JAMRIS 2016 Vol 10 No 2

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Articles 1

JOURNAL OF AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS

Publisher:Industrial Research Institute for Automation and Measurements PIAP

Editor-in-ChiefJanusz Kacprzyk

(Polish Academy of Sciences, PIAP, Poland)

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Kaoru Hirota (Japan Society for the Promotion of Science, Beijing Office)

Jan Jabłkowski (PIAP, Poland)

Witold Pedrycz (ECERF, University of Alberta, Canada)

Co-Editors:Roman Szewczyk (PIAP, Warsaw University of Technology)

Oscar Castillo (Tijuana Institute of Technology, Mexico)

Marek Zaremba (University of Quebec, Canada)

(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)

Articles2

Technical B-H Saturation Magnetization Curve Models for SPICE, FEM and MoM SimulationsRoman SzewczykDOI: 10.14313/JAMRIS_2-2016/10

The Application of Mobile Robots for Building Safety ControlBarbara Siemiątkowska, Bogdan Hrasymowicz-Boggio, Mateusz WiśniowskiDOI: 10.14313/JAMRIS_2-2016/11

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

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

Control Methods Design for a Model of Asymmetrical QuadrocopterRyszard Beniak; Oleksandr GudzenkoDOI: 10.14313/JAMRIS_2-2016/14

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

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

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

JOURNAL OF AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 10, N° 2, 2016

DOI: 10.14313/JAMRIS_2-2016

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CONTENTS

Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 10, N° 2 2016

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Technical B-H Saturation Magnetization Curve Models for SPICE, FEM and MoM Simulations

Roman Szewczyk

Submitted: 2nd February 2016; accepted 29th February 2016

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 sophis-ticated, for technical purposes, such curve may be mod-elled by simplified equations.Paper presents the quantitative assessment of the qual-ity of four technical models of BH saturation magneti-zation curve performed for four modern magnetic ma-terials: constructional corrosion resistant steel, Mn-Zn ferrite, amorphous alloy with perpendicular anisotropy as well as Finemet-type nanocrystalline magnetic mate-rial. 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, perpen-dicular anisotropy, Mn-Zn ferrite, Finemet

1. IntroductionMethods 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 for-based 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.

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

mi0=4.*pi.*1e–7;

Bmodel=Hmeas.*mi0.*mi;

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


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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 ModelsValidation 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.

4. Validation ProcedureProcedure 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

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

Model Material Parameter Value

Model 1:Linear model

Steel 3H13Bs 1.05 T

µa 464

Mn-Zn ferriteBs 0.36 T

µa 8737

Amorphous alloyBs 1.31 T

µa 4 494

Nanocrystalline Finemet

Bs 1.15 T

µa 326 743

Model 2:Langevin model

Steel 3H13Bs 1.25 T

a 544


a 10.0


a 58.3


Bs 1.24 T

a 0.68

Model 3:atan-function basedmodel

Steel 3H13k 0.0098

µa 641

Mn-Zn ferritek 0.053

µa 12 404

Amorphous alloyk 0.0096

µa 7223


k 0.83

µa 527 485

Model 4:exponential–function model

Steel 3H13Bs 1.09 T

µa 564


µa 11 005


µa 5 850


Bs 1.16 T

µa 412 879

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].


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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

– 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

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

Parameter

Model Material emax (%) s(%) R2

Model 1: Linear model

Steel 3H13 74 20 0.92

Mn-Zn ferrite 27 7.8 0.98

Amorphous alloy 7.4 1.0 0.9998

Nanocrystalline Finemet 78 14 0.97


Steel 3H13 78 21 0.92


Amorphous alloy 10 2.8 0.997


Model 3:atan-based model

Steel 3H13 80 21 0.92


Amorphous alloy 12 3.1 0.996


Model 4:expoten-tial model

Steel 3H13 78 21 0.92


Amorphous alloy 7.3 1.8 0.9992


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

Model Calculation time (s)

Model 1:Linear model

0.41


0.59

Model 3:atan-basedmodel

0.41

Model 4:expotentialmodel

0.45

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


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

5. ResultsThe 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

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) amor-phous 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) amor-phous alloy, d) Finemet-type nanocrystalline alloy


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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. ConclusionsPresented 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, [email protected]

*Corresponding author

REFERENCES

[1] L. Rayleigh, “On the behaviour of iron and steel under the operation of feeble magnetic forces” Philosophical Magazine, vol. 23, no. 142, 1887, 225–245. DOI: 10.1080/14786448708628000.

[2] D. C. Jiles, D. Atherton, “Theory of ferromagnetic hysteresis”, J. Magn. Magn. Mater, vol. 61, 1986, 48. DOI: 10.1016/0304-8853(86)90066-1.

[3] 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.

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

[5] 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: 10.1016/0304-8853(94)01603-8

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


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[6] 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.

[7] 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.

[8] 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

[9] F. J. Perez-Cebolla, A. Martinez-Iturbe, B. Martin-del-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

[10] Y. Takahashi, C. Matsumoto, S. Wakao, “Large-Scale 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.

[10] 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.

[11] N.C. Pop, O.F. Caltun, “Jiles-Atherton magnetic hysteresis parameters identification”, Acta Phys. Pol. A, 2011, 120, 491–496.

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

[13] 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.

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

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

Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 10, N 2 2016

T A M R B S CT A M R B S CT A M R B S CT A M R B S C

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 mo-bile robots for control of building safety parameters. In-doormobile robots are becoming a reality and their avail-ability and applica ons are expected to grow rapidly inthe near future. Such robots are usually equipped withcameras and laser range finders, which could be usedto detect hazardous situa ons in their opera ng envi-ronment, such as evacua on route obstruc ons, emer-gency sign occlusions or accumula on of dangerous ma-terials. We demonstrate how these safety-related aug-menta ons of a mobile robot system can be achievedwith few addi onal resources and validate experimen-tally the concept using an indoor robot for emergencysign and evacua on route control.

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

1. Introduc onThe range of potential applications for mobile

robots is enormous. However, their current real lifeusage is limited mainly to: delivery robots [1], guidingrobots [2] and cleaning robots [3]. In this article wepresent anewpossible applicationof an indoormobilerobot for safety control. We discuss a method whichintegrates different algorithm used in mobile roboticsand allows the robot to perform building safety con-trol during day-to-day work, such as cleaning, deliv-ery.

A mobile robot can be perceived as a kind of anagent [4]. It obtains the information about an environ-ment and performs some actions. In order to act inthe environment themobile robot requires navigationmodule [5].

A typical robotic system consists of the followingparts:- Perception [6] – data obtained from sensors (cam-eras, laser range- inders) are analysed and repre-sented in a suitable form.

- Mapping [7,8] allows us to build the representationof the environment. Usually metric maps are built.We can distinguish between occupancy grid and fea-ture–basedmaps. Occupancy grid represents the en-vironment as a grid of cells. To each cell a numer-ical value which represents the possibility that thecorresponding area is occupied by an obstacles is at-tached. This kind of a map allows us fast generationof a collision-free path but it requires a huge amount

ofmemory. In feature-based representations the en-vironment is described as a set of features: lines, cor-ners, etc. Such representation is very useful duringprocess of localization but path-planning based onthis kind of map is time consuming. In robotics alsonon-metric representations of the environment areused. Topological maps [9] represent the environ-ment in the form of a graph. Each node representsthe a part of the environment – for example a roomor a corridor. Two nodes are connected if there is re-lation between distinctive parts in the environment.Semanticmaps [10], contain data about themeaningand functionality of the detected objects and places.

- Localization [8, 11] allows the robot to determinethe position in a given coordinate system. Usuallyodometry is an important source of informationabout the robot position. It is inexpensive and pro-vides a good short time accuracy, but if the robottravels for a prolonged period of time errors in de-termining its position increase so additional meth-ods are used. In the literature Kalman ilters or par-ticle ilters are used to estimate the robot’s position.In these methods, encoder readings are used as aninput and sensors measurements as observations.

- Path planning – the aim of path planning is to indoptimum collision-free path to the target location[12, 13]. We can distinguish between global and lo-cal methods. Global methods require the map of thewhole environment and are time consuming. In thecase of localmethodspath is plannedon informationabout nearby obstacles. Themethod is fast but it canbe trapped in local minima.

- Traveling along a planned path the angular and lin-ear velocities of the robot are computed. Usually Dy-namic Window Approach (DWA) [14] is used.During the navigation additional actions can be

performed. In the case of safety control the robot hasto detect and recognize the emergency-signs and todetect obstacles.

Human safety is a crucial aspect in the design andmaintenance of any building. This topic involves mul-tiple risk factors, some of the major being related toire, explosions and earthquakes. In order to mini-mize such hazards, building codes are used, whichare state-level sets of rules specifying minimum stan-dards of building construction and maintenance [15–17]. Building codes in different countries share manyprinciples regarding human safety, and most impor-tantly, rules regarding hazard prevention and evacu-ation procedures. These codes (e.g. the ICC Interna-

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Fig. 1. The system architecture

tional Property Maintenance Code) specify the num-bers of exits, exit capacities, visibility of exit signs,emergency escapes, corridor and stairway parame-ters, accumulation of garbage or other materials inpassageways, doors, windows, ire escapes, stairways,accumulation of lammable or hazardous materials,ire detectors, ire alarms, extinguishers, and manyotherparameters of thebuilding. Failure to ful ill theseand other requirements results in conditions whichare a potential threat to human safety thus periodicinspections are required. In this work we explore thepossibility to use mobile robots for control of vari-ous building safety measures. We can recognize threedifferent anomalous situations: the emergency sign isput in the wrong place, the emergency exit is occludedand the emergency exit is encumbered. Autonomous,service mobile robots are used increasingly often inlarge buildings for such tasks as cleaning loors, trans-porting items or guiding visitors. According to recentforecasts, within the next ten years mobile robotic as-sistants are expected to become common in house-holds as well. Most advanced applications of servicerobots require these machines to be equipped withmultiple sensors such as colour cameras, range ind-ers and depth cameras. These sensors are used forself-localization and for performing particular tasksinvolving objects and places that are recognized bythe robot. Modern indoor navigation techniques usean internal map and representation of the currentstate of the world held in the robot’s memory, whichis subject to constant updates. Taking these qualitiesof mobile robots into consideration it would be veryconvenient to engage them into additional, non-time-consuming, safety-related activities, which could savehuman labour as well as shorten the reaction timeto hazardous conditions and increase overall buildingsafety.

In buildings where mobile robots are alreadypresent,many safety control tasks could be automatedat virtually no cost andwould require only a small frac-tion of the robots’ computational effort. Robots withon-board cameras and localization systems can easilycheck for the presence or visibility of arbitrary signsand other items at their required position (speci ied

on the robot’s internal map), as well as detect haz-ardous obstructions of evacuation routes, doors andstairs.Mobile robots operating inside the building on aregular basis could detect such conditions and prompthuman reaction in a fast and reliable manner.

In order to test this conceptwe design a safety con-trol system integrated with the regular software andhardware of an indoor robot. In the experimental sec-tion we validate several elements of this system usinga large robot for transport tasks operating in the build-ing of Faculty of Mechatronics.

2. The System ArchitectureThe system architecture is presented in Fig. 1. The

system operates on the ROS platform [18]. The RobotOperating System is an open source project that func-tions as a bridge between hardware and other infras-tructure (communication modules, operator panels,different computers), state-of-the-art algorithms andmethods developed and used by our team. This high-level software includes: navigation algorithms whichallows for autonomous localization, and locomotion ofthe robot, automatic task and actions planning, com-munication with the robot by different interfaces (e.g.Graphical User Interfaces, gestures and voice com-mands recognition etc.), data processing.2.1. Sensors

The robot (Fig. 2) used in our system is equippedwith sensors which allow us to acquire informationabout the environment without installation of any ad-ditional devices in the building. The proposed sen-sor system includes: Proprioceptive sensors – usedin order to estimate (not determine) the robot posi-tion relative to a starting location. The laser scanner[19] is anoptical, non-contact distancemeasuring sen-sor. The method of measurement is based on point-ing a laser beams onto the environment and calcu-lating a distance from each received re lection. Typi-cal sensor of this kind is a 2D scanner with 180 or270 angle range. The rotation of high speed inter-nal mirror enables high (50 Hz) frequency of scan-ning with the range of 25 m or higher. The indus-trial laser scanners are reliable and could be usedwith

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concern of the safety regulations (Machinery Direc-tive 2006/42/EC [20]). Laser scanners are designed tooperate faultlessly even in harsh environment (directsunlight or black night, the variety of weather condi-tions, dynamic objects). Mounting a laser scanner onthe tilt mechanism is a common solution to make highaccuracy 3D scanning. It provides a dense lat scan infront of the robot, which allows to automatically de-tect objects in prede ined zones. Inmost cases the twozones are de ined: warning zone in which occurrenceof an obstacle limits robot velocity andprotective zonewhich forces the robot to stop. The robot continues itstask after object disappearance.

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

The Kinect [21] is amotion controller designed fortheXbox360 console. This inexpensive device is a verygood 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 gathervisual information and 3D point cloud which containsa set of points, laying on surfaces around, with de ined3D coordinates and RGB values. Such information al-lows to reproduce a 3D digital model (with colour in-formation) of a scene seen by the sensor. Vision cam-eras are used to detect and recognize lat visual signsplaced on the the walls.

The robot is also equipped with communicationlink with building network and simple Human-Robotinterface (HRI).

2.2. Mapping

In our system we assumed that the map of the en-vironment is known. It is obtained based on documen-tation of the building [22]. Themap is de ined as graphG = N,E,M,P,L, where: N is a set of nodes, each noderepresents some area (room) in the building, E is a setof edges, M is a set of grid-based maps of the areas(nodes), L is a set of landmarks. A grid-based map isattached to each node (room). Fig. 3 presents the ideabehind our approach.

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

2.3. Path PlanningOur hybrid path planing system works in the fol-

lowing steps: A start node (the initial position of therobot) is de ined; The goal node is indicated; Thetopological path between the start and goal nodes isplanned; Themetric path between nodes is generated;Details of themapping and path planningmodules aredescribed in our articles [23,24].

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

2.4. Localiza onIn large buildings there are areas inwhich the clas-

sic approach to metric localization (i.e. based on laserand odometry data) can fail. A good example of sucharea is a long corridor, where the laser range inderperceives only two parallel walls. Since there are noshapes to be unambiguously matched to the knownmap, the metric localization system relies exclusivelyon odometry data. Odometry alone, however, tends toaccumulate errors over time, since it can detect onlyrelative displacements of the robot. This accumulatederrors increase up to several meters depending on thelength of the corridor, which is unacceptable if therobot is supposed to reach some speci ic point of thecorridor (e.g. a selected room door). To prevent thissituation, we have developed a localization methodthat relies on visual data captured with a colour cam-era. In order to use this method, a semantic map of thebuilding must be provided. It is expected that, at its

11


lowest level, this map contains qualitative and quan-titative information about the spaces or subspaces ofthe building. Assuming that the robot is equippedwitha laser range inder, we irst run a conventional MonteCarlo localization algorithm to obtain an approximatemetric position, suf icient for localization on the se-mantic map (based on knowledge of the subspacesboundaries).When the robotnavigates inside aknownsubspace, the algorithm constantly searches the cam-era input for a set of known visual templates (pro-vided by the semantic map), such as room numbers,emergency signs, boards, wall patterns, etc. Once oneof these templates is found, the robot position relativeto the template is calculated. Since the templates haveixed positions and orientations, the calculation of theposition of the robot in the map is trivial. The positionis updated and the conventional localization algorithmcontinues to run until another template is found.

Fig. 5. The samples of emergency signs

Thesenatural templates canbedetectedusing con-volutionalmatching:We irst detect thewalls capturedin the range inder data data using Hough line detec-tion. Then we apply a perspective transform from theoriginal camera input to a normalized viewpoint (i.e.the viewpoint of a virtual camera with its optical axisnormal to thewall, positioned at a given distance fromit). The resulting image is used for template matchingusing a fast heuristic algorithm [25].

The presented localization algorithmwas tested ina long corridor using a mobile robot equipped with arange inder and a colour camera (with its optical axisdeviated60degrees from the robot’s axis). The corri-dor was divided into four subspaces, two of them hadno characteristic features visible by the laser rangescanner. For each of these subspaces, three naturaltemplates were captured. While navigating, on eachtemplate detection, the robot’s displacement alongthe corridor was corrected by 0.5–2 m, depending onthe magnitude of the accumulated odometry errors.A visualization of the combined localization systemhas been shown in Fig. 4. The cubes on thewalls repre-sent natural templates. The white dots are wall pointsperceived by the range inder, the cloud of arrowsrepresents possible robot positions randomly spreadaround the position calculated based on template de-tection (used by the Monte Carlo metric localization).The attached image (top right) shows the transformedcamera input with a bounding box on the detectedtemplate.

Beyond localization on a corridor, the proposedalgorithm provides, obviously, information about the

a) b)

Fig. 6. A corridor with objects blocking the exit. Simpleanalysis of the data from laser scanner indicatespossible evacua on problem.

presence of any known templates of interest. Thus, aspresented in the next section, this method can be eas-ily developed to accomplish the task of veri ication ofthe visibility of emergency and warning signs, doornumbers, information boards and other important latfeatures of the building.

3. Emergency-routes and Signs Verifica onThe emergency management signi icance is un-

questionable in all public buildings. The complexityof the emergency routes is rising nowadays, alongsidewith sophisticated public infrastructures architecture.The design of faultless emergency-routes should beconsidered as most important. The escape routes ver-i ication and its further analysis has to be done toimprove building safety. A safety plan is designedand approved by professionals and should not bechanged unless the building structure was modi ied.The veri ication process should be performed period-ically to avoid unauthorized changes. Typically onlyemergency-signs existence is checked. In most cases,the signs position and quality should also be consid-ered. The accumulation of objects on the escape-pathshould not occur to prevent clashes during evacuation.Fig. 6a presents the image of the sample environment.The obstacles (a box and bottles) which are placednear the exit door can be easily using a laser rangeinder (Fig. 6b). It is possible to easily checked if thewidth of the corridors and emergency exits complieswith the safety standards.

3.1. A Proof-of-concept ExperimentWehaveperformedaproof-of-concept experiment

using our mobile robot (Kurier). The experiment wasplanned as follows: the programmed path re lectedone of the emergency escape paths from our labora-tory to the indicated emergency exit. Alongside thisemergency-route a set of safety-signs existence wasveri ied and the width of the corridors or emergencyexitswasmeasured. The emergency-routes plan couldbe presented as a list joint segments whose nodes(vertices) are emergency signs, and the edges areevacuation paths. In the irst run, a remotely con-trolled robot was programmed to record positions ofall emergency signs (positions were con irmed by the

12


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 posi-tions became nodes of our representation graph. Theexperiment goal was an autonomous run to the emer-gency exit, during which the veri ication of all signsexistence was performed. Before veri ication experi-ment, we have occluded one of the signs and put abig obstacle in the corridor. Then an autonomous nav-igation 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 icationof the actual passage width in the certain space. Thewidth could be set for each space independently (e.g.corridors, doorways, emergency exits). The robot isaware of the room typedue to the building representa-tion and its localization. If the width of the measuredcorridor is lower than a threshold for actual space, atrajectory is recorded with a warning label, and de-noted in the report as gray dotted line. At the sametime, the signs detection algorithm veri ies the exis-tence and positions of all emergency-signs recordedpreviously. If any sign is missing, also a warning la-bel is recorded. In the ig. 7, a positive sign detectionis represented with black circles and a negative withblack square. Such a generated report could be used asconcise aid for professionals. Their focus could be putonmost important aspects to check, expediting wholeveri ication process.

3.2. Experimental Results

In the carried experiment, the robot’s laser range-inder and odometric sensors were used to localizethe robot based on a particle ilter method. The range-inder was used to repeatedly measure the corridorwidth. The 2D template detection program was ac-tive during the whole experiment serving 2 functions:providing an additional source of localization (rele-vant only in the corridor) and reporting the foundtemplates for later off-line emergency sign veri ica-tion. The template detection algorithm performedwell most of the time. All the visible signs were de-tected. However, we found dif iculties detecting tem-plates viewed from a sharp angle (skewed by morethan 60 degrees form the frontal view), as well as de-

tecting templates with metal, glass or other re lect-ing materials (in the experiment no such templateswere used). Since the system is designed to work inlarge buildings with arti icial light, we did not inten-tionally modify the standard lighting during the test -however, it should be noted that this would probablynegatively impact performance. The proposedmethodwould present problems for either small robots (withcameras close to the ground) or narrow corridors,since the robot would be too close to the wall, limitingthe wall area covered by the ield of view. The maindif iculties during the experimental run were encoun-tered during autonomous navigation through the rel-atively narrow laboratory doorway - the limited preci-sion of the drives and control system caused multiplepath re-planning iterations, and thus some unneces-sary adjusting movements of the robot.

4. ConclusionWhen an emergency occurs within a building, it

is crucial to guide the people towards exits. It is re-quired that the accumulation of objects on the escape-path does not occur to prevent clashes during evacu-ation and and thus, evacuation signs must be placedalong the path. In this article we have presented anew possible application of an indoor mobile robot -safety control.We show that this task can be solve dur-ing day-to-day work of a mobile robot such as clean-ing or delivery. We strongly believe that a coopera-tion between Architecture-Engineering-Constructionand Mobile Robotics domains could considerably ac-celerate the development of mobile service robots in-tended for public buildings.

AUTHORSBarbara Siemiatkowska∗ – Warsaw Universityof Technology, (22) 2348647, e-mail: [email protected], www: Faculty of Mecha-tronics.Bogdan Hrasymowicz-Boggio∗ – Warsaw Uni-versity of Technology, (22) 2348647, e-mail: [email protected], www: Faculty of Mechatronics.Mateusz Wisniowski∗ – Warsaw University

13


of Technology, (22) 2348647, e-mail: [email protected], www: Faculty of Mechatronics.∗Corresponding author

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”Development of a hospital mobile platformfor logistics tasks”. In: Digital Communica-tions and Networks, vol. 1, 2015, 102–111.DOI:10.1016j.dcan.2015.03.001.

[2] S. Thurn, M. Bennewitz, W. Burgard,etal.,”MINERVA: A SecondGeneration Mu-seum Tour-Guide Robot”. In: Proceed-ings IEEE International Conference onRobotics and Automation, 1999, 1999–2005.DOI:10.1109ROBOT.1999.770401.

[3] K. Al-Wahedi, A. Darwish, B. Kodiah, ”Cost BasedNavigation for Autonomous Vacuum Cleaners”.In: Robot Intelligence Technology and Applica-tions 2012: An Edition of the Presented Pa-pers from the 1st International Conference onRobot Intelligence Technology and Applications,DOI:10.1007978-3-642-37374-9_40.

[4] Stuart J. Russell, Peter Norvig, Arti icial Intelli-gence: A Modern Approach, 2 ed., Pearson Educa-tion, ISBN: 9780130803023, 2003.

[5] B. Siemiatkowska et al., ”Towards semantic nav-igation system”. In: Recent Advances in Intelli-gent Information Systems. Proceedings, ed. by: M.Klopotek et al. Exit, 2009,711–720, ISBN978-83-7051-580-5.

[6] https://willowgarage.com/pages/research/perception

[7] R. B. Rusu et al., ”Towards 3D point cloudbased object maps for household envi-ronment”, Journal of Robotics and Au-tonomous Systems, 2008, vol. 56, 927–941,http:dx.doi.org10.1016j.robot.2008.08.005.

[8] S. Thrun, W. Burgard, D. Fox, ProbabilisticRobotics (Intelligent Robotics and AutonomousAgents), The MIT Press 2005, ISBN 0-262-20162-3.

[9] E. Remolina, B. Kuipers, ”Towards a gen-eral theory of topological maps”, Arti icialIntelligence, 2004, vol. 152, no. 1, 47–104.DOI:10.1016S0004-3702(03)00114-0.

[10] O. M. Mozos et al., ”Supervised semanticlabeling of places using information ex-tracted from sensor data”, Robotics andAutonomous Systems, vol. 5,2007, 392–402.DOI:10.1016/j.robot.2006.12.003.

[11] M. P ingsthorn, B. Slamet, A. Visser, ”A scalablehybrid multi-robot SLAM method for highly de-tailedmaps”. In: Proceedings of the 11th RoboCupInternational Symposium, 2007.

[12] H. Chu, H. A. Eimaraghy, ”Real-time multi-robotpath planner based on a heuristic approach”.In Proc. of the IEEE International Conference onRobotics & Automation (ICRA), 1992.

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

[14] D. Fox, W. Burgard, S. Thrun, ”The DynamicWindow Approach to Collision Avoidance”, IEEERobotics 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 EnvironmentalStudies”,vol.16, 2007, No. 4B.

[20] http://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32006L0042&rid=6

[21] Kinect. www.xbox.com/pl-PL/Kinect[22] W. Turek, K. Cetnarowicz, M. Multan, T. Sosnicki,

A. Borkowski. ”Modeling buildings in the contextof Mobile Robotics”. In: Proceedings of the 13thPolish Conference on Robotics (KKR 2014), 2014.

[23] B. Siemiatkowska, B. Piekarski Bartosz, B.Harasymowicz-Boggio, ”Navigating robots in-side building”. In: Prace Naukowe PolitechnikiWarszawskiej. Elektronika, O icynaWydawniczaPolitechniki Warszawskiej, 2014, 503–512.

[24] M. Przybylski, D. Koguciuk, B. Siemiatkowska, B.Harasymowicz-Boggio, L. Chechlinski: ”Integra-tion of Qualitative and Quantitative Spatial Datawithin a Semantic Map for Service Robots”.In:Progress in Automation, Robotics and Measur-ing Techniques. Volume 2 Robotics.Series: Ad-vances in Intelligent Systems and Computing,vol. 351, 2015, 223–232, DOI:10.1007/978-3-319-15847-1_22.

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

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Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 10, N∘ 2 2016

P S P H M M C MA J O O C




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 us-ing low frequencies with the goal to compensate normalfric on effects in drive trains that nega vely influence theperformance of e.g. a standard PID controller. The firstcontrol scheme uses pulse height modula on to espe-cially compensate s ck slip effects but increases vibra onand noise in the drive train. To reduce such side effectsa modified phase shi ed pulse height control schemebased onmul ple actuated joints is introduced. Both con-trol schemes are compared with a standard linear con-troller as reference and evaluated by using six quality cri-teria.

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

1. Introduc onIn many actuation applications, motors are com-

bined with a gearbox to change transmission. This in-creases friction effects and leads to a notable reduc-tion in control quality when using conventional con-trollers like PID control. Effects such as delayed re-sponse and non-linear control torque relationshipscan especially be observed when using low rotationalspeeds or when driving high loads.

Various compensation methods have been pro-posed that attempt to create friction models to pre-dict the actual friction of the system and improvemotor controllers based on them. Le-Tien and Albu-Schaffer [4] use a static friction model describingCoulomb, viscose and load dependent friction. Olsenet. al. [7] investigate the behaviors of the dynamic Lu-Gre and Bliman-Sorine friction models to derive com-pensation methods and show results from practicalexperiments. Swevers et. al. [12] extend these mod-els by incorporating hysteresis with non-local mem-ory and achieve improved accuracy in compensation.

Usually, parameters for both static or dynamicmodels have to be adapted to each controlled systemseparately and this can prove to be cumbersome. Con-sequently, parameter identi ication methods [14] oradaptive compensation through learning such as neu-ral networks [9] or adaptive fuzzy systems have beenproposed [13].

Instead of using active compensation, differentmodel-free methods have been investigated that re-duce friction outside of the controller. Well knowntechniques include dither [17], [8] and pulsed motor

control [15], [16]. Dither is the introduction of a high-frequency noise signal into non-linear systems in or-der tohelp the systemstabilize. It is employed success-fully in hydraulic and pneumatic use cases while it hassome detrimental properties when used with direct-drive actuators.

Pulsed or impulsivemotor control is amethod thatreduces the system’s stiction by pulsing the motorcontrol with pulses large enough to overcome stictiononlywhen themotor is in lowspeeds. This is especiallyuseful when starting and stopping often and operat-ing with low rotation speeds, as is usually the case inrobotics applications. The optimal length and ampli-tude of the pulses depend on the system properties.Appropriatemethods to obtain themarediscussed e.g.in [15].

Drawbacks of both methods adding arti icial oscil-lation to the systemarepossible noise andhigherwearof mechanic components. By using multiply actuatedjoints, some of these can be alleviated through com-bined control upon thebehavior of pulsed controlwithonly one motor. Multiply actuated joints are also em-ployed e.g. in robotics to achieve regulated joint stiff-ness [1]. Further bene its are increased ef iciency, re-dundancy and fail-safety [3], [11], [2].

In this article we employ an adapted pulse basedmotor control method to reduce friction effects. To re-duce the disadvantageous side-effects of those meth-ods, namely vibration and noise, we propose a novelcontrol method based on the use of multiple parallelactuators for one joint and emphasize its propertiesin reducing those effects. In the following, two vari-ations of motor control methods for multiple actua-tors are compared with a standard linear controlleras reference. The behavior of a motor unit is evalu-ated by means of six quality criteria. The describedcontrol methods are closely related to results given inSiedel [10] which focuses on the methods applied forrobotic purposes. Various igures are also taken fromthere.

2. Mul ply Actuated Motor UnitThe motor unit employs multiple identical actu-

ators for which the servo unit Dynamixel RX-28 byROBOTISwas used. In this article, only the use of iden-tical actuators is investigated. Further options can alsobe gained by combining actuators with different gainratios and output powers, cf. [6]. The unit consists ofa DC motor, a spur gear and electrical components formotor drive, communication and control. The DC mo-tor is an RE-max 17 fromMaxon (see [5] for speci ica-

15


Fig. 1. The coupling gearbox with two RX-28s. Theservo units are coupled to the measuring sha with thespur gear. The lower part of the picture shows themeasuring sha that is coupled to the a achedmeasuring devices using a balancer coupling

1

6

7

2

4

3

5

Fig. 2. Test setup enclosed in sound isola ng foampanels. Le : Measuring microphone (1) at 0.5mdistance to the motor unit. Middle: drive test bench (2)with motor unit (3), pendulum (4) and controlelectronics (5). Right: Two-channel oscilloscope (6) andtemperature sensor (7)

tions) and is operating on 12 to 16V. An STMicroelec-tronics L6201 H-bridge is used as the power driverfor the motor. On the low level, the motor is drivenwith a pulse width modulated voltage at a frequencyof 15.6 kHz. The ive-stage spur gear has a gear ratioof 1:195. With the exception of the output socket gearthat is mounted on double ball bearing, all gears aremounted on plain bearing. In the motor unit, two mo-tors are coupled directly with themeasuring shaft at aratio of 1:1, see Figure 1. Thewhole drive train – servogearbox and coupling – has a backlash of 0.8degrees.

3. Experimental SetupThe 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 anoptical angle sensor. The sensors are assembled in acoaxial manner, preventing the introduction of addi-tional mechanical backlash. A microcontroller boardwith an STM32 processor is used for motor controland data communication to and from a connectedcomputer.

Attached to the end of themeasuring shaft is a pen-dulum with the following properties:- mass𝑚 = 1168 g- length 𝑙 = 0.282m- rotatory inertia 𝐽 = 99 gmThe pendulum mass and lever arm length are chosenin such a way that the maximum drive torque is notsuf icient to de lect the pendulum to the horizontalplane at slow speeds. This also avoids overshoot.

Figure 2 also shows additional measuring deviceswhich are used to evaluate the drive characteristicsbased on the previously de ined quality criteria. AHAMEG HM1008-2 digital oscilloscope measures mo-tor current and voltage to get the motor power con-sumption while an RS-components digital thermome-ter gets the current motor temperature. Since the os-cilloscope uses discrete sampling, an LC ilter is added.The temperature readings are only used to maintaincomparable starting conditions for each test run andare not recorded. An HDM M30 measurement micro-phone is placed in a 50 cm distance to provide soundpressure measurements. The whole setup is enclosedby special acoustic foam panels to reduce ambientnoise and to improve the sound measurement quality.

3.1. Test SequenceAll of the following experiments are carried out un-

der uniform conditions regarding motor temperature,operating voltage and ambient noise. Each test takes20 seconds. The actuators are controlled without anyfeedback.

Tests are startedwith the pendulumhanging downso that no torque is applied to either measurementshaft or the motor unit. Over the irst 10 seconds themotor reference voltage 𝑈 for both motors is grad-ually increased from 0V up to 14.8V which raises thependulum inonedirection.During thenext 10 secondsthe power is gradually decreased down to 0V again sothe pendulum is lowered again. The progression of thereference voltage can be seen in the upper part of Fig-ure 3.

The duration of 20 seconds for each test is cho-sen so that the motors do not heat up considerablyand at the same time dynamical effects can be disre-garded because of rather slow raising and lowering ofthe pendulum. To minimize the in luence of measure-ment scattering, each test is repeated ive times andthe values are averaged over all ive runs.

4. Reference ControlThe irst test is intended to show the current sit-

uation with gearbox induced friction effects and willserve as reference measurement. For readability, thevector

𝑈(𝑡) = 𝑈 (𝑡)𝑈 (𝑡) (1)

is the combined vector of the voltages of both motors𝑈 (𝑡) and𝑈 (𝑡). For the referencemeasurement, both

16


servos get the same voltage signal,𝑈 (𝑡) = 𝑈 (𝑡) = 𝑈 (𝑡) (2)

so both motors always apply the same torque. Thelower part of Figure 3 displays the movement (or tra-jectory) of the pendulum for each of the ive tests. Thesingle test trajectories as well as the averaged trajec-tory over all tests both show extensive non-linearitiesand hysteresis effects. These are caused by friction inthe gearboxes of both servo units.

0

5

10

15

0 2 4 6 8 10 12 14 16 18 200.0

0.4

0.8

1.2

Uref

[V]

y[r

ad]

t [s]

Fig. 3. Top: reference voltage 𝑈 sweep over me.Bo om: single trajectories of the pendulum (grey lines)resul ng from the reference voltage from subsequenttrials and averaged trajectory (black). The dashed linelabels the turning point of the reference voltage

In the irst half of the test runs, the in luence ofthewell known stick-slip effect is clearly visible, whichraises the pendulum with stuttering motion. Follow-ing the law of Coulomb, both friction types are pro-portional to the motor drive torque. This leads to theobservation that the relation between the static fric-tion and the dry friction is independent of the appliedtorque. With the static friction coef icient 𝜇 and thedry friction coef icient 𝜇 ,

𝜇𝜇 = constant. (3)

Also see [4] formore details on torque dependent fric-tion effects.

As dynamic effects can be neglected, it is possibleto compare the start and end of each plateau of therising trajectory and conclude on the difference of thefriction coef icients. Startingwith the dry friction coef-icient 𝜇 , it can be seen that in this test case the stickyfriction is 38.8% stronger than the dry friction (witha standard deviation of 12.8%). The driving torquetherefore has to be increased by 38.8% to further liftthe pendulum after it came to a standstill. In general,this value is greatly dependent on the type of gearboxand the ratio of transmission but also on various otherfactors such as lubrication, operating temperature andwear.

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

This area is shifted forward on the time axis, whilesome jumps are even happening only after the max-imum voltage has been reached at 10 seconds. Thedifference between the maximum angle that the pen-dulum reaches in single runs is up to 0.41 rad. Un-til reaching the maximum reference voltage 𝑈 thewhole drive train is in drive mode. Drive mode is hereused to name one of the modes of four quadrant op-eration of servo motors. In drive mode, motor torqueand turning direction are the same whereas in brakemode, the directions are opposing each other. Themo-tor moment is acting in the turning direction and thefriction torque is acting against the motor torque, ineffect reducing the effective torque. When the refer-ence voltage is decreased, the motor moment is alsoreduced without changing the motor direction. Thependulum should be lowering. The motor unit how-ever is now in brake mode, so motor torque and fric-tion torque are adding up. The pendulum is also notmoving since static friction is still effective which islarger than dry friction. As a result, the pendulum isonly lowering after the driving voltage falls short of22.2%of themaximum reference voltage. The pendu-lum then starts suddenly and is moving quickly in thebeginning.

This test using conventional control shows whichnon-linear and hysteretic effects can occur in the driveunit. These effects are of course also dependent on theparameters of the pendulum and the behavior of thereference voltage.

5. Pulse Height Modulated Motor ControlIn order to mitigate these effects, a common tech-

nique is to use pulsating motor drive rather than con-tinuousmotor drive [15], allowing themotor to alwaysmove even for very small target values. Usually pulsewidthmodulated (PWM)drive is usedwhich alters theduty cycle to achieve the desired drive voltage on aver-age. Using the similar pulse height modulated (PHM)motor control which changes the height of equally dis-tributed pulses has the bene it of a much higher pos-sible resolution when the temporal resolution of thedigital circuitry in use is not very high. The basic ideaof these techniques is to change between the frictiontypes (stiction and dry friction) as well as the motormodes (drive and brakemode) in a controlledmannerwith comparably low frequency in order to linearizethe drive behavior.

First, the operation of the PHM drive will be de-scribed. It is implemented in three steps: creating thebase oscillation, modifying the oscillation amplitudein relation to the reference voltage and combining thereference voltage with the oscillation. The base oscil-lation is derived froman analytic representation of thetriangle oscillation

𝑓 (𝑥) = 2𝜋 sin sin(2 𝜋 𝑥) . (4)

which is modi ied to alter the amplitude as necessary.The triangle oscillation 𝑓 (𝑥) is expandedwith the pa-rameters 𝑎 to alter the amplitude, and 𝑓 to alter the

17


Um

Um /2

U[V

]

t [s]

Fig. 4. Envelope curve (grey) of the modulatedoscilla on in rela on to the reference signal 𝑈 (blackline). The dashed lines indicate half of the maximumvoltage and effec ve maximum voltage 𝑈 (14.8V).

frequency

𝑈 (𝑡) = 𝑎 𝑓 (𝑓 𝑡) (5)

with 𝑡 being the time variable to receive the base fre-quency voltage 𝑈 (𝑡). The amplitude 𝑎 of the basefrequency is modi ied in relation to the reference volt-age within a range of 0 to 𝑈 /2V. A reference volt-age of 𝑈 = 0.00V relates to an amplitude of0.00V while 𝑈 = 𝑈 /2 relates to an amplitudeof 𝑈 /2. When the reference voltage assumes val-ues greater than 𝑈 /2 up to 𝑈 , the amplitude isfalling down to 0.00V to not receive values above themaximum drive voltage. Figure 4 shows the referencevoltage (black) and the envelope of the pulse modu-lated voltage (grey) as it is set in the later test runs.

The amplitude progression in relation to the refer-ence voltage can be given as

𝑎 = 2𝑈 at ≤𝑈 ≤2𝑈 − 2𝑈 at 𝑈 < ∧𝑈 >

(6)

Finally, the amplitude modulated base oscillationis added to the reference voltage and the resulting sig-nal is passed to both motors similarly.

𝑈 (𝑡) = 𝑈 (𝑡) = 𝑈 (𝑡) + 𝑈 (𝑡) (7)

In order to determinewhich pulse frequency deliv-ers closest to ideal behavior, ive different frequenciesfor the base oscillation were tested. The frequencieswere chosen with regard to the frequency of the mo-tor control loopwhich isworkingwith100Hz. Accord-ingly, the highest possible frequency is 50Hz of which25Hz, 12.5Hz, 6.25Hz and 3.125Hz are derived.

As mentioned, each frequency test run is repeatedive times and averaged. The obtained trajectories aredisplayed in Figure 5 which also includes the trajec-tories from the previous test (reference control) forcomparison.

Comparing the results of the reference control, allPMC trajectories more or less display the followingproperties: The spread of the trajectories is lower, aplateau forms at higher values of the reference voltageand the stick-slip effect is minimized as can be seen inas smooth motion in place of the previous jumps and

0 4 8 12 16 200.0

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Reference PMC 3.125 Hz

y[r

ad]

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ad]

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ad]

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ad]

y[r

ad]

t [s] t [s]

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t [s] t [s]

PMC 12.5 Hz

PMC 25 Hz PMC 50 Hz

Fig. 5. Trajectories of the pendulum with referencecontrol and with PMC at five different base frequencies(3.125Hz to 50Hz). The separate trajectories aredrawn in grey, the averages for each frequency in red.The black line shows the averaged trajectory for thereference control

stuttering. Finally, the hysteresis is greatly reducedand the trajectory is closer to being symmetrical.

At 3.125Hz, in luence of the frequency is clearlyvisible as a modulated frequency during upwards anddownwardsmotion. Higher frequencies don’t producethis behavior anymore as the pendulum’s inertia actsas a low pass ilter that removes higher frequencies.The frequencies from 6.25Hz to 50Hz do not producesigni icant differences between the trajectories, sug-gesting low in luence of the speci ic frequency.

6. Phase Shi ed Pulse Height Modulated Mo-tor ControlA possible modi ication of the previously de-

scribed PMC is to have each motor be pulsed at differ-ent points in time with phase shifted modulated pulseoscillations. In the case of two motors, the signals areshifted by 𝜋, phase shift for any other number of mo-torswill be looked at in the next section. The expandedequation

𝑈 (𝑡) = 𝑎 𝑓 (𝑓 𝑡 + 𝜑) (8)

yields a phase shifted oscillation with 𝜑 as the phaseshift parameter. Using 𝜑 = 0 and 𝜑 = 𝜋 for 𝜑, thebase oscillations 𝑈 (𝑡) and 𝑈 (𝑡) are obtained and,written as a vector

𝑈(𝑡) = 𝑈 (𝑡) + 𝑈 (𝑡)𝑈 (𝑡) + 𝑈 (𝑡) , (9)

are fed to both the motors. With this altered controlmethod, again ive test runs per base frequency areperformed. The resulting trajectories are shown inFigure 6 which again includes the resulting trajecto-ries of the reference control test for comparison.

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

18


0 4 8 12 16 200.0

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t [s] t [s]

t [s] t [s]

PPMC 3.125 Hz

PPMC 6.25 Hz PPMC 12.5 Hz

PPMC 25 Hz PPMC 50 Hz

Fig. 6. Trajectories of the pendulum with referencecontrol and with PPMC at five different basefrequencies (3.125Hz to 50Hz). The separatetrajectories are drawn in grey, the averages for eachfrequency in red. The black line shows the averagedtrajectory for the reference control

the maximum voltage. Then the results are scattered.The plateaus of all ive trajectories are more or lessof the same length but clearly shifted forward on thetime axis, showing the in luence of the motor drivemodes. At a frequency of 6.25Hz no jumps in the tra-jectories are visible anymore, but the transitions to theplateaus aremore smooth. As with the PMC, the PPMCshows onlymarginal differences at higher frequencies(12.5Hz, 25Hz and 50Hz)

7. ResultsThe visual examination of the trajectories only al-

lows limited conclusions about the properties of theparticular control methods, of which PMC and PPMCalso show frequency dependent behavior. Apart fromthe movement alone shown by the trajectory of thependulum, other quality measures are important tojudge each control method appropriately. These in-clude the torque behavior, energy consumption, hys-teresis, linearity, maximum torque, operating smooth-ness, energy ef iciency and operational noise. Thesecan either be measured directly or derived from thetaken measures trajectory, torque, energy consump-tion and operating noise. A signi icance analysis wasmade to make sure that, regarding the standard de-viation over the repeated measurements, the best re-sults are signi icantly better. The results of the bestand second best test run within each quality measureare tested for signi icant difference with a two-samplet-test. The chosen signi icance criterion for differenceis 𝑝 > 5%.7.1. Hysteresis

Starting with determining the hysteresis, we lookat the trajectory. In the present case, the hysteresis de-scribes the difference of the trajectory when increas-ing (𝑡 = 0 to 10 seconds) and decreasing (𝑡 = 10 to 20

seconds) height. The quality is determined as follows:irst, the second half of the trajectory is mirrored atthe middle axis (at 10 seconds) onto the irst half. Thediagrams that are produced in this way are shown inFigure 7. The direction of movement is labelled witharrows. It is well visible howmuch ascent and descentoverlap and therefore exhibit a certain hysteresis.

A numerical value for the hysteresis is obtainedas the sum of the squared errors. The correspondingequation is

𝐸 (𝑦) =/

𝑦(𝑡) − 𝑦(𝑇/2 − 𝑡) (10)

where the error is assumed as the vertical distancebetween the ascent and descent curves for each timestep.

0 2 4 6 8 100.0

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Fig. 7. Comparison between hysteresis and linearity ofaveraged trajectories. The individual graphs showascent and descent of averaged trajectories with thedifferent control methods. The direc on of mo on islabeled with arrows while the addi onal curvesbetween the trajectories show the ideal course of therespec ve trajectory assuming no fric on. It iscomputed using a motor model, using parametersderived from the actual trajectories

A good result for the hysteresis is achieved if 𝐸is minimal and therefore the area between rising andfalling trajectory is as small as possible. The best resultwas measured for the PMC at 12.5Hz. Comparing tothe second best result, the PPMC at 12.5Hz, the bestresult is signi icantly better.

19


7.2. LinearityThe next step is to evaluate the linearity of the tra-

jectory, which covers the extent to which the drive be-havior in luenced by the real friction approaches theideal friction free behavior. To evaluate the drive char-acteristics we have another look at the individual pen-dulum trajectories. In order to picture an ideal trajec-tory graph, the physical properties of the pendulumand a simpleDC servomotormodel is employedwhichassumes a directly proportional relation betweenmo-tor current and torque and ignores friction or dynam-ical effects. Except for the torque constant, all param-eters for the model are known. The actual operatingbehavior is heavily dependent on the control methodused (Reference, PMC and PPMC) which is why thetorque constant has to be determined for each test runseparately. This is donewith the least squaresmethod.The resulting ideal trajectories based on the modelare adjusted to the real trajectories. Since the upperplateaus of the trajectories are distorting the values,only the data up to 90% of the maximum angle areused. The determined ideal trajectories can be seenin Figure 7 between the trajectories of ascent and de-scent.

For these areas of each measurement, it is nowpossible to get the linearity deviation of the averagedreal course 𝑦 to the ideal course 𝑦 . This is done againby computing the sum of squared errors.

𝐸 =%

𝑦 (𝑡) − 𝑦 (𝑡) . (11)

A positive result for Linearity is achieved if𝐸 is small.The best result was obtained for the PMC at 6.25Hzwhich is signi icant in comparison to the PPMC at6.25Hz.

7.3. Maximum Torque and Vibra onsFor assessing the torque behavior of the motor

unit we look at the effective torque that is measuredwith the torque sensor between drive train and thependulum. The quality of the behavior is determinedby the criteria maximum torque and occurring vibra-tions. The maximum torque that can be delivered isdetermined by averaging the torque values around thepoint at which the pendulum is at maximum height.Averaging is done, as the dynamic behavior of thependulum might result in high torque values for ashort time which however can not be maintainedover longer time spans. The best result for maximumtorque was achieved for the PPMC at 50Hz which issigni icantly better in comparison to the PMC at 50Hz.

Vibrations are on the one hand produced by oscil-lations resulting from the sticky friction in the gear-box but on the other hand because of the pulsated con-trol signal. Figure 8 shows exemplary torquemeasure-ments for trajectories for each control method. Thereference control on the left shows vibrations becauseof the alternating friction types. The PMC howevershows vibrations resulting from the pulsed control.

Reference PMC 6.25 Hz PPMC 6.25 Hz

M[Nm]

t [s] t [s] t [s]1612840 20 1612840 20 1612840 20

1.0

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Fig. 8. Exemplary effec ve torque courses for threedifferent control methods. Le : Reference, Middle:PMC (6.25Hz), Right: PPMC (6.25Hz)

Using the same base frequency, the PPMC shows no-ticeably less vibrations because of phase shifted con-trol of both motor units where pulses can compensatefor each other inside of the coupling gear.

The vibrations can be quantized by fast Fourier-transforming (FFT) the torque signal, yielding a fre-quency spectrum for each torque signal. The value ofthe frequency with the maximum overall level is as-sumed as the vibration level. The best result for vibra-tions was achieved for the PPMC at 25Hzwhich is sig-ni icantly better in comparison to the PMC at 25Hz.7.4. Energy Consump on

For the energy consumption, the measurementsfor current and voltage are used which are obtainedusing the oscilloscope. Figure 9 shows exemplary datafor the power consumption for each of the controlmethods. Reference control and PPMC produce simi-lar curves. Only in the middle of ascent and descent,small luctuations are visible. However, when usingPMC, noticeable luctuations are produced that havethe sameheight as the amplitude of the oscillation thatis modulated onto the reference voltage 𝑈 .

0

10

20

30

40

Reference PPMC 6.25 Hz

P[W

]

t [s] t [s] t [s]200 161284 200 161284 200 161284

PMC 6.25 Hz

Fig. 9. Exemplary power consump on courses for threedifferent control methods. Le : Reference, Middle:PMC (6.25Hz), Right: PPMC (6.25Hz)

Regarding the energy consumption, no immediatedisadvantage is to be expected from the luctuations,as long as the input power is not exceeding the con-trol electronics. Crucial to the quality of the controlmethod rather is the overall energy usage of the ac-tuators per trial run, determined by the time integral

𝐸 = 𝑈(𝑡)𝐼(𝑡)𝑑𝑡 (12)

of the product of operating voltage 𝑈(𝑡), current 𝐼(𝑡)and𝑇 being the duration of the test run. Smaller valuesof 𝐸 are better. The best result was achieved with the

20


reference controlwhich is signi icant in comparison tothe PMC and PPMC at all frequencies.

7.5. Opera ng NoiseThe sound produced is measured at a sample rate

of 48 kHz (mono) to evaluate the controlmethod qual-ity. As before for vibrations, a frequency analysis isdone and the level of the maximal frequency is ob-tained as an indicator for the amount of noise gener-ated.

Here, the best result was obtained for the PMCat 3.125Hz which is signi icant in comparison to thePPMC at 3.125Hz.

7.6. Overall Ra ngSummarizing the results for each of the criteria,

the values arenormalized andmapped to a color rangeand arranged in an overview matrix as seen in Fig-ure 10. Each result is marked by color and addition-ally an “x” or “o” to show if it is above or below aver-age. The best result for each criterion is marked witha white triangle.

Hys

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Fig. 10. Color coded ra ng matrix for the three controlmethods Reference, PMC and PPMC by means of thequality criteria Hysteresis, Linearity, Maximum Torque,Vibra on, Energy Consump on and Noise Genera on(column 1 to 6). Column 7 displays the average valueover the criteria and gives an overall ra ng for theindividual control method. The chosen weights relateto the influence each criterion has for the overall ra ng(3: strong, 2: medium and 1: low). White trianglesmark the best result for each criterion

While determining the quality criteria, it was no-ticed that the results for each criterion have differentvariance, i.e. the differences between best and worstresult vary. Also, the results show that each criteriondoes not have the same importance for the evaluation.Therefore three different steps of weighting were in-troduced (see Figure 10). The overall results on theright side show a weighted sum of quality criteria.

It is noteworthy that the best results are not dis-tributed equally. None of the rows – meaning none ofthe control methods – achievesmore than one best re-sult. The same can be said for the worst results. The

best overall result is producedwith the PPMCat 25Hz.This control method shows the best single value in vi-bration behavior and overall good values for the othercriteria. The next best result is the PPMC at 50Hz andthe worst after the reference control is the PPMC at3.125Hz.

In conclusion, the PPMC shows the bene its ofmultiple actuators and the possibilities of therebyenhanced control methods. The improvement of theoverall drive behavior is considerable compared tothe conventional reference control method. The onlydisadvantage, albeit small, is the energy consumptionthat is about 3% higher for PPMC at 25Hz.

8. PPMC Conversion for an Arbitrary Amountof Coupled ActuatorsIn order to expand the PPMC to more than two

actuators, the phase shift 𝜑 for all motors has to bedistributed equally to the whole cycle duration of thecontrol signal, aiming for minimal vibration and noisegeneration. The same principle applies to combustionengines, where the succession of the ignition for eachcylinder is equally distributed to one turn of the crankshaft.

Since discrete signal processing does not allowevery desired phase shift, the distribution has to bealigned to what is possible. In the following method,odd or even amounts of actuators are dealt with sepa-rately.8.1. Even Number of Actuators

As described in Section 6 the base oscillation forthe second motor is being shifted by 𝜋, so that bothcompensate for the torque impulses of the other. Tokeep this principle for more motors, the phase shiftsover the irst half of the cycle duration (0 ≤ 𝜑 < 𝜋)have to be identical to those over the second half (𝜋 ≤𝜑 < 2𝜋). Furthermore, to reach as even a distributionas possible of all phase shifts over the cycle durationin order to minimize vibrations and noise, the PPMCfor𝑁 actuators is adjusted as follows.

In the beginning, the amount of work cycles of thecontrol circuitry during one period of the base oscilla-tion is determined. The amount 𝑍 is

𝑍 = 𝑓𝑓 , (13)

where 𝑓 is the working frequency of the control cir-cuitry and 𝑓 is the frequency of the base oscillation.

A possible base oscillation with frequency 𝑓 forthe PPMC following Section 6 necessitates that 𝑍 iseven. Otherwise the maxima of the base oscillationcould be lost in the discretized representation.

Since the amount 𝑁 of actuators is not necessarilya multiple of 𝑍 , it has to be determined which sub-divisions of 𝑍 are possible. For that, let𝑊 be the setof even dividers of 𝑍 as de ined by

𝑊 ∶= x ∈ ℕ ∣ 𝑥 divides 𝑍 ∧ 𝑥 even . (14)

The set is only allowed to contain even dividers as oth-erwise an equal distribution of the actuators across

21


both half periods would not be possible. In the follow-ing, 𝑊 will be regarded as a set that is sorted in de-scending order, where𝑊 is the 𝑖’th element of that setand 𝑧 is the number of elements of𝑊, 𝑧 = |𝑊|. Since𝑍 is even in any case, the last element of𝑊 is

𝑊 = 2. (15)

For better illustration, the elements of𝑊 can alsobe understood as layers where the 𝑖’th layer 𝑊 con-tains subdivisions of the cycle duration. Figure 11 dis-plays an exemplary phase shift matrix with the possi-ble phase shifts𝑊 for each layer drawn as rectangles.The amount of actuators that is resulting from the sub-divisions is then listed in each of the rectangles.

Fig. 11. Exemplary phase shi matrix for the values:𝑍 = 8 and𝑊 = 8, 4, 2. The rectangles indicate thepossible phase shi s

It is now possible for each subdivision or layer 𝑊 todetermine how many sets of actuators 𝑞 it can hold.The amount of actuators for each layer is

𝑁 = 𝑞 𝑊 . (16)

In order to keep the difference between the phaseshifts and thereby the temporal distance between thepower peaks as small as possible, the actuators are as-signed to the layers of the matrix in ascending order,i. e. the lowest layer with the inest time grid is alwaysilled irst. The following scheme is used:

𝑞 = 𝑁𝑊

𝑞 = 𝑁 − 𝑁𝑊

𝑞 = 𝑁 − 𝑁 − 𝑁𝑊

⋮

𝑞 = ⎛⎜

⎝

𝑁 − ∑ 𝑁

𝑊⎞⎟

⎠

(17)

It therefore has to hold true that

𝑁 = 𝑁 + 𝑁 +𝑁 +⋯+𝑁 . (18)

The distribution of the actuators is now given. Forthat we have to determine the associated phase shifts.We irst get the step size of the phase shifts

Δ𝜑 = 2𝜋𝑊 (19)

in relation to the 𝑖’th layer, whereas the cycle durationis normalized to 2𝜋. In total, there are 𝑁 actuators,let 𝑘 ∈ 1, … , 𝑁 . The phase shift𝜑 assigned to thestep size Δ𝜑 for the 𝑘’th actuator can be determinedfor all actuators by

𝜑 = ((𝑘 − 1) mod 𝑊)Δ𝜑 . (20)

Fig. 12. Distribu on of the actuators in a phase shimatrix that is generated from 𝑍 = 8. Solely an evenamount of actuators 𝑁 = 2, 4, … , 24 is used

This method obtains the phase shift values for anarbitrary even amount of actuators and examples canbe seen in Figure 12. All actuators can be distributedequally on the lowest layer of thematrix (𝑊 ). Becauseof the high pulse frequencies possible, the lowest vi-brations are expected. Furthermore, the power max-ima are always at the same level if all actuators are puton the same level of the matrix which should have apositive in luence on the operating smoothness.

8.2. Uneven Number of ActuatorsWith the intent to evenlydistribute thephase shifts

over both halves 0 ≤ 𝜑 < 𝜋 and 𝜋 ≤ 𝜑 < 2𝜋 of thecycle duration of the control voltage, the distribution

22


for an odd number of actuators can not be symmetri-cal. This can also be checked with equation 18. If it ishowever necessary to use an uneven amount of actua-tors, the phase shift for an actuator can be determinedusing

𝜑 = 𝑍2 − 1 2𝜋

𝑊 (21)

and

𝜑 = (𝑍 − 1) 2𝜋𝑊 . (22)

That way it is at least ensured that for the accordingpoints in time the torque load of inside the drive trainis minimal in the case that the other actuators can notbe distributed solely on the lowest layer and thereforethe maximum torques are unequal. The calculation ofthephase shift values for the remaining actuators𝑁−1can be done following themethod as described before.It is to be noted that when using an uneven amount ofactuators, the asymmetry has less of an effect themoreactuators are used.

9. ConclusionThe present article has pointed out some of the

gearbox induced friction effects that occur when twoactuators and a conventional controlmethod are used.The effects are resulting from sticky friction and dryfriction (producing among others the stick-slip effect),and effects that are based on the differences betweenthe effective torque of the twomotor operatingmodes(drive mode and brake mode) and the associated hys-teresis of the drive behavior. In order to compen-sate for those effects, irst a modi ied variant of clas-sic pulse modulated control (PMC) was examined. Itshowed typical side-effects such as increased vibra-tion and noise. Based upon the PMC, the novel phaseshifted pulse modulated control (PPMC) was intro-duced. The PPMC is derived from the PMC and lever-ages the possibilities of multiple actuators to improveon the PMC.

After presenting the experimental setup and thetest procedure, the trajectories of the test load (pen-dulum) that are resulting from the individual test runsfor the control methods reference control, PMC andPPMC have been discussed. Only looking at the tra-jectories gives irst clues to the bene its of using ei-ther PMC or PPMC over the reference method. A de-tailed evaluation of the control quality was given insubsequent sections based on the six quality criteriaenergy consumption, hysteresis, linearity, maximumtorque, operating smoothness, energy ef iciency andoperational noise. The various properties of the con-trol methods were then compared.

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

typical hysteresis effects from the conventional con-trol method can be compensated almost completelyand the linearity could be improved noticeably. More-over, because of the phase shifted control of both mo-tors, the PPMC exhibits signi icantly less vibration andnoise generation compared to the PMC. The only mea-sured disadvantage is the slightly increased energyconsumption that is3%higher than theunpulsed con-trol method.

Subsequently it has been described how the PPMCcan be generalized to drive trains with an arbitraryamount of parallel actuators. For this, each servo unitneeds its own phase shifted modulated pulse signalcontrolling themotor in succession to the others.Withthe obtained distribution of the phase shifts, a low-noise and low-vibration operation is ensured.

Both operating methods PMC and PPMC are espe-cially well suited for slow movements. When increas-ing the speed of movement, the in luence of the stick-slip effect to hysteresis is decreasing. Switching be-tween pulsed and unpulsed controllers is possible torely on the properties on other control methods.

AUTHORSTorsten Siedel – Space Applications Services NV/SA,Leuvensesteenweg 325, 1932 Zaventem, Belgium,e-mail: [email protected] Bethge – Department of Advanced Robotics,Istituto Italiano di Tecnologia, Via Morego 30, 16163Genoa, Italy, e-mail: [email protected] Hild – Neurorobotics Research Lab-oratory, Beuth-Hochschule fur Technik Berlin,e-mail: [email protected], www:http://neurorobotics.eu/.

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C A D M PP 𝛼- F R 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 offuzzy rela ons during aggrega on process is considered.It means that proper es of fuzzy rela ons 𝑅 ,… , 𝑅 ona set 𝑋 are compared with proper es of the aggregatedfuzzy rela on 𝑅 = 𝐹(𝑅 ,… , 𝑅 ), where 𝐹 is a func onof the type 𝐹 ∶ [0, 1] → [0, 1]. There are discussed𝛼-proper es (which may be called graded proper es -to some grade 𝛼) as reflexivity, irreflexivity, symmetry,asymmetry, an symmetry, connectedness and transi v-ity, where 𝛼 ∈ [0, 1]. Fuzzy rela ons with a given gradedproperty are analyzed (there may be diverse grades ofthe same property) and the obtained grade of the ag-gregated fuzzy rela on is provided. There is also dis-cussed the „converse” problem. Namely, rela on 𝑅 =𝐹(𝑅 ,… , 𝑅 ) is assumed to have a graded property andthe proper es of rela ons 𝑅 ,… , 𝑅 are examined (pos-sibly with some assump ons on 𝐹). Presented here con-sidera ons have possible applica ons in decision makingalgorithms. This is why interpreta on of the consideredgraded proper es and possible poten al in decisionmak-ing is presented.

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

1. Introduc onSince Zadeh has introduced de inition of fuzzy re-

lations [38], [39], the theory of themwas developed byseveral authors. Thanks to the „fuzzy environment”wemay discuss diverse types of fuzzy relation properties.For example, gradedproperties of fuzzy relationswereobserved in [23] and 𝛼-properties were introduced in[10]. These properties may be understood as proper-ties to some grade 𝛼, where 𝛼 ∈ [0, 1].

Aggregation functions, including means [24], arenow widely investigated and there are a few mono-graphes devoted to this topic, e.g. [2], [7], [22]. Aggre-gation is a fundamental process in multicriteria deci-sion making and in other scienti ic disciplines wherethe fusion of different pieces of information for ob-taining the inal result is important. For example, inthe multicriteria decision making a inite set of alter-natives 𝑋 = 𝑥 ,… , 𝑥 and a inite set of criteriaon the base of which the alternatives are evaluated𝐾 = 𝑘 ,… , 𝑘 may be considered. Fuzzy relations𝑅 ,… , 𝑅 on a set 𝑋 corresponding to each criterionare provided. With the use of a function 𝐹 the aggre-gated fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is obtainedand it is supposed to help decision makers to make

up their mind. It is useful to knowwhich properties offuzzy relations𝑅 ,… , 𝑅 are transposed to the relation𝑅. There are several works contributed to the problemof preservation of properties of fuzzy relations duringaggregation process, e.g. [21], [31], [32], [34].

In this paper theproblemofpreservationof gradedproperties of fuzzy relations (cf. [14], [16], [18]) is ex-amined. A inite number of fuzzy relations having agiven graded property is considered (there can be di-verse grades of the same property) and the obtainedgrade of the aggregated fuzzy relation is provided.There are discussed several graded properties: re-lexivity, irre lexivity, symmetry, asymmetry, antisym-metry, connectedness and transitivity. There is alsoconsidered another problem. Namely, relation 𝑅 =𝐹(𝑅 ,… , 𝑅 ) is assumed to have a graded propertyand relations 𝑅 ,… , 𝑅 are examined whether theyhave the same property. Appropriate assumptions on𝐹 to ful ill the required property are proposed. Pre-sented in this paper results may have applications indecision making problems what is more widely de-scribed in Section 3. Moreover, the interpretation ofthe graded properties in the context of decision mak-ing is provided.

The aimof this paper is also to compare three algo-rithms which follow from the theoretical results pre-sented here. These algorithms (their complexity) andtheoretical results (assumptions on functions used inaggregation process) are compared in order to obtainthemost useful practically result. The assumptions on𝐹 which are used to aggregate 𝑅 ,… , 𝑅 are the mini-mal ones, i.e. we do not necessarily consider aggrega-tion functions 𝐹 but just functions 𝐹 ∶ [0, 1] → [0, 1],which were recently called ’fusion functions’ [6]. If itcomes to complexity, it turned out that it is the samefor each presented algorithm (for a given property). Inthe case of assumptions on fusion functions𝐹 the situ-ation may be different what is analyzed in Section 7.1.

In Section 2, useful de initions are collected. InSection 3, motivation from real-life situations to con-sider such theoretical problem is presented. In Sec-tion 4, diverse dependencies and interpretation of 𝛼-properties are discussed. In Section 5, graded prop-erties: re lexivity, irre lexivity, symmetry, asymmetry,antisymmetry, connectedness and transitivity are ex-amined one by one, in the context of their preservationin aggregation process. In Section 6, reciprocity prop-erty and other concepts and properties connectedwith decision making algorithms are mentioned. Fi-nally, in Section 7 comparison of algorithms based onthe theoretical studies presented in this paper are pro-

25


vided.

2. PreliminariesNowwe recall some de initionswhichwill be help-

ful in our investigations.De inition 1 ([38]). A fuzzy relation in𝑋 ≠ ∅ is a func-tion 𝑅 ∶ 𝑋 × 𝑋 → [0, 1]. The family of all fuzzy relationsin 𝑋 is denoted by ℱℛ(𝑋).

With the use of 𝑛-argument functions 𝐹 we aggre-gate given fuzzy relations 𝑅 ,… , 𝑅 for a ixed 𝑛 ∈ .De inition 2 ([25]). Let 𝐹 ∶ [0, 1] → [0, 1],𝑅 ,… , 𝑅 ∈ ℱℛ(𝑋). 𝑅 ∈ ℱℛ(𝑋), where

𝑅 (𝑥, 𝑦) = 𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)), 𝑥, 𝑦 ∈ 𝑋,

will be called an aggregated fuzzy relation. A function𝐹 preserves a property of fuzzy relations if for every re-lation 𝑅 ,… , 𝑅 ∈ ℱℛ(𝑋) having this property, 𝑅 alsohas this property.

Example 1. Projections 𝑃 (𝑡 , … , 𝑡 ) = 𝑡 , 𝑘 ∈1,… , 𝑛 preserve each property of fuzzy relations be-cause for 𝐹 = 𝑃 we get 𝑅 = 𝑅 .

De inition 3 ([7]). Let 𝑛 ⩾ 2. A function 𝐹 ∶[0, 1] → [0, 1] is called an aggregation function, if itis increasing with respect to any variable, i.e. for any𝑠 , … , 𝑠 , 𝑡 , … , 𝑡 ∈ [0, 1]

( ∀⩽ ⩽

𝑠 ⩽ 𝑡 ) ⇒ 𝐹(𝑠 , … , 𝑠 ) ⩽ 𝐹(𝑡 , … , 𝑡 ) (1)

and 𝐹(0,… , 0) = 0, 𝐹(1,… , 1) = 1.De inition 4 ([15]). An operation 𝐶 ∶ [0, 1] → [0, 1]is called a fuzzy conjunction if it is increasing and

𝐶(1, 1) = 1, 𝐶(0, 0) = 𝐶(0, 1) = 𝐶(1, 0) = 0.

An operation 𝐷 ∶ [0, 1] → [0, 1] is called a fuzzy dis-junction if it is increasing and

𝐷(0, 0) = 0, 𝐷(1, 1) = 𝐷(0, 1) = 𝐷(1, 0) = 1.

Fuzzy conjunctions and disjunctions are examplesof binary aggregation functions. Conversely, if a binaryaggregation function has a zero element 𝑧 = 0 (as inthe case of the geometric mean), then it is a fuzzy con-junction. Similarly, if a binary aggregation function hasa zero element 𝑧 = 1, then we get a fuzzy disjunction.De inition 5. A fuzzy conjunction which has a neutralelement 1 is called a t-seminorm [20] (a semicopula [1],a conjunctor [9]). A fuzzy disjunction which has a neu-tral element 0 is called a t-semiconorm.

Corollary 1. If an operation 𝐵 ∶ [0, 1] → [0, 1] isincreasing and has a neutral element 1 (neutral ele-ment 0), then it is a fuzzy conjunction ful illing property𝐵(𝑥, 𝑦) ⩽ min(𝑥, 𝑦) (fuzzy disjunction ful illing prop-erty 𝐵(𝑥, 𝑦) ⩾ max(𝑥, 𝑦)).

Triangular norms and conorms are examples ofconjunctions and disjunctions having neutral element1 or 0, respectively.

De inition 6 ([28]). A triangular norm 𝑇 ∶ [0, 1] →[0, 1] (a triangular conorm 𝑆 ∶ [0, 1] → [0, 1]) is anarbitrary associative, commutative, increasing in bothvariables function having a neutral element 𝑒 = 1 (𝑒 =0).

Basic triangular norms and conorms are presentedbelow.Example 2 ([28], p. 6). For arbitrary 𝑠, 𝑡 ∈ [0, 1] wehave functions:• lattice, 𝑇 (𝑠, 𝑡) = min(𝑠, 𝑡), 𝑆 (𝑠, 𝑡) = max(𝑠, 𝑡),• Łukasiewicz, 𝑇 (𝑠, 𝑡) = max(𝑠 + 𝑡 − 1, 0),𝑆 (𝑠, 𝑡) = min(𝑠 + 𝑡, 1),• product, 𝑇 (𝑠, 𝑡) = 𝑠𝑡, 𝑆 (𝑠, 𝑡) = 𝑠 + 𝑡 − 𝑠𝑡,

• drastic, 𝑇 (𝑠, 𝑡) =0, 𝑠, 𝑡 < 1𝑠, 𝑡 = 1𝑡, 𝑠 = 1

,

𝑆 (𝑠, 𝑡) =1, 𝑠, 𝑡 > 0𝑠 𝑡 = 0𝑡, 𝑠 = 0

.

Thanks to the associativity property triangularnorms and conorms may be extended to 𝑛-argumentfunctions. Special case of aggregation functions are theones which are idempotent.Lemma 1 ([25], Proposition 5.1). Every function 𝐹 ∶[0, 1] → [0, 1] increasing in each variable and idem-potent

∀∈[ , ]

𝐹(𝑡, … , 𝑡) = 𝑡 (2)

ful ils for any 𝑡 , … , 𝑡 ∈ [0, 1]

min(𝑡 , … , 𝑡 ) ⩽ 𝐹(𝑡 , … , 𝑡 ) ⩽ max(𝑡 , … , 𝑡 ). (3)

Herewe present examples of functionswhich ful il(3).Example 3. Let 𝜑 ∶ [0, 1] → be a continuous, strictlymonotonic function. A quasi-linear mean (cf. [25], p.112) is the function

𝐹(𝑡 , … , 𝑡 ) = 𝜑 ( 𝑤 𝜑(𝑡 )), 𝑡 , … , 𝑡 ∈ [0, 1],

where ∑ 𝑤 = 1,𝑤 ∈ [0, 1]. Particularly, we obtain

weighted arithmetic means

𝐹(𝑡 , … , 𝑡 ) = 𝑤 𝑡 , 𝑡 , … , 𝑡 ∈ [0, 1],

where ∑ 𝑤 = 1,𝑤 ∈ [0, 1]. An aggregation function

𝐹(𝑡 , … , 𝑡 ) = 𝑝 max⩽ ⩽

𝑡 + (1 − 𝑝) min⩽ ⩽

𝑡 (4)

is idempotent, where 𝑝 ∈ (0, 1) is a parameter.

There are some connections between functions.For example, we may consider relation of dominanceof one function over another.

26


De inition 7 (cf. [36], [34]). Let 𝑚, 𝑛 ∈ . A function𝐹 ∶ [0, 1] → [0, 1] dominates a function 𝐺 ∶ [0, 1] →[0, 1] ( 𝐹 ≫ 𝐺), if for arbitrary matrix [𝑎 ] = 𝐴 ∈[0, 1] × we have

𝐹(𝐺(𝑎 ,… , 𝑎 ), … , 𝐺(𝑎 , … , 𝑎 )) ⩾

𝐺(𝐹(𝑎 ,… , 𝑎 ), … , 𝐹(𝑎 ,… , 𝑎 )). (5)

Lemma 2. Let 𝐺 ∶ [0, 1] → [0, 1] be increasing,𝑚 =2 (cf. (5)). Thus min ≫ 𝐺 ([34], p. 16) and 𝐺 ≫ max(cf. [11], Theorem 2), so for 𝑠 , ..., 𝑠 , 𝑡 , ..., 𝑡 ∈ [0, 1]wehave respectively

min(𝐺(𝑠 , ..., 𝑠 ), 𝐺(𝑡 , ..., 𝑡 )) ⩾

𝐺(min(𝑠 , 𝑡 ), ..., min(𝑠 , 𝑡 )) (6)and

𝐺(max(𝑠 , 𝑡 ), ..., max(𝑠 , 𝑡 )) ⩾

max(𝐺(𝑠 , ..., 𝑠 ), 𝐺(𝑡 , ..., 𝑡 )). (7)

Theorem 1. An increasing in each variable function𝐹 ∶ [0, 1] → [0, 1] dominates minimum (𝐹 ≫ min)if and only if

𝐹(𝑡 , … , 𝑡 ) = min(𝑓 (𝑡 ), … , 𝑓 (𝑡 )), 𝑡 , … , 𝑡 ∈ [0, 1],(8)

where functions 𝑓 ∶ [0, 1] → [0, 1] are increasing for𝑘 = 1,… , 𝑛 (cf. [34], Proposition 5.1).An increasing in each variable function 𝐹 ∶ [0, 1] →[0, 1] is dominated by maximum (max ≫ 𝐹) if and onlyif

𝐹(𝑡 , … , 𝑡 ) = max(𝑓 (𝑡 ), … , 𝑓 (𝑡 )), 𝑡 , … , 𝑡 ∈ [0, 1],(9)

where functions 𝑓 ∶ [0, 1] → [0, 1] are increasing for𝑘 = 1,… , 𝑛.

Example 4 (cf. [31]). Here are examples of functionsful illing (8):if 𝑓 (𝑡) = 𝑡, 𝑘 = 1,… , 𝑛, then 𝐹 = min,if for some 𝑘 ∈ 1,… , 𝑛, 𝑓 (𝑡) = 𝑡, 𝑓(𝑡) = 1 for 𝑖 ≠ 𝑘,then 𝐹 = 𝑃 ,if 𝑓 (𝑡) = max(1 − 𝑣 , 𝑡), 𝑣 ∈ [0, 1], 𝑘 = 1,… , 𝑛,max⩽ ⩽

𝑣 = 1, then 𝐹 is the weighted minimum

𝐹(𝑡 , … , 𝑡 ) = min⩽ ⩽

max(1 − 𝑣 , 𝑡 ), (10)

where 𝑡 = (𝑡 , … , 𝑡 ) ∈ [0, 1] .Here are examples of functions ful illing (9):if 𝑓 (𝑡) = 𝑡, 𝑘 = 1,… , 𝑛, then 𝐹 = max,if for some 𝑘 ∈ 1,… , 𝑛, 𝑓 (𝑡) = 𝑡, 𝑓(𝑡) = 0 for 𝑖 ≠ 𝑘,then 𝐹 = 𝑃 ,if 𝑓 (𝑡) = min(𝑣 , 𝑡), 𝑣 ∈ [0, 1], 𝑘 = 1,… , 𝑛,max⩽ ⩽

𝑣 = 1, then 𝐹 is the weighted maximum

𝐹(𝑡 , … , 𝑡 ) = max⩽ ⩽

min(𝑣 , 𝑡 ), (11)

where 𝑡 = (𝑡 , … , 𝑡 ) ∈ [0, 1] .

Lemma 3 (cf. [18]). If a function 𝐹 ∶ [0, 1] → [0, 1]is increasing in each variable and has a neutral element𝑒 = 1, i.e.

∀∈[ , ]

∀⩽ ⩽

𝐹(1,… , 1, 𝑡, 1, … , 1) = 𝑡, (12)

where 𝑡 is at the 𝑘-th position, then 𝐹 ⩽ min.If a function 𝐹 ∶ [0, 1] → [0, 1] is increasing in eachvariable and has a neutral element 𝑒 = 0, i.e.

∀∈[ , ]

∀⩽ ⩽

𝐹(0,… , 0, 𝑡, 0, … , 0) = 𝑡, (13)

where 𝑡 is at the 𝑘-th position, then 𝐹 ⩾ max.Here are recalled de initions of concepts con-

nected with fuzzy relations.De inition 8 (cf. [38]). Let 𝑅 ∈ ℱℛ(𝑋), 𝛼 ∈ [0, 1]. The𝛼-cut of a fuzzy relation 𝑅 is the relation

𝑅 = (𝑥, 𝑦) ∈ 𝑋 × 𝑋 ∶ 𝑅(𝑥, 𝑦) ⩾ 𝛼. (14)

The strict 𝛼-cut of a fuzzy relation 𝑅 is the relation

𝑅 = (𝑥, 𝑦) ∈ 𝑋 × 𝑋 ∶ 𝑅(𝑥, 𝑦) > 𝛼. (15)

De inition 9 (cf. [38]). Let 𝑅, 𝑆 ∈ ℱℛ(𝑋). The compo-sition of relations 𝑅 and 𝑆 is called the relation

(𝑅∘𝑆)(𝑥, 𝑧) = sup∈

min(𝑅(𝑥, 𝑦), 𝑆(𝑦, 𝑧)), (𝑥, 𝑧) ∈ 𝑋×𝑋.

(16)The power of a relation 𝑅 is called the sequence 𝑅 = 𝑅and 𝑅 = 𝑅 ∘ 𝑅 for 𝑛 ∈ .

Remark 1. If 𝑐𝑎𝑟𝑑 𝑋 = 𝑛, 𝑋 = 𝑥 ,… , 𝑥 , then arelation 𝑅 ∈ ℱℛ(𝑋)may be presented by a matrix 𝑅 =[𝑟 ], where 𝑟 = 𝑅(𝑥 , 𝑥 ), 𝑖, 𝑘 = 1,… , 𝑛.

3. Mo va onIn this section the idea ofmulticriteria (or similarly

multiagent) decision making is recalled. Presentedproblem is related to considerations provided in thispaper. Fuzzy relations in such setting represent thepreferences.

Let 𝑐𝑎𝑟𝑑 𝑋 = 𝑚,𝑚 ∈ , 𝑋 = 𝑥 ,… , 𝑥 be a set ofalternatives. In multicriteria decision making a deci-sion maker has to choose among the alternatives withrespect to a set of criteria. Let 𝐾 = 𝑘 ,… , 𝑘 be theset of criteria on the base of which the alternatives areevaluated. 𝑅 ,… , 𝑅 be fuzzy relations correspondingto each criterion represented by matrices, where𝑅 ∶ 𝑋 × 𝑋 → [0, 1], 𝑘 = 1,… , 𝑛, 𝑛 ∈ , 𝑅 (𝑥 , 𝑥 ) = 𝑟 ,1 ⩽ 𝑖, 𝑗 ⩽ 𝑚. We assume that for example:𝑟 – an intensity with which 𝑥 is better than 𝑥 under𝑘 ∈ 𝐾,𝑟 = 1 – „𝑥 is absolutely better than 𝑥 undercriterion 𝑘”,𝑟 = 0 – „𝑥 is absolutely better than 𝑥 undercriterion 𝑘”,𝑟 = 0.5 – „𝑥 is equally good as 𝑥 under criterion 𝑘”,so it is natural that 𝑟 = 0.5.

27


Similarly, if we consider multiagent decision mak-ing problems, relations𝑅 ,… , 𝑅 represent the prefer-ences of each agent and no criteria (certainly, we cancombine these two situations, i.e. many criteria andmany agents).

Relation𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is supposed tohelp thedecision maker to make up his/her mind. Some func-tions𝐹maybemore adequate for aggregation than theothers since they may (or not) preserve the requiredproperties of individual fuzzy relations 𝑅 ,… , 𝑅 . Ac-cording to some experimental works [40] weightedarithmetic mean and function (4) are the aggregationfunctions which occur themost often in the process ofhuman decision making. Such properties, if they areful illed by fuzzy relations, may be a form of measureof consistency of choices or may provide the interpre-tation of choices. This is why preservation of theseproperties may be interested and required in aggre-gation process formulticriteria ormultiagent decisionmaking problems.

Application of similar considerations by a numer-ical example is presented in [32] where the choice orranking problems of a set of alternatives evaluated byfuzzy preference relations using the aggregation func-tions are considered. It is shown howproperties of theaggregated fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ), depend-ing on the properties of the individual fuzzy relations𝑅 ,… , 𝑅 , help to solve the given problem. However, inthat paper it is stressed also another problem, namelythe sensitivity of the aggregation operators with re-spect to variations in their arguments. In that paperseveral weighted aggregation operators, i.e. operatorswhich use the importance of criteria, given as weights,are considered.

In the presented multicriteria or multiagent deci-sion making problems it is sometimes required thatthe given fuzzy relations representing the preferencesare reciprocal, i.e. fuzzy relation 𝑅 in 𝑋 is reciprocal if𝑅(𝑥, 𝑦) + 𝑅(𝑦, 𝑥) = 1 for 𝑥, 𝑦 ∈ 𝑋. However, if 𝑅 is notreciprocal, there aremethods to transform it to the re-ciprocal one [3].

4. Graded Proper es of Fuzzy Rela onsNow, dependencies related to 𝛼-properties in the

context of aggregation process, between relations𝑅 ,… , 𝑅 on a set 𝑋 and the aggregated fuzzy rela-tion𝑅 = 𝐹(𝑅 ,… , 𝑅 )will be investigated. Moreover,some previous results will be recalled.

De inition 10 ([10], p. 75, [18]). Let 𝛼 ∈ [0, 1]. A re-lation 𝑅 ∈ ℱℛ(𝑋) is:- 𝛼-re lexive, if ∀

∈𝑅(𝑥, 𝑥) ⩾ 𝛼,

- 𝛼-irre lexive, if ∀∈

𝑅(𝑥, 𝑥) ⩽ 1 − 𝛼,

- totally 𝛼-connected, if ∀, ∈

max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) ⩾𝛼,

- 𝛼-connected, if ∀, , ∈

max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) ⩾ 𝛼,

- 𝛼-asymmetric, if ∀, ∈

min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) ⩽ 1−𝛼,

- 𝛼-antisymmetric, if ∀, , ∈

min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) ⩽1 − 𝛼,

- 𝛼-symmetric, if ∀, ∈

𝑅(𝑥, 𝑦) ⩾ 1 − 𝛼 ⇒ 𝑅(𝑦, 𝑥) ⩾𝑅(𝑥, 𝑦),

- 𝛼-transitive, if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑧)) ⩾ 1 − 𝛼 ⇒ 𝑅(𝑥, 𝑧) ⩾min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑧)).

Let us notice that conditions for 𝛼-symmetry and𝛼-transitivity may be written in a more convenientway.

Corollary 2. Let 𝛼 ∈ [0, 1]. A relation 𝑅 ∈ ℱℛ(𝑋) is𝛼-symmetric if and only if

∀, ∈

𝑅(𝑥, 𝑦) ⩾ 1 − 𝛼 ⇒ 𝑅(𝑦, 𝑥) = 𝑅(𝑥, 𝑦). (17)

Corollary 3 (cf. [13], Theorem 10). Let 𝑅 ∈ ℱℛ(𝑋),𝛼 ∈ [0, 1]. Relation 𝑅 is 𝛼-transitive if and only if

𝑅 ⩾ 1 − 𝛼 ⇒ 𝑅 ⩾ 𝑅 . (18)

Corollary 4. Let 𝑅 ∈ ℱℛ(𝑋), 𝛽 ∈ [0, 1]. If relation 𝑅 is𝛽-𝑃, then it is 𝛼-𝑃 for any 𝛼 ∈ [0, 𝛽], where 𝑃: re lexiv-ity, irre lexivity, symmetry, asymmetry, antisymmetry,connectedness, total connectedness, transitivity.

Proof. Let 𝛼 ⩽ 𝛽. We use the fact, which is easy to seefor each property 𝛼-𝑃, where 𝑃: re lexivity, irre lex-ivity, symmetry, asymmetry, antisymmetry, connect-edness, total connectedness, transitivity, that if 𝑅 ∈ℱℛ(𝑋) is 𝛽-𝑃, then it is 𝛼-𝑃.

If we have a reciprocal fuzzy relation 𝑅 describ-ing preferences, then the properties in De inition 10may provide some practical information according tothe preferences over the given set of alternatives. Forexample, the 0.5-asymmetry of a reciprocal fuzzy re-lation guarantees that at least one of the alternatives𝑥 or 𝑥 is preferred to the other one with the fuzzyvalue lower than or equal to 0.5 (or these alterna-tives are indifferent), which means that if 𝑥 is pre-ferred to 𝑥 , then it is not true that 𝑥 is preferred to𝑥 . This interpretation of 0.5-asymmetry for a recipro-cal fuzzy relation is analogous to the one of asymme-try for crisp relations (i.e., if element 𝑥 is in relationwith 𝑥 , then it is not true that 𝑥 is in relation with𝑥 [33]). Similarly, we can interpret the other proper-ties. For 𝛼-connectedness, the greater the value of 𝛼(namely, the closer it is to the value1), the choice of thealternative is more precise (con ident/sure). For pref-erence relations, if it comes to 𝛼-re lexivity, practicallyonly 0.5-re lexivity occurs and with the given de ini-tion, if 𝑅 is 0.5-re lexive then it is automatically 0.5-irre lexive. Moreover, reciprocal preference relationis always totally 0.5-connected and 0.5-asymmetric.Since the ixed value of 0.5 on the diagonal may un-derstate or in late the value of 𝛼 for these proper-ties, it makes sense to distinguish total connectednessand connectedness and similarly, asymmetry and an-tisymmetry. 𝑅 ∈ ℱℛ(𝑋) has the highest value of 𝛼-symmetry for preference relation 𝑅 in the case when

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all elements in the set of alternatives𝑋 are indifferent.In fact, in such situation relation 𝑅 is symmetric (it is𝛼-symmetric for𝛼 ∈ [0, 1]).Moreover,we have the fol-lowing statement: if 𝑅 ∈ ℱℛ(𝑋) is reciprocal, then 𝑅is totally 𝛼-connected (𝛼-connected) if and only if 𝑅 is𝛼-asymmetric (𝛼-antisymmetric). That is not the casefor𝑅 ∈ ℱℛ(𝑋)which is not reciprocal (cf. Example 5).In the sequelwewill present the results in general set-ting of fuzzy relations, sometimes with the commentson reciprocal preference relations.

For practical reasons it is useful to ind the great-est value of 𝛼 for which 𝑅 ∈ ℱℛ(𝑋) is 𝛼–𝑃 for a givenproperty 𝑃: re lexivity, irre lexivity, symmetry, asym-metry, antisymmetry, connectedness, total connect-edness, transitivity. Applying de initions of the givenproperties and Corollary 4 one can ind this value inthe following way.Corollary 5. Let 𝑅 ∈ ℱℛ(𝑋),

𝛼 = 1 − sup, ∈

min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)),

𝛽 = 1 − supmin(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)),

𝛾 = inf, ∈

max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)),

𝛿 = inf max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)),

𝜇 = inf∈𝑅(𝑥, 𝑥),

𝜈 = inf∈(1 − 𝑅(𝑥, 𝑥)) = 1 − sup

∈𝑅(𝑥, 𝑥).

Thus a relation 𝑅 is: 𝛼–asymmetric for 𝛼 ∈ [0, 𝛼 ],𝛽–antisymmetric for 𝛽 ∈ [0, 𝛽 ], totally 𝛾–connectedfor 𝛾 ∈ [0, 𝛾 ], 𝛿–connected for 𝛿 ∈ [0, 𝛿 ], 𝜇–re lexivefor 𝜇 ∈ [0, 𝜇 ] and 𝜈–irre lexive for 𝜈 ∈ [0, 𝜈 ].

For symmetry and transitivity we have adequatehalf-closed intervals. Moreover, for checking the 𝛼-transitivity of a fuzzy relation 𝑅, the composition of 𝑅by itself will be useful.Corollary 6. Let 𝑅 ∈ ℱℛ(𝑋). Thus 𝑅 is 𝛼-symmetricfor 𝛼 ∈ [0, 1] if 𝑅(𝑥, 𝑦) = 𝑅(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋 or 𝑅 is𝛼-symmetric for 𝛼 ∈ [0, 𝛼 ) if there exist 𝑥, 𝑦 ∈ 𝑋 suchthat 𝑅(𝑥, 𝑦) ≠ 𝑅(𝑦, 𝑥), where

𝛼 = 1 − sup( , ) ( , ), , ∈

𝑅(𝑥, 𝑦).

𝑅 is 𝛽-transitive for 𝛽 ∈ [0, 1] if 𝑅 (𝑥, 𝑦) ⩽ 𝑅(𝑥, 𝑦) forall 𝑥, 𝑦 ∈ 𝑋 or 𝑅 is 𝛽-transitive for 𝛽 ∈ [0, 𝛽 ) if thereexist 𝑥, 𝑦 ∈ 𝑋 such that 𝑅(𝑥, 𝑦) < 𝑅 (𝑥, 𝑦), where

𝛽 = 1 − sup( , ) ( , ), , ∈

𝑅 (𝑥, 𝑦).

Example 5. Let 𝑐𝑎𝑟𝑑 𝑋 = 2, 𝑅 ∈ ℱℛ(𝑋), where

𝑅 = 0.7 0.20.5 0.4 .

The relation 𝑅 is totally 𝛼–connected and 𝛼-re lexivefor 𝛼 ∈ [0, 0.4] and 𝛼–connected for 𝛼 ∈ [0, 0.5]. Itis 𝛼-asymmetric and 𝛼-irre lexive for 𝛼 ∈ [0, 0.3] and

𝛼-antisymmetric for 𝛼 ∈ [0, 0.8]. 𝑅 is 𝛼-symmetric for𝛼 ∈ [0, 0.5) and 𝛼-transitive for 𝛼 ∈ [0, 1] (it followsfrom the fact that 𝑅 = 𝑅).Let 𝑐𝑎𝑟𝑑 𝑋 = 3. We consider 𝑅 ∈ ℱℛ(𝑋) which is re-ciprocal, where

𝑅 =0.5 0.8 0.30.2 0.5 0.40.7 0.6 0.5

, 𝑅 =0.5 0.5 0.40.4 0.5 0.40.5 0.7 0.5

.

The relation 𝑅 is totally 𝛼–connected and 𝛼-re lexivefor 𝛼 ∈ [0, 0.5] and 𝛼–connected for 𝛼 ∈ [0, 0.6]. Itis 𝛼-asymmetric and 𝛼-irre lexive for 𝛼 ∈ [0, 0.5] and𝛼-antisymmetric for 𝛼 ∈ [0, 0.6]. 𝑅 is 𝛼-symmetric for𝛼 ∈ [0, 0.2) and 𝛼-transitive for 𝛼 ∈ [0, 0.3).

Remark 2. The presented 𝛼-properties (graded prop-erties) for 𝛼 = 1 become the basic properties of fuzzyrelations [39]. Graded properties are „fuzzy versions” ofproperties introduced by Zadeh. It means that, if a fuzzyrelation, e.g. is not re lexive, it may be re lexive to somegrade 𝛼, where 𝛼 ∈ [0, 1].

Remark 3. Taking into account 𝛼 = 0, each fuzzyrelation is 0-re lexive, 0-irre lexive, 0-asymmetric, 0-antisymmetric, 0-connected and totally 0-connected.However, it is not true for graded symmetry and tran-sitivity. If in Corollary 6, 𝛼 = 0 (or similarly 𝛽 = 0),then 𝑅 is not 𝛼-symmetric for any 𝛼 ∈ [0, 1] (𝑅 is not𝛽-transitive for any 𝛽 ∈ [0, 1]).

Example 6. Let card 𝑋 = 3, relations 𝑅, 𝑆 ∈ ℱℛ(𝑋) bepresented by matrices:

𝑅 =0 0 10 0 01 0 0

, 𝑆 =0 0 00 0 01 0 0

.

The relation 𝑅 is not 0-transitive becausemin(𝑟 , 𝑟 ) = 1 but 0 = 𝑟 < min(𝑟 , 𝑟 ) = 1.The relation 𝑆 is not 0-symmetric because 𝑠 = 1 and0 = 𝑠 < 𝑠 = 1.

Notions of𝛼-properties have their connectionwithcuts and strict cuts of a fuzzy relation.

Theorem 2 (cf. [17]). Let 𝛼 ∈ [0, 1], 𝑅 ∈ ℱℛ(𝑋).A fuzzy relation 𝑅 is totally 𝛼-connected (𝛼-connected,𝛼-re lexive) if and only if relation 𝑅 is totally con-nected (connected, re lexive). A fuzzy relation 𝑅 is 𝛼-asymmetric (𝛼-antisymmetric, 𝛼-irre lexive) if and onlyif relation 𝑅 is asymmetric (antisymmetric, irre lex-ive). If a fuzzy relation 𝑅 is 𝛼-transitive, then relation𝑅 is transitive. If a fuzzy relation 𝑅 is 𝛼-symmetric,then relation 𝑅 is symmetric.

Similar characterizations for other properties forfuzzy relations one may ind in [12] (Theorem 1). Theconditions for 𝛼-symmetry and 𝛼-transitivity are onlythe suf icient ones.

Example 7 (cf. [17]). Let card 𝑋 = 2, 𝑅 ∈ ℱℛ(𝑋),

𝑅 = 0.3 0.50.7 0.4 .

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The cuts 𝑅 are symmetric for 𝛽 ∈ [0, 0.5] ∪ (0.7, 1],so the cuts 𝑅 have this property for 𝛼 ⩾ 0.5 and𝛼 < 0.3. Relation 𝑅 is 𝛼-symmetric for 𝛼 ∈ [0, 0.3), as aresult for 𝛼 = 0.5, the cut 𝑅 . is symmetric, while 𝑅 isnot 0.5-symmetric.

Let 𝑅 ∈ ℱℛ(𝑋), card 𝑋 = 3,

𝑅 =0.7 0 00.8 0.9 00.6 0.9 0.8

, 𝑆 = 𝑅 =0.7 0 00.8 0.9 00.8 0.9 0.8

.

The cuts 𝑅 are transitive for 𝛽 ∈ [0, 0.6] ∪ (0.8, 1],so the cuts 𝑅 have this property for 𝛼 ∈ [0, 0.2) ∪[0.4, 1]. Since 0.8 = 𝑠 ⩾ 1 − 𝛼 for 𝛼 ∈ [0.4, 1] and𝑠 = 0.8 > 0.6 = 𝑟 , relation 𝑅 is not 𝛼-transitivefor 𝛼 ∈ [0.4, 1] (it is 𝛼-transitive for 𝛼 ∈ [0, 0.2), seeCorollary 6).

Other results describing gradedproperties one canind in [10] (p. 78–79).

5. Aggrega on of Fuzzy Rela onsIn this section we will present 𝛼-properties of

fuzzy relations and diverse approaches of aggregatingsuch relations. There will be presented the followingtype of theorems for aggregated fuzzy relation 𝑅 :- aggregation of𝑅 ,… , 𝑅 all having the same grade ofa given 𝛼-property to obtain𝑅 with the same grade𝛼,

- aggregation of 𝑅 ,… , 𝑅 with possible diversegrades 𝛼 ,… , 𝛼 of a given graded property toobtain 𝑅 with the suitable grade 𝛼,

- starting from 𝑅 having some grade 𝛼 and check-ing whether 𝑅 ,… , 𝑅 all have the same grade 𝛼 ofa given graded property.

5.1. ReflexivityGraded re lexivity was considered by many au-

thors, e.g. [8], [10].

Theorem 3 ([16]). Let 𝛼 ∈ [0, 1]. 𝐹 ∶ [0, 1] → [0, 1]preserves 𝛼-re lexivity of fuzzy relations, if and only if

𝐹|[ , ] ⩾ 𝛼.

Theorem 4 ([16]). 𝐹 ∶ [0, 1] → [0, 1] preserves 𝛼-re lexivity of fuzzy relations for arbitrary 𝛼 ∈ [0, 1] ifand only if 𝐹 ⩾ min.

By Lemma 1 we know that every increasing andidempotent function preserves 𝛼-re lexivity of fuzzyrelations for arbitrary 𝛼 ∈ [0, 1]. In particular, we get

Corollary 7. Quasi-linearmeans preserve 𝛼-re lexivityof fuzzy relations for any 𝛼 ∈ [0, 1].

Theorem 5. Let 𝛼 ,… , 𝛼 ∈ [0, 1], a function 𝐹 ∶[0, 1] → [0, 1] be increasing in each variable. If re-lations 𝑅 ∈ ℱℛ(𝑋) are 𝛼 -re lexive for 𝑖 = 1,… , 𝑛,then relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-re lexive for 𝛼 =𝐹(𝛼 ,… , 𝛼 ).

Proof. Let 𝛼 ,… , 𝛼 ∈ [0, 1], a function 𝐹 ∶ [0, 1] →[0, 1] be increasing in each variable, 𝑅 ∈ ℱℛ(𝑋) be𝛼 -re lexive for 𝑖 = 1,… , 𝑛, 𝑥 ∈ 𝑋. Then

𝑅(𝑥, 𝑥) = 𝐹(𝑅 (𝑥, 𝑥), … , 𝑅 (𝑥, 𝑥)) ⩾ 𝐹(𝛼 ,… , 𝛼 ),

so relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-re lexive for 𝛼 =𝐹(𝛼 ,… , 𝛼 ).

Each aggregation function is increasing, so we get

Corollary 8. Let 𝛼 ,… , 𝛼 ∈ [0, 1], 𝐹 ∶ [0, 1] → [0, 1]be an aggregation function. If relations 𝑅 ∈ ℱℛ(𝑋)are 𝛼 -re lexive for 𝑖 = 1,… , 𝑛, then relation 𝑅 =𝐹(𝑅 ,… , 𝑅 ) is 𝛼-re lexive for 𝛼 = 𝐹(𝛼 ,… , 𝛼 ).

Theorem6. Let𝛼 ∈ [0, 1] and𝐹 ⩽ min. If a fuzzy rela-tion 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-re lexive, then all relations𝑅 ,… , 𝑅 are 𝛼-re lexive.

Proof. Let 𝛼 ∈ [0, 1], 𝐹 ⩽ min, 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) be𝛼-re lexive, 𝑥 ∈ 𝑋, 𝑘 ∈ 1,… , 𝑛. Then

𝑅 (𝑥, 𝑥) ⩾ min⩽ ⩽

𝑅 (𝑥, 𝑥) ⩾

𝐹(𝑅 (𝑥, 𝑥), … , 𝑅 (𝑥, 𝑥)) ⩾ 𝛼.As a result relation 𝑅 is 𝛼-re lexive.

In virtue of Lemma 3 we get

Corollary 9. Let 𝛼 ∈ [0, 1], 𝐹 be a t-seminormor a t-norm. If a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 )is 𝛼-re lexive, then all relations 𝑅 ,… , 𝑅 are also 𝛼-re lexive.

The next example shows that the condition pre-sented in Theorem 6 is only suf icient.

Example 8. Let 𝑐𝑎𝑟𝑑 𝑋 = 2. We consider fuzzy rela-tions with matrices:

𝑅 = 0 11 1 , 𝑆 = 1 1

1 0 ,

𝑊 = max(𝑅, 𝑆) = 1 11 1 ,

𝑊 = 𝑅 + 𝑆2 = 0.5 1

1 0.5 .

Relation 𝑊 is 𝛼-re lexive for 𝛼 ∈ [0, 1], 𝑊 for 𝛼 ∈[0, 0.5], but relations 𝑅, 𝑆 do not have this property forany 𝛼 ∈ (0, 1].

5.2. IrreflexivityFor irre lexivity, generallywe get dual results to re-

lexivity.

Theorem 7 ([16]). Let 𝛼 ∈ [0, 1]. A function 𝐹 ∶[0, 1] → [0, 1] preserves 𝛼-irre lexivity of fuzzy rela-tions if and only if

𝐹|[ , ] ⩽ 1 − 𝛼.

Theorem 8 ([16]). A function 𝐹 ∶ [0, 1] → [0, 1]preserves 𝛼-irre lexivity of fuzzy relations for arbitrary𝛼 ∈ [0, 1] if and only if 𝐹 ⩽ max.

30


Corollary 10. Quasi–linear means preserve 𝛼-irre lexivity of fuzzy relations for arbitrary 𝛼 ∈ [0, 1].De inition 11 (cf. [7]). A function 𝐹 ∶ [0, 1] → [0, 1]is super additive, if for all 𝑖 = 1,… , 𝑛 and all 𝑥 , 𝑦 , 𝑥 +𝑦 ∈ [0, 1]𝐹(𝑥 + 𝑦 ,… , 𝑥 + 𝑦 ) ⩾ 𝐹(𝑥 ,… , 𝑥 ) + 𝐹(𝑦 ,… , 𝑦 ).

(19)Example 9. Weighted arithmetic means andminimumare super additive functions.Theorem 9. Let 𝛼 ,… , 𝛼 ∈ [0, 1], 𝐹 ∶ [0, 1] → [0, 1]be a super additive aggregation function. If relations𝑅 ∈ ℱℛ(𝑋) are 𝛼 -irre lexive for 𝑖 = 1,… , 𝑛, thenrelation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-irre lexive for 𝛼 =𝐹(𝛼 ,… , 𝛼 ).Proof. Let 𝛼 ,… , 𝛼 ∈ [0, 1], 𝐹 ∶ [0, 1] → [0, 1] be asuper additive aggregation function, 𝑅 ∈ ℱℛ(𝑋) be𝛼 -irre lexive for 𝑖 = 1,… , 𝑛, 𝑥 ∈ 𝑋. Then 𝑅 (𝑥, 𝑥) +𝛼 ⩽ 1, so

𝐹(𝑅 (𝑥, 𝑥), … , 𝑅 (𝑥, 𝑥)) + 𝐹(𝛼 ,… , 𝛼 )⩽ 𝐹(𝑅 (𝑥, 𝑥)+𝛼 ,… , 𝑅 (𝑥, 𝑥)+𝛼 ) ⩽ 𝐹(1,… , 1) = 1.As a result

𝐹(𝑅 (𝑥, 𝑥), … , 𝑅 (𝑥, 𝑥)) ⩽ 1 − 𝐹(𝛼 ,… , 𝛼 ),so 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-irre lexive for 𝛼 =𝐹(𝛼 ,… , 𝛼 ).Corollary 11. Let 𝛼 ,… , 𝛼 ∈ [0, 1]. If relations 𝑅 ∈ℱℛ(𝑋) are 𝛼 -irre lexive for 𝑖 = 1,… , 𝑛, then relation

𝑅 = ∑ 𝑤 𝑅 is 𝛼-irre lexive, where ∑ 𝑤 = 1, 𝑤 ∈

[0, 1] and 𝛼 = ∑ 𝑤 𝛼 .

Analogously to re lexivity we obtain the followingresult.Theorem 10. Let 𝛼 ∈ [0, 1] and 𝐹 ⩾ max. If a fuzzyrelation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-irre lexive, then all re-lations 𝑅 ,… , 𝑅 are also 𝛼-irre lexive.

In virtue of Lemma 3 we getCorollary 12. Let 𝛼 ∈ [0, 1], 𝐹 be a t-conorm or at-semiconorm. If a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 )is 𝛼-irre lexive, then all relations 𝑅 ,… , 𝑅 are also 𝛼-irre lexive.

The next example shows that the condition givenin Theorem 10 is only suf icient.Example 10. Let 𝑐𝑎𝑟𝑑 𝑋 = 2. We consider fuzzy rela-tions with matrices:

𝑅 = 0 11 1 , 𝑆 = 1 1

1 0 ,

𝑊 = min(𝑅, 𝑆) = 0 11 0 ,

𝑊 = 𝑅 + 𝑆2 = 0.5 1

1 0.5 .

Relation 𝑊 is 𝛼-irre lexive for 𝛼 ∈ [0, 1], 𝑊 for 𝛼 ∈[0, 0.5], but relations 𝑅, 𝑆 do not have this property forany 𝛼 ∈ (0, 1].

5.3. ConnectednessHere graded connectedness and total connected-

ness will be examined. The total 0.5-connectednesswas regarded in [32] (p. 619). In that paper this prop-erty is called weak comparability. It was shown therethat maximum preserves the total 0.5-connectedness([32], Table 1).Theorem 11 ([16]). Let 𝛼 ∈ [0, 1], 𝑐𝑎𝑟𝑑 𝑋 ⩾ 2.A function 𝐹 ∶ [0, 1] → [0, 1] preserves total 𝛼-connectedness (𝛼-connectedness) of fuzzy relations, ifand only if for any 𝑠, 𝑡 ∈ [0, 1]

( ∀⩽ ⩽

max(𝑠 , 𝑡 ) ⩾ 𝛼) ⇒ max(𝐹(𝑠), 𝐹(𝑡)) ⩾ 𝛼.

Theorem 12 ([16]). Let 𝑐𝑎𝑟𝑑 𝑋 ⩾ 2. A function𝐹 ∶ [0, 1] → [0, 1] preserves total 𝛼-connectedness(𝛼-connectedness) of fuzzy relations for arbitrary 𝛼 ∈[0, 1], if and only if

∀, ∈[ , ]

max(𝐹(𝑠), 𝐹(𝑡)) ⩾ min⩽ ⩽

max(𝑠 , 𝑡 ).

Corollary 13. Maximum and the weighted maximumpreserve total 𝛼-connectedness (𝛼-connectedness) offuzzy relations for arbitrary 𝛼 ∈ [0, 1].Theorem 13. Let 𝛼 ,… , 𝛼 ∈ [0, 1], a function 𝐹 ∶[0, 1] → [0, 1] be increasing in each variable andmax ≫ 𝐹. If relations 𝑅 ∈ ℱℛ(𝑋) are totally 𝛼 -connected (𝛼 -connected) for 𝑖 = 1,… , 𝑛, then rela-tion 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is totally 𝛼-connected (𝛼-connected) for 𝛼 = 𝐹(𝛼 ,… , 𝛼 ).Proof. Let 𝛼 ,… , 𝛼 ∈ [0, 1], a function 𝐹 ∶ [0, 1] →[0, 1]be increasing in each variable,max ≫ 𝐹 and𝑅 ∈ℱℛ(𝑋)be𝛼 -connected for 𝑖 = 1,… , 𝑛, 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦.Then by Lemma 2 and by the fact that max ≫ 𝐹 weobtain

max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) =max(𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)),𝐹(𝑅 (𝑦, 𝑥), … , 𝑅 (𝑦, 𝑥))) ⩾𝐹(max(𝑅 (𝑥, 𝑦), 𝑅 (𝑥, 𝑦)), … ,max(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑥))) ⩾

𝐹(𝛼 ,… , 𝛼 ) = 𝛼.It means that a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is𝛼-connected for 𝛼 = 𝐹(𝛼 ,… , 𝛼 ). Proof for total 𝛼-connectedness is analogous.

We can also compute the value of 𝛼 for which afuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-connected (to-tally 𝛼-connected) for concrete functions 𝐹 in anotherway than it is presented in Theorem 13. It is shown inthe following example.Example 11. Let 𝛼 ,… , 𝛼 ∈ [0, 1]. If relations 𝑅 ∈ℱℛ(𝑋) are𝛼 –connected (totally𝛼 –connected) for 𝑖 =1,… , 𝑛, then relation 𝑅 ∈ ℱℛ(𝑋) is 𝛼-connected (to-tally 𝛼-connected), where

𝑅 = 1𝑛 𝑅 , 𝛼 = 1

𝑛 max⩽ ⩽

𝛼 .

31


Note that the arithmeticmean is not dominated bymax-imum.

Theorem 14. Let 𝛼 ∈ [0, 1] and 𝐹 ⩽ min. If a fuzzyrelation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is totally 𝛼-connected (𝛼-connected), then all fuzzy relations𝑅 ,… , 𝑅 are totally𝛼-connected (𝛼-connected).Proof. Let 𝛼 ∈ [0, 1], 𝐹 ⩽ min and a fuzzyrelation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) be 𝛼-connected,𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, 𝑘 ∈ 1,… , 𝑛. As a re-sult we have max(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑥)) ⩾ 𝛼, so𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) = 𝑅(𝑥, 𝑦) ⩾ 𝛼 or𝐹(𝑅 (𝑦, 𝑥), … , 𝑅 (𝑦, 𝑥)) = 𝑅(𝑦, 𝑥) ⩾ 𝛼. Let usconsider the irst case. Since 𝐹 ⩽ min, we get

𝑅 (𝑥, 𝑦) ⩾ min⩽ ⩽

𝑅 (𝑥, 𝑦) ⩾

𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) ⩾ 𝛼.It means that max(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑥)) ⩾ 𝛼. Similarlywe may consider the second case, i.e. 𝑅(𝑦, 𝑥) ⩾ 𝛼.Thus relations 𝑅 are 𝛼-connected for 𝑖 ∈ 1, … , 𝑛.The proof for total 𝛼-connectedness is analogous.

By Lemma 3 we getCorollary 14. Let 𝛼 ∈ [0, 1], 𝐹 be a t-norm or a t-seminorm. If a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) istotally 𝛼-connected (𝛼-connected), then all fuzzy rela-tions 𝑅 ,… , 𝑅 are totally 𝛼-connected (𝛼-connected).Example 12. The condition given in Theorem14 is only suf icient. For total 𝛼-connectednessit is enough to consider relations from Exam-ple 8. Relation 𝑊 is totally 𝛼-connected for𝛼 ∈ [0, 1], 𝑊 for 𝛼 ∈ [0, 0.5], but relations 𝑅, 𝑆do not have this property for any 𝛼 ∈ (0, 1]. For𝛼-connectedness let us take 𝑅 = [𝑟 ], with 𝑟 = 1 and𝑆 = [𝑠 ], with 𝑠 = 0 for 𝑖, 𝑗 = 1,… , 𝑛. Then relation𝑊 = max(𝑅, 𝑆) = 𝑅 and 𝑅, 𝑊 are 𝛼-connected for𝛼 ∈ [0, 1], while 𝑆 is not 𝛼-connected for any 𝛼 ∈ (0, 1].5.4. Asymmetry

Now graded asymmetry and antisymmetry willbe discussed. The obtained results are dual to theones obtained for total 𝛼-connectedness and 𝛼-connectedness, respectively. It is worth mentioningthat in [32] (p. 619) the 0.5-asymmetry was consid-ered. However, in that paper this property is calledweak asymmetry. It was shown there that minimumpreserves the 0.5-asymmetry ([32], Table 1).Theorem15 ([16]). Let 𝛼 ∈ [0, 1], card𝑋 ⩾ 2. A func-tion 𝐹 ∶ [0, 1] → [0, 1] preserves 𝛼-asymmetry (𝛼-antisymmetry) of fuzzy relations, if and only if for any𝑠, 𝑡 ∈ [0, 1]

( ∀⩽ ⩽

min(𝑠 , 𝑡 ) ⩽ 1−𝛼) ⇒ min(𝐹(𝑠), 𝐹(𝑡)) ⩽ 1−𝛼.

Theorem 16 ([16]). Let 𝑐𝑎𝑟𝑑 𝑋 ⩾ 2. A function𝐹 ∶ [0, 1] → [0, 1] preserves 𝛼-asymmetry (𝛼-antisymmetry) of fuzzy relations for arbitrary 𝛼 ∈[0, 1], if and only if

∀, ∈[ , ]

min(𝐹(𝑠), 𝐹(𝑡)) ⩽ max⩽ ⩽

min(𝑠 , 𝑡 ).

Corollary 15. The minimum and the weighted mini-mum (11) preserve 𝛼-asymmetry (𝛼-antisymmetry) offuzzy relations for arbitrary 𝛼 ∈ [0, 1].

Dually to graded connectedness properties, byLemma 2, similarly to the proof of Theorem 9 we mayprove

Theorem 17. Let 𝛼 ,… , 𝛼 ∈ [0, 1], a function 𝐹 ∶[0, 1] → [0, 1] be a super additive increasing in eachvariable function and 𝐹 ≫ min. If relations 𝑅 ∈ℱℛ(𝑋) are totally 𝛼 -asymmetric (𝛼 -antisymmetric)for 𝑖 = 1,… , 𝑛, then relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-asymmetric (𝛼-antisymmetric) for 𝛼 = 𝐹(𝛼 ,… , 𝛼 ).

In Theorem 1 we have the characterization ofincreasing functions which dominate minimum. Ap-propriate examples are presented in Example 4 andamong them minimum is a super additive function(because, by Lemma 2, it dominates any increasingfunction which coincides with the inequality (19)).

We can also compute the value of 𝛼 for which afuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-asymmetric(𝛼-antisymmetric) for concrete functions𝐹 in anotherway than it is presented in Theorem 17. It is shown inthe following example.

Example 13. Let 𝛼 ,… , 𝛼 ∈ [0, 1]. If relations 𝑅 ∈ℱℛ(𝑋) are 𝛼 –asymmetric (𝛼 –antisymmetric) for 𝑖 =1,… , 𝑛, then relation 𝑅 ∈ ℱℛ(𝑋) is 𝛼-asymmetric (𝛼-antisymmetric), where

𝑅 = 1𝑛 𝑅 , 𝛼 = 1

𝑛 min⩽ ⩽

𝛼 .

Note that the arithmetic mean does not dominate min-imum.

Dually to Theorem 14 we may prove

Theorem 18. Let 𝛼 ∈ [0, 1] and 𝐹 ⩾ max. If afuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-asymmetric (𝛼-antisymmetric), then also all relations 𝑅 ,… , 𝑅 are 𝛼-asymmetric (𝛼-antisymmetric).

By Lemma 3 we obtain

Corollary 16. Let 𝛼 ∈ [0, 1], 𝐹 be a t-conorm or at-semiconorm. If a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 )is 𝛼-asymmetric (𝛼-antisymmetric), then all relations𝑅 ,… , 𝑅 are 𝛼-asymmetric (𝛼-antisymmetric).

Example 14. The condition given in Theorem 18 isonly suf icient. For 𝛼-asymmetry it is enough to con-sider relations from Example 10. The relation 𝑊 is 𝛼-asymmetric for 𝛼 ∈ [0, 1],𝑊 for 𝛼 ∈ [0, 0.5], but rela-tions 𝑅, 𝑆 do not have this property for any 𝛼 ∈ (0, 1].For 𝛼-antisymmetry let us take 𝑅 = [𝑟 ], with 𝑟 = 1and 𝑆 = [𝑠 ], with 𝑠 = 0 for 𝑖, 𝑗 = 1,… , 𝑛. Then rela-tion𝑊 = min(𝑅, 𝑆) = 𝑆 and 𝑆,𝑊 are 𝛼-antisymmetricfor 𝛼 ∈ [0, 1], while 𝑅 is not 𝛼-antisymmetric for any𝛼 ∈ (0, 1].

32


5.5. SymmetryNow graded symmetry will be discussed.

Theorem 19 ([18]). Let 𝛼 ∈ [0, 1]. If a function 𝐹 ∶[0, 1] → [0, 1] ful ils

𝐹|[ , ] ⧵[ , ] < 1 − 𝛼,then it preserves 𝛼-symmetry of relations 𝑅 ,… , 𝑅 ∈ℱℛ(𝑋).Theorem 20 ([18]). If a function 𝐹 ∶ [0, 1] → [0, 1]ful ils condition 𝐹 ⩽ min, then it preserves 𝛼-symmetryof fuzzy relations for arbitrary 𝛼 ∈ [0, 1].Corollary 17. Any triangular norm or a t-seminormpreserves 𝛼-symmetry of fuzzy relations for arbitrary𝛼 ∈ [0, 1].Example 15. Since any projection 𝑃 , 𝑘 ∈ , preservesthe 𝛼-symmetry for each 𝛼 ∈ [0, 1] but it is not truethat 𝑃 ⩽ min, then Theorem 20 gives only a suf icientcondition for preservation of the 𝛼-symmetry for any𝛼 ∈ [0, 1].Theorem 21. Let 𝛼 ,… , 𝛼 ∈ [0, 1], 𝐹 ⩽ min. If rela-tions 𝑅 ∈ ℱℛ(𝑋) are 𝛼 -symmetric for 𝑖 = 1,… , 𝑛,then relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is 𝛼-symmetric for𝛼 = 𝐹(𝛼 ,… , 𝛼 ).Proof. Let relations 𝑅 be 𝛼 -symmetric for 𝑖 = 1,… , 𝑛and 𝑥, 𝑦 ∈ 𝑋. If 𝑅(𝑥, 𝑦) = 𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) ⩾1 − 𝛼 and 𝐹 ⩽ min, then for 𝑘 = 1, ..., 𝑛

𝑅 (𝑥, 𝑦) ⩾ min(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) ⩾1 − 𝛼 = 1 − 𝐹(𝛼 ,… , 𝛼 ).

Moreover, for 𝑘 = 1, ..., 𝑛1 − 𝐹(𝛼 ,… , 𝛼 ) ⩾ 1 −min(𝛼 ,… , 𝛼 ) ⩾ 1 − 𝛼 .

As a result 𝑅 (𝑥, 𝑦) ⩾ 1 − 𝛼 for 𝑘 = 1, ..., 𝑛. Itmeans that 𝑅 (𝑥, 𝑦) = 𝑅 (𝑦, 𝑥) for 𝑘 = 1, ..., 𝑛, so𝑅(𝑥, 𝑦) = 𝑅(𝑦, 𝑥) and 𝑅 is 𝛼-symmetric for 𝛼 =𝐹(𝛼 ,… , 𝛼 ).

If it comes to the „converse problem” for 𝛼-symmetrywehave several counter-examples. Observethat diverse functions were applied for aggregation offuzzy relations, namely greater (smaller) thanor equalto minimum (maximum).Example 16. Let 𝑐𝑎𝑟𝑑 𝑋 = 2. We consider fuzzyrelations with matrices:

𝑅 = 0 10 0 , 𝑆 = 0 0

1 0 ,

𝑊 = min(𝑅, 𝑆) = 𝑅 ⋅ 𝑆 = 0 00 0 ,

𝑊 = max(𝑅, 𝑆) = 𝑅 + 𝑆 − 𝑅 ⋅ 𝑆 = 0 11 0 ,

𝑊 = 𝑅 + 𝑆2 = 0 0.5

0.5 0 .

Relations 𝑊 ,𝑊 ,𝑊 are 𝛼-symmetric for 𝛼 ∈ [0, 1],but relations 𝑅, 𝑆 do not have this property for any 𝛼 ∈[0, 1].

Remark 4. Let 𝛼 ∈ [0, 1]. If we would assume that 𝐹 isidempotent, increasing and injective with respect to allarguments, then if a fuzzy relation 𝑅 = 𝐹(𝑅 ,… , 𝑅 )is 𝛼-symmetric, then also all relations 𝑅 ,… , 𝑅 are 𝛼-symmetric. However, idempotency and injectivnesswithrepsect to all arguments makes a contraposition (if𝐹(𝑥, 𝑥) = 𝑥, then for the remaining arguments thereare no values). Moreover, injectivness with repsect to allarguments, as a property itself, is not so easy to be ful-illed (arithmetic mean, minimum, maximum, geomet-ric mean, uninorms – including t-norms and t-conorms,are not injective with respect to all arguments). As-suming injectivness with respect to a ixed variable, i.e.𝐹(𝑥 , 𝑦) = 𝐹(𝑥 , 𝑦) ⇒ 𝑥 = 𝑥 for all 𝑦 ∈ [0, 1], ingeneral (𝐹 should bewithout a zero element) is not con-tradictorywith idempotency of𝐹, but this assumption isnot enough to obtain the required result which is shownby the counter-example above (relations 𝑅, 𝑆,𝑊 in Ex-ample 16, where the arithmeticmean is idempotent andinjective with a ixed variable).

5.6. Transi vityIn [31] a special case of the graded transitivity is

considered. Namely, this is the 0.5-transitivity (therethis property is called moderate transitivity). How-ever, the problem of preservation of this property dur-ing aggregation process is not discussed. The propertyof the 0.5-transitivity is also known as one of the typesof a stochastic transitivity (e.g. [19]).

Theorem 22 ([18]). Let 𝛼 ∈ [0, 1]. If an increasingfunction 𝐹 ∶ [0, 1] → [0, 1] ful ils

𝐹|[ , ] ⧵[ , ] < 1 − 𝛼,

and 𝐹 ≫ min, then it preserves 𝛼-transitivity of fuzzyrelations.

Example 17 ([18]). Let 𝑎 ∈ (0, 1] and 𝐹 ∶ [0, 1] →[0, 1] be of the form

𝐹(𝑠, 𝑡) = 0, (𝑠, 𝑡) ∈ [0, 𝑎) × [0, 𝑎)min(𝑠, 𝑡), otherwise

𝐹 is a 𝑡–norm and 𝐹|[ , ] ⧵[ , ] < 1 − 𝛼 but it doesnot dominate minimum. However, the function 𝐹 pre-serves the 𝛼-transitivity for each 𝛼 ∈ [0, 1) and 𝛼 ⩽1 − 𝑎. As a result conditions for preservation of the 𝛼-transitivity stated in Theorem 22 are only suf icient.

Theorem 23 ([18]). If a function 𝐹 ∶ [0, 1] → [0, 1]is increasing in each variable, ful ils 𝐹 ≫ min and 𝐹 ⩽min, then it preserves 𝛼-transitivity of fuzzy relationsfor any 𝛼 ∈ [0, 1].

Corollary 18. Minimum and the aggregation function

𝐴 (𝑡 , … , 𝑡 ) = 1, (𝑡 , … , 𝑡 ) = (1,… , 1)0, otherwise

preserve the𝛼-transitivity of fuzzy relations for any𝛼 ∈[0, 1] (because both functions ful il assumptions of The-orem 23).

33


Theorem 24. Let 𝛼 ,… , 𝛼 ∈ [0, 1], 𝐹 ⩽ min, 𝐹 ≫min and a function 𝐹 be increasing. If relations 𝑅 ∈ℱℛ(𝑋) are 𝛼 -transitive for 𝑖 = 1,… , 𝑛, then relation𝑅 = 𝐹(𝑅 ,… , 𝑅 ) is𝛼-transitive for𝛼 = 𝐹(𝛼 ,… , 𝛼 ).

Proof. Let relations 𝑅 be 𝛼 -transitive for 𝑖 = 1,… , 𝑛and 𝑥, 𝑦, 𝑧 ∈ 𝑋. If

min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑧)) =

min(𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)), 𝐹(𝑅 (𝑦, 𝑧), … , 𝑅 (𝑦, 𝑧)))⩾ 1 − 𝛼

and 𝐹 ⩽ min, then by the monotonicity of minimumwe get

min(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑧)) ⩾min(min(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)),min(𝑅 (𝑦, 𝑧), … , 𝑅 (𝑦, 𝑧))) ⩾1 − 𝛼 = 1 − 𝐹(𝛼 ,… , 𝛼 )

for 𝑘 = 1, ..., 𝑛. Moreover, for 𝑘 = 1, ..., 𝑛

1 − 𝐹(𝛼 ,… , 𝛼 ) ⩾ 1 −min(𝛼 ,… , 𝛼 ) ⩾ 1 − 𝛼 .

As a result min(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑧)) ⩾ 1 − 𝛼for 𝑘 = 1, ..., 𝑛. By assumptions it means thatmin(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑧)) ⩽ 𝑅 (𝑥, 𝑧) for 𝑘 = 1, ..., 𝑛.Since 𝐹 ≫ min and 𝐹 is increasing, one obtains

min(𝑅(𝑥, 𝑦), 𝑅(𝑦, 𝑧)) =

min(𝐹(𝑅 (𝑥, 𝑦), ..., 𝑅 (𝑥, 𝑦)), 𝐹(𝑅 (𝑦, 𝑧), ..., 𝑅 (𝑦, 𝑧))) ⩽𝐹(min(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑧)), ..., min(𝑅 (𝑥, 𝑦), 𝑅 (𝑦, 𝑧))) ⩽

𝐹(𝑅 (𝑥, 𝑧), ..., 𝑅 (𝑥, 𝑧)) = 𝑅(𝑥, 𝑧)which proves the𝛼-transitivity of a relation𝑅 for𝛼 =𝐹(𝛼 ,… , 𝛼 ).

If we look for functions 𝐹 which ful il both condi-tions 𝐹 ≫ min and 𝐹 ⩽ min we see that 𝐹 = minwhich is an aggregation function, ful ils these condi-tions. Moreover, we have the following property

Corollary 19 ([18]). For a function𝐹 ∶ [0, 1] → [0, 1]which has a neutral element 𝑒 = 1 the following holdstrue: F is increasing in each variable, 𝐹 ≫ 𝑚𝑖𝑛 and 𝐹 ⩽min if and only if 𝐹 = min.

It means that the only t-seminorm that ful ils con-ditions of Corollary 19 is minimum. If it comes to the„converse problem” for𝛼-transitivityweobtained sev-eral counter-examples. In the following example di-verse functions were applied to aggregate fuzzy rela-tions, namely greater (smaller) than or equal to mini-mum (maximum).

Example 18. Let card 𝑋 = 3. For fuzzy relationsdescribed by matrices:

𝑅 =0 1 11 1 00 0 1

, 𝑆 =1 0 00 1 11 1 0

we have the following aggregated fuzzy relations

min(𝑅, 𝑆) = 𝑅 ⋅ 𝑆 =0 0 00 1 00 0 0

,

max(𝑅, 𝑆) = 𝑅 + 𝑆 − 𝑅 ⋅ 𝑆 =1 1 11 1 11 1 1

,

𝑅 + 𝑆2 =

0.5 0.5 0.50.5 1 0.50.5 0.5 0.5

,

which are 𝛼-transitive for each 𝛼 ∈ [0, 1], while rela-tions𝑅 and 𝑆 do not have this property for any𝛼. For ex-ample for 𝛼 = 1 and relation𝑅we havemin(𝑟 , 𝑟 ) =1 ⩾ 0, but 0 = 𝑟 < min(𝑟 , 𝑟 ) = 1.

Remark5. Let𝛼 ∈ [0, 1]. If𝐹 is idempotent, increasingand injective, then if a fuzzy relation𝑅 = 𝐹(𝑅 ,… , 𝑅 )is 𝛼-transitive, then also all relations 𝑅 ,… , 𝑅 are 𝛼-transitive. However, these assumptions on 𝐹 are contra-dictory (cf. Remark 4).

6. Reciprocity Property and Other ConceptsRelated to Decision Making ProblemsIn this section we present notions, concepts and

concerns which occur in decision making algorithms.6.1. Reciprocity

Preservation of reciprocity may be useful in aggre-gation of fuzzy relations. Sometimes this property isrequired in such situations, so we present adequateassumptions on functions to preserve this property.

De inition 12 (cf. [4]). Relation 𝑅 ∈ ℱℛ(𝑋) is calledreciprocal if for any 𝑥, 𝑦 ∈ 𝑋 it holds𝑅(𝑥, 𝑦)+𝑅(𝑦, 𝑥) =1.

De inition 13 (cf. [7], p. 31). Let 𝐹 ∶ [0, 1] → [0, 1].A function 𝐹 is called a dual function to 𝐹, if for all𝑥 ,… , 𝑥 ∈ [0, 1]

𝐹 (𝑥 , … , 𝑥 ) = 1 − 𝐹(1 − 𝑥 ,… , 1 − 𝑥 ).

𝐹 is called a self-dual function, if it holds 𝐹 = 𝐹 .

Theorem 25. Let 𝐹 ∶ [0, 1] → [0, 1]. 𝐹 is self-dual ifand only if 𝐹 preserves the reciprocity property of fuzzyrelations.

Proof. Let 𝑥, 𝑦 ∈ 𝑋, 𝑅 ∈ ℱℛ(𝑋) for 𝑖 = 1,… , 𝑛 bereciprocal fuzzy relations,𝐹 be self-dual, whichmeansthat 𝐹 = 𝐹 . Thus 𝑅 (𝑦, 𝑥) = 1 − 𝑅 (𝑥, 𝑦) and

1 − 𝑅 (𝑦, 𝑥) = 1 − 𝐹(𝑅 (𝑦, 𝑥), … , 𝑅 (𝑦, 𝑥)) =

1 − 𝐹(1 − 𝑅 (𝑥, 𝑦), … , 1 − 𝑅 (𝑥, 𝑦)) =𝐹 (𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) =

𝐹(𝑅 (𝑥, 𝑦), … , 𝑅 (𝑥, 𝑦)) = 𝑅 (𝑥, 𝑦).

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The dual functions to fuzzy conjunctions are fuzzydisjunctions and vice versa, and since these twoclasses are disjoint, it follows that neither a fuzzy con-junction nor a fuzzy disjunction (including t-normsand t-conorms) is a self-dual function. For any binaryfunction 𝐹, if it is without zero divisors (with the zeroelement 0), then 𝐹 cannot be self-dual (see [5]). Anyself-dual and commutative binary aggregation func-tion 𝐹 satis ies 𝐹(𝑥,1 − 𝑥) = for all 𝑥 ∈ [0, 1]. Theconcept of self-duality is especially developed for ag-gregation functions. Interesting properties and char-acterizations of self-dual aggregation functions onecan ind in [29]. A weighted arithmetic mean, medianand all quasi-linearmeans forwhich𝜑 ∶ [0, 1] → [0, 1]ful ils 𝜑(1 − 𝑥) = 1 − 𝜑(𝑥), are self-dual aggregationfunctions.

If a relation 𝑅 ∈ ℱℛ(𝑋) is not reciprocal (deci-sion makers were not informed to make such choices)there exist the ways to make it reciprocal. We presentsuch a formula for a inite case, since practically, in de-cision making problems, we have inite set of alterna-tives. Let 𝑅 ∈ ℱℛ(𝑋), where 𝑋 = 𝑥 ,… , 𝑥 . We mayobtain from 𝑅 a normalized fuzzy reciprocal relation𝑅∗ ∈ 𝐹𝑅(𝑋) in the following way

𝑅∗ =if 𝑅 + 𝑅 ≠ 0

0 otherwise .

If we have a reciprocal relation, then we get speci icinterpretation of properties of this relation (see Sec-tion 4, pages 7-8). Reciprocity is in a sense a form ofconsistency or clearness of choices of decision mak-ers. However, not every function which preserves agiven 𝛼-property, preserves also reciprocity. To sim-plify the considered algorithms, we do not considerthe requirement of reciprocity at any stage. We con-centrate on𝛼-properties and their behaviour in aggre-gation process, which is the main topic of this paper.6.2. Improving Judgements of Decision Makers

Itmay happen that someof the individual relationswill not have the required property, for example 𝛼-transitivity for some 𝛼. In such situation we may as-sume two options in the algorithms. The irst one willbe not to run algorithm in such a case. The second onewill be to improve a little preferences of decisionmak-ers to obtain more ’regular’ results, i.e. to obtain allrelations with the required property. >From mathe-matical point of view, if it comes to standard fuzzy re-lation properties, there are known some results howto improve the relation (for example, to make it tran-sitive, if it is not transitive, cf. [35]). Namely, if rela-tion 𝑅 is not re lexive it is enough to consider 𝑅 ∨ 𝐼,where 𝐼 ∈ ℱℛ(𝑋) is the identity relation. 𝑅 ∨ 𝐼 is ob-viously re lexive. If 𝑅 is not symmetric we may con-sider symmetric closure𝑅∨𝑅 or symmetric interior𝑅 ∧ 𝑅 , which are symmetric relations. For asymme-try (antisymmetry) and there is no appropriate clo-sure/interior, so there is no unique method to createthe asymmetric (antisymmetric) or connected (totalconnected) relation from the given one. However, forexample the relation 𝑅 ⧵ 𝑅 is asymmetric. To obtain

the transitive relation from the given 𝑅 we may con-sider its closure as the sum of powers of 𝑅, but oftensuch closure is the full relation (𝑅 ≡ 1), so it is notuseful from practical point of view. However, there arealso othermethods to obtain a transitive relation fromthe given non-transitive one, which are not so differ-ent from the original relation (cf. [37]). Moreover, theproposed in that paper concept enables to determinerelations which are transitive only for a part of the el-ements under consideration.

If it comes to 𝛼-properties, thanks to the gradual-ness of these properties we may, in most of the cases,obtain some grade of𝛼 towhich a relation has the con-sidered property.Wemay treat a given𝛼-property as ameasure of this property of the fuzzy relation 𝑅. How-ever, it may happen that we get 𝛼 = 0, and in thecase of 𝛼-symmetry and 𝛼-transitivity it may be noneof 𝛼 ∈ [0, 1] (cf. Remark 3). If we need some improve-ments of the grade of the given property, wemay havetwo approaches. The irst one is to use the existingmethods for obtaining standard properties and whatis equivalent, in the same way, to obtain 𝛼-propertiesfor any 𝛼 ∈ [0, 1] (cf. Remark 2, Corollary 4). The sec-ondone is to increase the gradeof𝛼 (not necessarily tothemaximumpossible value) towhich a given relationhas the considered 𝛼-property. To change the grade of𝛼 for these properties we may apply Corollaries 5 and6 in an adequateway (using the properties of in imumand supremum and changing the values of the givenrelation 𝑅). Moreover, for reciprocal relations, takinginto account properties connected with the diagonal(𝛼-re lexivity, 𝛼-irre lexivity) there is no need tomakeany improvements, since by de inition, the values onthe diagonal are ixed. Furthermore, for reciprocal re-lations we rather would like to obtain asymmetricthan symmetric relations and reciprocal relation isalways 0.5-asymmetric and totally 0.5-connected, sopractically the following 𝛼-properties may be consid-ered: antisymmetry, connectedness, transitivity.

6.3. Methods to Obtain the Final Order of Alterna vesThere exist diverse methods to ind an alterna-

tive as solution from a given 𝑅 ∈ ℱℛ(𝑋). One of themost widely used is the weighted vote (see [26, 27]).If we have a given preference relation 𝑅 ∈ ℱℛ(𝑋),where 𝑋 = 𝑥 ,… , 𝑥 , then the weighted vote strat-egy means taking as the preferred alternative the so-lution of

arg 𝑚𝑎𝑥,⋯,

𝑅 . (20)

However, in some situations this method does not al-low us to choose an alternative as solution in a uniqueway (cf. [3]). When this happens, sometimes it is ad-visable to apply a different method. One of the mostwidely used methods is the one given by Orlovskyin 1978 and called nondominance method [30]. Thismethod extracts as the solution the least dominatedalternative/alternatives of the fuzzy decision makingproblem starting froma fuzzy preference relation. Themaximal nondominated elements of a fuzzy prefer-ence relation 𝑅 are calculated by means of the follow-

35


ing operations:1) Compute the fuzzy strict preference relation

𝑅 = 𝑅 − 𝑅 if 𝑅 > 𝑅0 otherwise (21)

2) Compute the nondominance degree of each alter-native 𝑁𝐷 = 1 − ⋁

,…,𝑅 , so we get a fuzzy set

𝑁𝐷 = [(𝑥 , 𝑁𝐷(𝑥 )) ∶ 𝑥 ∈ 𝑋].3) Select as alternative: 𝑎𝑟𝑔 max

,…,𝑁𝐷 .

However, with this method we may also not obtainthe clear unique result. In such situation there is alsoa method to obtain an interval-valued fuzzy relationfrom the given fuzzy relation and then use one of themany possible linear orders for interval-valued fuzzysetting [3], which allows us to obtain the unique alter-native from a given set of alternatives 𝑋.

7. Comparison of AlgorithmsWe present here three algorithms to obtain the

inal solution from a given set of alternatives. Weuse here theoretical results presented in the paper.Our aim is to compare these approaches for decisionmaking problems. Here, we do not pay attention tothe reciprocity requirements (as it was explainedbefore) and ways of obtaining the best alternative.This is why, in the algorithms, we omit this inal stepof inding the best alternative.

For all presented algorithms, we will have thefollowing inputs:𝑋 = 𝑥 , ..., 𝑥 , 𝐹 - aggregation function,𝑅 , ..., 𝑅 ∈ ℱℛ(𝑋). The given 𝛼-property will bedenoted for short 𝛼 − 𝑃.

In Algorithm 1 we assume aggregation of fuzzyrelations with the common grade of 𝛼 for the givenproperty 𝛼 − 𝑃. Function 𝐹 is one of those whichpreserves such 𝛼-property.

Algorithm 1 – the steps:1) Check the grade of the property 𝛼 − 𝑃 of each 𝑅

for 𝑘 = 1, ..., 𝑛2) Fix the common grade of the property𝛼−𝑃 of each

𝑅 for 𝑘 = 1, ..., 𝑛3) Determine the relation 𝑅 with the use of aggrega-

tion function 𝐹Output: the aggregated fuzzy relation 𝑅 with the

property 𝛼 − 𝑃.

In Algorithm 2 we aggregate fuzzy relations withpossible diverse grades of 𝛼 for the given property𝛼 − 𝑃, i.e. 𝛼 -𝑃, 𝛼 -𝑃, ..., 𝛼 -𝑃. Function 𝐹 is oneof those which preserves such 𝛼 -𝑃, 𝛼 -𝑃, ..., 𝛼 -𝑃properties.

Algorithm 2 – the steps:1) Check the grade of the property 𝛼 − 𝑃 of each 𝑅

for 𝑘 = 1, ..., 𝑛

2) Determine the relation 𝑅 with the use of aggrega-tion function 𝐹

3) Determine the value 𝛼 = 𝐹(𝛼 ,… , 𝛼 )Output: the aggregated fuzzy relation 𝑅 with the

property 𝛼-𝑃, where 𝛼 = 𝐹(𝛼 ,… , 𝛼 ).

Note that, if for a given function 𝐹 the grade 𝛼 instep 3 is different from 𝐹(𝛼 ,… , 𝛼 ), then it is enoughto put in this step appropriate value of 𝛼 (cf. Example13).

In Algorithm 3 we do not determine the gradesof 𝛼 for individual fuzzy relations, but we do it forthe inal result, i.e. the aggregated fuzzy relation.Algorithm 3, with applied appropriate aggregationfunction 𝐹, guarantees that 𝑅 ,… , 𝑅 have the samegrade of 𝛼-𝑃, for a given property 𝑃.

Algorithm 3 - the steps:1) Determine the relation 𝑅 with the use of aggrega-

tion function 𝐹2) Check the grade of 𝛼-𝑃 for 𝑅

Output: the aggregated fuzzy relation 𝑅 and𝑅 ,… , 𝑅 with the same property 𝛼-𝑃.

In the following subsections we will perform com-parison of complexity and usefulness of functions𝐹 preserving diverse properties (more useful prac-tically, weaker assumptions, losing less informationetc.).7.1. Comparing Assump ons on Func ons Used for Ag-

grega onWe will consider re lexivity property only in

the case of general fuzzy relations (not necessarilyreciprocal ones). The results for the other propertiescan be analyzed in a similar way (with similar conclu-sions). Comparing assumptions on 𝐹 for the case ofre lexivity we cannot conclude clearly which way isbetter (Algorithm 1 or Algorithm 2), it depends on thevalues of fuzzy relations. Let us see some examples.

Let card 𝑋 = 2, 𝑅 , 𝑅 ∈ ℱℛ(𝑋), 𝑅 = ,where

𝑅 = 0.8 00 0.6 , 𝑅 = 0.7 0

0 0.9 ,

𝑅 = 0.75 00 0.75 .

𝑅 is 0.6-re lexive and 𝑅 is 0.7-re lexive, 𝑅 is 0.75-re lexive. Considering Algorithm 1, the common valueof re lexivity of 𝑅 and 𝑅 is 0.6. If we take 𝐹 ⩾ min(which is for example the arithmetic mean) we havethe guarantee that 𝐹 preserves 0.6-re lexivity (cf. The-orem 4). However, 𝑅 may also have the greater levelof 𝛼-re lexivity, which is the case for our examples.Considering Algorithm 2, and taking as an aggregatingfunction any increasing 𝐹 (cf. Theorem 5) we get forthe arithmetic mean 𝛼 = 𝐹(0.6, 0.7) = 0.65, so𝑅 is 0.65-re lexive but it may have higher value of

36


re lexivity, which is the case in our situation.

Let card 𝑋 = 2, 𝑅 , 𝑅 ∈ ℱℛ(𝑋), 𝑅 = ,where

𝑅 = 0.8 00 0.6 , 𝑅 = 0.9 0

0 0.7 ,

𝑅 = 0.85 00 0.65 .

𝑅 is 0.6-re lexive and 𝑅 is 0.7-re lexive, 𝑅 is 0.65-re lexive. Considering Algorithm 1, the common valueof re lexivity of 𝑅 and 𝑅 is 0.6. The arithmetic mean(as explained above) preserves 0.6-re lexivity. How-ever,𝑅 may have higher level of𝛼-re lexivity, which isthe case in our example. Considering Algorithm 2, weget for the arithmetic mean 𝛼 = 𝐹(0.6, 0.7) = 0.65,so 𝑅 is 0.65-re lexive and this coincides with the realvalue of re lexivity in the considered example.

If it comes to Algorithm 3, it is enough to check thegrade of𝛼-re lexivity of the fuzzy relation𝑅 and if ag-gregating function ful ils the property 𝐹 ⩽ min, thenwe know that all aggregated relations 𝑅 ,… , 𝑅 are ofthe same grade of 𝛼-re lexivity (cf. Theorem 6). Thereis also the risk of loosing information about the realgrade of 𝛼 of particular fuzzy relations involved in theprocess of aggregation. Lut us see the example, where𝑅 , 𝑅 ∈ 𝐹𝑅(𝑋), 𝐹 = min and

𝑅 = 0.6 00 0.8 , 𝑅 = 0.2 0

0 0.3 ,

𝑅 = 0.2 00 0.3 .

𝑅 and 𝑅 , 𝑅 are 0.2-re lexive, but 𝑅 is in fact 0.6-re lexive.

To sumup, using thesemethodswe should remem-ber that the grade of 𝛼-property of the aggregatedfuzzy relation𝑅 (Algorithms 1 and 2) and input fuzzyrelations𝑅 ,… , 𝑅 (Algorithm3) ’is theminimal of themaximum possible’ to be obtained (all depends on theform of fuzzy relations).

It is also worth mentioning that assumptionson functions 𝐹 to preserve 𝛼-transitivity are ratherstrong. However, if in de inition of𝛼-transitivitywe re-placeminwith arbitrary binary operation ∗ ∶ [0, 1] →[0, 1], then we may weaker in a signi icant way theassumptions on function 𝐹 to preserve such 𝛼-∗-transitivity. Thus it is enough if 𝐹 ≫ ∗ and 𝐹 ⩽ min,and we have many examples of such functions 𝐹 (cf.[17]).

For aggregation of fuzzy relations with diversegrades, some assumptions seem to be strong. In Theo-rem 17 for preservation of asymmetry we have strongassumption 𝐹 ≫ min, but for concrete functions 𝐹,like for example the arithmetic mean in Example 13,we may compute the value of 𝛼 for the property 𝛼-𝑃(without following assumptions of Theorem 17). Notethat in this example 𝛼 ≠ 𝐹(𝛼 ,… , 𝛼 ).

If it comes to the converse problem, for 𝛼-symmetry and𝛼-transitivity it is not clear if such func-tions do exist (cf. Remarks 4 and 5).

7.2. Comparison of Complexity of the Algorithms

We will present the time complexity of the sepa-rate steps and operations in the presented algorithmsand thenwewill give the complexity of each algorithmfor each property.

Fuzzy relations𝑅 ,… , 𝑅 are de ined in a set X con-sisting of 𝑚 elements, so complexity will depend onthe variable 𝑚 (the size of a matrix representing 𝑅).We get the following time complexities:- determining the grade 𝛼 of re lexivity (irre lexivity)is 𝑂(𝑚), since this is determining the minimal (max-imal) value of a list of 𝑚 non-ordered elements (cf.Corollary 5),- determining the grade𝛼 of connectedness (total con-nectedness, asymmetry, antisymmetry) is 𝑂(𝑚 ) (cf.Corollary 5),- determining the grade 𝛼 of symmetry is 𝑂(𝑚 ) (cf.Corollary 6),- determining the grade 𝛼 of transitivity is 𝑂(𝑚 ) (cf.Corollary 6).

Taking into account that the remaining operationsin Algorithms1, 2, 3 should be performed atmaximum𝑛 times (𝑛 is the number of fuzzy relations) we get thefollowing time complexities.

Corollary 20. Re lexivity and irre lexivity: Algorithms1, 2 and 3 take 𝑂(𝑚) computational time complexity.Connectedness, total connectedness, symmetry, asym-metry and antisymmetry: Algorithms 1, 2 and 3 take𝑂(𝑚 ) computational time complexity. Transitivity: Al-gorithms 1, 2 and 3 take 𝑂(𝑚 ) computational timecomplexity.

8. ConclusionIn this paper preservation of basic classes of 𝛼-

properties of fuzzy relations in the context of aggrega-tion process were discussed. Mutual dependencies re-lated to these properties, between relations 𝑅 ,… , 𝑅on a set 𝑋 and the aggregated fuzzy relation 𝑅 =𝐹(𝑅 ,… , 𝑅 )were examined. Suf icient conditions forfunctions 𝐹 ∶ [0, 1] → [0, 1] to ful ill the given prop-erty were provided (regarding three possible casesof approach to aggregation procedure). Moreover, di-verse ’regularities’ and interpretation of 𝛼-propertieswere discussed, also in the context of reciprocal re-lations and decision making problems. Finally, com-parison of obtained results, including suitable deci-sion making algorithms were provided (there wereanalyzed the time complexities of the presented algo-rithms and assumptions on fusion functions useful toobtain the required results). All algorithms were im-plemented and tested in Java programming language.

ACKNOWLEDGEMENTSThis work was partially supported by the Centre forInnovation and Transfer of Natural Sciences and Engi-neering Knowledge in Rzeszow, through Project Num-ber RPPK.01.03.00-18-001/10.

37


AUTHORSUrszula Bentkowska∗ – University of Rzeszow,Interdisciplinary Centre for Computational Mod-elling, ul. Pigonia 1, 35-310 Rzeszow, Poland, e-mail:[email protected] Balicki – University of Rzeszow, Inter-disciplinary Centre for Computational Modelling, ul.Pigonia 1, 35-310 Rzeszow, Poland, e-mail: [email protected].∗Corresponding author

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40

Control Methods Design for a Model of Asymmetrical Quadrocopter

Ryszard Beniak, Oleksandr Gudzenko

Submitted: 9th February 2016; accepted: 19th May 2016

DOI: 10.14313/JAMRIS_2-2016/14

Abstract:The paper describes the results of quadrocopters mo-tion 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 dynam-ics, quadrocopter, dynamic mode

1. IntroductionRecently 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 ma-neuverable, 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 con-figuration 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 wide-spread use they have not received. The most popu-lar quadrocopters obtained with using UAVs. Today quadrocopters are used in various fields of human ac-tivity.

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

An analysis of the literature proved that math-ematical models can be classified as follows:

1. Linear model. Used for simple maneuvers [3] or for the calculation of the control algorithm by com-plex 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 co-incides 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 com-plex maneuvers that require high control precision. HoverBike is an example of this kind asymmetric con-struction [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 nec-essary 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], back-stepping 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 linear-quadratic 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 quadro-copter within two trajectories: a circle and an eight-shaped 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 ModelMost manufacturers simplify their tasks by devel-

oping 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 com-ply with such symmetry. Manufactured devices differ significantly since the center of gravity with geomet-ric center and the arm with the motors may be posi-tioned 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). 1l is the distance from the edge of the second platform to the intersection with the center of the first platform, lll 221 =+ . sl is the distance from the edge of the platform to the center of the motor. In Fig. 2, a dotted line shows a quadrocopter symmet-ric 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 coordi-nates 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 cccc ZYXo is re-lated to the center of mass of quadrocopters (CSM), and the coordinate system gggg ZYXo 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

pppc ZYXo (MCS). MCS starts at the center of mass of quadrocopters. The axis pc xO is connected with one of the arms of a quadrocopter, axis pc yO lies in the plane of a quadrocopter, axis pc zO is upwardly direct-ed relative to a quadrocopter. The angular position of a quadrocopter is defined in MCS by Euler angles

T),,( ψθφη = : roll φ , pitch θ and yaw ψ .

Fig. 1. Geometric model of quadrocopter

Fig. 2. Design quadrocopters

The center of mass of a quadrocopter is defined by vector T)z,y,x(X = 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. Ro-tation matrix from CSM to the ICS has the form [14]:

,ccscs

sccssssscccscscsscsssccc

)(Rot

−−+

+−=

φθφθθ

φψφθψφθψφψθψ

φθψφψφψφθψθψ

η

where The connection between the linear speed in the

ICS and the CSM has the form (1).

cV)(RotX ⋅= η (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).

ηηΛΩ

θφφ

θφφ

θ

⋅

−

−=⋅=

ccscscs

00

01 (2)

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

Fig. 3. Coordinate systems


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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]

The platform1M 1h,b,a ( );X c 0001 =

The unit with equipment 2M 2h,d,c

;hhX c

+−=

200 21

2

The first arm3M 32 h,l,l ∆

;hhXc

+=

200 31

3

The second arm4M ( 34 MM = ) 32 h,l,l ∆

;hh)sin(ll)cos(llXc

+−−=

222312121

4 αα

The motor 15M 522 h,r,r ( )

);hhh(.h

;hllX*

*sc

531

5

250

0

++=

−=

The motor 26M ( 56 MM = ) 522 h,r,r ( );h)ll(X *

sc 06 −−=

The motor 37M ( 57 MM = ) 522 h,r,r ( );h)sin()ll()cos()ll(X *

ssc αα −−−−= 227

The motor 48M ( 58 MM = ) 522 h,r,r ( );h)sin()ll()cos()ll(X *

ssc αα −−= 118

ters in CSM T),,( 321 θθθΘ = were selected as general-ized coordinates.

=

Xq

Θ; ; (3)

(4)

where Q is generalized force, M is the quadrocopter mass, ∑

=

=8

1jjMM , I is the inertia tensor of a quadrocop-

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

Quadrocopters inertia tensor I can be written as:

+−−−+−−−+

+=22

22

22

mmmmmm

mmmmmm

mmmmmm

g

yxzyzxzyzxyxzxyxzy

MII ,

where gI is the inertia tensor in CSG, mX is the vector coordinates of the mass center,

T

mmmm )z,y,x(X = ,

The inertia tensor gI is described by the relation

∑=

=8

1kk,gg II , where k,gI is the inertia tensor of the

structure element k in the CSG. Table 2 shows the for-mulas for calculating the inertia tensor of quadrocop-ter’s elements. For simplicity, arms, the platform and the equipment unit are treated as rectangular paral-lelepiped elements. The motors are treated in the cal-culation 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).

In Table 2

m

m

m

CA

A

000000

is the inertia tensor of motors relative to the principal axes of inertia.

The generalized force Q can be represented in the form T

FM )Q,Q(Q = , where MQ is a generalized torque in the rotational motion, FQ is a component of gener-alized force in translational motion. The main compo-nents of the generalized force can be written as (5) and (6).

girM MUQ += ; (5)


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resmguF FFFQ ++= , (6)

where U is the vector of the rotational force caused by the operation of motors,

T)U,U,U(U 321= , girM is the gyroscopic moment, uF is the traction of motors, Fmg is the force of gravity acting on the quadrocopter,

resF is resistance force, , S is the aerodynamic force coefficient vector [2].

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

(7)

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

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

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

(8)

The system of equations (8) should be supple-mented with the equations describing the forces and


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

Fig. 4. The lifting force and torque motors

The lifting force and torque are directly propor-tional to the square of the rotation speed [4]. Formu-las for traction and torque are of the form (9).

(9)

where 1k and 2k are constant coefficients, iω is the angular velocity of rotation of the motor i , )xx( ci − and )yy( ci − 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 quadrocop-ters rotational speed and motors kinetic torque:

T

iimmgir CKM

⋅×=×= ∑

=

4

1

00 ωΩΩ (10)

The dynamic equations of motion of quadrocop-ters (8) equations of traction, torque (9) and gyro-scopic 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 follow-ing form ΛΛ IJ T= . After simplification we obtain dy-namic equations of quadrocopters motion in the final form (11).

(11)

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 fol-lowing criteria were used for the comparison of se-lected 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 en-ergy 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 DynamicsThe 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 grav-ity center tr

cX and the yaw angle trψ . This method of defining the desired trajectory is the most informa-tive for the controlled quadrocopter operator.

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


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, (12)

where 1C , 2C are matrixes of known feedback coeffi-cients.

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

(13)

Similarly, the controls of angular acceleration of the angular position are:

(14)

where 3C , 4C are known feedback coefficients.From the dynamic equations of motion around the

gravity center we can determine the control torques.

, (15)

where , .

To determine the angular velocities of motors we can use a system of equations (9). This system is lin-ear relative to 2

i)(sign i ωω , coefficient matrix is con-

stant for the configuration and does not degenerate. It means that the system of equations (9) provides a unique solution.

3.2. LQR MethodThe 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 meth-od. The system of equations (11) can be written as (16). So the linearized system has the form (17).

(16)

(17)

where 00 W,Y are the state vector and control vector at some point.

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

(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).

(19)

, R(T) = 0 (20)

(21)

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 equa-tions (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 in-verse dynamics. The functional (18) is the criterion for the process convergence.

4. SimulationBased 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

41500150 ,..,i,i =<< ω . 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], l s = 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 sec-ond 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 52 /πθ = rad/s (24). For the second trajectory, the quadrocopter flew along “eight-


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

=−===<≤

;)t(;)t(z;)t(y;)t(x;.t

II

II

0100100

ψ (22)

=−−=

==+=<≤

;)t(;).t()t(z

;)t(y;)t(x;.t;tt.

IIII

IIIIss

01105

0010510

ψππ (23)

=−=

=−−=≤

;)t());tt(sin()t(z

;)t(y));tt(cos()t(x;tt

oIIIs

oIII

oIIIs

oIIIs

05

2

05

21

ψπ

π

(24)

=−=

=−−=≤

∞∞

∞∞

;)t());tt(sin()t(z

;)t(y));tt(cos()t(x;tt

IIIsIII

IIIsIIIs

05

4

05

21

ψπ

π

(25)

The results of the simulation are shown in Figs. 6–9. Table 4 shows the numerical value of the evalu-ation criteria.

In Figs. 6–9 for the state vector dotted line indi-cates 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 con-clude that both methods are able to solve the prob-lem of control successfully. The control algorithm obtained by both methods is within the predeter-mined limits. The most difficult phase to control is the transition from the second to the third stage. This

Fig. 5. Motion trajectories of quadrocopters

Table 4. Properties of quadrocopters motion

Trajectory “circle” Trajectory “eight-shaped”

Criteria LQR ID LQR ID

Functional )W,Y(Φ 2.65 12.00 4.50 17.15

Part )Y(Φ of the functional 0.27 4.57 0.68 6.10

Part )W(Φ of the functional 2.39 7.43 3.82 11.05

Standard deviation [m] 0.06 0.05 0.10 0.14

Maximum deviation of the position [m] 0.15 0.17 0.38 0.65


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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 tra-jectory, 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. ConclusionsThe control problem of asymmetric quadrocop-

ters was illustrated an example of complex trajecto-ries “circle” and “eight-shaped”. The new mathemati-cal model which takes into account the asymmetry of

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


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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

the synthesized control algorithms allow us to consid-er the choice of solution methods for conditions va-riety. LQR method involves large computational costs and requires a prior knowledge of the motion trajec-tory, but it allows decreasing the energy costs and ob-taining 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 mo-ments of trajectory discontinuity the system may lose stability especially in aggressive manoeuvers.

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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AUTHORSRyszard Beniak* – Opole University of Technology, Faculty of Electrical Engineering, Automatic Control and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland. E-mail: [email protected]

Oleksandr Gudzenko - Opole University of Technol-ogy, Faculty of Electrical Engineering, Automatic Con-trol and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland. E-mail: [email protected]


REFERENCES

[1] Erdinc Altug, James P. Ostrowski, Robert Maho-ny, “Control of a Quadrotor Helicopter Using Vi-sual Feedback”. In: Proceedings of the 2002 IEEE, International Conference on Robotics & Automa-tion, Washington, DC, May 2002, 72–77.

[2] J. Gordon Leishman, Principles of Helicopter Aerodynamics, 2nd edition, Cambridge University Press, 2006, 25–33.

[3] H. Bolandi, M. Rezaei, R. Mohsenipour, H. Nemati, S. Smailzadeh, “Attitude Control of a Quadrotor with Optimized PID Controller”, Intelligent Con-trol and Automation, vol. 4, no. 3, 2013, 335–342.

DOI: 10.4236/ica.2013.43039.[4] H. Voos, “Nonlinear State-Dependent Riccati

Equation Control of a Quadrotor UAV”. In: Con-ference: Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Sympo-sium on Intelligent Control, Munich, Germany, 2006, 2547–2552. DOI: 10.1109/CACSD-CCA-ISIC.2006.4777039.

[5] Wei Dong, Guo-Ying Gu, Xiangyang Zhu, Han Ding, “Solving the Boundary Value Problem of an Under-Actuated Quadrotor with Subspace Sta-bilization Approach”, Journal of Intelligent and Robotic Systems, vol. 80, no. 2, November 2015, 299–311. DOI: 10.1007/s10846-014-0161-3.

[6] Weihua Zhaoa, Tiauw Hiong Go, “Quadcopter for-mation flight control combining MPC and robust feedback linearization”, Journal of the Franklin Institute, vol. 351, no. 3, March 2014, 1335–1355. DOI: 10.1016/ j.jfranklin.2013.10.021.

[7] İ. Can Dikmen, Aydemir Arısoy, Hakan Temeltaş, “Attitude Control of a Quadrotor”, Recent Ad-vances in Space Technologies, Istanbul, 2009, 722–727. DOI: 10.1109/ RAST.2009.5158286.

[8] Erdinç Altu˘g, James P. Ostrowski, Camillo J. Tay-lor, “Control of a Quadrotor Helicopter Using Dual CameraVisual Feedback”, International Journal of Robotics Research vol. 24, no. 5, May 2005, 329–341. DOI: 10.1177/0278364905053804.

[9] A. Bemporad, C. Rocchi, “Decentralized Hybrid Model Predictive Control of a Formation of Un-

manned Aerial Vehicles”, Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, 2011, 7488–7493. DOI: 10.1109/CDC.2011. 6160521.

[10] http://www.hover-bike.com/ - The official web-site of The Hoverbike.

[11] Abdelhamid Tayebi, Stephen McGilvray, “Attitude Stabilization of a VTOL Quadrotor Aircraft”, IEEE Transactions on Control Systems Technology, vol. 14, no. 3, May 2006, 562–571. DOI: 10.1109/TCST.2006. 872519

[12] Hanoch Efraim, Amir Shapiro, Gera Weiss, “Quadrotor with a Dihedral Angle: on the Effects of Tilting the Rotors Inwards”, Journal of Intel-ligent & Robotic Systems, November 2015, vol. 80, no. 2, 313–324. DOI: 10.1007/s10846-015-0176-4.

[13] K. M. Zemalache, L. Beji, H. Maaref, “Control of a drone: study and analysis of the robustness”, Journal of Automation, Mobile Robotics & Intelli-gent Systems, vol. 2, no. 1, 2008, 33–42.

[14] Thomas S. Alderete, “Simulator aero model implementation”, NASA Ames Research Center, Moffett Field, California.

[15] Mark W. Spong, Seth Hutchinson, M. Vidyasagar, Robot Dynamics and Control, 2nd edition, 2004.

[16] Krzystof Piotr Jankowski, “Inverse dynamics control in robotics applications”, Canada by Traf-ford Publishing, Ltd., Victoria, British Columbia, 2004. DOI:10.13140/RG.2.1. 1015. 3683.

[17] M. Athans, P.L. Falb, Sterowanie optymalne, Warszawa, 1966 (in Polish).

[18] Gurianov A.E. “Control Simulation of quadrocop-ters”, Electronic Science and Technology Journal Engineering Journal, Moscow, BMSTU, 2014, 522-534, ISSN 2307–0595.


50

Any-angle Global Path Planning for Skid-Steered Mobile Robots on Heterogeneous Terrain

Piotr Jaroszek

Submitted: 25th January 2016; accepted: 20th April 2016

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 operat-ing on heterogeneous terrain. Methods of path search-ing 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. Opera-tion 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. IntroductionOne of the most important tasks associated with

autonomy of robot motion is global path planning. Ev-ery type of robot requires individual approach to this problem, because robot motion capabilities depend on kinematic structure of the mobile platform. An-other equally important component of the global path planning is adequate representation of the environ-ment. Most often the robot operates in heterogeneous environment, which can be characterized by diverse properties, and taking this fact into account is defi-nitely beneficial in the path planning process. A desir-able solution returns path optimal according to the adopted criteria and possible for realization by the ro-bot 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 plan-ning is presented for one of the more frequently used types of mobile robots – the platform with non-steered wheels. To this end, the any-angle path plan-ning 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 plan-ning, 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 sig-nificantly slower and difficult in description and im-plementation. 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. Di-jkstra [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 pos-sible 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 dis-tance between the nodes, though in general the travel cost does not have to be associated with length. By ap-propriately choosing search politics and cost models, it is possible to obtain solution of the optimal path planning task for any criterion. The essential prob-lem 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 of-ten 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 unre-alistic and is not the shortest one in the real continu-ous 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* algo-rithm (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, compris-es also several unnecessary turns which may intro-duce 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 ma-neuver. 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 discretiza-tion, 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 com-putation 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 com-plex 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 ap-propriate 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 com-pared to other A* family algorithms. The Theta* al-gorithm 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 fac-tor. The important factor, besides optimality of found paths, is the possibility of representation of heterog-enous terrain which directly affects the cost accord-ing to each of the assumed optimality criteria. Author

believes that Theta* is one of the most suitable algo-rithms for the mobile robot global optimal path plan-ning 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 pur-pose 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 accord-ing to any cost model that includes longitudinal mo-tion of a robot and its (pivot) turning on the known heterogeneous terrain.

3. Robot Model and Environment RepresentationThe map. The search space is represented in the

form of a weighted graph with nodes as the admis-sible robot locations. The following assumptions con-cerning terrain map discretization were introduced:• the terrain map is divided into square-shaped ele-

ments, 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 autono-mous 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, pt

y – 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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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 mo-tion along Δsi for a dense grid introduces a small error, which was allowed for the sake of significant simplifi-cation of the cost calculation:

(5)

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 mo-tion 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 corre-sponding to the graph node deemed the start node for a given edge. The start node of the edge is defined

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(px

t,pyt) along the edge i and ends mo-

tion at the node n(pxt+1,py

t+1), one can write:

, (6)

(7)

where T(pxt,py

t) is terrain type property mentioned earlier. Thus, the weights assigned to edges depend directly on: nodes they connect, direction of robot mo-tion 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 orienta-tion Δθi with respect to terrain map are equal to:

, (9)

, (10)

where S > i > 0 and for i = 1, the py0 px

0 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 in-stance, models of energy consumption by robot drives during straight-line motion and during turning. Ad-ditionally, 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* MethodStarting 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 exclu-sively 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 pos-sibilities 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)

The main idea and difference of Theta* as com-pared 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 mak-ing 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 neigh-bor 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 dur-ing traverse, the visibility ceases to be a sufficient condition of optimality of the found path – replace-ment 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.Fig. 3. Pseudocode of the reference Theta* [7]

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


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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.

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

ColorSet of μ

Traversability#1 #2 #3

A 0.1 0.1 0.5 Yes

B 0.3 0.4 2.5 Yes

C 0.5 0.8 5 Yes

D 0.8 1.6 12 Yes

Black N/A N/A N/A No

a) b)

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%

#2 6.84% 12.84% 2.24%

#3 7.22% 12.07% 2.67%

Table. 3. Summary of results of path lengthPath length

Set of μ Average gain Maximum gain Minimum gain

#1 1.84% 8.86% -20.63%

#2 1.76% 8.30% -13.30%

#3 1.51% 8.13% -8.83%

6. ConclusionUse of the appropriate motion cost models en-

ables solution of the global path planning task during operation of real mobile robots according to arbitrary

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

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 longitudi-nal 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 aux-iliary constants that modify values of the appropri-ate costs.

Fig. 5. Cells found using Bresenham algorithm for a ter-rain homogeneity check

5. Simulation ResultsA number of simulations of path planning using

the proposed Theta* modification and the standard version of A* with the same models of cost and envi-ronment 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 tri-als 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 ac-cording to the colors shown in Fig. 7.

Fig. 7. A map used for algorithm testing


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criterion which depends on heterogeneous terrain. Adjustment of the any-angle algorithm to the consid-ered 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.

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

E-mail: [email protected]

REFERENCES

[1] 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.

[2] Chung, W. K., Xu, Y., “A generalized 3-D path plan-ning 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.

[3] Cormen, T. H., Stein, C., Rivest, R. L., Leiser-son, C. E., “Introduction to Algorithms”, 2nd ed. McGraw-Hill Higher Education, 2001.

[4] Dijkstra, E. W., “A note on two problems in con-nexion with graphs”, Numerische Mathema-tik, vol. 1, no. 1, 1959, 269–271. DOI:10.1007/BF01386390.

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

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

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

[8] Yap, P. K. Y., Burch, N., Holte, R. C., Schaeffer, J., “Abstract: Block A* and Any-Angle Path-Plan-ning”. In: Fourth Annual Symposium on Combina-torial Search, July 5, 2011.

[9] Ferguson, D., Stentz, A., “Field D*: An Inter-polation-Based Path Planner and Replan-ner”. In: Robotics Research, 2007, 239–253. DOI:10.1007/978-3-540-48113-3_22.

[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 Interna-tional 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.


56

E2LP Remote Laboratory. New Challenges in System Development

Rafał Kłoda, Jan Piwiński, Kacper Kurzejamski

Submitted: 19th March 2016; accepted: 30th May 2016

DOI: 10.14313/JAMRIS_2-2016/16

Abstract:The embedded engineering became the key success fac-tor 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 ap-proach 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 Engi-neering Learning Platform (E2LP) – 7th Framework Pro-gramme 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 ex-periments. The whole concept was evaluated at partner univer-sities 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 inte-gration, embedded systems

1. IntroductionAs embedded software systems have grown in

number, complexity, and importance in the modern world, a corresponding need to teach computer sci-ence students how to effectively engineer such sys-tems has arisen [1].

Early exposure to embedded computing systems is crucial for students to be prepared for the embed-ded 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 ca-reers [2].

Focusing the necessary changes in computer en-gineering and computer science education, we are

meeting the next dilemma – what are generic prin-ciples which should be covered in the embedded engineering education for variety of using fields. An intuitive approach is to ask industry about needed en-gineering 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 engi-neering education). Today, technology cycles are even shorter than the engineering education time, which means that the industry needs today could only ex-press 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 pro-cesses 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 be-tween 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 Plat-

form (E2LP) is a European FP7 project of three years duration, started in September 2012. [10]. The proj-ect’s motivations and project research goals are wide-ly 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 cre-ative learning environment and knowledge manage-ment that encourage use of ICT. New learning model is challenging the education of engineers in embed-ded 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 ad-dition to the technological approach, the use of cog-


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FRONT PANEL

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E2LP BOARD

NI PCI-6509

USBLAB SERVER

RS-232

RS-232

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LCD JOY LED SW

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RS-232

E2LP REMOTE LABORATORY FRAMEWORK

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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 implemen-tation 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 differ-ent technologies and system solution optimizations. As a result, the produced embedded computer engi-neering learning platform – 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 sys-tems, thereby ensuring that our strongholds in auto-motive, avionics, industrial automation, mobile com-munications, telecoms and medical systems are able to develop.

One of the man objective in E2LP project was to provide unified Remote Laboratory for embedded en-gineering education. In E2LP project a Remote Labo-ratory 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 (E-learning portal). The RL development and implemen-tation 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.

RL framework (Fig. 1) consists of three main ele-ments:

1. E-learning portal [13] This part of RL pro-vides 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 ex-perimental 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 applica-tions/services/interfaces based on several IT technol-ogies, 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 applica-tions and services in E2LP server, as well as in several cases enable user to individually configure the com-munication with a given device.

The main advantage of proposed E2LP RL frame-work is a possibility for students to interact with the real E2LP platform interfaces, implemented as a web services in Moodle [14] (acronym for Modular Object-Oriented 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 prac-tice embedded solutions are implemented on differ-ent platforms (hardware and software). The variety and dynamics of used platforms cannot be covered by a unified study program. Therefore, some kind of ab-straction and generalization in engineering education has to be applied. A possible overcoming approach

Fig. 1. E2LP Remote Laboratory framework


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is proposed in [15] trying to leverage orthogonality, theoretical background and necessary education plat-forms. Having in mind a crucial point of the embed-ded systems approach, all problems should be solved by software approach, this platform broadly uses the unified API (application programming interface) ap-proach.

Virtual and remote laboratories (VRLs) are e-learning resources that enhance the accessibility of experimental setups providing a distance teach-ing framework which meets the student’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 re-mote laboratories into several courses, in various con-texts and using various methodologies. These inte-grations, 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 di-rectly with the equipment by performing physical ac-tions (e.g. manipulating with the hands, pressing but-tons, 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 stu-dent and course supervisor. This type of systems can isolate students and reduce their motivation in learn-ing process. Furthermore, students could not receive instant feedback from their questions and cannot talk in real-time about results obtained in the learning ac-tivities with the teacher. Finally students are required to demonstrate their final projects on the actual hard-ware, the Massive Open Online Course (MOOC) plat-forms 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 equip-ment and execute remote operations to carry out ex-ercises. 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 pur-pose 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 pro-vided 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 lab-oratory exercise. The second role of e-learning portal

is management of users, which means enable them ac-cess to the laboratory hardware and software (book-ing 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 flexibil-ity. 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 ele-ments 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, im-proving, creating science-based resources, and sup-porting the software that is used all over the world. There are many reasons to use a Virtual Learning En-vironment (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 learn-ing can take place anywhere.

Monitor the progress and keep records of your students learning.

Extending the classroom by providing online dis-cussion, testing, activities, and, most importantly, al-lowing 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 continu-

ously refreshed interfaces and signals from the real board and enable users to remotely control and pro-gram the board directly from their computer at home, having instant visual feedback.

To achieve this, it is necessary to forward data di-rectly 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 en-able 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, tu-torials 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 im-plemented into Moodle based platform for e-learning course (Basic information, Theoretical explanations, Instructions, Configure Platform, Feedback, Discus-sion on results questionnaire for lab evaluation).

3. After booking in a given time slot users could remotely program given set of exercises over the In-ternet and simultaneously, in real time, could moni-tor 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 con-trol, 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.

6. New Challenges in RL Development 6.1 Modularity of Moodle Platform

The use of Moodle open source platform as an us-er-friendly interface for virtual laboratories provides benefits for every participant in the developer-teach-er-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 labora-tory. 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 LaboratoryThe progress in the field of remote teaching pro-

vides easier access to the laboratory for greater num-ber of students. That, being a great advantage of such systems, also generates problems connected with oc-cupation 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 sys-tem, 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 ser-vice. 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 de-pending on the device. That way, establishing a con-nection (from the Moodle layer) with a specific path allows for control of a certain device. During the proj-ect, a specific schematic was introduced:

http://server_address/Device1http://server_address/Device2The main advantage of such approach is the pos-

sibility of quick and easy change of the number of cur-

Fig. 2. Multi-board selection


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rently available devices without the need of modifica-tion 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 updat-ed 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 possibil-ity 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 ac-tion. 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 administra-tion 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.

6.3. “Check Solution” ModuleAs 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,

dedicated for the hardware platform the Moodle in-terface 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 data-base, for the needs of further revision.

The procedure of checking the students’ solu-tions is based on the definition of system’s required reactions for an occurrence of a certain user’s ac-tion. It operates on the basis of a dynamic compari-son 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 com-parison 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 responsibil-ity for the correctness of the procedure is removed and he is able to fully focus on the subject of the ex-ercise.

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 CheckingThe 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 in-

crease 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 syn-ergic 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

, LC 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 OY: 1 1 1 1 1 , LC D : H e l l o , R E S E T : 1 , P O W E R : 0 , E X T E R N A L : 0

Fig.3. E2LP RL board front panel


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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 OY: 1 1 1 1 1 , LC D : H e l l o , R E S E T : 1 , P O W E R : 0 , E X T E R N A L : 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 manipu-lation 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 test-ing 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 OY: 1 1 1 11,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 ,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 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 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 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 re-quired action-reaction scheme .

Due to the parallel nature of operations in the FPGA technology the teacher has to define a pa-rameter defining the expected response time. If it is shorter than the processing of a web request, the re-sponse 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 CheckingThe 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 algo-rithm checks if that succeeded. That approach creates the possibility of applying custom text for each user

making the teaching process personal and more inter-active (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 dis-played 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 pat-tern to fully automate the checking procedure, yet the exercises would not be personalized that way. Anoth-er idea is to integrate the LCD text pattern with the Moodle user personal information (e.g. name, user-name or email).

7. Evaluation and Discussion on ResultsSystem integration of many hardware and soft-

ware modules in embedded systems makes testing and verification the key success factors in embed-ded solutions [24]. Therefore, testing and verification methodology reflects as a big challenge for embedded engineering education.

Evaluation and assessment of engineering educa-tion is an important point closing the improvement loop of education process. Here, a dilemma is what and how to measure – achieved knowledge level, us-ability 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. Re-sults in the first two years of usage have been evalu-ated using the established evaluation tools showing visible improvements of study efficiency. Develop-ment and evaluation results are published at confer-ences 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 or-ganized workshops.

The established E2LP consortium prepared a framework for promotion and exploitation of the developed learning platform for embedded engineer-ing. A preparation of a follow-up project is in prog-ress addressing necessary updates and extensions for a broader usage of the E2LP, from introduction cours-es at high schools up to expert courses for lifelong en-gineering education.

The E2LP project involved the development, im-plementation and evaluation of an advanced learn-ing 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 Labo-ratory, 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 sys-tem capabilities and engaging them in contribution in testing the developed RL platform as an additional val-ue to study programs in WUT.

A study was completed under “Intelligent Measure-ment Devices” – a new course in the Electronic Measure-ment Systems specialization on engineering degree.

During this course students gain comprehensive skills: knowledge about the intelligent sensors, mea-surements devices and systems operation rules, com-petence 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 to-wards presented system, the quantitative on-line analy-sis 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 us-ers 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 read-ability 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 er-ror messages and poor programming experience. The low mark might be a reason of very low student’s initial knowledge level of FPGA systems.

8. ConclusionsThis 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

Results presented in the paper confirms that in-troduction 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 Moo-dle Platform enable student to acquire desired knowl-edge 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 meth-odology 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 litera-ture and practice of engineering education, especially in the context of developing and evaluating new tech-nology-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 Uni-versity of Technology.

AUTHORSRafał Kłoda*– Industrial Research Institute for Au-tomation and Measurements PIAP, al. Jerozolimskie 202, 02-486, Warsaw, Poland, [email protected].

Jan Piwiński– Industrial Research Institute for Au-tomation and Measurements PIAP, al. Jerozolimskie 202, 02-486, Warsaw, Poland, [email protected].

Kacper Kurzejamski – Institute of Metrology and Biomedical Engineering, Warsaw University of Tech-nology, Św. A. Boboli 8, 02-525 Warsaw, Poland, [email protected].


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N B A N W M RN B A N W M RN B A N W M RN B A N W M R

Submi ed: 16th February 2016; accepted: 18th June 2016

Mariam Al-Sagban, Rached Dhaouadi

DOI: 10.14313/JAMRIS_2-2016/17

Abstract:This paper presents a novel reac ve naviga on algo-rithm for wheeled mobile robots under non-holonomicconstraints and in unknown environments. Two tech-niques are proposed: a geometrical based technique anda neural network based technique. Themobile robot trav-els to a pre-defined goal posi on safely and efficientlywithout any prior map of the environment by modulat-ing its steering angle and turning radius. The dimen-sions and shape of the robot are incorporated to deter-mine the set of all possible collision-free steering angles.The algorithm then selects the best steering angle can-didate. In the geometrical naviga on technique, a safeturning radius is computed based on an equa on derivedfrom 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 func-onwhichminimizes the traveled distance to the goal po-

si on while avoiding obstacles. The experimental resultsdemonstrate that the algorithms are capable of drivingthe robot safely across a variety of indoor environments.

Keywords: reac ve naviga on, obstacle avoidance, au-tonomous ground robots, recurrent neural networks

1. Introduc onMobile robots have rapidly evolved over the past

years to encompass a wide spectrum of applications:Robots are assisting in driving vehicles, aiding inmed-ical tasks, and taking charge in hazardous rescue mis-sions. Autonomous navigation is a key feature in allof these applications. It deals with the problem ofnavigating to a target location while avoiding colli-sion with obstacles that may be present in the envi-ronment. One approach to autonomous navigation ismodel-based approach. It uses a model of the envi-ronment to generate a safe path to the target loca-tion [11]. Classicalmethods for this type include: Roadmaps [7], cell decomposition, and potential ields [8].Although these methods may produce ef icient paths,a global and accurate map of the environment is notavailable when the environment is unknown or is dy-namic. Hence, model based methods are used only inarti icially controlled environment. A complementaryapproach to autonomous navigation is obstacle avoid-ance (also known as reactive navigation). No prior in-formation is required about the environment. Instead,obstacles are discovered in real timewhile the robot isexecuting its mission. Themain challenges in develop-ing such methods are: computational complexity, sen-

sors uncertainties, robot geometrical shape, and kine-matic and dynamic constraints. In addition, becausea global map of the environment is not available therobot may produce inef icient paths or converge to alocal minimum (trap situation) [5].

Neural Networks-based techniques have been pro-posed in the literature to solve themotion problem. In[6] the path planning problem is viewed as two sub-problems: ind space and ind path. Two neural net-works connected in cascade are used. The irst neu-ral network is responsible for inding the C-free spacewhich is the set of all possible robot con igurationsthat avoids collisionwith obstacles. The second neuralnetwork guides the robot through the free space seg-ments to the target location. The navigationalmethod-ology used in [3] is a Probabilistic Neural Network(PNN). This type of network facilitates the trainingprocess, allows a faster time of response and has a lowcomputational cost. The output of the network repre-sents the steering direction which takes three forms:forward, right turn, and left turn. The autonomousnavigation of a mobile robot is considered as a clas-si ication problem. The motion of the robot and thesensorial information are the patterns to be classi ied.The obstacle avoidance problem in [10] is divided intothree subproblems: General obstacle avoidance, cor-ridor and wall following, and passing through a door.A separate neural network is designed for each oneof those subproblems. The algorithm uses the acces-sible space as the input to the neural network. Theoutput of the network is the steering angle and veloci-ties. The number of neurons in the hidden layer is de-termined by using a Bayesian framework which com-putes the evidence of a set of neural networkswith dif-ferent hidden nodes. The accurate usable accessiblespace is computed by incorporating the wheel chairdimensions, laser information, and encoder data. Thealgorithm choice is interesting because it is able to op-timize the number of hidden neurons for a given prob-lem. However, the algorithm is considered incompletefrom the autonomous navigation perspective becauseit does not provide a framework that is able to recog-nize the relevant situation and selects the correspond-ing output. The authors divide the obstacle avoidanceproblem into sub-problems because they claim that asingle neural network could not provide the desiredperformance. However, It is not clearwhether it is nec-essary to separate the avoiding obstacle and passingthrough a door tasks since they do not produce a con-lict in training the network. A training con lict is cre-ated when the same input pattern is mapped to two

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different output values. Another arti icial intelligencetechnique is introduced in [4]. The path planning isdone using particle swarm optimization. The desiredpath is designed to avoid obstacles while maintaininga smooth continuous path. The path is expressed asa 5th order polynomial. Two of the polynomial coef-icients are estimated via particle swarm optimizationsuch that obstacles are avoided while the other coef-icients are chosen such that a smooth path is gen-erated. The particle swarm optimization is requiredto estimate the best polynomial coef icients as wellas other parameters called the critical points. The al-gorithm was veri ied in simulation. However, non-holonomic constrains are not considered. Overall, ar-ti icial neural networks provides an interesting plat-form for the obstacle avoidance problem because oftheir generalization ability, ability to learn from exam-ples and the ability to extract temporal dependencies.In this paper, we present an obstacle avoidance tech-nique based on recurrent neural networks that takesinto consideration the kinematic constraints of differ-ential drive robots. While the common trend is to usemore than one neural network, we use a single dy-namic neural network. The obstacle avoidance prob-lem is a dynamic problem that should be solved us-ing dynamic methods. In order to guarantee optimalconvergence, we require the neural network learningenvironment to satisfy certain conditions that are de-rived using Lyapunov stability method. Also, the ear-lier presented neural networks techniques generateda dataset by manually driving the robot across dif-ferent scenarios. We automate the process by gener-ating a sub-optimal dataset using a computer algo-rithm. For optimum performance, the neural networkis trained using the real-time recurrent learning algo-rithm along with a customized objective function toequip the robot with the capability of improving itslearning while in motion.

2. Autonomous Naviga on Methodology2.1. Geometrical Naviga on Algorithm

The main steps in the obstacles avoidance algo-rithm are: Identify a reference steering angle, modelthe environment, compute the con iguration space,and select the desired steering angle and radius of cur-vature [1]. The reference steering angle, 𝛾𝑟𝑒𝑓 repre-sents the steering angle that the robot takes in the ab-sence of obstacles. It is an intermediate reference an-gle that will later help us ind 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑.

In this paper, we consider a mobile robot with adifferential drive con iguration as shown in Fig. 1. Letthe robot con iguration be:

𝑞𝑟 = (𝑥𝑟, 𝑦𝑟, 𝜃𝑟), (1)

where (𝑥𝑟, 𝑦𝑟) is the position of the robot in the 𝑥 − 𝑦plane, and 𝜃𝑟 ∈ [0, 2𝜋) is the robot’s orientation withrespect to the x axis.Let the target con iguration be:

𝑞𝑡𝑎𝑟𝑔𝑒𝑡 = (𝑥𝑡𝑎𝑟𝑔𝑒𝑡, 𝑦𝑡𝑎𝑟𝑔𝑒𝑡, 𝜃𝑡𝑎𝑟𝑔𝑒𝑡). (2)

Fig. 1. Target and robot coordinates

Let 𝑒 be the vector connecting the robot referencepoint to the target location. The phase angle of 𝑒 isgiven by:

𝛼 = arctan 𝑦𝑡𝑎𝑟𝑔𝑒𝑡 − 𝑦𝑟𝑥𝑡𝑎𝑟𝑔𝑒𝑡 − 𝑥𝑟

. (3)

To correct the error in orientation, the robot shouldturn by a reference steering angle 𝛾𝑟𝑒𝑓 . The instanta-neous turning radius 𝑟𝑐 can be evaluated by:

𝑟𝑐 = 𝐿𝑣𝑟 + 𝑣𝑙𝑣𝑟 − 𝑣𝑙

, (4)

where 𝑣𝐿 and 𝑣𝑅 are the translational velocities of theleft and rightwheels and𝐿 is the distance between thewheels.

The second main step is to model the surround-ing environment. A partial polarmapof theworkspaceis constructed in the robot local frame. The robot isequippedwith a laser range inder that is programmedto scan the 200 ∘ front view of the robot in 20 sectors,with 10 ∘ angular resolution. The sensor returns a setof points:

𝒫(𝑞(𝑡𝑖)) = 𝑝1, 𝑝2, ..., 𝑝𝑗, ..., 𝑝20. (5)

A point 𝑝𝑗 is expressed by a pair (𝑑𝑗, 𝛽𝑗) where 𝑑𝑗 isthedistance between the robot and the obstacle at sec-tor 𝑗. 𝛽𝑗 is the orientation of the 𝑗𝑡ℎ sector, 𝑆𝑗, withrespect to the local x axis. The subset ofworkspace ob-stacles seen at con iguration 𝑞(𝑡𝑖) is identi ied by ap-plying a threshold on 𝑑𝑗:

𝒪(𝑞(𝑡𝑖)) = 𝑝𝑗 ∈ 𝒫(𝑞(𝑡𝑖))|𝑑𝑗 ≤ 𝑅𝑠𝑎𝑓𝑒. (6)

The third main step of the algorithm is to computethe con iguration space𝐶𝑜𝑏𝑠𝑡. First, consider the casewhere only a point obstacle exists in the workspace:𝒪 = 𝑝𝑗. To ind 𝒞𝑜𝑏𝑠𝑡, we slide the robot around 𝑝𝑗and trace the con igurations it went through, as illus-trated in Fig. 2. Hence, 𝒞𝑜𝑏𝑠𝑡 is enclosed by a circle 𝐶𝑗of radius 𝑅 and center 𝐼𝑗 = (𝐼𝑗,𝑥, 𝐼𝑗,𝑦):

𝒞𝑜𝑏𝑠𝑡 = 𝑞 ∈ 𝒞|(𝑥 − 𝐼𝑗,𝑥)2 + (𝑦 − 𝐼𝑗,𝑦)2 ≤ 𝑅2,(7)

𝐼𝑗,𝑥 = 𝑅 + 𝑑𝑗 cos 𝛽𝑗,𝐼𝑗,𝑦 = 𝑑𝑗 sin 𝛽𝑗, −100∘ ≤ 𝛽𝑗 ≤ 100∘.

65


Fig. 2. C-space Algorithm

Next, we ind 𝐿𝑖 which is the radial distance betweenthe robot and the boundary of 𝒞𝑗 at angle 𝛽𝑖:

𝐿𝑖 = min𝜌𝑗 cos (𝛽𝑖 − 𝜙𝑗) ± √𝑅2 − 𝜌2𝑗 sin2 (𝛽𝑖 − 𝜙𝑗),

𝛼𝑚𝑖𝑛 ≤ 𝛽𝑖 ≤ 𝛼𝑚𝑎𝑥, 𝛼𝑚𝑖𝑛 = min𝜙𝑗 ± sin 𝑅𝜌𝑗

,𝛼𝑚𝑎𝑥 = max𝜙𝑗 ± sin 𝑅𝜌𝑗

(8)

If 𝒪 includes 𝑚 obstacle points, then 𝒞𝑜𝑏𝑠𝑡 =⋃

1≤𝑗≤𝑚𝒞𝑗. Now, to select the desired steering angle,

the sectors in 𝒞 are classi ied as free or occupied. The𝑗𝑡ℎ sector 𝑆𝑗 is occupied if 𝐿𝑗 ≤ 𝑅𝑠𝑎𝑓𝑒; otherwiseit is free. Adjacent free sectors are grouped togetherto form gaps. Then, the gaps are classi ied as wide,medium, and narrow. Every gap edge is a candidate forthe desired steering angle. A cost function is de ined toaid in the selection process:

𝐶𝑜𝑠𝑡 = 𝑐1(𝛾𝑟𝑒𝑓 − 𝛽𝑗) + 𝑐2𝛽𝑗. (9)The irst term in equation (9) represents how closethe desired steering direction is to the goal locationand the second term represents how close the currentsteering direction is to the current robot heading.Theangle of the gap edge that has the minimum cost is se-lected as 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑. The candidate of the desired steer-ing angle is irst considered within the wide gaps. Ifnone is available the search is performed within themedium gaps. The inal choice is the narrow gaps.

The inal step of the algorithm is to determine thedesired radius of curvature of the robot trajectory.Due to the kinematic constraints, the robot can notachieve the desired steering angle instantly. Instead,the robot follows a circular arc if thewheels’ velocitiesare constant. The path from the initial con iguration tothe inal con igurationmay intersectwith𝒞𝑜𝑏𝑠𝑡(𝑞(𝑡𝑖))causing a collision. Therefore, using a radius of curva-ture that is a function of the surrounding obstacles issafer than using a ixed radius for all obstacle scenar-ios. Let 𝑆0 be the sector that contains the local x axisand let 𝑆𝑑𝑒𝑠𝑖𝑟𝑒𝑑 be the sector that contains the de-sired steering angle 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑. Let 𝐿𝑚

𝑗 be the distancebetween the obstacle point 𝑜𝑗 and the reference point𝑚 shown in Fig. 3. The relationship between 𝐿𝑗 and𝐿𝑚

𝑗 is given by:

𝐿𝑚𝑗 = √(𝐿𝑗𝑐𝑜𝑠𝛽𝑗 + 𝑎)2 + (𝐿𝑗𝑠𝑖𝑛𝛽𝑗)2, (10)

Fig. 3. Turning Radius Selec on

where 𝑎 is the distance between the robot referencepoint and the reference point 𝑚. De ine 𝐿𝑚𝑖𝑛 as thedistance of the nearest obstacle point that exists any-where between 𝑆0 and 𝑆𝑑𝑒𝑠𝑖𝑟𝑒𝑑. The turning radius𝑟𝑐 is chosen such that the trajectory passes throughthe point (𝐿𝑚𝑖𝑛, 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑) as shown in Fig. 3. Fromthe geometry, the turning radius is obtained as:

𝑟𝑐 = 𝐿𝑚𝑖𝑛2𝑠𝑖𝑛(𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑) . (11)

To include a safety buffer, the turning ra-dius is designed to pass through the point(𝐿𝑚𝑖𝑛 − 𝑑𝑠𝑎𝑓𝑒2, 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑) instead. Also, the turningradius 𝑟𝑐 saturates if it is greater than a thresholdvalue 𝑟𝑙𝑎𝑟𝑔𝑒.2.2. Neural Naviga on Algorithm

A neural network is proposed to work in collab-oration with the reactive navigation algorithm devel-oped earlier to optimize the navigational capabilitiesof the overall system.While the navigational algorithmavoids obstacles, it doesnot contain any constraints onthe length of the path to the target position. The neu-ral network is incorporated into the system to mini-mize the length of the path taken. The neural networkselects the optimum trajectory based on the obstaclesinformation, target con iguration, and the turning ra-dius proposed by the navigational algorithm. The neu-ral network outputs the most promising turning ra-dius of the robot trajectory.

This paper uses a Diagonal Recurrent Neural Net-work (DRNN) which despite its simple structure, hasthe ability to adapt the learning rates such that thenetwork convergence is guaranteed. Fig. 4 depictsthe network architecture. For the neural network toachieve this objective, it needs to overcome few chal-lenges. One of those challenges is that the ‘correct’turning radius is not available. Hence, the training can-not be conducted in a supervised manner where adataset is available to help the network form a mapbetween the input data and the desired value. To over-come this obstacle, a hybird training scheme is pro-posed. First, the neural network will receive super-vised training based on a sub-optimal dataset. Second,the neural network will be trained on-line to adjust its

66


Fig. 4. Neural Network Architecture

weights in order to produce an optimum value basedon an evaluation function. In the irst training phase,the network is trained to map the input values to adesired value. The dataset is generated using the re-active navigation algorithm. The training is based onbackpropogation and is done off-line. The purpose ofthis training phase is to provide an adequate initial setof weights for the next phase as opposed of startingphase 2 from random variables. In the second train-ing phase, the optimum turning radius is unknown.However, the radius taken by the robot can be eval-uated. The network receieves feedback about its per-formance based on an evaluation function describedas:

𝐽 = 12𝑒2 = 1

2(𝑒2𝑥 + 𝑒2

𝑦) (12)𝑒𝑥 = 𝑥𝑡 − 𝑥, 𝑒𝑦 = 𝑦𝑡 − 𝑦

where (𝑥𝑡, 𝑦𝑡) is the target position. The derivative ofthe evaluation function with respect to the weights isgiven by:

𝜕𝐽𝜕𝑊 = −(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦) 𝜕𝑟

𝜕𝑊 (13)

where𝐽𝑥 and𝐽𝑦 are the sensitivities of the systemandare given by:

𝐽𝑥 = 𝜕𝑥𝜕𝑟 = 𝑠𝑖𝑔𝑛(𝑥(𝑡) − 𝑥(𝑡 − 1)

𝑟(𝑡) − 𝑟(𝑡 − 1) ) (14)

𝐽𝑦 = 𝜕𝑦𝜕𝑟 = 𝑠𝑖𝑔𝑛(𝑦(𝑡) − 𝑦(𝑡 − 1)

𝑟(𝑡) − 𝑟(𝑡 − 1) ) (15)

𝜕𝑟𝜕𝑊 is estimated online using Real Time RecurrentLearning (RTRL) [1]. The neural network weights areupdated according to the gradient descent technique:

Δ𝑊 = −𝜂 𝜕𝐽𝜕𝑊 (16)

Subsituting eq.12 into 16 gives:

Δ𝑊 = −𝜂𝑒 𝜕𝑒𝜕𝑊 (17)

The training phases are shown in Fig. 5 and 6.

Fig. 5. Phase 1 Training

Fig. 6. Phase 2 Training

2.3. Adap ve learning and stability analysisIn order to maintain the system stability, we put a

restriction on the learning rate using Lyapunov theo-rem. De ine the lyapunov function 𝑉 (𝑘) = 1

2 𝑒2(𝑘) ≥0. To prove system stability, we need to show thatΔ𝑉 = 𝑉 (𝑘 + 1) − 𝑉 (𝑘) ≤ 0. Hence,

Δ𝑉 = 12(𝑒2(𝑘 + 1) − 𝑒2(𝑘)) (18)

Subsitute 𝑒(𝑘 + 1) = 𝑒(𝑘) + Δ𝑒(𝑘) in eq.18:

Δ𝑉 = 𝑒(𝑘)Δ𝑒(𝑘) + 12Δ2𝑒(𝑘) (19)

= Δ𝑒(𝑘)(𝑒(𝑘) + 12Δ𝑒(𝑘)) (20)

Δ𝑒(𝑘) is given by:

Δ𝑒(𝑘) = [𝜕𝑒(𝑘)𝜕𝑊 ]

𝑇Δ𝑊 (21)

Subsituting eq.17 in 21 gives:

Δ𝑒(𝑘) = −𝜂𝑒∥ 𝜕𝑒𝜕𝑊 ∥

2(22)

In order to ind 𝜕𝑒(𝑘)𝜕𝑊 ,the chain rule is used:

𝜕𝑒𝜕𝑊 = 𝜕𝑟

𝜕𝑊 [ 𝜕𝑒𝜕𝑥

𝜕𝑥𝜕𝑟 + 𝜕𝑒

𝜕𝑦𝜕𝑦𝜕𝑟 ] (23)

𝜕𝑒𝜕𝑥 = −𝑒𝑥

𝑒 , 𝜕𝑒𝜕𝑦 = −𝑒𝑦

𝑒(24)

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Hence eq 23 becomes:

𝜕𝑒𝜕𝑊 = − 𝜕𝑟

𝜕𝑊 (𝑒1𝐽𝑥 + 𝑒2𝐽𝑦𝑒 ) (25)

Substituting 25 in 21, gives:

Δ𝑒(𝑘) = −𝜂𝑒 ∥ 𝜕𝑟

𝜕𝑊 ∥2(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦)2 (26)

Substituting 26 in 20:

Δ𝑉 =𝜂∥ 𝜕𝑟𝜕𝑊 ∥

2(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦)2∗

(−1 + 𝜂2𝑒2 ∥ 𝜕𝑟

𝜕𝑊 ∥2(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦)2) (27)

the term 𝜂‖ 𝜕𝑟𝜕𝑊 ‖2(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦)2 ≥ 0, hence in order

to have Δ𝑉 ≤0, 𝜂 should be chosen as:

𝜂 ≤ 2𝑒2

‖ 𝜕𝑟𝜕𝑊 ‖2(𝑒𝑥𝐽𝑥 + 𝑒𝑦𝐽𝑦)2

(28)

3. Experimental ResultsThe performance of the proposed algorithm is ver-

i ied over a variety of real unstructured indoor en-vironments using an autonomous mobile robot plat-form [2]. Themobile robot platform is designed to op-erate in an indoor environment with a solid lat sur-face. A differential steering system is employed to gen-erate forward and steered motion. The platform pro-vides a rich computing environment consisting of asingle board computer and amicrocontroller. It is alsoequippedwith a laser range sensor andultrasonic sen-sors for obstacle detection as well as a compass andwheel encoders for localization. The laser range sen-sor is calibrated to scan the 200 degrees front viewof the robot in 20 sectors with a 10 degrees angularresolution, Themobile robot platform is shown in Fig-ure 7.The platform has a cylindrical structure with a35cm diameter and approximately 30cm height. For

Fig. 7. Target and robot coordinates

all the testing scenarios, the data acquisition is per-formed with a sample time T=1s. The measured vari-ables consist of the current robot position and orien-tation (𝑥𝑟, 𝑦𝑟, 𝜃) and the twenty-sector readings of thelaser sensor.

In all experiments, the robots initial position is(0,0) while the goal position is at (1.5,-1.5). In the irstexperiment, an obstacle is placed along the robot di-rect path, which is the straight line that connects therobot’s initial con iguration to the target con iguration(𝑒). There were also other obstacles surrounding therobot as shown in Figure 8. Fig. 8b depicts the tra-jectory obtained from the neural network based con-troller. The length of the trajectory is 2.4157m. Figures8c - 8e are snapshot of the intermediate robot param-eters at different instances in time. The obstacles seenby the robot at each instant of time is shown in theCartesian coordinates as black dots. The solid yellowline is an approximation to the obstacle contour. Thepolar histogram shows how each sector is classi ied toeither free or occupied. The reference steering angle,𝛾𝑟𝑒𝑓 , is illustrated as a solid red line while the desiredsteering angle, 𝛾𝑑𝑒𝑠𝑖𝑟𝑒𝑑, is shown as a dashed greenline.

In the second experiment shown in Fig. 9a, therobot successfully avoids the 2 obstacles, and drivesits way to 𝑞𝑡𝑎𝑟𝑔𝑒𝑡 as shown Fig. 9b. The robot veloc-ities and control actions are shown in Fig. 9c and Fig.9d respectively. The length of the trajectory is 2.3016m and it takes 106 s to execute.

In the third experiment shown in Fig. 10a, the gapbetween the obstacles is reduced. The robot correctlyidenti ied the gap as navigable and went in between.However, the left side of the robot touched the obsta-cle. The trajectory is shown in Fig. 10b and the robotmotion and control action are shown in Fig. 10c andFig. 10d. The length of the trajectory is 2.1349 m andis completed in 90 s.

In the fourth experiment shown in Fig. 11a, therobot avoids the narrow gap by contouring the obsta-cles, discovers a blocked path, reverses direction andprogresses to the target con iguration. The trajectoryis depicted in Fig. 11b and the robot velocities and con-trol action are depicted in Fig. 11c and 11d. The lengthof the trajectory is 7.4457 m and the time it take is 227s.

In the ifth experiment shown in Fig. 12a, the robotdiscovers the dead end and turns around the obstaclesto reach 𝑞𝑡𝑎𝑟𝑔𝑒𝑡. The trajectory is depicted in Fig. 12b.The robot velocities and control action are depicted inFig. 12c and Fig. 12d. The trajectory exhibited someluctuations. The length of the trajectory is 5.4778𝑚and is completed in 169𝑠.

There are several metrics that can be used to eval-uate the performance of a navigation system [9]. Thefollowing performance metrics are used to evaluatethe quality of the trajectory while considering the se-curity or proximity to obstacles and the smoothness ofthe trajectory relative to the control effort.1) Security Metric-1 (SM1): Mean distance between

the robot and the obstacles through the entiremis-sion measured by the laser sensor (20 sectors).

2) Security Metric-2 (SM2): Mean minimum-distanceto obstacles. This is taken from the average of thelowest value of the laser sensor data (20 sectors).

3) Path length: distance traveled by the robot to ac-

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(a) (b)

(c)

(d) (e)

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 thetarget position.

𝑃𝐿 =𝑛

∑𝑖=1

√(𝑥𝑖 − 𝑥𝑖−1)2 + (𝑦𝑖 − 𝑦𝑖−1)2 (29)

where, (𝑥𝑖, 𝑦𝑖), 𝑖 = 1, 2, … , 𝑛 are the n-pointCartesian coordinates of the robot along the giventrajectory .

4) Time 𝑇𝐴: time taken to accomplish the task.5) Smoothness of the trajectory relative to control ef-

fort.𝑇 𝐵𝐸 =

𝑛∑𝑖=1

𝑘2(𝑥𝑖, 𝑦𝑖) (30)

where 𝑘(𝑥𝑖, 𝑦𝑖) is the curvature at any point(𝑥𝑖, 𝑦𝑖) across the trajectory.

𝑘(𝑥𝑖, 𝑦𝑖) = 𝑓″(𝑥𝑖)[1 + (𝑓′(𝑥𝑖))2] 3

2(31)

Table 1 summarizes the experimental results obtainedfrom the 5 different scenarios. The results show thatthe neural navigation algorithm allows the robot totransit through narrow zones keeping a safe distancefrom the obstacles while generating smooth trajecto-ries. In the neural navigation algorithm, a threshold of

Tab. 1. Performance Metrics

Scenario Performance MetricSM1 (m) SM2 (m) 𝑃𝐿 (m) 𝑇𝐴 (s) 𝑇 𝐵𝐸

1 0.4688 0.1973 2.4157 82 2.3186e-022 0.4563 0.1588 2.3016 106 9.2964e-053 0.4439 0.1474 2.7349 90 6.8655e-024 0.4478 0.1890 5.4778 169 1.1589e-015 0.4727 0.2704 7.4458 227 2.4902e-02

0.5m was placed on the maximum distance that therobot can view from the laser sensor data. The devi-ation of the security metrics SM1 from this maximumvalue (0.5m) is relatively low, which means that thechosen routes passed always through an obstacle freearea. The SM2 index gives also an idea about the risktaken by the robot through the different missions intermsof proximity to obstacles. The values of the𝑇 𝐵𝐸index are also low, which is desirable, since the energyrequirements are increased according to the increasein the curvature of the trajectory. Scenario 2 showsa very low TBE because the corresponding trajectoryis straighter that the other scenarios. Fig. 9b shows asmaller change in the orientation during each controlperiod, with consequent energy saving and less struc-tural effort on the robot.

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(a) (b)

(c) (d)

Fig. 9. (a) shows the robot ini al posi on and the surrounding obstacles; (b) shows the obstacle points in green, thearea occupied by the robot at each instance in me in red, and the reference point trajectory in blue; (c) describes therobot 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 obstaclepoints in green, the area occupied by the robot at each instance in me in red, and the reference point trajectory inblue; (c) describes the robot veloci es; (d) describes the robot control vector.

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(a) (b)

(c) (d)


(a) (b)

(c) (d)


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4. ConclusionsThe paper presents two reactive navigation al-

gorithms for a wheeled mobile robot under non-holonomic constraints and in unknownenvironments.The mobile robot travels to a pre-de ined goal posi-tion safely and ef icientlywithout any priormap of theenvironment. The irst method is based on a reactivenavigation algorithm which incorporates the dimen-sions and shape of the robot to determine the set ofall possible collision-free steering angles. The steer-ing angle that falls in the widest gap and is closest tothe target is selected. The algorithm also takes into ac-count the non-holonomic constraints of differentiallysteered robots by computing circular trajectorieswithadaptive radius of curvature. The second navigationalgorithm introduces a neural network based reactivenavigation. The algorithm aims to generate an opti-mized path by using a user-de ined objective functionwhich minimizes the traveled distance to the goal po-sition while avoiding obstacles. To this end, a diago-nal recurrent neural network (DRNN) has been em-ployed to achieve the necessary generalization capa-bility across a variety of indoor environments. Thenetwork is trained through off-line learning followedby an on-line learning algorithmwith guaranteed con-vergence. The performances of the algorithms are ver-i ied over a variety of real unstructured indoor en-vironments using an autonomous mobile robot plat-form. The results demonstrated that the algorithm iscapable of driving the robot safely through differentobstacle arrangements.

AUTHORSMariam Al-Sagban∗ – American University of Shar-jah, UAE, e-mail: [email protected] Dhaouadi∗ – American University of Sharjah,UAE, e-mail: [email protected].∗Corresponding author

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[2] M. Al-Sagban. “Autonomous robot navigationbased on recurrent neural networks”. Master’sthesis, American University of Sharjah, 2012.

[3] V. Castro, J. Neira, C. Rueda, J. Villamizar, andL. Angel, “Autonomous navigation strategies formobile robots using a probabilistic neural net-work (pnn)”, 33rd Annual Conference of the IEEEIndustrial Electronics Society, 2007, 2795–2800.

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