JAMRIS 2016 Vol 10 No 2

VOLUME 10 N°2 2016 www.jamris.org pISSN 1897-8649 (PRINT) / eISSN 2080-2145 (ONLINE)

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JOURNAL OF AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS

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Jacek Salach (Warsaw University of Technology, Poland) Maciej Trojnacki (PIAP, Poland)

(Polish Academy of Sciences, PIAP, Poland)

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Dimitar Filev (Research & Advenced Engineering, Ford Motor Company, USA) Kaoru Hirota (Japan Society for the Promotion of Science, Beijing Office) Jan Jabłkowski (PIAP, Poland) Witold Pedrycz (ECERF, University of Alberta, Canada)

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Editorial Board: Chairman - Janusz Kacprzyk (Polish Academy of Sciences, PIAP, Poland) Plamen Angelov (Lancaster University, UK) Adam Borkowski (Polish Academy of Sciences, Poland) Wolfgang Borutzky (Fachhochschule Bonn-Rhein-Sieg, Germany) Chin Chen Chang (Feng Chia University, Taiwan) Jorge Manuel Miranda Dias (University of Coimbra, Portugal) Andries Engelbrecht (University of Pretoria, Republic of South Africa) Pablo Estévez (University of Chile) Bogdan Gabrys (Bournemouth University, UK) Fernando Gomide (University of Campinas, São Paulo, Brazil) Aboul Ella Hassanien (Cairo University, Egypt) Joachim Hertzberg (Osnabrück University, Germany) Evangelos V. Hristoforou (National Technical University of Athens, Greece) Ryszard Jachowicz (Warsaw University of Technology, Poland) Tadeusz Kaczorek (Bialystok University of Technology, Poland) Nikola Kasabov (Auckland University of Technology, New Zealand) Marian P. Kazmierkowski (Warsaw University of Technology, Poland) Laszlo T. Kóczy (Szechenyi Istvan University, Gyor and Budapest University of Technology and Economics, Hungary) Józef Korbicz (University of Zielona Góra, Poland) Krzysztof Kozłowski (Poznan University of Technology, Poland) Eckart Kramer (Fachhochschule Eberswalde, Germany) Rudolf Kruse (Otto-von-Guericke-Universität, Magdeburg, Germany) Ching-Teng Lin (National Chiao-Tung University, Taiwan) Piotr Kulczycki (AGH University of Science and Technology, Cracow, Poland) Andrew Kusiak (University of Iowa, USA)

Mark Last (Ben-Gurion University, Israel) Anthony Maciejewski (Colorado State University, USA) Krzysztof Malinowski (Warsaw University of Technology, Poland) Andrzej Masłowski (Warsaw University of Technology, Poland) Patricia Melin (Tijuana Institute of Technology, Mexico) Fazel Naghdy (University of Wollongong, Australia) Zbigniew Nahorski (Polish Academy of Sciences, Poland) Nadia Nedjah (State University of Rio de Janeiro, Brazil) Duc Truong Pham (Birmingham University, UK) Lech Polkowski (Polish-Japanese Institute of Information Technology, Poland) Alain Pruski (University of Metz, France) Rita Ribeiro (UNINOVA, Instituto de Desenvolvimento de Novas Tecnologias, Caparica, Portugal) Imre Rudas (Óbuda University, Hungary) Leszek Rutkowski (Czestochowa University of Technology, Poland) Alessandro Saffiotti (Örebro University, Sweden) Klaus Schilling (Julius-Maximilians-University Wuerzburg, Germany) Vassil Sgurev (Bulgarian Academy of Sciences, Department of Intelligent Systems, Bulgaria) Helena Szczerbicka (Leibniz Universität, Hannover, Germany) Ryszard Tadeusiewicz (AGH University of Science and Technology in Cracow, Poland) Stanisław Tarasiewicz (University of Laval, Canada) Piotr Tatjewski (Warsaw University of Technology, Poland) Rene Wamkeue (University of Quebec, Canada) Janusz Zalewski (Florida Gulf Coast University, USA) Teresa Zielinska (Warsaw University of Technology, Poland)

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JOURNAL OF AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 10, N° 2, 2016 DOI: 10.14313/JAMRIS_2-2016

CONTENTS 3

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Technical B-H Saturation Magnetization Curve Models for SPICE, FEM and MoM Simulations Roman Szewczyk DOI: 10.14313/JAMRIS_2-2016/10 9

Control Methods Design for a Model of Asymmetrical Quadrocopter Ryszard Beniak; Oleksandr Gudzenko DOI: 10.14313/JAMRIS_2-2016/14 50

The Application of Mobile Robots for Building Safety Control Barbara Siemiątkowska, Bogdan Hrasymowicz-Boggio, Mateusz Wiśniowski DOI: 10.14313/JAMRIS_2-2016/11

Any-angle Global Path Planning for Skid-Steered Mobile Robots on Heterogeneous Terrain Piotr Jaroszek DOI: 10.14313/JAMRIS_2-2016/15 56

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Phase Shifted Pulse Height Modulated Motor Control for Multiply Actuated Joints to Optimize Operating Characteristics Torsten Siedel, Stefan Bethge, Manfred Hild DOI: 10.14313/JAMRIS_2-2016/12 25

Comparison of Algorithms for Decision Making Problems and Preservation of α-properties of Fuzzy Relations in Aggregation Process Urszula Bentkowska, Krzysztof Balicki DOI: 10.14313/JAMRIS_2-2016/13

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E2LP Remote Laboratory. New Challenges in System Development Rafał Kłoda, Jan Piwiński, Kacper Kurzejamski DOI: 10.14313/JAMRIS_ 2-2016/16 64

Neural Based Autonomous Navigation of Wheeled Mobile Robots Mariam Al-Sagban, Rached Dhaouadi DOI: 10.14313/JAMRIS_ 2-2016/17

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Technical B-H Saturation Magnetization Curve Models for SPICE, FEM and MoM Simulations Submitted: 2nd February 2016; accepted 29th February 2016

Roman Szewczyk DOI: 10.14313/JAMRIS_2-2016/10 Abstract: Recent development of SPICE, FEM and MoM software often requires the fast and reliable description of BH saturation magnetization curve. In spite of the fact that physical models of BH saturation curve are very sophisticated, for technical purposes, such curve may be modelled by simplified equations.

Paper presents the quantitative assessment of the quality of four technical models of BH saturation magnetization curve performed for four modern magnetic materials: constructional corrosion resistant steel, Mn-Zn ferrite, amorphous alloy with perpendicular anisotropy as well as Finemet-type nanocrystalline magnetic material. Presented results confirm reliability of the model as well as indicate that high-speed calculation may be done using arctangent function.

Keywords: saturation, magnetization curve, perpendicular anisotropy, Mn-Zn ferrite, Finemet

1. Introduction Methods of modeling of the magnetic hysteresis loop are developed for over one hundred years [1].

However, process of magnetization of magnetic material is one of the most sophisticated problems connected with contemporary physics. As a result, in spite of many different approaches [2, 3, 4, 5] and the use of the most advanced numerical methods [6, 7], problem of quantitative modeling of magnetic hysteresis loop remains unsolved.

On the other hand, technical simulations oriented on simulation programs with integrated circuits emphasis (SPICE) [8], finite element method (FEM) [9] or method of the moments (MoM) [10] don’t require sophisticated analyses of the shape of the hysteresis loops. To be useful for technological simulations, the model of the magnetic hysteresis loop should provide fast and reliable reproduction of the shape of B-H magnetization curve.

Such models were proposed and implemented for calculations in electrical engineering since early thirties of 20th century [10]. However, in spite of wide use for numerical simulations, quantitative analysis of quality of the most popular models seems to be still not presented from the point of view of modeling the properties of modern magnetic materials.

This paper is an approach to fill this gap. Four the most popular models of B-H saturation curve

were analyzed from the point of view of quality of the modeling of modern magnetic materials: constructional corrosion resistant steel, soft ferrite, amorphous and nanocrystalline alloy. As a result, the quality of the modeling together with its efficiency was assessed by quantitative parameters.

2. Technical B-H Saturation Magnetization Curve Models

During the investigation four of the most popular models of B-H magnetization curve were tested.

Model 1: Linear model considering the amplitude permeability µa and saturation flux density Bs. This model is given by the following equation: (1) Problem connected with this model is a non-linear derivative. Moreover, in the case of use of linear model in Octave/Matlab it is very important to avoid forbased loop to introduce saturation to linear model.

Instead of for-based loop, the vectorisation method is recommended, as about 20 times faster solution. The example of vectorisation of equation 1 implemented in Octave is presented in the Fig. 1. mi0=4.*pi.*1e–7;

Bmodel=Hmeas.*mi0.*mi; Bmodel = max(Bmodel, (–1).*Bs); Bmodel = min(Bmodel, Bs);

Fig. 1. Linear model implementation. Vectorisation of saturation up to Bs implemented in Octave Model 2: Model given by the Langevin function describing the B-H magnetization curve in paramagnetic material [11]. This model is determined by the saturation flux density Bs and parameter a. Langevin function based model is given by the following equation: (2)

It should be indicated, that parameters of this model describes physical properties only for isotropic materials. However, Langevin curve can be used for

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modeling of any material. In such a case parameter a doesn’t describe domain wall density [12].

Model 3: Model based on the shape of arctangent function described by amplitude permeability µa and parameter k. Model is given by the following equation [13]: (3)

It should be stressed that this model hasn’t physical interpretation, however can quite well reproduce the shape of saturation B-H curve and has continuous derivative.

Model 4: exponential function based model [14] using saturation flux density Bs and amplitude permeability µa given by the following equation: (4) Model hasn’t physical interpretation, it just reproduces the shape and has continuous derivative.

Moreover, typographical mistake in stating equation occurred in [14].

3. Materials for Validation of the Models

Validation of the four models was performed for the four modern magnetic materials commonly used in the industry: Material 1: corrosion resistant martensitic steel 3H13 (X30Cr13). Such steel is used for critical components in the energetic industry.

Material 2: manganese-zinc Mn0.51Zn0.44Fe2.05O4 high permeability ferrite for power conversion applications.

Material 3: M680 type core produced by Magnetec Company, made of amorphous alloy with possibility of nanocrystallization, the NANOPERM LM (Fe73.5Cu1Nb3Si15.5B7). This amorphous alloy, which exhibits strong perpendicular anisotropy is especially useful for current transformers.

Material 4: high permeability nanocrystalline Fe73.5Cu1Nb3Si13.5B9 Finemet-type alloy for electronic industry All samples were ring-shaped to avoid demagnetization.

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the differences between the results of modeling Bmodel and experimental data Bmeas, given by the following equation: (5)

The minimization of the target function F, was carried out by the simplex search method [15].

Table 1. Results of the identification of the parameters for technical models of B-H saturation curve Model

Material

Steel 3H13 Model 1: Linear model

Model 2: Langevin model

Model 3: atanfunction based model

Model 4: exponential –function model

Parameter

Value

Bs

1.05 T

Bs

0.36 T

Bs

1.31 T

Bs

1.15 T

µa

Mn-Zn ferrite

µa

Amorphous alloy

µa

Nanocrystalline Finemet Steel 3H13 Mn-Zn ferrite Amorphous alloy

4. Validation Procedure

Procedure of validation of the models consists of several steps, covering both experimental measurements as well as mathematical modeling: Step 1. Experimental measurements of magnetic hysteresis loops for all four materials were carried out. For measurements the digitally controlled hysteresisgraph was used. Measurements were performed on ring-shaped samples, which were wound by magnetizing and sensing winding.

Measurement uncertainty of measurements using this system was assessed as 5%.

Step 2. Parameters of the models were determined in optimization process. Target function F for optimization was determined as the sum of squares of

N° 2

Nanocrystalline Finemet

464

8737

4 494

µa

326 743

a

544

Bs

1.25 T

Bs

0.44 T

Bs

1.46 T

Bs

1.24 T

k

0.0098

k

0.053

a a a

10.0 58.3 0.68

Steel 3H13

µa

Mn-Zn ferrite

µa

Amorphous alloy

12 404

µa

7223

Nanocrystalline Finemet Steel 3H13

k

k

641

0.0096 0.83

µa

527 485

µa

564

Bs Bs

1.09 T 0.38 T

Mn-Zn ferrite

µa

Amorphous alloy

11 005

µa

5 850

Nanocrystalline Finemet

Bs

Bs µa

1.32 T 1.16 T

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Table 2. Quality of the models for different soft magnetic materials Parameter

Model

Material

Model 1: Steel 3H13 Linear model Mn-Zn ferrite

Model 2: Langevin model

Model 3: atan-based model

Model 4: expoten-tial model

emax (%)

s(%)

27

7.8

74

R2

20

0.92

0.98

Amorphous alloy

7.4

1.0

0.9998

Steel 3H13

78

21

0.92

Nanocrystalline Finemet Mn-Zn ferrite

Amorphous alloy

Nanocrystalline Finemet Steel 3H13

Mn-Zn ferrite

Amorphous alloy

Nanocrystalline Finemet Steel 3H13

Mn-Zn ferrite

Amorphous alloy

Nanocrystalline Finemet

78

14

26

8.2

87

14

10

80

0.997

21

0.92

8.1

88

14

78 25

7.3 83

0.98

2.8

26

12

0.97

0.97

0.996

21

0.92

1.8 14

Step 3. Parameters determining the quality of the models were calculated for different soft magnetic materials. During the assessment following parameters were calculated: – emax (%) – maximal difference (given in percents) between the results of modeling and results of measurements

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Table 3. Calculation time for B-H models (calculated for 106 points) Model

Model 1: Linear model Model 2: Langevin model

Model 3: atan-based model

Model 4: expotential model

Calculation time (s) 0.41 0.59 0.41 0.45

0.98

3.1

8.1

N° 2

0.97

0.98

0.9992 0.97

– σ (%) – mean square root of the difference (given in percents) between the results of modeling and results of measurements – R2 – determination coefficient.

Mathematical modeling was carried out using open-source OCTAVE 4.0.0 with optim toolbox 1.4.1.

However, developed code is fully compatible with

Fig. 1. Results of the fitting of linear model (model 1) for: a) steel 3H13, b) Mn-Zn ferrite, c) amorphous alloy, d) Finemettype nanocrystalline alloy Articles

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MATLAB. To enable verification of the validation process, scripts used in presented research are available at: http://zsisp.mchtr.pw.edu.pl/BHmodels

5. Results

The results of the determination of model’s parameters using the minimization of the target function F are presented in the Table 1. It should be indicated, that parameters, which are physically

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justified (such as amplitude permeability µa or saturation flux density Bs) are coherent for different models. Results of the fitting of different models for tested materials may be also seen in the Figures 1–4.

Parameters determining the quality of the models for different magnetic materials are presented in the Table 2. It can be seen, that the quality of the model is mainly determined by the wideness of the hysteresis loops, whereas influence of the shape of

Fig. 2. Results of the fitting of Langevin function based model (model 2) for: a) steel 3H13, b) Mn-Zn ferrite, c) amorphous alloy, d) Finemet-type nanocrystalline alloy

Fig. 3. Results of the fitting of arctangent function based model (model 3) for: a) steel 3H13, b) Mn-Zn ferrite, c) amorphous alloy, d) Finemet-type nanocrystalline alloy 6

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Fig. 4. Results of the fitting of exponential function based model (model 4) for: a) steel 3H13, b) Mn-Zn ferrite, c) amorphous alloy, d) Finemet-type nanocrystalline alloy

BH curve is negligible. This is especially seen in the case of amorphous alloy, where the highest values of R2 coefficient and the best fitting was obtained.

Assessment of calculation time for different models is given in the table 3. Assessment was done for 106 test points and is presented in seconds.

Tests were performed for Octave 4.0.0 working at MINGW32_NT-6.1 Windows 7 Service Pack 1 i686, with i5-2400 3.1GHz core.

It can be seen in Table 3, that all four models are very effective in calculation of large numbers of points of BH curve. Moreover, both linear model (model 1) and atan-based model (model 3) gives similar time for calculation.

6. Conclusions

Presented results indicate, that all four models of B-H saturation magnetization curves enables fast and reliable modeling. However, arctangent function based model (model 2) achieved similar efficiency of calculation as a linear model, which makes it especially suitable for technical simulations oriented on simulation programs with integrated circuits emphasis, finite element method or method of the moments.

Accuracy of the results of the modeling by all four models is similar and is determined mainly by the lack of the representation of magnetic hysteresis loop in B-H saturation magnetization curve. However, very good results were obtained for linear model, which makes it very useful, due to its simplicity and efficiency of calculations.

AUTHORS Roman Szewczyk* – Industrial Research Institute for Automation and Measurements, Al. Jerozolimskie 202, 02-486 Warsaw, Poland, rszewczyk@onet.pl *Corresponding author

REFERENCES

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A. Globus, „Some physical consideration about the domain wall size. Theory of magnetization mechanism”, J. de Physique C1, 1977, C1-1. DOI: 10.1051/jphyscol:1977101.

Vadja F., Della Torre E., “Measurement of output dependent Preisach functions”, IEEE Trans. Magn., vol. 27, 1991, 4757. DOI: 10.1109/20.278938.

Augustyniak B., Degauque J., “New approach to hysteresis process investigation using mechanical and magnetic Barkhausen effects”, J. Magn. Magn. Mater., vol. 140–144, part 3, Feb. 1995, 187–189. DOI: 10.1016/03048853(94)01603-8 Articles

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Kucuk I., “Prediction of hysteresis loop in magnetic cores using neural network and genetic algorithm”, J. Magn. Magn. Mater., vol. 305, 2006, 423. DOI: 10.1016/j.jmmm.2006.01.137.

Wilson P. R., Ross J. N., Brown A. D., “Optimizing the Jiles-Atherton model of hysteresis by a genetic algorithm”, IEEE Trans. Magn., vol. 37, 2001, 989. DOI:10.1109/20.917182.

A. Maxim, D. Andreu, J. Boucher, “A new analog Behavioral SPICE macromodel of magnetic components”. In: Proceedings of the IEEE International Symposium on Industrial Electronics, ISIE ’97, 1997, 183. DOI: 10.1109/ ISIE.1997.648925 F. J. Perez-Cebolla, A. Martinez-Iturbe, B. Martindel-Brio, E. Laloya, S. Mendez, C. E. Montano, “3D FEM characterization of a switched reluctance motor from direct experimental determination of the material magnetization curve”. In: IEEE International Conference on Industrial Technology (ICIT), 2012, 19–21 March 2012, 971–976. DOI: 10.1109/ICIT.2012.6210065 Y. Takahashi, C. Matsumoto, S. Wakao, “LargeScale and Fast Nonlinear Magnetostatic Field Analysis by the Magnetic Moment Method With the Adaptive Cross Approximation”, IEEE Transactions on Magnetics, vol. 43, no. 4, April 2007, 1277–1280. DOI: 10.1109/ TMAG.2006.890973.

J. P. Barton, “Empirical Equations for the Magnetization Curve”, Transactions of the American Institute of Electrical Engineers, vol. 52, no. 2, June 1933, 659–664. DOI: 10.1109/T-AIEE.1933.5056367.

N.C. Pop, O.F. Caltun, “Jiles-Atherton magnetic hysteresis parameters identification”, Acta Phys.

Pol. A, 2011, 120, 491–496.

D. C. Jiles, D. L. Atherton “Ferromagnetic hysteresis”, IEEE Trans. Magn., vol. 19, 1983, 2183. DOI: 10.1109/TMAG.1983.1062594.

M. M. Ponjavic, M. R. Duric “Nonlinear modeling of the self-oscillating fluxgate current sensor”, IEEE Sensors Journal, vol. 7, 2007, 1546. DOI: 10.1109/JSEN.2007.908234.

G. Mirsky, “Magnetic-Core Modeling Offers Insight into Behavior, Operating Range, Saturation”, Electronic Design, Sep 9, 2015.

Lagarias, J.C., J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence Properties of the NelderMead Simplex Method in Low Dimensions”, SIAM Journal of Optimization, vol. 9, no. 1, 112–147, 1998. DOI:10.1137/S1052623496303470.

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Submi ed: 9th February 2016; accepted: 5th May 2016

Barbara Siemiatkowska, Bogdan Hrasymowicz-Boggio, Mateusz Wisniowski DOI: 10.14313/JAMRIS_2-2016/11 Abstract: In this ar cle we propose the applica on of service mobile robots for control of building safety parameters. Indoor mobile robots are becoming a reality and their availability and applica ons are expected to grow rapidly in the near future. Such robots are usually equipped with cameras and laser range finders, which could be used to detect hazardous situa ons in their opera ng environment, such as evacua on route obstruc ons, emergency sign occlusions or accumula on of dangerous materials. We demonstrate how these safety-related augmenta ons of a mobile robot system can be achieved with few addi onal resources and validate experimentally the concept using an indoor robot for emergency sign and evacua on route control.

Keywords: mapping, classifica on, control, mobile robot, safety

1. Introduc on The range of potential applications for mobile robots is enormous. However, their current real life usage is limited mainly to: delivery robots [1], guiding robots [2] and cleaning robots [3]. In this article we present a new possible application of an indoor mobile robot for safety control. We discuss a method which integrates different algorithm used in mobile robotics and allows the robot to perform building safety control during day-to-day work, such as cleaning, delivery.

A mobile robot can be perceived as a kind of an agent [4]. It obtains the information about an environment and performs some actions. In order to act in the environment the mobile robot requires navigation module [5].

A typical robotic system consists of the following parts: - Perception [6] – data obtained from sensors (cameras, laser range- inders) are analysed and represented in a suitable form.

- Mapping [7, 8] allows us to build the representation of the environment. Usually metric maps are built.

We can distinguish between occupancy grid and feature–based maps. Occupancy grid represents the environment as a grid of cells. To each cell a numerical value which represents the possibility that the corresponding area is occupied by an obstacles is attached. This kind of a map allows us fast generation of a collision-free path but it requires a huge amount

of memory. In feature-based representations the environment is described as a set of features: lines, corners, etc. Such representation is very useful during process of localization but path-planning based on this kind of map is time consuming. In robotics also non-metric representations of the environment are used. Topological maps [9] represent the environment in the form of a graph. Each node represents the a part of the environment – for example a room or a corridor. Two nodes are connected if there is relation between distinctive parts in the environment.

Semantic maps [10], contain data about the meaning and functionality of the detected objects and places.

- Localization [8, 11] allows the robot to determine the position in a given coordinate system. Usually odometry is an important source of information about the robot position. It is inexpensive and provides a good short time accuracy, but if the robot travels for a prolonged period of time errors in determining its position increase so additional methods are used. In the literature Kalman ilters or particle ilters are used to estimate the robot’s position.

In these methods, encoder readings are used as an input and sensors measurements as observations.

- Path planning – the aim of path planning is to ind optimum collision-free path to the target location [12, 13]. We can distinguish between global and local methods. Global methods require the map of the whole environment and are time consuming. In the case of local methods path is planned on information about nearby obstacles. The method is fast but it can be trapped in local minima.

- Traveling along a planned path the angular and linear velocities of the robot are computed. Usually Dynamic Window Approach (DWA) [14] is used.

During the navigation additional actions can be performed. In the case of safety control the robot has to detect and recognize the emergency-signs and to detect obstacles.

Human safety is a crucial aspect in the design and maintenance of any building. This topic involves multiple risk factors, some of the major being related to ire, explosions and earthquakes. In order to minimize such hazards, building codes are used, which are state-level sets of rules specifying minimum standards of building construction and maintenance [15– 17]. Building codes in different countries share many principles regarding human safety, and most importantly, rules regarding hazard prevention and evacuation procedures. These codes (e.g. the ICC Interna9

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Fig. 1. The system architecture

tional Property Maintenance Code) specify the numbers of exits, exit capacities, visibility of exit signs, emergency escapes, corridor and stairway parameters, accumulation of garbage or other materials in passageways, doors, windows, ire escapes, stairways, accumulation of lammable or hazardous materials, ire detectors, ire alarms, extinguishers, and many other parameters of the building. Failure to ful ill these and other requirements results in conditions which are a potential threat to human safety thus periodic inspections are required. In this work we explore the possibility to use mobile robots for control of various building safety measures. We can recognize three different anomalous situations: the emergency sign is put in the wrong place, the emergency exit is occluded and the emergency exit is encumbered. Autonomous, service mobile robots are used increasingly often in large buildings for such tasks as cleaning loors, transporting items or guiding visitors. According to recent forecasts, within the next ten years mobile robotic assistants are expected to become common in households as well. Most advanced applications of service robots require these machines to be equipped with multiple sensors such as colour cameras, range inders and depth cameras. These sensors are used for self-localization and for performing particular tasks involving objects and places that are recognized by the robot. Modern indoor navigation techniques use an internal map and representation of the current state of the world held in the robot’s memory, which is subject to constant updates. Taking these qualities of mobile robots into consideration it would be very convenient to engage them into additional, non-timeconsuming, safety-related activities, which could save human labour as well as shorten the reaction time to hazardous conditions and increase overall building safety.

In buildings where mobile robots are already present, many safety control tasks could be automated at virtually no cost and would require only a small fraction of the robots’ computational effort. Robots with on-board cameras and localization systems can easily check for the presence or visibility of arbitrary signs and other items at their required position (speci ied 10

on the robot’s internal map), as well as detect hazardous obstructions of evacuation routes, doors and stairs. Mobile robots operating inside the building on a regular basis could detect such conditions and prompt human reaction in a fast and reliable manner.

In order to test this concept we design a safety control system integrated with the regular software and hardware of an indoor robot. In the experimental section we validate several elements of this system using a large robot for transport tasks operating in the building of Faculty of Mechatronics.

2. The System Architecture The system architecture is presented in Fig. 1. The system operates on the ROS platform [18]. The Robot Operating System is an open source project that functions as a bridge between hardware and other infrastructure (communication modules, operator panels, different computers), state-of-the-art algorithms and methods developed and used by our team. This highlevel software includes: navigation algorithms which allows for autonomous localization, and locomotion of the robot, automatic task and actions planning, communication with the robot by different interfaces (e.g.

Graphical User Interfaces, gestures and voice commands recognition etc.), data processing.

2.1. Sensors The robot (Fig. 2) used in our system is equipped with sensors which allow us to acquire information about the environment without installation of any additional devices in the building. The proposed sensor system includes: Proprioceptive sensors – used in order to estimate (not determine) the robot position relative to a starting location. The laser scanner [19] is an optical, non-contact distance measuring sensor. The method of measurement is based on pointing a laser beams onto the environment and calculating a distance from each received re lection. Typical sensor of this kind is a 2D scanner with 180◦ or 270◦ angle range. The rotation of high speed internal mirror enables high (50 Hz) frequency of scanning with the range of 25 m or higher. The industrial laser scanners are reliable and could be used with

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concern of the safety regulations (Machinery Directive 2006/42/EC [20]). Laser scanners are designed to operate faultlessly even in harsh environment (direct sunlight or black night, the variety of weather conditions, dynamic objects). Mounting a laser scanner on the tilt mechanism is a common solution to make high accuracy 3D scanning. It provides a dense lat scan in front of the robot, which allows to automatically detect objects in prede ined zones. In most cases the two zones are de ined: warning zone in which occurrence of an obstacle limits robot velocity and protective zone which forces the robot to stop. The robot continues its task after object disappearance.

Fig. 3. The map of the environment [23]

2.3. Path Planning Our hybrid path planing system works in the following steps: A start node (the initial position of the robot) is de ined; The goal node is indicated; The topological path between the start and goal nodes is planned; The metric path between nodes is generated; Details of the mapping and path planning modules are described in our articles [23, 24].

Fig. 2. The robot Kurier with sensors: 1 – Vision camera, 2 – Kinect 3D camera, 3 – Laser scanner

The Kinect [21] is a motion controller designed for the Xbox 360 console. This inexpensive device is a very good replacement for costly advanced laser scanners.

The device includes a vision camera and depth sensor.

With the use of this one device it is possible to gather visual information and 3D point cloud which contains a set of points, laying on surfaces around, with de ined 3D coordinates and RGB values. Such information allows to reproduce a 3D digital model (with colour information) of a scene seen by the sensor. Vision cameras are used to detect and recognize lat visual signs placed on the the walls.

The robot is also equipped with communication link with building network and simple Human-Robot interface (HRI).

2.2. Mapping In our system we assumed that the map of the environment is known. It is obtained based on documentation of the building [22]. The map is de ined as graph G = {N,E,M,P,L}, where: N is a set of nodes, each node represents some area (room) in the building, E is a set of edges, M is a set of grid-based maps of the areas (nodes), L is a set of landmarks. A grid-based map is attached to each node (room). Fig. 3 presents the idea behind our approach.

Fig. 4. Example of visual localiza on on a metric map.

2.4. Localiza on In large buildings there are areas in which the classic approach to metric localization (i.e. based on laser and odometry data) can fail. A good example of such area is a long corridor, where the laser range inder perceives only two parallel walls. Since there are no shapes to be unambiguously matched to the known map, the metric localization system relies exclusively on odometry data. Odometry alone, however, tends to accumulate errors over time, since it can detect only relative displacements of the robot. This accumulated errors increase up to several meters depending on the length of the corridor, which is unacceptable if the robot is supposed to reach some speci ic point of the corridor (e.g. a selected room door). To prevent this situation, we have developed a localization method that relies on visual data captured with a colour camera. In order to use this method, a semantic map of the building must be provided. It is expected that, at its 11

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lowest level, this map contains qualitative and quantitative information about the spaces or subspaces of the building. Assuming that the robot is equipped with a laser range inder, we irst run a conventional Monte Carlo localization algorithm to obtain an approximate metric position, suf icient for localization on the semantic map (based on knowledge of the subspaces boundaries). When the robot navigates inside a known subspace, the algorithm constantly searches the camera input for a set of known visual templates (provided by the semantic map), such as room numbers, emergency signs, boards, wall patterns, etc. Once one of these templates is found, the robot position relative to the template is calculated. Since the templates have ixed positions and orientations, the calculation of the position of the robot in the map is trivial. The position is updated and the conventional localization algorithm continues to run until another template is found.

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Fig. 6. A corridor with objects blocking the exit. Simple analysis of the data from laser scanner indicates possible evacua on problem.

presence of any known templates of interest. Thus, as presented in the next section, this method can be easily developed to accomplish the task of veri ication of the visibility of emergency and warning signs, door numbers, information boards and other important lat features of the building.

3. Emergency-routes and Signs Verifica on

Fig. 5. The samples of emergency signs These natural templates can be detected using convolutional matching: We irst detect the walls captured in the range inder data data using Hough line detection. Then we apply a perspective transform from the original camera input to a normalized viewpoint (i.e. the viewpoint of a virtual camera with its optical axis normal to the wall, positioned at a given distance from it). The resulting image is used for template matching using a fast heuristic algorithm [25].

The presented localization algorithm was tested in a long corridor using a mobile robot equipped with a range inder and a colour camera (with its optical axis deviated 60 degrees from the robot’�s axis). The corridor was divided into four subspaces, two of them had no characteristic features visible by the laser range scanner. For each of these subspaces, three natural templates were captured. While navigating, on each template detection, the robot’�s displacement along the corridor was corrected by 0.5–2 m, depending on the magnitude of the accumulated odometry errors.

A visualization of the combined localization system has been shown in Fig. 4. The cubes on the walls represent natural templates. The white dots are wall points perceived by the range inder, the cloud of arrows represents possible robot positions randomly spread around the position calculated based on template detection (used by the Monte Carlo metric localization).

The attached image (top right) shows the transformed camera input with a bounding box on the detected template.

Beyond localization on a corridor, the proposed algorithm provides, obviously, information about the 12

The emergency management signi icance is unquestionable in all public buildings. The complexity of the emergency routes is rising nowadays, alongside with sophisticated public infrastructures architecture.

The design of faultless emergency-routes should be considered as most important. The escape routes veri ication and its further analysis has to be done to improve building safety. A safety plan is designed and approved by professionals and should not be changed unless the building structure was modi ied.

The veri ication process should be performed periodically to avoid unauthorized changes. Typically only emergency-signs existence is checked. In most cases, the signs position and quality should also be considered. The accumulation of objects on the escape-path should not occur to prevent clashes during evacuation.

Fig. 6a presents the image of the sample environment.

The obstacles (a box and bottles) which are placed near the exit door can be easily using a laser range inder (Fig. 6b). It is possible to easily checked if the width of the corridors and emergency exits complies with the safety standards.

3.1. A Proof-of-concept Experiment We have performed a proof-of-concept experiment using our mobile robot (Kurier). The experiment was planned as follows: the programmed path re lected one of the emergency escape paths from our laboratory to the indicated emergency exit. Alongside this emergency-route a set of safety-signs existence was veri ied and the width of the corridors or emergency exits was measured. The emergency-routes plan could be presented as a list joint segments whose nodes (vertices) are emergency signs, and the edges are evacuation paths. In the irst run, a remotely controlled robot was programmed to record positions of all emergency signs (positions were con irmed by the

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Fig. 7. Emergency-route verifica on experiment. 1 – narrow passage, 2 – emergency signs, 3 – corridor with obstacle, 4 – emergency–exit, 5 – start posi on, 6 – recorded path, 7 – EXIT posi on.

operator). Those recorded templates and their positions became nodes of our representation graph. The experiment goal was an autonomous run to the emergency exit, during which the veri ication of all signs existence was performed. Before veri ication experiment, we have occluded one of the signs and put a big obstacle in the corridor. Then an autonomous navigation was executed - starting from our laboratory, heading for the certain emergency exit. In the ig. 8, a solid continuous path leading to the exit is shown.

The solid black colour represents positive veri ication of the actual passage width in the certain space. The width could be set for each space independently (e.g. corridors, doorways, emergency exits). The robot is aware of the room type due to the building representation and its localization. If the width of the measured corridor is lower than a threshold for actual space, a trajectory is recorded with a warning label, and denoted in the report as gray dotted line. At the same time, the signs detection algorithm veri ies the existence and positions of all emergency-signs recorded previously. If any sign is missing, also a warning label is recorded. In the ig. 7, a positive sign detection is represented with black circles and a negative with black square. Such a generated report could be used as concise aid for professionals. Their focus could be put on most important aspects to check, expediting whole veri ication process.

3.2. Experimental Results In the carried experiment, the robot’s laser rangeinder and odometric sensors were used to localize the robot based on a particle ilter method. The rangeinder was used to repeatedly measure the corridor width. The 2D template detection program was active during the whole experiment serving 2 functions: providing an additional source of localization (relevant only in the corridor) and reporting the found templates for later off-line emergency sign veri ication. The template detection algorithm performed well most of the time. All the visible signs were detected. However, we found dif iculties detecting templates viewed from a sharp angle (skewed by more than 60 degrees form the frontal view), as well as de-

tecting templates with metal, glass or other re lecting materials (in the experiment no such templates were used). Since the system is designed to work in large buildings with arti icial light, we did not intentionally modify the standard lighting during the test however, it should be noted that this would probably negatively impact performance. The proposed method would present problems for either small robots (with cameras close to the ground) or narrow corridors, since the robot would be too close to the wall, limiting the wall area covered by the ield of view. The main dif iculties during the experimental run were encountered during autonomous navigation through the relatively narrow laboratory doorway - the limited precision of the drives and control system caused multiple path re-planning iterations, and thus some unnecessary adjusting movements of the robot.

4. Conclusion When an emergency occurs within a building, it is crucial to guide the people towards exits. It is required that the accumulation of objects on the escapepath does not occur to prevent clashes during evacuation and and thus, evacuation signs must be placed along the path. In this article we have presented a new possible application of an indoor mobile robot safety control. We show that this task can be solve during day-to-day work of a mobile robot such as cleaning or delivery. We strongly believe that a cooperation between Architecture-Engineering-Construction and Mobile Robotics domains could considerably accelerate the development of mobile service robots intended for public buildings.

AUTHORS

Barbara Siemiatkowska∗ – Warsaw University of Technology, (22) 2348647, e-mail: siemiatkowskab@gmail.com, www: Faculty of Mechatronics.

Bogdan Hrasymowicz-Boggio∗ – Warsaw University of Technology, (22) 2348647, e-mail: mysticdrow@gmail.com, www: Faculty of Mechatronics.

Mateusz Wisniowski∗ – Warsaw University 13

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of Technology, (22) 2348647, e-mail: qmpstudios@gmail.com, www: Faculty of Mechatronics. ∗ Corresponding author

[11] M. P ingsthorn, B. Slamet, A. Visser, ”A scalable hybrid multi-robot SLAM method for highly detailed maps”. In: Proceedings of the 11th RoboCup International Symposium, 2007.

REFERENCES

[12] H. Chu, H. A. Eimaraghy, ”Real-time multi-robot path planner based on a heuristic approach”.

In Proc. of the IEEE International Conference on Robotics & Automation (ICRA), 1992.

[1] C. A. A. Calderon, E. R. Mohan,B. Sin Ng., ”Development of a hospital mobile platform for logistics tasks”. In: Digital Communications and Networks, vol. 1, 2015, 102–111.

DOI:10.1016j.dcan.2015.03.001.

[2] S. Thurn, M. Bennewitz, W. Burgard,et al.,”MINERVA: A SecondGeneration Museum Tour-Guide Robot”. In: Proceedings IEEE International Conference on Robotics and Automation, 1999, 1999–2005.

DOI:10.1109ROBOT.1999.770401.

[3] K. Al-Wahedi, A. Darwish, B. Kodiah, ”Cost Based Navigation for Autonomous Vacuum Cleaners”.

In: Robot Intelligence Technology and Applications 2012: An Edition of the Presented Papers from the 1st International Conference on Robot Intelligence Technology and Applications, DOI:10.1007978-3-642-37374-9_40.

[4] Stuart J. Russell, Peter Norvig, Arti icial Intelligence: A Modern Approach, 2 ed., Pearson Education, ISBN: 9780130803023, 2003.

[5] B. Siemiatkowska et al., ”Towards semantic navigation system”. In: Recent Advances in Intelligent Information Systems. Proceedings, ed. by: M. Klopotek et al. Exit, 2009,711–720, ISBN 978-837051-580-5.

[6] https://willowgarage.com/pages/ research/perception [7] R. B. Rusu et al., ”Towards 3D point cloud based object maps for household environment”, Journal of Robotics and Autonomous Systems, 2008, vol. 56, 927–941, http:dx.doi.org10.1016j.robot.2008.08.005.

[8] S. Thrun, W. Burgard, D. Fox, Probabilistic Robotics (Intelligent Robotics and Autonomous Agents), The MIT Press 2005, ISBN 0-26220162-3.

[9] E. Remolina, B. Kuipers, ”Towards a general theory of topological maps”, Arti icial Intelligence, 2004, vol. 152, no. 1, 47–104.

DOI:10.1016S0004-3702(03)00114-0.

[10] O. M. Mozos et al., ”Supervised semantic labeling of places using information extracted from sensor data”, Robotics and Autonomous Systems, vol. 5,2007, 392–402.

DOI:10.1016/j.robot.2006.12.003.

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[13] J. C. Latombe, Robot Motion Planning, Kluwer Academic Publishers. MA Boston 1992.

[14] D. Fox, W. Burgard, S. Thrun, ”The Dynamic Window Approach to Collision Avoidance”, IEEE Robotics and Automation, vol. 4, no. 1, 1997, 23–33.DOI: 10.1109/100.580977.

[15] International Building Code http://publicecodes.cyberregs.com/icod/ibc/ [16] Polish Building Code http://prawo.ws/budowlane/ [17] Canadian Codes Centre http://www.nrc-cnrc.gc.ca/eng/solutions/ advisory/codes_centre_index.html [18] ROS - Robot Operating System - http://ros.org [19] A. Typiak, ”Terrain Mapping Method for Fast Mobile Vehicle, Polish Journal of Environmental Studies”,vol.16, 2007, No. 4B. [20] http://eur-lex.europa.eu/legalcontent/EN/TXT/PDF /?uri=CELEX:32006L0042&rid=6 [21] Kinect. www.xbox.com/pl-PL/Kinect [22] W. Turek, K. Cetnarowicz, M. Multan, T. Soś nicki, A. Borkowski. ”Modeling buildings in the context of Mobile Robotics”. In: Proceedings of the 13th Polish Conference on Robotics (KKR 2014), 2014.

[23] B. Siemia̧ tkowska, B. Piekarski Bartosz, B. Harasymowicz-Boggio, ”Navigating robots inside building”. In: Prace Naukowe Politechniki Warszawskiej. Elektronika, O icyna Wydawnicza Politechniki Warszawskiej, 2014, 503–512.

[24] M. Przybylski, D. Koguciuk, B. Siemiatkowska, B. Harasymowicz-Boggio, L. Chechliń ski: ”Integration of Qualitative and Quantitative Spatial Data within a Semantic Map for Service Robots”.In: Progress in Automation, Robotics and Measuring Techniques. Volume 2 Robotics.Series: Advances in Intelligent Systems and Computing, vol. 351, 2015, 223–232, DOI:10.1007/978-3319-15847-1_22.

[25] OpenCV - wiki. [online], 2005. Tristen Georgiou Fast Match Template.

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Submi ed: 18th December 2015; accepted: 19th May 2016

Torsten Siedel, Stefan Bethge, Manfred Hild DOI: 10.14313/JAMRIS_2-2016/12 Abstract: In this ar cle, we propose two model free control schemes that are based on pulse height modula on using low frequencies with the goal to compensate normal fric on effects in drive trains that nega vely influence the performance of e.g. a standard PID controller. The first control scheme uses pulse height modula on to especially compensate s ck slip effects but increases vibra on and noise in the drive train. To reduce such side effects a modified phase shi ed pulse height control scheme based on mul ple actuated joints is introduced. Both control schemes are compared with a standard linear controller as reference and evaluated by using six quality criteria.

Keywords: Mul ple Actuators, Model-free, Fric on Compensa on, Pulse Modula on

1. Introduc on In many actuation applications, motors are combined with a gearbox to change transmission. This increases friction effects and leads to a notable reduction in control quality when using conventional controllers like PID control. Effects such as delayed response and non-linear control torque relationships can especially be observed when using low rotational speeds or when driving high loads.

Various compensation methods have been proposed that attempt to create friction models to predict the actual friction of the system and improve motor controllers based on them. Le-Tien and AlbuSchä ffer [4] use a static friction model describing Coulomb, viscose and load dependent friction. Olsen et. al. [7] investigate the behaviors of the dynamic LuGre and Bliman-Sorine friction models to derive compensation methods and show results from practical experiments. Swevers et. al. [12] extend these models by incorporating hysteresis with non-local memory and achieve improved accuracy in compensation.

Usually, parameters for both static or dynamic models have to be adapted to each controlled system separately and this can prove to be cumbersome. Consequently, parameter identi ication methods [14] or adaptive compensation through learning such as neural networks [9] or adaptive fuzzy systems have been proposed [13].

Instead of using active compensation, different model-free methods have been investigated that reduce friction outside of the controller. Well known techniques include dither [17], [8] and pulsed motor

control [15], [16]. Dither is the introduction of a highfrequency noise signal into non-linear systems in order to help the system stabilize. It is employed successfully in hydraulic and pneumatic use cases while it has some detrimental properties when used with directdrive actuators.

Pulsed or impulsive motor control is a method that reduces the system’s stiction by pulsing the motor control with pulses large enough to overcome stiction only when the motor is in low speeds. This is especially useful when starting and stopping often and operating with low rotation speeds, as is usually the case in robotics applications. The optimal length and amplitude of the pulses depend on the system properties.

Appropriate methods to obtain them are discussed e.g. in [15].

Drawbacks of both methods adding arti icial oscillation to the system are possible noise and higher wear of mechanic components. By using multiply actuated joints, some of these can be alleviated through combined control upon the behavior of pulsed control with only one motor. Multiply actuated joints are also employed e.g. in robotics to achieve regulated joint stiffness [1]. Further bene its are increased ef iciency, redundancy and fail-safety [3], [11], [2].

In this article we employ an adapted pulse based motor control method to reduce friction effects. To reduce the disadvantageous side-effects of those methods, namely vibration and noise, we propose a novel control method based on the use of multiple parallel actuators for one joint and emphasize its properties in reducing those effects. In the following, two variations of motor control methods for multiple actuators are compared with a standard linear controller as reference. The behavior of a motor unit is evaluated by means of six quality criteria. The described control methods are closely related to results given in Siedel [10] which focuses on the methods applied for robotic purposes. Various igures are also taken from there.

2. Mul ply Actuated Motor Unit The motor unit employs multiple identical actuators for which the servo unit Dynamixel RX-28 by ROBOTIS was used. In this article, only the use of identical actuators is investigated. Further options can also be gained by combining actuators with different gain ratios and output powers, cf. [6]. The unit consists of a DC motor, a spur gear and electrical components for motor drive, communication and control. The DC motor is an RE-max 17 from Maxon (see [5] for speci ica15

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Attached to the end of the measuring shaft is a pendulum with the following properties: - mass đ?‘š = 1168 g - length đ?‘™ = 0.282 m - rotatory inertia đ??˝ = 99 gm

Fig. 1. The coupling gearbox with two RX-28s. The servo units are coupled to the measuring sha with the spur gear. The lower part of the picture shows the measuring sha that is coupled to the a ached measuring devices using a balancer coupling 2

5

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The pendulum mass and lever arm length are chosen in such a way that the maximum drive torque is not suf icient to de lect the pendulum to the horizontal plane at slow speeds. This also avoids overshoot.

Figure 2 also shows additional measuring devices which are used to evaluate the drive characteristics based on the previously de ined quality criteria. A HAMEG HM1008-2 digital oscilloscope measures motor current and voltage to get the motor power consumption while an RS-components digital thermometer gets the current motor temperature. Since the oscilloscope uses discrete sampling, an LC ilter is added.

The temperature readings are only used to maintain comparable starting conditions for each test run and are not recorded. An HDM M30 measurement microphone is placed in a 50 cm distance to provide sound pressure measurements. The whole setup is enclosed by special acoustic foam panels to reduce ambient noise and to improve the sound measurement quality.

3.1. Test Sequence

4 7

Fig. 2. Test setup enclosed in sound isola ng foam panels. Le : Measuring microphone (1) at 0.5 m distance to the motor unit. Middle: drive test bench (2) with motor unit (3), pendulum (4) and control electronics (5). Right: Two-channel oscilloscope (6) and temperature sensor (7) tions) and is operating on 12 to 16 V. An STMicroelectronics L6201 H-bridge is used as the power driver for the motor. On the low level, the motor is driven with a pulse width modulated voltage at a frequency of 15.6 kHz. The ive-stage spur gear has a gear ratio of 1:195. With the exception of the output socket gear that is mounted on double ball bearing, all gears are mounted on plain bearing. In the motor unit, two motors are coupled directly with the measuring shaft at a ratio of 1:1, see Figure 1. The whole drive train – servo gearbox and coupling – has a backlash of 0.8 degrees.

3. Experimental Setup The center of Figure 2 shows the drive test bench which contains the motor unit with two servo units, a torque sensor that is rotationally decoupled and an optical angle sensor. The sensors are assembled in a coaxial manner, preventing the introduction of additional mechanical backlash. A microcontroller board with an STM32 processor is used for motor control and data communication to and from a connected computer.

16

All of the following experiments are carried out under uniform conditions regarding motor temperature, operating voltage and ambient noise. Each test takes 20 seconds. The actuators are controlled without any feedback.

Tests are started with the pendulum hanging down so that no torque is applied to either measurement shaft or the motor unit. Over the irst 10 seconds the motor reference voltage đ?‘ˆ for both motors is gradually increased from 0 V up to 14.8 V which raises the pendulum in one direction. During the next 10 seconds the power is gradually decreased down to 0 V again so the pendulum is lowered again. The progression of the reference voltage can be seen in the upper part of Figure 3.

The duration of 20 seconds for each test is chosen so that the motors do not heat up considerably and at the same time dynamical effects can be disregarded because of rather slow raising and lowering of the pendulum. To minimize the in luence of measurement scattering, each test is repeated ive times and the values are averaged over all ive runs.

4. Reference Control The irst test is intended to show the current situation with gearbox induced friction effects and will serve as reference measurement. For readability, the vector đ?‘ˆ(đ?‘Ą) =

đ?‘ˆ (đ?‘Ą) đ?‘ˆ (đ?‘Ą)

(1)

is the combined vector of the voltages of both motors đ?‘ˆ (đ?‘Ą) and đ?‘ˆ (đ?‘Ą). For the reference measurement, both

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servos get the same voltage signal, đ?‘ˆ (đ?‘Ą) = đ?‘ˆ (đ?‘Ą) = đ?‘ˆ

(đ?‘Ą)

(2)

so both motors always apply the same torque. The lower part of Figure 3 displays the movement (or trajectory) of the pendulum for each of the ive tests. The single test trajectories as well as the averaged trajectory over all tests both show extensive non-linearities and hysteresis effects. These are caused by friction in the gearboxes of both servo units.

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Bo om: single trajectories of the pendulum (grey lines) resul ng from the reference voltage from subsequent trials and averaged trajectory (black). The dashed line labels the turning point of the reference voltage In the irst half of the test runs, the in luence of the well known stick-slip effect is clearly visible, which raises the pendulum with stuttering motion. Following the law of Coulomb, both friction types are proportional to the motor drive torque. This leads to the observation that the relation between the static friction and the dry friction is independent of the applied torque. With the static friction coef icient đ?œ‡ and the dry friction coef icient đ?œ‡ , đ?œ‡ = constant.

(3) đ?œ‡ Also see [4] for more details on torque dependent friction effects.

As dynamic effects can be neglected, it is possible to compare the start and end of each plateau of the rising trajectory and conclude on the difference of the friction coef icients. Starting with the dry friction coeficient đ?œ‡ , it can be seen that in this test case the sticky friction is 38.8 % stronger than the dry friction (with a standard deviation of 12.8 %). The driving torque therefore has to be increased by 38.8 % to further lift the pendulum after it came to a standstill. In general, this value is greatly dependent on the type of gearbox and the ratio of transmission but also on various other factors such as lubrication, operating temperature and wear.

The topmost plateau marks the maximum angle the pendulum has reached in that particular test run.

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This area is shifted forward on the time axis, while some jumps are even happening only after the maximum voltage has been reached at 10 seconds. The difference between the maximum angle that the pendulum reaches in single runs is up to 0.41 rad. Until reaching the maximum reference voltage đ?‘ˆ the whole drive train is in drive mode. Drive mode is here used to name one of the modes of four quadrant operation of servo motors. In drive mode, motor torque and turning direction are the same whereas in brake mode, the directions are opposing each other. The motor moment is acting in the turning direction and the friction torque is acting against the motor torque, in effect reducing the effective torque. When the reference voltage is decreased, the motor moment is also reduced without changing the motor direction. The pendulum should be lowering. The motor unit however is now in brake mode, so motor torque and friction torque are adding up. The pendulum is also not moving since static friction is still effective which is larger than dry friction. As a result, the pendulum is only lowering after the driving voltage falls short of 22.2 % of the maximum reference voltage. The pendulum then starts suddenly and is moving quickly in the beginning.

This test using conventional control shows which non-linear and hysteretic effects can occur in the drive unit. These effects are of course also dependent on the parameters of the pendulum and the behavior of the reference voltage.

5. Pulse Height Modulated Motor Control In order to mitigate these effects, a common technique is to use pulsating motor drive rather than continuous motor drive [15], allowing the motor to always move even for very small target values. Usually pulse width modulated (PWM) drive is used which alters the duty cycle to achieve the desired drive voltage on average. Using the similar pulse height modulated (PHM) motor control which changes the height of equally distributed pulses has the bene it of a much higher possible resolution when the temporal resolution of the digital circuitry in use is not very high. The basic idea of these techniques is to change between the friction types (stiction and dry friction) as well as the motor modes (drive and brake mode) in a controlled manner with comparably low frequency in order to linearize the drive behavior.

First, the operation of the PHM drive will be described. It is implemented in three steps: creating the base oscillation, modifying the oscillation amplitude in relation to the reference voltage and combining the reference voltage with the oscillation. The base oscillation is derived from an analytic representation of the triangle oscillation đ?‘“ (đ?‘Ľ) =

2 sin đ?œ‹

sin(2 đ?œ‹ đ?‘Ľ) .

(4)

which is modi ied to alter the amplitude as necessary.

The triangle oscillation đ?‘“ (đ?‘Ľ) is expanded with the parameters đ?‘Ž to alter the amplitude, and đ?‘“ to alter the 17

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đ?‘ˆ (đ?‘Ą) = đ?‘Ž đ?‘“ (đ?‘“ đ?‘Ą)

(5)

with đ?‘Ą being the time variable to receive the base frequency voltage đ?‘ˆ (đ?‘Ą). The amplitude đ?‘Ž of the base frequency is modi ied in relation to the reference volt/2 V. A reference voltage within a range of 0 to đ?‘ˆ age of đ?‘ˆ = 0.00 V relates to an amplitude of 0.00 V while đ?‘ˆ = đ?‘ˆ /2 relates to an amplitude of đ?‘ˆ /2. When the reference voltage assumes values greater than đ?‘ˆ /2 up to đ?‘ˆ , the amplitude is falling down to 0.00 V to not receive values above the maximum drive voltage. Figure 4 shows the reference voltage (black) and the envelope of the pulse modulated voltage (grey) as it is set in the later test runs.

The amplitude progression in relation to the reference voltage can be given as đ?‘Ž =

2đ?‘ˆ 2đ?‘ˆ − 2đ?‘ˆ

at at đ?‘ˆ

â&#x2030;¤đ?&#x2018;&#x2C6; <

â&#x2030;¤ â&#x2C6;§đ?&#x2018;&#x2C6;

> (6)

Finally, the amplitude modulated base oscillation is added to the reference voltage and the resulting signal is passed to both motors similarly. đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) = đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) = đ?&#x2018;&#x2C6;

(đ?&#x2018;Ą) + đ?&#x2018;&#x2C6; (đ?&#x2018;Ą)

(7)

In order to determine which pulse frequency delivers closest to ideal behavior, ive different frequencies for the base oscillation were tested. The frequencies were chosen with regard to the frequency of the motor control loop which is working with 100 Hz. Accordingly, the highest possible frequency is 50 Hz of which 25 Hz, 12.5 Hz, 6.25 Hz and 3.125 Hz are derived.

As mentioned, each frequency test run is repeated ive times and averaged. The obtained trajectories are displayed in Figure 5 which also includes the trajectories from the previous test (reference control) for comparison.

Comparing the results of the reference control, all PMC trajectories more or less display the following properties: The spread of the trajectories is lower, a plateau forms at higher values of the reference voltage and the stick-slip effect is minimized as can be seen in as smooth motion in place of the previous jumps and 18

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t [s]

1.2 0.8 0.4 0.0

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PMC 50 Hz y [rad]

0.4

0

PMC 12.5 Hz

20

0.8

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t [s]

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frequency

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PMC 25 Hz y [rad]

Fig. 4. Envelope curve (grey) of the modulated oscilla on in rela on to the reference signal đ?&#x2018;&#x2C6; (black line). The dashed lines indicate half of the maximum voltage and eďŹ&#x20AC;ec ve maximum voltage đ?&#x2018;&#x2C6; (14.8 V).

0.8

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y [rad]

y [rad]

PMC 6.25 Hz

t [s]

2016

PMC 3.125 Hz y [rad]

U [V]

Um / 2

y [rad]

Reference

Nâ&#x2C6;&#x2DC; 2

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t [s]

1.2 0.8 0.4 0.0

0

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8

12

16

20

t [s]

Fig. 5. Trajectories of the pendulum with reference control and with PMC at ďŹ ve diďŹ&#x20AC;erent base frequencies (3.125 Hz to 50 Hz). The separate trajectories are drawn in grey, the averages for each frequency in red.

The black line shows the averaged trajectory for the reference control stuttering. Finally, the hysteresis is greatly reduced and the trajectory is closer to being symmetrical.

At 3.125 Hz, in luence of the frequency is clearly visible as a modulated frequency during upwards and downwards motion. Higher frequencies donâ&#x20AC;&#x2122;t produce this behavior anymore as the pendulumâ&#x20AC;&#x2122;s inertia acts as a low pass ilter that removes higher frequencies.

The frequencies from 6.25 Hz to 50 Hz do not produce signi icant differences between the trajectories, suggesting low in luence of the speci ic frequency.

6. Phase Shi ed Pulse Height Modulated Motor Control A possible modi ication of the previously described PMC is to have each motor be pulsed at different points in time with phase shifted modulated pulse oscillations. In the case of two motors, the signals are shifted by đ?&#x153;&#x2039;, phase shift for any other number of motors will be looked at in the next section. The expanded equation đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) = đ?&#x2018;&#x17D; đ?&#x2018;&#x201C; (đ?&#x2018;&#x201C; đ?&#x2018;Ą + đ?&#x153;&#x2018;)

(8)

yields a phase shifted oscillation with đ?&#x153;&#x2018; as the phase shift parameter. Using đ?&#x153;&#x2018; = 0 and đ?&#x153;&#x2018; = đ?&#x153;&#x2039; for đ?&#x153;&#x2018;, the base oscillations đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) and đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) are obtained and, written as a vector đ?&#x2018;&#x2C6;(đ?&#x2018;Ą) =

đ?&#x2018;&#x2C6; đ?&#x2018;&#x2C6;

(đ?&#x2018;Ą) + đ?&#x2018;&#x2C6; (đ?&#x2018;Ą) (đ?&#x2018;Ą) + đ?&#x2018;&#x2C6; (đ?&#x2018;Ą)

,

(9)

are fed to both the motors. With this altered control method, again ive test runs per base frequency are performed. The resulting trajectories are shown in Figure 6 which again includes the resulting trajectories of the reference control test for comparison.

At the lowest frequency of 3.125 Hz, the pendulum is raised almost continuously to just before reaching

Journal of Automation, Mobile Robotics & Intelligent Systems

Reference

PPMC 3.125 Hz 1.2

0.8 0.4 0.0

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16

0.8 0.4 0

4

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16

20

t [s]

8

12

16

20

t [s]

0.8 0.4 0.0

20

0.8

4

1.2

t [s]

1.2

0

PPMC 12.5 Hz

0

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16

PPMC 50 Hz y [rad]

y [rad]

PPMC 25 Hz

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PPMC 6.25 Hz

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t [s]

2016

seconds) height. The quality is determined as follows: irst, the second half of the trajectory is mirrored at the middle axis (at 10 seconds) onto the irst half. The diagrams that are produced in this way are shown in Figure 7. The direction of movement is labelled with arrows. It is well visible how much ascent and descent overlap and therefore exhibit a certain hysteresis.

A numerical value for the hysteresis is obtained as the sum of the squared errors. The corresponding equation is /

1.2

đ??¸ (đ?&#x2018;Ś) =

0.8

đ?&#x2018;Ś(đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;Ś(đ?&#x2018;&#x2021;/2 â&#x2C6;&#x2019; đ?&#x2018;Ą)

(10)

0.4 0.0

0

4

8

12

16

20

t [s]

Fig. 6. Trajectories of the pendulum with reference control and with PPMC at ďŹ ve diďŹ&#x20AC;erent base frequencies (3.125 Hz to 50 Hz). The separate trajectories are drawn in grey, the averages for each frequency in red. The black line shows the averaged trajectory for the reference control

where the error is assumed as the vertical distance between the ascent and descent curves for each time step.

Reference y [rad]

y [rad]

y [rad]

1.2

Nâ&#x2C6;&#x2DC; 2

VOLUME 10,

0.8 0.4 0.0

0

2

4

6

8

10

7.1. Hysteresis Starting with determining the hysteresis, we look at the trajectory. In the present case, the hysteresis describes the difference of the trajectory when increasing (đ?&#x2018;Ą = 0 to 10 seconds) and decreasing (đ?&#x2018;Ą = 10 to 20

y [rad]

y [rad]

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PMC 12.5 Hz

2

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6

8

0.4 0.0

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8

10

t [s]

2

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8

10

t [s]

0

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8

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0.8 0.4 0.0

10

10

t [s]

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t [s]

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0

PPMC 25 Hz

0

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4

6

8

PPMC 50 Hz y [rad]

y [rad]

PMC 50 Hz

8

0.8

0.0

10

0.8

6

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t [s] y [rad]

y [rad]

PMC 25 Hz

4

PPMC 12.5 Hz y [rad]

0

2

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0.0

10

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0

PPMC 6.25 Hz

t [s]

0.8

0.0

0.4 0.0

10

0.8

0.0

0.8

t [s] y [rad]

y [rad]

PMC 6.25 Hz

7. Results The visual examination of the trajectories only allows limited conclusions about the properties of the particular control methods, of which PMC and PPMC also show frequency dependent behavior. Apart from the movement alone shown by the trajectory of the pendulum, other quality measures are important to judge each control method appropriately. These include the torque behavior, energy consumption, hysteresis, linearity, maximum torque, operating smoothness, energy ef iciency and operational noise. These can either be measured directly or derived from the taken measures trajectory, torque, energy consumption and operating noise. A signi icance analysis was made to make sure that, regarding the standard deviation over the repeated measurements, the best results are signi icantly better. The results of the best and second best test run within each quality measure are tested for signi icant difference with a two-sample t-test. The chosen signi icance criterion for difference is đ?&#x2018;? > 5%.

0.8

0.0

y [rad]

the maximum voltage. Then the results are scattered.

The plateaus of all ive trajectories are more or less of the same length but clearly shifted forward on the time axis, showing the in luence of the motor drive modes. At a frequency of 6.25 Hz no jumps in the trajectories are visible anymore, but the transitions to the plateaus are more smooth. As with the PMC, the PPMC shows only marginal differences at higher frequencies (12.5 Hz, 25 Hz and 50 Hz)

t [s]

PPMC 3.125 Hz

PMC 3.125 Hz

10

t [s]

0.8 0.4 0.0

0

2

4

6

8

10

t [s]

Fig. 7. Comparison between hysteresis and linearity of averaged trajectories. The individual graphs show ascent and descent of averaged trajectories with the diďŹ&#x20AC;erent control methods. The direc on of mo on is labeled with arrows while the addi onal curves between the trajectories show the ideal course of the respec ve trajectory assuming no fric on. It is computed using a motor model, using parameters derived from the actual trajectories A good result for the hysteresis is achieved if đ??¸ is minimal and therefore the area between rising and falling trajectory is as small as possible. The best result was measured for the PMC at 12.5 Hz. Comparing to the second best result, the PPMC at 12.5 Hz, the best result is signi icantly better.

19

Journal of Automation, Mobile Robotics & Intelligent Systems

%

đ?&#x2018;Ś (đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;Ś (đ?&#x2018;Ą)

.

(11)

A positive result for Linearity is achieved if đ??¸ is small.

The best result was obtained for the PMC at 6.25 Hz which is signi icant in comparison to the PPMC at 6.25 Hz. 7.3. Maximum Torque and Vibra ons For assessing the torque behavior of the motor unit we look at the effective torque that is measured with the torque sensor between drive train and the pendulum. The quality of the behavior is determined by the criteria maximum torque and occurring vibrations. The maximum torque that can be delivered is determined by averaging the torque values around the point at which the pendulum is at maximum height.

Averaging is done, as the dynamic behavior of the pendulum might result in high torque values for a short time which however can not be maintained over longer time spans. The best result for maximum torque was achieved for the PPMC at 50 Hz which is signi icantly better in comparison to the PMC at 50 Hz. Vibrations are on the one hand produced by oscillations resulting from the sticky friction in the gearbox but on the other hand because of the pulsated control signal. Figure 8 shows exemplary torque measurements for trajectories for each control method. The reference control on the left shows vibrations because of the alternating friction types. The PMC however shows vibrations resulting from the pulsed control.

20

M [Nm]

The next step is to evaluate the linearity of the trajectory, which covers the extent to which the drive behavior in luenced by the real friction approaches the ideal friction free behavior. To evaluate the drive characteristics we have another look at the individual pendulum trajectories. In order to picture an ideal trajectory graph, the physical properties of the pendulum and a simple DC servo motor model is employed which assumes a directly proportional relation between motor current and torque and ignores friction or dynamical effects. Except for the torque constant, all parameters for the model are known. The actual operating behavior is heavily dependent on the control method used (Reference, PMC and PPMC) which is why the torque constant has to be determined for each test run separately. This is done with the least squares method.

The resulting ideal trajectories based on the model are adjusted to the real trajectories. Since the upper plateaus of the trajectories are distorting the values, only the data up to 90 % of the maximum angle are used. The determined ideal trajectories can be seen in Figure 7 between the trajectories of ascent and descent.

For these areas of each measurement, it is now possible to get the linearity deviation of the averaged real course đ?&#x2018;Ś to the ideal course đ?&#x2018;Ś . This is done again by computing the sum of squared errors.

PMC 6.25 Hz

Reference

4.0

2016

PPMC 6.25 Hz

3.0 2.0 1.0 0.0

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8

12

16

20 0

4

t [s]

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20 0

t [s]

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t [s]

Fig. 8. Exemplary eďŹ&#x20AC;ec ve torque courses for three diďŹ&#x20AC;erent control methods. Le : Reference, Middle: PMC (6.25 Hz), Right: PPMC (6.25 Hz) Using the same base frequency, the PPMC shows noticeably less vibrations because of phase shifted control of both motor units where pulses can compensate for each other inside of the coupling gear.

The vibrations can be quantized by fast Fouriertransforming (FFT) the torque signal, yielding a frequency spectrum for each torque signal. The value of the frequency with the maximum overall level is assumed as the vibration level. The best result for vibrations was achieved for the PPMC at 25 Hz which is signi icantly better in comparison to the PMC at 25 Hz. 7.4. Energy Consump on For the energy consumption, the measurements for current and voltage are used which are obtained using the oscilloscope. Figure 9 shows exemplary data for the power consumption for each of the control methods. Reference control and PPMC produce similar curves. Only in the middle of ascent and descent, small luctuations are visible. However, when using PMC, noticeable luctuations are produced that have the same height as the amplitude of the oscillation that is modulated onto the reference voltage đ?&#x2018;&#x2C6; .

Reference P [W]

7.2. Linearity

đ??¸ =

Nâ&#x2C6;&#x2DC; 2

VOLUME 10,

PMC 6.25 Hz

PPMC 6.25 Hz

40 30 20 10 0

0

4

8

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20 0

t [s]

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Fig. 9. Exemplary power consump on courses for three diďŹ&#x20AC;erent control methods. Le : Reference, Middle: PMC (6.25 Hz), Right: PPMC (6.25 Hz) Regarding the energy consumption, no immediate disadvantage is to be expected from the luctuations, as long as the input power is not exceeding the control electronics. Crucial to the quality of the control method rather is the overall energy usage of the actuators per trial run, determined by the time integral đ??¸ =

đ?&#x2018;&#x2C6;(đ?&#x2018;Ą) đ??ź(đ?&#x2018;Ą) đ?&#x2018;&#x2018;đ?&#x2018;Ą

(12)

of the product of operating voltage đ?&#x2018;&#x2C6;(đ?&#x2018;Ą), current đ??ź(đ?&#x2018;Ą) and đ?&#x2018;&#x2021; being the duration of the test run. Smaller values of đ??¸ are better. The best result was achieved with the

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 10,

reference control which is signi icant in comparison to the PMC and PPMC at all frequencies.

7.5. Opera ng Noise The sound produced is measured at a sample rate of 48 kHz (mono) to evaluate the control method quality. As before for vibrations, a frequency analysis is done and the level of the maximal frequency is obtained as an indicator for the amount of noise generated.

Here, the best result was obtained for the PMC at 3.125 Hz which is signi icant in comparison to the PPMC at 3.125 Hz. 7.6. Overall Ra ng

6.25 Hz PMC PPMC 12.5 Hz 25 Hz 50 Hz

PMC PPMC PMC PPMC PMC

x o x o o o o o o o o

x o o o x o o o o o o

o x o x o x o x o o o

o o x o o o o o o x x

o o o o o x o x o o o

3

2

2

1

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PPMC Weight 3

ng Rati rall

x o x o o o o o o o o

Ove

rity Max Torq ue Vibr atio n Ene rgy usag e Nois e

Line a

Reference PMC 3.125 Hz PPMC

Hys

tere sis

Summarizing the results for each of the criteria, the values are normalized and mapped to a color range and arranged in an overview matrix as seen in Figure 10. Each result is marked by color and additionally an â&#x20AC;&#x153;xâ&#x20AC;? or â&#x20AC;&#x153;oâ&#x20AC;? to show if it is above or below average. The best result for each criterion is marked with a white triangle.

good

x x x o x x o x o o o

o

x

bad

Fig. 10. Color coded ra ng matrix for the three control methods Reference, PMC and PPMC by means of the quality criteria Hysteresis, Linearity, Maximum Torque, Vibra on, Energy Consump on and Noise Genera on (column 1 to 6). Column 7 displays the average value over the criteria and gives an overall ra ng for the individual control method. The chosen weights relate to the inďŹ&#x201A;uence each criterion has for the overall ra ng (3: strong, 2: medium and 1: low). White triangles mark the best result for each criterion While determining the quality criteria, it was noticed that the results for each criterion have different variance, i.e. the differences between best and worst result vary. Also, the results show that each criterion does not have the same importance for the evaluation.

Therefore three different steps of weighting were introduced (see Figure 10). The overall results on the right side show a weighted sum of quality criteria.

It is noteworthy that the best results are not distributed equally. None of the rows â&#x20AC;&#x201C; meaning none of the control methods â&#x20AC;&#x201C; achieves more than one best result. The same can be said for the worst results. The

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best overall result is produced with the PPMC at 25 Hz. This control method shows the best single value in vibration behavior and overall good values for the other criteria. The next best result is the PPMC at 50 Hz and the worst after the reference control is the PPMC at 3.125 Hz. In conclusion, the PPMC shows the bene its of multiple actuators and the possibilities of thereby enhanced control methods. The improvement of the overall drive behavior is considerable compared to the conventional reference control method. The only disadvantage, albeit small, is the energy consumption that is about 3 % higher for PPMC at 25 Hz.

8. PPMC Conversion for an Arbitrary Amount of Coupled Actuators In order to expand the PPMC to more than two actuators, the phase shift đ?&#x153;&#x2018; for all motors has to be distributed equally to the whole cycle duration of the control signal, aiming for minimal vibration and noise generation. The same principle applies to combustion engines, where the succession of the ignition for each cylinder is equally distributed to one turn of the crank shaft.

Since discrete signal processing does not allow every desired phase shift, the distribution has to be aligned to what is possible. In the following method, odd or even amounts of actuators are dealt with separately.

8.1. Even Number of Actuators As described in Section 6 the base oscillation for the second motor is being shifted by đ?&#x153;&#x2039;, so that both compensate for the torque impulses of the other. To keep this principle for more motors, the phase shifts over the irst half of the cycle duration (0 â&#x2030;¤ đ?&#x153;&#x2018; < đ?&#x153;&#x2039;) have to be identical to those over the second half (đ?&#x153;&#x2039; â&#x2030;¤ đ?&#x153;&#x2018; < 2 đ?&#x153;&#x2039;). Furthermore, to reach as even a distribution as possible of all phase shifts over the cycle duration in order to minimize vibrations and noise, the PPMC for đ?&#x2018; actuators is adjusted as follows.

In the beginning, the amount of work cycles of the control circuitry during one period of the base oscillation is determined. The amount đ?&#x2018;? is đ?&#x2018;? =

đ?&#x2018;&#x201C; , đ?&#x2018;&#x201C;

(13)

where đ?&#x2018;&#x201C; is the working frequency of the control circuitry and đ?&#x2018;&#x201C; is the frequency of the base oscillation.

A possible base oscillation with frequency đ?&#x2018;&#x201C; for the PPMC following Section 6 necessitates that đ?&#x2018;? is even. Otherwise the maxima of the base oscillation could be lost in the discretized representation.

Since the amount đ?&#x2018; of actuators is not necessarily a multiple of đ?&#x2018;? , it has to be determined which subdivisions of đ?&#x2018;? are possible. For that, let đ?&#x2018;&#x160; be the set of even dividers of đ?&#x2018;? as de ined by đ?&#x2018;&#x160; â&#x2C6;ś= {x â&#x2C6;&#x2C6; â&#x201E;&#x2022; â&#x2C6;Ł đ?&#x2018;Ľ divides đ?&#x2018;? â&#x2C6;§ đ?&#x2018;Ľ even} .

(14)

The set is only allowed to contain even dividers as otherwise an equal distribution of the actuators across 21

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 10,

both half periods would not be possible. In the following, đ?&#x2018;&#x160; will be regarded as a set that is sorted in descending order, where đ?&#x2018;&#x160; is the đ?&#x2018;&#x2013;â&#x20AC;&#x2122;th element of that set and đ?&#x2018;§ is the number of elements of đ?&#x2018;&#x160;, đ?&#x2018;§ = |đ?&#x2018;&#x160;|. Since đ?&#x2018;? is even in any case, the last element of đ?&#x2018;&#x160; is đ?&#x2018;&#x160; = 2.

(15)

For better illustration, the elements of đ?&#x2018;&#x160; can also be understood as layers where the đ?&#x2018;&#x2013;â&#x20AC;&#x2122;th layer đ?&#x2018;&#x160; contains subdivisions of the cycle duration. Figure 11 displays an exemplary phase shift matrix with the possible phase shifts đ?&#x2018;&#x160; for each layer drawn as rectangles.

The amount of actuators that is resulting from the subdivisions is then listed in each of the rectangles.

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The distribution of the actuators is now given. For that we have to determine the associated phase shifts.

We irst get the step size of the phase shifts Î&#x201D;đ?&#x153;&#x2018; =

2đ?&#x153;&#x2039; đ?&#x2018;&#x160;

(19)

in relation to the đ?&#x2018;&#x2013;â&#x20AC;&#x2122;th layer, whereas the cycle duration is normalized to 2 đ?&#x153;&#x2039;. In total, there are đ?&#x2018; actuators, let đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018; }. The phase shift đ?&#x153;&#x2018; assigned to the step size Î&#x201D;đ?&#x153;&#x2018; for the đ?&#x2018;&#x2DC;â&#x20AC;&#x2122;th actuator can be determined for all actuators by đ?&#x153;&#x2018;

= ((đ?&#x2018;&#x2DC; â&#x2C6;&#x2019; 1) mod đ?&#x2018;&#x160; ) Î&#x201D;đ?&#x153;&#x2018; .

(20)

Fig. 11. Exemplary phase shi matrix for the values: đ?&#x2018;? = 8 and đ?&#x2018;&#x160; = {8, 4, 2}. The rectangles indicate the possible phase shi s It is now possible for each subdivision or layer đ?&#x2018;&#x160; to determine how many sets of actuators đ?&#x2018;&#x17E; it can hold.

The amount of actuators for each layer is đ?&#x2018; =đ?&#x2018;&#x17E; đ?&#x2018;&#x160;.

(16)

In order to keep the difference between the phase shifts and thereby the temporal distance between the power peaks as small as possible, the actuators are assigned to the layers of the matrix in ascending order, i. e. the lowest layer with the inest time grid is always illed irst. The following scheme is used: đ?&#x2018; đ?&#x2018;&#x17E; = đ?&#x2018;&#x160; đ?&#x2018; â&#x2C6;&#x2019;đ?&#x2018; đ?&#x2018;&#x17E; = đ?&#x2018;&#x160; đ?&#x2018; â&#x2C6;&#x2019;đ?&#x2018; â&#x2C6;&#x2019;đ?&#x2018; đ?&#x2018;&#x17E; = đ?&#x2018;&#x160;

Fig. 12. Distribu on of the actuators in a phase shi matrix that is generated from đ?&#x2018;? = 8. Solely an even amount of actuators đ?&#x2018; = {2, 4, â&#x20AC;Ś , 24} is used

(17)

â&#x2039;Ž â&#x17D;&#x203A; đ?&#x2018;&#x17E; =â&#x17D;&#x153;

đ?&#x2018; â&#x2C6;&#x2019; â&#x2C6;&#x2018; đ?&#x2018; đ?&#x2018;&#x160;

â&#x17D;?

â&#x17D;&#x17E; â&#x17D;&#x; â&#x17D;

8.2. Uneven Number of Actuators

It therefore has to hold true that đ?&#x2018; =đ?&#x2018; +đ?&#x2018; +đ?&#x2018; +â&#x2039;Ż+đ?&#x2018; .

22

This method obtains the phase shift values for an arbitrary even amount of actuators and examples can be seen in Figure 12. All actuators can be distributed equally on the lowest layer of the matrix (đ?&#x2018;&#x160; ). Because of the high pulse frequencies possible, the lowest vibrations are expected. Furthermore, the power maxima are always at the same level if all actuators are put on the same level of the matrix which should have a positive in luence on the operating smoothness.

(18)

With the intent to evenly distribute the phase shifts over both halves 0 â&#x2030;¤ đ?&#x153;&#x2018; < đ?&#x153;&#x2039; and đ?&#x153;&#x2039; â&#x2030;¤ đ?&#x153;&#x2018; < 2 đ?&#x153;&#x2039; of the cycle duration of the control voltage, the distribution

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 10,

for an odd number of actuators can not be symmetrical. This can also be checked with equation 18. If it is however necessary to use an uneven amount of actuators, the phase shift for an actuator can be determined using đ?&#x2018;? â&#x2C6;&#x2019;1 2

2đ?&#x153;&#x2039; đ?&#x2018;&#x160;

(21)

đ?&#x153;&#x2018; = (đ?&#x2018;? â&#x2C6;&#x2019; 1)

2đ?&#x153;&#x2039; . đ?&#x2018;&#x160;

(22)

đ?&#x153;&#x2018; = and

That way it is at least ensured that for the according points in time the torque load of inside the drive train is minimal in the case that the other actuators can not be distributed solely on the lowest layer and therefore the maximum torques are unequal. The calculation of the phase shift values for the remaining actuators đ?&#x2018; â&#x2C6;&#x2019;1 can be done following the method as described before.

It is to be noted that when using an uneven amount of actuators, the asymmetry has less of an effect the more actuators are used.

9. Conclusion The present article has pointed out some of the gearbox induced friction effects that occur when two actuators and a conventional control method are used.

The effects are resulting from sticky friction and dry friction (producing among others the stick-slip effect), and effects that are based on the differences between the effective torque of the two motor operating modes (drive mode and brake mode) and the associated hysteresis of the drive behavior. In order to compensate for those effects, irst a modi ied variant of classic pulse modulated control (PMC) was examined. It showed typical side-effects such as increased vibration and noise. Based upon the PMC, the novel phase shifted pulse modulated control (PPMC) was introduced. The PPMC is derived from the PMC and leverages the possibilities of multiple actuators to improve on the PMC. After presenting the experimental setup and the test procedure, the trajectories of the test load (pendulum) that are resulting from the individual test runs for the control methods reference control, PMC and PPMC have been discussed. Only looking at the trajectories gives irst clues to the bene its of using either PMC or PPMC over the reference method. A detailed evaluation of the control quality was given in subsequent sections based on the six quality criteria energy consumption, hysteresis, linearity, maximum torque, operating smoothness, energy ef iciency and operational noise. The various properties of the control methods were then compared.

To produce comparable results, the tests used two coupled standard DC servomotors (Dynamixel RX-28) in all cases. The separate comparison results as well as the overall ratings showed that the PPMC at a base frequency of 25 Hz has the best overall result, where it delivered good to very good results for all criteria. The

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typical hysteresis effects from the conventional control method can be compensated almost completely and the linearity could be improved noticeably. Moreover, because of the phase shifted control of both motors, the PPMC exhibits signi icantly less vibration and noise generation compared to the PMC. The only measured disadvantage is the slightly increased energy consumption that is 3 % higher than the unpulsed control method.

Subsequently it has been described how the PPMC can be generalized to drive trains with an arbitrary amount of parallel actuators. For this, each servo unit needs its own phase shifted modulated pulse signal controlling the motor in succession to the others. With the obtained distribution of the phase shifts, a lownoise and low-vibration operation is ensured.

Both operating methods PMC and PPMC are especially well suited for slow movements. When increasing the speed of movement, the in luence of the stickslip effect to hysteresis is decreasing. Switching between pulsed and unpulsed controllers is possible to rely on the properties on other control methods.

AUTHORS Torsten Siedel â&#x20AC;&#x201C; Space Applications Services NV/SA, Leuvensesteenweg 325, 1932 Zaventem, Belgium, e-mail: torsten.siedel@spaceapplications.com.

Stefan Bethge â&#x20AC;&#x201C; Department of Advanced Robotics, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genoa, Italy, e-mail: stefan.bethge@iit.it.

Manfred Hild â&#x20AC;&#x201C; Neurorobotics Research Laboratory, Beuth-Hochschule fuĚ&#x2C6; r Technik Berlin, e-mail: hild@beuth-hochschule.de, www: http://neurorobotics.eu/.

REFERENCES [1] M. G. Catalano, G. Grioli, M. Garabini, F. Bonomo, M. Mancinit, N. Tsagarakis, and A. Bicchi, â&#x20AC;&#x153;VSACubeBot: a Modular Variable Stiffness Platform for Multiple Degrees of Freedom Robotsâ&#x20AC;?. In: International Conference on Robotics and Automation (ICRA), 2011, 5090 â&#x20AC;&#x201C; 5095.

[2] J. Davies, R. Dixon, R. M. Goodall, and T. Steffen, â&#x20AC;&#x153;Multi-agent Control of High Redundancy Actuationâ&#x20AC;?, International Journal of Automation and Computing, vol. 11, no. 1, 2014, 1 â&#x20AC;&#x201C; 9.

[3] R. Dixon, T. Steffen, J. Davies, A. Zolotas, J. Pearson, and X. Du, â&#x20AC;&#x153;HRA-Intrinsically Fault Tolerant Actuation Through High Redundancyâ&#x20AC;?, 2009.

[4] L. Le-Tien and A. Albu-SchaĚ&#x2C6; ffer, â&#x20AC;&#x153;Adaptive Friction Compensation in Trajectory Tracking Control of DLR Medical Robots with Elastic Jointsâ&#x20AC;?.

In: International Conference on Intelligent Robots and Systems (IROS), 2012.

[5] Maxon Motor AG. Datasheet, RE-max 17, April 2012.

[6] J. B. Morrell and J. K. Salisbury, â&#x20AC;&#x153;Parallel-Coupled Micro-Macro Actuatorsâ&#x20AC;?, The International Journal of Robotics Research, 1998.

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[7] H. Olsson, K. J. Aströ m, C. Canudas de Wit, M. Gä fvert, and P. Lischinsky, “Friction Models and Friction Compensation”, European Journal of Control, vol. 4, 1998, 176 – 195.

[8] A. A. Pervozvanski and C. Canudas-De-Wit, “Asymptotic analysis of the dither effect in systems with friction”, Automatica, vol. 38, no. 1, 2002, 105–113.

[9] R. R. Selmic and F. L. Lewis, “Neural-network approximation of piecewise continuous functions: application to friction compensation”, Neural Networks, IEEE Transactions on, vol. 13, no. 3, 2002, 745–751.

[10] M. T. Siedel. Hybride Steuerung parallel gekoppelter Aktoren am Beispiel des humanoiden Roboters Myon. PhD thesis, Institut fü r Informatik, Humboldt-Universitä t zu Berlin, 2015.

[11] T. Steffen, R. Dixon, R. M. Goodall, and A. Zolotas.

“Requirements Analysis for High Redundancy Actuation”. Technical report, Department of Electronic and Electric Engineering, Loughborough University, 2007.

[12] J. Swevers, F. Al-Bender, C. Ganseman, and T. Projogo, “An integrated friction model structure

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with improved presliding behavior for accurate friction compensation”, Automatic Control, IEEE Transactions on, vol. 45, no. 4, 2000, 675–686.

[13] Y. Wang, D. Wang, and T. Chai, “Extraction and adaptation of fuzzy rules for friction modeling and control compensation”, Fuzzy Systems, IEEE Transactions on, vol. 19, no. 4, 2011, 682–693.

[14] Z. Wenjing, “Parameter identi ication of lugre friction model in servo system based on improved particle swarm optimization algorithm”.

In: Control Conference, 2007. CCC 2007. Chinese, 2007, 135–139.

[15] T. Wescott. “Controlling Motors in the Presence of Friction and Backlash”. http://www. wescottdesign.com/, 2010. [Online; retrieved March 8th, 2013].

[16] T. M. Yang Sangsik, “Adaptive pulse width control for precise positioning under the in luence of stiction and coulomb friction.”, ASME Journal of Dynamical Systems, Measurement and Control, vol. 110, 1988, 221–227.

[17] G. Zames and N. A. Shneydor, “Dither in nonlinear systems”, IEEE Transactions on Automatic Control, vol. 21, no. 5, 1976, 660––667.

Journal of Automation, Mobile Robotics & Intelligent Systems

C P

VOLUME 10,

A đ?&#x203A;ź-

D F

M R

Nâ&#x2C6;&#x2DC; 2

2016

P A

P

Submi ed: 25th January 2016; accepted: 16th May 2016

Urszula Bentkowska, Krzysztof Balicki DOI: 10.14313/JAMRIS_2-2016/13 Abstract: In the paper the problem of preserva on of proper es of fuzzy rela ons during aggrega on process is considered.

It means that proper es of fuzzy rela ons đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; on a set đ?&#x2018;&#x2039; are compared with proper es of the aggregated fuzzy rela on đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ), where đ??š is a func on of the type đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1]. There are discussed đ?&#x203A;ź-proper es (which may be called graded proper es to some grade đ?&#x203A;ź) as reďŹ&#x201A;exivity, irreďŹ&#x201A;exivity, symmetry, asymmetry, an symmetry, connectedness and transi vity, where đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. Fuzzy rela ons with a given graded property are analyzed (there may be diverse grades of the same property) and the obtained grade of the aggregated fuzzy rela on is provided. There is also discussed the â&#x20AC;&#x17E;converseâ&#x20AC;? problem. Namely, rela on đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is assumed to have a graded property and the proper es of rela ons đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are examined (possibly with some assump ons on đ??š). Presented here considera ons have possible applica ons in decision making algorithms. This is why interpreta on of the considered graded proper es and possible poten al in decision making is presented.

Keywords: decision making algorithms, fuzzy rela ons, proper es of fuzzy rela ons, aggrega on func ons

1. Introduc on Since Zadeh has introduced de inition of fuzzy relations [38], [39], the theory of them was developed by several authors. Thanks to the â&#x20AC;&#x17E;fuzzy environmentâ&#x20AC;? we may discuss diverse types of fuzzy relation properties.

For example, graded properties of fuzzy relations were observed in [23] and đ?&#x203A;ź-properties were introduced in [10]. These properties may be understood as properties to some grade đ?&#x203A;ź, where đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Aggregation functions, including means [24], are now widely investigated and there are a few monographes devoted to this topic, e.g. [2], [7], [22]. Aggregation is a fundamental process in multicriteria decision making and in other scienti ic disciplines where the fusion of different pieces of information for obtaining the inal result is important. For example, in the multicriteria decision making a inite set of alternatives đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ } and a inite set of criteria on the base of which the alternatives are evaluated đ??ž = {đ?&#x2018;&#x2DC; , â&#x20AC;Ś , đ?&#x2018;&#x2DC; } may be considered. Fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; on a set đ?&#x2018;&#x2039; corresponding to each criterion are provided. With the use of a function đ??š the aggregated fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is obtained and it is supposed to help decision makers to make

up their mind. It is useful to know which properties of fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are transposed to the relation đ?&#x2018;&#x2026;. There are several works contributed to the problem of preservation of properties of fuzzy relations during aggregation process, e.g. [21], [31], [32], [34].

In this paper the problem of preservation of graded properties of fuzzy relations (cf. [14], [16], [18]) is examined. A inite number of fuzzy relations having a given graded property is considered (there can be diverse grades of the same property) and the obtained grade of the aggregated fuzzy relation is provided.

There are discussed several graded properties: relexivity, irre lexivity, symmetry, asymmetry, antisymmetry, connectedness and transitivity. There is also considered another problem. Namely, relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is assumed to have a graded property and relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are examined whether they have the same property. Appropriate assumptions on đ??š to ful ill the required property are proposed. Presented in this paper results may have applications in decision making problems what is more widely described in Section 3. Moreover, the interpretation of the graded properties in the context of decision making is provided.

The aim of this paper is also to compare three algorithms which follow from the theoretical results presented here. These algorithms (their complexity) and theoretical results (assumptions on functions used in aggregation process) are compared in order to obtain the most useful practically result. The assumptions on đ??š which are used to aggregate đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are the minimal ones, i.e. we do not necessarily consider aggregation functions đ??š but just functions đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1], which were recently called â&#x20AC;&#x2122;fusion functionsâ&#x20AC;&#x2122; [6]. If it comes to complexity, it turned out that it is the same for each presented algorithm (for a given property). In the case of assumptions on fusion functions đ??š the situation may be different what is analyzed in Section 7.1.

In Section 2, useful de initions are collected. In Section 3, motivation from real-life situations to consider such theoretical problem is presented. In Section 4, diverse dependencies and interpretation of đ?&#x203A;źproperties are discussed. In Section 5, graded properties: re lexivity, irre lexivity, symmetry, asymmetry, antisymmetry, connectedness and transitivity are examined one by one, in the context of their preservation in aggregation process. In Section 6, reciprocity property and other concepts and properties connected with decision making algorithms are mentioned. Finally, in Section 7 comparison of algorithms based on the theoretical studies presented in this paper are pro25

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vided.

2. Preliminaries Now we recall some de initions which will be helpful in our investigations.

De inition 1 ([38]). A fuzzy relation in đ?&#x2018;&#x2039; â&#x2030; â&#x2C6;&#x2026; is a function đ?&#x2018;&#x2026; â&#x2C6;ś đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x2039; â&#x2020;&#x2019; [0, 1]. The family of all fuzzy relations in đ?&#x2018;&#x2039; is denoted by â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;).

With the use of đ?&#x2018;&#x203A;-argument functions đ??š we aggregate given fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; for a ixed đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; .

De inition 2 ([25]). Let đ??š â&#x2C6;ś [0, 1] đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;). đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), where

â&#x2020;&#x2019;

[0, 1],

đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) = đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, will be called an aggregated fuzzy relation. A function đ??š preserves a property of fuzzy relations if for every relation đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) having this property, đ?&#x2018;&#x2026; also has this property.

Example 1. Projections đ?&#x2018;&#x192; (đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = đ?&#x2018;Ą , đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;} preserve each property of fuzzy relations because for đ??š = đ?&#x2018;&#x192; we get đ?&#x2018;&#x2026; = đ?&#x2018;&#x2026; .

De inition 3 ([7]). Let đ?&#x2018;&#x203A; ⊞ 2. A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is called an aggregation function, if it is increasing with respect to any variable, i.e. for any đ?&#x2018; , â&#x20AC;Ś , đ?&#x2018; , đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] ( â&#x2C6;&#x20AC;

⊽ ⊽

đ?&#x2018; ⊽ đ?&#x2018;Ą ) â&#x2021;&#x2019; đ??š(đ?&#x2018; , â&#x20AC;Ś , đ?&#x2018; ) ⊽ đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą )

(1)

and đ??š(0, â&#x20AC;Ś , 0) = 0, đ??š(1, â&#x20AC;Ś , 1) = 1.

De inition 4 ([15]). An operation đ??ś â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is called a fuzzy conjunction if it is increasing and đ??ś(1, 1) = 1,

đ??ś(0, 0) = đ??ś(0, 1) = đ??ś(1, 0) = 0.

An operation đ??ˇ â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is called a fuzzy disjunction if it is increasing and đ??ˇ(0, 0) = 0,

2016

De inition 6 ([28]). A triangular norm đ?&#x2018;&#x2021; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] (a triangular conorm đ?&#x2018;&#x2020; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1]) is an arbitrary associative, commutative, increasing in both variables function having a neutral element đ?&#x2018;&#x2019; = 1 (đ?&#x2018;&#x2019; = 0).

Basic triangular norms and conorms are presented below.

Example 2 ([28], p. 6). For arbitrary đ?&#x2018; , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] we have functions: â&#x20AC;˘ lattice, đ?&#x2018;&#x2021; (đ?&#x2018; , đ?&#x2018;Ą) = min(đ?&#x2018; , đ?&#x2018;Ą), đ?&#x2018;&#x2020; (đ?&#x2018; , đ?&#x2018;Ą) = max(đ?&#x2018; , đ?&#x2018;Ą), â&#x20AC;˘ Ĺ ukasiewicz, đ?&#x2018;&#x2021; (đ?&#x2018; , đ?&#x2018;Ą) = max(đ?&#x2018; + đ?&#x2018;Ą â&#x2C6;&#x2019; 1, 0), đ?&#x2018;&#x2020; (đ?&#x2018; , đ?&#x2018;Ą) = min(đ?&#x2018; + đ?&#x2018;Ą, 1), â&#x20AC;˘ product, đ?&#x2018;&#x2021; (đ?&#x2018; , đ?&#x2018;Ą) = đ?&#x2018; đ?&#x2018;Ą, đ?&#x2018;&#x2020; (đ?&#x2018; , đ?&#x2018;Ą) = đ?&#x2018; + đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018; đ?&#x2018;Ą, 0, đ?&#x2018; , đ?&#x2018;Ą < 1 â&#x20AC;˘ drastic, đ?&#x2018;&#x2021; (đ?&#x2018; , đ?&#x2018;Ą) = đ?&#x2018; , đ?&#x2018;Ą = 1 , đ?&#x2018;Ą, đ?&#x2018; = 1 1, đ?&#x2018; , đ?&#x2018;Ą > 0 đ?&#x2018;&#x2020; (đ?&#x2018; , đ?&#x2018;Ą) = đ?&#x2018; đ?&#x2018;Ą = 0 . đ?&#x2018;Ą, đ?&#x2018; = 0 Thanks to the associativity property triangular norms and conorms may be extended to đ?&#x2018;&#x203A;-argument functions. Special case of aggregation functions are the ones which are idempotent.

Lemma 1 ([25], Proposition 5.1). Every function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] increasing in each variable and idempotent â&#x2C6;&#x20AC; đ??š(đ?&#x2018;Ą, â&#x20AC;Ś , đ?&#x2018;Ą) = đ?&#x2018;Ą (2) â&#x2C6;&#x2C6;[ , ]

ful ils for any đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] min(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) ⊽ đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) ⊽ max(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ). (3) Here we present examples of functions which ful il (3).

Example 3. Let đ?&#x153;&#x2018; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; be a continuous, strictly monotonic function. A quasi-linear mean (cf. [25], p.

112) is the function

đ??ˇ(1, 1) = đ??ˇ(0, 1) = đ??ˇ(1, 0) = 1.

Fuzzy conjunctions and disjunctions are examples of binary aggregation functions. Conversely, if a binary aggregation function has a zero element đ?&#x2018;§ = 0 (as in the case of the geometric mean), then it is a fuzzy conjunction. Similarly, if a binary aggregation function has a zero element đ?&#x2018;§ = 1, then we get a fuzzy disjunction.

De inition 5. A fuzzy conjunction which has a neutral element 1 is called a t-seminorm [20] (a semicopula [1], a conjunctor [9]). A fuzzy disjunction which has a neutral element 0 is called a t-semiconorm.

26

Nâ&#x2C6;&#x2DC; 2

đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = đ?&#x153;&#x2018;

(

đ?&#x2018;¤ đ?&#x153;&#x2018;(đ?&#x2018;Ą )), đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1],

where â&#x2C6;&#x2018; đ?&#x2018;¤ = 1, đ?&#x2018;¤ â&#x2C6;&#x2C6; [0, 1]. Particularly, we obtain weighted arithmetic means đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) =

đ?&#x2018;¤ đ?&#x2018;Ą , đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1],

where â&#x2C6;&#x2018; đ?&#x2018;¤ = 1, đ?&#x2018;¤ â&#x2C6;&#x2C6; [0, 1]. An aggregation function

Corollary 1. If an operation đ??ľ â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is increasing and has a neutral element 1 (neutral element 0), then it is a fuzzy conjunction ful illing property đ??ľ(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊽ min(đ?&#x2018;Ľ, đ?&#x2018;Ś) (fuzzy disjunction ful illing property đ??ľ(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ max(đ?&#x2018;Ľ, đ?&#x2018;Ś)).

is idempotent, where đ?&#x2018;? â&#x2C6;&#x2C6; (0, 1) is a parameter.

Triangular norms and conorms are examples of conjunctions and disjunctions having neutral element 1 or 0, respectively.

There are some connections between functions.

For example, we may consider relation of dominance of one function over another.

đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = đ?&#x2018;? max đ?&#x2018;Ą + (1 â&#x2C6;&#x2019; đ?&#x2018;?) min đ?&#x2018;Ą ⊽ ⊽

⊽ ⊽

(4)

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De inition 7 (cf. [36], [34]). Let đ?&#x2018;&#x161;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; . A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] dominates a function đ??ş â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] ( đ??š â&#x2030;Ť đ??ş), if for arbitrary matrix [đ?&#x2018;&#x17D; ] = đ??´ â&#x2C6;&#x2C6; [0, 1] Ă&#x2014; we have đ??š(đ??ş(đ?&#x2018;&#x17D; , â&#x20AC;Ś , đ?&#x2018;&#x17D; ), â&#x20AC;Ś , đ??ş(đ?&#x2018;&#x17D; đ??ş(đ??š(đ?&#x2018;&#x17D; , â&#x20AC;Ś , đ?&#x2018;&#x17D;

,â&#x20AC;Ś,đ?&#x2018;&#x17D;

), â&#x20AC;Ś , đ??š(đ?&#x2018;&#x17D; , â&#x20AC;Ś , đ?&#x2018;&#x17D;

)).

(5)

Lemma 2. Let đ??ş â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be increasing, đ?&#x2018;&#x161; = 2 (cf. (5)). Thus min â&#x2030;Ť đ??ş ([34], p. 16) and đ??ş â&#x2030;Ť max (cf. [11], Theorem 2), so for đ?&#x2018; , ..., đ?&#x2018; , đ?&#x2018;Ą , ..., đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] we have respectively

â&#x2C6;&#x20AC;

â&#x2C6;&#x20AC;

⊽ ⊽

đ??š(1, â&#x20AC;Ś , 1, đ?&#x2018;Ą, 1, â&#x20AC;Ś , 1) = đ?&#x2018;Ą,

(12)

where đ?&#x2018;Ą is at the đ?&#x2018;&#x2DC;-th position, then đ??š ⊽ min.

If a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is increasing in each variable and has a neutral element đ?&#x2018;&#x2019; = 0, i.e. â&#x2C6;&#x20AC;

â&#x2C6;&#x2C6;[ , ]

â&#x2C6;&#x20AC;

⊽ ⊽

đ??š(0, â&#x20AC;Ś , 0, đ?&#x2018;Ą, 0, â&#x20AC;Ś , 0) = đ?&#x2018;Ą,

(13)

where đ?&#x2018;Ą is at the đ?&#x2018;&#x2DC;-th position, then đ??š ⊞ max.

Here are recalled de initions of concepts connected with fuzzy relations.

min(đ??ş(đ?&#x2018; , ..., đ?&#x2018; ), đ??ş(đ?&#x2018;Ą , ..., đ?&#x2018;Ą )) ⊞ đ??ş(min(đ?&#x2018; , đ?&#x2018;Ą ), ..., min(đ?&#x2018; , đ?&#x2018;Ą ))

2016

Lemma 3 (cf. [18]). If a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is increasing in each variable and has a neutral element đ?&#x2018;&#x2019; = 1, i.e. â&#x2C6;&#x2C6;[ , ]

)) ⊞

Nâ&#x2C6;&#x2DC; 2

(6)

and

De inition 8 (cf. [38]). Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. The đ?&#x203A;ź-cut of a fuzzy relation đ?&#x2018;&#x2026; is the relation đ?&#x2018;&#x2026; = {(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x2039; â&#x2C6;ś đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ đ?&#x203A;ź}.

đ??ş(max(đ?&#x2018; , đ?&#x2018;Ą ), ..., max(đ?&#x2018; , đ?&#x2018;Ą )) ⊞

(14)

The strict đ?&#x203A;ź-cut of a fuzzy relation đ?&#x2018;&#x2026; is the relation max(đ??ş(đ?&#x2018; , ..., đ?&#x2018; ), đ??ş(đ?&#x2018;Ą , ..., đ?&#x2018;Ą )).

(7)

Theorem 1. An increasing in each variable function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] dominates minimum (đ??š â&#x2030;Ť min) if and only if đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = min(đ?&#x2018;&#x201C; (đ?&#x2018;Ą ), â&#x20AC;Ś , đ?&#x2018;&#x201C; (đ?&#x2018;Ą )), đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1], (8) where functions đ?&#x2018;&#x201C; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] are increasing for đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A; (cf. [34], Proposition 5.1).

An increasing in each variable function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is dominated by maximum (max â&#x2030;Ť đ??š) if and only if đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = max(đ?&#x2018;&#x201C; (đ?&#x2018;Ą ), â&#x20AC;Ś , đ?&#x2018;&#x201C; (đ?&#x2018;Ą )), đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1], (9) where functions đ?&#x2018;&#x201C; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] are increasing for đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;.

Example 4 (cf. [31]). Here are examples of functions ful illing (8): if đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = đ?&#x2018;Ą, đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then đ??š = min, if for some đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;}, đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = đ?&#x2018;Ą, đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = 1 for đ?&#x2018;&#x2013; â&#x2030; đ?&#x2018;&#x2DC;, then đ??š = đ?&#x2018;&#x192; , if đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = max(1 â&#x2C6;&#x2019; đ?&#x2018;Ł , đ?&#x2018;Ą), đ?&#x2018;Ł â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, max đ?&#x2018;Ł = 1, then đ??š is the weighted minimum ⊽ ⊽

đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = min max(1 â&#x2C6;&#x2019; đ?&#x2018;Ł , đ?&#x2018;Ą ), ⊽ ⊽

(10)

where đ?&#x2018;Ą = (đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) â&#x2C6;&#x2C6; [0, 1] .

Here are examples of functions ful illing (9): if đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = đ?&#x2018;Ą, đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then đ??š = max, if for some đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;}, đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = đ?&#x2018;Ą, đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = 0 for đ?&#x2018;&#x2013; â&#x2030; đ?&#x2018;&#x2DC;, then đ??š = đ?&#x2018;&#x192; , if đ?&#x2018;&#x201C; (đ?&#x2018;Ą) = min(đ?&#x2018;Ł , đ?&#x2018;Ą), đ?&#x2018;Ł â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, max đ?&#x2018;Ł = 1, then đ??š is the weighted maximum ⊽ ⊽

đ??š(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = max min(đ?&#x2018;Ł , đ?&#x2018;Ą ), ⊽ ⊽

(11)

đ?&#x2018;&#x2026; = {(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x2039; â&#x2C6;ś đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) > đ?&#x203A;ź}.

(15)

De inition 9 (cf. [38]). Let đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;). The composition of relations đ?&#x2018;&#x2026; and đ?&#x2018;&#x2020; is called the relation (đ?&#x2018;&#x2026;â&#x2C6;&#x2DC;đ?&#x2018;&#x2020;)(đ?&#x2018;Ľ, đ?&#x2018;§) = sup min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2020;(đ?&#x2018;Ś, đ?&#x2018;§)), (đ?&#x2018;Ľ, đ?&#x2018;§) â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;Ă&#x2014;đ?&#x2018;&#x2039;. â&#x2C6;&#x2C6;

(16) The power of a relation đ?&#x2018;&#x2026; is called the sequence đ?&#x2018;&#x2026; = đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026; = đ?&#x2018;&#x2026; â&#x2C6;&#x2DC; đ?&#x2018;&#x2026; for đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; .

Remark 1. If đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = đ?&#x2018;&#x203A;, đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ }, then a relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) may be presented by a matrix đ?&#x2018;&#x2026; = [đ?&#x2018;&#x; ], where đ?&#x2018;&#x; = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ , đ?&#x2018;Ľ ), đ?&#x2018;&#x2013;, đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;.

3. Mo va on In this section the idea of multicriteria (or similarly multiagent) decision making is recalled. Presented problem is related to considerations provided in this paper. Fuzzy relations in such setting represent the preferences.

Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = đ?&#x2018;&#x161;, đ?&#x2018;&#x161; â&#x2C6;&#x2C6; , đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ } be a set of alternatives. In multicriteria decision making a decision maker has to choose among the alternatives with respect to a set of criteria. Let đ??ž = {đ?&#x2018;&#x2DC; , â&#x20AC;Ś , đ?&#x2018;&#x2DC; } be the set of criteria on the base of which the alternatives are evaluated. đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; be fuzzy relations corresponding to each criterion represented by matrices, where đ?&#x2018;&#x2026; â&#x2C6;ś đ?&#x2018;&#x2039; Ă&#x2014; đ?&#x2018;&#x2039; â&#x2020;&#x2019; [0, 1], đ?&#x2018;&#x2DC; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ , đ?&#x2018;Ľ ) = đ?&#x2018;&#x; , 1 ⊽ đ?&#x2018;&#x2013;, đ?&#x2018;&#x2014; ⊽ đ?&#x2018;&#x161;. We assume that for example: đ?&#x2018;&#x; â&#x20AC;&#x201C; an intensity with which đ?&#x2018;Ľ is better than đ?&#x2018;Ľ under đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ??ž, đ?&#x2018;&#x; = 1 â&#x20AC;&#x201C; â&#x20AC;&#x17E;đ?&#x2018;Ľ is absolutely better than đ?&#x2018;Ľ under criterion đ?&#x2018;&#x2DC;â&#x20AC;?, đ?&#x2018;&#x; = 0 â&#x20AC;&#x201C; â&#x20AC;&#x17E;đ?&#x2018;Ľ is absolutely better than đ?&#x2018;Ľ under criterion đ?&#x2018;&#x2DC;â&#x20AC;?, đ?&#x2018;&#x; = 0.5 â&#x20AC;&#x201C; â&#x20AC;&#x17E;đ?&#x2018;Ľ is equally good as đ?&#x2018;Ľ under criterion đ?&#x2018;&#x2DC;â&#x20AC;?, so it is natural that đ?&#x2018;&#x; = 0.5.

where đ?&#x2018;Ą = (đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) â&#x2C6;&#x2C6; [0, 1] .

27

Journal of Automation, Mobile Robotics & Intelligent Systems

Similarly, if we consider multiagent decision making problems, relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; represent the preferences of each agent and no criteria (certainly, we can combine these two situations, i.e. many criteria and many agents).

Relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is supposed to help the decision maker to make up his/her mind. Some functions đ??š maybe more adequate for aggregation than the others since they may (or not) preserve the required properties of individual fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; . According to some experimental works [40] weighted arithmetic mean and function (4) are the aggregation functions which occur the most often in the process of human decision making. Such properties, if they are ful illed by fuzzy relations, may be a form of measure of consistency of choices or may provide the interpretation of choices. This is why preservation of these properties may be interested and required in aggregation process for multicriteria or multiagent decision making problems.

Application of similar considerations by a numerical example is presented in [32] where the choice or ranking problems of a set of alternatives evaluated by fuzzy preference relations using the aggregation functions are considered. It is shown how properties of the aggregated fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ), depending on the properties of the individual fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; , help to solve the given problem. However, in that paper it is stressed also another problem, namely the sensitivity of the aggregation operators with respect to variations in their arguments. In that paper several weighted aggregation operators, i.e. operators which use the importance of criteria, given as weights, are considered.

In the presented multicriteria or multiagent decision making problems it is sometimes required that the given fuzzy relations representing the preferences are reciprocal, i.e. fuzzy relation đ?&#x2018;&#x2026; in đ?&#x2018;&#x2039; is reciprocal if đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) + đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) = 1 for đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. However, if đ?&#x2018;&#x2026; is not reciprocal, there are methods to transform it to the reciprocal one [3].

4. Graded Proper es of Fuzzy Rela ons Now, dependencies related to đ?&#x203A;ź-properties in the context of aggregation process, between relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; on a set đ?&#x2018;&#x2039; and the aggregated fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) will be investigated. Moreover, some previous results will be recalled.

De inition 10 ([10], p. 75, [18]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. A relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is: - đ?&#x203A;ź-re lexive, if â&#x2C6;&#x20AC; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ) ⊞ đ?&#x203A;ź, â&#x2C6;&#x2C6;

- đ?&#x203A;ź-irre lexive, if â&#x2C6;&#x20AC; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ) ⊽ 1 â&#x2C6;&#x2019; đ?&#x203A;ź, â&#x2C6;&#x2C6;

- totally đ?&#x203A;ź-connected, if â&#x2C6;&#x20AC;

, â&#x2C6;&#x2C6;

đ?&#x203A;ź, - đ?&#x203A;ź-connected, if

, ,

â&#x2C6;&#x20AC;

- đ?&#x203A;ź-asymmetric, if â&#x2C6;&#x20AC;

, â&#x2C6;&#x2C6;

28

â&#x2C6;&#x2C6;

max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊞

max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊞ đ?&#x203A;ź,

min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊽ 1â&#x2C6;&#x2019;đ?&#x203A;ź,

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- đ?&#x203A;ź-antisymmetric, if

, ,

â&#x2C6;&#x20AC;

â&#x2C6;&#x2C6;

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min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊽

1 â&#x2C6;&#x2019; đ?&#x203A;ź, - đ?&#x203A;ź-symmetric, if â&#x2C6;&#x20AC;

, â&#x2C6;&#x2C6;

đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź â&#x2021;&#x2019; đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) ⊞

đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), - đ?&#x203A;ź-transitive, if for all đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;§)) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;§)).

â&#x2021;&#x2019; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;§) ⊞

Let us notice that conditions for đ?&#x203A;ź-symmetry and đ?&#x203A;ź-transitivity may be written in a more convenient way.

Corollary 2. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. A relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is đ?&#x203A;ź-symmetric if and only if â&#x2C6;&#x20AC; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź â&#x2021;&#x2019; đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś).

, â&#x2C6;&#x2C6;

(17)

Corollary 3 (cf. [13], Theorem 10). Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. Relation đ?&#x2018;&#x2026; is đ?&#x203A;ź-transitive if and only if đ?&#x2018;&#x2026; ⊞1â&#x2C6;&#x2019;đ?&#x203A;ź â&#x2021;&#x2019;đ?&#x2018;&#x2026; ⊞đ?&#x2018;&#x2026; .

(18)

Corollary 4. Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, 1]. If relation đ?&#x2018;&#x2026; is đ?&#x203A;˝-đ?&#x2018;&#x192;, then it is đ?&#x203A;ź-đ?&#x2018;&#x192; for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, đ?&#x203A;˝], where đ?&#x2018;&#x192;: re lexivity, irre lexivity, symmetry, asymmetry, antisymmetry, connectedness, total connectedness, transitivity.

Proof. Let đ?&#x203A;ź ⊽ đ?&#x203A;˝. We use the fact, which is easy to see for each property đ?&#x203A;ź-đ?&#x2018;&#x192;, where đ?&#x2018;&#x192;: re lexivity, irre lexivity, symmetry, asymmetry, antisymmetry, connectedness, total connectedness, transitivity, that if đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is đ?&#x203A;˝-đ?&#x2018;&#x192;, then it is đ?&#x203A;ź-đ?&#x2018;&#x192;.

If we have a reciprocal fuzzy relation đ?&#x2018;&#x2026; describing preferences, then the properties in De inition 10 may provide some practical information according to the preferences over the given set of alternatives. For example, the 0.5-asymmetry of a reciprocal fuzzy relation guarantees that at least one of the alternatives đ?&#x2018;Ľ or đ?&#x2018;Ľ is preferred to the other one with the fuzzy value lower than or equal to 0.5 (or these alternatives are indifferent), which means that if đ?&#x2018;Ľ is preferred to đ?&#x2018;Ľ , then it is not true that đ?&#x2018;Ľ is preferred to đ?&#x2018;Ľ . This interpretation of 0.5-asymmetry for a reciprocal fuzzy relation is analogous to the one of asymmetry for crisp relations (i.e., if element đ?&#x2018;Ľ is in relation with đ?&#x2018;Ľ , then it is not true that đ?&#x2018;Ľ is in relation with đ?&#x2018;Ľ [33]). Similarly, we can interpret the other properties. For đ?&#x203A;ź-connectedness, the greater the value of đ?&#x203A;ź (namely, the closer it is to the value 1), the choice of the alternative is more precise (con ident/sure). For preference relations, if it comes to đ?&#x203A;ź-re lexivity, practically only 0.5-re lexivity occurs and with the given de inition, if đ?&#x2018;&#x2026; is 0.5-re lexive then it is automatically 0.5irre lexive. Moreover, reciprocal preference relation is always totally 0.5-connected and 0.5-asymmetric.

Since the ixed value of 0.5 on the diagonal may understate or in late the value of đ?&#x203A;ź for these properties, it makes sense to distinguish total connectedness and connectedness and similarly, asymmetry and antisymmetry. đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) has the highest value of đ?&#x203A;źsymmetry for preference relation đ?&#x2018;&#x2026; in the case when

Journal of Automation, Mobile Robotics & Intelligent Systems

all elements in the set of alternatives đ?&#x2018;&#x2039; are indifferent.

In fact, in such situation relation đ?&#x2018;&#x2026; is symmetric (it is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]). Moreover, we have the following statement: if đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is reciprocal, then đ?&#x2018;&#x2026; is totally đ?&#x203A;ź-connected (đ?&#x203A;ź-connected) if and only if đ?&#x2018;&#x2026; is đ?&#x203A;ź-asymmetric (đ?&#x203A;ź-antisymmetric). That is not the case for đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) which is not reciprocal (cf. Example 5).

In the sequel we will present the results in general setting of fuzzy relations, sometimes with the comments on reciprocal preference relations.

For practical reasons it is useful to ind the greatest value of đ?&#x203A;ź for which đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is đ?&#x203A;źâ&#x20AC;&#x201C;đ?&#x2018;&#x192; for a given property đ?&#x2018;&#x192;: re lexivity, irre lexivity, symmetry, asymmetry, antisymmetry, connectedness, total connectedness, transitivity. Applying de initions of the given properties and Corollary 4 one can ind this value in the following way.

Corollary 5. Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x203A;ź = 1 â&#x2C6;&#x2019; sup min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)), , â&#x2C6;&#x2C6;

đ?&#x203A;˝ = 1 â&#x2C6;&#x2019; sup min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)), đ?&#x203A;ž = inf max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)), , â&#x2C6;&#x2C6;

đ?&#x203A;ż = inf max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)), đ?&#x153;&#x2021; = inf đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ), â&#x2C6;&#x2C6;

đ?&#x153;&#x2C6; = inf (1 â&#x2C6;&#x2019; đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ)) = 1 â&#x2C6;&#x2019; sup đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ). â&#x2C6;&#x2C6;

â&#x2C6;&#x2C6;

Thus a relation đ?&#x2018;&#x2026; is: đ?&#x203A;źâ&#x20AC;&#x201C;asymmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, đ?&#x203A;ź ], đ?&#x203A;˝â&#x20AC;&#x201C;antisymmetric for đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, đ?&#x203A;˝ ], totally đ?&#x203A;žâ&#x20AC;&#x201C;connected for đ?&#x203A;ž â&#x2C6;&#x2C6; [0, đ?&#x203A;ž ], đ?&#x203A;żâ&#x20AC;&#x201C;connected for đ?&#x203A;ż â&#x2C6;&#x2C6; [0, đ?&#x203A;ż ], đ?&#x153;&#x2021;â&#x20AC;&#x201C;re lexive for đ?&#x153;&#x2021; â&#x2C6;&#x2C6; [0, đ?&#x153;&#x2021; ] and đ?&#x153;&#x2C6;â&#x20AC;&#x201C;irre lexive for đ?&#x153;&#x2C6; â&#x2C6;&#x2C6; [0, đ?&#x153;&#x2C6; ].

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đ?&#x203A;ź-antisymmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.8]. đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5) and đ?&#x203A;ź-transitive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] (it follows from the fact that đ?&#x2018;&#x2026; = đ?&#x2018;&#x2026;).

Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = 3. We consider đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) which is reciprocal, where đ?&#x2018;&#x2026;=

0.5 0.2 0.7

0.8 0.5 0.6

0.3 0.4 0.5

,

đ?&#x2018;&#x2026; =

0.5 0.4 0.5

0.5 0.5 0.7

0.4 0.4 0.5

.

The relation đ?&#x2018;&#x2026; is totally đ?&#x203A;źâ&#x20AC;&#x201C;connected and đ?&#x203A;ź-re lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5] and đ?&#x203A;źâ&#x20AC;&#x201C;connected for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.6]. It is đ?&#x203A;ź-asymmetric and đ?&#x203A;ź-irre lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5] and đ?&#x203A;ź-antisymmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.6]. đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.2) and đ?&#x203A;ź-transitive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.3).

Remark 2. The presented đ?&#x203A;ź-properties (graded properties) for đ?&#x203A;ź = 1 become the basic properties of fuzzy relations [39]. Graded properties are â&#x20AC;&#x17E;fuzzy versionsâ&#x20AC;? of properties introduced by Zadeh. It means that, if a fuzzy relation, e.g. is not re lexive, it may be re lexive to some grade đ?&#x203A;ź, where đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Remark 3. Taking into account đ?&#x203A;ź = 0, each fuzzy relation is 0-re lexive, 0-irre lexive, 0-asymmetric, 0antisymmetric, 0-connected and totally 0-connected.

However, it is not true for graded symmetry and transitivity. If in Corollary 6, đ?&#x203A;ź = 0 (or similarly đ?&#x203A;˝ = 0), then đ?&#x2018;&#x2026; is not đ?&#x203A;ź-symmetric for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] (đ?&#x2018;&#x2026; is not đ?&#x203A;˝-transitive for any đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, 1]).

Example 6. Let card đ?&#x2018;&#x2039; = 3, relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) be presented by matrices: đ?&#x2018;&#x2026;=

0 0 0 0 1 0

1 0 0

,

đ?&#x2018;&#x2020;=

0 0 0 0 1 0

0 0 0

.

For symmetry and transitivity we have adequate half-closed intervals. Moreover, for checking the đ?&#x203A;źtransitivity of a fuzzy relation đ?&#x2018;&#x2026;, the composition of đ?&#x2018;&#x2026; by itself will be useful.

The relation đ?&#x2018;&#x2026; is not 0-transitive because min(đ?&#x2018;&#x; , đ?&#x2018;&#x; ) = 1 but 0 = đ?&#x2018;&#x; < min(đ?&#x2018;&#x; , đ?&#x2018;&#x; ) = 1.

The relation đ?&#x2018;&#x2020; is not 0-symmetric because đ?&#x2018; = 1 and 0 = đ?&#x2018; < đ?&#x2018; = 1.

Corollary 6. Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;). Thus đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] 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; or đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, đ?&#x203A;ź ) if there exist đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; such that đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) â&#x2030; đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ), where

Notions of đ?&#x203A;ź-properties have their connection with cuts and strict cuts of a fuzzy relation.

đ?&#x203A;ź =1â&#x2C6;&#x2019;

sup ( , )

đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś).

( , ), , â&#x2C6;&#x2C6;

đ?&#x2018;&#x2026; is đ?&#x203A;˝-transitive for đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, 1] if đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊽ đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) for all đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; or đ?&#x2018;&#x2026; is đ?&#x203A;˝-transitive for đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, đ?&#x203A;˝ ) if there exist đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; such that đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) < đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), where đ?&#x203A;˝ =1â&#x2C6;&#x2019;

sup ( , )

đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś).

( , ), , â&#x2C6;&#x2C6;

Example 5. Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = 2, đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), where đ?&#x2018;&#x2026;=

Theorem 2 (cf. [17]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;).

A fuzzy relation đ?&#x2018;&#x2026; is totally đ?&#x203A;ź-connected (đ?&#x203A;ź-connected, đ?&#x203A;ź-re lexive) if and only if relation đ?&#x2018;&#x2026; is totally connected (connected, re lexive). A fuzzy relation đ?&#x2018;&#x2026; is đ?&#x203A;źasymmetric (đ?&#x203A;ź-antisymmetric, đ?&#x203A;ź-irre lexive) if and only if relation đ?&#x2018;&#x2026; is asymmetric (antisymmetric, irre lexive). If a fuzzy relation đ?&#x2018;&#x2026; is đ?&#x203A;ź-transitive, then relation đ?&#x2018;&#x2026; is transitive. If a fuzzy relation đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric, then relation đ?&#x2018;&#x2026; is symmetric.

0.7 0.2 0.5 0.4

.

The relation đ?&#x2018;&#x2026; is totally đ?&#x203A;źâ&#x20AC;&#x201C;connected and đ?&#x203A;ź-re lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.4] and đ?&#x203A;źâ&#x20AC;&#x201C;connected for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5]. It is đ?&#x203A;ź-asymmetric and đ?&#x203A;ź-irre lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.3] and

Similar characterizations for other properties for fuzzy relations one may ind in [12] (Theorem 1). The conditions for đ?&#x203A;ź-symmetry and đ?&#x203A;ź-transitivity are only the suf icient ones.

Example 7 (cf. [17]). Let card đ?&#x2018;&#x2039; = 2, đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x2018;&#x2026;=

0.3 0.5 0.7 0.4

.

29

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The cuts đ?&#x2018;&#x2026; are symmetric for đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, 0.5] â&#x2C6;Ş (0.7, 1], so the cuts đ?&#x2018;&#x2026; have this property for đ?&#x203A;ź ⊞ 0.5 and đ?&#x203A;ź < 0.3. Relation đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.3), as a result for đ?&#x203A;ź = 0.5, the cut đ?&#x2018;&#x2026; . is symmetric, while đ?&#x2018;&#x2026; is not 0.5-symmetric.

Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), card đ?&#x2018;&#x2039; = 3,

đ?&#x2018;&#x2026;=

0.7 0 0.8 0.9 0.6 0.9

0 0 0.8

, đ?&#x2018;&#x2020;=đ?&#x2018;&#x2026; =

0.7 0.8 0.8

0 0.9 0.9

0 0 0.8

The cuts đ?&#x2018;&#x2026; are transitive for đ?&#x203A;˝ â&#x2C6;&#x2C6; [0, 0.6] â&#x2C6;Ş (0.8, 1], so the cuts đ?&#x2018;&#x2026; have this property for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.2) â&#x2C6;Ş [0.4, 1]. Since 0.8 = đ?&#x2018; ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź for đ?&#x203A;ź â&#x2C6;&#x2C6; [0.4, 1] and đ?&#x2018; = 0.8 > 0.6 = đ?&#x2018;&#x; , relation đ?&#x2018;&#x2026; is not đ?&#x203A;ź-transitive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0.4, 1] (it is đ?&#x203A;ź-transitive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.2), see Corollary 6).

Other results describing graded properties one can ind in [10] (p. 78â&#x20AC;&#x201C;79).

5. Aggrega on of Fuzzy Rela ons In this section we will present đ?&#x203A;ź-properties of fuzzy relations and diverse approaches of aggregating such relations. There will be presented the following type of theorems for aggregated fuzzy relation đ?&#x2018;&#x2026; : - aggregation of đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; all having the same grade of a given đ?&#x203A;ź-property to obtain đ?&#x2018;&#x2026; with the same grade đ?&#x203A;ź, - aggregation of đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; with possible diverse grades đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź of a given graded property to obtain đ?&#x2018;&#x2026; with the suitable grade đ?&#x203A;ź, - starting from đ?&#x2018;&#x2026; having some grade đ?&#x203A;ź and checking whether đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; all have the same grade đ?&#x203A;ź of a given graded property.

Theorem 3 ([16]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;ź-re lexivity of fuzzy relations, if and only if đ??š|[

, ]

⊞ đ?&#x203A;ź.

Theorem 4 ([16]). đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;źre lexivity of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] if and only if đ??š ⊞ min.

By Lemma 1 we know that every increasing and idempotent function preserves đ?&#x203A;ź-re lexivity of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. In particular, we get Corollary 7. Quasi-linear means preserve đ?&#x203A;ź-re lexivity of fuzzy relations for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Theorem 5. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be increasing in each variable. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -re lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-re lexive for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

30

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Proof. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be increasing in each variable, đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) be đ?&#x203A;ź -re lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. Then đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ľ) = đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)) ⊞ đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ), so relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-re lexive for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ). .

Each aggregation function is increasing, so we get Corollary 8. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be an aggregation function. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -re lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-re lexive for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Theorem 6. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] and đ??š ⊽ min. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-re lexive, then all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are đ?&#x203A;ź-re lexive.

Proof. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š ⊽ min, đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) be đ?&#x203A;ź-re lexive, đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;}. Then đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ) ⊞ min đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ) ⊞ ⊽⊽

đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)) ⊞ đ?&#x203A;ź.

As a result relation đ?&#x2018;&#x2026; is đ?&#x203A;ź-re lexive.

In virtue of Lemma 3 we get Corollary 9. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š be a t-seminorm or a t-norm. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-re lexive, then all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are also đ?&#x203A;źre lexive.

The next example shows that the condition presented in Theorem 6 is only suf icient.

Example 8. Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = 2. We consider fuzzy relations with matrices:

5.1. ReďŹ&#x201A;exivity Graded re lexivity was considered by many authors, e.g. [8], [10].

Nâ&#x2C6;&#x2DC; 2

đ?&#x2018;&#x2026;=

0 1 1 1

,

đ?&#x2018;&#x2026;+đ?&#x2018;&#x2020; = 2

1 1

1 0

,

1 1

,

0.5 1 1 0.5

.

đ?&#x2018;&#x160; = max(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x160; =

1 1

đ?&#x2018;&#x2020;=

Relation đ?&#x2018;&#x160; is đ?&#x203A;ź-re lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x160; for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5], but relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1].

5.2. IrreďŹ&#x201A;exivity For irre lexivity, generally we get dual results to relexivity.

Theorem 7 ([16]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;ź-irre lexivity of fuzzy relations if and only if đ??š|[

,

]

⊽ 1 â&#x2C6;&#x2019; đ?&#x203A;ź.

Theorem 8 ([16]). A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;ź-irre lexivity of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] if and only if đ??š ⊽ max.

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Corollary 10. Quasiâ&#x20AC;&#x201C;linear means preserve đ?&#x203A;źirre lexivity of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

De inition 11 (cf. [7]). A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is super additive, if for all đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A; and all đ?&#x2018;Ľ , đ?&#x2018;Ś , đ?&#x2018;Ľ + đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1] đ??š(đ?&#x2018;Ľ + đ?&#x2018;Ś , â&#x20AC;Ś , đ?&#x2018;Ľ + đ?&#x2018;Ś ) ⊞ đ??š(đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ ) + đ??š(đ?&#x2018;Ś , â&#x20AC;Ś , đ?&#x2018;Ś ).

(19) Example 9. Weighted arithmetic means and minimum are super additive functions.

Theorem 9. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be a super additive aggregation function. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -irre lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-irre lexive for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Proof. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be a super additive aggregation function, đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) be đ?&#x203A;ź -irre lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. Then đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ) + đ?&#x203A;ź ⊽ 1, so

⊽ đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)+đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)+đ?&#x203A;ź ) ⊽ đ??š(1, â&#x20AC;Ś , 1) = 1.

As a result đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)) ⊽ 1 â&#x2C6;&#x2019; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ), so đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-irre lexive for đ?&#x203A;ź đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

=

Corollary 11. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -irre lexive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = â&#x2C6;&#x2018; đ?&#x2018;¤ đ?&#x2018;&#x2026; is đ?&#x203A;ź-irre lexive, where â&#x2C6;&#x2018; đ?&#x2018;¤ = 1, đ?&#x2018;¤ â&#x2C6;&#x2C6; [0, 1] and đ?&#x203A;ź = â&#x2C6;&#x2018; đ?&#x2018;¤ đ?&#x203A;ź .

Analogously to re lexivity we obtain the following result.

Theorem 10. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] and đ??š ⊞ max. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-irre lexive, then all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are also đ?&#x203A;ź-irre lexive.

Corollary 12. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š be a t-conorm or a t-semiconorm. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-irre lexive, then all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are also đ?&#x203A;źirre lexive.

The next example shows that the condition given in Theorem 10 is only suf icient.

Example 10. Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = 2. We consider fuzzy relations with matrices: 0 1

1 1

,

1 1 1 0

đ?&#x2018;&#x2020;=

đ?&#x2018;&#x160; = min(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) =

0 1

1 0

, ,

đ?&#x2018;&#x2026;+đ?&#x2018;&#x2020; 0.5 1 = .

1 0.5 2 Relation đ?&#x2018;&#x160; is đ?&#x203A;ź-irre lexive for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x160; for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5], but relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1]. đ?&#x2018;&#x160; =

Theorem 11 ([16]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; ⊞ 2.

A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves total đ?&#x203A;źconnectedness (đ?&#x203A;ź-connectedness) of fuzzy relations, if and only if for any đ?&#x2018; , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] ( â&#x2C6;&#x20AC;

⊽ ⊽

max(đ?&#x2018; , đ?&#x2018;Ą ) ⊞ đ?&#x203A;ź) â&#x2021;&#x2019; max(đ??š(đ?&#x2018; ), đ??š(đ?&#x2018;Ą)) ⊞ đ?&#x203A;ź.

Theorem 12 ([16]). Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; ⊞ 2. A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves total đ?&#x203A;ź-connectedness (đ?&#x203A;ź-connectedness) of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], if and only if â&#x2C6;&#x20AC;

max(đ??š(đ?&#x2018; ), đ??š(đ?&#x2018;Ą)) ⊞ min max(đ?&#x2018; , đ?&#x2018;Ą ). ⊽ ⊽

Corollary 13. Maximum and the weighted maximum preserve total đ?&#x203A;ź-connectedness (đ?&#x203A;ź-connectedness) of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Theorem 13. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be increasing in each variable and max â&#x2030;Ť đ??š. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are totally đ?&#x203A;ź connected (đ?&#x203A;ź -connected) for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is totally đ?&#x203A;ź-connected (đ?&#x203A;źconnected) for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Proof. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be increasing in each variable, max â&#x2030;Ť đ??š and đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) be đ?&#x203A;ź -connected for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x2018;Ľ â&#x2030; đ?&#x2018;Ś.

Then by Lemma 2 and by the fact that max â&#x2030;Ť đ??š we obtain max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) = max(đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ))) ⊞ đ??š(max(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), â&#x20AC;Ś ,

In virtue of Lemma 3 we get

đ?&#x2018;&#x2026;=

2016

5.3. Connectedness Here graded connectedness and total connectedness will be examined. The total 0.5-connectedness was regarded in [32] (p. 619). In that paper this property is called weak comparability. It was shown there that maximum preserves the total 0.5-connectedness ([32], Table 1).

, â&#x2C6;&#x2C6;[ , ]

đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ľ)) + đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź )

Nâ&#x2C6;&#x2DC; 2

max(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ))) ⊞ đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) = đ?&#x203A;ź.

It means that a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-connected for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ). Proof for total đ?&#x203A;źconnectedness is analogous.

We can also compute the value of đ?&#x203A;ź for which a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-connected (totally đ?&#x203A;ź-connected) for concrete functions đ??š in another way than it is presented in Theorem 13. It is shown in the following example.

Example 11. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź â&#x20AC;&#x201C;connected (totally đ?&#x203A;ź â&#x20AC;&#x201C;connected) for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is đ?&#x203A;ź-connected (totally đ?&#x203A;ź-connected), where đ?&#x2018;&#x2026;=

1 đ?&#x2018;&#x203A;

đ?&#x2018;&#x2026;,

đ?&#x203A;ź=

1 max đ?&#x203A;ź . đ?&#x2018;&#x203A; ⊽⊽ 31

Journal of Automation, Mobile Robotics & Intelligent Systems

Note that the arithmetic mean is not dominated by maximum.

Nâ&#x2C6;&#x2DC; 2

2016

Corollary 15. The minimum and the weighted minimum (11) preserve đ?&#x203A;ź-asymmetry (đ?&#x203A;ź-antisymmetry) of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Theorem 14. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] and đ??š ⊽ min. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is totally đ?&#x203A;ź-connected (đ?&#x203A;źconnected), then all fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are totally đ?&#x203A;ź-connected (đ?&#x203A;ź-connected).

Dually to graded connectedness properties, by Lemma 2, similarly to the proof of Theorem 9 we may prove

Proof. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š ⊽ min and a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) be đ?&#x203A;ź-connected, đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x2018;Ľ â&#x2030; đ?&#x2018;Ś, đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;}. As a result we have max(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊞ đ?&#x203A;ź, so đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ đ?&#x203A;ź or đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ)) = đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) ⊞ đ?&#x203A;ź. Let us consider the irst case. Since đ??š ⊽ min, we get

Theorem 17. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be a super additive increasing in each variable function and đ??š â&#x2030;Ť min. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are totally đ?&#x203A;ź -asymmetric (đ?&#x203A;ź -antisymmetric) for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;źasymmetric (đ?&#x203A;ź-antisymmetric) for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ min đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ ⊽⊽

đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) ⊞ đ?&#x203A;ź.

It means that max(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ)) ⊞ đ?&#x203A;ź. Similarly we may consider the second case, i.e. đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) ⊞ đ?&#x203A;ź.

Thus relations đ?&#x2018;&#x2026; are đ?&#x203A;ź-connected for đ?&#x2018;&#x2013; â&#x2C6;&#x2C6; {1, â&#x20AC;Ś , đ?&#x2018;&#x203A;}.

The proof for total đ?&#x203A;ź-connectedness is analogous.

By Lemma 3 we get Corollary 14. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š be a t-norm or a tseminorm. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is totally đ?&#x203A;ź-connected (đ?&#x203A;ź-connected), then all fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are totally đ?&#x203A;ź-connected (đ?&#x203A;ź-connected).

Example 12. The condition given in Theorem 14 is only suf icient. For total đ?&#x203A;ź-connectedness it is enough to consider relations from Example 8. Relation đ?&#x2018;&#x160; is totally đ?&#x203A;ź-connected for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x160; for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5], but relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1]. For đ?&#x203A;ź-connectedness let us take đ?&#x2018;&#x2026; = [đ?&#x2018;&#x; ], with đ?&#x2018;&#x; = 1 and đ?&#x2018;&#x2020; = [đ?&#x2018; ], with đ?&#x2018; = 0 for đ?&#x2018;&#x2013;, đ?&#x2018;&#x2014; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;. Then relation đ?&#x2018;&#x160; = max(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026;, đ?&#x2018;&#x160; are đ?&#x203A;ź-connected for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], while đ?&#x2018;&#x2020; is not đ?&#x203A;ź-connected for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1].

5.4. Asymmetry Now graded asymmetry and antisymmetry will be discussed. The obtained results are dual to the ones obtained for total đ?&#x203A;ź-connectedness and đ?&#x203A;źconnectedness, respectively. It is worth mentioning that in [32] (p. 619) the 0.5-asymmetry was considered. However, in that paper this property is called weak asymmetry. It was shown there that minimum preserves the 0.5-asymmetry ([32], Table 1).

Theorem 15 ([16]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], card đ?&#x2018;&#x2039; ⊞ 2. A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;ź-asymmetry (đ?&#x203A;źantisymmetry) of fuzzy relations, if and only if for any đ?&#x2018; , đ?&#x2018;Ą â&#x2C6;&#x2C6; [0, 1] ( â&#x2C6;&#x20AC;

⊽ ⊽

min(đ?&#x2018; , đ?&#x2018;Ą ) ⊽ 1â&#x2C6;&#x2019;đ?&#x203A;ź) â&#x2021;&#x2019; min(đ??š(đ?&#x2018; ), đ??š(đ?&#x2018;Ą)) ⊽ 1â&#x2C6;&#x2019;đ?&#x203A;ź.

Theorem 16 ([16]). Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; ⊞ 2. A function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] preserves đ?&#x203A;ź-asymmetry (đ?&#x203A;źantisymmetry) of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], if and only if â&#x2C6;&#x20AC;

, â&#x2C6;&#x2C6;[ , ]

32

VOLUME 10,

min(đ??š(đ?&#x2018; ), đ??š(đ?&#x2018;Ą)) ⊽ max min(đ?&#x2018; , đ?&#x2018;Ą ). ⊽ ⊽

In Theorem 1 we have the characterization of increasing functions which dominate minimum. Appropriate examples are presented in Example 4 and among them minimum is a super additive function (because, by Lemma 2, it dominates any increasing function which coincides with the inequality (19)).

We can also compute the value of đ?&#x203A;ź for which a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-asymmetric (đ?&#x203A;ź-antisymmetric) for concrete functions đ??š in another way than it is presented in Theorem 17. It is shown in the following example.

Example 13. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź â&#x20AC;&#x201C;asymmetric (đ?&#x203A;ź â&#x20AC;&#x201C;antisymmetric) for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is đ?&#x203A;ź-asymmetric (đ?&#x203A;źantisymmetric), where

đ?&#x2018;&#x2026;=

1 đ?&#x2018;&#x203A;

đ?&#x2018;&#x2026;,

đ?&#x203A;ź=

1 min đ?&#x203A;ź . đ?&#x2018;&#x203A; ⊽⊽

Note that the arithmetic mean does not dominate minimum.

Dually to Theorem 14 we may prove Theorem 18. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] and đ??š ⊞ max. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-asymmetric (đ?&#x203A;źantisymmetric), then also all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are đ?&#x203A;źasymmetric (đ?&#x203A;ź-antisymmetric).

By Lemma 3 we obtain Corollary 16. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š be a t-conorm or a t-semiconorm. If a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-asymmetric (đ?&#x203A;ź-antisymmetric), then all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are đ?&#x203A;ź-asymmetric (đ?&#x203A;ź-antisymmetric).

Example 14. The condition given in Theorem 18 is only suf icient. For đ?&#x203A;ź-asymmetry it is enough to consider relations from Example 10. The relation đ?&#x2018;&#x160; is đ?&#x203A;źasymmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ?&#x2018;&#x160; for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 0.5], but relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1].

For đ?&#x203A;ź-antisymmetry let us take đ?&#x2018;&#x2026; = [đ?&#x2018;&#x; ], with đ?&#x2018;&#x; = 1 and đ?&#x2018;&#x2020; = [đ?&#x2018; ], with đ?&#x2018; = 0 for đ?&#x2018;&#x2013;, đ?&#x2018;&#x2014; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;. Then relation đ?&#x2018;&#x160; = min(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2020; and đ?&#x2018;&#x2020;, đ?&#x2018;&#x160; are đ?&#x203A;ź-antisymmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], while đ?&#x2018;&#x2026; is not đ?&#x203A;ź-antisymmetric for any đ?&#x203A;ź â&#x2C6;&#x2C6; (0, 1].

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 10,

5.5. Symmetry Now graded symmetry will be discussed.

Theorem 19 ([18]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] ful ils đ??š|[

, ] ⧾[

, ]

< 1 â&#x2C6;&#x2019; đ?&#x203A;ź,

then it preserves đ?&#x203A;ź-symmetry of relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;).

â&#x2C6;&#x2C6;

Theorem 20 ([18]). If a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] ful ils condition đ??š ⊽ min, then it preserves đ?&#x203A;ź-symmetry of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Corollary 17. Any triangular norm or a t-seminorm preserves đ?&#x203A;ź-symmetry of fuzzy relations for arbitrary đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Example 15. Since any projection đ?&#x2018;&#x192; , đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; , preserves the đ?&#x203A;ź-symmetry for each đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] but it is not true that đ?&#x2018;&#x192; ⊽ min, then Theorem 20 gives only a suf icient condition for preservation of the đ?&#x203A;ź-symmetry for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Theorem 21. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š ⊽ min. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -symmetric for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-symmetric for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Proof. Let relations đ?&#x2018;&#x2026; be đ?&#x203A;ź -symmetric for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A; and đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. If đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź and đ??š ⊽ min, then for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;

1 â&#x2C6;&#x2019; đ?&#x203A;ź = 1 â&#x2C6;&#x2019; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Remark 4. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If we would assume that đ??š is idempotent, increasing and injective with respect to all arguments, then if a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-symmetric, then also all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are đ?&#x203A;źsymmetric. However, idempotency and injectivness with repsect to all arguments makes a contraposition (if đ??š(đ?&#x2018;Ľ, đ?&#x2018;Ľ) = đ?&#x2018;Ľ, then for the remaining arguments there are no values). Moreover, injectivness with repsect to all arguments, as a property itself, is not so easy to be fulilled (arithmetic mean, minimum, maximum, geometric mean, uninorms â&#x20AC;&#x201C; including t-norms and t-conorms, are not injective with respect to all arguments). Assuming injectivness with respect to a ixed variable, i.e. đ??š(đ?&#x2018;Ľ , đ?&#x2018;Ś) = đ??š(đ?&#x2018;Ľ , đ?&#x2018;Ś) â&#x2021;&#x2019; đ?&#x2018;Ľ = đ?&#x2018;Ľ for all đ?&#x2018;Ś â&#x2C6;&#x2C6; [0, 1], in general (đ??š should be without a zero element) is not contradictory with idempotency of đ??š, but this assumption is not enough to obtain the required result which is shown by the counter-example above (relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;, đ?&#x2018;&#x160; in Example 16, where the arithmetic mean is idempotent and injective with a ixed variable).

5.6. Transi vity In [31] a special case of the graded transitivity is considered. Namely, this is the 0.5-transitivity (there this property is called moderate transitivity). However, the problem of preservation of this property during aggregation process is not discussed. The property of the 0.5-transitivity is also known as one of the types of a stochastic transitivity (e.g. [19]).

đ??š|[

Moreover, for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A; 1 â&#x2C6;&#x2019; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) ⊞ 1 â&#x2C6;&#x2019; min(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź .

As a result đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;. It means that đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) = đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ) for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;, so đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś) = đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) and đ?&#x2018;&#x2026; is đ?&#x203A;ź-symmetric for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

If it comes to the â&#x20AC;&#x17E;converse problemâ&#x20AC;? for đ?&#x203A;źsymmetry we have several counter-examples. Observe that diverse functions were applied for aggregation of fuzzy relations, namely greater (smaller) than or equal to minimum (maximum).

Example 16. Let đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039; = 2. We consider fuzzy relations with matrices: 0 0

1 0

,

đ?&#x2018;&#x2020;=

đ?&#x2018;&#x160; = min(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2026; â&#x2039;&#x2026; đ?&#x2018;&#x2020; =

0 0 1 0 0 0

0 1

đ?&#x2018;&#x160; = max(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2026; + đ?&#x2018;&#x2020; â&#x2C6;&#x2019; đ?&#x2018;&#x2026; â&#x2039;&#x2026; đ?&#x2018;&#x2020; = đ?&#x2018;&#x2026;+đ?&#x2018;&#x2020; = đ?&#x2018;&#x160; = 2

0 0.5 0.5 0

, ] ⧾[

, ]

< 1 â&#x2C6;&#x2019; đ?&#x203A;ź,

and đ??š â&#x2030;Ť min, then it preserves đ?&#x203A;ź-transitivity of fuzzy relations.

Example 17 ([18]). Let đ?&#x2018;&#x17D; â&#x2C6;&#x2C6; (0, 1] and đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] be of the form đ??š(đ?&#x2018; , đ?&#x2018;Ą) =

0, min(đ?&#x2018; , đ?&#x2018;Ą),

(đ?&#x2018; , đ?&#x2018;Ą) â&#x2C6;&#x2C6; [0, đ?&#x2018;&#x17D;) Ă&#x2014; [0, đ?&#x2018;&#x17D;) otherwise

đ??š is a đ?&#x2018;Ąâ&#x20AC;&#x201C;norm and đ??š|[ , ] ⧾[ , ] < 1 â&#x2C6;&#x2019; đ?&#x203A;ź but it does not dominate minimum. However, the function đ??š preserves the đ?&#x203A;ź-transitivity for each đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1) and đ?&#x203A;ź ⊽ 1 â&#x2C6;&#x2019; đ?&#x2018;&#x17D;. As a result conditions for preservation of the đ?&#x203A;źtransitivity stated in Theorem 22 are only suf icient.

Theorem 23 ([18]). If a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] is increasing in each variable, ful ils đ??š â&#x2030;Ť min and đ??š ⊽ min, then it preserves đ?&#x203A;ź-transitivity of fuzzy relations for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

, 0 0

2016

Theorem 22 ([18]). Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If an increasing function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] ful ils

đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) ⊞ min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) ⊞

đ?&#x2018;&#x2026;=

Nâ&#x2C6;&#x2DC; 2

, 1 0

,

.

Relations đ?&#x2018;&#x160; , đ?&#x2018;&#x160; , đ?&#x2018;&#x160; are đ?&#x203A;ź-symmetric for đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], but relations đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1].

Corollary 18. Minimum and the aggregation function đ??´ (đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) =

1, 0,

(đ?&#x2018;Ą , â&#x20AC;Ś , đ?&#x2018;Ą ) = (1, â&#x20AC;Ś , 1) otherwise

preserve the đ?&#x203A;ź-transitivity of fuzzy relations for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] (because both functions ful il assumptions of Theorem 23).

33

Journal of Automation, Mobile Robotics & Intelligent Systems

Theorem 24. Let đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], đ??š ⊽ min, đ??š â&#x2030;Ť min and a function đ??š be increasing. If relations đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) are đ?&#x203A;ź -transitive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;, then relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-transitive for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

min(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2026; â&#x2039;&#x2026; đ?&#x2018;&#x2020; =

0 0 0 1 0 0

max(đ?&#x2018;&#x2026;, đ?&#x2018;&#x2020;) = đ?&#x2018;&#x2026; + đ?&#x2018;&#x2020; â&#x2C6;&#x2019; đ?&#x2018;&#x2026; â&#x2039;&#x2026; đ?&#x2018;&#x2020; =

min(đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§))) ⊞1â&#x2C6;&#x2019;đ?&#x203A;ź and đ??š ⊽ min, then by the monotonicity of minimum we get min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§)) ⊞ min(min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), min(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§))) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź = 1 â&#x2C6;&#x2019; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;. Moreover, for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A; 1 â&#x2C6;&#x2019; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) ⊞ 1 â&#x2C6;&#x2019; min(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź .

As a result min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§)) ⊞ 1 â&#x2C6;&#x2019; đ?&#x203A;ź for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;. By assumptions it means that 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;§) for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A;.

Since đ??š â&#x2030;Ť min and đ??š is increasing, one obtains

đ?&#x2018;&#x2026;+đ?&#x2018;&#x2020; = 2

0 0 0

1 1 1 1 1 1

0.5 0.5 0.5 0.5 1 0.5 0.5 0.5 0.5

,

1 1 1

,

,

which are đ?&#x203A;ź-transitive for each đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1], while relations đ?&#x2018;&#x2026; and đ?&#x2018;&#x2020; do not have this property for any đ?&#x203A;ź. For example for đ?&#x203A;ź = 1 and relation đ?&#x2018;&#x2026; we have min(đ?&#x2018;&#x; , đ?&#x2018;&#x; ) = 1 ⊞ 0, but 0 = đ?&#x2018;&#x; < min(đ?&#x2018;&#x; , đ?&#x2018;&#x; ) = 1.

Remark 5. Let đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1]. If đ??š is idempotent, increasing and injective, then if a fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) is đ?&#x203A;ź-transitive, then also all relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are đ?&#x203A;źtransitive. However, these assumptions on đ??š are contradictory (cf. Remark 4).

6. Reciprocity Property and Other Concepts Related to Decision Making Problems In this section we present notions, concepts and concerns which occur in decision making algorithms.

6.1. Reciprocity

min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;§)) = min(đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), ..., đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)), đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§), ..., đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§))) ⊽ đ??š(min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§)), ..., min(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;§))) ⊽ đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;§), ..., đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;§)) = đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;§)

Preservation of reciprocity may be useful in aggregation of fuzzy relations. Sometimes this property is required in such situations, so we present adequate assumptions on functions to preserve this property.

which proves the đ?&#x203A;ź-transitivity of a relation đ?&#x2018;&#x2026; for đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

De inition 12 (cf. [4]). Relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is called reciprocal if for any đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; it holds đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś)+đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;Ľ) = 1.

If we look for functions đ??š which ful il both conditions đ??š â&#x2030;Ť min and đ??š ⊽ min we see that đ??š = min which is an aggregation function, ful ils these conditions. Moreover, we have the following property

De inition 13 (cf. [7], p. 31). Let đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1].

A function đ??š is called a dual function to đ??š, if for all đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1]

Corollary 19 ([18]). For a function đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] which has a neutral element đ?&#x2018;&#x2019; = 1 the following holds true: F is increasing in each variable, đ??š â&#x2030;Ť đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; and đ??š ⊽ min if and only if đ??š = min.

It means that the only t-seminorm that ful ils conditions of Corollary 19 is minimum. If it comes to the â&#x20AC;&#x17E;converse problemâ&#x20AC;? for đ?&#x203A;ź-transitivity we obtained several counter-examples. In the following example diverse functions were applied to aggregate fuzzy relations, namely greater (smaller) than or equal to minimum (maximum).

Example 18. Let card đ?&#x2018;&#x2039; = 3. For fuzzy relations described by matrices: đ?&#x2018;&#x2026;=

34

2016

we have the following aggregated fuzzy relations

Proof. Let relations đ?&#x2018;&#x2026; be đ?&#x203A;ź -transitive for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A; and đ?&#x2018;Ľ, đ?&#x2018;Ś, đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. If min(đ?&#x2018;&#x2026;(đ?&#x2018;Ľ, đ?&#x2018;Ś), đ?&#x2018;&#x2026;(đ?&#x2018;Ś, đ?&#x2018;§)) =

Nâ&#x2C6;&#x2DC; 2

VOLUME 10,

0 1 1 1 0 0

1 0 1

,

đ?&#x2018;&#x2020;=

1 0 1

0 1 1

0 1 0

đ??š (đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ ) = 1 â&#x2C6;&#x2019; đ??š(1 â&#x2C6;&#x2019; đ?&#x2018;Ľ , â&#x20AC;Ś , 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ ). đ??š is called a self-dual function, if it holds đ??š = đ??š .

Theorem 25. Let đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1]. đ??š is self-dual if and only if đ??š preserves the reciprocity property of fuzzy relations.

Proof. Let đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) for đ?&#x2018;&#x2013; = 1, â&#x20AC;Ś , đ?&#x2018;&#x203A; be reciprocal fuzzy relations, đ??š be self-dual, which means that đ??š = đ??š . Thus đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś) and 1 â&#x2C6;&#x2019; đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ś, đ?&#x2018;Ľ)) = 1 â&#x2C6;&#x2019; đ??š(1 â&#x2C6;&#x2019; đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , 1 â&#x2C6;&#x2019; đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) = đ??š (đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) = đ??š(đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś), â&#x20AC;Ś , đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś)) = đ?&#x2018;&#x2026; (đ?&#x2018;Ľ, đ?&#x2018;Ś).

Journal of Automation, Mobile Robotics & Intelligent Systems

The dual functions to fuzzy conjunctions are fuzzy disjunctions and vice versa, and since these two classes are disjoint, it follows that neither a fuzzy conjunction nor a fuzzy disjunction (including t-norms and t-conorms) is a self-dual function. For any binary function đ??š, if it is without zero divisors (with the zero element 0), then đ??š cannot be self-dual (see [5]). Any self-dual and commutative binary aggregation function đ??š satis ies đ??š(đ?&#x2018;Ľ, 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ) = for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; [0, 1]. The concept of self-duality is especially developed for aggregation functions. Interesting properties and characterizations of self-dual aggregation functions one can ind in [29]. A weighted arithmetic mean, median and all quasi-linear means for which đ?&#x153;&#x2018; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] ful ils đ?&#x153;&#x2018;(1 â&#x2C6;&#x2019; đ?&#x2018;Ľ) = 1 â&#x2C6;&#x2019; đ?&#x153;&#x2018;(đ?&#x2018;Ľ), are self-dual aggregation functions.

If a relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is not reciprocal (decision makers were not informed to make such choices) there exist the ways to make it reciprocal. We present such a formula for a inite case, since practically, in decision making problems, we have inite set of alternatives. Let đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), where đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ }. We may obtain from đ?&#x2018;&#x2026; a normalized fuzzy reciprocal relation đ?&#x2018;&#x2026;â&#x2C6;&#x2014; â&#x2C6;&#x2C6; đ??šđ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;) in the following way đ?&#x2018;&#x2026;â&#x2C6;&#x2014; =

if đ?&#x2018;&#x2026; + đ?&#x2018;&#x2026; â&#x2030; 0 0

otherwise .

If we have a reciprocal relation, then we get speci ic interpretation of properties of this relation (see Section 4, pages 7-8). Reciprocity is in a sense a form of consistency or clearness of choices of decision makers. However, not every function which preserves a given đ?&#x203A;ź-property, preserves also reciprocity. To simplify the considered algorithms, we do not consider the requirement of reciprocity at any stage. We concentrate on đ?&#x203A;ź-properties and their behaviour in aggregation process, which is the main topic of this paper.

6.2. Improving Judgements of Decision Makers It may happen that some of the individual relations will not have the required property, for example đ?&#x203A;źtransitivity for some đ?&#x203A;ź. In such situation we may assume two options in the algorithms. The irst one will be not to run algorithm in such a case. The second one will be to improve a little preferences of decision makers to obtain more â&#x20AC;&#x2122;regularâ&#x20AC;&#x2122; results, i.e. to obtain all relations with the required property. >From mathematical point of view, if it comes to standard fuzzy relation properties, there are known some results how to improve the relation (for example, to make it transitive, if it is not transitive, cf. [35]). Namely, if relation đ?&#x2018;&#x2026; is not re lexive it is enough to consider đ?&#x2018;&#x2026; â&#x2C6;¨ đ??ź, where đ??ź â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;) is the identity relation. đ?&#x2018;&#x2026; â&#x2C6;¨ đ??ź is obviously re lexive. If đ?&#x2018;&#x2026; is not symmetric we may consider symmetric closure đ?&#x2018;&#x2026; â&#x2C6;¨đ?&#x2018;&#x2026; or symmetric interior đ?&#x2018;&#x2026; â&#x2C6;§ đ?&#x2018;&#x2026; , which are symmetric relations. For asymmetry (antisymmetry) and there is no appropriate closure/interior, so there is no unique method to create the asymmetric (antisymmetric) or connected (total connected) relation from the given one. However, for example the relation đ?&#x2018;&#x2026; ⧾ đ?&#x2018;&#x2026; is asymmetric. To obtain

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the transitive relation from the given đ?&#x2018;&#x2026; we may consider its closure as the sum of powers of đ?&#x2018;&#x2026;, but often such closure is the full relation (đ?&#x2018;&#x2026; â&#x2030;Ą 1), so it is not useful from practical point of view. However, there are also other methods to obtain a transitive relation from the given non-transitive one, which are not so different from the original relation (cf. [37]). Moreover, the proposed in that paper concept enables to determine relations which are transitive only for a part of the elements under consideration.

If it comes to đ?&#x203A;ź-properties, thanks to the gradualness of these properties we may, in most of the cases, obtain some grade of đ?&#x203A;ź to which a relation has the considered property. We may treat a given đ?&#x203A;ź-property as a measure of this property of the fuzzy relation đ?&#x2018;&#x2026;. However, it may happen that we get đ?&#x203A;ź = 0, and in the case of đ?&#x203A;ź-symmetry and đ?&#x203A;ź-transitivity it may be none of đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] (cf. Remark 3). If we need some improvements of the grade of the given property, we may have two approaches. The irst one is to use the existing methods for obtaining standard properties and what is equivalent, in the same way, to obtain đ?&#x203A;ź-properties for any đ?&#x203A;ź â&#x2C6;&#x2C6; [0, 1] (cf. Remark 2, Corollary 4). The second one is to increase the grade of đ?&#x203A;ź (not necessarily to the maximum possible value) to which a given relation has the considered đ?&#x203A;ź-property. To change the grade of đ?&#x203A;ź for these properties we may apply Corollaries 5 and 6 in an adequate way (using the properties of in imum and supremum and changing the values of the given relation đ?&#x2018;&#x2026;). Moreover, for reciprocal relations, taking into account properties connected with the diagonal (đ?&#x203A;ź-re lexivity, đ?&#x203A;ź-irre lexivity) there is no need to make any improvements, since by de inition, the values on the diagonal are ixed. Furthermore, for reciprocal relations we rather would like to obtain asymmetric than symmetric relations and reciprocal relation is always 0.5-asymmetric and totally 0.5-connected, so practically the following đ?&#x203A;ź-properties may be considered: antisymmetry, connectedness, transitivity.

6.3. Methods to Obtain the Final Order of Alterna ves There exist diverse methods to ind an alternative as solution from a given đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;). One of the most widely used is the weighted vote (see [26, 27]).

If we have a given preference relation đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), where đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , â&#x20AC;Ś , đ?&#x2018;Ľ }, then the weighted vote strategy means taking as the preferred alternative the solution of arg đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ đ?&#x2018;&#x2026; .

(20) ,â&#x2039;Ż,

However, in some situations this method does not allow us to choose an alternative as solution in a unique way (cf. [3]). When this happens, sometimes it is advisable to apply a different method. One of the most widely used methods is the one given by Orlovsky in 1978 and called nondominance method [30]. This method extracts as the solution the least dominated alternative/alternatives of the fuzzy decision making problem starting from a fuzzy preference relation. The maximal nondominated elements of a fuzzy preference relation đ?&#x2018;&#x2026; are calculated by means of the follow35

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ing operations: 1) Compute the fuzzy strict preference relation đ?&#x2018;&#x2026; â&#x2C6;&#x2019;đ?&#x2018;&#x2026; đ?&#x2018;&#x2026; = 0

if đ?&#x2018;&#x2026; > đ?&#x2018;&#x2026; otherwise

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2) Determine the relation đ?&#x2018;&#x2026; with the use of aggregation function đ??š (21)

2) Compute the nondominance degree of each alternative đ?&#x2018; đ??ˇ = 1 â&#x2C6;&#x2019; â&#x2039; đ?&#x2018;&#x2026; , so we get a fuzzy set ,â&#x20AC;Ś,

đ?&#x2018; đ??ˇ = [(đ?&#x2018;Ľ , đ?&#x2018; đ??ˇ(đ?&#x2018;Ľ )) â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;].

3) Determine the value đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ) Output: the aggregated fuzzy relation đ?&#x2018;&#x2026; with the property đ?&#x203A;ź-đ?&#x2018;&#x192;, where đ?&#x203A;ź = đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

Note that, if for a given function đ??š the grade đ?&#x203A;ź in step 3 is different from đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ), then it is enough to put in this step appropriate value of đ?&#x203A;ź (cf. Example 13).

3) Select as alternative: đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D; max {đ?&#x2018; đ??ˇ }. ,â&#x20AC;Ś,

However, with this method we may also not obtain the clear unique result. In such situation there is also a method to obtain an interval-valued fuzzy relation from the given fuzzy relation and then use one of the many possible linear orders for interval-valued fuzzy setting [3], which allows us to obtain the unique alternative from a given set of alternatives đ?&#x2018;&#x2039;.

7. Comparison of Algorithms We present here three algorithms to obtain the inal solution from a given set of alternatives. We use here theoretical results presented in the paper.

Our aim is to compare these approaches for decision making problems. Here, we do not pay attention to the reciprocity requirements (as it was explained before) and ways of obtaining the best alternative.

This is why, in the algorithms, we omit this inal step of inding the best alternative.

For all presented algorithms, we will have the following inputs: đ?&#x2018;&#x2039; = {đ?&#x2018;Ľ , ..., đ?&#x2018;Ľ }, đ??š - aggregation function, đ?&#x2018;&#x2026; , ..., đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;). The given đ?&#x203A;ź-property will be denoted for short đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192;.

In Algorithm 1 we assume aggregation of fuzzy relations with the common grade of đ?&#x203A;ź for the given property đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192;. Function đ??š is one of those which preserves such đ?&#x203A;ź-property.

Algorithm 1 â&#x20AC;&#x201C; the steps: 1) Check the grade of the property đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192; of each đ?&#x2018;&#x2026; for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A; 2) Fix the common grade of the property đ?&#x203A;ź â&#x2C6;&#x2019;đ?&#x2018;&#x192; of each đ?&#x2018;&#x2026; for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A; 3) Determine the relation đ?&#x2018;&#x2026; with the use of aggregation function đ??š Output: the aggregated fuzzy relation đ?&#x2018;&#x2026; with the property đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192;.

In Algorithm 2 we aggregate fuzzy relations with possible diverse grades of đ?&#x203A;ź for the given property đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192;, i.e. đ?&#x203A;ź -đ?&#x2018;&#x192;, đ?&#x203A;ź -đ?&#x2018;&#x192;, ..., đ?&#x203A;ź -đ?&#x2018;&#x192;. Function đ??š is one of those which preserves such đ?&#x203A;ź -đ?&#x2018;&#x192;, đ?&#x203A;ź -đ?&#x2018;&#x192;, ..., đ?&#x203A;ź -đ?&#x2018;&#x192; properties.

Algorithm 2 â&#x20AC;&#x201C; the steps: 1) Check the grade of the property đ?&#x203A;ź â&#x2C6;&#x2019; đ?&#x2018;&#x192; of each đ?&#x2018;&#x2026; for đ?&#x2018;&#x2DC; = 1, ..., đ?&#x2018;&#x203A; 36

In Algorithm 3 we do not determine the grades of đ?&#x203A;ź for individual fuzzy relations, but we do it for the inal result, i.e. the aggregated fuzzy relation.

Algorithm 3, with applied appropriate aggregation function đ??š, guarantees that đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; have the same grade of đ?&#x203A;ź-đ?&#x2018;&#x192;, for a given property đ?&#x2018;&#x192;.

Algorithm 3 - the steps: 1) Determine the relation đ?&#x2018;&#x2026; with the use of aggregation function đ??š 2) Check the grade of đ?&#x203A;ź-đ?&#x2018;&#x192; for đ?&#x2018;&#x2026; Output: the aggregated fuzzy relation đ?&#x2018;&#x2026; đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; with the same property đ?&#x203A;ź-đ?&#x2018;&#x192;.

and

In the following subsections we will perform comparison of complexity and usefulness of functions đ??š preserving diverse properties (more useful practically, weaker assumptions, losing less information etc.).

7.1. Comparing Assump ons on Func ons Used for Aggrega on We will consider re lexivity property only in the case of general fuzzy relations (not necessarily reciprocal ones). The results for the other properties can be analyzed in a similar way (with similar conclusions). Comparing assumptions on đ??š for the case of re lexivity we cannot conclude clearly which way is better (Algorithm 1 or Algorithm 2), it depends on the values of fuzzy relations. Let us see some examples.

Let card đ?&#x2018;&#x2039; = 2, đ?&#x2018;&#x2026; , đ?&#x2018;&#x2026; where đ?&#x2018;&#x2026; =

â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x2018;&#x2026; =

0.8 0 0 0.6

, đ?&#x2018;&#x2026; =

đ?&#x2018;&#x2026; =

0.75 0 0 0.75

0.7 0 0 0.9

,

,

.

đ?&#x2018;&#x2026; is 0.6-re lexive and đ?&#x2018;&#x2026; is 0.7-re lexive, đ?&#x2018;&#x2026; is 0.75re lexive. Considering Algorithm 1, the common value of re lexivity of đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026; is 0.6. If we take đ??š ⊞ min (which is for example the arithmetic mean) we have the guarantee that đ??š preserves 0.6-re lexivity (cf. Theorem 4). However, đ?&#x2018;&#x2026; may also have the greater level of đ?&#x203A;ź-re lexivity, which is the case for our examples.

Considering Algorithm 2, and taking as an aggregating function any increasing đ??š (cf. Theorem 5) we get for the arithmetic mean đ?&#x203A;ź = đ??š(0.6, 0.7) = 0.65, so đ?&#x2018;&#x2026; is 0.65-re lexive but it may have higher value of

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VOLUME 10,

re lexivity, which is the case in our situation.

Let card đ?&#x2018;&#x2039; = 2, đ?&#x2018;&#x2026; , đ?&#x2018;&#x2026; where đ?&#x2018;&#x2026; =

, đ?&#x2018;&#x2026; =

đ?&#x2018;&#x2026; =

0.85 0 0 0.65

0.9 0 0 0.7

,

,

.

đ?&#x2018;&#x2026; is 0.6-re lexive and đ?&#x2018;&#x2026; is 0.7-re lexive, đ?&#x2018;&#x2026; is 0.65re lexive. Considering Algorithm 1, the common value of re lexivity of đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026; is 0.6. The arithmetic mean (as explained above) preserves 0.6-re lexivity. However, đ?&#x2018;&#x2026; may have higher level of đ?&#x203A;ź-re lexivity, which is the case in our example. Considering Algorithm 2, we get for the arithmetic mean đ?&#x203A;ź = đ??š(0.6, 0.7) = 0.65, so đ?&#x2018;&#x2026; is 0.65-re lexive and this coincides with the real value of re lexivity in the considered example.

If it comes to Algorithm 3, it is enough to check the grade of đ?&#x203A;ź-re lexivity of the fuzzy relation đ?&#x2018;&#x2026; and if aggregating function ful ils the property đ??š ⊽ min, then we know that all aggregated relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are of the same grade of đ?&#x203A;ź-re lexivity (cf. Theorem 6). There is also the risk of loosing information about the real grade of đ?&#x203A;ź of particular fuzzy relations involved in the process of aggregation. Lut us see the example, where đ?&#x2018;&#x2026; , đ?&#x2018;&#x2026; â&#x2C6;&#x2C6; đ??šđ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;), đ??š = min and đ?&#x2018;&#x2026; =

0.6 0 0 0.8 đ?&#x2018;&#x2026; =

, đ?&#x2018;&#x2026; =

0.2 0 0 0.3

0.2 0 0 0.3

.

2016

7.2. Comparison of Complexity of the Algorithms

â&#x2C6;&#x2C6; â&#x201E;ąâ&#x201E;&#x203A;(đ?&#x2018;&#x2039;), đ?&#x2018;&#x2026; =

0.8 0 0 0.6

Nâ&#x2C6;&#x2DC; 2

,

đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026; , đ?&#x2018;&#x2026; are 0.2-re lexive, but đ?&#x2018;&#x2026; is in fact 0.6re lexive.

To sum up, using these methods we should remember that the grade of đ?&#x203A;ź-property of the aggregated fuzzy relation đ?&#x2018;&#x2026; (Algorithms 1 and 2) and input fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; (Algorithm 3) â&#x20AC;&#x2122;is the minimal of the maximum possibleâ&#x20AC;&#x2122; to be obtained (all depends on the form of fuzzy relations).

It is also worth mentioning that assumptions on functions đ??š to preserve đ?&#x203A;ź-transitivity are rather strong. However, if in de inition of đ?&#x203A;ź-transitivity we replace min with arbitrary binary operation â&#x2C6;&#x2014; â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1], then we may weaker in a signi icant way the assumptions on function đ??š to preserve such đ?&#x203A;ź-â&#x2C6;&#x2014;transitivity. Thus it is enough if đ??š â&#x2030;Ť â&#x2C6;&#x2014; and đ??š ⊽ min, and we have many examples of such functions đ??š (cf.

[17]).

For aggregation of fuzzy relations with diverse grades, some assumptions seem to be strong. In Theorem 17 for preservation of asymmetry we have strong assumption đ??š â&#x2030;Ť min, but for concrete functions đ??š, like for example the arithmetic mean in Example 13, we may compute the value of đ?&#x203A;ź for the property đ?&#x203A;ź-đ?&#x2018;&#x192; (without following assumptions of Theorem 17). Note that in this example đ?&#x203A;ź â&#x2030; đ??š(đ?&#x203A;ź , â&#x20AC;Ś , đ?&#x203A;ź ).

If it comes to the converse problem, for đ?&#x203A;źsymmetry and đ?&#x203A;ź-transitivity it is not clear if such functions do exist (cf. Remarks 4 and 5).

We will present the time complexity of the separate steps and operations in the presented algorithms and then we will give the complexity of each algorithm for each property.

Fuzzy relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; are de ined in a set X consisting of đ?&#x2018;&#x161; elements, so complexity will depend on the variable đ?&#x2018;&#x161; (the size of a matrix representing đ?&#x2018;&#x2026;).

We get the following time complexities: - determining the grade đ?&#x203A;ź of re lexivity (irre lexivity) is đ?&#x2018;&#x201A;(đ?&#x2018;&#x161;), since this is determining the minimal (maximal) value of a list of đ?&#x2018;&#x161; non-ordered elements (cf.

Corollary 5), - determining the grade đ?&#x203A;ź of connectedness (total connectedness, asymmetry, antisymmetry) is đ?&#x2018;&#x201A;(đ?&#x2018;&#x161; ) (cf.

Corollary 5), - determining the grade đ?&#x203A;ź of symmetry is đ?&#x2018;&#x201A;(đ?&#x2018;&#x161; ) (cf.

Corollary 6), - determining the grade đ?&#x203A;ź of transitivity is đ?&#x2018;&#x201A;(đ?&#x2018;&#x161; ) (cf.

Corollary 6).

Taking into account that the remaining operations in Algorithms 1, 2, 3 should be performed at maximum đ?&#x2018;&#x203A; times (đ?&#x2018;&#x203A; is the number of fuzzy relations) we get the following time complexities.

Corollary 20. Re lexivity and irre lexivity: Algorithms 1, 2 and 3 take đ?&#x2018;&#x201A;(đ?&#x2018;&#x161;) computational time complexity.

Connectedness, total connectedness, symmetry, asymmetry and antisymmetry: Algorithms 1, 2 and 3 take đ?&#x2018;&#x201A;(đ?&#x2018;&#x161; ) computational time complexity. Transitivity: Algorithms 1, 2 and 3 take đ?&#x2018;&#x201A;(đ?&#x2018;&#x161; ) computational time complexity.

8. Conclusion In this paper preservation of basic classes of đ?&#x203A;źproperties of fuzzy relations in the context of aggregation process were discussed. Mutual dependencies related to these properties, between relations đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; on a set đ?&#x2018;&#x2039; and the aggregated fuzzy relation đ?&#x2018;&#x2026; = đ??š(đ?&#x2018;&#x2026; , â&#x20AC;Ś , đ?&#x2018;&#x2026; ) were examined. Suf icient conditions for functions đ??š â&#x2C6;ś [0, 1] â&#x2020;&#x2019; [0, 1] to ful ill the given property were provided (regarding three possible cases of approach to aggregation procedure). Moreover, diverse â&#x20AC;&#x2122;regularitiesâ&#x20AC;&#x2122; and interpretation of đ?&#x203A;ź-properties were discussed, also in the context of reciprocal relations and decision making problems. Finally, comparison of obtained results, including suitable decision making algorithms were provided (there were analyzed the time complexities of the presented algorithms and assumptions on fusion functions useful to obtain the required results). All algorithms were implemented and tested in Java programming language.

ACKNOWLEDGEMENTS This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge in RzeszoĚ w, through Project Number RPPK.01.03.00-18-001/10.

37

Journal of Automation, Mobile Robotics & Intelligent Systems

AUTHORS

Urszula Bentkowska∗ – University of Rzeszó w, Interdisciplinary Centre for Computational Modelling, ul. Pigonia 1, 35-310 Rzeszó w, Poland, e-mail: ududziak@ur.edu.pl.

Krzysztof Balicki – University of Rzeszó w, Interdisciplinary Centre for Computational Modelling, ul.

Pigonia 1, 35-310 Rzeszó w, Poland, e-mail: kbalicki@ur.edu.pl. ∗

Corresponding author

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[37] Wagenknecht M., „On pseudo-transitive approximations of fuzzy relations”, Fuzzy Sets Syst., vol.

44, 1991, 45–55.

[38] Zadeh L. A., „Fuzzy sets”, Inform. Control, vol. 8, 1965, 338–353.

[39] Zadeh L. A., „Similarity relations and fuzzy orderings”, Inform. Sci., vol. 3, 1971, 177–200.

[40] Zimmermann H. J., Zysno P., „Decisions and evaluations by hierarchical aggregation of information”, Fuzzy Sets Syst., vol. 10, 1983, 243–260.

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Control Methods Design for a Model of Asymmetrical Quadrocopter Submitted: 9th February 2016; accepted: 19th May 2016

Ryszard Beniak, Oleksandr Gudzenko DOI: 10.14313/JAMRIS_2-2016/14 Abstract: The paper describes the results of quadrocopters motion properties for the control based on the inverse dynamics method and optimal control method with synthesis linear-quadratic regulator (LQR). Motion of quadrocopters is tested for composite trajectories. The new model of asymmetrical quadrocopters, taking into account the rotation and shift of one arm relative to the other, was developed. A few criteria for evaluation of the effectiveness of control methods of quadrocopters are presented in this paper. An analysis of the results allows selecting a method for solving the problem of quadrocopters control and making recommendations for the formation of trajectories.

Keywords: linear-quadratic regulator, inverse dynamics, quadrocopter, dynamic mode

1. Introduction

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Recently the development of unmanned aerial vehicles (UAV) has been started. Quadrocopter is an example of such vehicle. Quadrocopter is a vehicle with four rotors, which are rigidly fixed to the body [1]. These features include the fact that they are maneuverable, can still be over a given point in space and carry additional equipment. However, there are several problems associated with using this type of construction. The main problem is calculation of the effective control of quadrocopters.

The first prototype of the aircraft with this configuration appeared in 1907 [2]. Vehicle was operated with a complex transmission, which make it difficult to control. The first full quadrocopter was developed in the 50s [2]. For a number of characteristics of these models gave way to aircraft and helicopters, so widespread use they have not received. The most popular quadrocopters obtained with using UAVs. Today quadrocopters are used in various fields of human activity.

Modern quadrocopters and most of the research on them based on simple construction, models, and, therefore, use simple control algorithm. In most cases, it reduces the effectiveness of control and is not always reasonable.

An analysis of the literature proved that mathematical models can be classified as follows:

1. Linear model. Used for simple maneuvers [3] or for the calculation of the control algorithm by complex methods of high computational cost (LQR, model predictive control) [4], [ 5], [6].

2. Non-linear symmetrical model. By symmetric model we mean a model, whose center of gravity coincides with the geometric center. By the geometric center of the construction we understand the point of intersection of center lines of the arm. These models allow implementing the regulator by on-line methods [1], [6–9].

3. Asymmetric model. Consider a model with such precision is necessary for the implementation of complex maneuvers that require high control precision.

HoverBike is an example of this kind asymmetric construction [10].

The new asymmetric model of quadrocopters is presented in this paper, as having the biggest number of perspectives. However, the efficiency of this model will not be improved if it uses the control algorithm for a symmetric or linear model. Therefore, it is necessary to analyze the control methods for this model.

Quadrocopters control most commonly uses the following: PD/PID – regulators [3], [6], [7], [11], [12], LQR [4], [5], model predictive control [6], [9], backstepping control [7], [8], sliding mode control [7] and inverse control [5], [7], [13]. In this paper, the methods are chosen to control the synthesis of linearquadratic regulator (LQR method), and the method of inverse dynamics.

The purpose of this paper is to develop a new mathematical model of quadrocopters and analyze the algorithms and principles of control for various kinds of trajectories, manoeuvers, and conditions.

The mathematical model has to take into account the asymmetry of the design and the effects of external influences. The problem was solved by the example of motion along a predetermined path.

The paper consists of three main sections and conclusions. The first section describes the design of quadrocopters and obtained dynamic equations of motion of asymmetrical quadrocopters. The second section describes a synthesis of control algorithm for quadrocopters using LQR method and the method of inverse dynamics. The third section presents the results of motion simulation of asymmetric quadrocopter within two trajectories: a circle and an eightshaped figure. These trajectories are described in the third section in details. To be concise, with respect to the trajectories, we will use the terms „circle” and „eight-shaped”.

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2. Development of a Mathematical Model Most manufacturers simplify their tasks by developing symmetry with respect to frame design. This greatly simplifies the mathematical description of the motion of quadrocopters, but on the other hand, it is necessary to use additional equipment to comply with such symmetry. Manufactured devices differ significantly since the center of gravity with geometric center and the arm with the motors may be positioned at any angle relative to each other.

This paper presents a model of quadrocopters which has the center of gravity structure shifted, one of the arms is also shifted relative to the geometric center of quadrocopters and rotated at an angle α , generally not a right angle, relative to the other arm (Fig. 1). l1 is the distance from the edge of the second platform to the intersection with the center of the first platform, l1 + l2 = 2l . l s is the distance from the edge of the platform to the center of the motor. In Fig. 2, a dotted line shows a quadrocopter symmetric model.

The main elements of quadrocopters are (Fig. 2): the basic platform, two arms, four motors, unit with electrical system and accessories. The geometrical dimensions, weight and the center of gravity coordinates in the coordinate system associated with the quadrocopters geometric center are shown in Table 1.

Quadrocopter moves relative to the fixed inertial coordinate system (ICS) ( oXYZ ). Axis 0x, 0y and 0z form an orthogonal right-handed coordinate system.

Axis 0z is in the opposite direction to the vector of gravity (Fig. 3). Introduce two auxiliary coordinate systems (CS). The coordinate system oc X cYc Z c is related to the center of mass of quadrocopters (CSM), and the coordinate system og X gYg Z g associated with the quadrocopters geometric center (CSG). The axis of the coordinate system are parallel to the axes of the inertial coordinate system. The quadrocopter related with the right movable orthogonal coordinate system oc X pYp Z p (MCS). MCS starts at the center of mass of quadrocopters. The axis Oc x p is connected with one of the arms of a quadrocopter, axis Oc y p lies in the plane of a quadrocopter, axis Oc z p is upwardly directed relative to a quadrocopter. The angular position of a quadrocopter is defined in MCS by Euler angles η = ( φ ,θ ,ψ )T : roll φ , pitch θ and yaw ψ .

Fig. 2. Design quadrocopters

Fig. 3. Coordinate systems The center of mass of a quadrocopter is defined by vector X = ( x , y , z )T in ICS. The linear velocity vector of a quadrocopter is defined as Vc = (vxc, vyc, vzc)T and the angular velocity vector as W = (p, q, r)T in CSM. Rotation matrix from CSM to the ICS has the form [14]:  cψ cθ  Rot (η ) =  sψ cθ   − sθ

sψ sφ + cψ sθ cφ   sψ sθ cφ − cψ sφ  ,  cθ c φ 

cψ sθ sφ − sψ cφ cψ cφ + sψ sθ sφ cθ sφ

where The connection between the linear speed in the ICS and the CSM has the form (1).

X = Rot (η ) ⋅ Vc

(1)

The transition matrix Λ for the angular velocity of the CSM to the MCS is described in [14]. The angular velocities connection in the form (2). 1 0  Ω = Λ ⋅η = 0 cφ 0 − sφ 

Fig. 1. Geometric model of quadrocopter

− sθ   sφ cθ  ⋅η cφ cθ 

(2)

We use Köenig’s theorem and Lagrange equation (3) for obtaining the dynamic equations of quadrocopters motion [15]. We form the kinetic energy of the system T . Vector coordinates of the center of mass X and the angular orientation of quadrocopArticles

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Table 1. Description of the geometric dimensions, weight and center of gravity coordinates of structural elements Structural component

Weight of the structural element [kg]

Length, width, thickness [m]

The coordinates of the gravity center [m]

M1

a , b , h1

The unit with equipment

M2

c , d , h2

X c 1 = (0 0 0);

The first arm

M3

2l , ∆l , h3

M 4 ( M 4 = M3 )

2l , ∆l , h3

The motor 1

M5

2r ,2r , h5

The motor 2

M 6 ( M 6 = M5 )

2r ,2r , h5

The motor 3

M7 ( M7 = M5 )

2r ,2r , h5

X c 7 = − ( l2 − l s ) cos( α ) − ( l2 − l s ) sin( α ) h* ;

The motor 4

M 8 ( M 8 = M5 )

2r ,2r , h5

X c 8 = ( l1 − l s ) cos( α ) ( l1 − l s ) sin( α ) h* ;

The platform

The second arm

ters in CSM Θ = ( θ 1 ,θ 2 ,θ 3 )T were selected as generalized coordinates. Θ  q= ; X 

;

(3) (4)

where Q is generalized force, M is the quadrocopter 8

mass, M = ∑ M j , I is the inertia tensor of a quadrocopj =1

ter, Ω is the angular velocity vector in the CSM, Ω = Θ .

Quadrocopters inertia tensor I can be written as:

 y m2 + z m2  I = I g + M  − x m ym   − x m zm

− x m ym x m2 + z m2 − ym zm

− x m zm   − ym zm  ,  x m2 + y m2 

where I g is the inertia tensor in CSG, X m is the vector coordinates of the mass center, 42

X m = ( x m , ym , z m )T ,

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h + h2   X c2 =  0 0 − 1 ; 2   h +h   X c3 =  0 0 1 3  ; 2    l −l X c 4 =  1 2 cos( α )  2

(

X c5 = l − l s *

l1 − l2 sin( α ) 2

)

h1 + h3  ; 2 

0 h* ;

h = 0.5( h1 + 2h3 + h5 );

(

)

X c 6 = − ( l − l s ) 0 h* ;

(

(

)

)

The inertia tensor I g is described by the relation 8

I g = ∑ I g ,k , where I g ,k is the inertia tensor of the k =1

structure element k in the CSG. Table 2 shows the formulas for calculating the inertia tensor of quadrocopter’s elements. For simplicity, arms, the platform and the equipment unit are treated as rectangular parallelepiped elements. The motors are treated in the calculation inertial tensor as cylinders.

In Table 2 xi , yi , zi are the components Xci of the center of gravity of the structural element (Table 1).

0   Am 0   In Table 2  0 Am 0  is the inertia tensor of  0 0 C m  

motors relative to the principal axes of inertia.

The generalized force Q can be represented in the form Q = ( QM ,QF )T , where QM is a generalized torque in the rotational motion, QF is a component of generalized force in translational motion. The main components of the generalized force can be written as (5) and (6).

QM = U + M gir ;

(5)

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Table 2. The inertia tensor of the structural elements The structural elements

The inertia tensor

The platform

The unit with equipment

The first arm

The second arm

The motor j

QF = Fu + Fmg + Fres ,

(6)

where U is the vector of the rotational force caused by the operation of motors, U = ( U 1 ,U 2 ,U 3 )T , M gir is the gyroscopic moment, Fu is the traction of motors, Fmg is the force of gravity acting on the quadrocopter, , S is the Fres is resistance force, aerodynamic force coefficient vector [2].

Project the generalized forces (6) on the base q and get the form (7).

(7)

where U 0 is lift force in the MCS, g is the acceleration of gravity.

After inserting (4), (5), (7) in (3) and adding supplement system kinematic relations (2) we finally obtain (8):

(8)

The system of equations (8) should be supplemented with the equations describing the forces and Articles

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torques in quadrocopter motors. Areas of vectors of forces and moments in the CSG are shown in Fig. 4.

(11) Fig. 4. The lifting force and torque motors The lifting force and torque are directly proportional to the square of the rotation speed [4]. Formulas for traction and torque are of the form (9).

(9)

where k1 and k2 are constant coefficients, ωi is the angular velocity of rotation of the motor i , ( x i − x c ) and ( y i − y c ) are distances from center of the motor i to the quadrocopter gravity center for axis Ox and Oy respectively.

The gyroscopic torque depends on the quadrocopters rotational speed and motors kinetic torque: 4   M gir = Ω × K m = Ω × 0 0 C m ⋅ ∑ ωi  i =1  

44

T

(10)

The dynamic equations of motion of quadrocopters (8) equations of traction, torque (9) and gyroscopic torque (10) create the system of quadrocopter equation. For further convenience, the equations of motion around the center of gravity of quadrocopters shift to the base η . The inertia tensor has the following form J = ΛT IΛ . After simplification we obtain dynamic equations of quadrocopters motion in the final form (11).

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In (11) the second equation describes the motion of the center of gravity of quadrocopters, and the first equation describes the motion around the center of gravity in the MCS. The main differences between symmetrical and asymmetrical models are the form of the equations (9). For asymmetrical model the equations (9) become more complicated and require additional analysis.

3. Control Development

The synthesis of the control algorithm was carried out by methods LQR and inverse dynamics. The following criteria were used for the comparison of selected methods: 1. The value of functional use in LQR method. We consider the part of functionality associated with the state vector, which is responsible for the achievement of the control objectives, and part of the functional related to the control, which is proportional to the energy costs as separate.

2. The standard deviation of the gravity center of predetermined trajectory.

3. The maximum deviation from the given position in absolute value.

3.1. Inverse Dynamics

The method of the inverse dynamics is used to find the forces acting on an object of known trajectory. The method of inverse dynamics is unstable. In practice, various modifications of the method were used to guarantee the stability of the closed system [15, 16].

Assume that the desired trajectory is defined as analytical functions of the vector position of the gravity center X ctr and the yaw angle ψ tr . This method of defining the desired trajectory is the most informative for the controlled quadrocopter operator.

The equation form for control of quadrocopters acceleration is as follows:

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(12)

where C1 , C 2 are matrixes of known feedback coefficients.

From the dynamic equations of gravity center motion it is possible to determine the thrust U 0 and the ~ values of roll φ and pitch θ~ angles for the realization of different maneuvers.

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(18) where Y tr is the vector of the desired trajectory of movement, Q and P are constant positive definite symmetric matrix.

The control is determined by formulas (19–21).

, R(T) = 0

(13)

(19)

(20)

(21)

Similarly, the controls of angular acceleration of the angular position are: (14) where C3 , C 4 are known feedback coefficients.

From the dynamic equations of motion around the gravity center we can determine the control torques. where

,

,

.

(15)

To determine the angular velocities of motors we can use a system of equations (9). This system is linear relative to sign( ωi )ω i2, coefficient matrix is constant for the configuration and does not degenerate. It means that the system of equations (9) provides a unique solution.

3.2. LQR Method

The LQR method is described in detail in [17].

Apply an algorithm to solve this problem. System of equations (11) was linearized and used in this method. The system of equations (11) can be written as (16). So the linearized system has the form (17).

(16)

(17) where Y0 ,W0 are the state vector and control vector at some point.

Suppose, the criterion of control quality [17] has the functional form (18).

The optimal control (19) is determined for a given trajectory with regard to minimizing the functional (18). The particularity of this method is that the equations (20) and (21) are integrated in the reverse time and require high computational cost.

Considering the fact that the original system is not linear, to solve the original problem with this method it is necessary to know the matrix of the system at all points of the trajectory. For this, we need to know the trajectory of the object, which is set by the operator, and the planned control, which is unknown. We used an iterative approach to solve this problem. As a first approximation selected control obtained by the inverse dynamics. The functional (18) is the criterion for the process convergence.

4. Simulation

Based on the obtained mathematical models and control algorithms, a mathematical complex has been developed by using MATLAB R2014b. Quadrocopter AR.Drone 1.0 was taken as a basis [18]. The angular rotation speed of motors is limited to the equation 150 < ωi < 500 , i = 1 ,..,4. The model parameters are: a = 0.2 [m], b = 0.2 [m], c = 0.1 [m], d = 0.1 [m], l = 0.3 [m], l1 = 0.33 [m], ls = 0.03 [m], Δl = 0.1 [m], r = 0.05 [m], h1 = 0.02 [m], h2 = 0.05 [m], h3 = 0.02 [m], h5 = 0.02 [m], α = 75°, M1 = 0.2 [kg], M2 = 0.1 [kg], M3 = 0.1 [kg], M5 = 0.05 [kg], C1 = 16, C2 = 16, C3 = 225, C4 = 40, Am = 0.005 [m2kg], Cm = 0.001 [m2kg], k1 = 0.7426·10-6 [m2kg], k2 = 0.1485·10-6 [m2kg], g = 9.8 [m/s2], Sx = 0.0024 [kg/m], Sy = 0.0072 [kg/m], Sz = 0.0072 [kg/m].

For the comparison of the efficiency of the control algorithms with the new model, two trajectories were selected. Both trajectories consisted of three stages.

In the first stage, a quadrocopter hovered motionless at a given point in space during 0.1 s. (22). In the second stage, the quadrocopter rose straight up and picked up speed for a maneuver (23). In the third stage, the quadrocopter was doing the maneuver. For the first trajectory, the quadrocopter flew around the ring in a vertical plane with a radius of 1 m and the angular speed θ = 2π / 5 rad/s (24). For the second trajectory, the quadrocopter flew along “eightArticles

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Fig. 5. Motion trajectories of quadrocopters

shaped” with a loop radius of 1 m and a time period of 5 s. (25). Trajectories are shown in Fig. 5. and described in detail in (22–25).

0 ≤ t < 0.1 ; x I ( t ) = 0 ; y I ( t ) = 0 ;  z I ( t ) = −1 ;ψ I ( t ) = 0 ;

(22)

5  0.1 ≤ t < t s ; t s = + 0.1 ; x I ( t ) = 0 ; y I ( t ) = 0 ;  π (23)  π z ( t ) = ( t − 0.1 ) − 1 ;ψ ( t ) = 0 ; I I 5 

2π  o o t ≤ t ; x III ( t ) = 1 − cos( ( t − t s )); y III ( t ) = 0;  s 5 (24)  z o ( t ) = sin( 2π ( t − t ));ψ o ( t ) = 0 ; III s III  5 

2π  ∞ ∞ t ≤ t ; x III ( t ) = 1 − cos( ( t − t s )); y III ( t ) = 0;  s 5 (25)  z ∞ ( t ) = sin( 4π ( t − t ));ψ ∞ ( t ) = 0 ; III s III  5 

The results of the simulation are shown in Figs.

6–9. Table 4 shows the numerical value of the evaluation criteria.

In Figs. 6–9 for the state vector dotted line indicates the desired trajectory. It should be noted that the deviation from the predetermined trajectory in the plane YZ equals less than 1.5 mm for absolute value in all cases.

According to the simulation results, we can conclude that both methods are able to solve the problem of control successfully. The control algorithm obtained by both methods is within the predetermined limits. The most difficult phase to control is the transition from the second to the third stage. This

Table 4. Properties of quadrocopters motion

Criteria

LQR

ID

Trajectory “eight-shaped” LQR

ID

Functional Φ ( Y ,W )

2.65

12.00

4.50

17.15

Part Φ ( W ) of the functional

2.39

7.43

3.82

11.05

0.38

0.65

Part Φ ( Y ) of the functional Standard deviation [m]

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Trajectory “circle”

Maximum deviation of the position [m] Articles

0.27

0.06

0.15

4.57

0.05

0.17

0.68

0.10

6.10

0.14

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Fig. 6. The state vector and the control vector to the trajectory “circle” by the LQR method

Fig. 7 .The state vector and the control vector to the trajectory “circle” by the inverse dynamics is caused by discontinuity of the desired state vector, namely discontinuity of the angular position. This is particularly well illustrated by the trajectory “circle” (Figs. 6–7). LQR method was implemented smoothly around that time, as can be seen from the Fig. 6 and Fig. 8. The method of inverse dynamics (ID) could not do it smoothly. To continue the flight along the trajectory, it is necessary to create high moments, it is shown by the peaks in control in Fig. 7 and Fig. 9.

An analysis of the imposed criteria shows that the mean deviation and the maximum deviation of the gravity center from predetermined trajectory are

approximately the same. However, the energy cost is higher in inverse dynamics.

The advantages of the inverse dynamics method mainly consist of their simplicity, computational speed in the calculation and the ability of application in on-line tasks.

5. Conclusions

The control problem of asymmetric quadrocopters was illustrated an example of complex trajectories “circle” and “eight-shaped”. The new mathematical model which takes into account the asymmetry of Articles

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Fig. 8. The state vector and the control vector to the trajectory “eight-shaped” by the LQR method

Fig. 9. The state vector and the control vector to the trajectory “eight-shaped” by the inverse dynamics

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the quadrocopter design has been developed. In order to study the characteristics, of the new model, control algorithms by LQR method and inverse dynamics for the motion along a predetermined trajectory have been synthesized. We suppose that the asymmetric construction properties improve maneuverability.

Therefore, in the control algorithms it is necessary to considered construction characteristics. This paper presents the solution algorithm for nonlinear model.

The criteria entered for the efficiency evaluation of Articles

the synthesized control algorithms allow us to consider the choice of solution methods for conditions variety. LQR method involves large computational costs and requires a prior knowledge of the motion trajectory, but it allows decreasing the energy costs and obtaining a smooth motion of trajectory. The method of inverse dynamics can be used in on-line mode, it does not require high computational costs, but at the moments of trajectory discontinuity the system may lose stability especially in aggressive manoeuvers.

Journal of Automation, Mobile Robotics & Intelligent Systems

AUTHORS Ryszard Beniak* – Opole University of Technology, Faculty of Electrical Engineering, Automatic Control and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland.

E-mail: r.beniak@po.opole.pl

Oleksandr Gudzenko - Opole University of Technology, Faculty of Electrical Engineering, Automatic Control and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland.

E-mail: o.gudzenko@doktorant.po.edu.pl *Corresponding author

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Any-angle Global Path Planning for Skid-Steered Mobile Robots on Heterogeneous Terrain Submitted: 25th January 2016; accepted: 20th April 2016

Piotr Jaroszek DOI: 10.14313/JAMRIS_2-2016/15 Abstract: The paper is concerned with selection of the algorithm of path planning for a skid-steered mobile robot operating on heterogeneous terrain. Methods of path searching were reviewed and their applicability to particular kinematic structure of a robot was assessed. The Theta* graph search algorithm was selected, because of its property of returning any-angle paths. Because in this method variable terrain type is not considered, necessary changes in algorithm structure were proposed to check homogeneity of the terrain. In order to enable choice of arbitrary optimization criterion, the model of cost dependent on terrain properties was introduced, which includes both longitudinal motion and turning. Operation of the modified algorithm with the introduced cost model was verified by means of simulation against A* reference algorithm often used in path planning tasks.

Keywords: spath planning, skid-steered mobile robot, Theta*, any-angle path planning, A*

1. Introduction

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One of the most important tasks associated with autonomy of robot motion is global path planning. Every type of robot requires individual approach to this problem, because robot motion capabilities depend on kinematic structure of the mobile platform. Another equally important component of the global path planning is adequate representation of the environment. Most often the robot operates in heterogeneous environment, which can be characterized by diverse properties, and taking this fact into account is definitely beneficial in the path planning process. A desirable solution returns path optimal according to the adopted criteria and possible for realization by the robot in its workspace. There is no single method of path planning which would be appropriate for all tasks in robotics, and thus in order to get the best results for a given problem, the methods have to be modified.

In this article, the approach of optimal path planning is presented for one of the more frequently used types of mobile robots â&#x20AC;&#x201C; the platform with nonsteered wheels. To this end, the any-angle path planning method with original modifications that include terrain non-homogeneity and generic robot motion model was used.

2. Selection of Path Planning Method for a robot with Non-steered Wheels on Heterogeneous Terrain There exist several methods of global path planning, whose usefulness varies depending on target application. The potential field methods [1] are fast, but they suffer from the local minima problem, the genetic methods have big potential [2], but are significantly slower and difficult in description and implementation. Both those groups are characterized by frequent lack of algorithm convergence, so sometimes the returned path is not optimal. Additionally, serious difficulties in representation of complex models of the environment and robot can be encountered. The alternative are graph search algorithms [3], where belong admissible non-heuristic algorithms, e.g. Dijkstra [4], and heuristic algorithms like A* [5]. The search space in those methods has to be discretized to the form of a weighted graph whose nodes and edges represent respectively available locations and possible movements between them. Moreover, the search algorithms make use of the strategy of choice of the search direction, which can be modified in order to achieve desirable motion behaviors that are possible for the chosen kinematic model of the robot.

In case of the graph search algorithms, the terrain where the robot operates is most often discretized to the grid of elements of identical size, based on which the graph is constructed. Weight of each edge of the graph depends in an unique way on properties of nodes which it connects. When the nodes represent locations, most often edge weight is equal to a distance between the nodes, though in general the travel cost does not have to be associated with length. By appropriately choosing search politics and cost models, it is possible to obtain solution of the optimal path planning task for any criterion. The essential problem which remains to be solved is how the path looks in the discretized terrain, which directly depends on graph representation, and the path appearance is often different than would be expected in continuous environment.

The path found on the terrain grid is optimal from the point of view of the graph, so it usually looks unrealistic and is not the shortest one in the real continuous environment. In [6], author presented analysis of the problem of path length dependency on various environment models and search methods on regular grids. Example of difference in the path length found with A* method, and the shortest one possible for the continuous environment is shown in Fig. 1.

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a)

b)

Fig. 1. The shortest path found with A* method in case of discrete environment (a) and in case of continuous environment (b) As one may notice, the path found by the A* algorithm (Fig. 1a), despite being optimal in the graph space (assuming path length as optimality criterion) does not reflect in a good enough degree the path which is optimal in the real continuous environment (Fig. 1b). The path, let alone it is longer than the shortest one in the continuous environment, comprises also several unnecessary turns which may introduce additional cost of robot motion. The robot will waste time and energy by turning in place, and time and energy affect the majority of optimality criteria traditionally used in robotics. It is possible to claim undoubtedly that for a skid-steered robot, turning in place has to be considered because of significant amounts of energy which are necessary for this maneuver. Therefore, the solution may consist in change of the environment representation or modification of the planning method to obtain paths more alike the real ones. The change of environment discretization, for example, to the visibility graph would solve the problem of path outlook, but it would also make representation of heterogeneity of the environment more difficult and would noticeably extend the computation time of the algorithm [7].

This problem can be also resolved by modification of the algorithm rather than the representation of the data. Example of this approach are any-angle path planning methods [6], that is, the methods for which the ultimate path outlook does not depend strictly on edges of the graph based on which the path was found.

Many approaches to this problem were proposed, from smoothing paths found by the A* to more complex modifications, that is, Theta* [7], Block A* [8], Field D* [9]. Based on [8] and [10] one may come to the conclusion that out of the previously mentioned methods, the Theta* algorithm will be the most appropriate for the mobile robot global path planning task. If path length is the chosen optimality criterion, then the found path is usually the shortest one and has smaller (or comparable) number of turns compared to other A* family algorithms. The Theta* algorithm yields to the other algorithms mainly as far as speed of computation is concerned, however, in the present case speed is not the most important factor. The important factor, besides optimality of found paths, is the possibility of representation of heterogenous terrain which directly affects the cost according to each of the assumed optimality criteria. Author

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believes that Theta* is one of the most suitable algorithms for the mobile robot global optimal path planning task because: • it returns the optimal path if it exists, • shape of the found path complies with the assumption of continuous environment, • it works fine with discrete representation of terrain.

Authors in [11] presented generalization of the Theta* toward maps with non-uniform cost of each cell. The introduced modifications are general purpose in the sense, they do not take into account model of robot kinematics.

The aim of this work is modification of the Theta* method so as to solve the problem of finding optimal path for a robot with all wheels non-steered according to any cost model that includes longitudinal motion of a robot and its (pivot) turning on the known heterogeneous terrain.

3. Robot Model and Environment Representation

The map. The search space is represented in the form of a weighted graph with nodes as the admissible robot locations. The following assumptions concerning terrain map discretization were introduced: • the terrain map is divided into square-shaped elements, which form the so-called occupancy grid, • length of side of an individual element l = 0.1 m, • the search space has the form of a graph with m nodes nj j=1,2,3…m, • each node nj stores information about terrain properties T(x,y) = μxy and about location on the map (x,y) which it represents, • the cost value is assigned to every edge depending on the properties of nodes connected by this edge.

The robot. The assumed mobile platform is equipped with non-steered or caster wheels and has the capability of: • moving along a straight line, • (pivot or in-place) turning through arbitrary angle.

It is assumed that during motion the robot does not turn along an arc, that is, forward motion and turning do not occur simultaneously. In the autonomous operation of this kind of robots it is favorable to avoid the combined motion, because the combined motion additionally increases possibility of wheel slip and other unpredicted motions. The point-to-point motion is the simplest to realize.

State of the robot at the time instant t is described as: ,

(1)

where ptx, pty – are respectively x and y coordinates of robot position on the map, θt – robot orientation with respect to the Cartesian coordinate system of the map at the time instant t (Fig. 2).

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by the search strategy and it depends on the previous position of the robot. After assuming that at the time instant t the robot has the state xt and it starts motion from the node n(pxt,pyt) along the edge i and ends motion at the node n(pxt+1,pyt+1), one can write: ,

Fig. 2. Robot pose in map coordinates The cost of robot motion can be represented in various ways, depending on the assumed criteria of path optimality. In order to allow arbitrary choice of the criteria, the total cost of motion C includes both longitudinal motion along a straight line and the pivot turning: (2)

where: CR – turning cost and CM – forward motion cost.

Motion along the path of length s can be divided into S elements of lengths Δsi for which elementary costs Ci will be equal to: (3)

After assuming that the elementary length Δsi is identical with the graph edge i, one may write that the total cost of path described on this graph is equal to: (4)

The cost of longitudinal motion and turning can be further determined based on terrain properties and the motion realized by the robot. Terrain property µi for the graph edge is equal to the value assigned to the node which was encountered earlier during robot motion. Use of only one node for the elementary motion along Δsi for a dense grid introduces a small error, which was allowed for the sake of significant simplification of the cost calculation: (5)

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where μi is the function parameter describing terrain property assigned to the start node of the i-th graph edge, R is the cost function for turning whose value depends on change of the angle of orientation of the robot Δθi, M is the cost function for robot forward motion whose value depends on the travelled distance Δsi. The turning cost for a robot with steered wheels can be much smaller than for the skid-steered robot.

The μi value is equal to the property of the discrete element of coordinates Xi and Yi on the map corresponding to the graph node deemed the start node for a given edge. The start node of the edge is defined Articles

(6)

(7)

where T(pxt,pyt) is terrain type property mentioned earlier. Thus, the weights assigned to edges depend directly on: nodes they connect, direction of robot motion and its orientation. In view of that, partial cost of robot motion between graph nodes can be written as: ,

(8)

where elementary length Δsi and change of orientation Δθi with respect to terrain map are equal to: ,

(9)

,

(10)

where S > i > 0 and for i = 1, the py0 px0 and θ0 are values of the robot initial state.

Equation (8) can be transformed into general form of the partial cost as a function of the robot variable state: (11)

Functions R and M can have arbitrary forms. For energy optimization of the path, they can be, for instance, models of energy consumption by robot drives during straight-line motion and during turning. Additionally, in order to keep the algorithm admissible, both functions have to be linear with respect to Δsi and Δθi, which are time dependent.

4. Global Path Planning with the Modified Theta* Method

Starting from a certain start node, the A* algorithm searches the state space graph, successively “closing” its “visited” nodes which are situated at the so-called frontier. During visiting the node nj, the cost of path necessary to reach this node is calculated based on the cost function F: (12)

,

(13)

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where: G(ni) – is the cost from the start node to the nj node, g(nj-1,nj) – partial cost of robot motion between nodes nj-1 and nj, l – number of nodes which form the path from n0 to nj, H(nj) – estimation of the cost of the remaining not-yet-found path from the node nj to the target.

Nodes which are subsequently visited from the currently closed location (i.e., node) depend exclusively on the chosen search policy and most often they are the neighboring nodes. For a graph based on the occupancy grid, the algorithm – if the node is not at the map boundary – most often tries to visit from 4 to 8 neighbors of the current node (including possibilities of diagonal motion). Each node additionally stores information about its parent, from where it was visited. If at the moment of visiting the node, the total cost F(ni) is smaller than previously determined (the same nodes can be visited from different directions), the pointer to the parent of this node is updated. If the node considered target node becomes closed, then the path is found and it is generated based on the pointers to parents.

Assuming that the robot at a given time instant can be in one location only (a graph node), the introduced earlier partial cost can be now assumed as follows:

(14)

Fig. 3. Pseudocode of the reference Theta* [7]

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Fig. 4. Theta* for the robot with non-steered wheels in the heterogeneous environment

The main idea and difference of Theta* as compared to A* is determination of node parents based on mutual visibility by means of the line-of-sight check (the pseudocode shown in Fig. 3, described in detail in [7]). During searching the graph, the parent-child relations are updated for any successive nodes making the path that mutually “see” each other. Pointer to parent is set at every successive node to the furthest visible node which belongs to the path. This leads to a frequent situation where the parent is not neighbor to its child in the sense of being connected by the graph edge. Consequently, it is not possible to use the cost model like that directly in the Theta* algorithm, because partial value in the heterogeneous terrain is not identical along the whole path between parent and child. When the cost is not homogeneous during traverse, the visibility ceases to be a sufficient condition of optimality of the found path – replacement of the visibility condition with the terrain homogeneity condition becomes necessary (Fig.4).

This condition is checked by iterative review of all cells lying on a line between the start point and end point under test (Fig. 5). Selection of the mentioned cells is carried out using the Bresenham algorithm [12]. If any of the cells is untraversable or type of terrain of the successive cells is not the same, then the terrain along the line connecting the chosen endpoints is heterogeneous. Otherwise, when the terrain is homogeneous, it is possible to use the derived earlier cost model because the μ parameter is constant.

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In Fig. 4 necessary changes in the ComputeCost function within Theta* algorithm as well as additional formulas are shown, which include arbitrary models of cost for longitudinal motion and turning as well as terrain heterogeneity testing. Example longitudinal motion CM and turning CR cost models are shown in Fig. 6. In those models, the example cost of longitudinal motion depends on travelled distance and the cost of turning, on absolute value of change of the angle of robot orientation. The M and R quantities are auxiliary constants that modify values of the appropriate costs.

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Table. 1. Sets of terrain properties used for algorithm testing and their assignment to areas on the map

5. Simulation Results A number of simulations of path planning using the proposed Theta* modification and the standard version of A* with the same models of cost and environment in order to compare quality of the obtained results were conducted. On a map of dimensions 512 x 512 cells (51.2 m x 51.2 m) shown in Fig. 7, 100 trials of searching of paths for each of three different sets of terrain parameters were carried out. Values of terrain properties for three variants are shown in Table 1 and they are assigned to cells of the map according to the colors shown in Fig. 7.

Traversability

#2

#3

A

0.1

0.1

0.5

Yes

C

0.5

0.8

5

Yes

N/A

No

0.3

D

a)

Set of μ

#1

B

Fig. 6. Example motion cost models for longitudinal motion C_M and turning C_R

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Values of R and M constants in the cost models are assumed respectively R = 5 and M = 1. Example paths found are shown in Fig. 8. The noticeable large number of turns in the path found by the modified Theta* is caused by high diversity of terrain and by not including costs associated with acceleration and stopping. Summary of results from 300 trials in total is shown in Tables 2 and 3. Gain in cost value and path length in case of Theta* as compared to the reference version of A* is presented.

Color

Fig. 5. Cells found using Bresenham algorithm for a terrain homogeneity check

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0.4

0.8

1.6

N/A

N/A

2.5

Yes

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Yes

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Fig. 8. Example paths found by A* (a) and Theta* (b) Table. 2. Summary of results of path cost Path cost Set of μ

Average gain

Maximum gain

Minimum gain

#1

5.69%

10.73%

1.28%

#3

7.22%

12.07%

2.67%

#2

6.84%

12.84%

2.24%

Table. 3. Summary of results of path length Path length Set of μ

Average gain

Maximum gain

Minimum gain

#1

1.84%

8.86%

-20.63%

#2

#3

1.76%

1.51%

8.30%

8.13%

-13.30% -8.83%

6. Conclusion Fig. 7. A map used for algorithm testing 54

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criterion which depends on heterogeneous terrain.

Adjustment of the any-angle algorithm to the considered robot model results in reduced cost of the found path as compared to the standard A* algorithm and the differences may reach over a dozen percent. It was also noticed that the gain is greater on the terrain where differences of properties are larger. Moreover, the length of path found is shorter on average, despite the used optimality criterion was disparate from the shortest path criterion. The changes proposed in the Theta* algorithm give opportunity of finding optimal path on heterogeneous terrain which can be realized by a robot with non-steered or caster wheels.

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[10] Uras, T., Koenig, S., “An Empirical Comparison of Any-Angle Path-Planning Algorithms”. In: Eighth Annual Symposium on Combinatorial Search, May 14, 2015.

[11] Choi, S., Yu, W., “Any-angle path planning on non-uniform costmaps”. In: 2011 IEEE International Conference on Robotics and Automation (ICRA), May 2011, 5615–5621. DOI:10.1109/ ICRA.2011.5979769.

[12] Bresenham, J. E., “Algorithm for computer control of a digital plotter”, IBM Systems Journal, vol. 4, no. 1, 1965, 25–30. DOI:10.1147/sj.41.0025.

AUTHOR

Piotr Jaroszek – Industrial Research Institute for Automation and Measurements (PIAP), Warsaw, 02-486, Poland.

E-mail: pjaroszek@piap.pl

REFERENCES

[1] [2]

[3] [4] [5]

[6] [7]

[8] [9]

Hwang, Y. K., Ahuja, N., “A potential field approach to path planning”, IEEE Transactions on Robotics and Automation, vol. 8, no. 1, 1992, 23–32. DOI:10.1109/70.127236.

Chung, W. K., Xu, Y., “A generalized 3-D path planning method for robots using Genetic Algorithm with an adaptive evolution process”. In: 2010 8th World Congress on Intelligent Control and Automation (WCICA), July 2010, 1354–1360.

DOI:10.1109/WCICA.2010.5554851.

Cormen, T. H., Stein, C., Rivest, R. L., Leiserson, C. E., “Introduction to Algorithms”, 2nd ed.

McGraw-Hill Higher Education, 2001.

Dijkstra, E. W., “A note on two problems in connexion with graphs”, Numerische Mathematik, vol. 1, no. 1, 1959, 269–271. DOI:10.1007/ BF01386390.

Hart, P. E., Nilsson, N. J., Raphael, B., “A Formal Basis for the Heuristic Determination of Minimum Cost Paths”, IEEE Transactions on Systems Science and Cybernetics, vol. 4, no. 2, 1968, 100– 107. DOI:10.1109/TSSC.1968.300136.

Nash, A., “Any-angle path planning”. University of Southern California, 2012.

Nash, A., Daniel, K., Koenig, S., Feiner, A., “Theta*: Any-angle Path Planning on Grids”. In: Proceedings of the 22Nd National Conference on Artificial Intelligence - Volume 2, Vancouver, British Columbia, Canada, 2007, 1177–1183.

Yap, P. K. Y., Burch, N., Holte, R. C., Schaeffer, J., “Abstract: Block A* and Any-Angle Path-Planning”. In: Fourth Annual Symposium on Combinatorial Search, July 5, 2011.

Ferguson, D., Stentz, A., “Field D*: An Interpolation-Based Path Planner and Replanner”. In: Robotics Research, 2007, 239–253.

DOI:10.1007/978-3-540-48113-3_22.

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E2LP Remote Laboratory. New Challenges in System Development Submitted: 19th March 2016; accepted: 30th May 2016

Rafał Kłoda, Jan Piwiński, Kacper Kurzejamski DOI: 10.14313/JAMRIS_2-2016/16 Abstract: The embedded engineering became the key success factor in many segments of human activities. Therefore, the needs for embedded engineers rapidly grew in last years. The engineering education in embedded systems is facing new challenges with the interdisciplinary approach and the high development dynamics of many specific components. The paper presents the results of Remote Laboratory (RL) service for distance learning for Embedded Systems, developed under Embedded Engineering Learning Platform (E2LP) – 7th Framework Programme funded project.

This project provides an unified learning platform for embedded engineering over-coming an important problem of the high introduction overload by separate courses typical for embedded engineering studies.

The developed E2LP includes the unified platform for typical courses (digital system design, computer system design) accompanied with basic set of exercises. The study efficiency is improved using the RL functionality.

E2LP RL delivered secure and open access e-learning portal, which allowed to create full course and provide alternative teaching methods through the real-time experiments.

The whole concept was evaluated at partner universities as well as at Warsaw University of Technology, where we introduced the new learning model in Digital System Design course.

Keywords: E2LP, remote laboratory, Curriculum integration, embedded systems

1. Introduction

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As embedded software systems have grown in number, complexity, and importance in the modern world, a corresponding need to teach computer science students how to effectively engineer such systems has arisen [1].

Early exposure to embedded computing systems is crucial for students to be prepared for the embedded computing demands of today’s world. However, exposure to systems knowledge often comes too late in the curriculum to stimulate students’ interests and to provide a meaningful difference in how they direct their choice of electives for future education and careers [2].

Focusing the necessary changes in computer engineering and computer science education, we are

meeting the next dilemma – what are generic principles which should be covered in the embedded engineering education for variety of using fields. An intuitive approach is to ask industry about needed engineering profiles where educated engineers will act for their lifetime. A joint work between academy and industry certainly improves engineering education programs as it is mentioned in [3] and [4]. But, this approach is facing an issue – the imbalance between technology cycle times and working life times. When technology cycle times, like in the past, are longer than the people working times, industry needs could perfectly define requirements for education which takes more than 10 years (about 12 years for basic and high school education plus about 5 years engineering education). Today, technology cycles are even shorter than the engineering education time, which means that the industry needs today could only express educational needs in the past [5]. Therefore, an inter-active approach between industry and academy with improving loops inside education times will be needed. A concurrent evaluation of systems and processes mentioned in [6] could be a right answer to this challenge. Even an early impact during basic and high school education could help to overcome this race between education and development [7], [8] and [9].

Those aforementioned issues were a genesis to create an unified learning platform, customized to embedded systems curriculum and was the main goal of the E2LP project.

2. E2LP Research Objectives

Embedded Computer Engineering Learning Platform (E2LP) is a European FP7 project of three years duration, started in September 2012. [10]. The project’s motivations and project research goals are widely discuss in Kastelan et al. [11].

The main idea behind this project is to provide a unified platform which will cover a complete process for embedded systems learning. A modular approach is considered for skills practice through supporting individualization in learning. This platform facilitates a novel development of universal approach in creative learning environment and knowledge management that encourage use of ICT. New learning model is challenging the education of engineers in embedded systems design through real-time experiments that stimulate curiosity with ultimate goal to support students to under-stand and construct their personal conceptual knowledge based on experiments. In addition to the technological approach, the use of cog-

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nitive theories on how people learn helps students to achieve a stronger and smarter adaptation of the subject. Applied methodology was evaluated from the scientific point of view in parallel with the implementation in order to feedback results to the R&D. Embedded systems are invisible electronics and corresponding software that bring intelligence to objects, processes and devices. The main challenge in engineering education for embedded systems is a complex interdisciplinary approach which includes: understanding of various systems based on different technologies and system solution optimizations.

As a result, the produced embedded computer engineering learning platform â&#x20AC;&#x201C; E2LP ensures a sufficient number of compatible educated future engineers in Europe, capable of designing complex systems and maintaining a leader-ship in the area of the supply and embedding of electronic components and systems, thereby ensuring that our strongholds in automotive, avionics, industrial automation, mobile communications, telecoms and medical systems are able to develop.

One of the man objective in E2LP project was to provide unified Remote Laboratory for embedded engineering education. In E2LP project a Remote Laboratory is an experiment, demonstration and a process running locally to design and control an experiment board based on a FPGA device, but with the ability to be monitored and controlled over the Internet (Elearning portal). The RL development and implementation phases are presented in detail by Kloda and Piwinski [12].

In the base case, the RL can be an experiment board connected to a computer through a standard interface and with the host computer connected to the Internet, which provides a remote access. The client can be any computer connected to the Internet with an ability to see the same interface as the local host as well as has the same programs, interfaces and modules.

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RL framework (Fig. 1) consists of three main elements: 1. E-learning portal [13] This part of RL provides an access to knowledge (on-line exercises, data sheets) as well as remote operations with E2LP main board through a web user interfaces.

2. Laboratory hardware. Main element is E2LP experimental board with programming cable device and other equipment to conduct remote learning process (E2LP server, digital card, serial port server).

3. Laboratory software. It includes the necessary software to programming board and other applications/services/interfaces based on several IT technologies, which provide proper functioning of the whole Remote Laboratory and their hardware components.

Here there are also a number of communication ports, which provide flawless operation of specific applications and services in E2LP server, as well as in several cases enable user to individually configure the communication with a given device.

The main advantage of proposed E2LP RL framework is a possibility for students to interact with the real E2LP platform interfaces, implemented as a web services in Moodle [14] (acronym for Modular ObjectOriented Dynamic Learning Environment) and work with software applications, on the same operational level like they are actually operating the same tools and instruments in classic lesson in laboratory.

3. Moodle Based Platform for E-learning Course Development

Talking about embedded engineering studies an important question is the learning platform. In practice embedded solutions are implemented on different platforms (hardware and software). The variety and dynamics of used platforms cannot be covered by a unified study program. Therefore, some kind of abstraction and generalization in engineering education has to be applied. A possible overcoming approach

Serial Server

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ETHERNET

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Fig. 1. E2LP Remote Laboratory framework Articles

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is proposed in [15] trying to leverage orthogonality, theoretical background and necessary education platforms. Having in mind a crucial point of the embedded systems approach, all problems should be solved by software approach, this platform broadly uses the unified API (application programming interface) approach.

Virtual and remote laboratories (VRLs) are elearning resources that enhance the accessibility of experimental setups providing a distance teaching framework which meets the studentâ&#x20AC;&#x2122;s hands-on learning needs [16] They have been considered as one of the five major shifts in a century of engineering education, thanks to the influence of information and computational technologies [17].

An important study of the implementation of VRLs into learning courses was reported here [18]. This study presents the results of integrating the open remote laboratories into several courses, in various contexts and using various methodologies. These integrations, all related to higher education engineering, were designed by teachers with different perspectives to achieve a range of learning outcomes.

In a traditional laboratory, the user interacts directly with the equipment by performing physical actions (e.g. manipulating with the hands, pressing buttons, turning knobs) and receiving sensory feedback (visual and audio). However, equipping a laboratory is a major expense and its maintenance can be difficult [19].

It should be stressed that Remote Laboratories cannot replace the classical education course. There are of course drawbacks of implementation such tools, mainly in lack of communication between student and course supervisor. This type of systems can isolate students and reduce their motivation in learning process. Furthermore, students could not receive instant feedback from their questions and cannot talk in real-time about results obtained in the learning activities with the teacher. Finally students are required to demonstrate their final projects on the actual hardware, the Massive Open Online Course (MOOC) platforms enable them to prepare for this at home, and then to be able to demonstrate valid hardware results in the laboratory [20] In E2LP project a Remote Laboratory is a service, which enable students to access the laboratory equipment and execute remote operations to carry out exercises. The main goal of RL was implementation of instant feedback from remote E2LP board in a way that user would operate with the real board as if it was connected locally. This functionality was a purpose to develop the GUI web interface of E2LP board front panel that exactly reflects the real board, which has connections to real signals from the real board.

Connection with the Remote Laboratory is provided via e-learning portal, which is based on Apache server, PHP and SQL server. It provides an access to knowledge (exercises, data sheets) and laboratory hardware through a web user interfaces to enable user to have the full experience of working on the laboratory exercise. The second role of e-learning portal Articles

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is management of users, which means enable them access to the laboratory hardware and software (booking functionality and authorization).

In E2LP project the e-learning platform is based on Moodle Platform, which is one of the most popular open source learning management systems.

Moodle is becoming increasingly popular in schools worldwide due to its ease of use and flexibility. Science is the perfect subject to benefit from the features of Moodle as students will find it the easiest to learn with the help of interactive content, rather than reading textbooks. Using Moodle teachers can easily construct richly-textured web-based courses.

A course can consist of a number of lessons, with each lesson including reading materials; activities such as quizzes, tests, surveys, and projects; and social elements that encourage interaction and group work between students. Moodle is relatively easy to install, but the real challenge lies in developing a learning process that leverages its power and maps effectively onto the established learning situation.

This platform is a huge community developing, improving, creating science-based resources, and supporting the software that is used all over the world.

There are many reasons to use a Virtual Learning Environment (VLE) such as Moodle to enhance teaching methods. These include the following: Being able to give your students access to course materials 24/7 in a controlled environment, so learning can take place anywhere.

Monitor the progress and keep records of your students learning.

Extending the classroom by providing online discussion, testing, activities, and, most importantly, allowing collaboration and communication for learning.

Make use of exciting multimedia and web-based content, allowing pupils with different learning styles to access the curriculum.

Helping science teachers collaborate, share, and store teaching resources, releasing them to students at your own pace.

4. E2LP RL Functionalities

RL is a gate which provides an access to continuously refreshed interfaces and signals from the real board and enable users to remotely control and program the board directly from their computer at home, having instant visual feedback.

To achieve this, it is necessary to forward data directly to the server over common interfaces or over local network by using dedicated hardware solutions and specified proper router configuration.

The E2LP RL should allow users to do following actions over an Internet connection, which are the list of E2LP Remote Laboratory main functionalities: 1. Dedicated software and hardware solutions provide an access to laboratory equipment and enable students to set them up and operate them at the required level to carry out selected exercises.

2. Users could access the essential data sheets, tutorials and software tools, which are available on the E-learning portal as an introduction to the course.

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Each laboratory exercise is presented in transparent form to the user through tabs and such division is implemented into Moodle based platform for e-learning course (Basic information, Theoretical explanations, Instructions, Configure Platform, Feedback, Discussion on results questionnaire for lab evaluation).

3. After booking in a given time slot users could remotely program given set of exercises over the Internet and simultaneously, in real time, could monitor the evolution of the experiment on implemented dedicated Graphical User interface (GUI) of the Front Panel of the real E2LP board .

4. Automatic verification of course assignments will allow an advanced management of assignments and submissions together with feedback information mechanisms for both teachers and students, which will verify, whether the students designs are correct or not according to the specifications.

5. RL Implementation

RL presents fully operational and tested system, which is enriched with dedicated modules to E2LP Mother Board, which provide real-time remote control, monitoring and programming. Below we show the main advantages of the system: • The final laboratory exercise on the web has sections (tabs) to enable user to have the full experience of working on the laboratory exercise.

These are Digital System Design course exercises, which aim is to control Switches, JOY Push Buttons, LEDs, LCD output in the front panel of the E2LP board as well as RS-232 port are available for remote operations.

• Advance booking system, which enables to reserve a time slot for individual remotely tests of the solution for a given exercise. Booking functionality enables to access up to 4 remote E2LP boards.

• The fast bit file loading module enables remote configuration and immediate respond of the successful E2LP board configuration, without a requirement for users to have a specialized Xilinx software to do it.

• The user friendly Graphical User Interface of the Front Panel, which reflects to the same panel on the real E2LP board, enables user to monitor and control remotely each switch, button, LED and LCD output. The GUI is enriched with the checking correctness of the solution module, which compares the students solution with a master, created by the teacher.

• Automatic verification module, which is based on regular expressions, checks the correctness of the users solution. The pattern for solution is prepared by the teacher or course creator. After comparison the user is informed visually about correctness of his solution.

• The ‘Discussion on results’ functionality module consist the output information from check correctness solution module, by showing the log records output from the E2LP board Front Panel and enable Teacher and user to exchange information about given exercise.

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Fig. 2. Multi-board selection

6. New Challenges in RL Development 6.1 Modularity of Moodle Platform

The use of Moodle open source platform as an user-friendly interface for virtual laboratories provides benefits for every participant in the developer-teacher-student chain. It allows for high level integration of hardware and software with highly configurable control procedures, in the same time supporting the teacher with a variety of administrative tools and the student with an accessible control panel. During the development of the E2LP project the modularity of Moodle platform was used to create custom blocks and modules [21], dedicated for the virtual laboratory. Some of the ideas incorporated an innovative module for the verification of student’s solution and the implementation of a multi-device system.

6.2. Multi-device Laboratory

The progress in the field of remote teaching provides easier access to the laboratory for greater number of students. That, being a great advantage of such systems, also generates problems connected with occupation of the physical components. Even though the platform may handle many students simultaneously, the hardware itself is restricted to a one-to-one work.

Therefore, it is crucial to develop a multi-device system, incorporating multiple hardware components into a unified platform based on Moodle.

The control of physical signals of the hardware (in the case of E2LP an FPGA Embedded System Board) operates on the basis of an active LabVIEW web service. Hence, for a multi-board approach the main web service was replicated, remaining in the exact same form as the original, only with a varying address depending on the device. That way, establishing a connection (from the Moodle layer) with a specific path allows for control of a certain device. During the project, a specific schematic was introduced: http://server_address/Device1 http://server_address/Device2 The main advantage of such approach is the possibility of quick and easy change of the number of curArticles

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Fig.3. E2LP RL board front panel rently available devices without the need of modification of the universal web service.

During the reservation of a time slot, students choose from the available boards (the system checks their availability for current time). On the submission of the reservation form, the Moodle database is updated with an information containing all the necessary data, including the device number. $enroll->enroll_date = $ctime; $enroll->course_id = (int)$data[‘course_id’]; $enroll->board = (int)$data[‘board_select’]; $DB->insert_record(‘wa_enroll_calendar’, $enroll, true); It is further used to dynamically generate correct user’s interface after logging in for a particular course, taking the device number into account while selecting appropriate web service path. That way any possibility of conflict is eliminated, as the user can only gain access to the board he had made a reservation for.

Again, a connection with the database is established, retrieving the device number and appending it to the web service URL, without the need of any user’s action. The default webservice_url for any activity is in the form http://server_address/Board, so if the device id is for example 2, the resulting URL is set as http:// server_address/Board2. $result = wafrontpanel_get_board_id(); $device_id=(int)($result->board); $modified_url = preg_replace(‘/ Board/’,’Board’.$device_id,$activity->webservice_url); That way the database values are translated into a dynamic creation of URL. Moreover, the default path is provided by the teacher via Moodle administration tools, so if the web service address is changed, no modification of the source code is needed. Below is shown the interface of the E2LP platform after the introduction of the multi-device approach.

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As the project is focused on university teaching methods, it needs an implementation of student’s progress verification. Moreover, it should create the possibility of revision of student’s performance in order to provide formative assessment from the teacher to maximize teaching efficiency [22]. Moodle platform’s custom modules development allows to create a dynamic comparison of predefined patterns, Articles

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dedicated for the hardware platform the Moodle interface is integrated with. This approach is open for universal applications, as the teachers are given a simple schematic of answer pattern definition that can be applied in any similar e-learning platform.

The student’s solution is also recorded in the database, for the needs of further revision.

The procedure of checking the students’ solutions is based on the definition of system’s required reactions for an occurrence of a certain user’s action. It operates on the basis of a dynamic comparison of a predefined answer patterns (generated by the teacher) and the log of user’s actions generated while fulfilling an exercise. This method also allows for personalized interaction between the user and the platform (via LCD interface). It is one step ahead of previous solutions suggested for student progress validation, that were based on a one-to-one comparison of the answer patterns, strictly defining the check solution procedure. This time the user is free to test his program in any order and configuration, as many times as he wants, and only the crucial parts of the log will be extracted and compared with the answer pattern. This way the student’s responsibility for the correctness of the procedure is removed and he is able to fully focus on the subject of the exercise.

The implemented method comprises two main fields: • Log checking – monitoring the whole progress of the exercise, and the system’s responses for certain actions (e.g. on Button 1 flash diodes three times).

• LCD checking – comparison of the example text and the text that the student managed to display on the LCD.

6.4. Log Checking

The previous version of the system was based on a one-to-one comparison between the log and the predefined answer pattern. This approach caused two main disadvantages: • In order to achieve a positive grade, student’s result had to be exactly the same as the answer pattern.

• The correct solution could have been marked as wrong if the student had performed required actions in a different order or the status of insignificant ports had been different (e.g. the position of switches when only LEDs are used).

In order to surpass those disadvantages and increase the algorithm’s efficiency an action – result approach was implemented. Instead of a one-to-one comparison, after the occurrence of a specific action (e.g. pressed button), an appropriate result is being searched for (e.g. LED sequence). It bases on a synergic combination of regular expressions and array data manipulation.

Sample log text: L E D : 1 1 1 0 0 0 0 0 , S W: 0 0 0 0 0 0 0 0 , J OY: 1 1 1 1 1 , L C D : H e l l o , R E S E T: 1 , P O W E R : 0 , E X T E R NA L : 0 L E D : 0 1 1 0 0 0 0 0 , S W : 0 0 0 0 0 0 0 0 , J O Y: 1 1 1 1 1 , L CD:Hello,RESET:1,POWER:0,EXTERNAL:0

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L E D : 0 0 1 0 0 0 0 0 , S W : 0 0 0 0 0 0 0 0 , J O Y: 1 1 1 1 1 , L CD:Hello,RESET:1,POWER:0,EXTERNAL:0 LED:00000000,SW:00000000,JOY:11101,LCD:Hello, RESET:1,POWER:0,EXTERNAL:0 Code responsible for translating the text into an approachable form: //change all \r\n or \r or \n to commas $log = preg_replace(‘/[\r\n]|\r|\n/’, ‘,’, $log); //change multiple commas to a single one $log = preg_replace(‘/\,+/’, ‘,’, $log); $log_split = array_reverse(preg_split(‘/(,LED:|,SW:|, JOY:|,LCD:|,RESET:|,POWER:|,EXTERNAL:|,BOX:|[\r\ n])/’,$log)); The use of regex has three main purposes here: cleanup of the log text (from unnecessary line breaks, carriage returns and commas), identification of log parts and division into blocks/arrays according to their content [23]. The position BOX is appended to the log as the input from the LCD pattern example.

Next step is the implementation of array manipulation in order to isolate only the required parts of the log for comparison. For example, stepping to the same parameter value (e.g. LED) in the next line is executed as incrementation of the array index by 7 (as there are seven elements in each line). That way, for testing LED response as a reaction for JOY input, only the highlighted sequence will be searched for in the log, ignoring other parameters: L E D : 1 1 1 0 0 0 0 0 , S W : 0 0 0 0 0 0 0 0 , J O Y: 1 1 1 1 1,LCD:Hello,RESET:1,POWER:0,EXTERNAL:0 L E D : 0 1 1 0 0 0 0 0 , S W : 0 0 0 0 0 0 0 0 , J O Y: 1 1 1 1 1 , LCD:Hello,RESET:1,POWER:0,EXTERNAL:0 L E D : 0 0 1 0 0 0 0 0 , S W : 0 0 0 0 0 0 0 0 , J O Y: 1 1 1 1 1 , LCD:Hello,RESET:1,POWER:0,EXTERNAL:0 LED:00000000,SW:00000000,JOY:11101,LCD:Hello, RESET:1,POWER:0,EXTERNAL:0 In the administration panel, the teacher can define blocks of such patterns, defining monitored inputs and outputs, hence allowing the user to freely test all functions of his program in any configuration, as many times as he wants. The introduced algorithm is highly efficient – redundant and insignificant parts of the user’s performance are ignored and the checking procedure is finished after the first occurrence of required action-reaction scheme .

Due to the parallel nature of operations in the FPGA technology the teacher has to define a parameter defining the expected response time. If it is shorter than the processing of a web request, the response will be shown in the same line as its trigger.

Therefore, the course creator defines if the algorithm should begin in the trigger line or in the next one. As a result, the platform becomes adjustable for parallel and serial operating systems (or systems with varying response time for different operations).

6.5. LCD Checking

The LCD pattern text is appended to the log and processed as such. The user is required to fill in the text that is expected to appear on the LCD, and the algorithm checks if that succeeded. That approach creates the possibility of applying custom text for each user

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making the teaching process personal and more interactive (e.g. displaying each student’s name on the LCD).

In order to prevent cheating, double verification of the LCD exercises is supported: Algorithmic comparison of user input with the displayed text.

The solution is saved to the Moodle database, so that the teacher may check if it meets the exercise passing criteria.

The introduction of the LCD module opens way for further development of this interactive platform.

It would be possible to use a predefined LCD text pattern to fully automate the checking procedure, yet the exercises would not be personalized that way. Another idea is to integrate the LCD text pattern with the Moodle user personal information (e.g. name, username or email).

7. Evaluation and Discussion on Results

System integration of many hardware and software modules in embedded systems makes testing and verification the key success factors in embedded solutions [24]. Therefore, testing and verification methodology reflects as a big challenge for embedded engineering education.

Evaluation and assessment of engineering education is an important point closing the improvement loop of education process. Here, a dilemma is what and how to measure – achieved knowledge level, usability of acquired knowledge, soft capabilities (team work, project management) as well as capability for innovations. Some approaches for education quality evaluation and assessment are proposed in [25], [26] and [27].

The E2LP has been evaluated at 8 universities in running study programs. Four of them are project partners and the other four universities outside the project have also adopted this learning platform. Results in the first two years of usage have been evaluated using the established evaluation tools showing visible improvements of study efficiency. Development and evaluation results are published at conferences and in the journal as well as summarized in a book of Springer special edition. Promotions have been organized at two fairs and in two externally organized workshops.

The established E2LP consortium prepared a framework for promotion and exploitation of the developed learning platform for embedded engineering. A preparation of a follow-up project is in progress addressing necessary updates and extensions for a broader usage of the E2LP, from introduction courses at high schools up to expert courses for lifelong engineering education.

The E2LP project involved the development, implementation and evaluation of an advanced learning platform for computers and embedded systems engineering education. Beyond the development of hardware and software, the project also included the development of an inventory of 65 experiments and lab assignments for students at three levels: exercises, problems and projects.

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This chapter presents the preliminary student’s practical validation of developed E2LP Remote Laboratory, which was performed at Warsaw University of Technology (WUT) at Mechatronics Faculty during the evaluation stage of the E2LP project. The main purpose of performed evaluation was showing to students system capabilities and engaging them in contribution in testing the developed RL platform as an additional value to study programs in WUT. A study was completed under “Intelligent Measurement Devices” – a new course in the Electronic Measurement Systems specialization on engineering degree.

During this course students gain comprehensive skills: knowledge about the intelligent sensors, measurements devices and systems operation rules, competence in signal processing and the methodology of novel apparatus construction.

One of the main purpose of the evaluation and the E2LP project was the enrich aforementioned skills with understanding the different digital logic circuits and their operation, implementation of Boolean functions using digital logic circuits, understand the Xilinx ISE software environment and tools as well as understand VHDL description of digital logic circuits.

To get the summative feedback from students towards presented system, the quantitative on-line analysis was conducted, which was prepared by other E2LP project partner Ben-Gurion University of the Negev, based on Computer System Usability Questionnaire (CSUQ) [28]. It includes many aspects that refer to the usage RL platform and its and user acceptance. The questionnaire results are presented and discussed by Kloda et al. [29].

Our E2LP Remote Laboratory is innovative learning platform, which easy customizes to any course needs and doesn’t require any cost for teachers, namely they don’t need any specialize software and hardware. Since the students were beginners in VHDL language and Xilinx environments we prepared separate set of easy exercises, which were in line with the curriculum of the subject.

Regarding the e-learning platform for FPGA, the users confirm, that proposed solution are powerful and efficiently improved by using RL. In this sense, students declare they somewhat agree with the idea that remote work is possible without the need to work with the real board.

Considering the aspects related to the user graphic interface, users proof that it is easy to use and its readability increase significantly. Moreover students stress that they are able to quick learning.

The biggest encountered problem was connected with using Xilinx ISE software, which was source of error messages and poor programming experience. The low mark might be a reason of very low student’s initial knowledge level of FPGA systems.

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This paper has discussed all the features provided

by E2LP RL for its implementation and deployment in embedded systems engineering education along with feedback from the universities that had deployed it in their learning curricula Articles

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Results presented in the paper confirms that introduction of RL into curriculum and new learning model is challenging in the education of engineers in embedded systems. Student, who has never had any practice with Xilinx ISE environment and any FPGA board configuration needs really precise procedure what to do in current exercise.

Proposed solutions based on integrated together Remote Laboratory components and e-learning Moodle Platform enable student to acquire desired knowledge about digital systems and significantly support learning process. During the evaluation it occurred that remote operations through real-time experiments stimulate the students curiosity and productivity.

In summary, we believe that the evaluation methodology and tools developed in this research were not just an important ingredient in the E2LP project, but could also contribute a meaningful layer to the literature and practice of engineering education, especially in the context of developing and evaluating new technology-based curricula.

ACKNOWLEDGEMENTS

E2LP Remote Laboratory development were performed in the Industrial Research Institute for Automation and Measurements PIAP. E2LP Remote Laboratory evaluation were made in the Institute of Metrology and Biomedical Engineering, Warsaw University of Technology.

AUTHORS

Rafał Kłoda*– Industrial Research Institute for Automation and Measurements PIAP, al. Jerozolimskie 202, 02-486, Warsaw, Poland, rkloda@piap.pl.

Jan Piwiński– Industrial Research Institute for Automation and Measurements PIAP, al. Jerozolimskie 202, 02-486, Warsaw, Poland, jpiwinski@piap.pl.

Kacper Kurzejamski – Institute of Metrology and Biomedical Engineering, Warsaw University of Technology, Św. A. Boboli 8, 02-525 Warsaw, Poland, kacp. erk@op.pl. *Corresponding author

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Mariam Al-Sagban, Rached Dhaouadi DOI: 10.14313/JAMRIS_2-2016/17 Abstract: This paper presents a novel reac ve naviga on algorithm for wheeled mobile robots under non-holonomic constraints and in unknown environments. Two techniques are proposed: a geometrical based technique and a neural network based technique. The mobile robot travels to a pre-defined goal posi on safely and efficiently without any prior map of the environment by modulating its steering angle and turning radius. The dimensions and shape of the robot are incorporated to determine the set of all possible collision-free steering angles.

The algorithm then selects the best steering angle candidate. In the geometrical naviga on technique, a safe turning radius is computed based on an equa on derived from the geometry of the problem. On the other hand, the neural-based technique aims to generate an op mized trajectory by using a user-defined objec ve funcon which minimizes the traveled distance to the goal posi on while avoiding obstacles. The experimental results demonstrate that the algorithms are capable of driving the robot safely across a variety of indoor environments.

Keywords: reac ve naviga on, obstacle avoidance, autonomous ground robots, recurrent neural networks

1. Introduc on Mobile robots have rapidly evolved over the past years to encompass a wide spectrum of applications: Robots are assisting in driving vehicles, aiding in medical tasks, and taking charge in hazardous rescue missions. Autonomous navigation is a key feature in all of these applications. It deals with the problem of navigating to a target location while avoiding collision with obstacles that may be present in the environment. One approach to autonomous navigation is model-based approach. It uses a model of the environment to generate a safe path to the target location [11]. Classical methods for this type include: Road maps [7], cell decomposition, and potential ields [8].

Although these methods may produce ef icient paths, a global and accurate map of the environment is not available when the environment is unknown or is dynamic. Hence, model based methods are used only in arti icially controlled environment. A complementary approach to autonomous navigation is obstacle avoidance (also known as reactive navigation). No prior information is required about the environment. Instead, obstacles are discovered in real time while the robot is executing its mission. The main challenges in developing such methods are: computational complexity, sen64

sors uncertainties, robot geometrical shape, and kinematic and dynamic constraints. In addition, because a global map of the environment is not available the robot may produce inef icient paths or converge to a local minimum (trap situation) [5].

Neural Networks-based techniques have been proposed in the literature to solve the motion problem. In [6] the path planning problem is viewed as two subproblems: ind space and ind path. Two neural networks connected in cascade are used. The irst neural network is responsible for inding the C-free space which is the set of all possible robot con igurations that avoids collision with obstacles. The second neural network guides the robot through the free space segments to the target location. The navigational methodology used in [3] is a Probabilistic Neural Network (PNN). This type of network facilitates the training process, allows a faster time of response and has a low computational cost. The output of the network represents the steering direction which takes three forms: forward, right turn, and left turn. The autonomous navigation of a mobile robot is considered as a classi ication problem. The motion of the robot and the sensorial information are the patterns to be classi ied.

The obstacle avoidance problem in [10] is divided into three subproblems: General obstacle avoidance, corridor and wall following, and passing through a door.

A separate neural network is designed for each one of those subproblems. The algorithm uses the accessible space as the input to the neural network. The output of the network is the steering angle and velocities. The number of neurons in the hidden layer is determined by using a Bayesian framework which computes the evidence of a set of neural networks with different hidden nodes. The accurate usable accessible space is computed by incorporating the wheel chair dimensions, laser information, and encoder data. The algorithm choice is interesting because it is able to optimize the number of hidden neurons for a given problem. However, the algorithm is considered incomplete from the autonomous navigation perspective because it does not provide a framework that is able to recognize the relevant situation and selects the corresponding output. The authors divide the obstacle avoidance problem into sub-problems because they claim that a single neural network could not provide the desired performance. However, It is not clear whether it is necessary to separate the avoiding obstacle and passing through a door tasks since they do not produce a conlict in training the network. A training con lict is created when the same input pattern is mapped to two

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different output values. Another arti icial intelligence technique is introduced in [4]. The path planning is done using particle swarm optimization. The desired path is designed to avoid obstacles while maintaining a smooth continuous path. The path is expressed as a 5th order polynomial. Two of the polynomial coeficients are estimated via particle swarm optimization such that obstacles are avoided while the other coeficients are chosen such that a smooth path is generated. The particle swarm optimization is required to estimate the best polynomial coef icients as well as other parameters called the critical points. The algorithm was veri ied in simulation. However, nonholonomic constrains are not considered. Overall, arti icial neural networks provides an interesting platform for the obstacle avoidance problem because of their generalization ability, ability to learn from examples and the ability to extract temporal dependencies.

In this paper, we present an obstacle avoidance technique based on recurrent neural networks that takes into consideration the kinematic constraints of differential drive robots. While the common trend is to use more than one neural network, we use a single dynamic neural network. The obstacle avoidance problem is a dynamic problem that should be solved using dynamic methods. In order to guarantee optimal convergence, we require the neural network learning environment to satisfy certain conditions that are derived using Lyapunov stability method. Also, the earlier presented neural networks techniques generated a dataset by manually driving the robot across different scenarios. We automate the process by generating a sub-optimal dataset using a computer algorithm. For optimum performance, the neural network is trained using the real-time recurrent learning algorithm along with a customized objective function to equip the robot with the capability of improving its learning while in motion.

2.1. Geometrical Naviga on Algorithm The main steps in the obstacles avoidance algorithm are: Identify a reference steering angle, model the environment, compute the con iguration space, and select the desired steering angle and radius of curvature [1]. The reference steering angle, đ?&#x203A;žđ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C; represents the steering angle that the robot takes in the absence of obstacles. It is an intermediate reference angle that will later help us ind đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; .

In this paper, we consider a mobile robot with a differential drive con iguration as shown in Fig. 1. Let the robot con iguration be: (1)

where (đ?&#x2018;Ľđ?&#x2018;&#x; , đ?&#x2018;Śđ?&#x2018;&#x; ) is the position of the robot in the đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;Ś plane, and đ?&#x153;&#x192;đ?&#x2018;&#x; â&#x2C6;&#x2C6; [0, 2đ?&#x153;&#x2039;) is the robotâ&#x20AC;&#x2122;s orientation with respect to the x axis.

Let the target con iguration be: đ?&#x2018;&#x17E;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą = (đ?&#x2018;Ľđ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą , đ?&#x153;&#x192;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą ).

(2)

2016

Fig. 1. Target and robot coordinates Let đ?&#x2018;˘â&#x192;&#x2014;đ?&#x2018;&#x2019; be the vector connecting the robot reference point to the target location. The phase angle of đ?&#x2018;˘â&#x192;&#x2014;đ?&#x2018;&#x2019; is given by: đ?&#x203A;ź = arctan

đ?&#x2018;Śđ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x; . đ?&#x2018;Ľđ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x;

(3)

To correct the error in orientation, the robot should turn by a reference steering angle đ?&#x203A;žđ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C; . The instantaneous turning radius đ?&#x2018;&#x;đ?&#x2018;? can be evaluated by: đ?&#x2018;&#x;đ?&#x2018;? = đ??ż

đ?&#x2018;Łđ?&#x2018;&#x; + đ?&#x2018;Łđ?&#x2018;&#x2122; , đ?&#x2018;Łđ?&#x2018;&#x; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2122;

(4)

where đ?&#x2018;Łđ??ż and đ?&#x2018;Łđ?&#x2018;&#x2026; are the translational velocities of the left and right wheels and đ??ż is the distance between the wheels.

The second main step is to model the surrounding environment. A partial polar map of the workspace is constructed in the robot local frame. The robot is equipped with a laser range inder that is programmed to scan the 200 â&#x2C6;&#x2DC; front view of the robot in 20 sectors, with 10 â&#x2C6;&#x2DC; angular resolution. The sensor returns a set of points: đ?&#x2019;Ť(đ?&#x2018;&#x17E;(đ?&#x2018;Ąđ?&#x2018;&#x2013; )) = {đ?&#x2018;?1 , đ?&#x2018;?2 , ..., đ?&#x2018;?đ?&#x2018;&#x2014; , ..., đ?&#x2018;?20 }.

2. Autonomous Naviga on Methodology

đ?&#x2018;&#x17E;đ?&#x2018;&#x; = (đ?&#x2018;Ľđ?&#x2018;&#x; , đ?&#x2018;Śđ?&#x2018;&#x; , đ?&#x153;&#x192;đ?&#x2018;&#x; ),

Nâ&#x2C6;&#x2DC; 2

(5)

A point đ?&#x2018;?đ?&#x2018;&#x2014; is expressed by a pair (đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; , đ?&#x203A;˝đ?&#x2018;&#x2014; ) where đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; is the distance between the robot and the obstacle at sector đ?&#x2018;&#x2014;. đ?&#x203A;˝đ?&#x2018;&#x2014; is the orientation of the đ?&#x2018;&#x2014;đ?&#x2018;Ąâ&#x201E;&#x17D; sector, đ?&#x2018;&#x2020;đ?&#x2018;&#x2014; , with respect to the local x axis. The subset of workspace obstacles seen at con iguration đ?&#x2018;&#x17E;(đ?&#x2018;Ąđ?&#x2018;&#x2013; ) is identi ied by applying a threshold on đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; : đ?&#x2019;Ş(đ?&#x2018;&#x17E;(đ?&#x2018;Ąđ?&#x2018;&#x2013; )) = {đ?&#x2018;?đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2019;Ť(đ?&#x2018;&#x17E;(đ?&#x2018;Ąđ?&#x2018;&#x2013; ))|đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; â&#x2030;¤ đ?&#x2018;&#x2026;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x201C;đ?&#x2018;&#x2019; }.

(6)

The third main step of the algorithm is to compute the con iguration space đ??śđ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą . First, consider the case where only a point obstacle exists in the workspace: đ?&#x2019;Ş = {đ?&#x2018;?đ?&#x2018;&#x2014; }. To ind đ?&#x2019;&#x17E;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą , we slide the robot around đ?&#x2018;?đ?&#x2018;&#x2014; and trace the con igurations it went through, as illustrated in Fig. 2. Hence, đ?&#x2019;&#x17E;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą is enclosed by a circle đ??śđ?&#x2018;&#x2014; of radius đ?&#x2018;&#x2026; and center đ??źđ?&#x2018;&#x2014; = (đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ľ , đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ś ): đ?&#x2019;&#x17E;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą = {đ?&#x2018;&#x17E; â&#x2C6;&#x2C6; đ?&#x2019;&#x17E;|(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ľ )2 + (đ?&#x2018;Ś â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ś )2 â&#x2030;¤ đ?&#x2018;&#x2026;2 }, (7) đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ľ = đ?&#x2018;&#x2026; + đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; cos đ?&#x203A;˝đ?&#x2018;&#x2014; , đ??źđ?&#x2018;&#x2014;,đ?&#x2018;Ś = đ?&#x2018;&#x2018;đ?&#x2018;&#x2014; sin đ?&#x203A;˝đ?&#x2018;&#x2014; , â&#x2C6;&#x2019;100â&#x2C6;&#x2DC; â&#x2030;¤ đ?&#x203A;˝đ?&#x2018;&#x2014; â&#x2030;¤ 100â&#x2C6;&#x2DC; .

65

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Fig. 2. C-space Algorithm Next, we ind đ??żđ?&#x2018;&#x2013; which is the radial distance between the robot and the boundary of đ?&#x2019;&#x17E;đ?&#x2018;&#x2014; at angle đ?&#x203A;˝đ?&#x2018;&#x2013; :

Fig. 3. Turning Radius Selec on

đ??żđ?&#x2018;&#x2013; = min{đ?&#x153;&#x152;đ?&#x2018;&#x2014; cos (đ?&#x203A;˝đ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x153;&#x2122;đ?&#x2018;&#x2014; ) Âą â&#x2C6;&#x161;đ?&#x2018;&#x2026;2 â&#x2C6;&#x2019; đ?&#x153;&#x152;đ?&#x2018;&#x2014;2 sin2 (đ?&#x203A;˝đ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x153;&#x2122;đ?&#x2018;&#x2014; )},

where đ?&#x2018;&#x17D; is the distance between the robot reference point and the reference point đ?&#x2018;&#x161;. De ine đ??żđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; as the distance of the nearest obstacle point that exists anywhere between đ?&#x2018;&#x2020;0 and đ?&#x2018;&#x2020;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; . The turning radius đ?&#x2018;&#x;đ?&#x2018;? is chosen such that the trajectory passes through the point (đ??żđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; , đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; ) as shown in Fig. 3. From the geometry, the turning radius is obtained as:

đ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2030;¤ đ?&#x203A;˝đ?&#x2018;&#x2013; â&#x2030;¤ đ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ , đ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; = min{đ?&#x153;&#x2122;đ?&#x2018;&#x2014; Âą sin ,đ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ = max{đ?&#x153;&#x2122;đ?&#x2018;&#x2014; Âą sin

đ?&#x2018;&#x2026; } đ?&#x153;&#x152;đ?&#x2018;&#x2014;

đ?&#x2018;&#x2026; } đ?&#x153;&#x152;đ?&#x2018;&#x2014; (8)

If đ?&#x2019;Ş includes đ?&#x2018;&#x161; obstacle points, then đ?&#x2019;&#x17E;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą = â&#x2039;&#x192; đ?&#x2019;&#x17E;đ?&#x2018;&#x2014; . Now, to select the desired steering angle,

1â&#x2030;¤đ?&#x2018;&#x2014;â&#x2030;¤đ?&#x2018;&#x161;

the sectors in đ?&#x2019;&#x17E; are classi ied as free or occupied. The đ?&#x2018;&#x2014;đ?&#x2018;Ąâ&#x201E;&#x17D; sector đ?&#x2018;&#x2020;đ?&#x2018;&#x2014; is occupied if đ??żđ?&#x2018;&#x2014; â&#x2030;¤ đ?&#x2018;&#x2026;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x201C;đ?&#x2018;&#x2019; ; otherwise it is free. Adjacent free sectors are grouped together to form gaps. Then, the gaps are classi ied as wide, medium, and narrow. Every gap edge is a candidate for the desired steering angle. A cost function is de ined to aid in the selection process: đ??śđ?&#x2018;&#x153;đ?&#x2018; đ?&#x2018;Ą = đ?&#x2018;?1 (đ?&#x203A;žđ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C; â&#x2C6;&#x2019; đ?&#x203A;˝đ?&#x2018;&#x2014; ) + đ?&#x2018;?2 đ?&#x203A;˝đ?&#x2018;&#x2014; .

(9)

The irst term in equation (9) represents how close the desired steering direction is to the goal location and the second term represents how close the current steering direction is to the current robot heading.The angle of the gap edge that has the minimum cost is selected as đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; . The candidate of the desired steering angle is irst considered within the wide gaps. If none is available the search is performed within the medium gaps. The inal choice is the narrow gaps.

The inal step of the algorithm is to determine the desired radius of curvature of the robot trajectory.

Due to the kinematic constraints, the robot can not achieve the desired steering angle instantly. Instead, the robot follows a circular arc if the wheelsâ&#x20AC;&#x2122; velocities are constant. The path from the initial con iguration to the inal con iguration may intersect with đ?&#x2019;&#x17E;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;Ą (đ?&#x2018;&#x17E;(đ?&#x2018;Ąđ?&#x2018;&#x2013; )) causing a collision. Therefore, using a radius of curvature that is a function of the surrounding obstacles is safer than using a ixed radius for all obstacle scenarios. Let đ?&#x2018;&#x2020;0 be the sector that contains the local x axis and let đ?&#x2018;&#x2020;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; be the sector that contains the desired steering angle đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; . Let đ??żđ?&#x2018;&#x161; đ?&#x2018;&#x2014; be the distance between the obstacle point đ?&#x2018;&#x153;đ?&#x2018;&#x2014; and the reference point đ?&#x2018;&#x161; shown in Fig. 3. The relationship between đ??żđ?&#x2018;&#x2014; and đ??żđ?&#x2018;&#x161; đ?&#x2018;&#x2014; is given by: 2 2 đ??żđ?&#x2018;&#x161; đ?&#x2018;&#x2014; = â&#x2C6;&#x161;(đ??żđ?&#x2018;&#x2014; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x203A;˝đ?&#x2018;&#x2014; + đ?&#x2018;&#x17D;) + (đ??żđ?&#x2018;&#x2014; đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x203A;˝đ?&#x2018;&#x2014; ) ,

66

(10)

đ?&#x2018;&#x;đ?&#x2018;? =

đ??żđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; .

2đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; )

(11)

To include a safety buffer, the turning radius is designed to pass through the point (đ??żđ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x201C;đ?&#x2018;&#x2019;2 , đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; ) instead. Also, the turning radius đ?&#x2018;&#x;đ?&#x2018;? saturates if it is greater than a threshold value đ?&#x2018;&#x;đ?&#x2018;&#x2122;đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019; .

2.2. Neural Naviga on Algorithm A neural network is proposed to work in collaboration with the reactive navigation algorithm developed earlier to optimize the navigational capabilities of the overall system. While the navigational algorithm avoids obstacles, it does not contain any constraints on the length of the path to the target position. The neural network is incorporated into the system to minimize the length of the path taken. The neural network selects the optimum trajectory based on the obstacles information, target con iguration, and the turning radius proposed by the navigational algorithm. The neural network outputs the most promising turning radius of the robot trajectory.

This paper uses a Diagonal Recurrent Neural Network (DRNN) which despite its simple structure, has the ability to adapt the learning rates such that the network convergence is guaranteed. Fig. 4 depicts the network architecture. For the neural network to achieve this objective, it needs to overcome few challenges. One of those challenges is that the â&#x20AC;&#x2DC;correctâ&#x20AC;&#x2122; turning radius is not available. Hence, the training cannot be conducted in a supervised manner where a dataset is available to help the network form a map between the input data and the desired value. To overcome this obstacle, a hybird training scheme is proposed. First, the neural network will receive supervised training based on a sub-optimal dataset. Second, the neural network will be trained on-line to adjust its

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Fig. 5. Phase 1 Training Fig. 4. Neural Network Architecture weights in order to produce an optimum value based on an evaluation function. In the irst training phase, the network is trained to map the input values to a desired value. The dataset is generated using the reactive navigation algorithm. The training is based on backpropogation and is done off-line. The purpose of this training phase is to provide an adequate initial set of weights for the next phase as opposed of starting phase 2 from random variables. In the second training phase, the optimum turning radius is unknown.

However, the radius taken by the robot can be evaluated. The network receieves feedback about its performance based on an evaluation function described as: 1 2 1 đ?&#x2018;&#x2019; = (đ?&#x2018;&#x2019;2đ?&#x2018;Ľ + đ?&#x2018;&#x2019;2đ?&#x2018;Ś ) 2 2 đ?&#x2018;&#x2019;đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;Ľ, đ?&#x2018;&#x2019;đ?&#x2018;Ś = đ?&#x2018;Śđ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;Ś đ??˝=

2.3. Adap ve learning and stability analysis (12)

where (đ?&#x2018;Ľđ?&#x2018;Ą , đ?&#x2018;Śđ?&#x2018;Ą ) is the target position. The derivative of the evaluation function with respect to the weights is given by: đ?&#x153;&#x2022;đ??˝ đ?&#x153;&#x2022;đ?&#x2018;&#x; = â&#x2C6;&#x2019;(đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś ) đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(13)

where đ??˝đ?&#x2018;Ľ and đ??˝đ?&#x2018;Ś are the sensitivities of the system and are given by: đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;Ľ(đ?&#x2018;Ą â&#x2C6;&#x2019; 1) = đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x201D;đ?&#x2018;&#x203A;( ) đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x2018;&#x;(đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;&#x;(đ?&#x2018;Ą â&#x2C6;&#x2019; 1) đ?&#x2018;Ś(đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;Ś(đ?&#x2018;Ą â&#x2C6;&#x2019; 1) đ?&#x153;&#x2022;đ?&#x2018;Ś = đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x201D;đ?&#x2018;&#x203A;( ) đ??˝đ?&#x2018;Ś = đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x2018;&#x;(đ?&#x2018;Ą) â&#x2C6;&#x2019; đ?&#x2018;&#x;(đ?&#x2018;Ą â&#x2C6;&#x2019; 1)

đ??˝đ?&#x2018;Ľ =

(14)

đ?&#x153;&#x2022;đ??˝ đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(15)

(16)

Subsituting eq.12 into 16 gives: Î&#x201D;đ?&#x2018;&#x160; = â&#x2C6;&#x2019;đ?&#x153;&#x201A;đ?&#x2018;&#x2019;

đ?&#x153;&#x2022;đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;&#x160;

The training phases are shown in Fig. 5 and 6.

In order to maintain the system stability, we put a restriction on the learning rate using Lyapunov theorem. De ine the lyapunov function đ?&#x2018;&#x2030; (đ?&#x2018;&#x2DC;) = 21 đ?&#x2018;&#x2019;2 (đ?&#x2018;&#x2DC;) â&#x2030;Ľ 0. To prove system stability, we need to show that Î&#x201D;đ?&#x2018;&#x2030; = đ?&#x2018;&#x2030; (đ?&#x2018;&#x2DC; + 1) â&#x2C6;&#x2019; đ?&#x2018;&#x2030; (đ?&#x2018;&#x2DC;) â&#x2030;¤ 0. Hence, Î&#x201D;đ?&#x2018;&#x2030; =

1 2 (đ?&#x2018;&#x2019; (đ?&#x2018;&#x2DC; + 1) â&#x2C6;&#x2019; đ?&#x2018;&#x2019;2 (đ?&#x2018;&#x2DC;)) 2

(18)

Subsitute đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC; + 1) = đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) + Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) in eq.18: 1 Î&#x201D;đ?&#x2018;&#x2030; = đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;)Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) + Î&#x201D;2 đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) 2 1 = Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;)(đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) + Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;)) 2

(19) (20)

Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) is given by:

đ?&#x153;&#x2022;đ?&#x2018;&#x; is estimated online using Real Time Recurrent đ?&#x153;&#x2022;đ?&#x2018;&#x160; Learning (RTRL) [1]. The neural network weights are updated according to the gradient descent technique: Î&#x201D;đ?&#x2018;&#x160; = â&#x2C6;&#x2019;đ?&#x153;&#x201A;

Fig. 6. Phase 2 Training

(17)

Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) = [

đ?&#x153;&#x2022;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) đ?&#x2018;&#x2021; ] Î&#x201D;đ?&#x2018;&#x160; đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(21)

Subsituting eq.17 in 21 gives: Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) = â&#x2C6;&#x2019;đ?&#x153;&#x201A;đ?&#x2018;&#x2019;â&#x2C6;Ľ In order to ind

đ?&#x153;&#x2022;đ?&#x2018;&#x2019; 2 â&#x2C6;Ľ đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(22)

đ?&#x153;&#x2022;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) ,the chain rule is used: đ?&#x153;&#x2022;đ?&#x2018;&#x160;

đ?&#x153;&#x2022;đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x153;&#x2022;đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x153;&#x2022;đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;Ś = [ + ] đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x2018;&#x2019;đ?&#x2018;Ś đ?&#x153;&#x2022;đ?&#x2018;&#x2019; đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;&#x2019; = â&#x2C6;&#x2019; đ?&#x2018;Ľ, =â&#x2C6;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x2018;&#x2019;

(23)

(24) 67

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Hence eq 23 becomes: đ?&#x153;&#x2022;đ?&#x2018;&#x; đ?&#x2018;&#x2019;1 đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;2 đ??˝đ?&#x2018;Ś đ?&#x153;&#x2022;đ?&#x2018;&#x2019; =â&#x2C6;&#x2019; ( ) đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x2018;&#x2019;

(25)

Substituting 25 in 21, gives: đ?&#x153;&#x201A; đ?&#x153;&#x2022;đ?&#x2018;&#x; 2 â&#x2C6;Ľ (đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś )2 Î&#x201D;đ?&#x2018;&#x2019;(đ?&#x2018;&#x2DC;) = â&#x2C6;&#x2019; â&#x2C6;Ľ đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(26)

Substituting 26 in 20: đ?&#x153;&#x2022;đ?&#x2018;&#x; 2 â&#x2C6;Ľ (đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś )2 â&#x2C6;&#x2014; đ?&#x153;&#x2022;đ?&#x2018;&#x160; đ?&#x153;&#x201A; đ?&#x153;&#x2022;đ?&#x2018;&#x; 2 â&#x2C6;Ľ (đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś )2 ) (â&#x2C6;&#x2019;1 + 2 â&#x2C6;Ľ 2đ?&#x2018;&#x2019; đ?&#x153;&#x2022;đ?&#x2018;&#x160;

Î&#x201D;đ?&#x2018;&#x2030; =đ?&#x153;&#x201A;â&#x2C6;Ľ

(27)

đ?&#x153;&#x2022;đ?&#x2018;&#x; 2 â&#x20AC;&#x2013; (đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś )2 â&#x2030;Ľ 0, hence in order đ?&#x153;&#x2022;đ?&#x2018;&#x160; to have Î&#x201D;đ?&#x2018;&#x2030; â&#x2030;¤0, đ?&#x153;&#x201A; should be chosen as:

the term đ?&#x153;&#x201A;â&#x20AC;&#x2013;

2đ?&#x2018;&#x2019;2

đ?&#x153;&#x201A;â&#x2030;¤ â&#x20AC;&#x2013;

đ?&#x153;&#x2022;đ?&#x2018;&#x; 2 â&#x20AC;&#x2013; (đ?&#x2018;&#x2019;đ?&#x2018;Ľ đ??˝đ?&#x2018;Ľ + đ?&#x2018;&#x2019;đ?&#x2018;Ś đ??˝đ?&#x2018;Ś )2 đ?&#x153;&#x2022;đ?&#x2018;&#x160;

(28)

3. Experimental Results The performance of the proposed algorithm is veri ied over a variety of real unstructured indoor environments using an autonomous mobile robot platform [2]. The mobile robot platform is designed to operate in an indoor environment with a solid lat surface. A differential steering system is employed to generate forward and steered motion. The platform provides a rich computing environment consisting of a single board computer and a microcontroller. It is also equipped with a laser range sensor and ultrasonic sensors for obstacle detection as well as a compass and wheel encoders for localization. The laser range sensor is calibrated to scan the 200 degrees front view of the robot in 20 sectors with a 10 degrees angular resolution, The mobile robot platform is shown in Figure 7.The platform has a cylindrical structure with a 35cm diameter and approximately 30cm height. For

Fig. 7. Target and robot coordinates all the testing scenarios, the data acquisition is performed with a sample time T=1s. The measured variables consist of the current robot position and orientation (đ?&#x2018;Ľđ?&#x2018;&#x; , đ?&#x2018;Śđ?&#x2018;&#x; , đ?&#x153;&#x192;) and the twenty-sector readings of the laser sensor.

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In all experiments, the robots initial position is (0,0) while the goal position is at (1.5,-1.5). In the irst experiment, an obstacle is placed along the robot direct path, which is the straight line that connects the robotâ&#x20AC;&#x2122;s initial con iguration to the target con iguration (đ?&#x2018;˘â&#x192;&#x2014;đ?&#x2018;&#x2019; ). There were also other obstacles surrounding the robot as shown in Figure 8. Fig. 8b depicts the trajectory obtained from the neural network based controller. The length of the trajectory is 2.4157 m. Figures 8c - 8e are snapshot of the intermediate robot parameters at different instances in time. The obstacles seen by the robot at each instant of time is shown in the Cartesian coordinates as black dots. The solid yellow line is an approximation to the obstacle contour. The polar histogram shows how each sector is classi ied to either free or occupied. The reference steering angle, đ?&#x203A;žđ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C; , is illustrated as a solid red line while the desired steering angle, đ?&#x203A;žđ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018; , is shown as a dashed green line.

In the second experiment shown in Fig. 9a, the robot successfully avoids the 2 obstacles, and drives its way to đ?&#x2018;&#x17E;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą as shown Fig. 9b. The robot velocities and control actions are shown in Fig. 9c and Fig. 9d respectively. The length of the trajectory is 2.3016 m and it takes 106 s to execute.

In the third experiment shown in Fig. 10a, the gap between the obstacles is reduced. The robot correctly identi ied the gap as navigable and went in between.

However, the left side of the robot touched the obstacle. The trajectory is shown in Fig. 10b and the robot motion and control action are shown in Fig. 10c and Fig. 10d. The length of the trajectory is 2.1349 m and is completed in 90 s.

In the fourth experiment shown in Fig. 11a, the robot avoids the narrow gap by contouring the obstacles, discovers a blocked path, reverses direction and progresses to the target con iguration. The trajectory is depicted in Fig. 11b and the robot velocities and control action are depicted in Fig. 11c and 11d. The length of the trajectory is 7.4457 m and the time it take is 227 s.

In the ifth experiment shown in Fig. 12a, the robot discovers the dead end and turns around the obstacles to reach đ?&#x2018;&#x17E;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;Ą . The trajectory is depicted in Fig. 12b.

The robot velocities and control action are depicted in Fig. 12c and Fig. 12d. The trajectory exhibited some luctuations. The length of the trajectory is 5.4778đ?&#x2018;&#x161; and is completed in 169đ?&#x2018; .

There are several metrics that can be used to evaluate the performance of a navigation system [9]. The following performance metrics are used to evaluate the quality of the trajectory while considering the security or proximity to obstacles and the smoothness of the trajectory relative to the control effort.

1) Security Metric-1 (SM1): Mean distance between the robot and the obstacles through the entire mission measured by the laser sensor (20 sectors).

2) Security Metric-2 (SM2): Mean minimum-distance to obstacles. This is taken from the average of the lowest value of the laser sensor data (20 sectors).

3) Path length: distance traveled by the robot to ac-

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Fig. 8. (b) Robot trajectory using neural network algorithm; (c) Intermediate robot parameters at sample = 8 seconds; (d) Intermediate robot parameters at sample = 29 seconds; (e) Intermediate robot parameters at sample = 51 seconds. complish the task from the initial position to the target position.

Tab. 1. Performance Metrics Scenario

đ?&#x2018;&#x203A;

đ?&#x2018;&#x192;đ??ż = â&#x2C6;&#x2018; â&#x2C6;&#x161;(đ?&#x2018;Ľđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 )2 + (đ?&#x2018;Śđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 )2

(29)

đ?&#x2018;&#x2013;=1

where, (đ?&#x2018;Ľđ?&#x2018;&#x2013; , đ?&#x2018;Śđ?&#x2018;&#x2013; ), đ?&#x2018;&#x2013; = 1, 2, â&#x20AC;Ś , đ?&#x2018;&#x203A; are the n-point Cartesian coordinates of the robot along the given trajectory .

1 2 3 4 5

SM1 (m) 0.4688 0.4563 0.4439 0.4478 0.4727

Performance Metric SM2 (m) đ?&#x2018;&#x192;đ??ż (m) đ?&#x2018;&#x2021;đ??´ (s) 0.1973 2.4157 82 0.1588 2.3016 106 0.1474 2.7349 90 0.1890 5.4778 169 0.2704 7.4458 227

đ?&#x2018;&#x2021; đ??ľđ??¸ 2.3186e-02 9.2964e-05 6.8655e-02 1.1589e-01 2.4902e-02

4) Time đ?&#x2018;&#x2021;đ??´ : time taken to accomplish the task.

5) Smoothness of the trajectory relative to control effort. đ?&#x2018;&#x203A;

đ?&#x2018;&#x2021; đ??ľđ??¸ = â&#x2C6;&#x2018; đ?&#x2018;&#x2DC;2 (đ?&#x2018;Ľđ?&#x2018;&#x2013; , đ?&#x2018;Śđ?&#x2018;&#x2013; )

(30)

đ?&#x2018;&#x2013;=1

where đ?&#x2018;&#x2DC;(đ?&#x2018;Ľđ?&#x2018;&#x2013; , đ?&#x2018;Śđ?&#x2018;&#x2013; ) is the curvature at any point (đ?&#x2018;Ľđ?&#x2018;&#x2013; , đ?&#x2018;Śđ?&#x2018;&#x2013; ) across the trajectory. đ?&#x2018;&#x2DC;(đ?&#x2018;Ľđ?&#x2018;&#x2013; , đ?&#x2018;Śđ?&#x2018;&#x2013; ) =

đ?&#x2018;&#x201C; â&#x20AC;ł (đ?&#x2018;Ľđ?&#x2018;&#x2013; ) 3 [1 + (đ?&#x2018;&#x201C; â&#x20AC;˛ (đ?&#x2018;Ľđ?&#x2018;&#x2013; ))2 ] 2

(31)

Table 1 summarizes the experimental results obtained from the 5 different scenarios. The results show that the neural navigation algorithm allows the robot to transit through narrow zones keeping a safe distance from the obstacles while generating smooth trajectories. In the neural navigation algorithm, a threshold of

0.5m was placed on the maximum distance that the robot can view from the laser sensor data. The deviation of the security metrics SM1 from this maximum value (0.5m) is relatively low, which means that the chosen routes passed always through an obstacle free area. The SM2 index gives also an idea about the risk taken by the robot through the different missions in terms of proximity to obstacles. The values of the đ?&#x2018;&#x2021; đ??ľđ??¸ index are also low, which is desirable, since the energy requirements are increased according to the increase in the curvature of the trajectory. Scenario 2 shows a very low TBE because the corresponding trajectory is straighter that the other scenarios. Fig. 9b shows a smaller change in the orientation during each control period, with consequent energy saving and less structural effort on the robot.

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Fig. 9. (a) shows the robot ini al posi on and the surrounding obstacles; (b) shows the obstacle points in green, the area occupied by the robot at each instance in me in red, and the reference point trajectory in blue; (c) describes the robot veloci es; (d) describes the robot control vector.

(a)

(b)

(c)

(d)

Fig. 10. (a) shows the robot ini al posi on, target posi on, and the surrounding obstacles; (b) shows the obstacle points in green, the area occupied by the robot at each instance in me in red, and the reference point trajectory in blue; (c) describes the robot veloci es; (d) describes the robot control vector.

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Fig. 11. (a) shows the robot ini al posi on and the surrounding obstacles; (b) shows the obstacle points in green, the area occupied by the robot at each instance in me in red, and the reference point trajectory in blue; (c) describes the robot veloci es; (d) describes the robot control vector.

(a)

(b)

(c)

(d)

Fig. 12. (a) shows the robot ini al posi on and the surrounding obstacles; (b) shows the obstacle points in green, the area occupied by the robot at each instance in me in red, and the reference point trajectory in blue; (c) describes the robot veloci es; (d) describes the robot control vector.

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4. Conclusions The paper presents two reactive navigation algorithms for a wheeled mobile robot under nonholonomic constraints and in unknown environments.

The mobile robot travels to a pre-de ined goal position safely and ef iciently without any prior map of the environment. The irst method is based on a reactive navigation algorithm which incorporates the dimensions and shape of the robot to determine the set of all possible collision-free steering angles. The steering angle that falls in the widest gap and is closest to the target is selected. The algorithm also takes into account the non-holonomic constraints of differentially steered robots by computing circular trajectories with adaptive radius of curvature. The second navigation algorithm introduces a neural network based reactive navigation. The algorithm aims to generate an optimized path by using a user-de ined objective function which minimizes the traveled distance to the goal position while avoiding obstacles. To this end, a diagonal recurrent neural network (DRNN) has been employed to achieve the necessary generalization capability across a variety of indoor environments. The network is trained through off-line learning followed by an on-line learning algorithm with guaranteed convergence. The performances of the algorithms are veri ied over a variety of real unstructured indoor environments using an autonomous mobile robot platform. The results demonstrated that the algorithm is capable of driving the robot safely through different obstacle arrangements.

AUTHORS Mariam Al-Sagban∗ – American University of Sharjah, UAE, e-mail: g00006931@aus.edu.

Rached Dhaouadi∗ – American University of Sharjah, UAE, e-mail: rdhaouadi@aus.edu. ∗

Corresponding author

REFERENCES [1] M. Al-Sagban and R. Dhaouadi, “Neural-based navigation of a differential-drive mobile robot”, 12th International Conference on Control Automation Robotics Vision, 2012, 353–358.

[2] M. Al-Sagban. “Autonomous robot navigation based on recurrent neural networks”. Master’s thesis, American University of Sharjah, 2012.

[3] V. Castro, J. Neira, C. Rueda, J. Villamizar, and L. Angel, “Autonomous navigation strategies for mobile robots using a probabilistic neural network (pnn)”, 33rd Annual Conference of the IEEE Industrial Electronics Society, 2007, 2795–2800.

[4] X. Chen and Y. Li, ““smooth formation navigation of multiple mobile robots for avoiding moving obstacles””, International Journal of Control, Automation, and Systems, vol. 4, no. 4, 2006, 466–479.

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[5] W. Chung, L.-C. Fu, and S.-H. Hsu. “Motion control”. In: “Springer Handbook of Robotics”, 133–159. 2008.

[6] D. Janglová , “Neural networks in mobile robot motion”, Inernational Journal of Advanced Robotic Systems, vol. 1, no. 1, 2004, 15–22.

[7] L. Kavraki, P. Svestka, J.-C. Latombe, and M. Overmars, “Probabilistic roadmaps for path planning in high-dimensional con iguration spaces”, IEEE Transactions on Robotics and Automation, vol. 12, no. 4, 1996, 566 –580.

[8] O. Khatib, “Real-time obstacle avoidance for manipulators and mobile robots”, International Journal of Robotics Research, vol. 5, no. 1, 1986, 90–98.

[9] N. D. Munoz, J. A. Valencia, and N. Londono, “Evaluation of navigation of an autonomous mobile robot”, Proceedings of the 2007 Performance Metrics for Intelligent Systems Workshop, Maryland USA, 2007, 15–21.

[10] H. T. Trieu, H. T. Nguyen, and K. Willey, “Advanced obstacle avoidance for a laser based wheelchair using optimised bayesian neural networks”, 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2008, 3463 –3466.

[11] J. Yen and N. P luger, “A fuzzy logic based extension to payton and rosenblatt’s command fusion method for mobile robot navigation”, IEEE Transactions on Systems, Man and Cybernetics, vol. 25, no. 6, 1995, 971 –978.