

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;.

Again, (8) is satis ied by re lexivity of đ?&#x2018;&#x2026; and Property 3.3(a).

Assume that for three weakening modi iers: rather, quite, and more or less semantic entailment is such that Then

đ?&#x2018;Ľ â&#x2030;¤ 22 22 < đ?&#x2018;Ľ < 29 đ?&#x2018;Ľ â&#x2030;Ľ 29 đ?&#x2018;Ľ â&#x2030;¤ 24 24 < đ?&#x2018;Ľ < 33 đ?&#x2018;Ľ â&#x2030;Ľ 33

On the other hand, â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160; and â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160; represent more or less warm and rather warm, respectively.

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192;

đ?&#x2018;Ľ â&#x2030;¤ 20 20 < đ?&#x2018;Ś < 25 đ?&#x2018;Ľ â&#x2030;Ľ 25

Now, fuzzy sets ([đ?&#x2018;&#x2026;] đ?&#x2018;&#x160; and [đ?&#x2018;&#x2026;] [đ?&#x2018;&#x2026;] đ?&#x2018;&#x160; given below, represent terms very warm and de initely warm, respectively.

Scheme 1.w: Take two t-norms, â&#x160;&#x2014; and â&#x160;&#x2122;, satisfying â&#x160;&#x2014; ⊽ â&#x160;&#x2122;. Then

rather P â&#x160;&#x2020; quite P â&#x160;&#x2020; more or less P.

|đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;Ś| 2

Note that đ?&#x2018;&#x2026; is re lexive and symmetric, but not sup-â&#x160;&#x2014;transitive for any t-norm â&#x160;&#x2014;. Indeed, we have

extremely đ?&#x2018;&#x192; â&#x160;&#x2020; de initely đ?&#x2018;&#x192; â&#x160;&#x2020; very đ?&#x2018;&#x192;.

đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;) â&#x160;&#x2020; đ?&#x2018;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x153;đ?&#x2018;&#x2018; (đ?&#x2018;&#x192;).

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Example 5.1 Let đ?&#x2018;&#x2039; = [0, +â&#x2C6;&#x17E;) be a universe of temperatures and let a resemblance relation on đ?&#x2018;&#x2039; be given by đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = min 1, max 0 , 2 â&#x2C6;&#x2019;

The inclusion (7) also holds by re lexivity of đ?&#x2018;&#x2026; and Property 3.3(a).

For example, the intuition dictates that from among intensifying modi iers extremely, de initely, and very, the irst one is the strongest modi ier, while the last one is the the weakest one. Hence, for any fuzzy predicate đ?&#x2018;&#x192;, we expect

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đ?&#x2018;Ľ â&#x2030;¤ 18 18 < đ?&#x2018;Ľ < 23 đ?&#x2018;Ľ â&#x2030;Ľ 23 đ?&#x2018;Ľ â&#x2030;¤ 16 16 â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ 23 đ?&#x2018;Ľ â&#x2030;Ľ 23

Note that [đ?&#x2018;&#x2026;]

[đ?&#x2018;&#x2026;]

đ?&#x2018;&#x160; â&#x160;&#x2020; [đ?&#x2018;&#x2026;]

đ?&#x2018;&#x160; â&#x160;&#x2020; đ?&#x2018;&#x160; â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160; â&#x160;&#x2020; â&#x;¨đ?&#x2018;&#x2026;â&#x;Š đ?&#x2018;&#x160;.

These membership functions are depicted in Fig. 1. â&#x2013;Ą Consider now the following three expressions: (E1) x is rather very P; (E2) x is P; (E3) x is de initely quite P.

rather đ?&#x2018;&#x192; = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ?&#x2018;&#x192; quite đ?&#x2018;&#x192; = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ?&#x2018;&#x192; more or less đ?&#x2018;&#x192; = â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ?&#x2018;&#x192; where â&#x160;&#x2014; and â&#x160;&#x2122; are t-norms such that â&#x160;&#x2014; ⊽ â&#x160;&#x2122;.

Note that in (E1) a weaker emphasis is put on the fact represented by very P, but stronger than in the statement (E2) â&#x20AC;&#x201C; consequently, the expressivity in somewhere in between. Similarly, (E3) is a stronger expression that x is quite P, but weaker than (E2), so 17

Journal of Automation, Mobile Robotics & Intelligent Systems

Fig. 1. Membership func ons for linguis c terms (a) Membership func ons for linguis c terms (from le to right): â&#x;¨ â&#x;Š (rather warm), â&#x;¨ â&#x;Š (more or less warm), (warm) 1.0

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(b) Membership func ons for linguis c terms (from le to right): (warm), [ ] (very warm), [ ] [ ] (deďŹ nitely warm)

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6. Concluding Remarks In this paper we have presented two applications of fuzzy modal operators. First, we have shown how these operators may be used for fuzzy set approximations. Basing on the observation that fuzzy set approximations maybe viewed as intuitionistic fuzzy sets, we have presented the application of these operators in the problem of skills matching for selecting research projects. Also, we have pointed out how fuzzy possibility and fuzzy necessity operators can be used for modeling linguistic hedges. This representation is based on the observation that linguistic hedges may be viewed as speci ic kind of modal expressions. The presented approach re lects the contextual meaning of these modi iers which is, in our opinion, intuitively justi ied.

1.0

AUTHOR Anna Maria Radzikowska â&#x20AC;&#x201C; Warsaw University of Technology, Faculty of Mathematics and Information Science, Koszykowa 75, 00-662 Warsaw, Poland, e-mail: A.Radzikowska@mini.pw.edu.pl.

0.8

0.6

0.4

0.2

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REFERENCES its expressive power is intermediate. In our framework the statements (E1) and (E3) can be represented by â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; and [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ?&#x2018;&#x192;, respectively, where â&#x160;&#x2014; (resp. â&#x160;&#x2122;) is a left-continuous t-norm and â&#x2020;&#x2019; (resp. â&#x2021;&#x2019;) is its residual implication. By Corollary 3.1, we have â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x2020;&#x2019; đ?&#x2018;&#x192; â&#x160;&#x2020; đ?&#x2018;&#x192; â&#x160;&#x2020; [đ?&#x2018;&#x2026;]â&#x2021;&#x2019; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2122; đ?&#x2018;&#x192;, which coincides with our intuition.

There is a kind of dualism between some linguistic hedges. Namely, let the following expressions be given: (E4a) rather not P; (E5a) not very P. In particular, if we say that a temperature outside is rather not warm, it obviously cannot be treated as extremely warm, whence rather not P â&#x160;&#x2020; not very P. By Property 3.3(c) this can be modeled by operators (2) for (E4a) and by (1) for (E5a) using a left-continuous t-norm â&#x160;&#x2014;, its residual implication â&#x2020;&#x2019;, and the negation ÂŹ induced by â&#x160;&#x2014;. However, if in some cases rather not P=not very P is required, one can apply Ĺ ukasiewicz connectives.

Similarly, consider the following expressions: (E4b) de initely not P; (E5b) not quite P. For example, if one says that outside is not even quite warm, the intuition dictates that it is de initely not warm, thus not quite P â&#x160;&#x2020; de initely not P. As before, by Property 3.3(c) this case may be supported by choosing a left-continuous t-norm â&#x160;&#x2014;, its residual implication, and the negation induced by â&#x160;&#x2014;. Ĺ ukasiewicz connectives are to be applied whenever equality is desired.

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[2] K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physicaâ&#x20AC;&#x201C;Verlag, 1999, DOI: 10.1007/978-3-7908-1870-3.

[3] M. BaczynĚ ski and B. Jayaram, Fuzzy Implications, Studies in Fuzziness and Soft Computing, Springer-Verlag Berlin Heidelberg, 2008, DOI: 10.1007/978-3-540-69082-5.

[4] M. De Cock and E. E. Kerre, â&#x20AC;&#x153;Fuzzy modi iers based on fuzzy relationsâ&#x20AC;?, Information Sciences, vol. 160, 2004, 173â&#x20AC;&#x201C;199, DOI: 10.1016/j.ins.2003.09.002.

[5] M. De Cock, A. M. Radzikowska, and E. E. Kerre. â&#x20AC;&#x153;Modelling Linguistic Modi iers using FuzzyRough Structuresâ&#x20AC;?. In: Proceedings of 8th International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems IPMU, June 2000, Madrid, Spain, 1381â&#x20AC;&#x201C; 1388. Consejo Superior de Investigaciones Cienti icas, 2000. ISBN 84-95479-02-8.

[6] M. De Cock, A. M. Radzikowska, and E. E. Kerre. â&#x20AC;&#x153;A Fuzzy-Rough Approach to the Representation of Linguistic Hedgesâ&#x20AC;?. In: B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena, and R. R. Yager, eds., Technologies of Constructing Intelligent Systems, volume 2, 33â&#x20AC;&#x201C;43. Physica-Verlag, Heidelberg, New York, 2002. DOI: 10.1007/978-37908-1797-3_3.

[7] S. Demri and E. OrĹ&#x201A;owska, Incomplete Information: Structure, Inference, Complexity, EATCS Monographs in Theoretical Computer Science,

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[15] E. Orłowska, A. M. Radzikowska, and I. Rewitzky, Dualities for Structures of Applied Logics, volume 56 of Mathematical Logic and Foundations, College Publications, 2015, ISBN: 978-84890181-0.

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[22] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzy rough sets”, Fuzzy Sets and Systems, vol. 126, 2002, 137–155, DOI: 10.1016/S0165-0114(01)00032-X. [23] A. M. Radzikowska and E. E. Kerre. “A Fuzzy Generalization of Information Relations”. In: E. Orłowska and M. Fitting, eds., Beyond Two: Theory and Applications of Multiple-Valued Logics, volume 114 of Studies in Fuzziness and Soft Computing, 287–312. Physica-Verlag Heidelberg, 2002. DOI: 10.1007/978-3-7908-1769-0.

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“Algebraic Characterisations of Some Fuzzy Information Relations”. In: O. Hryniewicz, J. Kacprzyk, and D. Kuchta, eds., Soft Computing: Foundations and Theoretical Aspects, 71–86. Akademicka O icyna Wydawnicza EXIT, Warsaw, 2005. ISBN: 83-87674-97-4.

[28] A. M. Radzikowska and E. E. Kerre, “Characterisations of main classes of fuzzy relations using fuzzy modal operators”, Fuzzy Sets and Systems, vol. 152, no. 2, 2005, 223–247, DOI: 10.1016/j.fss.2004.09.005.

[29] A. M. Radzikowska and E. E. Kerre. “Fuzzy Information Relations and Operators: An Algebraic Approach Based on Residuated Lattices”.

In: H. de Swart, E. Orłowska, G. Schmidt, and 19

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[30] E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets”, Fuzzy Sets and Systems, vol. 114, no. 3, 2000, 505–518, DOI: 10.1016/S0165-0114(98)00244-9.

[31] R. Wille. “Restructuring lattice theory: An approach based on hierarchies of concepts”. In: I. Rival, ed., Ordered sets, volume 83 of NATO Advanced Studies Institute, 445–470. Springer Netherlands, 1982. DOI: 0.1007/978-94-0097798-3_15.

[32] L. A. Zadeh, “Fuzzy sets”, Information and Control, vol. 8, 1965, 338–353, DOI: 10.1016/S00199958(65)90241-X. [33] L. A. Zadeh, “A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges”, Journal of Cybernetics, vol. 2:3, 1972, 4–34, DOI: 10.1080/01969727208542910.

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D Submi ed: 13th January 2017; accepted: 16th February 2017

Anna Maria Radzikowska DOI: 10.14313/JAMRIS_1-2017/3 Abstract: In this paper we present an applica on of fuzzy approxima on operators in suppor ng medical diagnosis. These operators are composi ons of fuzzy modal operators. The underlying idea is based on the observa on that approxima ons of fuzzy sets may be viewed as intui onis c fuzzy sets. Reasoning scheme is determined by distances between intui onis c fuzzy sets proposed by Szmidt and Kacprzyk.

Keywords: Fuzzy modal operators, Fuzzy set approximaons, Intui onis c fuzzy sets, Medical diagnosis

1. Introduc on In real-life problems we deal with information which is usually incomplete. The reasons are generally twofold. First, it follows from the fact that only partial data about the problem under consideration can be obtained. Second, the available data are often given in an imprecise form, for example when expressed using linguistic terms like â&#x20AC;&#x153;quite goodâ&#x20AC;? or â&#x20AC;&#x153;rather coldâ&#x20AC;?. Therefore, new information derived from incomplete data is in general uncertain.

In many applications the available information has a form of a set of objects and a set of their properties. Formal methods of analysis of such information were extensively developed within rough set theory (see, for example, Demri and OrĹ&#x201A;owska [5] and OrĹ&#x201A;owska [8]). While descriptions of objects are explicit information, relationships between objects/properties are new data that can be derived and constitute implicit information about domains in discourse. Such relationships are represented by information relations.

A typical example of an information relation is an indiscernibility relation: two objects are indiscernible whenever they have the same selected properties. Approximation techniques, usually based on modal (or modal-like) operators, are applied in reasoning schemes.

Fuzzy set theory, originally introduced by Zadeh [33], offers a variety of methods for representing and processing imprecise (or vague, fuzzy) information.

Therefore, when information of such a kind is admitted, fuzzy generalizations of traditional techniques are to be applied. Fuzzy information relations were widely investigated by Radzikowska and Kerre [19, 25, 27].

Logical systems capable to reason about these relations were considered by Radzikowska [14]. Comprehensive expositions of logical and algebraic aspects of information relations and knowledge and approx-

imation operators were presented by OrĹ&#x201A;owska, Radzikowska, and Rewitzky [9].

Fuzzy sets allow for representation of graded information in the sense that degrees of memberships are given, yet one is unaware to what extend nonmembership refers. For instance, if we know that a patient đ?&#x2018;? suffers from pneumonia up to the degree 0.7, we can only say that this disease is excluded for đ?&#x2018;? at most to the degree 0.3. Atanassov [1, 2] generalized fuzzy sets by providing two parameters for each element of the universe in discourse: the degrees of membership and the degree of non-membership. This allows us for stating that, e.g., đ?&#x2018;? suffers from pneumonia up to the degree 0.7 and pneumonia is excluded for đ?&#x2018;? up to the degree 0.1, thus 0.2 is the hesitation degree i.e., our lack of knowledge. In consequence, we obtain a more lexible tool for representing vague information.

In this paper we present an application of relationbased approximation techniques to medical diagnosis problem. Assume that we are given a set đ?&#x2018;&#x192; of patients, a set đ??ˇ of some diseases, and a set đ?&#x2018;&#x2020; of symptoms of diseases from đ??ˇ. Each patient đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; is characterized by symptoms đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;, and each disease đ?&#x2018;&#x2018; â&#x2C6;&#x2C6; đ??ˇ is described in terms of its symptoms đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;. Our aim is to derive the proper medical diagnosis for each patient. For this purpose it is necessary to use information obtained from medical tests made for patients as well as medical knowledge about diseases. Medical knowledge actually occurs in two forms: as an explicit information given in the form of descriptions of diseases, and implicit knowledge that can be derived from these descriptions. Our methodology involves approximation methods that allow us to determine to what extend particular patient (resp. disease) at least and at most shows (resp. is characterized by) particular symptoms. Having applied these techniques we determine two intuitionistic fuzzy relations representing descriptions of patients and diseases, respectively, in terms of their symptoms. Following the idea proposed by Szmidt and Kacprzyk [29, 32], medical diagnosis is determined by distances between intuitionistic fuzzy sets.

The paper is organized as follows. In Section 2 we recall basic notions of fuzzy sets, fuzzy relations, fuzzy logical connectives and intuitionistic fuzzy sets. Next, in Section 3, we present fuzzy approximation operators based on fuzzy relations. An application of these operations for supporting medical diagnosis is discussed in Section 4. Concluding remarks complete the paper.

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2. Preliminaries

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â&#x20AC;&#x201C; the Kleene-Dienes implication

In this section we recall basic notions of fuzzy set theory which are used in our presentation.

2.1. Fuzzy Sets Let đ?&#x2018;&#x2039; be a non-empty domain. A fuzzy set in đ?&#x2018;&#x2039; is any mapping đ??š â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; [0, 1]. For every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ??š(đ?&#x2018;Ľ) is the degree to which đ?&#x2018;Ľ belongs to đ??š. Given two fuzzy set đ??´, đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;), â&#x20AC;&#x201C; đ??´ is (totally) included in đ??ľ, written đ??´ â&#x160;&#x2020; đ??ľ, if đ??´(đ?&#x2018;Ľ) ⊽ đ??ľ(đ?&#x2018;Ľ) for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;; â&#x20AC;&#x201C; đ??´ is (totally) equal to đ??ľ, written đ??´ = đ??ľ, if đ??´(đ?&#x2018;Ľ) = đ??ľ(đ?&#x2018;Ľ) for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;.

The family of all fuzzy sets in đ?&#x2018;&#x2039; will be denoted by â&#x201E;ą(đ?&#x2018;&#x2039;).

A fuzzy relation in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; is a fuzzy set in đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x152;.

For đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and for đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ľ is đ?&#x2018;&#x2026;-related with đ?&#x2018;Ś. A fuzzy relation đ?&#x2018;&#x2026; in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; is called crisp if đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2C6;&#x2C6; {0, 1} for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and for all đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;. The family of all fuzzy relations in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; will be written â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;). For đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), the converse relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x152;, đ?&#x2018;&#x2039;) is de ined as đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś). For every đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, we write đ?&#x2018;Ľđ?&#x2018;&#x2026; to denote the fuzzy set in đ?&#x2018;&#x152; de ined as (đ?&#x2018;Ľđ?&#x2018;&#x2026;)(đ?&#x2018;Ś) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś).

Analogously, for any đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x2018;&#x2026;đ?&#x2018;Ś â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) is de ined as (đ?&#x2018;&#x2026;đ?&#x2018;Ś)(đ?&#x2018;Ľ) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś). A fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x2039;) is a fuzzy relation on đ?&#x2018;&#x2039;.

2.2. Fuzzy Logical Connec ves Fuzzy logical connectives are generalizations of logical connectives of classical logic. Triangular norms generalize classical conjunction. Speci ically, a triangular norm (t-norm, for short) is a mapping â&#x160;&#x2014; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1], commutative (đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś = đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;Ľ, đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1]), associative (đ?&#x2018;Ľ â&#x160;&#x2014; (đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;§) = (đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś) â&#x160;&#x2014; đ?&#x2018;§, đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1]), increasing in both arguments (đ?&#x2018;Ľ ⊽ đ?&#x2018;§ implies đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;Ś ⊽ đ?&#x2018;§ â&#x160;&#x2014; đ?&#x2018;Ś and đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;Ľ ⊽ đ?&#x2018;Ś â&#x160;&#x2014; đ?&#x2018;§, đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1]), and satisfying the boundary condition đ?&#x2018;Ľ â&#x160;&#x2014; 1 = đ?&#x2018;Ľ for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1]. The most popular t-norms are: â&#x20AC;&#x201C; the standard t-norm (the largest t-norm) đ?&#x2018;Ľâ&#x160;&#x2014; đ?&#x2018;Ś = min(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x20AC;&#x201C; the product operation đ?&#x2018;Ľâ&#x160;&#x2014; đ?&#x2018;Ś = đ?&#x2018;Ľđ?&#x2018;Ś â&#x20AC;&#x201C; the Ĺ ukasiewicz t-norm đ?&#x2018;Ľâ&#x160;&#x2014; đ?&#x2018;Ś = max(0, đ?&#x2018;Ľ + đ?&#x2018;Ś â&#x2C6;&#x2019; 1).

A t-norm â&#x160;&#x2014; is left-continuous whenever it is leftcontinuous on both arguments. For the extended studies on t-norms we refer a reader to Klement, Mesiar and Pap [7].

A fuzzy implication is a [0, 1] â&#x2C6;&#x2019; [0, 1] map â&#x2020;&#x2019; with decreasing 1 and increasing 2 partial mappings (đ?&#x2018;Ľ ⊽ đ?&#x2018;§ implies đ?&#x2018;§ â&#x2020;&#x2019; đ?&#x2018;Ś ⊽ đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś and đ?&#x2018;Ś â&#x2020;&#x2019; đ?&#x2018;Ľ ⊽ đ?&#x2018;Ś â&#x2020;&#x2019; đ?&#x2018;§ for all đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1]) and satisfying 1 â&#x2020;&#x2019; 1 = 0 â&#x2020;&#x2019; 0 = 0 â&#x2020;&#x2019; 1 = 1 and 1 â&#x2020;&#x2019; 0 = 0. The most popular fuzzy implications are 22

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đ?&#x2018;Ľâ&#x2020;&#x2019;

đ?&#x2018;Ś = max(1 â&#x2C6;&#x2019; đ?&#x2018;Ľ, đ?&#x2018;Ś)

â&#x20AC;&#x201C; the Ĺ ukasiewicz implication đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś = min(1, 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ + đ?&#x2018;Ś) â&#x20AC;&#x201C; the GoĚ&#x2C6; del implication đ?&#x2018;Ľâ&#x2020;&#x2019; đ?&#x2018;Ś =

1 đ?&#x2018;Ś

for đ?&#x2018;Ľ ⊽ đ?&#x2018;Ś elsewhere.

A special class of fuzzy implications are residual implications: given a left-continuous t-norm â&#x160;&#x2014; its residual implication (also called the residuum of â&#x160;&#x2014;) is de ined for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;Ľ â&#x2020;&#x2019; đ?&#x2018;Ś = sup{đ?&#x2018;§ â&#x2C6;&#x2C6; [0, 1] â&#x2C6;ś đ?&#x2018;Ľ â&#x160;&#x2014; đ?&#x2018;§ ⊽ đ?&#x2018;Ś}.

The Ĺ ukasiewicz and the GoĚ&#x2C6; del implications are examples of residual implications based on â&#x160;&#x2014; and â&#x160;&#x2014; , respectively, while Kleene-Dienes implication is not a residual one. Fuzzy implications were extensively investigated by BaczynĚ ski and Jayaram [3].

A fuzzy negation is a mapping ÂŹ â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1], decreasing and satisfying ÂŹ0 = 1 and ÂŹ1 = 0. The standard fuzzy negation is ÂŹ đ?&#x2018;Ľ = 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1]. Residual implications lead to fuzzy negations: ÂŹđ?&#x2018;Ľ = đ?&#x2018;Ľ â&#x2020;&#x2019; 0. The Ĺ ukasiewicz implication induces the standard fuzzy negation, that is ÂŹ = ÂŹ , and the GoĚ&#x2C6; del implication induces the fuzzy negation ÂŹ đ?&#x2018;Ľ = 0 for đ?&#x2018;Ľ â&#x2030; 1 and ÂŹ 1 = 0.

Given a fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) and a fuzzy negation ÂŹ, we write ÂŹđ??´ to denote the ÂŹ-complementation of đ??´, that is the fuzzy set in đ?&#x2018;&#x2039; de ined for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, (ÂŹđ??´)(đ?&#x2018;Ľ) = ÂŹđ??´(đ?&#x2018;Ľ).

2.3. Intui onis c Fuzzy Sets Now, let us recall basic notions of intuitionistic fuzzy set theory (see Atanassov [1]). Let a nonempty domain đ?&#x2018;&#x2039; be given. An intuitionistic fuzzy set in đ?&#x2018;&#x2039; is given by đ??´ = {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;}, where đ?&#x153;&#x2021; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) and đ?&#x153;&#x2C6; â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) are called a membership and a non-membership function, respectively, and satisfy đ?&#x153;&#x2021; (đ?&#x2018;Ľ)+đ?&#x153;&#x2C6; (đ?&#x2018;Ľ) ⊽ 1 for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. The value đ?&#x153;&#x2039; (đ?&#x2018;Ľ) = 1â&#x2C6;&#x2019; đ?&#x153;&#x2021; (đ?&#x2018;Ľ)â&#x2C6;&#x2019;đ?&#x153;&#x2C6; (đ?&#x2018;Ľ), đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, is called a hesitation margin which re lects the lack of knowledge of membership or nonmembership of đ?&#x2018;Ľ to đ??´. The family of all intuitionistic fuzzy sets in đ?&#x2018;&#x2039; will be denoted by â&#x201E;?â&#x201E;ą(đ?&#x2018;&#x2039;).

Clearly, any fuzzy sets đ??´ in đ?&#x2018;&#x2039; is a speci ic intuitionistic fuzzy set đ??´ = {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), 1 â&#x2C6;&#x2019; đ?&#x153;&#x2021; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;}.

An intuitionistic fuzzy relation in đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; is an intuitionistic fuzzy set in đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x152;, i.e., it is given by đ?&#x2018;&#x2026; = {((đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x153;&#x2021; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;} with đ?&#x153;&#x2021; , đ?&#x153;&#x2C6; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) satisfying đ?&#x153;&#x2021; (đ?&#x2018;Ľ, đ?&#x2018;Ś) + đ?&#x153;&#x2C6; (đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2030;¤ 1 for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;. Accordingly, đ?&#x153;&#x2021; (đ?&#x2018;Ľ, đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; is đ?&#x2018;&#x2026;â&#x20AC;&#x201C;related with đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x153;&#x2C6; (đ?&#x2018;Ľ, đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ľ and đ?&#x2018;Ś are not đ?&#x2018;&#x2026;-related, and đ?&#x153;&#x2039; (đ?&#x2018;Ľ, đ?&#x2018;Ś) = 1â&#x2C6;&#x2019;đ?&#x153;&#x2021; (đ?&#x2018;Ľ, đ?&#x2018;Ś)â&#x2C6;&#x2019;đ?&#x153;&#x2C6; (đ?&#x2018;Ľ, đ?&#x2018;Ś) is a hesitation margin.

For the extensive studies of the theory of intuitionistic fuzzy sets we refer to [2].

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Traditionally, a distance between two intuitionistic fuzzy sets in đ?&#x2018;&#x2039; is de ined with respect to two parameters, that is the degrees of membership and the degrees of non-membership. The drawback of this approach was pointed out by Szmidt and Kacprzyk in [29] and a novel de inition was proposed where all three parameters, that is including hesitation regions, are taken into account. More speci ically, let đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ } and let đ??´ = {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} and đ??ľ = {(đ?&#x2018;Ľ, đ?&#x153;&#x2021; (đ?&#x2018;Ľ), đ?&#x153;&#x2C6; (đ?&#x2018;Ľ)) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} be two intuitionistic fuzzy sets in đ?&#x2018;&#x2039;. Then â&#x20AC;&#x201C; the normalized Hamming distance between đ??´ and đ??ľ: đ?&#x203A;ż (đ??´, đ??ľ) =

1 2đ?&#x2018;&#x203A;

đ?&#x2018;&#x2018;

(1)

where |

( )| + |

( )

( )| + |

( )

( )

( )|.

â&#x20AC;&#x201C; the normalized Euclidean distance between đ??´ and đ??ľ:

đ?&#x203A;ż (đ??´, đ??ľ) =

1 2đ?&#x2018;&#x203A;

đ?&#x2018;&#x2019;

(2)

where (

( )

( )) (

( )

( )) (

( )

( )) .

3. Rela on-based Fuzzy Set Approxima ons Let đ?&#x2018;&#x2039; and đ?&#x2018;&#x152; be two non-empty universes and let đ?&#x2018;&#x2026; â&#x160;&#x2020; đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x152; be a relation on đ?&#x2018;&#x2039; and đ?&#x2018;&#x152;. Intuitively, đ?&#x2018;&#x2039; may be viewed as a set of objects, đ?&#x2018;&#x152; is treated as a set of their properties, and for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; and for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, đ?&#x2018;Ľđ?&#x2018;&#x2026;đ?&#x2018;Ś states that an object đ?&#x2018;Ľ has the property đ?&#x2018;Ś.

Note that any set đ??´ â&#x160;&#x2020; đ?&#x2018;&#x2039; may be viewed as a representation of an expert decision concerning objects from đ?&#x2018;&#x2039;, or as a representation of some feature (not necessarily from đ?&#x2018;&#x152;) characterizing particular objects from the set đ?&#x2018;&#x2039;. Analogously, any set đ??ľ â&#x160;&#x2020; đ?&#x2018;&#x152; may represent characterization of some object (not necessarily from đ?&#x2018;&#x2039;) in terms of properties from the set đ?&#x2018;&#x152;.

Any relation đ?&#x2018;&#x2026; â&#x160;&#x2020; đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x152; allows us to derive some implicit information about objects from đ?&#x2018;&#x2039;, and properties from đ?&#x2018;&#x152;. Speci ically, we can infer about links between objects (resp. properties) basing on their properties (resp. objects having these properties). In the terminology well-known in rough set theory (see, e.g., Demri and OrĹ&#x201A;owska [5]) these links are formalized by information relations. Here let us recall two of such relations:

â&#x20AC;˘ for objects: objects đ?&#x2018;Ľ and đ?&#x2018;Ľ are compatible if they share some common property đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;; â&#x20AC;˘ for properties: properties đ?&#x2018;Ś and đ?&#x2018;Ś are compatible if some object đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; has both properties.

2017

â&#x20AC;&#x201C; relevance (also called inclusion, or forward inclusion) â&#x20AC;˘ for objects: an object đ?&#x2018;Ľ is relevant to an object đ?&#x2018;Ľ if all properties of đ?&#x2018;Ľ are also properties of đ?&#x2018;Ľ ; â&#x20AC;˘ for properties: a property đ?&#x2018;Ś is relevant to a property đ?&#x2018;Ś if all objects having the property đ?&#x2018;Ś have also the property đ?&#x2018;Ś .

In the following we will not indicate directly whether compatibility (resp. relevance) refers to objects or properties since it will clearly follow from the context they are used in.

Example 3.1 Let us consider a set đ?&#x2018;&#x192; of four people: Al, Bob, Joe, and Ted and a set đ?&#x2018;&#x2020; of ive symptoms of diseases these patients suffer from: Temperature, Headache, Stomach pain, Cough, and Chest pain. Tab. 1 represents characterization of the patients in terms of their symptoms given by a binary (crisp) relation.

Note that Al and Joe are compatible since they both have temperature, while Temperature and Headache are compatible since Al shows both symptoms. Moreover, Joe is relevant to Al and Cough is relevant to Temperature. â&#x2014;ť Information, as given in Example 3.1, although sometimes useful, in many real-life problems is practically meaningless. In particular, it is unknown how strong Alâ&#x20AC;&#x2122;s headache is, whether indeed nobody shows chest pain, or may be some patients suffer from a very slight one, etc. If medical diagnosis is to be determined, we essentially need to know to what extend patients show particular symptoms. These leads us to fuzzy structures which are commonly used for representation of graded information. In Tab. 2 a fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x192;, đ?&#x2018;&#x2020;) shows to what degree particular patients show speci ic symptoms.

When imprecise data are involved, we actually have fuzzy information relations, in particular a fuzzy compatibility and a fuzzy relevance. Fuzzy information relations were extensively investigated by Radzikowska and Kerre [16, 17, 19, 20, 22, 25].

In order to infer about relationships between object/properties in the environment of fuzzy information, fuzzy modal operators are useful. These operators were investigated and widely discussed by Radzikowska and Kerre [10â&#x20AC;&#x201C;12, 24, 26, 27]. Let us recall some basic facts. Given a t-norm â&#x160;&#x2014; and its residual implication â&#x2020;&#x2019;, the following two â&#x201E;ą(đ?&#x2018;&#x152;) â&#x2C6;&#x2019; â&#x201E;ą(đ?&#x2018;&#x2039;) operators are de ined for every fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), for any fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), and for every đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, ([đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??´)(đ?&#x2018;Ľ) = inf (đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2020;&#x2019; đ??´(đ?&#x2018;Ś))

(3)

(â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´)(đ?&#x2018;Ľ) = sup(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x160;&#x2014; đ??´(đ?&#x2018;Ś))

(4)

â&#x2C6;&#x2C6;

â&#x20AC;&#x201C; compatibility:

Nâ&#x2C6;&#x2DC; 1

â&#x2C6;&#x2C6;

The operators (3) and (4) are called fuzzy necessity and fuzzy possibility, respectively. Assume that đ?&#x2018;&#x2026; is a fuzzy relation on đ?&#x2018;&#x2039;, for instance, a fuzzy similarity relation which re lects similarities of object determined 23

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Nâ&#x2C6;&#x2DC; 1

2017

Tab. 1. Pa ents and their symptoms Temperature 1 0 1 1

Al Bob Joe Ted

Headache 1 0 1 0

Stomach pain 0 1 0 0

Cough 1 0 0 1

Chest pain 0 0 0 0

Tab. 2. Symptoms characteris c for the pa ents considered đ?&#x2018;&#x2026;

Temperature

Headache

Stomach pain

Cough

Chest pain

Al Bob Joe Ted

0.8 0.0 0.8 0.6

0.6 0.4 0.8 0.5

0.2 0.6 0.0 0.3

0.6 0.1 0.2 0.7

0.1 0.1 0.0 0.3

by their properties. Then (3) and (4) are fuzzy lower and fuzzy upper rough approximation operators extensively studied by Radzikowska and Kerre [13, 18, 21, 23].

The intuitive meaning of (3) and (4) is the following: given đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) representing characterization of some object, and đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, â&#x20AC;&#x201C; ([đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??´)(đ?&#x2018;Ľ) is the degree to which the object đ?&#x2018;Ľ is relevant to đ??´; â&#x20AC;&#x201C; (â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´)(đ?&#x2018;Ľ) is the degree to which the object đ?&#x2018;Ľ is compatible with đ??´.

Analogously, taking đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x2039;) representing an expert decision, for any đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, ([đ?&#x2018;&#x2026; ]â&#x160;&#x2014; đ??ľ)(đ?&#x2018;Ś) is the degree to which the property đ?&#x2018;Ś is relevant to the decision đ??ľ and (â&#x;¨đ?&#x2018;&#x2026; â&#x;Šâ&#x160;&#x2014; đ??ľ)(đ?&#x2018;Ś) is the degree to which đ?&#x2018;Ś is compatible with đ??ľ. Now, basing on the above operators let us deine two mappings â&#x2013;˛â&#x160;&#x2014; , â&#x2013;źâ&#x160;&#x2014; â&#x2C6;ś â&#x201E;ą(đ?&#x2018;&#x152;) â&#x2020;&#x2019; â&#x201E;ą(đ?&#x2018;&#x152;) for every đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), â&#x2013;˛â&#x160;&#x2014; đ??´ = â&#x;¨đ?&#x2018;&#x2026;

â&#x;Šâ&#x160;&#x2014; [đ?&#x2018;&#x2026;]â&#x160;&#x2014; đ??´

(5)

â&#x2013;źâ&#x160;&#x2014; đ??´ = [đ?&#x2018;&#x2026;

]â&#x160;&#x2014; â&#x;¨đ?&#x2018;&#x2026;â&#x;Šâ&#x160;&#x2014; đ??´.

(6)

These operators are fuzzy generalizations of the respective operators investigated by DuĚ&#x2C6; ntsch and Gediga in [6]. Intuitively, for any đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and for any đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, â&#x20AC;&#x201C; (â&#x2013;˛â&#x160;&#x2014; đ??´)(đ?&#x2018;Ś) is the degree to which some object characterized by the property đ?&#x2018;Ś is relevant to the object đ??´; â&#x20AC;&#x201C; (â&#x2013;źâ&#x160;&#x2014; đ??´)(đ?&#x2018;Ľ) is the degree to which all objects characterized by the property đ?&#x2018;Ś are compatible with the object đ??´.

Radzikowska [12] showed that for every đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and for all đ??´, đ??ľ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), (P1) đ??´ â&#x160;&#x2020; đ??ľ implies â&#x2013;˛â&#x160;&#x2014; đ??´ â&#x160;&#x2020; â&#x2013;˛â&#x160;&#x2014; đ??ľ and â&#x2013;źâ&#x160;&#x2014; đ??´ â&#x160;&#x2020; â&#x2013;źâ&#x160;&#x2014; đ??ľ, (P2) â&#x2013;˛â&#x160;&#x2014; đ??´ â&#x160;&#x2020; đ??´ â&#x160;&#x2020; â&#x2013;źâ&#x160;&#x2014; đ??´, (P1) states that both operators are monotone and, due to (P2), they work as approximation operators: 24

â&#x2013;˛â&#x160;&#x2014; đ??´ is a lower bound of đ??´, whereas â&#x2013;źâ&#x160;&#x2014; đ??´ is an upper bound of đ??´. Therefore, for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, (â&#x2013;˛â&#x160;&#x2014; đ??´)(đ?&#x2018;Ś) can be viewed as the degree to which đ?&#x2018;Ś at least (certainly) belongs to đ??´, and (â&#x2013;źâ&#x160;&#x2014; đ??´)(đ?&#x2018;Ś) can be interpreted as the degree to which đ?&#x2018;Ś at most (possibly) belongs to đ??´. Hence, the value ÂŹâ&#x160;&#x2014; â&#x2013;źâ&#x160;&#x2014; đ??´(đ?&#x2018;Ś), where ÂŹâ&#x160;&#x2014; is the fuzzy negation induced by â&#x160;&#x2014;, is the degree to which đ?&#x2018;Ś certainly does not belong to đ??´. For any đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;), the pair (â&#x2013;˛â&#x160;&#x2014; đ??´), â&#x2013;źâ&#x160;&#x2014; đ??´) will be referred to (â&#x2013;˛â&#x160;&#x2014; , â&#x2013;źâ&#x160;&#x2014; )approximation of đ??´ with respect to đ?&#x2018;&#x2026; and â&#x160;&#x2014;.

Remark 3.1 1) Note that for any đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;, we have â&#x2013;˛â&#x160;&#x2014; (đ?&#x2018;Ľđ?&#x2018;&#x2026;) = đ?&#x2018;Ľđ?&#x2018;&#x2026;. In general, however, â&#x2013;źâ&#x160;&#x2014; (đ?&#x2018;Ľđ?&#x2018;&#x2026;) â&#x2030; đ?&#x2018;Ľđ?&#x2018;&#x2026;.

2) Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) and let đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) be given. For an arbitrary left-continuous t-norm â&#x160;&#x2014;, â&#x2013;˛â&#x160;&#x2014; đ??´(đ?&#x2018;Ś) + ÂŹâ&#x160;&#x2014; â&#x2013;źâ&#x160;&#x2014; đ??´(đ?&#x2018;Ś) â&#x2030;° 1.

Let đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) and let â&#x2013;˛ đ??´ and â&#x2013;ź đ??´ be its lower and upper bounds, respectively, determined by the Ĺ ukasiewicz t-norm â&#x160;&#x2014; . Since the negation ÂŹ induced by the Ĺ ukasiewicz t-norm is the standard fuzzy negation ÂŹ , the following condition holds for every đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;: (â&#x2013;˛ đ??´)(đ?&#x2018;Ś) + (ÂŹ â&#x2013;ź đ??´)(đ?&#x2018;Ś) ⊽ 1.

Consequently, the (â&#x2013;˛ , â&#x2013;ź )-approximation of đ??´ uniquely determines an intuitionistic fuzzy set in đ?&#x2018;&#x152; as the following observation states.

Observation 3.1 For every fuzzy relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;), the (â&#x2013;˛ , â&#x2013;ź )-approximation of any fuzzy set đ??´ â&#x2C6;&#x2C6; â&#x201E;ą(đ?&#x2018;&#x152;) determines an intuitionistic fuzzy set đ??´ = {(đ?&#x2018;Ś, đ?&#x153;&#x2021; (đ?&#x2018;Ś), đ?&#x153;&#x2C6; (đ?&#x2018;Ś)) â&#x2C6;ś đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x152;} in đ?&#x2018;&#x152; given by: đ?&#x153;&#x2021; (đ?&#x2018;Ś) = â&#x2013;˛ đ??´(đ?&#x2018;Ś) đ?&#x153;&#x2C6; (đ?&#x2018;Ś) = 1 â&#x2C6;&#x2019; â&#x2013;ź đ??´(đ?&#x2018;Ś).

This idea is the basis for the medical diagnosis problem presented in the next section.

Finally, it is worth noting that Ĺ ukasiewicz logical connectives are very useful in fuzzy generalizations of many structures. Radzikowska and Kerre [18] showed that these fuzzy connectives are the best ones for fuzzy generalization of traditional (crisp) rough sets.

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4. Medical Diagnosis Using Fuzzy Rela onbased Approxima ons Let us consider a set đ?&#x2018;&#x192; of patients, a set đ?&#x2018;&#x2020; of symptoms of some diseases, and a set đ??ˇ of medical diagnosis. On the basis of medical knowledge each diagnosis is characterized by particular symptoms. Also, having made some medical tests each patient is described by symptoms he shows. Our aim is to determine a proper diagnosis for each patient.

As noted before, a fuzzy approach is highly justi ied for this problem. Szmidt and Kacprzyk [30, 32] assumed that a given information is represented by intuitionistic fuzzy relations. Similar representation was earlier given by De, Biswas, and Roy [4]. In these approaches it is required to know to what extend particular symptoms characterize given diagnosis (resp. patients) as well as to what extent symptoms they do not characterize given diagnosis (resp. patients). Here we assume that the available information is more restricted: all we know about symptoms is to what extent they characterize diagnosis (resp. patients). Then we have two fuzzy relations: a relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x192;, đ?&#x2018;&#x2020;) which provides descriptions of particular patients in terms of symptoms they show and a relation đ?&#x2018;&#x201E; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ??ˇ, đ?&#x2018;&#x2020;) which describes particular diagnosis in terms of their characteristic symptoms. For a set đ?&#x2018;&#x192; of patients and a set đ?&#x2018;&#x2020; of symptoms given in Section 3, and a set đ??ˇ = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem} of diagnosis, examples of fuzzy relation đ?&#x2018;&#x2026; and đ?&#x2018;&#x201E; are presented in Tab. 2 and Tab. 3, respectively. Similar data were presented by Szmidt and Kacprzyk [30] and De, Biswas, and Roy [4].

Clearly, in order to make a proper diagnosis one needs the possibly broadest medical knowledge. Note that a relation đ?&#x2018;&#x201E; represents only partial medical knowledge which may be referred to as explicit knowledge.

New information that are derived from đ?&#x2018;&#x201E; constitutes an implicit knowledge which should be taken into account in the process of medical diagnosing.

The simplest solution of our problem is based on distances of fuzzy sets. For each patient đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; we consider a distance between his/her description đ?&#x2018;?đ?&#x2018;&#x2026; and characterization of particular diagnosis đ?&#x2018;&#x2018;đ?&#x2018;&#x201E;. The proper diagnosis is pointed out by the shortest distance.

However, this method does not take into account implicit medical knowledge that can be derived from a relation đ?&#x2018;&#x201E; which, in turn, may lead to highly doubtful results. Another solution, originally proposed by Sanchez ( [28] and later on developed by De, Biswas, and Roy [4]), is based on a composition of fuzzy relations đ?&#x2018;&#x2026; and đ?&#x2018;&#x201E;. Namely, we determine the fuzzy relation đ?&#x2018;&#x2021; = đ?&#x2018;&#x2026; â&#x2C6;&#x2DC; đ?&#x2018;&#x201E; â&#x2C6;&#x2C6; â&#x201E;&#x203A;(đ?&#x2018;&#x192;, đ??ˇ) de ined for every đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; and for every đ?&#x2018;&#x2018; â&#x2C6;&#x2C6; đ??ˇ, đ?&#x2018;&#x2021;(đ?&#x2018;?, đ?&#x2018;&#x2018;) = sup min(đ?&#x2018;&#x2026;(đ?&#x2018;?, đ?&#x2018; ), đ?&#x2018;&#x201E;

(đ?&#x2018;&#x2018;, đ?&#x2018; )).

â&#x2C6;&#x2C6;

In the context of intuitionistic fuzzy sets De, Biswas, and Roy [4], as well as Szmidt and Kacprzyk [32], pointed out that this method has an essential drawback since it prefers dominating symptoms which could make the diagnosis incorrect.

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Here we present another approach using approximation operators (5) and (6) with the underlying relation đ?&#x2018;&#x201E;. This way the approximations are determined by medical knowledge, both explicit and implicit. Firstly, for each patient đ?&#x2018;? we approximate his/her characterization in terms of symptoms, i.e., the fuzzy set đ?&#x2018;?đ?&#x2018;&#x2026;. Next, in analogous way each diagnosis description đ?&#x2018;&#x201E;đ?&#x2018;&#x2018; is approximated. Using Observation 3.1, two IF-relations are then obtained: patient-symptoms relation and symptom-diagnosis relation. Finally, following the idea proposed by Szmidt and Kacprzyk [30â&#x20AC;&#x201C; 32], we calculate distances between the IF-set style description of patients and the IF-set style description of diagnosis. For each patient the shortest distance points out his proper medical diagnosis.

Concretely, as in Radzikowska [15], the following procedure is proceeded.

Step 1: For each patient đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;, approximate his symptoms with respect to the diagnosis-symptom relation đ?&#x2018;&#x201E;: determine the (â&#x2013;˛ , â&#x2013;ź )-approximation of đ?&#x2018;?đ?&#x2018;&#x2026;.

Step 2: For each diagnose đ?&#x2018;&#x2018; â&#x2C6;&#x2C6; đ??ˇ, approximate its symptoms with respect to the diagnosissymptom characterization đ?&#x2018;&#x201E;: calculate (â&#x2013;˛ , â&#x2013;ź )approximation of đ?&#x2018;&#x2018;đ?&#x2018;&#x201E;.

Step 3: Determine IF-relations đ?&#x2018;&#x2026; (patientâ&#x20AC;&#x201C;symptom) and đ?&#x2018;&#x201E; (diagnosis-symptom) on the basis of approximations obtained in Step 1 and 2: calculate đ?&#x153;&#x2021; (đ?&#x2018;?, đ?&#x2018; ) =â&#x2013;˛ (đ?&#x2018;?đ?&#x2018;&#x2026;) đ?&#x153;&#x2C6; (đ?&#x2018;?, đ?&#x2018; ) =1 â&#x2C6;&#x2019; â&#x2013;ź (đ?&#x2018;?đ?&#x2018;&#x2026;) đ?&#x153;&#x2039; (đ?&#x2018;?, đ?&#x2018; ) =â&#x2013;ź (đ?&#x2018;?đ?&#x2018;&#x2026;) â&#x2C6;&#x2019; â&#x2013;˛ (đ?&#x2018;?đ?&#x2018;&#x2026;) and đ?&#x153;&#x2021; (đ?&#x2018;&#x2018;, đ?&#x2018; ) =â&#x2013;˛ (đ?&#x2018; đ?&#x2018;&#x201E;) đ?&#x153;&#x2C6; (đ?&#x2018;&#x2018;, đ?&#x2018; ) =1 â&#x2C6;&#x2019; â&#x2013;ź (đ?&#x2018; đ?&#x2018;&#x201E;) đ?&#x153;&#x2039; (đ?&#x2018;&#x2018;, đ?&#x2018; ) =â&#x2013;ź (đ?&#x2018; đ?&#x2018;&#x201E;) â&#x2C6;&#x2019; â&#x2013;˛ (đ?&#x2018; đ?&#x2018;&#x201E;), respectively.

Step 4: For each đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; and for each diagnosis đ?&#x2018;&#x2018; â&#x2C6;&#x2C6; đ??ˇ, calculate distance between đ?&#x2018;?đ?&#x2018;&#x2026; and đ?&#x2018;&#x2018;đ?&#x2018;&#x201E; â&#x20AC;&#x201C; the lowest distance points out the proper diagnosis.

Tab. 4 presents lower and upper bounds of đ?&#x2018;?đ?&#x2018;&#x2026; for every patient đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192;. Note that due to medical tests Al suffers from cough up to the degree 0.6 (see Tab. 2).

Having taken into account his other symptoms and medical knowledge about speci icity of cough for particular diseases (Tab. 3), it was estimated that Al has cough at least to the degree 0.6 and at most to the degree 0.8. Using linguistic terms one can say that his cough is estimated between rather strong and strong.

Also, it turns out that his headache and stomach pain totally coincide with what was established by medical tests (Tab. 4), that is, he shows both symptoms at least and at most up to the same degree.

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Tab. 3. Symptoms characteris c for the diagnosis considered đ?&#x2018;&#x201E; Viral fever Malaria Typhoid Stomach problem Chest problem

Temperature 0.4 0.7 0.3 0.1 0.1

Headache 0.3 0.1 0.6 0.2 0.0

stomach pain 0.1 0.0 0.2 0.8 0.2

Cough 0.4 0.7 0.2 0.2 0.2

Chest pain 0.1 0.1 0.1 0.2 0.8

Tab. 4. Approximated symptoms characteris c for the pa ents

Al Bob Joe Ted

Temperature

Headache

Stomach pain

Cough

Chest pain

(0.6,0.8) (0.0,0.3) (0.2,0.8) (0.6,0.7)

(0.6,0.6) (0.3,0.4) (0.4,0.8) (0.5,0.5)

(0.2,0.2) (0.6,0.8) (0.0,0.2) (0.3,0.3)

(0.6,0.8) (0.0,0.3) (0.2,0.8) (0.6,0.7)

(0.1,0.2) (0.1,0.2) (0.0,0.2) (0.3,0.3)

Tab. 5. Approximated symptoms characteris c for the diagnosis Temperature

Headache

Stomach pain

Cough

Chest pain

(0.4,0.4) (0.7,0.7) (0.3,0.3) (0.1,0.3) (0.1,0.3)

(0.3,0.4) (0.2,0.4) (0.6,0.7) (0.2,0.4) (0.0,0.4)

(0.1,0.2) (0.6,0.7) (0.2,0.2) (0.8,0.8) (0.2,0.2)

(0.4,0.4) (0.0,0.2) (0.2,0.3) (0.2,0.3) (0.2,0.3)

(0.1,0.2) (0.7,0.7) (0.1,0.2) (0.2,0.2) (0.8,0.8)

Viral fever Malaria Typhoid Stomach problem Chest problem

Tab. 6. Pa ent-symptom intui onis c fuzzy rela on đ?&#x2018;&#x2026;

Temperature

Headache

Stomach pain

Cough

Chest pain

Al Bob Joe Ted

(0.6,0.2,0.2) (0.0,0.7,0.3) (0.2,0.2,0.6) (0.6,0.3,0.1)

(0.6,0.4,0,0) (0.3,0.6,0.1) (0.4,0.2,0.4) (0.5,0.5,0.0)

(0.2,0.8,0.0) (0.6,0.2,0.2) (0.0,0.8,0.2) (0.3,0.7.0.0)

(0.6,0.2,0.2) (0.0,0.7,0.3) (0.2,0.2,0.6) (0.6,0.3,0.1)

(0.1,0.8,0,1) (0.1,0.8,0.1) (0.0,0.8,0.2) (0.3,0.7,0.0)

Tab. 7. Diagnosis-symptom intui onis c fuzzy rela on đ?&#x2018;&#x201E;

Temperature

Headache

Stomach pain

Cough

Chest pain

Viral fever Malaria Typhoid Stomach problem Chest problem

(0.4,0.6,0) (0.7,0.3,0.0) (0.3,0.7,0.0) (0.1,0.7,0.2) (0.1,0.7,0.2)

(0.3,0.6,0.1) (0.2,0.6,0.2) (0.6,0.3,0.1) (0.2,0.6,0.2) (0.0,0.6,0.4)

(0.1,0.8,0.1) (0.0,0.8,0.2) (0.2,0.8,0.0) (0.8,0.2,0.0) (0.2,0.8,0.0)

(0.4,0.6,0.0) (0.7,0.3,0.0) (0.2,0.7,0.1) (0.2,0.7,0.1) (0.2,0.7,0.1)

(0.1,0.8,0.1) (0.1,0.8,0.1) (0.1,0.8,0.1) (0.2,0.8,0.0) (0.8,0.2,0.0)

Tab. 8. The normalized Hamming distances for pa ents from the possible diagnosis

Al Bob Joe Ted

Viral fever 0.24 0.28 0.36 0.24

Malaria 0.20 0.42 0.34 0.20

Typhoid 0.22 0.28 0.32 0.26

Next, characteristics of particular diagnosis given in Tab. 3 are approximated using medical knowledge represented in the fuzzy relation đ?&#x2018;&#x201E;. The results are presented in Tab. 5. In particular, for Temperature, its lower and upper bounds coincide for Viral fever, Malaria, and Typhoid, so the relation đ?&#x2018;&#x201E; itself precisely characterizes this symptom for these diseases. For the two

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Stomach problem 0.42 0.14 0.48 0.36

Chest problem 0.46 0.38 0.48 0.40

remaining diseases, however, Temperature can be stated only approximately in view of the derived information.

On the basis of approximations given in Tab. 4 and Tab. 5, two intuitionistic fuzzy relations are calculated and the results are given in Tab. 6 and Tab. 7, respectively. For example, Joe has a headache up to the degree

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Tab. 9. The normalized Euclidean distances for pa ents from the possible diagnosis

Al Bob Joe Ted

Viral fever . . . .

Malaria 0.044 . .

0.038

Typhoid . .

0.136 .

0.4 and at the same time this symptom is slightly excluded (to the degree 0.2), so it is unknown whether he actually suffers from this pain up to the degree 0.4. Similarly, Headache is not a characteristic symptom for Chest problem, but it is excluded for this diagnosis only up to the degree 0.6 â&#x20AC;&#x201C; it is then undetermined whether this symptom is speci ic for this diagnosis up to the degree 0.4.

Now, taking into account data from Tab. 6 and Tab. 7 we calculate distances between intuitionistic fuzzy set đ?&#x2018;?đ?&#x2018;&#x2026; and đ?&#x2018;&#x2018;đ?&#x2018;&#x201E; for every đ?&#x2018;? â&#x2C6;&#x2C6; đ?&#x2018;&#x192; and for every đ?&#x2018;&#x2018; â&#x2C6;&#x2C6; đ??ˇ.

For the normalized Hamming distance the results are shown in Tab. 8. The shortest distance points out the proper diagnosis. Namely, Al and Ted suffer from malaria, Bob from stomach problem, and Joe has typhoid.

For the normalized Euclidean distance the results are similar as shown in Tab. 9.

5. Concluding Remarks In this paper we have shown an application of fuzzy approximation operators in supporting medical diagnosis. These operators are compositions of fuzzy necessity and fuzzy possibility modal operators wellknown in fuzzy modal logics. Our approach is based on the observation that approximations of fuzzy sets lead to intuitionistic fuzzy sets. Then, given two fuzzy relations representing characterizations of patients and diseases, respectively, in terms of their symptoms, we have obtained two respective intuitionistic fuzzy relations. Following the idea proposed by Szmidt and Kacprzyk [30], proper medical diagnosis are determined basing on distances between intuitionistic fuzzy sets.

AUTHOR Anna Maria Radzikowska â&#x20AC;&#x201C; Warsaw University of Technology, Faculty of Mathematics and Information Science, Koszykowa 75, 00-662 Warsaw, Poland, e-mail: A.Radzikowska@mini.pw.edu.pl.

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[17] A. M. Radzikowska and E. E. Kerre. “Towards Studying of Fuzzy Information Relations”. In: J. M. Garibaldi and R. I. John, eds., Proceedings of the 2nd International Conference in Fuzzy Logics and Technology EUSFLAT, Leicester, United Kingdom, September 5-7, 2001, 365–369. De Montfort University, Leicester, UK, 2001. http://dblp.unitrier.de/rec/bib/conf/eus lat/RadzikowskaK01.

[18] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzzy rough sets”, Fuzzy Sets and Systems, vol. 126, 2002, 137–155, DOI: 10.1016/S0165-0114(01)00032-X. [19] A. M. Radzikowska and E. E. Kerre. “A Fuzzy Generalization of Information Relations”. In: E. Orłowska and M. Fitting, eds., Beyond Two: Theory and Applications of Multiple-Valued Logics, volume 114 of Studies in Fuzziness and Soft Computing, 287–312. Physica-Verlag Heidelberg, 2002. DOI: 10.1007/978-3-7908-1769-0.

[20] A. M. Radzikowska and E. E. Kerre, “Algebraic Characterizations of Some Fuzzy Information Relations”. In: Proceedings of the 13rd IEEE International Conference on Fuzzy Systems FUZZ– IEEE, July 25-29, 2004, Budapest, Hungary, vol. 1, 2004, 115–120, ISBN 0-7803-8353-2.

[21] A. M. Radzikowska and E. E. Kerre. “Fuzzy Rough Sets based on Residuated Lattices”. In: J. F. Peters, A. Skowron, D. Dubois, J. W. GrzymałaBusse, M. Inuiguchi, and L. Polkowski, eds., Transactions on Rough Sets II: Rough Sets and Fuzzy Sets, volume 3135 of Lecture Notes in Computer Science, 278–296. Springer-Verlag Berlin Heidelberg, 2004. DOI: 10.1007/978-3-540-277781_14.

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[28] E. Sanchez. “Solutions in composite fuzzy relation equation. application to medical diagnosis in Brouwerian Logic”. In: M. M. Gupta, G. N. Saridis, and B. R. Gaines, eds., Fuzzy Automata and Decision Process, 221–234. Elsevier, North Holland, 1977.

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Multi-Strategy Navigation for a Mobile Data Acquisition Platform Using Genetic Algorithms Submitted: 5th December 2016; accepted: 25th January 2017

Fadi Halal, Marek B. Zaremba DOI: 10.14313/JAMRIS_1-2017/4 Abstract: Monitoring of biological and chemical pollutants in large bodies of water requires the acquisition of a large number of in-situ measurements by a mobile sensor platform.

Critical to this problem is an efficient path planning method, easily adaptable to different control strategies that ensure the collection of data of the greatest value. This paper proposes a deliberative path planning algorithm, which features the use of waypoints for a ship navigation trajectory that are generated by Genetic Algorithm (GA) based procedures. The global search abilities of Genetic Algorithms are combined with the heuristic local search in order to implement a navigation behaviour suitable to the required data collection strategy. The adaptive search system operates on multi-layer maps generated from remote sensing data, and provides the capacity for dealing with multiple classes of water pollutants. A suitable objective function was proposed to handle different sampling strategies for the collection of samples from multiple water pollutant classes. A region-of-interest (ROI) component was introduced to deal effectively with the large scale of search environments by pushing the search towards ROI zones. This resulted in the reduction of the search time and the computing cost, as well as good convergence to an optimal solution. The global path planning performance was further improved by multipoint crossover operators running in each GA generation.

The system was developed and tested for inland water monitoring and trajectory planning of a mobile sample acquisition platform using commercially available satellite data.

Keywords: genetic algorithms, path planning, monitoring system, remote sensing, navigation control, heuristic search

1. Introduction

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Acquisition of a large number of in-situ measurements by a mobile platform is a basic task in the process of monitoring biological and chemical pollutants in large bodies of water. Monitoring of environmental phenomena in inland waters requires measuring a variety of physical processes, such as nutrient concentration, wind effects, and solar radiation [26]. Remote sensing (RS) techniques provide significant advantages in terms of spatial and temporal coverage and cost-efficiency. The maps of large environment areas are often obtained through the processing of satellite

imagery. The multi-spectral data can subsequently be used to obtain models of water pollutants, such as the concentration of chlorophyll (Chl-a) or total suspended sediments (TSS) [17], by applying such measures as the maximum chlorophyll index (MCI) [10] or the ocean chlorophyll 4 algorithm (OC4v4) [21]. In many situations the remote sensing data have to be augmented and updated by in situ measurements. This is due to the need for precise local measurements, for the calibration of satellite imagery in varying water conditions, and for the purpose of precise local decision making.

Critical to this sample acquisition problem is an efficient path planning method, easily adaptable to different control strategies that ensure the collection of data of the greatest value. Acquisition of different types of samples may require appropriate behaviours that implement different collection strategies. Designing a multi behaviour search system for a mobile sample acquisition platform requires answering the following questions. Which is the suitable navigation mode for a specific water pollutant? How to compute the cost of the solution? How can the solution of the path planning problem deal with multiple patches of high concentration of the pollutant?

In general, the path planning procedure designs a trajectory that visits a given set of points such that the optimisation process minimises the total travel distance. This task can be defined in terms of a combinatorial optimization problem with a globally optimal solution that satisfies all hard and soft constraints.

The optimal solution or a set of globally optimal solutions minimises or maximises the objective function. The path finding problem is typically defined in terms of the Travelling Salesman Problem (TSP) [7] or a more general Vehicle Routing Problem (VRP) [4]. Determining the optimal solution is an NP-hard problem, so the size of problems that can be solved optimally is limited [3]. In the situation of environment monitoring systems, the problem is even more complex because exact positions of the sampling points are not known a priori. In practice, therefore, solutions to optimal path planning problems have to incorporate heuristic methods.

A variety of heuristic methods have been investigated. Evolutionary algorithms have been employed in many variants. In [6] an ant colony optimization system was presented to solve the problem of designing an optimal trajectory for a mobile data acquisition platform. Luo et al. [20]an intelligent mobile vehicle is required to reach multiple goals with a shortest path

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that, in this paper, is capable of being implemented in TSP (Traveling Salesman Problem proposed a hybrid GA and D* algorithm for real-time map building and navigation for multiple goals purpose. Yoshikawa and Terai [32] proposed a car navigation system using hybrid genetic algorithms and D algorithm. Their system finds a route which has several passing points before arriving at the final destination. In [18] the path planning problem for a submarine navigation application was solved using the artificial bee colony algorithm. The use of a cultural hybrid algorithm to solve the mission planning was reported in [33]. An improved simulated annealing artificial network to plan the path for a mobile robot was employed in [8].

Genetic algorithms have been frequently used in NP-hard problems due to their flexibility and high quality of the search results [25]. They can provide a solution without any advance knowledge about the environment, and are largely unconstrained by the limitations of the classical search methods [24]. By mimicking natural evolution processes, they have the ability to adaptively search large spaces in near-optimal ways. In practical terms, GA methods are easy to interface with simulation models. An important feature that should be considered in implementing GA techniques is that they are problem specific. Due to the constraints of a particular problem and the operation of crossover and mutation mechanisms, feasible offsprings often cannot be obtained by applying exclusively genetic algorithms. In order to ensure the feasibility, additional algorithms should be incorporated.

For example, [34] developed an improved genetic algorithm, where an obstacle avoidance algorithm and the distinguish algorithm are combined with a GA algorithm to select only the feasible paths and to improve the path planning efficiency. The distinguish algorithm is designed for distinguishing whether the path is feasible or not.

In this paper we present a hybrid GA-based method developed to optimize path planning and navigation using pollutant maps generated from RS imagery.

The power of the global GA search is combined with the speed of the local optimizer. Both optimizers work cooperatively to find the optimal solution, where GA determines the optimal region, and then the local optimizer takes over to find the best position for acquiring water samples [13]. In order to deal effectively with the large-scale environment, the following modifications to the state-of-the-art approaches were introduced. In the first place, this paper implements an improved combination of a GA with an obstacle avoidance algorithm and the distinguish algorithm proposed initially in [34]. This algorithm puts a feasible path in the feasible group and deletes an infeasible path or keeps it in the infeasible group, which markedly improves the efficiency of the path planning. The big family pool was adopted in our system, which consists of all old-generation solutions and currentgeneration offsprings obtained after mutation and crossover operations combined with different metaheuristic solutions. Based on the Cooperative Genetic Optimization Algorithm [14], it offers a greater search selection diversity and gives the system the ability to

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save the elite searching experience from one population to the next one.

Multi-layered maps were employed to generate spatial and functional properties of the environment.

Those maps enable the planning system to perceive and interpret environments according to different environment features. ROI maps can be extracted from the multi-layer map as additional layers. The ROI approach facilitates the planning system in directing the search toward desirable patches by paying additional attention to desired regions, and assuring at the same time the generation of feasible solutions [11]easily adaptable to different control strategies that ensure the collection of data of the greatest value. This paper proposes a hybrid Genetic Algorithm (GA. In general, each optimization problem to be solved by a GA method requires a unique fitness function that represents a performance criterion used in the evaluation of the performance of all chromosomes in the population. Many functions, such as travelling distance, time window and the sample values (weights) should be optimized simultaneously. This may involve a combination of maximization and minimization criteria [5]. Individual objective functions are usually combined into a single composite function by weighting the objectives with a weight vector. The result of the optimization should reach a reasonable solution that compromises multiple objectives [23]. For mission planning of an unmanned aerial vehicle (UAV), [29] used the distance, the hazard, and the maneuvering of the route as components of their cost function.

Each component has a weight factor assigned according to the objectives of the mission. The hazard is related to the existence of obstacles near the path, and the maneuvering refers to the maneuvers required to perform target tracking. For efficient determination and search of the best flight (UAV) routes, an objective function was created in [27] which involves the timeliness and the smoothness of the path. The objective function discussed in [9] included several components: the cost of the motion from the start node to the current node, the heuristically estimated value of getting from the current node to the goal, the terrain traversability component, the direction change cost, and the cost of navigating in shadow areas. Each component has a corresponding coefficient factor used to weight the objective function components according to its importance to the mission. An optimized path planning for skid-steered mobile robots [16] uses a cost function which consists of the terrine properties, longitudinal motion and turning of the robot. In this work, an objective function proposed to deal with the experiment conditions comprises the following components: the samples value, the ROI award, the distance, and the sampling time.

The waypoint technique was used in the path planning process as an approach appropriate for large monitoring environments [30]. Waypoints are defined as abstract points [15] used to determine local positions [28] through which a mobile platform can navigate, reach its region-of-interest destination, and collect the water pollutant samples [22]. In the application discussed in this paper, waypoints correArticles

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spond to sampling points. In order to deal with multiple sampling areas, multi-point crossover (MPC) was implemented. The MPC operator works to build the final solution which consists of valuable segments of local paths from many search strategies. The mutation operator improves the local search and helps the population to avoid local minima. The evolution process optimizes the path planning by designing new chromosomes which consist of best value samples from many global paths.

Experiments were conducted on data from Lake Winnipeg located in Manitoba, Canada. The adaptive search techniques presented in the paper were applied to optimize the location of the sampling points for different pollution indices and behaviours: the concentration of individual pollutants and their combinations, and the maximum gradient of pollutant concentration.

The structure of the paper is as follows. Section 2 addresses the sample acquisition problem using remote sensing data. A discussion of the proposed hybrid GA-based architecture for path planning and the optimisation of the multi-behaviour sample acquisition is presented in Section 3. Experimental results are discussed in Section 4.

2. Multi-Strategy Sample Acquisition Mission 2.1. Problem Statement

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The problem addressed in this paper consists in planning a trajectory for precise acquisition of water pollutants by a mobile platform, when the planning process is guided by prior rudimentary information about the distribution of pollutants obtained from remote sensing data. The acquisition mission should incorporate different acquisition strategies.

The sample acquisition mission is performed within a more general procedure consisting of the following phases: 1) Determination of water regions and their types, sample location zones, and water pollutants to be sampled; 2) Identification of the pollutant detection indices, coverage methods (e.g., uniform coverage, maximum concentration gradient) and the number of samples; 3) Selection of the sources of remote sensing data and their calibration methods; 4) Selection of the ancillary data from in situ sensors (e.g. wind, temperature); 5) Determination of the acquisition mission parameters (e.g., total mission time).

Most of the above factors and conditions affect the strategies that have to be incorporated in the planning procedure. Mission strategies can be classified in two categories: (1) Water pollutant concentration strategies In this type of strategies the aquatic acquisition platform collects the most valuable samples from different pollutant classes and their combinations, such as • Chl-a, • Chl-a & (TSS), • Chl-a & Dissolved Organic Carbon (DOC), • Chl-a & TSS & DOC. Articles

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In this class of strategies, specific samples should be collected while neglecting other samples within a certain time window. Time windows can be imposed because of the deterioration of the quality of samples over a period of time. Time requirements for Chl-a concentration sampling are discussed in [12].

With respect to the types of pollutants, the RS data have to be pre-classified. The final path maximizes the value of the collected samples along a trajectory that traverses regions of different distributions of the pollutant concentration. As a result, the planning algorithm works on many maps created to represent different concentration levels for different water pollutant classes. The optimal strategy directs the path to the best Region of Interest (ROI) zone. The samples values (weights) vary depending on the mission objective.

(2) Local coverage strategies: In this mode the platform executes a specific navigation and collection behaviour depending on the shape of the sample spatial distribution. We distinguish here such sampling strategies as the uniform coverage of high-concentration areas, sampling at local concentration maxima, and sampling along maximum gradient lines, which is of interest in many environment monitoring applications [36]. The sampling process can be different in each patch to comply with the general and local mission goals.

Both types of strategies execute under some specific constraints. Time window constraints can be imposed on certain pollutant patches, and travel distance constraints on other patches. Also, a certain number of samples have to be collected in a specific patch before heading to another one.

2.2. GA-Based Planning System

Due to the complexity of the mission trajectory optimization problem, a hybrid GA/Adaptive Search system is proposed and investigated in this paper. The general architecture of the planning system is based on the deliberative architecture model [19]. As illustrated in Fig. 1, the deliberative level comprises the

Data sources

Multilayered map

Feature Generation Global

Deliberative model Reactive model

Planning module Behaviour selector

Acquisition platform

Environment

Fig. 1. General architecture of the GA- based planning system

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environment modelling level, which operates on the remote sensing data and the ancillary information, and the adaptive GA-based trajectory generation level.

Water wave reflection can be exploited to determine the concentration of water pollutants. Examples of spectral signatures for different samples of chlorophyll pigment and TSS are shown in Fig. 2.

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The maps provide, for each spatial point (pixel), the numerical values NLi of the measured pollutants. The spatial resolution of the maps corresponds to the resolution of satellite images. Figure 3 shows the following layers: bathymetric map (L1), Chlorophyll-a (L2), TSS (L3), and the maximum gradient of chlorophyll-a (L4).

The overall goal of the acquisition mission is to maximize the quantity and the quality of the collected water pollutant samples V during the mission:

Fig. 2. Spectral signatures: a) Chl-a, b) TSS

N° 1

(3)

where V is the value of the sample, Nj is the number of the samples for each pollutant, and M is the number of water pollutant classes.

3. GA Method for Path Planning The following two models were applied to measure the concentration of TSS [17] and Chl-a [10], [1] using different spectral bands of satellite images:

(1)

3.1. Genetic Algorithm Architecture

The basic operation of the proposed GA-based path planning procedure can be summarized as follows (Fig. 4). The sampling points correspond to the Start

Initial waypoint pool

where Lxxx is the radiance value of the band at wavelength xxx, and

Objective function

MCI = L709 − L681 − 0.389 (L753 − L681) (2)

The factor 0.389 is calculated as the wavelength ratio (709–681) / (753–681).

The input data structure used to generate the information required for multi-strategy path planning is implemented in the form of a multi-layer map (Fig. 3), which consists of a set of overlaying grid-based maps.

Adaptive search system Navigation Strategy 1

Navigation Strategy 2

Navigation Strategy 3

Sort big search family and generate next generation No

Termination criteria Yes Stop

Fig. 4. Genetic Algorithm based path planning Radiance 1

Radiance 6

Radiance 9 .

Radiance 15 Satellite radiance

Fig. 3. Multi-layer map

L1, L2, L3, L4

waypoints of the global path of the mobile platform.

Thus, the global path consists of several local paths, which are the arcs between two waypoints with a directed connection between them. The initial population of waypoints is pruned to generate collision free paths, subsequently stored in the initial chromosome pool population. Unfeasible solutions are deleted.

The adaptive search (AS) system improves the elite paths (the best 10 solutions) and returns efficient paths adapted to the local navigation behaviour. The big family pool consists of all old-generation solutions and current-generation offsprings obtained after the mutation and crossover operations combined with AS solutions. It gives the system the ability to save the elite search experience from one population to the next one [14]. The big family search results are sorted and pruned to form the next generation (Fig. 5).

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search

A more detailed description of individual steps of the algorithm follows below.

operators

Best

Best

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Heading angle

b)

heading

Fig. 5. Big family search [14]

c)

3.2. Path Planning and Initial Waypoint Population In the GA-based path planning procedure the population is represented, as in the Vehicle Routing Problem, by ordered sets of waypoints. Each feasible set is considered to be an individual in the population. Each waypoint, which is a sample candidate, represents a location in the environment (x,y). The initial genotype can be represented by a cell array, where each pair of cells represents the local path length and the heading angle towards the subsequent waypoint.

The path planning generator works as follows: 1) Determine the first waypoint in the path, i.e., the starting point, with the initial angle equals to zero.

2) While the path planning doesn’t reach the desired target, generate a random number of L, the path length, between Lmin and Lmax, and a random heading angle β between βmin and βmax obtaining the next waypoints[31]. A maximum number of waypoints is given for each search strategy.

3) Different strategies are applied to water pollutant patches by adjusting L and β. Each path planning strategy handles different number of samples depending on the search path. x

XR

YR

YR

YR

L2 L1

Water L3 Sample XR

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4) Continue with another patch or return to the starting point, depending on constraints, such as the maximum travel distance or the maximum number of water samples.

Figure 6 illustrates the path planning generator.

The chromosomes are encoded as an integer string. Each gene consists of two variables, the local path length and the heading angle as shown in Fig. 7a. Depending on the start point and the chromosome, the waypoint generation produces records as in Fig. 7b. The path planning waypoints are represented in the form of a long array as depicted in Fig. 7c. The GA search finds the waypoints between the starting point of the mission and the destination point. a)

Worst

N° 1

Starting point

Travel distance

Heading angle

travel distance

Node 1

travel distance

previous waypoint (start point)

Node 2

.

……..

..

Heading angle

travel distance

waypoint1 x coordinate (Latitude)

Waypoint1 y coordinate (Longitude)

Node n

Destination point

Fig. 7. Chromosome and waypoint array. a) GA chromosome; b) Waypoint representation; c) Waypoint array An obstacle free path planning algorithm [35] was adopted to deal with spatial constraints. It produces a feasible path that satisfies the conditions that the waypoints should be located outside the obstacles, in the sampling space, and the local path should not intersect with the obstacles.

In order to comply with the feasibility constraints and to enhance the efficiency of the path, a certain number of the waypoints in the elite solutions can be modified for each generation by applying three possible operations: waypoint deletion, insertion, or replacement [2] a tabu search system model is designed and a tabu search planner algorithm for solving the path planning problem is proposed. A comprehensive simulation study is conducted using the proposed model and algorithm, in terms of solution quality and execution time. A comparison between our results with those of A* and genetic algorithms (GA. Waypoint deletion eliminates all waypoints in the clear water body. The waypoint insertion operation explores the neighbourhood and inserts a new waypoint, according to a predefined behaviour for each water pollutant type. After deleting and inserting the waypoints the algorithm evaluates the path, conducts a neighbourhood search to replace the lowest waypoint value with a new one, and builds another feasible path Pn that satisfies the mission constraints.

3.3. Fitness Function XR Start point y

Fig. 6. Waypoint generation scheme 34

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The fitness function is a particular type of the objective function that quantifies the optimality of a solution and evaluates the suitability of a solution with respect to the overall goal. In our navigation problem, it maximizes the collected information, directs the ro-

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bot towards the ROI, and incorporates distance and time penalties.

The proposed fitness function F consists of 4 components, calculated for each candidate sample

3.4. Multi-Behaviour Operation

Behaviour 1– Short local path and high sample values.

The sampling process selects the best sample according to equation

F = SV + ROI + DIS + ST (4) where: SV – data set value, which determines the value of acquired samples according to Eq. 5; (5)

where sample values are calculated as the values of V in Eq. 3.

ROI – the region of interest award, introduced in order to optimize the convergence of the search for quality samples (Eq. 6):

(6)

where: DIS – distance factor; ST – sampling time factor.

Two objective functions with different forms of DIS and ST factors were tested to assess their impact on the effectiveness of the sample acquisition mission: Objective function 1 linearly maximizes the sample value and the ROI award and exponentially minimizes the sampling time and the mission travel distance. The distance and the time become, as the sample acquisition mission progresses, quadratically more expensive.

Objective function 2 linearly maximizes the sample value as well as the sampling time and the ROI award, and linearly minimises the mission travel distance.

Objective function 1

Objective function 2

The basic idea of the multi-strategy GA-based path planning is that the acquisition platform explores water pollutant patches using different behavioural characteristics depending on the sampling requirements in each patch. The behaviours affect the local search optimization where the best evaluated neighbour is selected according to the adopted behaviour.

The following behaviours represent different sampling strategies.

(7)

where i is the departure waypoint, j is the destination waypoint, and is the chlorophyll concentrations in cell ( x,y) of the MCI layer.

Behaviour 2 – Maximum gradient (MG) sampling.

Valuable samples (bigger than a given threshold number) are selected along a short local path according to the following equation:

(8)

The sampling behaviour for other samples maximizes the local path according to equation

(9)

Behaviour 3 – Multiple pollutant patches.

The AS procedure selects the best sample value (Eq. 10), with the maximum local path range distance and the highest sample weight.

(10)

where and are Chl-a and TSS concentrations in cell (x,y) taken from the MCI and TSS maps.

Behaviour 4: Long local path and TSS sampling The AS procedure selects the best sample value as defined by equation (11), where the value sample corresponds to the maximum local path range distance and the highest sample weight;

(11)

An example of water pollutant patches obtained for different behaviours from a 3-layer map (MCI, TSS and MG) is shown in Fig. 8.

3.5. Multi-point Crossover

Fig. 8. Linear and nonlinear DIS and ST components of the fitness function

Multi-point crossover is used to enhance the process of selecting valuable samples located in distant zones. The crossover procedure is explained in Fig. 10. Parent chromosomes, P1 and P2, are cut at multiple random locations, and the portions of the chromosomes between the cuts are swapped. The Articles

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TSS

Figure 11 represents the overall architecture of the developed adaptive GA-based mission planning system. The mission objective is defined and accompanied with a strategy definition to achieve the mission goal. A multi-layer map is generated to interpret the global environment and to weight the importance of different water pollutants in the sampling strategy.

A set of ROIs is generated to guide the search toward specific patches associated with their acquisition strategies.

An adaptive search algorithm improves the multistrategy path planning in different patches employing local search optimising procedures. A suitable fitness function evaluates the chromosome in the search for maximizing the mission goal.

Behavior 2 zone

4. Experimental Results

4.1. Experimental Framework

Multi-strategy sample collection

Multi-layer map

Fig. 9. Water pollutant zones for multi-behaviour navigation

High value water Samples patch

B

C

D

E

F

B

D

C

High value water Samples patch

B

C

D

E

E

C

D

F

High value water Samples patch

F

G Target point

Start point Crossover point

B

G Target point

Start point Crossover point

E

F

High value water Samples patch

Fig. 10. Multi-point crossover. a) A two-chromosome and two-point crossover. b) Two offsprings 36

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3.6. Planning Process

Behavior 3- 4 zone

Maximum Chl-a gradient concentration

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result is a pair of offsprings I1 and I2. The crossover is applied on the best-fitness chromosomes chosen from the pool. Due to the difference in the chromosome length, the crossover point should be applied to the shorter chromosome.

Behavior 1 zone

Chl-a

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The experiments were carried out using satellite data from the northern basin of Lake Winnipeg for a path starting at the point located at longitude (99˚02’08”) W and latitude (55˚35’18”) N and the destination point at longitude (96˚ 50’ 24”) W and latitude (51˚55’51”) N. The direct distance between the start point and the target is around 236 km. The maps used in the experiments were in the form of a raster grid, where the dimensions of cells corresponded to the resolution of the MERIS satellite sensor, i.e., 260 m × 300 m. Each cell had an associated value Vx,y obtained from the multi-layer map as discussed in Section 2.

ROI maps guide the multi-strategy sampling to orient the acquisition platform toward the valuable samples in the ROI zones using the penalty/award mechanism. Figures 12 a) b) and c) show regions of interest for MCI, TSS and the maximum gradient of the chlorophyll concentration. The regions are defined as the concentration of TSS bigger than 0.3 from the normalised TSS model, and the concentration of chlorophyll-a bigger than 0.5 from the MCI normalised model. Figure 12d represents the overall ROI formed from the MCI and TSS zones. Figure 12e illustrates three ROI zones, which are MCI, TSS and maximum gradient chlorophyll concentration, used in the experiments.

Matlab Genetic Algorithm Optimization Toolbox (GAOT) was used to program the proposed hybrid system. Table 1 shows the Genetic Algorithm parameters chosen for the optimization process.

Four experiments were conducted with two objective functions (Fig. 8) tested. Objective function 2 (linear optimization) was incorporated in the fitness function used in experiments 1 and 2, and objective function 1 (exponential optimization) in experiments 3 and 4. Hard distance and time constraints were implemented in the first two experiments. The mission

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Table 1. Parameters of the Genetic Algorithm Genetic Parameters

Magnitude

Population size

120

Number of generations

150

Crossover rate

60% randomly and the elite

Type of crossover

Single-point and multi-point crossover

Selection type

Roulette Wheel

Mutation rate

Type of mutation

5% randomly and the elite

a)

b)

4 point random & 4 maximum points c)

Modelling and pruning

Mission e

d)

Strategy

e)

Fig. 12. a) Chl-a ROI (MCI > 0.5); b) TSS ROI (TSS > 0.3); c) Chl-a Max Gradient ROI; d) Combined Chl-a & TSS regions of interest, and e) Combined Chl-a & TSS & MG regions of interest time was bounded by the value of 12 hours, and the travel distance was limited to 400 km. In experiments 3 and 4, the mission time had to be less than 9 hours, and the travel distance was limited to 330 km.

Desired ROIs

4.2. Path Planning Experiments In the first experiment, the sample value (SV) was the sum of the TSS and Chl-a sample values. The results show that the path includes 10 samples from the clear water zone (outside the ROI zone), as shown in Fig. 13. The obtained results provide the rationale for hybridising the GA-based search for optimal samples.

Path waypoints

BH_1 ROI_1

Search algorithm BH_2 ROI_

BH_3 ROI_

trajectory

Fig. 11. Adaptive GA-Based Navigation System

Fig. 13. Sample acquisition paths: Experiment 1 Articles

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Experiment 2.

A simple adaptive search, consisting in limiting the search to ROIs, was introduced in the second experiment. However, no specific behaviour guided the waypoint generation. Figure 14 presents the path generated by the modified system. The sampling area is located entirely in the ROI. Table 2 compares the performance of the two experiments.

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performance in the two different patches, as shown in Fig. 15.

Table 2. Results of experiments 1 and 2

Sampling time

path length (m) Samples value ROI award

Experiment 1 (GA)

Experiment 2 (ROI-optimized GA)

3.9989e+005

3.4364e+005

0.3675

0.5550

0.475 @ 38 samples 0.475 @ 38 samples 0.7004

0.8304

Fig. 14. Sample acquisition paths: Experiment 2 The path in the second experiment was approximately 56 km shorter and the value of the samples increased by about 13 percent, while keeping the number of samples at the same level.

4.3. Multi-Behaviour Navigation

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In order to assess the multi-behaviour performance of the system and to further improve the path quality – in the context of the GA methodology – different behaviours were introduced to the local adaptive search the next two experiments. The third experiment explores the local behaviour optimization which performs two collection strategies depending on the types of the samples. Therefore, the ROI set consists of two zones, Chl-a and TSS. The search minimises the local path in the MCI patch according to Eq. 7, and maximises the local path in the TSS patch according to Eq. 11. The neighbourhood of a solution is explored, and the best neighbor is selected according to the adopted behaviour in each patch. Objective function 1 was used to optimise this experiment. The multi behaviour navigation shows good sampling Articles

Fig. 15. Sample acquisition path from experiment 3 The mission collects 22 pure chl-a samples and 6 TSS samples along a 282 km long path. The samples value is 0.645, and ROI award equals to 0.6125. The distances between the chlorophyll samples are shorter than between the TSS samples, which is a consequence of applying the behaviour equation (Eq. 7) and high award for the Chl-a ROI. The longer local path between the six TSS samples results from the behaviour equation (Eq. 11). The total mission time is 8 hours and 54 minutes. The travel time is 7 hours and 14 minutes.

In the fourth experiment, the zone of the maximum gradient of chlorophyll concentration was introduced, which produced three separate patches with three different local search behaviours. Due to the behaviour conflict between the maximum gradient and the maximum value of the chlorophyll concentration, a new ROI zone was created. Thus, the three separate ROIs were generated as follows: the Chl-a zone, the maximum gradient of chlorophyll concentration, and the chlorophyll and TSS concentration zone. Figure 16 depicts the ROI map which was used in this experiment. The Chl-a samples were treated as the highest value samples with the shortest local path in the search algorithm (Eq. 7). In the maximum gradient zone, the search made the acquisition platform navi-

Fig. 16. Multi behaviour sampling for different patches

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gate in adaptive way to follow the maximum gradient curve, using Eq. 8 and Eq. 9, and to maintain a proper distance between the samples. In the chlorophyll and

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The results without the enhancement are shown in Fig. 18a. Both the quality of the solution and the speed of the optimization are enhanced by an order of magnitude by applying the improved operations (Fig. 18b).

The repeatability of the results is depicted, for experiments 3 and 4, in Figures 19a and 19b respectively. The convergence of both the best solution and the average solution is high.

Fig. 19. Convergence in experiment 3 & 4 Fig. 17. Sample acquisition path from experiment 4

5. Conclusions TSS zone, the behaviour model as in Eq. 10 was adopted. All behaviour optimization algorithms explored the neighbourhood and selected new waypoints in order to enhance the quality of the solution. Figure 17 shows an example of the planned path.

The path planning algorithm produced 28 samples as follows: 9 samples from the TSS & Chl-a zone; 5 samples from the MG zone; 14 samples from Chl-a zone including the start waypoint. The samples were collected along a path 285 km long. The normalized sample value was 0.5040 with the ROI award equal to 0.5650.

4.4. Convergence Analysis

To improve the convergence of the GA-based search, two crossover and two mutation operations were employed. The solutions to these operators were divided into two categories as follows: the first one consists of the elite solutions, and randomly selected solutions represent the second category.

The simulation results show that: (1) The new procedure effectively enhanced the global search ability and improved the local searching ability; (2) High convergence rate was obtained.

In this paper, hybrid genetic algorithms were proposed for navigation in a partly known environment, where the objective of the planning task is to find the optimal path for a mobile sample acquisition platform. The total quantity and quality of water samples is to be maximized according to navigation goals specified for each acquisition zone. Sampling in each patch may be guided by different patterns of behaviour for different purposes. Thus, the acquisition system is able to execute different behaviours along the global path. A hybrid genetic search was developed to deal with such a complex environment. The adaptive search algorithm models behaviours in different surrounding areas and executes them in each generation at the level of local path navigation. The locality of the navigation was defined in terms of regions of interest (ROI). In the process of generating the waypoints, the adaptive search deletes and inserts new waypoints in each solution depending on the ROI behaviour. This enhances the flexibility and the efficiency of path planning. The ROI component was introduces also in the fitness function, greatly speeding up the convergence of the planning process. Tests were conducted using medium-resolution satellite imagery. Multi-layered maps provided a rich context to the adaptive search system to perform flexible local search behaviours.

The experiments performed on large area environment show that the adaptive GA-based path planning method offers robust search capabilities and supports different sample acquisition strategies, ensuring the collection of meaningful data over multiple areas of interest.

ACKNOWLEDGEMENTS

Fig. 18. Convergence in experiment 1 & 2

The authors acknowledge funding from the International Science and Technology Partnership (ISTP – Canada) and the Natural Sciences and Engineering Research Council of Canada, grant 9227.

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AUTHORS Fadi Halal*, Marek B. Zaremba – Département d’informatique et d’ingénierie Université du Québec en Outaouais Gatineau, QC, J8X 3X7, Canada.

E-mail: half02@uqo.ca; zaremba@uqo.ca *Corresponding author

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Application the GPS Observations in SPP Method for Aircraft Positioning in Flight Experiment in Dęblin, Poland (01.06.2010) Submitted: 27th October 2016; accepted: 22nd January 2017

Kamil Krasuski DOI: 10.14313/JAMRIS_1-2017/5 Abstract: In this article, the results of GPS positioning in civil aviation are presented. The flight test was conducted using Cessna 172 aircraft in Dęblin on 1st of June 2010. The aircraft position was determinated using Single Point Positioning method for GPS code observations. The numerical computations were executed in Aircraft Positioning Software (APS) in Scilab 5.4.1 language. The average accuracy of aircraft position is higher than 11 m in horizontal coordinates and about 13 m in vertical plane, respectively.

Keywords: GPS, SPP method, HPL, VPL, least square estimation

1. Introduction

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Since few years the development of GNSS technique in precise air navigation is visible in Poland.

Especially, the GNSS technique is implemented in air navigation to improve the aircraft position in real time or post-processing. The implementation of GNSS sensor is focused on applied of SBAS data (e. g. EGNOS correction) in civil aviation [2, 3, 5, 6]. The EGNOS system is utilized in civil aviation for non-precision approach (NPA) or approach (APV-I) to landing procedure [7].

The typical accuracy of NPA procedure is equal to 220 m in horizontal plane whereas NPA procedure for vertical plane is not available. In case of the APV-I procedure, the accuracy of aircraft position amounts to 16 m for horizontal plane and 20 m for vertical plane [8].

The requirement of EGNOS system is also concerned to integrity, time of alarm, continuity and availability parameters [1]. Only two parameters (accuracy and availability) are suitable in aviation procedures for the GPS system. Average accuracy of aircraft position in GPS system is recommended by ICAO annex and it is equals to 9 m for horizontal plane and 15 m for vertical plane respectively. The critical accuracy with probability 95% can reach up to 17 m for horizontal plane and 37 m for vertical plane respectively. The availability parameter should be more than 99% for all air operations for whole area of the Earth [10].

In this paper, the accuracy results of aircraft positioning in GPS system are presented. The flight experiment was conducted on 1st of June 2010 in Dęblin, Poland using Cessna 172 aircraft. The aircraft position was recovery based on Single Point Positioning (SPP) method for GPS code observations. The computations of aircraft position was executed in Aircraft Positioning Software (APS), which source code was written in

Scilab 5.4.1 language. The final results of aircraft position were compared with accuracy of NPA and APV-I procedures. The structure of article was divided into 5 sections: introduction, methodology of research, experiment and results, discussion and conclusions.

2. Mathematical Model for Designation the Aircraft’s Position Using GPS Observations

The SPP method is applied in standalone positioning in geodesy and navigation also. The basic equation of SPP method is described as below [17]:

(1) where: l – the pseudorange value (C/A code) at 1st frequency in GPS system, d – the geometric distance between satellite and receiver; include information about: the Earth rotation, the Sagnac effect, the Satellite and Receiver Phase Center Offset, time of pseudorange travelling through atmosphere, ,

(x, y, z) – aircraft’s coordinates in ECEF frame, (XGPS, YGPS, ZGPS) – GPS satellite coordinates, C – speed of light, dtr – receiver clock bias, dts – satellite clock bias, Ion – ionosphere delay, Trop – troposphere delay, Rel – relativistic effect, TGD – Time Group Delay, RDCBL1 – Receiver Differential Code Bias, referenced to L1 frequency, MC/A – multipath effect.

The unknown parameters (e.g. aircraft’s coordinates and receiver clock bias) from equation (1) are estimated using least square solution in stochastic processing, as follows [14]:

(2)

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m02priori

⋅ ml 2

The mathematical formulations from section (2) were utilized for determination of aircraft position in flight experiment on 1st June 2010 in Dęblin (see Figs. 1 and 2). The aircraft trajectory was recovery using GPS code observations from dual-frequency Topcon HiperPro receiver [4]. Raw GPS observations in RINEX file were collected in memory disc in the receiver which was installed in pilot’s cabin in Cessna 172 aircraft. The time of flight test was equal to 3361 measurements epochs with interval of 1 second.

[19],

m0priori – standard error of unit weight a priori, , ml – accuracy of pseudorange, [18],

ml0 – standard deviation of code C/A in GPS system, ml0 = 3 m [20], EL – elevation angle, L = AT ∙ p ∙ dl – misclosure vector, dl – vector include difference between observations and modeled parameters, m0post – standard error of unit weight a posteriori, n – number of observations, n>4, for each measurement epoch, k – number of unknown parameters, k = 4, for each measurement epoch, V – vector of residuals, CQx – covariance matrix, mQx – standard deviations for unknown parameters, parameter mQx is referenced to ECEF frame.

The parameters mQx can be expressed in geodetic frame BLh, as below [15]:

(3)

where: mBLh – covariance matrix in geodetic frame (BLh), mBLh = R ∙ mQx ∙ RT, R – transition matrix from geocentric (XYZ) to geodetic frame (BLh), mB – standard deviation of Latitude, mL – standard deviation of Longitude, mh – standard deviation of ellipsoidal height.

The mathematical scheme in equation (2) is solved in iterative procedure in adjustment processing for each measurement epoch. In addition, the results from equation (2) are checked and controlled using , as follows [19]: global test

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3. The Experiment and Results

Qx – vector with unknown parameters, N = AT ∙ p ∙ A – matrix of normal equation frame, A – full rank matrix, p – matrix of weights, p=

N° 1

Fig. 1. The horizontal trajectory of Cessna 172 aircraft The aircraft’s coordinates were calculated in Aircraft Positioning Software (APS) in Scilab 5.4.1 language. The APS program can be applied in post-processing mode for determination of aircraft position using GNSS data. Currently, the GPS, GLONASS and GPS/GLONASS observations are implemented for SPP (Single Point Positioning) module, IF LC (IonosphereFree linear combination) module, SD-BS (Single Difference Between-Satellites) module and Doppler module in APS program.

(4)

where: f – number of freedom degrees, f = n – k, (1 – α) – significance level, α = 0.05.

If m0post is much higher than 1, then blunder errors from pseudoranges are detected and removed.

In connection with this criterion, the adjustment processing of GPS observations is executed again, as in equation (2).

Fig. 2. The vertical trajectory of Cessna 172 aircraft In this paper, the SPP module in APS program was applied to obtained aircraft position in flight experiment in Dęblin. The basic parameters and input models of SPP module was configurated as below: Articles

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−− GNSS system: GPS system, −− type of observations: C/A code at 1st frequency, −− type of RINEX file: 2.10, −− source of satellite ephemeris data: precise ephemeris from the CODE Analysis Center [22], −− source of satellite clock data: precise ephemeris from the CODE Analysis Center [22], −− method of satellite position computation: 9-degrees Lagrange polynomial [14], −− method of satellite clock bias computation: 9-degrees Lagrange polynomial [14], −− satellite clock bias correction: satellite clock bias from precise ephemeris is corrected using Differential Code Biases in SPP method [18], −− effect of Earth rotation and time of pseudorange travelling through atmosphere: applied, −− relativistic effect: applied [21], −− ionosphere source: Klobuchar model [12], −− troposphere source: Simple model [13], −− Time Group Delays: applied [16], −− instrumental bias RDCBL1: not applied, −− multipath and measurement noise: not applied, −− satellite and receiver phase center offset: based on ANTEX file from IGS service, −− Sagnac effect: applied [14], −− cutoff elevation: 5° [10], −− positioning mode: kinematic, −− mathematical model of solution: least square estimation in iterative scheme, −− adjustment processing: applied, −− maximum number of iteration in single measurement epoch: N=10, −− number of unknown parameters: k=4, for each measurement epoch, −− number of observations: n>4, for each measurement epoch, −− interval of computations: 1 s, −− initial coordinates of aircraft position: based on header of RINEX file, −− receiver clock bias: estimated, −− time of GNSS system: GPS Time, −− reference frame: IGS’08, −− statistical test: test Chi-square, −− value of m0post after adjustment processing: m0post ∊ (0.9 : 1.1) , −− significance level: (1 – α) = 0.95, −− maximum value of DOP coefficients: DOP = 6. −− coefficients value for HPL and VPL level: kHPL = 6 and kVPL = 5.33 [9].

Figure 3 presents values of m0post and parameters as a final results of statistical test Chi-square.

The mean value of m0post equals to 1.002, with range between 0.915 and 1.099. The term m0post is less than parameter for all measurement epochs and it can be concluded that Chi-square test was obtained in the experiment.

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Fig. 3. The values of statistical test Chi-square Figure 4 presents PDOP and GDOP values for each measurement epoch in flight test in Dęblin. The PDOP parameter is a function of position errors only but GDOP term includes error of receiver clock bias as well. The mean value of PDOP term amount to 1.7 with range between 1.3 and 4.0. In case of GDOP term, the mean value equals 2.0 with range between 1.4 and 4.6. The minimum value of GDOP and PDOP parameters is available if number of GPS satellites reach up to 10. The maximum value of GDOP and PDOP parameters is available if number of GPS satellites decreases to 5. It should be noticed that both values of PDOP and GDOP terms are less than maximum value of DOP coefficient (e.g. DOP = 6).

Fig. 4. The values of DOP parameters Figure 5 presents the accuracy (e.g. standard deviation parameter) values of receiver clock bias for each measurement epoch. The standard deviation of receiver clock bias was calculated as in equation (5) [14]:

(5)

The average accuracy of receiver clock bias is about 38.8 ns (in meter unit: 11.6 m) with range between 3.9 ns and 97.6 ns (in meter unit: 1.1 m and 29.3 m). The accuracy of receiver clock bias is irregularly but still growing up almost to 100 ns.

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where: kHPL = 6 – for horizontal plane, kVPL = 5.33 – for vertical plane.

The values of kHP and kVPL parameters in equation (5) are referenced to landing approach APV-I in EGNOS system.

Table 1. The comparison of HPL/VPL value from GPS solution and NPA procedure HPL/VPL parameter

Fig. 5. The standard deviation of receiver clock bias Figure 6 presents the accuracy (e. g. standard deviation term) values of aircraft position in geodetic frame BLh for each measurement epoch. The average value of Latitude accuracy equals to 10.5 m with range between 1.0 m and 24.6 m. The average value of Longitude accuracy equals to 6.5 m with range between 0.5 m and 14.2 m. The average accuracy of each horizontal coordinates is higher than accuracy of ICAO standard (e. g. 17 m). The average accuracy of ellipsoidal height amounts to 13.1 m with magnitude order between 1.8 m and 31.9 m. The average accuracy of ellipsoidal height for Cessna 172 aircraft is higher than ICAO standard in vertical plane (e.g. 37 m).

Fig. 6. The standard deviations of aircraft’s position in geodetic frame BLh

4. Discussion In section (4), the accuracy of aircraft’ coordinates was verified with accuracy of NPA and APV-I procedures. The NPA and APV-I procedures are implemented in Polish aviation based on SoL (Safety of Life) service in EGNOS system. The majority of accuracy parameters in NPA and APV-I procedures are called HPL and VPL terms. The HPL and VPL parameters are calculated using formula (6) [11]:

(6)

HPL value VPL value

Average accuracy of HPL/VPL term based on GPS observations

Theoretical accuracy of HPL/VPL term in NPA procedure

VPL= 69.9 m

Not available

HPL = 74.2 m

556 m

In Table 1, the HPL/VPL results from APS program were compared with NPA standards. The average value of HPL term equals to 74.2 m based on GPS observations in flight test in Dęblin. The results shows that the average value of HPL term is not exceeded the theoretical accuracy of NPA procedure (e. g. 556 m) in horizontal plane and the average value of VPL term equals to 69.9 m. The theoretical accuracy of NPA is still not active in vertical plane. In connection with it, the comparison of VPL value from GPS solution and NPA procedure is currently impossible. This test shows that the range of HPL parameter is between 6.9 m and 170.5 m, whereas the range of VPL parameter is between 9.7 m and 170.3 m, respectively.

Fig. 7. The accuracy of HPL/VPL parameters in comparison to EGNOS APV-I standards The values of HPL/VPL terms from APS program was also compared with EGNOS APV-I standards (see Fig. 7). The accuracy of HPL/VPL terms in EGNOS APV-I procedure are equal to 40 m in horizontal plane and 50 m in vertical plane respectively. The values of HPL term in flight test are much more than EGNOS APV-I standard for horizontal plane (e. g. about 99% results of all measurement epochs). In case of the VPL parameter, about 31% results (e. g. 1045 measurement epochs) from APS program is less than EGNOS APV-I standard for vertical plane. The values of HPL/ VPL terms are still growing up for that experiment Articles

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Journal of Automation, Mobile Robotics & Intelligent Systems

and this is a negative situation in context of safety in air operations. The results of HPL/VPL parameters in flight test in Dęblin are showed that implementation of GPS system in approach APV-I in air navigation is still limited.

5. Conclusions

In this paper, the results of GPS positioning in Polish aviation were presented. The flight test was conducted in military airport in Dęblin on 1st of June 2010 using Cessna 172 aircraft. The aircraft position was recovery based on SPP method for GPS code observations with time interval of 1 s. The raw GPS observations were collected in dual-frequency Topcon HiperPro receiver which was installed in pilot’s cabin in Cessna 172 aircraft. The numerical computations of aircraft’s coordinates were executed in Aircraft Positioning Software (APS) in Scilab 5.4.1 language. The least square estimation in adjustment processing of GPS observations in applied in SPP module in APS program. Numbers of parameters were used in this article, such as the aircraft’s trajectory, standard error of unit weight a posteriori, test Chi-square, GDOP and PDOP, accuracy of aircraft position, accuracy of receiver clock bias, HPL/VPL which were presented in Figure 1 to 7. The results shows that the average accuracy of aircraft position is higher than theoretical accuracy of GPS system for ICAO standard. The values of HPL/VPL terms were compared with NPA and APV-I standards also. The average value of HPL term equals to 74.2 m and it is higher than accuracy of NPA procedure in horizontal plane. In case of the accuracy of EGNOS APV-I standards, the values of HPL parameters are much more than 40 m in horizontal plane.

The average value of VPL term equals to 69.9 m and it can be only compared with EGNOS APV-I standards.

The values of VPL term from 1045 measurement epochs are less than 50 m in vertical plane. Preliminary results analyzed in this paper, indicate that the GPS system must be still monitored in aspect of landing approach in EGNOS APV-I procedure.

ACKNOWLEDGEMENTS

The author would like to thanks for Ph.D. Henryk Jafernik (PAFA, Dęblin) for available RINEX files from flight experiment in Dęblin’2010.

AUTHOR Kamil Krasuski – Faculty of Geodesy, Cartography and Cadastre, District Office in Ryki, 08-500 Ryki, Poland.

E-mail: kk_deblin@wp.pl

REFERENCES

46

[1] K. Banaszek, M. Malarski, “Required navigation performance and risk of airport operations”, Problemy eksploatacji, 4, 2009, 71–78. (in Polish) [2] Ciećko A., Grunwald G., Kaźmierczak R., Grzegorzewski M., Oszczak S., Ćwiklak J., Bakuła M., Articles

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“Analysis of the accuracy and availability of ASGEUPOS services in air navigation and transport”, Logistyka, 3, 2014, 1091–1100. (in Polish) [3] Ciećko A., Grzegorzewski M., Oszczak S., Ćwiklak J., Grunwald G., Balint J., Szabo S., “Examination of EGNOS Safety-Of-Live Service in Eastern Slovakia”, Annual of Navigation, 22. DOI: 10.1515/ aon-2015-0021, 2015, 65–78.

[4] Ćwiklak J., Grzegorzewski M., Oszczak S., Jafernik H., Ciećko A., “The use of EGNOS system for air navigation in eastern Poland”, Problemy eksploatacji, 1, 2011, 57–64. (in Polish) [5] Fellner A., Jafernik H., “Airborne measurement system during validation of EGNOS/GNSS essential parameters in landing”, Reports on Geodesy and Geoinformatics, 96, 2014, 27–37. DOI: 10.2478/rgg-2014-0004.

[6] Fellner A., Fellner R., Piechoczek E., “Pre-flight validation RNAV GNSS approach procedures for EPKT in „EGNOS APV Mielec project””, Scientific Journal of Silesian University of Technology. Series Transport, 90, 2016, 37–46. DOI: 10.20858/ sjsutst.2016.90.4.

[7] Grunwald G., Ciećko A., Oszczak S., Kaźmierczak R., Grzegorzewski M., Ćwiklak J., “The application of EGNOS system in aircrafts monitoring reffered to Safety-Of-Life service activation“, Aparatura Badawcza i Dydaktyczna, 3, 2011, 133–142. (in Polish) [8] Grunwald G., Bakuła M., Ciećko A., Kaźmierczak R., Grzegorzewski M., Ćwiklak J., ”Integrity of satellite positioning in air transport”, Logistyka, 3, 2014, 2231–2238. (in Polish) [9] Grunwald G., Ciećko A., Bakuła M., Kaźmierczak R., ”Examination of GPS/EGNOS integrity in north-eastern Poland”, IET Radar, Sonar & Navigation, 10 (1), 2016, 114–121. DOI: 10.1049/ iet-rsn.2015.0053.

[10] International Civil Aviation Organization, ICAO standards and recommended practices (SARPS), Annex 10 volume I (Radio navigation aids), 2006, The paper is available at website: http://www.ulc.gov.pl/pl/prawo/prawomi%C4%99dzynarodowe/206-konwencje, current version: 27 January 2015. (in Polish) [11] Jokinen A., Feng S., Milner C., Schuster W., Ochieng W., Hide C., Moore T., Hill C., Precise Point Positioning and integrity monitoring with GPS and GLONASS, Paper presented at Conference: European Navigation Conference 2011 in London, United Kingdom, 2011.

[12] Klobuchar J. A., “Ionospheric time-delay algorithm for single-frequency GPS users”, IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, vol. AES-23, no. 3, 1987, 325– 331.

[13] Krasuski K., ”Accuracy analysis of aircraft’s positioning based on GLONASS observations”, Problemy Mechatroniki. Uzbrojenie, Lotnictwo, Inżynieria Bezpieczeństwa, 18 (4), 2014, 33–44. (in Polish) [14] Krasuski K., Wierzbicki D., ”Utilization L2C code for determination of user’s position”, Geodetski

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vestnik, 59 (4), 2015, 789-808. DOI: 10.15292/ geodetski-vestnik.2015.04.789-808.

[15] Osada E., Geodesy, Oficyna Wydawnicza Politechniki Wrocławskiej, ISBN 83-7085-663-2, 2001, Wrocław, 237–238. (in Polish) [16] Petrovski I. G., GPS, GLONASS, Galileo, and BeiDou for Mobile Devices, Published in the United States of America by Cambridge University Press, New York, ISBN 978-1-107-03584-3 Hardback, 2014, 71.

[17] Sanz Subirana J., Juan Zornoza J. Hernández-Pajares M., GNSS Data Processing, Volume I: Fundamentals and Algorithms, ESA Communications, ESTEC, Noordwijk, Netherlands, ISBN: 978-929221-886-7, 2013, 139–151.

[18] Schaer S., Mapping and predicting the Earth’s ionosphere using Global Positioning System, PhD thesis, Neunundfünfzigster Band volume 59, ISBN: 3-908440-01-7, 1999, Zürich, Switzerland, 70, 121–122.

[19] Schüler T., On ground-based GPS tropospheric delay estimation, PhD thesis, Heft 73, Universität der Bundeswehr München, Germany, ISSN: 0173–1009, 2001, 93–96.

[20] Spits J., Total Electron Content reconstruction using triple frequency GNSS signals, PhD thesis, 2011, Universitè de Liège, Belgium, 35.

[21] Xu G., GPS Theory, Algorithms and Applications2nd edition, Springer Berlin Heidelberg New York, Potsdam, Germany, ISBN: 978-3-54072714-9, 2007, 66.

[22] URL1: ftp://ftp.unibe.ch/aiub/CODE/2010/, current on 2016.

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Particle Swarm Optimization for Tuning PSS-PID Controller of Synchronous Generator Submitted: 13th May 2016, accepted: 18th November 2016

Amina Derrar, Abdelatif Naceri DOI: 10.14313/JAMRIS_1-2017/6 Abstract: In this paper the design an optimal PSS-PID controller for single machine connected to an infinite bus (SMIB). We presented a novel application of particle swarm optimization (PSO) for the optimal tuning of the new PSS-PID controller. The proposed approach has superior features, including easy implementation, stable convergence characteristic and good computational efficiency. The synchronous generator is modeled and the PSO algorithm is implemented in Simulink of Matlab. The obtained results have proved that (PSO) are a powerful tools for optimizing the PSS parameters, and more robustness of the system IEEE SMIB.

trol methodology based PSO technique, the proposed algorithm works as online auto tuning for the PSS-PID controller parameters on the real time without time consuming as well as no requiring for tedious efforts.

2. Mathematical Modeling of Power System

In this paper a simplified dynamic model of power system, namely, a single machine connected to an infinite bus (SMIB) is considered. It consists of a single synchronous generator connected through a parallel transmission line to a very large network approximated by an infinite bus as shown in Fig. 1.

Keywords: Synchronous Generator, PSS, particle swarm optimization, PID controller

1. Introduction

48

Over the past decade, more than 90% of industrial controllers are still implemented based on PID control algorithms as no other controller matches the simplicity, effectiveness, robustness, clear functionality and ease of implementation [3] The Power System Stabilizer (PSS-PID) is a device that improves the damping of generator electromechanical oscillations. Stabilizers have been employed on large generators for several decades; permitting techniques applied in the automatic excitation regulator of powerful synchronous generators: the robust stabilizer (PSS-PSO) and (PSS-PID) control schemes against system variation in the SMIB power system, with a test of robustness against parametric uncertainties of the synchronous machines (electric and mechanic), and make a comparative study between these two control techniques for PSS systems.

One of the most recent heuristic algorithms, the particle swarm optimization (PSO), is a population based stochastic optimization technology by Eberhart and Kennedy in 1995, inspired by social behavior of bird flocking and fish schooling. It is used for optimization of continuous nonlinear functions [1, 2].

The fundamental essence of the contribution of this work is to overcome the building of robust controller that has high order than that of the system where the controller is not easy to implement for this system in practical engineering application. This difficulty can be solved by the proposed algorithm that built a robust PSS-PID controller through applying the cognitive con-

Fig. 1. Block schematic diagram of the proposed SMIB Power system controller x1 = Δw

(1)

Let the state variable of interest be the machine’s rotor speed variation and the power system acceleration. x2 = ΔP = Pm – Pe (2)

Where x1 is the speed deviation and x2 is accelerating power, Pm and Pe represents respectively the mechanical and electrical power. It is possible to represent the power system in the following form:

x1 = Δw

α x2 = f (x1, x2) + g (x1, x2)u y = x1

(3) (4) (5)

Where α=1/2H and H is the per unit inertia constant of the machine. x=[x1x2] is the state vector of the system and f(x1,x2) and g(x1,x2) are nonlinear functions and u is the PSS (Power System Stabilizer) control signal.

The PSS-PID controller is well known and widely used to improve the dynamic response as well as to reduce or eliminate the steady state error. The derivative controller adds a finite zero to the open loop system transfer function and improves the transient response. The integral controller adds a pole at the origin, thus increasing system type by one and reducing the steady state error due to a step function to

Journal of Automation, Mobile Robotics & Intelligent Systems

zero. The transfer function of a PSS-PID controller is given in the s-domain as follows:

ΔEf = ω(p). [(ω0u(p) + ω1u(p))ΔU(p) +(ω0ω(p) + ω1u(p)) ωbf(p). Δωu(p) + ωif(p). Δif(p) + ωuf(p). Δuf(p)] (6)

Efd0 – value of the control signals in the steady state generator.

3. Particle Swarm Optimization (PSO) Algorithm

PSO is one of the optimization techniques first proposed by Eberhart and colleagues [4, 8].

The algorithm adopted uses a set of particles flying over a search space to locate a global optimum, where a swarm of n particles communicate either directly or indirectly with one another using search directions, in each iteration of PSO, each particle updates its position.

Based on three components, by determines its velocity using, previous velocity, best previous position, and the best previous position of its neighborhood [5, 7] Figure 2 illustrates the flow chart of PSO algorithm.

The basic concept of PSO lies in accelerating each particle toward the best position found by it so far (pbest) and the global best position (gbest) obtained so far by any particle, with a random weighted acceleration at each time step, this is done by the equations (7) and (8):

Vi(t+1) = ω ∙ Vi(t) + φ1 ∙ r1 ∙ (Pbi(t) – Xi(t)) + φ2 ∙ r2 ∙ (Pg(t) – Xi(t)) (7) Xi(t + 1) = Xi(t) + Vi(t) (8)

Where: Pg = Global Best Position. Pb = Self Best Position. Φ1 and φ2 =Acceleration Coefficients. w = Inertial Weight. Vi = Velocity. Xi = Particle.

Once the particle computes the new Xt it then evaluates its new location. If fitness (X t) is better than fitness (pb), then pb = Xt and fitness (pb) = fitness(X t), in the end of iteration the fitness (Pg) = the better fitness (pb) and Pg = pb

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the objective function, which in this case is the error criterion, and it is discussed in detail. For the PSS-PID controller design, the controller settings estimated results in a stable closed loop system are ensured.

This function must maximize the stability margin by increasing damping factors while minimizing the system real eigenvalues. Therefore, all eigenvalues are in the stability area, the multi-objective function calculating steps are: 1) Formulate the linear system in an open-loop (without PSS); 2) Locate the PSS and its parameters initialized by the PSO through an initial position and acceleration; 3) Calculate the closed loop system eigenvalues and take only the dominant modes:

4) Find the system eigenvalues real parts (σ) and damping factor ζ; 5) Determine the (ζ) minimum value and the (- σ) maximum value, which can be formulated, respectively, as: (minimum (ζ)) and (maximum ˗(σ)); 6) Gather both objective functions in a multi-objective function F as follows:

7) Return this Multi-objective function value the to the AG program to restart a new generation.

5. SIMULATION RESULT

To evaluate the effectiveness of the proposed PSSPSO to improve the stability of power system, the dynamic performance of the proposed PSS was examined under different loading conditions. The performance of the PSO based PSS is compared with the PSS. Randomly i nitialize population p ositions and velocities Evaluate fitness of particle If particle fitness> particle best fitness Update particle best If particle fitness> global best fitness Update global best

Update particle velocity

Fig. 2. Schematic diagram of the displacement of a particle

Update particle position

4. Implementation of PSO Based PSS-PID Controller The optima value of PSS-PID controller parameters K1w, K2w, T1, T2 are to be found using GA. All possible sets of controller parameters values are particles whose values are adjusted to minimize and maximize

end

Fig. 3. Flowcharts of PSO Articles

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Journal of Automation, Mobile Robotics & Intelligent Systems

TBB = 500, Q = 0.1896(pu), XL = 0,5(pu) Initialization Sized swarm = 20.

N

K1w

K2w

T1

T2

segma

1

8.1472

6.5574

0.0437

0.0751

-4.2137

3

1.2699

8.4913

0.0762

0.0505

-0.2249

2

9.0579

4

9.1338

5

6.3236

6

0.9754

7

2.7850

8

5.4688

9

9.5751

10

9.6489

11

1.5761

12

9.7059

13

9.5717

14

4.8538

15

8.0028

16

1.4189

17

4.2176

18

9.1574

19

7.9221

20

9.5949

0.3571

0.0380

9.3399

0.0791

6.7874

0.0186

7.5774

0.0487

7.4313

0.0443

3.9223

0.0643

6.5548

0.0706

1.7119

0.0751

7.0605

0.0275

0.3183

0.0676

2.7692

0.0652

0.4617

0.0162

0.9713

0.0118

8.2346

0.0496

6.9483

0.0955

3.1710

0.0339

9.5022

0.0582

0.3445

Number of iterations = 6.

0.0223

0.0255 0.0698 0.0890 0.0958 0.0547 0.0138 0.0149 0.0257 0.0840 0.0254 0.0813 0.0243 0.0928 0.0350 0.0196 0.0251 0.0615 0.0473

-2.6095 -2.4152 -1.6597 -2.8314 -2.9341 -0.4399 -2.5975 -3.0369 -1.0794 -2.8566 -0.3977 -1.1644 -1.9807 -3.8044 -2.0597

T1

T2

segma

1

8.1472

6.5574

0.0437

0.0751

-4.2137

3

7.8169

5.3741

0.0392

0.0705

-4.4529

4

7.8169

5

7.9242

6

7.9242

5.3741

5.7720 5.7720

0.0437

0.0392

0.0404 0.0404

0.0751

0.0705

0.0720 0.0720

Fig. 4 The iteration

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K2w

6.5574

2017

-0.1659

K1w

8.1472

N° 1

-1.9318

N

2

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-4.2137

-4.4529

-4.5472 -4.5472

The optimized parameters: K1W = 7.9242, K2W = 5.7720, T1 = 0.0404,T2 = 0.0720.

With segma = –4.5472.

Figures show examples of simulation results, respectively, “Ug” the stator terminal voltage; ‘Pe’ the electromagnetic power system, ‘g’ skid, ‘delta’ the internal angle. ξ

PSS-PSO

PSS-PID

NO variation

-0,03594

-0,3568

Mecanique variation

-0,005899

-0,3736

-0,1329

-1 ,544

Electrique variation

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Electrique and mecanique variation Articles

-0,1886

-0,3243

Fig. 5. Functioning system in the nominal regime used of TBB-500 connected to a long line with, PSS-PSO, PSSPID (Ug, Pe, skid, delta)

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5.1. Robustness Tests 1. Electric variation T = 3 s

2. Mechanical variation t = 5 s Electrical and mechanical variation t = 7 s

Fig. 7. Functioning system in the nominal regime used of TBB-500 connected to a long line with, PSS-PSO, PSS-PID (robustness tests) (Ug, Pe, skid, delta) The main advantages of the PSS-PSO controller are the robustness of the system whenever a disturbance occurred and in case of the uncertainty in the parameters.

From the simulation results, the effect of the controller can be realized from decrease of dynamic performances (static errors negligible so better precision, and very short setting time so very fast system, and we found that after few oscillations, the system returns to its equilibrium state even in critical situations (specially the under-excited regime) and granted the stability and the robustness of the studied system.

6. Conclusion

Fig. 6. Functioning system in the nominal regime used of TBB-500 connected to a long line with, PSS-PSO, PSSPID (robustness tests) (Ug, Pe, skid, delta)

In this work the PSO algorithm has been utilized to find the optimal parameters of conventional PSS. The result of PSO technique is compared with classical PID. The system becomes more robust and the dynamic performance of the PSS-PSO is superior than the conventionally tuned PSS under small as well large perturbation. Simulation of the response of the proposed PSS to various disturbances changes in network configuration and system loading have demonstrated the effectiveness of the PSS-PSO. Articles

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AUTHORS

Appendix

Amina Derrar*, Abdelatif Naceri – IRECOM Laboratory, Department of Electrical Engineering UDL – SBA University, BP 98, Sidi Bel Abbes – 22000, Algeria.

Parameters of the used Turbo-Alternators

*Corresponding author

REFERENCES

[1] Mohammed H. al-khafaji, Shatha S. Abdulla alkabragyi, “Particle Swarm Optimization for Total Operating Cost Minimization in Electrical Power System”, Eng. & Tech. Journal, vol. 29, no.12, 2011, 2539–2550.

[2] C. H. Chen, S. N. Yeh, “Particle Swarm Optimization for Economic Power Dispatch with ValvePoint Effects”. In: IEEE PES Transmission and Distribution Conference and Exposition Latin America, Venezuela 2006. DOI: 10.1109/TDCLA.2006.311397.

[3] Ahmed Sabah Al-Araji, “Applying Cognitive Methodology in Designing On-Line Auto-Tuning Robust PID Controller for the Real Heating System”, Journal of Engineering, vol. 20, no. 9, September 2014, 43–61.

[4] Fadhil A. Hassan, Lina J. Rashad, “Particle Swarm Optimization for Adapting Fuzzy Logic Controller of SPWM Inverter Fed 3-Phase I.M”, Eng. & Tech. Journal, vol. 29, no. 14, 2011, 2912–2925.

[5] Fadel Mohammed , Bassam Abdelelah Kidher Mahmood, “Design of Fractional Order PID Controller Based Particle SWARM”, Diyala Journalof Engineering Sciences, vol. 7, no. 4, December 2014, 24–39.

[6] Jin-Kao Hao, Philippe Galinier, Michel Habib, “Méthaheuristiques pour l’optimisation combinatoire et l’affectation sous contraintes”, LERIA, U.F.R. Sciences, Université d’Angers, 2 bd Lavoisier, 49045 Angers. Avalaible at : http://www.info. univ-angers.fr/pub/hao/papers/RIA.pdf.

[7] D. E. Ghourad et al., “Exploitation des techniques fréquentielles avancées dans le contrôle automatique d’excitation des machines synchrones”, Journal of Advanced Research in Science and Technology, 2014, 1(2), 58–77.

[8] K.E. Khoshmardan, M.R.Dastranj, M.O. Taleghani, A. Hajipoor, “Design a Fuzzy Logic Based Speed Controller for DC Motor with Particle Swarm Optimization “PSO” Algorithm”, Australian Journal of Basic and Applied Sciences.Azad University, Sabzevar, Iran.2011, 1283–1290.

[9] C. Thanga, S.P. Snvastava, P. Agarwal, “Particle Swarm and Fuzzy Logic Based Optimal Energy Control of Induction Motor for a Mine Hoist Load Diagram”, IAENG International Journal of Computer Science, vol. 36, no. 1, 2009.

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Parameters

TBB-500

Units of measure

Xd

1.869

p.u.

Nominal power Power Factor

500

MW

0.85

Xq

p.u.

1.5

Xs

p.u.

0.194

Xf

p.u.

1.79

Xsf

p.u.

0.115

Xsfd

p.u.

0.063

Xsf1q

p.u.

0.0487

Xsf2q Ra

p.u.

0.0487

p.u.

0.0481

p.u.

0.0055

Rf

0.000844

R2q

0.115

R1d R1q

p.u.

p.u.

0.061

p.u. p.u.

Parameters of the Regulator AVR Parameter

SG: TBB-500

K1ua

-7

T1u Te

K0ua

0.039 0.04 -50

Parameters of the used conventional PID-PSS Parameter

SG: TBB-500

K1ua

-8

T1u Te

K0ua Tfc

0.039 0.04 -15

0.07

T1 ω T0 ω K1 ω K0 ω Tif

0.026

Kuf

1

Kif

Tuf

Parameters of the used PSO PSO Property

*Size of the swarm *C1 *C2 *w * The range of adjustment parameters *Objective function

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N° 1

1 1 2

0.03 -1

0.05

Value/Method 100 0.4 0.4 0.1 5<K1W<10, 5<K2w<10 0.0005<T1<0.1, 0.0001<T2<0.1 σ)+min(ξ)

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Fuzzy Switching for Multiple Model Adaptive Control in Manipulator Robot Submitted: 26th August 2016; accepted: 17th February 2017

Behrouz Kharabian, Hosseyn Bolandi, Seyed Majid Smailzadeh, Seyed Kamaledin Mousavi Mashhadi DOI: 10.14313/JAMRIS_1-2017/7 Abstract: In this paper, fuzzy logic is used to perform switching controllers for Multiple Model Adaptive Control (MMAC) in manipulator robot. In the cases which uncertainty bounds of system’s parameters are large, the performance and stability issue of system are considerable concerns. Multiple Model Adaptive Control approach can be useful method to stabilize these kinds of systems. In this control method, the uncertainty bound is divided into several smaller bounds. As a result, the process of stabilization would be streamlined. In this regard, one estimation is obtained for uncertain parameter in every minor bound, and based on estimation errors designed controller can alter. In order to avoid switching controllers and pertinent challenges a summation of controllers with coefficient tuned by fuzzy logic is considered. Simulation results substantiate the efficacy of this method.

Keywords: manipulator robot, fuzzy logic, multiple model adaptive control, switching

1. Introduction In every system, uncertainty can have different origins like inaccuracy in measurement tools, changes in features of system and other factors related to the nature of system. If uncertainty bounds of system parameters are small, negative effects of uncertainty can be negligible. In contrast, if uncertainty bounds of parameters are large, instability and unacceptable performance can arise. Multiple Model Control approach is an efficacious control method to ameliorated unpleasant influences of uncertainty. In this control approach, the uncertainty bound is divided into multiple bounds and in every bound an estimation is obtained.

Estimation errors are used to alter controller and appropriate controller is inserted in closed loop system.

Kuipers and Ioannou [1] used conventional MMAC in two-cart most-spring damper system with uncertainty in spring parameter and delay in input signal. Jin and Li [2] utilized MMAC with weighted control signal which used a performance index based on errors of estimation. In this paper, fuzzy logic is used to tune coefficients and controllers. Fuzzy approach as an intelligent control approach has more flexibility and causes controllers’ coefficients to be adjusted based on desires of designer in a way that controller which has low estimation error has more influence on total

control signal. In prior weighting method, like what is in [3], excessive gains for corresponding controllers result in having high gain for total control signal.

This issue can eventuate in probability of instability.

In fuzzy weighing method, membership function of coefficients can be designed so that coefficient of inappropriate controller have less overlap; therefore, appropriate controller has noteworthy effect.

This paper is organized as follows: section 2 describes the dynamic model of manipulator robot. In section 4, fuzzy weighting method is explained. Section 5 includes simulation results. In final, conclusions are rendered in section 6.

2. System Dynamic

In this paper, a two DOF manipulator is considered as system. The first joint is Revolute and the second joint is Prismatic. Figure 1 shows this manipulator. In order to obtain dynamic model of system Lagrangian equation is used.

(1)

where (θ1, d2) and ( ) are displacement and derivative of displacement in joints. £ is Lagrangion of system. τ is exerted torque on joints. Therefore, dynamic equation of system is modeled as:

(2)

,

, .

where M (θ1, d2) is inertia matrix. is vector of the cariolis and centrifugal force. mi is mass of ith link. Ii is inertia tensor of ith link. θ1 is rotation of first joint. d2 is displacement of second joint with respect to first joint. l1 is distance of mass center of first link from first joint. In this system, uncertainty is presumed to be in l1.

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(5)

where c1 and c2 are positive constants. In this control method, switching process causes instability probability. Hence, appropriate measures is needed to reduce the possibility of unstable behavior.

Fig 1. Schematic of manipulator robot

4. Fuzzy Weighting Method in MMAC

3. Multiple Model Adaptive Control Multiple Model Adaptive Control is a versatile approach to stabilize system with large uncertainty bound. In systems which uncertainty bounds are large, dynamic characteristic of system alters drastically. As a consequence, system is prone to be unstable. In MMAC method uncertainty bounds are divided and in smaller bounds more accurate estimations are obtained. Estimation errors are used to select the controller block related to smallest estimation error and appropriate controller is inserted in closed loop system. Figure 2 depicts the configuration of MMAC.

In order to avoid encountering outcomes of switching in MMAC, a summation of weighted controllers is used as pertinent controller instead of switching among several controllers. In this approach, coefficients of controllers are tuned based on estimation errors by means of fuzzy logic. Consequently, total control signal is presented as:

(6)

where αi is coefficient of ith controller. The configuration of fuzzy weighting approach is shown in Figure 3.

Fig 3. Configuration of MMAC with fuzzy weighting approach Fig 2. Configuration of MMAC Sliding mode control is considered as control method. Hence, exerted torque is presented as:

τi= s – ki sat(s) (3)

where ki is positive constant which is obtained as

54

(4)

and s is sliding surface.

Articles

Membership functions for estimation errors is shown in Figure 4. Membership functions of controllers’ coefficients are shown in Figure 5. Fuzzy rule is considered as below to tune coefficients of controllers.

– If ej=i is small and s are medium or big, Then αj=i is big and s are small.

In this rule, ej=i is the smallest estimation error, ej=is are other estimation errors, αj=i is the coefficient of controller designed based on uncertain system which is in uncertainty bound with smallest estimation error, and s are other coefficients.

In order to analyze the stability of system by use of this control method energy function is considered as:

(7)

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(9)

Based on designed controller, inequality (8) is vindicated to confirm the stability situation of system as is shown in Figure 6.

Fig 4. Membership function of estimation errors

Fig 6. Vindication of stability based on the passivity method

Fig 5. Membership function of coefficients of controllers

Based on passivity theory for stability proof of the system the following inequality should be vindicated.

(8)

5. Simulation Results

Fig7. Quality of tracking the desired trajectory in Revolute joint

In this paper, manipulator robot is considered whose first joint is Revolute and the second joint is Prismatic. Uncertain parameter of system is length of vector between first joint and center of mass of first link. Dynamic parameters of manipulator robot are presented in Table 1.

Table 1. Dynamic parameters of system m1

1.2

m2

1.2

I1

0.1

I2

l1

0.1

0.6

l2

0.6

c1, c2, γ1 and γ2 are considered as the following matrices and k according to Equation (4) is obtained.

The uncertain parameter alters in the following interval.

Fig 8. Quality of tracking the desired trajectory in Prismatic joint Exerted torque as control law causes manipulator robot joints to track the desired trajectories. Figures 7 and 8 show the quality of tracking.

6. Conclusions

In this paper, fuzzy weighting method was used to substitute for switching method in Multiple Model Articles

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Journal of Automation, Mobile Robotics & Intelligent Systems

Adaptive Control. This method provides assurance regarding stable behavior of system with the presence of uncertain parameter with large bound. Fuzzy logic tunes coefficients of designed controllers forming total controller. Consequently, control system eliminates sharp jumps. Simulation results display appropriate outcomes obtained by proposed approach.

AUTHORS

Behrouz Kharabian*, Hosseyn Bolandi, Seyed Majid Smailzadeh and Seyed Kamaledin Mousavi Mashhadi – Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran. E-mail: beh16819@yahoo.com. *Corresponding author.

REFERENCES

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[1] M. Kuipers, P. Ioannou, „Practical Robust Adaptive Control: Benchmark Example”. In: American Control Conference, 2008. DOI: 10.1109/ ACC.2008.4587315.

[2] X. Z. Jin, Q. Li, „The Application of Multiple Model Adaptive Control to Superheated Steam Temperature”. In: Proceedings of the Eighth International Conference on Machine Learning and Cybernetics, Baoding, 2009, 12–15.

[3] N. Sadati, R. Ghadami, „Adaptive multi-model sliding mode control of robotic manipulators using soft computing”, Neurocomputing, vol. 71, no. 13–15, 2008, 2702–2710. DOI: 10.1016/j. neucom.2007.06.019.

[4] R. Vinodha, S. Abraham Lincoln, J. Prakash, „Multiple Model and Neural based Adaptive Multi-loop PID Controller for a CSTR Process”, International Journal of Electrical and Computer Engineering, vol. 2, no. 8, 2009, 412–417.

[5] S. Kamalasadan, A. A. Ghandakly, „A Novel Fuzzy Multiple Reference Model Adaptive Controller Design”, International Journal of Fuzzy Systems, vol. 8, no. 3, 2006.

[6] M. C. Turner, D. J. Walker, „Linear quadratic bumpless transfer”, Automatica, vol 36, no. 8, 2000, 1089–1101.

[7] L. Giovanini, A. W. Ordys, Michael J. Grimble, „Adaptive Predictive Control using Multiple Models, Switching and Tuning, Adaptive Predictive Control using Multiple Models, Switching and Tuning”, International Journal of Control, Automation, and Systems, vol. 4, no. 6, 2006, 669–681.

[8] S. S. Ge, F. Hongand T. H. Lee, „Adaptive Neural Control of Nonlinear Time-Delay Systems With Unknown Virtual Control Coefficients”, IEEE Tran. on systems, man, and cybernetics – part b: cybernetics, vol.34, no. 1, 2004, 499–516.

[9] Y. Jia, H. Kokame, J. Lunze, „Simultaneous Adaptive Decoupling and Model Matching Control of a Fluidized Bed Combustor for Sewage Sludge”, Articles

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[10] D. F. Wang, P. Han, G. Y. Wang, H.M. Lu, „MultipleModel Adaptive Predictive Functional Control and its Application”. In: Proceedings of the First International Conference on Machine Learning and Cybernetics, Beijing, 2002.

[11] Y. Zhang, T. Chai, Y. Fu, H. Niu, „Nonlinear Adaptive Control Method Based on ANFIS and Multiple Models”. In: Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, P.R. China, 16–18 Dec., 2009). DOI: 10.1109/ CDC.2009.5399518.

[12] B. M. Mirkin, P.O. Gutman, „Output feedbackmodel reference adaptive control for multi-input– multi-output plants with state delay”, Systems & Control Letters, vol. 54, no. 10, 2005, 961–972.

[13] A. Sassi, C. Ghordel, A. Abdelkrim, „Multiple model adaptive control of complex systems”.

In: Proceedings of International Conference on Control, Engineering & Information Technology (CEIT’14), 2014, 229–235.

[14] B. Chaudhuri, R. Majumder, B. C. Pal, „Application of Multiple-Model Adaptive Control Strategy for Robust Damping of Interarea Oscillations in Power System”, IEEE Tran. on Control Systems Technology, vol. 12, no. 5, 2004, 727–736.

[15] H. Ke, W. Li, „Adaptive Control Using Multiple Models without Switching”, Journal of Theoretical and Applied Information Technology, vol. 53, no. 2, 2013, 229–235.

[16] S. Blazic, I. Skrjanc, „A Robust Fuzzy Adaptive Control Algorithm for a Class of Nonlinear Systems”, Adaptive and Natural Computing Algorithms, vol. 7824, 2013, 297–306.

[17] D. Gao, Z. Sun, B. Xu, „Fuzzy Adaptive Control for Pure-feedback System Via Time Scale Separation”, International Journal of Control, Automation and Systems, vol. 11, no. 1, 2013, 147–158.

DOI: 10.1007/s12555-010-0011-4.

[18] X. Zhao, P. Shi, X. Zheng, „Fuzzy Adaptive Control Design and Discretization for a Class of Nonlinear Uncertain Systems”, IEEE Trans. Cybern, vol. 46, no, 6, 2016, 1476–1483. DOI: 10.1109/ TCYB.2015.2447153.

[19] A. Boulkroune, M. Tadjine, M. Msaad, M. Farza, „Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction”, Fuzzy Sets and Systems, vol. 161, no. 6, 2010, 797–820. DOI: 10.1016/j.fss.2009.04.011.

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An Efficiency No Adaptive Backstepping Speed Controller Based Direct Torque Control Submitted: 19th October 2016; accepted: 1st March 2017

Abdelkader Ghezouani, Brahim Gasbaoui, Jamel Ghouili, Asma Amal Benayed DOI: 10.14313/JAMRIS_1-2017/8 Abstract: The most problem of direct torque control are high torque ripple and Settling time to overcome this problem an efficiency Backstopping speed controller are proposed.

This paper makes a comparison of the effectiveness of three PI speed controller based direct torque control, the first one is the classical PI speed controller (CL-PISC), the second are no Adaptive Backstepping controller (NABACKSC), and the third type are adaptive fuzzy PI controller (AF-PISC). The parameters of adaptive fuzzy PI are dynamically adjusted with the assistance of fuzzy logic controller. The non-Adaptive Backstopping controller is designed based on the Lyapunov stability theorem. The direct torque control is very adapted for electric propulsion systems; we apply this new strategy for an 15 Kw induction motor. The proposed PI controllers are simulated in MATLAB SIMULINK environment. The simulation results confirmed that the NA-BACKSC, present robust and the best dynamic behavior on direct torque control compared to AF-PISC and CL-PISC. Keywords: backstepping, induction motor, DTC, PI controller, fuzzy controller

1. Introduction Last twenty years the Induction motor is one of the most widely used actuator for industrial applications because of its reliability, ruggedness and relatively low cost. The control of induction motor system is challenging, since the dynamical system is multivariable, coupled, and highly nonlinear. Among the most appropriate commands to the electric propulsion system is the direct torque control.

Direct torque control (DTC) is a closed-loop control technique for induction machine, which implementation is based on hysteresis comparators .In this method, control variables are torque and stator flux of induction machine. This technique was initially proposed in [1, 2]. The main advantages of DTC are robust and fast torque response, no requirements for PWM pulse generation and current regulators, as well as good steady-state and dynamic performances. In this work the design of Backstopping to control a winding system is proposed in order to improve the performances of the control system, which are coupled mechanically, and synthesis of the robust control and application to synchronize and to maintain a constant

mechanical tension between the conrollers of the system. The advantage of Backstepping control is its robustness and ability to handle the non-linear behavior of the system. The model of the winding system, and in particular the model of the mechanical coupling, are developed and presented in Section (2). Section (3) shows the direct field oriented control (FOC) of induction motor Section (4) shows the development of Backstepping technique control design. The Speed Control of Each induction machine by Backstepping controllers design is given in section (5). Simulation results using MATLAB SIMULINK of different studied cases is defined in Section (6). Finally, the conclusions are drawn in Section (7). In this work, a No Adaptive Backstepping controller was analysed and applied to the control of direct torque control of the asynchronous machine. Simulation tests showing a remarkable behavior of Non-Adaptive Backstepping controller in regulation and prosecution, a disturbance rejection significantly better than other regulators, very good performance and robustness.

2. Direct Torque Control Strategy

The basic DTC strategy is developed in 1986 by Takahashi [3]. It is based on the determination of instantaneous space vectors in each sampling period regarding desired flux and torque references. The block diagram of the original DTC strategy is shown in Figure 1. The reference speed is compared to the measured one. The obtained error is applied to the speed regulator PI whose output provides the reference torque. The estimated stator flux and torque are compared to the corresponding references. The errors are applied to the stator flux and torque hysteresis regulators, respectively. The outputs of the stator flux and torque regulators and the phase of the stator flux are applied to the space vector selection table block which generates the convenient combinations of the states (ON or OFF) of the inverter power switches. There are eight switching combinations, two of which correspond to zero voltage space vectors which are (000) and (111).

The stator flux is controlled by a tow-level hysteresis regulator, where the logical function takes ‘‘+‫‏‬1’’ to increase and ‘‘-1’’ to decrease it. The electromagnetic torque is controlled by its hysteresis regulator, where the logical function gives not only the states ‘‘+‫‏‬1’’ and ‘‘-1’’ (increase/decrease), but also ‘‘0’’ to hold [4].

The estimation value of flux and its phase angle is calculated in expression 2, 3, 4 and 5, respectively.

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(2)

(3)

(4)

(5)

(6)

(8)

(9)

Where: y(t) is the output of the control, e(t) is the input of the control, reference Ω*(t) is the reference speed, Kp and Ti are the parameter of the scale and of the integrator. The correspondent discrete equation is: (10)

(7)

Where: Cem is the electromagnetic Torque, Cr is a Load Torque, Ω is the phase rotor speed, J, p and B are the inertia, number of pairs of pole and fractional coefficient.

3. Controller Design

Where: y(k) is the output on the time of k the sampling, e(k) is the error on the time of k sampling, T is the cycle of the sampling, and

3.1. Adaptive Fuzzy PI Controller

Fuzzy controllers have been widely applied to industrial process. Especially, fuzzy controllers are effective techniques when either the mathematical

Inverter

Induction Motor Load

Sa

Sb

Sc

Clark Transformation

Switching Table

Stator flux

Torque Estimator

Estimator Sector Calculation

ϕe s t i

* ϕ ref

Flux hysterisis

Tem

comparator

Torque hysterisis comprateur

Ω Tem*

Fig. 1. Basic Direct Torque Control Scheme for AC Motor Drives 58

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model of the system is nonlinear or not the mathematical model exists. In this paper, the fuzzy control system adjusts the parameter of the PI control by the fuzzy rule. Dependent on the state of the system, the adaptive PI realized is no more a linear regulator according to this principle. In most of these studies, the Fuzzy controller used to drive the PI is defined by the control by the fuzzy rule.

Dependent on the state of the system, the adaptive PI realized is no more a linear regulator according to this principle. In most of these studies, the Fuzzy controller used to drive the PI is defined by the authors from a series of experiments [7]–[8]. The expression of the PI is given in the Equation (8):

Where: fsα, fsb are the a and b axes stator Flux, fs is the stator Flux, qs is the phase angle.

And the torque is controlled by three-level hysteresis. Its estimation value is calculated in expression (7).

N° 1

Backstepping speed controller

* Ω ref

(11)

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Simple transformations applied to equation (11) lead to:

The online tuning equation for Kp and Kpi are show above:

(13)

∆e Ω

*

e(Ω)

T

Speed PI Controller

Kp

∆ eT

∗ em

Ki

Tem

Fuzzy Controller

N

∆e(Ω)

ZE

ZE

P

B

S

B

S

B

B

B

Table 2. Fuzzy tuning rules of Ki

B

e(Ω)

N

∆e(Ω)

ZE

N

ZE

P

B

S

B

S

B

B

P

M

B

B

medium and small (tuning rules given in Table 1 and Table 2), and the membership function is illustrated in Figure 3 for gain Kp and Figure 5 for gain Ki. Using the settings given in Table (1 and 2) the fuzzy controllers were obtained and are given in Figure 4 and 6.

Ti

d dt

N

B

P

(14)

The frame of the fuzzy adaptive PI controller is illustrated in Figure. 2.

The linguistic variables are defines as {N, ZE, P, B, M, S} meaning negative, zero error, positive, big,

0.8 0.6 0.4 0.2 1

Adaptive Fuzzy PI controller

0 derr

-1

-1

1

0 err

(a)

(a)

Degree of membership

(c)

P

0.5 0 -1.5

1

-1

-0.5

N

0 err

0.5

1

1.5

Ze

(b)

P

0.5 0 -1.5

1

-1

-0.5

0 derr

0.5

1

S

1.5

(c)

B

0.5 0 -0.5

0

0.5 Ti

1

Degree of membership

Ze

1.5

Fig. 3. Membership function for (a): error e(k), (b) error derivate ∆e(k), (c) output Kp

Degree of membership

Degree of membership

(b)

N

Degree of membership

Fig. 4. View plot surface of fuzzy controller for Kp

Degree of membership

Fig. 2. PI gains online tuning by fuzzy logic controller

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Table 1. Fuzzy tuning rules of KP

(12)

3.2. Online Tuning

N° 1

1

N

Ze

P

0.5 0 -1.5

1

-1

-0.5

0 err

N

0.5

1

Ze

1.5

P

0.5 0 -1.5

-1

-0.5

0 derr

S

1

0.5

M

1

1.5

B

0.5 0 0

0.2

0.4

Ti

0.6

0.8

1

Fig. 5. Membership function, (a) error e(k), (b) error derivate ∆e(k), (c) output Ki Articles

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0.6

0 derr

-1

0

-1

1

err

3.2. Backstepping Speed Controller

(15)

where:

(16)

Are constant parameters which are related to the motor parameters. The first step of the Backstepping control is defined log error of the state variable by the following calculation. The speed error:

(17)

Then the derivative of speed track error can be represented as:

with:

(18)

(19)

(20)

Then:

Subsequently we define the Lyapunov function of the form:

In order to guarantee control input:

60

(21)

(22)

we select the following

By substting (22) into (23), we can obtain:

(23) (24)

From equation (24), we can conclude that the system is stable. By integrating equation (24), we can obtain: Articles

d dt

Ω ref

Its derivative gives:

(26)

The block diagram of the proposed non Adaptive Backstepping control system is shown in Figure 7.

From equation (15), it is not difficult to drive:

(25)

and is bounded. According Then, to Barbalet Lemma [5–6], we can conclude

Fig. 6. View plot surface of fuzzy controller for Ki

2017

From equation (23) the integrating of parameter of the equation (24) is less than infinite.

0.4 1

N° 1

1 a

eΩ

Limiter

K a −

Tref

1 Js + B

Ω

c a

Fig. 7. The No adaptive Backstepping speed Controller

4. Simulation Results The control scheme described in Figure 1 was tested by simulations and to evaluate the performance of the proposed structure, we have implemented on the Matlab / Simulink environment. To examine the performance and robustness of our controller we undergo our system to several test servers. The simulation results of the efficiency Non Adaptive Backstopping speed controller based DTC will be compared with adaptive Fuzzy PI speed controller and conventional PI speed control schema. The parameters of the induction motor used in the simulation are shown in Appendix.

4.1. Constant Speed Application

The simulation conditions are given as follows: the speed is 100 rd/s and the reference flux is 0.98 Wb; the initial load torque is 0 N m. According to the Figure 8., shown below, there is an excelling response time in setting time by NA-BACKSC (the speed reaches the reference value after t = 0.21 s for the Backstepping controller and t = 0.35 s for the other two types of controllers) which reduces the time of the transitional system, and improve the saveing energy.

You can also see a significant overshot (D = 2.5%) for the CL-PISC. By against the Steady-state errore converges to zero.

Figure 9. and Figure 10. shows the variation of electromagnetic torque and current, respectively.

4.2 Load Torque Application

To test the robustness of induction motor based DTC using three types of regulators, is to introduce a nominal load torque 35 Nm betwene t = 1 s and 1.5 s.

Journal of Automation, Mobile Robotics & Intelligent Systems

140

2

200

4 3

100 80

110

1

60

100

40

90

20

80

0 0

0.5

0.15

0.2

0.25

1

0.3

0.35

1.5

2

1

0.25

1

1.5

0.2

0

0

2

0.5

2

2.5

Temps [s]

Fig. 9. Elecromagnetic torque response. (1) CL-PISC, (2) Reference, (3) NA-BACKSC, (4) AF-PISC

Figure 13 illustrates clearly the robustness of the proposed PI controller more particularly for speed of response a reverse of speed responses of the reference there of to 100 rad/s â&#x20AC;&#x201C; 100 rad/s. The torque climbs to nearly 10 N m, when the motor starts and stabilizes rapidly when the motor reaches the reference value Figure 15 and 16 shows the variation of current. It can be concluding that the proposed non adaptive Backstepping controllers are robust. The stator current present slight ripple for reversing the direction of rotation of the speed.

Figure 17 Shows that the flux of the DTC controller offers the fast transient responses that mean the trajectory of stator flux established more quickly than that of the Conventional Direct Torque Control.

Figure 18 shown the stator flux trajectory for the different speed controller.

2

2.5

Load Torque

80

1

102

60

100

40

4

98

20

96 0.98

0 0.5

1

1.02 1.04 1.06 1.08

1

1.5

2

2.5

Temps [s]

Fig. 11. Speed reponse. (1) AF-PISC, (2) Reference, (3) CL-PISC, (4) NA-BACKSC 3

Torque [N.m]

100

60

2

50

40

1

20 0 0.98

1

1.02 1.04

0

4

-50

-100

0

0.5

1

1.5

2

2.5

Temps [s]

Fig. 12. Electromagnetic torque response. (1) CL-PISC, (2) Reference, (3) AF-PISC, (4) NA-BACKSC 200 50 150

Current [A]

4.3 Inverse Rotation Speed

1.5

100

0

According to Figure 11 the speed reponse stabilizes at the desired reference value and the same for the perturbation effect when applying the load torque it appears that a small decrease in speed (2.5 rad/s for a CL-PISC and 0.05 rad/s for NA-BACKSC).

The time necessary to eliminate the disturbance effect is faster with AF-PISC compared to the CL-PISC. It is very intersting to shows that NA-BACKSC are insensitive to this variation of the load torque Figure 11, the stator current increased proportionally to that applied load torque Figure 12. Furthermore, the electromagnetic torque acts very quickly to follow the load torque and has introduced a remarkable reduction of harmonics in the case of CL-PISC and AF-PISC Figure 13. and the introduction of perturbations is immediately rejected by the control system.

1

3

120

4 0.5

0.15

Fig. 10. Current response. (1) NA-BACKSC, (2) CL-PISC, (3) AF-PISC

0.3

-50

0

0.1

Temps [s]

0

-100

-20 50

140 0.2

0

100

-100

Speed [Rad/s]

Torque [N.m]

50

60 40 20 0 -20

2

20

1

-50

2.5

Temps [s]

3

2017

0.4

Fig. 8. Speed response (rad/s). (1) AF-PISC, (2) Reference (3) CL-PISC, (4) NA-BACKSC 100

N° 1

2&3

150

Current [A]

Speed [Rad/s]

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100

1

0 -50

50

1

1.2

1.4

1.6

0 -50 -100

0

0.5

1

1.5

2

2.5

Temps [s]

Fig. 13. Current response. (1) NA-BACKSC, (2) CL-PISC, (3) AF-PISC Articles

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(a)

2

100

1

50 -80 0

4

-90

-50

-100

-100

-110 0

1.6

1.8

0.5

2

3

1

1.5

2

0.5 0 -0.5 -1

1

0

4

-50

-100

0

0.5

1

1.5

2

0.5 0 -0.5 -1 -1.5

2.5

(c)

Courant [A]

150

2&3

50 0 -50 -100

0.5

1

1.5

2

0.5 0 -0.5 -1

2.5

Temps [s]

Fig. 17. Current response. (1) NA-BACKSC, (2) L-PISC, (3) AF-PISC

1

1

-1.5 0

0

1.5

Phis-Beta [Wb]

200

-1

Phis-Alpha [Wb]

Fig. 16. Elecromagnetic torque response. (1) CL-PISC, (2) Reference, (3) NA-BACKSC, (4) AF-PISC

1

1

1

Temps [s]

100

0

1.5

Phis-Beta [Wb]

Torque [N.m]

(b)

3 2

-1

Phis-Alpha [Wb]

Fig. 15. Speed reponse. (1) AF-PISC, (2) Reference, (3) CL-PISC, (4) NA-BACKSC

50

-1

0

1

Phis-Alpha [Wb]

Fig. 18 The stator flux circle. (a) CL-PISC, (b) NA-BACKSC, (c) AF-PISC

Table 4. Comparison of simulation results Performance Index Rise time Peak of Electromagnetic Torque [N·m] Current amplitude [A] Disturbance rejection Time [s] The time reverse speed [s] Overshot [rad/s] Design

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Controller Design Classical PI Controller

Fuzzy PI adaptive Controller

No Adaptive Backstopping Controller

0.264

0.263

0.209

122

122

172

0.5

0.5

Simple

Difficult

47

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-1.5

2.5

Temps [s]

100

N° 1

1.5

Phis-Beta [Wb]

Speed [Rad/s]

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0.5

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Simple

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4.4. Comparative study

AUTHORS

Table 4 shows a comparison stadies between the results obtained by direct torque control (DTC) shemas using classical PI controller, adaptive fuzzy PI controller and no adaptive backstepping PI speed controller. It is clearly that the no adaptive controller Backstepping offers better performances in both of the overshoot control and the tracking error and eliminate torque peaks. However, the adaptive Fuzzy PI controller remains average compared to non adapatatif Backstepping controller.

Abdelkader Ghezouani, Brahim Gasbaoui* and Asma Amal Benayed – Faculty of Sciences and Technology, Department of Electrical Engineering Bechar University B.P 417 BECHAR (08000), Algeria.

E-mail: gasbaoui_2009@yahoo.com.

5. Conclusion

The research outlined in this paper has demonstrated the feasibility of an effechency backstepping controller using direct torque control. The results obtained by simulation show that this structure permits the realization of the robust control based on Fuzzy inference system, with good dynamic and static performances for induction motor control. The proposed no adaptive Backstepping speed controller model improve the speed and torque reponses and gives a good riseing time and no overshot. From the foregoing results it’s clear that the No adapative Backstepping speed controller is effective for further instructions and disturbance rejection of the induction machine.

Appendix

Induction Motor Parameters Parameter name

Symbol

Value

Unit

Rotor Inductance

Lr

0.0651

Ls

0.0651

H

Mutual Inductance

Lm

0.0641

Rs

0.2147

W

Rotor Resistance

Rr

0.2205

W

Number of Pole Pairs

p

2

Motor-Load inertia

J

0.102

Rated Power

PN

15

U

380

Nominal Frequency

fN

50

B

0.009541

Stator Inductance

Stator Resistance

Rated Voltage

Viscous Friction coefficient

H 

kg · m2

Gamel Ghouili – Department of Electrical Engineering, Moncton University (Canada). *Corresponding author

REFERENCES

[1] Baader U., Depenbrock M., Gierse G., “Direct Self Control (DSC) of Inverter Fed Induction Machine.

A Basis for Speed Control without Speed Measurement”, IEEE Transaction on Industry Applications, vol. 28, no. 3, May/June 1992, 581–588.

[2] Kukrer O., “Discrete-time Current Control of Voltage-fed Three-phase PWM Inverter“, IEEE Transaction on Power Electronics, vol. 11, no. 2, March 1996, 260–269.

[3] Takahashi I., Noguchi T., “A New Quick-response and High Efficiency Control Strategy of an Induction Motor“, IEEE Transaction on Industrial Applications, vol. IA–22, No. 5, Sept. 1986, 820–827.

[4] Vasudevan M., Arumugam R., Paramasivam S., “High-performance Adaptive Intelligent Direct Torque Control Schemes for Induction Motor Drives“, Serbian Journal of Electrical Engineering, vol. 2, No. 1, May 2005, 93–116.

[5] Kristic M., Kanellakopoulos I., Kokotovic P. V., Nonlinear and Adaptive Control Design, New York: John Wiley and Sons Inc. 1995.

[6] Tao, G, “ Adaptive Control Design and Analysis“, (New Jersey: Wiley-Interscience, 2003.

[7] Gupta. A., Khambadkone. A. M., “A space vector pwm scheme for multilevel inverters based on two-level space vector pwm,” IEEE Transaction on Industrial Electronics, vol. 53, October 2006.

[8] Sun D., He Y., “Space vector modulated based constant switching frequency direct torque control for permanent magnet synchronous motor”.

In: Proceedings of the CSEE, vol. 25, no. 12, 2005, 112–116.

KW

Volt

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Acknowledgements This work was supported by Laboratory of Smart Grids & Renewable Energies (S.G.RE). Faculty of technology, Department of Electrical Engineering, Bechar University, Algeria.

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Self-sensing Teleoperation System Based on 1-dof Pneumatic Manipulator Submitted: 20th September 2016; accepted 16th March 2017

Mateusz Saków, Karol Miądlicki, Arkadiusz Parus DOI: 10.14313/JAMRIS_1-2017/9 Abstract: Paper presents a novel approach to a control design of bilateral teleoperation systems with force-feedback. The problem statement, analysis of research achievements to date, and the scope of the study are presented. The new design of a control unit for a master-slave system with force-feedback is presented on a simple and ideal 1-DOF bilateral teleoperation system. System control unit was based on an inverse model. The model was used to reduce value of force in the force-feedback communication channel, that the system might generate in freemotion. A substantial part of the paper is focused on the development of a mathematical model covering phenomena occurring in the investigated control scheme.

The new approach was validated on a test-stand of a rotating non-linear pneumatic manipulator arm. Two linear pneumatic actuators were used in the drive system.

The paper presents the modeling procedure of the experimental setup and the model used in the study. Three experiments are described to demonstrate the new control approach to master-slave objects with force-feedback.

The paper contains conclusions regarding the control system and the experimental setup.

Keywords: telerobotics, modeling, inverse problems, manipulators

1. Introduction

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Researcher’s attempts to ensure safe operation of various machines have led to the development of master-slave control systems with force-feedback.

Most master-slave systems are unilateral; i.e. a device that is being controlled (slave) should behave exactly as the device that controls it (master) [53–57]. However, as research continued, it was noticed that the operator that enters into interaction with the master subsystem/manipulator should also be able to feel the precise effect of the environment on the slave subsystem side. The problem poses significant challenges in its practical application, due to large distances and the inevitable time delay [1–19].

The applications of master-slave systems are widespread, including performing tasks in environments hostile to man, contaminated sites, in the depths of oceans and seas, radioactive interiors of nuclear power stations, and even other applications like medical

rehabilitation. This specific branch of robotics faces many challenges that have been tackled by scientists all over the world for many years.

The main problem that arises in the communication channel between actuation devices are delays that inhibit their communication. The problem is particularly pronounced while sending information over large distances. Another challenge is the stability of such systems, given known or unknown delays in the communication channel.

Scientist work is not only focused on teleoperation control schemes. The human hand is thought to be the most universal tool. Its universal nature poses many challenges for those designing actuation devices with force-feedback. Rakotondrabe et al. [20] proposed a novel method of fitting tie rods on an exoskeleton covering the human hand, which very closely resembles the solution observed in the natural world [2, 3, 11–13, 15, 21–27]. Fingertips are important elements of the human hand. They have also been modified regarding their mechanical structure. Zhang et al. worked on the structure of the fingertip of a manipulator for the human hand [12, 28].

Further in section 1.1 of the paper we are going to introduce what, in our opinion, are the most important papers regarding control schemes and algorithms that have pushed forward telerobotics, from 1966 to the present day.

1.1. Achievements in the Field of the Sensory Teleoperation Devices

Any analysis of researchers’ achievements in the field of master-slave systems would show how diversified the systems have become, as their type depends on the control system. This diversity will be discussed briefly below.

In 1966, William R. Ferrel introduced work about master-slave manipulators, where forces were encountered by the remote hand and were transmitted back to the operator. The author discovered that at very great distances there was a transmission delay between an operator’s movement and a resulting force. An investigation was made into the effect of long delays and differences in strategy on positioning time with force-feedback alone. Positioning could be accomplished, but delay, coupled with high loop gain, created serious instability [14].

Gunter Niemeyer, in his work [29], discussed problems of telerobotics. Author mentioned about many new potential uses of advanced telerobotic sys-

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tems that have recently been suggested or explored, such as safety applications or microsurgery. What is important is that this paper studied how the existence of transmission time delays affects the application of advanced robot control schemes for effective forcereflecting telerobotic systems, which would best exploit the presence of the human operator, while making full use of available robot control technology and computing power.

In 1992, Won S. Kim presented two papers [9, 10], the “Shared compliant control” and the “Developments of New Force Reflecting Control Schemes and an Application to a Teleoperation Training Simulator”.

In the first paper control scheme was incorporated into an advanced six degree of freedom force-reflecting telemanipulation system. The author investigated the effect of time delay on human telemanipulation task performance. Shared compliant control enabled the operator to control the telemanipulator by having a compliant hand, which softens contact forces between the robot hand and objects of environment.

Third and fourth novel schemes of force-reflecting control enabled high force reflection gain: positionerror-based force reflection and low-pass-filtered force reflection were both combined with shared compliance control from the previous work. Both presented control schemes enabled unprecedented high force reflection gains, with reduced bandwidth for dissimilar Master-Slave arms, when unity position scaling was used.

In 1993, Thomas B. Sheridan summarized thirty years of research into dealing with the effects of time delay in the control loop on human teleoperations in space [30]. The author presented experiments which showed the effects of the delay on human performance in task completion, along with demonstrations of predictive displays to help the person overcome the delay.

Dale A. Lawrence’s 1993 work involved space applications of telerobots, characterized by significant communication delays between operator commands and resulting robot actions at a remote site [31].

A high degree of telepresence was also desired to enable operators to conduct teleoperation tasks safely.

In 1994, Yasuyoshi Yokokohji [32] presented the analysis and design methods of master-slave teleoperation systems. The primary goal of this paper was to build a superior master-slave system that could provide good maneuverability. Author proposed new control schemes of master-slave manipulators that provide the ideal kinesthetic coupling, such the operator could maneuver the system as though they were directly manipulating the remote object by themselves.

In 1999, Jong Hyeon Park presented an important paper about bilateral teleoperation systems, connected to computer networks such as the Internet [33].

Control schemes have to deal with varying communication time delay. Based on this fact, it was easy for the entire to become unstable due to irregular time delays. In this paper, the author designed a slidingmode controller for the slave, and an impedance controller for the master. The author proposed a modi-

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fied sliding-mode controller, in which the nonlinear gain could be set independently of the time delay variation.

Wen-Hong Zhu, in 2000, presented an adaptive motion/force controller which was developed for unilateral or bilateral teleoperation systems [34]. This method could be applied in both position- and ratecontrol modes, with arbitrary motion or force scaling.

In 2001, Paula Arcara presented a number of control schemes [18, 19], proposed for telemanipulation robotic systems. Because of the intrinsically inconstant and large time delays, due to the communication channel, passivity was largely used in these schemes in order to achieve the stability of the overall teleoperation system.

Craig W. Alexander, in 2001, developed a tuning method which was compared to an adaptive Smith Predictor strategy [35]. The robustness of each method was considered for time-varying plant parameters.

Saghir Mumir’s and Wayne J. Book work in 2001, reintroduced wave-based teleoperation from a new point of view [36]. The authors were able to sort out the main disadvantage of this method, which previously relied on the performance deteriorating rapidly with increasing delay. This work focused on the use of a modified Smith Predictor, a Kalman filter, and an energy regulator to improve the performance of a wavebased teleoperator working through the Internet with varying time delay.

The most recent control scheme developed for bilateral teleoperation with force-feedback was published by Ilia G. Polushin in 2015 [4, 37]. This type of algorithm was a modified projection-based forcereflection algorithm, which has been demonstrated to substantially improve stability characteristics of bilateral teleoperators with communication delays.

A new type of PBFR algorithm was developed, which solved the aforementioned problem. The new algorithm was based on the idea of separating different frequency bands in the force-reflection signal, and applying the PBFR principle to the low-frequency component, while reflecting the high-frequency component directly.

This is yet another application of force-feedback systems. Another is the interaction and sensory feeling of a virtual environment; a method on which many scientists have been working for years. Researchers have suggested many control schemes for objects that are part of master-slave systems, including passive, predictive, adaptive, and wave-variable control, control with variable structure, or sliding-mode control. The decision as to which of the above would be the best when designing a new master-slave system is crucial, as it will result in either an improvement or a deterioration of the system performance [1–17, 38–40].

1.2. Sensorless Bilateral Teleoperation Systems

So far, domain of sensorless teleoperation belongs to piezoelectric crystals. Piezoelectric crystals can work at the same time as actuator, body and a force sensor, especially, when we are developing devices from a large group of single crystals. An advantage of Articles

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using piezoelectric crystals as actuators is that we can definitely add their speed and forces, even computed by single crystal. Disadvantages of using this type of crystals are that they use high voltage sources, and while it is important that they are really small, they are also very expensive. Researchers focus not only on control schemes aimed at better stability, but also on the quality of reflecting effect of the environment on actuators.

In 1998, Tadao Takigami introduced a self-sensing actuator for the first time, which was a new concept for intelligent materials, where a single piezoelectric element simultaneously performs as both a sensor and an actuator [28].

In 2006, Yuguo Cui discover that the displacement of a micro-motion worktable driven by a piezo-ceramic actuator could be measured by the self-sensing method in the absence of an independent sensor [41].

In 2007, Wei Tech Ang found that the effective employment of piezoelectric actuators in microscale dynamic trajectory-tracking applications was limited by two factors: the intrinsic hysteretic behavior of piezoelectric ceramic; and structural vibration as a result of the actuator’s own mass, stiffness, and damping properties [42].

Yusuke Ishikiriyama and T. Morita, in 2010, presented work about self-sensing control method of piezoelectric actuators that compensate for the hysteresis characteristics by using the linear relationship between the permittivity change and the piezoelectric displacement [43].

Also in 2010, Micky Rakotondrabe focused on the dynamic self-sensing of the motion of piezoelectric actuators [44]. The proposed measurement technique was subsequently used for a closed-loop control. Aiming to obtain a self-sensing scheme that estimates the transient and steady-state modes of the displacement, the author extended a previous static self-sensing scheme by adding a dynamic part.

Again in 2011, Micky Rakotondrabe, developed a new micro-gripper dedicated to micromanipulation and micro-assembly tasks [45]. Based on a new actuator, called a thermo-piezoelectric actuator, the micro-gripper presents both high-range and highpositioning resolution.

Finally, Micky Rakotondrabe continue his studies and, in 2015, presented his work about a self-sensing technique, using an actuator as a sensor at the same time [20, 46]. This was possible for most actuators with a physically reversible principle, such as piezoelectric materials.

Sensorless and self-sensing, large appliances are rare, even in scientific literature. There are only couple of paper, rising problem of inverse modelling used in self-sensing control unit of bilateral teleoperators.

This work and papers [25, 32, 40, 47–50], addresses this problem.

First paper [48], presents a method for the impedance control of a pneumatic linear actuator for tasks involving contact interaction. The method presented takes advantage of the natural compliance of pneumatic actuators such that a load cell, typically used in impedance control. The central notion of the method Articles

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is that by departing from a stiff actuation system, lowbandwidth acceleration measurements could be used in lieu of high-bandwidth force measurements.

Second paper [51], presents teleoperated minimally invasive surgery systems, measurement and display of sense of force to the operator was a problem. In this paper, it was proposed a master-slave system for laparoscopic surgery, which can provide force-feedback to the surgeon without using force sensors. Pneumatic cylinders were used as the actuator of the manipulators to achieve this.

Both papers are based on the same control methodology, the impedance control. In [48] control methodology contained an inner loop to control the pressure on two sides of a pneumatic cylinder, while an outer loop enforces an impedance relationship between external forces and motion and commands desired pressures to the inner loop. The inner loop enforces the natural compliance of the pneumatic actuator by controlling both the sum and difference of the pressures on both sides of the pneumatic actuator. In [51], a bilateral dynamic control system was designed using a neural network for acquisition of the inverse dynamics. The obtained inverse dynamics was used as a feedforward controller and to estimate the external force from the differential pressure of the cylinders.

1.3. Scope of Study

So far, the main presented control schemes for bilateral teleoperation systems with force-feedback have some defects. These defects mean the use a large number of sensors were required, mediating between the environment and the bodies of the slave manipulator, in rotary joints. A situation in which the environment affects one degree of freedom, in accordance with that degree of freedom, is relatively simple by using a single sensor. However, where the design of the manipulator depends on many degrees of freedom, and moves in the three-dimensional space, use of a single or multiple sensors could be considered as expensive, or not adequate for the proper operation of such a system.

This paper presents a novel approach for designing a control scheme for a master-slave system with force feed-back. The difference between figures presenting sensor methods thus far is that, in the case of the proposed control scheme, there are no sensors mediating between the manipulator body and the environment, relative to paper [1–19]. The same thing can be noticed in self-sensing piezo-ceramic microcontrollers used for micromanipulation and in [2, 20, 28, 32, 41, 43–46]. The only sensors used in the whole system are position encoders and pressure sensors. Where a simple pneumatic manipulator is an introduction to the work on the car cranes, which are actually much bigger than devices in the presented literature.

According to this project, operator needs to feel the crane load, but also the feeling of a contact is required. Contact between the object of environment must be realized in the way that, the system will push back the operator, in the contact situation with unmovable object. Introduction to work on much big-

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ger devices, means consideration of disadvantages like long hydraulic pipes, which are also included in the presented test-stand. The problem of high friction values and many other, which will occur during further work, have to be overcome during preliminary work on the test-stand.

Inverse model of the manipulator structure used in the control unit corresponds to the manipulator operation without any environmental impact on the slave subsystem. Based on this fact it is possible to obtain relatively accurate information about the environmental impact on the specific degree of freedom of the slave manipulator. This important feature eliminates the need to use a sensor (susceptible component) between the body of the manipulator and the environment, or between the actuator and the manipulator body.

An important feature of this approach on the design of the control system is that the value of the impact of the environment is transmitted to a specific master manipulator degree of freedom, as a response from the equivalent degree of freedom in the slave manipulator, but without using geometrical relationships resulting from the construction of the manipulator. Difference between impedance control [49, 52], in this paper system is that, the system is not controlling the pressure inside the actuator chamber, which measured pressure is being subtracted by the estimated pressure. This estimated pressure, is calculated by the inverse model of subsystem Slave.

2. Problems and Modeling of Self-sensing System I Bilateral Teleoperation

This section of work, discusses theoretical problems of teleoperation with force-feedback.

First important problem are the delays in the communication channel. The delays are the main feature causing the instability of such a system. Many designed control schemes focus on maintaining the stability of the telemanipulation system, see [1–19] for further details.

Second issue of described systems, is the transmittance of object felt in the force-feedback channel by the operator of subsystem Master, during free motion of the Slave subsystem. Where free motion of Slave manipulator is understood as a motion without interaction between the Slave subsystem and the environment.

Third and last problem, that will be discussed is the transmittance felt in the force-feedback channel by the operator of subsystem Master, during interaction between the Slave manipulator and the environment. This is a key factor, which tells the operator, how well he can feel the environmental impact on the Slave manipulator joint.

There are also many other characteristic features, which describe the robustness, stability and optimal control of telemanipulator systems. The perfect example is an inertia, a damping, a tracking, a stiffness and a mass felt in the force-feedback channel. Authors of [18, 19], have described, that the ideal telemanipulator should be stable for any value of time delay in the communication channel, present an inertia as low

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as possible, achieve zero tracking error, display the same stiffness at the master side as the one perceived in the interaction at the slave side, present no position drift during contact between Slave manipulator and the environment object.

2.1. Self-sensing Sensorless Estimation Method in Force Feedback Telemanipulation, 1-DOF Ideal Example

The remotely controlled system consists of two subsystems – the Master subsystem and the Slave subsystem. Both subsystems, the Master (a), and the Slave (b), are presented in Fig. 1, are a simple rigid objects described by their inertia.

Fig. 1. Graphical presentation of models: master subsystem (a), the slave subsystem (b) These manipulator bodies move in an environment described by the dissipative element he. The damper represents a center of air. The bodies of the manipulators move without the friction between them and the world frame.

Master subsystem acts as a motion scanner, which sends the information about its own position xm to the control unit of the slave manipulator. Master subsystem motion depends on two forces applied to the body of Master manipulator. The first is the gravity, described as Gm = Mm g, where g is the acceleration of gravity and Mm is the mass of the body. The second force is the force applied by the operator Fh to the body of Master manipulator.

2.2. Ideal Master-Slave System Transmittance Analysis

Executive Slave subsystem, is a duplicate of the Master subsystem, under conditions of kinematics, dimensions and mass. This subsystem also moves in the same environment as the Master subsystem, which is described by the damping parameter he. Slave manipulator is described by its mass – Ms, gravity force Gs, trajectory – xs, control force Fs, which is generated by the actuator and the environmental impact – by force Fe. The transfer function Bi, which describes dynamics of manipulators, can be presented as an equation (1):

(1)

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where i – index, m for Master subsystem, s for Slave subsystem, s – Laplace operator, Mi – mass. Further designation used in this work is included in Tab. 1.

Standard telemanipulation system using force sensor, can be represented as a block diagram in Fig. 2.

In Fig. 2, system senses the environmental force impact, by the force sensor and sends the value of force, back to the Master manipulator in the communication channel Fes. In presented work, system do not measure environmental force impact, but estimates its value based on the control signals of the slave controller and current Slave manipulator position. Modified structure of the telemanipulation system is presented in Fig. 3.

In presented method system has an additional block in the communication channel. The estimation block, calculates the force of environmental impact based on the force value computed by the model of the Slave subsystem. Force-feedback estimation

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Fh Fe Fs Fes Fsm Gm Gs

Bs e(s) K(s) Ms Mm xm xs

Fig. 3. Block diagram of the presented method with the force-feedback estimation block

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Description

Force applied by man to the body of Master manipulator

Environmental force impact on the slave manipulator Control force on the Slave side, which is generated by the actuator

Estimated value of the force-feedback, in the communication channel The estimated value of the force generated by the drive, during the free motion of manipulator a slave.

Gravity force attached to the Master manipulator Gravity force attached to the Slave manipulator

Transfer function, which describes dynamics of the Master manipulator Transfer function, which describes dynamics of the Slave manipulator

Position error in control unit of the Slave manipulator Transfer function, which describes regulator of Slave manipulator position Slave manipulator inertia

Master manipulator inertia

Master manipulator position Slave manipulator position

block, subtracts measured force of the drive, from estimated by model in free motion. Modified system is described in detail in Fig. 4.

One of the main problem of methods, which are using force sensors and rotary joints is that, the control unit need a huge amount of force sensors placed

Fig. 4. Block diagram of system in details, used for the proofs and analysis 68

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Tab. 1 Description of symbols used in section 2 of work

Bm

Fig. 2. Block diagram of standard sensor method

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on the manipulator arm. This feature is crucial to deliver correct value of environmental torque impact in each joint, using standard sensor control scheme for teleoperation. Presented method delivers values of environmental force impact on the slave manipulator to the operator, which is measured in the drive track in each joint of the Slave manipulator independently.

In the paper an analysis of the proposed method is presented. In results the system, based on the presented method of estimation in the force-feedback channel, will send to the Master manipulator zero value of force, during free-motion of Slave manipulator.

First characteristic transmittance, which describes Slave side of the telemanipulation system is a transmittance without impact of gravity force and environmental force on the Slave manipulator – Fig. 4. The gravity force and the environmental force are described as equation (2):

(2)

First step in finding the slave subsystem closedloop and the inverse model transmittance, is reducing the Slave subsystem transmittance – see Fig. 4, to a simple transfer function. This transfer function will be described by a relation of two signals xm, which is the position of Master, sent to Slave and the xs, which is the position of Slave. The transmittance is presented as equation (3):

(3)

Equation (3) describes the closed-loop system of the Slave manipulator, including transfer function of the position controller K(s). The controller transfer function is unknown for the transmittance analysis, because it is possible to use many structures of position regulators and it won’t change presented method result.

In the second step, the slave subsystem closedloop transfer function is determined as (3). The Second transmittance, including the inverse model of force-feedback estimation block and the closed-loop of slave subsystem, is defined by the ratio of the estimated value of the force generated by the drive, during the free motion of the Slave manipulator – named Fsm and the Master position – xm. Presented in equation (4):

(4)

Equation (4) describes one of the characteristic transfer functions, the function that is responsible for reducing the force in a force-feedback communication channel.

Third step requires finding the transmittance of closed-loop Slave system, which senses the control signal Fs from the regulator block K(s) output. Theoretically, this signal is just the control force, applied to the body of the Slave manipulator. In practice, the

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control signal on the Slave side could be a voltage, a current or like it is presented in the third part of this paper, an air pressure. To find this transfer function, it is required to find a solution of two equations presented as (5):

(5)

where e(s) – Slave subsystem position error, described as e(s) = xm(s) – xs(s). Looking for a solution of the equations (5) by ratio of F (s)/xm(s), we get equation (6):

(6)

exactly the same as transmittance (4). This means that subsystem Slave during free motion task, sends zero value in the force-feedback communication channel. This is confirmed by the transmittance difference equation (7):

(7)

For the operator of such a system, this situation is really comfortable, but requires very accurate inverse mathematical model of subsystem slave.

It is important to show, that the subsystem slave, which is under influence of the environmental force, sends to the operator exactly the force of the environmental impact.

In the second part of transmittance analysis (4) and (6), external forces are taken in to account. These forces were omitted during first analysis. We get two new equations (8) and (9), which describes Slave subsystem in the Fig. 4:

(8)

(9)

Subtracting equations (8) and (9), obtain equation (10):

(10)

After simplifying equation (10) we get (11):

Fs(s) – Fsm(s) = Fe, (11)

where the difference Fs(s) – Fsm(s), according to the scheme of Fig. 4, corresponds to the signal of forcefeedback communication channel (12):

Fes = Fe .

(12)

As it is seen, if we are able to build a high accurate mathematical model of Slave subsystem, it is possible to transmit the value of the environmental impact exArticles

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actly, to the operator by using the presented method.

Note, however, that getting a model that exactly corresponds to the actual object, is in practice very difficult or even impossible, so the value of its environment in a force-feedback presented system will strongly depend on the accuracy of this model.

2.3. Ideal Master-Slave Stability Analysis

Theoretical stability analysis of presented method, according to the ideal system presented in Fig. 4 is simple. The force-feedback communication channel, delivers to the operator, information about environmental impact on the Slave manipulator. In the ideal situation, inverse model of Slave subsystem is subtracting only some value from measured force signal. If the inverse model is stable and the delay in the communication channel is negligible, there is no need of stability analysis, because the operators force Fh, is applied to the body of manipulator, which is a second order transfer function. But when high value of delay T appears in the communication channel, system will send to the operator, value of the force-feedback with double delay T and the operator will be the main cause of the instability of this system, when he will be unaware of the delay in the communication channel (Exceptions are variable delays in the communication channel and different constant delays in different ways of communication channel, but this work focuses only on system where the delay is considered as negligible). The double delay in force-feedback communication channel comes from the problem of telemanipulation bilateralism. Systems send information in both directions with delays. So the force-feedback communication channel will be described as an equation (13):

(13)

where T is the value of delay in seconds, only in one direction, and in one communication channel.

At this point, appears a serious problem. What if, the model is slightly different than Slave subsystem?

The system will be described quite differently than it is presented in this work, but this problem will be discussed in another work.

3. The Experiment

3.1. Mechanical Structure of the Test-Stand

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The mechanical features of a slave and a master subsystem are completely identical. The exoskeleton subsystem master was attached to the operator’s elbow. Hence, it was not necessary to do the calculations of pressure in the feedback resulting from differences in the mass and dimensions of the master and the slave. The subsystem master was mounted on a strong and heavy table. The experimental setup is presented in Fig. 5.

The manipulator arm with its drive system which was taken into account in the mathematical inverse model of pressure in chambers. There is a stationary base plate, which is fixed to the table. The bending actuator and its extension bend the manipulator arm.

The straightening actuator and its extension straighten the manipulator arm.

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Fig. 5. Slave Manipulator As can be seen, the radius at which the actuator computes motion is not constant. This radius depends on the rotated position of the manipulator arm. This important feature makes the subsystem slave a nonlinear object. This feature caused problems in the control and modeling of the subsystem slave. Mounting pneumatic drives in this way is not accidental. Using two drives, affects the symmetry of the surface of the piston, which, as it turned out, considerably improved the quality and stability of the entire subsystem slave.

The manipulator body is made of an aluminum alloy, while the entire structure was permanently mounted on the table.

Two types of signals are used in the system. Output signals are analog signals for pressure measurement, and input/output discrete signals for the encoders and valves. Encoders that were used to build the test-stand had a number of pulses equal to 500 per revolution. The pressure gauge used to measure pressure in the system had a maximum measurement value of 10 bar, proportionally sensing the pressure as 1 to 10 V. Referring to the equations from part 2 of paper, those variables can be given a new designation that are more characteristic of pneumatics.

As you can see in Fig. 6, there are three control signals V1, V2 and SD (Cs in Fig. 4). They are summarized in Tab. 2.

Ps is an analog pressure sensor for a slave subsystem. Only one pressure sensor was used in the system, which had the task to bring a test-stand solution

Tab. 2 Signals in pneumatic control scheme – slave Theoretical Pneumatic Designation Designation

– discrete signal controlling left coil of the valve – discrete signal controlling right coil of the valve

Cs

Fs

Description

Ps

– PWM used to control the throttle

– analogue pressure sensor signal

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Fig. 6. Pneumatic scheme of slave closer to a bulky lifting equipment, like a car cranes.

Usually it is not possible to modify those structures on a large scale. Theoretically, the control signals correspond exactly to the same control structure. Namely, it is a fact that the previously discussed force was passed in the channels of communication in the theoretical model from Fig. 4. In the research bench is the pressure in the piston chamber at a given moment in the system. In the case of a master subsystem it was easy to use a pressure control valve (see Fig. 7), which controls the air pressure on the basis of the set value.

Then the pressure will only reach destined piston chambers using on/off valves V4 and V5.

Fig. 7. Pneumatic scheme of Master Use of Pz, the pressure control valve, in this manner increases the cost, and makes it difficult to build the entire system, but it reduces the effect of a stepped pressure rise when only 3/2 switching valves are administered, which are controlled by solenoids.

At the initial stage, one could say that the greatest impact on the results of the experiment will be the slave subsystem, it was required to make sure that its behavior is optimal. An important feature was also the design of the actuators. The cylinders, as shown in Fig. 5 and Fig. 6, are mounted in such a way, that their effective piston areas are symmetrical, resulting in a significant improvement in the stability of the entire system, as shown in Fig. 8.

The arrangement of cylinders was not the only modification; a throttle servo mechanism was also

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used, which controls the amount of an air flow based on the value of the error; the effect of both modifications is compared in Fig. 8. It can be seen that the improvement of the quality of work is relatively large. A system without identical piston areas and a throttle servo mechanism could not be considered to work properly, while with the introduction of modifications the system becomes stable, and the quality of the motion shows a significant improvement. Characteristics shown in Fig. 8 illustrate perfectly how this change affected the improvement of slave subsystem motion tracking was improved. A 25% difference in the surface of the pistons leads to significant oscillation and instability. This meant that the system was not fit for further work on it.

Fig. 8. Behavior of the system without and with throttle The introduction of the symmetry of the surface of the piston has other very important advantages: the system generates similar forces in both directions, at the same time. The necessity of conversion in the control pressure forces was thus omitted. The modifier that allows the improvement in the work object slave manipulator arm is also a servo throttle, and its absence also resulted in significant oscillations and instability of the system. The throttle led to the termination of system oscillations and to generating a trajectory, in a manner very similar to the reference even using simple PID control. The PID controller parameters were selected by an operator during system operation.

3.2. Modeling of Pressure in the Actuator Chamber

Based on Equations (1) to (12) from part II of the paper, it is possible to build a model of a subsystem slave, which describes the dynamics of the system, and also shows how the pressure is estimated, based on the input signal of the model and on the slave subsystem position.

First of all, there is a model of the geometrical structure of slave subsystem, which is required. The model is actually an inverse model of nonlinear manipulator arm. Based on the manipulator arm, a geometrical and dynamic model of the slave and master subsystem was built, as shown in Fig. 9.

The angles of mechanical structure are changing iteratively. The geometrical model of rotating arm is Articles

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Fig. 9. The geometrical relationships in manipulator structures considered in inverse modelling

Fig. 10. Base measurement for identification of the model pressure on the test bench

dependent on the dimensions of actuators. The dimensions of each actuator cause movement of the entire manipulator arm. To build a model which will behave exactly like the one in Fig. 5, requires the use of geometrical relationships among actuator, base, and rotational arm of the manipulator, as shown in Fig 9.

Model from Fig. 9 describe the estimated pressure in free motion, in the time domain by equation (14):

trolled throttle, Xs is the current position of the slave, and y1 is the measured pressure.

The fundamental measurement, which shows that one setting signal was generated from several signals.

It turned out during tests that this signal was more efficient in terms of modeling than providing several signals individually.

An attempt was made to identify many models with the runs shown in Fig. 10. However, it was difficult to find a model that would provide a minimum fit of 20%, compared to the reference runs. According to a criterion based on the function FIT (Fit curve or surface to data function) from Matlab and defined with Equation (16):

(14)

where A1 and A2 are the areas of pistons – first and second actuator, ε(t) is the angular acceleration of the manipulator arm, G1 and G2 are the gravity forces applied to the body of manipulator. Rest variable values are angles and radiuses used for derive the equation (14).

As it turned out during tests, simple geometric and mechanical model was not enough to properly estimate pressure. This model was incorporated to structure of nonlinear autoregressive model with exogenous input – NLARX. The nonlinear part of model NLARX was based on binary tree. This model has estimated relatively good the pressure, relative to the simple equation (14).

3.3. Inverse Dynamics Model on NLARX Structure

All the components of the slave responsible for initiating motion were taken into account while modeling, included valves and the mechanical structure of the manipulator. When the test-stand had been completed, the setting signals, the position, and pressure signals in the actuator chamber were recorded, as shown in Fig. 10.

The upper run y1 shows pressure changes during the system operation. The lower run, as presented in Fig. 10, shows a certain cluster of setting signals, based on Equation (15):

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where V1 and V2 are discrete signals of opening of valve 5/3, is the percentage of valve lift of the conArticles

(16)

where yh is the output from the identified model, ŷ is the average value of the real run, and y in the model is the measurement of pressure. The only model that had the potential of being relatively accurate was NLARX – pressure run and NLARX model response presented in Fig. 11.

NLARX can represent pressure changes in the real system relatively well and, according to the FIT criterion, it reaches 78% of probability. The result was so good that it had to be double-checked whether or not it was a coincidence. Therefore, another pressure measurement was made for the same trajectory and it was checked how the model would behave, given Measured and simulated model output

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Subsystem Position 40

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the same setting signals. The run turned out to be almost the same as that shown in Fig. 11, and it reached a very similar value of 77.5% of similarity between the reference and modeled pressure changes.

Based on all features, which were described in section 3 so far, slave side controller was modified to the form shown in Fig 12.

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transmit adequate information to the feedback with a relatively large time delay of 0.5 s. This is due to the compressibility of the medium in the system, and is not the fault of the control system whose clocking frequency was 10 kHz. Third and last test was focused on goal, if the system was able to feel the load of inertia, which was attached to the slave manipulator arm. Run can be seen in Fig. 15.

Pressure [bar]

Having identified the model of the slave subsystem, tests were conducted to verify the operation of the whole system. The aim of the first measurement was to check how the system would behave, given no interaction with the environment. The only interaction of the environment which occurs for the nonlinear manipulator arm is gravity and resistance to motion, and in this particular case, the friction and resistance of air surrounding the manipulator. However, even these component data were modeled within the structure of model NLARX. Owing to this, such data can be considered as negligible when conducting certain runs by the slave system of the manipulator, as they exert the same influence both on the real object and on the model. Some interesting runs of the first experiment are presented in Fig. 13.

The aim of the second experiment was to check if the system would show the maximum pressure at the moment when it would encounter an object it would not be able to move. The results of the experiment are shown in Fig. 14.

The contact phase can be seen in the upper and lower runs. The control system precisely mapped the maximum pressure of 2 bar. The maximum pressure of 2 bar in force feedback is the effective pressure, resulting from using the control method that relies on pressure changes in the system. The maximum pressure in the system is 6 bar. However, it is counteracted by the pressure of 4 bar, and the whole system stiffens.

The value of 2 bar means that the system was able to

20

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As it is presented, slave subsystem mechanical model used more signals, than it was initially established. This model used inputs and outputs of the slave device. The problem of long pneumatic tubes occured during analysis. But as it turn out operator could barely feel the delay of pressure feedback on the Master Manipulator, even if the delay was relatively large and it was 0.5 seconds.

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The estimated pressure, this time was seriously distorted, but around 15 second of lower run at steady state it delivered the information with only 5% error to the expected value. The main cause of distorted pressure feedback was simple PID controller and the disturbed position tracking ability with high change of manipulator arm inertia. Imperfect model also had an impact on the distortion of the value in the forcefeedback communication channel. In the future, the ability of position tracking will be improved, but on different hydraulic device.

AUTHORS

The aim of the experiment was to verify whether or not the operator is able to feel the effect of environment on the slave, using a pneumatic drives and relatively inexpensive control systems using an inverse dynamics model and based on a nonlinear object. It is naturally possible to confirm the assumption. However, it is biased with a relatively large error depending on the similarity between the model and the real object. A typical PID controller was used for position control, due to its simple implementation.

It was also important because of the low computing power of the system controller, whose computing capacity was focused on the mathematical model that simulated pressure in the drive system of the slave.

This work did not focus on position tracking ability, but on the self-sensing pressure estimation in the force-feedback communication channel.

Concerning the contact phenomenon, the presented system transfers stimuli from the outside environment until it attempts to grip, or until it comes into contact with the environment. When it attempts to grip, a problem arises, as the system is controlled with position error only, and does not have any deformable elements. This means that the object will be gripped with the maximum force, and this value will be sent over to the master; i.e. the operator. It can be dealt with by tuning the system to accommodate lower effective pressure. Consequently, the force with which the system will press on an object can be regulated. However, at this point, another drawback of the system is manifested. Even the smallest change in pressure will result in having to change the mathematical model that simulates pressure. This adds the additional challenge of having to find the right model, which is both difficult and time-consuming.

In the future, the ability of position tracking will be improved, but on a different hydraulic device. Also it is planned to close the self-sensing method inside the controller, without use of external sensors like the pressure sensor used in this paper.

Arkadiusz Parus – Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University, Szczecin, 70-310 Piastów 19. Poland.

E-mail: arkadiusz.parus@zut.edu.pl.

4. Conclusion

ACKNOWLEDGEMENTS

The work was carried out as part of PBS3/ A6/28/2015, “The use of augmented reality, interactive voice systems and operator interface to control a crane”, financed by NCBiR (The National Centre for Research and Development, Poland).

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Mateusz Saków* – Institute of Mechanical Technology, Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University, Szczecin, 70-310 Piastów 19, Poland.

E-mail: mateusz.sakow@zut.edu.pl.

Karol Miądlicki – Institute of Mechanical Technology, Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University, Szczecin, 70-310 Piastów 19, Poland.

E-mail: karol.miadlicki@zut.edu.pl.

*Corresponding author

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Implementation of Micro Airborne Radio Relay Submitted: 19th August 2016; accepted: 15th March 2017

Karol Niewiadomski, Grzegorz Kasprowicz DOI: 10.14313/JAMRIS_1-2017/10 Abstract: This paper presents results of Micro Airborne Radio Relay implementation designed to extend range of mobile robots in search and rescue missions. Architecture of routing device with hardware based on ARM development board is presented, as well as overview of based on mbed-rtos software. Selected radio solution with radio modem and antennas is described. Paper presents results of proposed system field tests.

Keywords: UAV, telemetry, telecommand, mobile robot, radio relay

1. Introduction Unmanned Aerial Vehicles are commonly used in various fields. UAVs come in variety of forms, sizes and degrees of autonomy. Some of them are remotely controlled by pilot with or without use of video stream from onboard camera. Others vehicles fly automatically or even semi autonomously. In all ceases operator of unmanned aircraft needs some sort of telecommand and telemetry to and from the vehicle.

Military, which is main user of Unmanned Aerial Vehicles, is in privileged position as it has access to various unavailable to civilian user means of telecommunication. Such technologies are for example: high bandwidth satellite links or high power, restricted frequency transmitters. These means of communication allow for seamlessly executed missions beyond visual line of sight (BVLOS).

Civilian users operate in much more restricted environment. They must comply witch many laws regulating operations of UAVs. One of them is regulation of radio equipment e.g. CLASS 1 devices [1]. In current legislation there are no specific rules for radio communication of unmanned aerial vehicles, so they operations are based on common rules. There is possibility of acquiring special permission for use of high performance radio communication system, but it is complicated because common rules are incompatible with requirements of UAV applications such as exclusiveness (there should be some specific fragment of radio band for specific operation) and mobile nature of UAVs operations.

1.1. Motivation

Currently, legislation in Poland allows for operating unmanned aircraft beyond visual line of

sight (BVLOS). With this type of mission one of the factors limiting the range is radio communication.

Radio waves carries the telemetry of aircraft such as altitude, speed and position as well as commands for the autopilot system and information specific to the mission goal e.g. acknowledgment of detected object.

Such communication is a prerequisite for safe conduct of the flight and successful completion of the task.

Radio communication range can be expanded using better radio modems, but this is not always the right solution. For example, further increasing of the transmitter power may be restricted by law or technical conditions of the plane – insufficient power supply or excessive weight.

For technical reasons, UHF band is used for communication with unmanned aerial systems (size of antennas, propagation conditions and the availability of bandwidth). Radio communication in this frequency band is strongly influenced by obstacles. Such an obstacle can completely prevent direct communication, even if in the line of sight coverage is sufficient. In some cases even curvature of Earth could incapacitate surface-to-surface communication.

Such issues are faced by MelAvio – team of students representing Warsaw University of Technology in UAV Challenge 2016 – Medical Express [2]. Script of the competition includes the necessity of landing unmanned aerial system at a distance of more than 10km from the ground station. Throughout the mission it is necessary to maintain continuous radio communications. MelAvio, during the start of in the UAV Challenge 2014 – Outback Rescue competition, was able to maintain the surface-to-surface communication at the distance of 5 km and surfaceto-air at the distance of 8 km. These results indicate that it will not be possible to meet the requirements using current method of direct radio communication, so airborne radio relay was proposed as a solution to the issue.

1.2. State of Knowledge

Today, radio communication is widely used in various fields of life. Television, mobile phones, wireless Internet access are all examples of radio technologies. Radio communication, like everything has restrictions. Some of them are physical, but some of them are legal based. This results often in worse than desired performance, like too low range. In some cases usage of radio relay could boost performance of radio link to sufficient level. One possibility of placing

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radio relay is embedding it in an aircraft. The main advantage of this approach is high mobility of aircraft.

Using airplanes as radio relays has long history.

One of earliest reported use cases of airborne radio relay is Battle of Ia Drang Valley in 1965 in Vietnam.

Since then airborne radio relay technology is in continuous development. Main field of operation is still military operations because of high cost of operation. Using small unmanned aerial vehicles as a carrier could significantly reduce cost in comparison with manned aircraft.

Total mass of aircraft is about proportional to it’s payload capacity. As CLASS 1 compatibility limits maximum performance of radio relay, it could be designed as light, as 1 kg without additional penalties.

It is fraction of human pilot mass, so purposely build unmanned aircraft could be at least order of magnitude lighter then small manned vehicle. Such reduction in weight reduces cost of operation and increases accessibility.

There are currently available radio relay modules build for UAVs such as STAR from Thales Communications[3]. This solution is limited to military users, as it uses restricted frequency.

There are also academic research of airborne radio relays. Some of them focuses on general network communication like ones used in computer communication. In those research IEEE 802.11 (WiFi) network devices are used. This results in high bandwidth at the expense of range [4], [5].

Other research focuses on similar topic to presented in this paper, but are limited to simulations without physical implementation [6].

2. The System Architecture 2.1. Frequency bands

Three frequency groups where considered for usage in radio relay: • 2400 MHz • 900 MHz • 450 MHz The 869MHz frequency band was chosen as final frequency. Receivers in this band has better sensitivity then ones for 2400 MHz (-110dBm vs. -100dBm).

Antenna size is still small enough (180mm for coaxial half-wave dipole with connector) to fit on sub 5kg UAV, where 450MHz antenna will be too big.

Higher frequency bands have also smaller Fresnel zone, which should by clear from obstructions for good communication. To achieve this, antennas should be placed high above the ground. On the relay UAV this is no issue as it could fly up to 120m above ground level (limited by law). On the ground we can use antenna mast, but to retain mobility it’s height must be limited to about 7m, which would be too low for 450MHz.

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to achieve maximum legal in Poland EIRP (29 dBm for Class 1 device [7]). This module has several additional features which are beneficial in radio relay application. Firstly they are compatible with MP-SiK firmware which enables them to work with more than two modems in network, but they don’t provide any routing capabilities, just conflict free operation. This is essential feature as airborne relay aircraft can work with just one radio modem to support communication channels: between itself and ground station and between itself and mobile robot executing primary mission. This also enhance reliability, as potentially any node can work as relay. To achieve similar results with simple point-to-point modems every node should have two modems (for three node network, as proposed in this paper).

Secondly RFD868+ uses FHSS technology which helps mitigating noise. It has built in two way diversity which enables connection of two antennas to one modem and automatical switchover to one which provides better signal.

This modem has also average in class sensitivity of -121 dBm at low data rates and about -110 dBm at data rates used in Micro Airborne Radio Relay.

Radio modems are configured to use 64 kbps FSK modulation. They use duty cycle of 10% to comply with Sub-class 30 of Class 1 device [7].

This solution should provide signal to noise ratio of 24 dBm at 15 km range.

2.3. Antennas

For the best usage of diversity function of RFD868+ modems it was chosen to use two half-wave dipole antennas in horizontal position mounted with 90° angle between them. This configuration results in horizontal polarization of radio waves and spatial diversity – each antenna supports two opposite quarters of horizon. This configuration is used in relay aircraft and in mobile robot. It results in 3 dBi gain in antenna subsystem.

On the ground there is horizontally mounted Yagi-Uda antenna with two directors – 6 dBi gain. It has usable cone of 60° (included angle) which main purpose is providing spatial selectivity on ground node. Antenna mast have no active antenna tracking capabilities, as manual aiming antenna turn out to be sufficient, because in most cases direction to region of interest is well defined. Second antenna on the ground is half-wave dipole to provide connectivity in ground node proximity and protect otherwise unused antenna channel from overload by reflections of unterminated connector.

2.4. Hardware

System requires at least three channels of communication: • To the vehicle autopilot; • To the radio modem; • To the payload.

The hardware is based on NUCLEO-F446RE development board and custom daughterboard.

The Nucleo board is based on STM32F446RET6 microcontroller with ARM Cortex-M4 core clocked at

Journal of Automation, Mobile Robotics & Intelligent Systems

180 MHz. This MCU provides all necessary interfaces (Fig. 1). Development board provides all necessary components for running micro controller – power regulator, decoupling capacitor network, clock source as well as embedded debugger. The daughterboard contains power preregulator which enables usage of 2S to 4S Lithium-Polymer battery packs commonly used for powering small mobile robots. Second function of the supplementary board is mechanical support for connecting wires. To ensure high reliability on board of research UAV each wire connected to board is firstly through-hole soldered, then passed through strain relief hole on PCB and tight to board in harness with zip tie. There are also three LED on the board for status indication.

The daughterboard contains also UART to USB bridge. It is built around FTDI FT230XS integrated circuit. It provides reliable mean of communication with payload computer.

As area of board is constrained by design of Nucleo baseboard, so several features was added in free space for future use: SPI bus for Ethernet connectivity or secondary radio module, footprint for I2C EEPROM, two standard servo outputs for antenna tracker functionality, additional UART with timer input for GPS (position and precise clock reference).

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Firstly, incoming data are received by HAL driver.

Then they are delivered to Packet Preparator. Each HAL driver has its own instance of Packet Preparator with own thread. It detects packet boundaries and identifies its type. It also checks correctness of data by use of CRC. Packet Preparator is responsible for wrapping data in Message structure which contains following information: • Type of message (e.g. Mavlink2 or Payload Control); • Length; • Source(which input device received it); • Unique fingerprint; • Priority; • Message content.

Fig. 1. Controller Block Diagram

2.5. Software Mbed platform was chosen for writing the software for the relay. It is cloud based IDE for ARM Cortex-M microcontrollers. Mbed is based on C++ language. It supports mainly development boards like ST NUCLEOF446RE used in this project. It provides complete SDK with compiler, HAL type peripheral library, RTOS and various third party libraries. This cloud IDE provides version control system based on Mercurial and export options for all major ARM toolchains.

Mavlink was chosen for use as telemetry protocol for its universality and commonness. It is telemetry and telecommand protocol used by many autopilots on the market, for example Pixhawk with PX4 software, VRBrain with Ardupilot, Parrot AR.Drone and many more. It is also supported by various ground control stations like Qgroundcontrol, MissionPlanner or Tower.

2.6. Relay algorithm

Relaying data is realized as a pipeline (Fig. 2). Each role in the pipeline is a class. Every object in pipeline has own thread for specific task. Communication between threads is provided by Mbed RTOS mail mechanism.

Fig. 2. Data Flow Pipeline

In the next step of pipeline message is Repetition Detector. It keeps track of 128 recently processed messages. If the message was already processed it would be discarded. Its role is to prevent looping of packets between nodes and uselessly wasting limited bandwidth. Single, common Repetition Detector is implemented in the pipeline.

Then message is moved to the Message Distributor that decides which transmitters should emit the message. The decision is made with use of routing table. It classifies the message by source and type.

There is single, common Message Distributor in the pipeline.

In the end the message is routed to one or couple Transmitters. They are responsible for feeding output HAL drivers with data. Transmitter may also drop the message if the output device is clogged up or if it is a command issued for other vehicle.

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Table 1. Mass of system components

3.1. System Properties

Component

Total mass of the system is 212.9 g (Table 1) with enough battery capacity for over 2 hours of continuous operation. Average current consumption of radio modem is 300 mA at 5 V. Router module consumes on average 120 mA also at 5 V. That makes average power consumption of 2.6 W from battery (including conversion losses). Dimensions of the complete router stack are 70 mm by 83 mm by 25 mm (WxDxH) (Fig. 3).

Mass

Daughterboard with harnesses

63.3 g

Radio modem

15.3 g

Nucleo board

34.3 g

Two half-wave dipole antennas

51 g

Battery pack (800 mAh, 7.4 V)

49 g

TOTAL

212.9 g

Router section of the system weights 111 g (including its share in battery mass). All UAVs needs telemetry and telecommand connection for safe operation, so already needs radio modem, antennas and power source for it. In this case implementation of relay functionality with this implementation is just addition of 111 g to existing aircraft. Required volume is more disturbing, as current implementation has volume of about 0.2 l, which in most UAVs will be mayor difficulty of implementation.

3.2. Range tests

Range tests were conducted in simulated scenario of Beyond Visual Line of Sight mission, because of real BVLOS flight requires special license and permission.

Instead, there were used two ground stations â&#x20AC;&#x201C; one with pilot and primary ground station operator, and

Fig. 3. Router module

Packet l os s ra o [1]

1 0,8 0,6

120 100 80 60 40

0,4 0,2 0

0

2

4

6

8

10

12

14

16

Di s ta nce ground s ta on - rel a y a i rcra [km]

Fig. 4. Packet loss ratio for different airborne relay flight altitudes [m]

Packet l os s ra o [1]

Mobile robot on the ground

Antenna mast

0,8 0,6 0,4 0,2 0

30

40

50

60

70

80

90

Rel ay a i rcra a l tude [m]

Fig. 5. Packet loss ratio at 4 km from airborne relay 80

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second simulated ground station with antenna mast and all required hardware. This station was set up in several locations and used to collect measurement data. Fig. 4 presents results of study on relay flight altitude on transmission range. Best results are achieved at 100 m above ground level. Altitude of 40 m was determined to be insufficient for communication over 10 km.

Fig. 5 shows result of comparison between on ground mast and proper antenna versus two halfwave dipole antennas 0.25 m above ground in mobile robot. Measurements were taken 4 km from relay aircraft. It is maximum usable range between relay and mobile robot on the ground, as it results in about 50% packet loss. High flight altitude of relay is clearly beneficial for link quality.

4. Conclusions and Future Work

Measurements results shows, that ability to elevate antenna to 100 m above ground helps achieve long communication rage. 200 g system could fly for an hour on board of 4 kg electric airplane at 100 m above ground. That kind of system could be built for about 10000 EUR and could operate at cost of electric energy to recharge batteries, while being mobile in opposition to 100 m antenna mast. It is also significantly cheaper than manned aircraft radio relay.

It is clearly competitive solution for communication ranges up to 15 km.

In the future system could be further improved in various ways: • Reduction of router module volume, • Support for authentication of passed data, • Addition of second, redundant transmission channel to increase reliability.

• Implementation of payload communication protocol for mobile robot, as current work was focused on telemetry and telecommand for autopilot systems.

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REFERENCES [1] “Urządzenia klasy 1 - UKE.” [Online]. Available: h t t p s : / / w w w. u k e . g o v. p l / u r z a d z e n i a klasy-1-15496.

[Accessed: 19-Aug-2016]. (in Polish) [2] “2016 Medical Express” [Online]. Available: https://uavchallenge.org/medical-express/2016medical-express/ [Accessed: 19-Aug-2016].

[3] “STAR’s First Flight.pdf.” [Online]. Available: http://www.thalescomminc.com/datasheets/ STAR’s%20First%20Flight.pdf. [Accessed: 21-Jan2016].

[4] S. Morgenthaler, T. Braun, Z. Zhao, T. Staub, M. Anwander, “UAVNet: A mobile wireless mesh network using Unmanned Aerial Vehicles”. In: 2012 IEEE Globecom Workshops (GC Wkshps), 2012, pp. 1603–1608.

[5] T. Johansen, A. Zolich, T. Hansen, A. J. Sorensen, et al., “Unmanned aerial vehicle as communication relay for autonomous underwater vehicle—Field tests”. In: Globecom Workshops (GC Wkshps), 2014, 2014, 1469–1474.

[6] S. Orn, U. Sterner, “Antenna gain requirements of airborne nodes in mobile core networks”. In: Military Communications and Information Systems Conference (MCC), 2013, p1–5.

[7] “https://www.uke.gov.pl/files/?id_plik=18904.”

[Online]. Available: https://www.uke.gov.pl/ files/?id_plik=18904. [Accessed: 19-Aug-2016].

ACKNOWLEDGEMENTS The research was performed in academic year 2015/16 under students individual research grant sponsored by Lockheed Martin Co. We thank MelAvio Students’ Research Club for support during test flights.

AUTHORS Karol Niewiadomski – Faculty of Power and Aeronautical Engineering, Warsaw University of Technology.

E-mail: niewiadomski.karol@gmail.com.

Grzegorz Kasprowicz* – Faculty of Electronics and Information Technology, Warsaw University of Technology.

E-mail: g.kasprowicz@elka.pw.edu.pl. *Corresponding author

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